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abstract: 'In this paper, we consider the nonlinear $\Psi$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $\Psi$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.'
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\
Kishor D. Kucche $^{1}$\
[email protected]\
Jyoti P. Kharade $^{2}$\
[email protected]\
J. Vanterler da C. Sousa $^{3}$\
[email protected]\
$^{1,2}$ Department of Mathematics, Shivaji University, Kolhapur-416 004, Maharashtra, India.\
$^{3}$ Department of Applied Mathematics, Imecc-Unicamp, 13083-859, Campinas, SP, Brazil.
**Key words:** $\Psi$–Hilfer fractional derivative; fractional differential equations; Impulsive; Nonlocal; Existence and Uniqueness; Fixed point theorem.\
**2010 Mathematics Subject Classification:** 26A33, 34A08, 34A12, 34G20.
Introduction
============
The fractional differential equations (FDEs) over the years have been the object of investigation by many researchers [@wang22]–[@zhou]. The fact is that certain natural phenomena by means of fractional differential equations are modeled and allows to better describe the real situation of the problem compared to the problem modeled by means of differential equations of whole order [@rica]–[@sousa6]. Recently, Sousa et al. [@sousa7] presented a fractional mathematical model by means of the time-fractional diffusion equation, which describes the concentration of nutrients in the blood and allows analyzing the solution of the model, better than the integer case. In addition, other mathematical models can be obtained in the literature involving fractional differential equations [@dipi4]–[@dipi3].
On the other hand, investigating the existence, uniqueness and stability of solutions of FDEs of the following types: functional, impulsive, evolution, with instantaneous and non-instantaneous impulses [@benchohra3]–[@wang2]. In this direction the subject has picked up strength and interest of the researchers, since the fractional derivatives allows the variation of the order of the differential equation that is straightforwardly associated with the solution of such FDEs.
Eminent mathematicians working in the field of FDEs, has been exhibiting critical and fascinating outcomes throughout the years that contribute significantly to the mathematical analysis of FDEs, few of them are: Balachandran ,Trujillo [@Krish], Zhou [@zhou], Wang [@wang2], Feckan [@fec], Benchohra [@benchohra3], O’Regan [@Ravi], Kilbas [@Kilbas], JinRong Wang [@Jinr], Agarwal [@agarwal], Diethelm [@Diethelm], Guo [@guo] and Mophou [@mophou].
The FDEs with impulsive effect play vital role in modeling real world physical phenomena involving in the study of population dynamics, biotechnology and chemical technology. Advancement in the theory of impulsive differential equations and its applications in the real world phenomena have been marvelously given in the monographs of Bainov and Simeonov [@bainov], Benchohra et al [@BenHer] and Samoilenko and Perestyuk [@LakBai].
In 2009, Benchohra and Slimani [@Benchohra] investigated various criterion for the existence of solutions for a class of initial value problems for impulsive fractional differential equations given by $$\label{imp01}
\begin{cases}
^c D^{\mu}y(t)=f(t, y(t)),~t \in [0,T], t\neq t_{k},\\
\Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^-)) \\
y(0)=y_{0} \in \mathbb{R},
\end{cases}$$ where $^c D^{\mu}(\cdot)$ is the Caputo fractional derivative of order $0< \mu \leq 1$, $f: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}$ is a given function, $I_{k}:\mathbb{R} \rightarrow \mathbb{R}$, $k=1,...,m$ , $0=t_{0}<t_{1}<\cdots <t_{m}<t_{m+1}=T$, $\Delta y |_{t=t{k}}=y(t_{k}^{+})-y(t_{k}^{-})$, $y(t^{+}_{k})=\lim_{h \rightarrow 0^-} y(t_{k}+h)$, $k=1,2,...,m$.
Benchohra and Seba [@benchohra] extended the study of existence for impulsive FDEs in the Banach spaces. The following year, Benchohra and Berhoun [@benchohra3], investigated sufficient conditions for the existence of solutions for impulsive FDEs with variable times.
In [@fec] Feckan et al. with the help of the examples it is demonstrated that the formula for the solutions of fractional impulsive FDEs considered in the few referred papers in [@fec] were incorrect. They have derived the valid formula for the solution of impulsive FDEs involving Caputo derivative and investigated the existence results for using Banach contraction principle and Leray-Schauder theorem.
In another interesting paper [@wang1], Wang and coauthor presented the idea of piecewise continuous solutions for Caputo fractional impulsive Cauchy problems and impulsive fractioanl boundary value problem. They acquired existence and uniqueness of solution and furthermore determined data dependence and Ulam stabilities of solutions by means of generalized singular Gronwall inequalities.
It is noticed that numerous works with refined and important mathematical tools have been published and others that are yet to come [@sousa; @sousa1; @Krish; @fec; @Ravi; @Jinr]. In any case, it is advantageous to utilize more broad fractional derivatives in which they hold a wide class of fractional derivatives as particular cases, particularly the traditional ones of Caputo and Riemann-Liouville (RL). Another fundamental advantage is the fact that the properties of the general fractional derivative viz, $\Psi$-Hilfer is the preservation of the properties of the respective cases, in particular, in the investigated property of a fractional differential equation, in this case, the existence and uniqueness of solutions [@sousa2]–[@sousa5].
In the present paper, we consider the following impulsive $\Psi$-Hilfer fractional differential equation ( impulsive $\Psi$-HFDE ) with initial condition $$\begin{aligned}
& ^H \mathbf{D}^{\mu,\, \nu; \, \Psi}_{a^+}u(t)=f(t, u(t)),~t \in \mathcal{\mathcal{J}}=[a,T]-\{t_1, t_2,\cdots ,t_m\},\label{e11}\\
&\Delta \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k)= \zeta_k \in \mathbb{R}, ~k = 1,2,\cdots,m, \label{e12}\\
& \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)=\delta \in \mathbb{R}, \label{e13}\end{aligned}$$ where $0<\mu<1,~0\leq\nu\leq 1, ~\varrho=\mu+\nu-\mu\nu$, $^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}(\cdot)$ is the $\Psi$-Hilfer fractional derivative of order $\mu $ and type $\nu$, $\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}$ is left sided $\Psi$-RL fractional integral operator, $ ~a=t_0< t_1 < t_2 < \cdots < t_m < t_{m+1}=T$, $
\Delta \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k)= \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^+)- \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^-)
$, $
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^+) = \lim_{\epsilon\to 0^+} \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k + \epsilon)$ $\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^-) = \lim_{\epsilon\to 0^-} \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k + \epsilon).
$
The main motivation for this work comes from the work highlighted above, with the purpose of investigating the existence and uniqueness of solution of impulsive $\Psi$-HFDEs and to provide new and more general results in the field of fractional differential equations.
We highlight here a rigorous analysis of Eq.(\[e11\])-Eq.(\[e13\]) regarding the main results and advantages obtained in this paper:
- With $\Psi(t)=t$ and taking the limits $\beta \rightarrow 0$ and $\beta \rightarrow 1$ of the Eq.(\[e11\])-Eq.(\[e13\]), we obtain the respective special cases for the differential equations, that is, the classical fractional derivatives of Riemann-Liouville and Caputo, respectively. In addition to the integer case, by choosing $\alpha$= 1. These are two special cases of fractional derivatives. However, a wide class of fractional derivatives can be obtained from the choice of the parameters $\beta $ and $ \Psi(t)$;
- Since it is possible to obtain a wide class of derivatives from the choice of $\beta$ and $\Psi(t)$; consequently, it is also possible to obtain a class of fractional differential equations with their respective fractional derivatives, as particular cases;
- A new class of solutions for impulsive $\Psi$-HFDEs;
- We investigate the existence and uniqueness results for the impulsive $\Psi$-HFDEs and extend it to the non-local impulsive $\Psi$-HFDEs.
Organization of Paper: In section 2, some definitions and results that are important for the development of the paper have been provided via Lemmas and Theorems. In section 3, we present a representation formula for the solution, i.e., we show that the problem (\[e11\])-(\[e13\]) is equivalent to the Volterra fractional integral equation. In section 4, we investigated the existence and uniqueness of the impulsive $\Psi$-HFDE. In Section 5, we will investigate the existence and uniqueness of a nonlocal impulse $\Psi$-HFDE. Concluding and remarks closing the paper.
Preliminaries
=============
In this section, we introduce preliminary facts that are utilized all through this paper.
Let $\mathcal{I}=[a,b ]$ and $\Psi\in C^{1}(\mathcal{I},\mathbb{R})$ an increasing function such that $\Psi'(x)\neq 0$, $\forall~ x\in \mathcal{I}$.
\[[@Kilbas]\] The $\Psi$-Riemann fractional integral of order $\mu>0$ of the function $h$ is given by $$\label{21}
\mathbf{I}_{a+}^{\mu ;\, \Psi }h\left( t\right) :=\frac{1}{\Gamma \left( \mu
\right) }\int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma )h\left( \sigma \right) d\sigma,$$ where $$\mathcal{L}_{\Psi}^{\mu}(t,\sigma )=\Psi ^{\prime }\left(\sigma \right) \left( \Psi \left(
t\right) -\Psi \left( \sigma \right) \right) ^{\mu-1}$$
Let $\mu>0$, $\nu>0$ and $\delta >0$. Then:
- $\mathbf{I}_{a^+}^{\mu;\, \Psi}\mathbf{I}_{a^+}^{\nu ;\, \Psi} h(t)=\mathbf{I}_{a^+}^{\mu+\nu;\, \Psi} h(t)$
- If $h(t)= (\Psi(t)-\Psi(a))^{\delta-1},$ then $\mathbf{I}_{a^+}^{\mu;\, \Psi}h(t)=\frac{\Gamma(\delta)}{\Gamma(\mu+\delta)}(\Psi(t)-\Psi(a))^{\mu + \delta-1}.$
\[\] The $\Psi$–Hilfer fractional derivative of function $h$ of order $\mu$, $(0<\mu<1)$ and of type $0\leq \nu \leq 1$, is defined by $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}h(t)= \mathbf{I}_{a^+}^{\nu ({1-\mu});\, \Psi} \left(\frac{1}{{\Psi}^{'}(t)}\frac{d}{dt}\right)^{'}\mathbf{I}_{a^+}^{(1-\nu)(1-\mu);\, \Psi} h(t).$$
\[ab\] If $h\in C^{1}(\mathcal{I}),$ $0<\mu<1$ and $0\leq\nu \leq 1 $, then
1. $\mathbf{I}_{a^+}^{\mu;\, \Psi}\, {^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}}h(t)= h(t)- \Omega_{\Psi}^{\varrho}(t,a)\mathbf{I}_{a^+}^{(1-\nu)(1-\mu);\Psi}h(a),$ where $\Omega_{\Psi}^{\varrho}(t,a)=\frac{(\Psi(t)-\Psi(a))^{\varrho-1}}{\Gamma(\varrho)}
$
2. ${^H\mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}}\,\mathbf{I}_{a^+}^{\mu;\, \Psi}h(t)=h(t).$
Consider the weighted space [@Sousa1] defined by $$C_{1-\varrho;\Psi}(\mathcal{I})=\left\{u:(a,b]\to\mathbb{R} : ~(\Psi(t)-\Psi(a))^{1-\varrho}u(t)\in C(\mathcal{I})\right\},0< \varrho\leq 1.$$ Define the weighted space of piecewise continuous functions as $$\begin{aligned}
\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{I},\mathbb{R}) =\{& u:(a,b]\to\mathbb{R} :u\in C_{1-\varrho;\Psi}((t_k,t_{k+1}],\mathbb{R}),k=0,1,2,\cdots, m,\\
& \mathbf{I}_{a^+}^{1-\varrho; \, \Psi} \, u(t_k^+), ~ \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}\, u(t_k^-) ~\mbox{exists and} ~\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}\, u(t_k^-)= \mathbf{I}_{a^+}^{1-\varrho; \, \Psi} \, u(t_k) \\
& ~\mbox{for} ~ \, k = 1,2,\cdots, m \}\end{aligned}$$ Clearly, $\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{I},\mathbb{R})$ is a Banach space with the norm $$\|u\|_{\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{I},\mathbb{R})} = \sup_ {t\in \mathcal{I}} \left|(\Psi(t)-\Psi(a))^{1-\varrho}u(t)\right|.$$ Note that for $\varrho =1$, we get $ \mathcal{PC}_{0; \, \Psi}(\mathcal{I},\mathbb{R})=PC(\mathcal{I},\mathbb{R})$ a particular case of the space $\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{I},\mathbb{R})$, whose details are given in [@Benchohra; @Wang; @Bai].
With suitable modification, the PC-type Arzela–Ascoli Theorem [@bainov; @wei] can be extended to the weighted space $\mathcal{PC}_{1-\varrho; \, \Psi}\left( I,\, \mathcal{X}\right)$, where $I$ is closed bounded interval.
\[pc\]
Let $\mathcal{X}$ be a Banach space and $\mathcal{W}_{1-\varrho;\, \Psi} \subset \mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{J},\mathcal{X}).$ If the following conditions are satisfied:
- $\mathcal{W}_{1-\varrho;\, \Psi}$ is uniformly bounded subset of $\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{J}, \mathcal{X})$;
- $\mathcal{W}_{1-\varrho;\, \Psi}$ is equicontinuous in $(t_k, t_{k+1}), k = 0, 1, 2,\cdots , m, where~ t_0 = a, t_{m+1} = T$ ;
- $\mathcal{W}_{1-\varrho;\, \Psi}(t) = \{u(t): u \in \mathcal{W}_{1-\varrho;\, \Psi}, ~t \in \mathcal{J} - {t_1, \cdots , t_m}\}, \mathcal{W}_{1-\varrho;\, \Psi}(t_k^+ ) = \{u(t_k^+ ): u \in \mathcal{W}_{1-\varrho;\, \Psi}\}~ \text{and}~ \mathcal{W}_{1-\varrho;\, \Psi}(t_k^- ) = \{u(t_k^- ): u \in \mathcal{W}_{1-\varrho;\, \Psi}\}$ are relatively compact subsets of X,
then $\mathcal{W}_{1-\varrho;\, \Psi}$ is a relatively compact subset of $\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{J}, X)$.
Let $\mathcal{W}_{1-\varrho; \, \Psi} \subset \mathcal{PC}_{1-\varrho; \, \Psi}\left( \mathcal{J},\,\mathcal{X}\right) $ satisfy the conditions (a) to (c). Let $\{z_n\}$ be any sequence in $\mathcal{W}_{1-\varrho; \, \Psi}$. Define $x_n(t)= (\Psi(t)-\Psi(a))^{1-\varrho}z_n(t), \forall\, n$. Then sequence $\{x_n\}\subset \mathcal{W}\subset PC(\mathcal{J},\mathcal{X})$, where $\mathcal{W}$ satisfy the conditions of Theorem 2.1 of [@wei]. Proceeding as in the proof of Theorem 2.1 of [@wei], there exist $x \in PC(\mathcal{J},\mathcal{X})$ such that $x_n \to x$ in $ PC(\mathcal{J},\mathcal{X})$ which in turn gives $z_n \to z$ in $ \mathcal{PC}_{1-\varrho; \, \Psi}\left( \mathcal{J},\,\mathcal{X}\right)$. This proves $\mathcal{W}_{1-\varrho;\, \Psi}$ is a relatively compact subset of $\mathcal{PC}_{1-\varrho; \, \Psi}(\mathcal{J}, \mathcal{X})$.
\[kf\] Let $\mathcal{M}$ be a closed, convex, and nonempty subset of a Banach space $\mathcal{X}$, and A, B the operators such that
- $ \mathcal{A}x + \mathcal{B}y \in \mathcal{M}$ whenever $x, y \in \mathcal{M}$;
- $\mathcal{A}$ is compact and continuous;
- $\mathcal{B}$ is a contraction mapping.
Then there exists $ z \in \mathcal{M} $ such that $z = \mathcal{A}z + \mathcal{B}z$.
Representation formula for the solution
=======================================
The following lemma play an important role in building an equivalent fractional integral equation of the impulsive $\Psi$-HFDE - .
Let $0<\mu<1$ and $0\leq\nu\leq 1,$ $\varrho=\mu+\nu-\mu\nu$ and $ h: \mathcal{J}\to \mathbb{R} $ be continuous.
Then for any $b\in \mathcal{J} $ a function $u\in C_{1-\varrho,\Psi}\left( \mathcal{J},\,\mathbb{R}\right) $ defined by $$\label{JK}
u(t)= \Omega_{\Psi}^{\varrho}(t,a)\left. \left\{\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(b)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=b} \right\}+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t)$$ is the solution of the $\Psi$–Hilfer fractional differential equation $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t),~t \in \mathcal{J}.$$
Applying $^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+} $ on both sides of the equation , we get $$\begin{aligned}
^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)
&= \left. \left\{\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(b)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=b} \right\} \, ^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}\Omega_{\Psi}^{\varrho}(t,a)\\
&\qquad + ^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), ~ t\in \mathcal{J}.\end{aligned}$$
Using the result ([@Sousa2], Page 10), $$\label{e32}
^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}(\Psi(t)-\Psi(a))^{\varrho-1}=0, ~ 0< \varrho <1,$$ and using the Theorem \[ab\], we get $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t),~t \in \mathcal{J}.$$
This completes the proof of the Lemma.
In the next result, utilizing the Lemma \[JK\], we obtain the equivalent fractional integral of the problem -.
\[fie\] Let $h: \mathcal{J}\to \mathbb{R}$ be a continuous function. Then a function $u \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$ is a solution of impulsive $\Psi$–HFDE $$\begin{aligned}
& ^H \mathbf{D}^{\mu,\, \nu; \, \Psi}_{a^+}u(t)=h(t),~t \in \mathcal{J}-\{t_1, t_2,\cdots ,t_m\},\label{e301}\\
&\Delta \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k)= \zeta_k \in \mathbb{R}, ~ ~k = 1,2,3,\cdots,m, \label{e302}\\
& \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)=\delta \in \mathbb{R}, \label{e303}\end{aligned}$$ if and only if u is a solution of the following fractional integral equation $$\label{e14}
u(t) =
\begin{cases}
\Omega_{\Psi}^{\varrho}(t,a)\, \delta+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t),~ \text{$t \in [a,t_1], $}\\
\Omega_{\Psi}^{\varrho}(t,a)\, \left(\delta+\sum_{i=1}^{k}\zeta_i\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), ~\text{ $t \in (t_k,t_{k+1}],~ k=1,2,\cdots,m $}.
\end{cases}$$
Assume that $u \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$ satisfies the impulsive $\Psi$–HFDE -.
If $t\in [a,t_1]$ then $$\label{e3.3}
\begin{cases}
^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t)\\
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)=\delta.
\end{cases}$$
Then the problem is equivalent to the following fractional integral [@Sousa2] $$\label{e15}
u(t)= \Omega_{\Psi}^{\varrho}(t,a) \, \delta+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t) ,\, \text{ $t \in [a,t_1] $.}$$
Now, if $t\in (t_1,t_2]$ then $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t), \, t\in (t_1,t_2]
~~\mbox{with}~~\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_1^+)- \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_1^-)=\zeta_1.$$
By Lemma \[JK\], we have $$\begin{aligned}
\label{e16}
u(t)\nonumber &= \Omega_{\Psi}^{\varrho}(t,a)\left. \left\{\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_1^+)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_1} \right\}+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t) \nonumber \\
&= \Omega_{\Psi}^{\varrho}(t,a) \left. \left\{\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_1^-)+\zeta_1-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_1} \right\}\nonumber \\
& \qquad +\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), ~ t\in (t_1,t_2].
\end{aligned}$$
Now, from , we have $$\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t)= \delta+\mathbf{I}_{a^+}^{{1-\varrho+\mu}; \, {\Psi}}h(t).$$
This gives $$\label{e17}
\left. \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_1^-)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_1}=\delta.$$
Using in , we obtain $$\label{e18}
u(t)=\Omega_{\Psi}^{\varrho}(t,a) (\delta+\zeta_1) +\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), ~ t\in (t_1,t_2].$$
Next, if $t\in (t_2,t_3]$ then $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t), \, t\in (t_2,t_3]~~\mbox{with}~~
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_2^+)- \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_2^-)=\zeta_2.$$
Again by Lemma \[JK\], we have $$\begin{aligned}
\label{e19}
u(t)&= \Omega_{\Psi}^{\varrho}(t,a) \, \left. \left\lbrace \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_2^+)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_2} \right \}+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t) \nonumber \\
&= \Omega_{\Psi}^{\varrho}(t,a) \,\left. \left\lbrace \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_2^-)+\zeta_2-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_2}\right \} \nonumber \\
&\qquad+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), \, t\in (t_2,t_3].\end{aligned}$$
From , we have $$\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t)= (\delta+\zeta_1)+\mathbf{I}_{a^+}^{{1-\varrho+\mu}; \, {\Psi}}h(t),$$ which gives $$\label{e20}
\left. \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t_2^-)-\mathbf{I}_{a^+}^{1-\varrho+\mu; \, \Psi}h(t)\right|_{t=t_2}=\delta+\zeta_1.$$
Using in , we get $$\label{e21}
u(t)=\Omega_{\Psi}^{\varrho}(t,a) \, (\delta+\zeta_1+\zeta_2)+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), \, t\in (t_2,t_3].$$
Continuing the above process, we obtain $$u(t)=\Omega_{\Psi}^{\varrho}(t,a) \, \left( \delta+\sum_{i=1}^{k}\zeta_i\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t), ~ \text{ $ t \in (t_k,t_{k+1}], ~k=1,2,\cdots,m$.}$$
Conversely, let $u \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$ satisfies the fractional integral equation . Then, for $t\in [a,t_1]$, we have $$u(t)= \Omega_{\Psi}^{\varrho}(t,a) \, \delta+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t).$$
Applying the $\Psi$-Hilfer fractional derivative operator $^H D^{\mu, \, \nu; \, \Psi}_{a^+}$ on both sides, we get $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=\delta \, {^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}}\Omega_{\Psi}^{\varrho}(t,a) + {^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t).$$
Utilizing and Theorem \[ab\], $$^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=h(t),~t \in [a,t_1].$$
Now, for $t\in (t_k,t_{k+1}],~ (k=1,2,\cdots,m)$, we have $$u(t)=\Omega_{\Psi}^{\varrho}(t,a) \, \left( \delta+\sum_{i=1}^{k}\zeta_i\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}h(t),~ \text{ $ t \in (t_k,t_{k+1}], ~k=1,2,\cdots,m$}.$$
Applying the operator $^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}(\cdot)$ on both sides and using and the Theorem \[ab\], we obtain $$\begin{aligned}
^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)&= \left\{\delta+\sum_{i=1}^{k}\zeta_i\right \}{^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}}\Omega_{\Psi}^{\varrho}(t,a) + {^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t)\\
&=h(t).\end{aligned}$$
We have proved that $u$ satisfies . Next, we prove that $u$ also satisfy the conditions and .
Applying the $\Psi$-RL fractional operator $\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}(\cdot)$ on both sides of , we get $$\begin{aligned}
\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t)&=\delta \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}\Omega_{\Psi}^{\varrho}(t,a)+ \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t)\\
&= \delta + \mathbf{I}_{a^+}^{{1-\varrho+\mu}; \, {\Psi}} h(t),\end{aligned}$$ and from which we obtain $$\label{e2}
\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(a)=\delta,$$ which is the condition .
Further, from equation , for $ t\in(t_k,t_{k+1}]$, we have $$\begin{aligned}
\label{e22}
\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t)&=\left\{\delta+\sum_{i=1}^{k}\zeta_i\right \}\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}\Omega_{\Psi}^{\varrho}(t,a)+ \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t) \nonumber\\
&= \delta+\sum_{i=1}^{k}\zeta_i + \mathbf{I}_{a^+}^{{1-\varrho+\mu}; \, {\Psi}} h(t), \end{aligned}$$ and for $ t\in(t_{k-1},t_k]$, we have $$\begin{aligned}
\label{e23}
\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}u(t)&= \left\{\delta+\sum_{i=1}^{k-1}\zeta_i\right \}\mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}}\Omega_{\Psi}^{\varrho}(t,a)+ \mathbf{I}_{a^+}^{{1-\varrho}; \, {\Psi}} \mathbf{I}_{a^+}^{\mu; \, \Psi}h(t) \nonumber\\
&= \delta+\sum_{i=1}^{k-1}\zeta_i + \mathbf{I}_{a^+}^{{1-\varrho+\mu}; \, {\Psi}} h(t),\end{aligned}$$
Therefore, from to , we obtain $$\label{e24}
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^+)- \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k^-)= \sum_{i=1}^{k}\zeta_i-\sum_{i=1}^{k-1}\zeta_i = \zeta_k$$ which condition . We have proved that $u$ satisfies the impulsive $\Psi$–HFDE -. This completes the proof.
Existence and Uniqueness results
================================
[**(Existence)**]{}\[ex\] Assume that the function $f:(a,T]\times \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the conditions:
- $f(\cdot,u(\cdot))\in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$ for any $u\in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right),$
- there exist a constant $0<L\leq \dfrac{\Gamma(\mu+\varrho)}{2\Gamma(\varrho)(\Psi(T)-\Psi(a))^\mu}$ satisfying $$|f(t,u)-f(t,v)|\leq L |u-v|, ~ t\in \mathcal{J}, ~u,v \in \mathbb{R}.$$
Then, the impulsive $\Psi$–HFDE - has at least one solution in ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$.
In the view of Lemma \[fie\], the equivalent fractional integral equation of the impulsive $\Psi$-HFDE - is given by $$\label{e41}
u(t) =
\Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k \right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t, u(t)), \text{ $t \in \mathcal{J} $}.$$
Consider the set $$\mathcal{B}_r = \left\{u\in \mathcal{PC}_{1-\varrho; \, \Psi}\left( \mathcal{J},\,\mathbb{R}\right) : \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)=\delta , ~\|u\|_{\mathcal{PC}_{1-\varrho; \, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} \leq r \right\},$$ where $$\mathcal{M} = \sup_{\sigma\in \mathcal{J}} |f(\sigma,0)|$$ and $$r \geq 2\left(\frac{1}{\Gamma(\varrho)}\left\{|\delta|+\sum_{i=1}^{m}|\zeta_i|\right\} + \frac{\mathcal{M}(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)} \right).$$
We define the operators $\mathcal {P}$ and $\mathcal {Q}$ on $\mathcal{B}_r$ by $$\begin{aligned}
&\mathcal {P}u(t)= \Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k \right), t\in \mathcal{J},\\
&\mathcal {Q}u(t)= \mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t)) ,\, t \in \mathcal{J}. \end{aligned}$$
Then the fractional integral equation can be written as operator equation $$u = \mathcal {P}u+\mathcal {Q}u , \, \, \, u \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right).$$ $\mathit{\rm \textbf{Step ~1:}}$ We prove that $ \mathcal {P}u+\mathcal {Q}v \in \mathcal{B}_r $ for any $u,v \in \mathcal{B}_r.$\
Let any $u,v \in \mathcal{B}_r$. Then using ($A_1$), for any $t\in \mathcal{J}$, we have $$\begin{aligned}
&\left|(\Psi(t)-\Psi(a))^{1-\varrho}(\mathcal {P}u(t)+\mathcal {Q}v(t))\right|\\
& = \left|(\Psi(t)-\Psi(a))^{1-\varrho}\left\{\Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,v(t))\right \}\right|\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k
|\right) + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,\left|f(\sigma,v(\sigma))\right|d\sigma\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+ \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\, |f(\sigma,v(\sigma))- f(\sigma,0)|d\sigma\\
& \qquad + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,0)|d\sigma\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+ \frac{L \, (\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)|v(\sigma)|d\sigma\\
& \qquad + \frac{\mathcal{M} \, (\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,d\sigma\\
&=\frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)\\
&\qquad+ \frac{L \, (\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,(\Psi(\sigma)-\Psi(a))^{\varrho-1}
\left|(\Psi(\sigma)-\Psi(a))^{1-\varrho}v(\sigma)\right|d\sigma \\
&\qquad + \frac{\mathcal{M}\, (\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \frac{(\Psi(t)-\Psi(a))^{\mu}}{\mu}\\
& \leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
L \,(\Psi(t)-\Psi(a))^{1-\varrho} \|v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} ~ \mathbf{I}_{a^+}^{\mu; \, \Psi}(\Psi(t)-\Psi(a))^{\varrho-1}\\
& \qquad + \frac{\mathcal{M}\, (\Psi(t)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}\\
& \leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
\frac{L\,\Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{\mu} \|v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} + \frac{\mathcal{M}\,(\Psi(t)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
\frac{L\,\Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(T)-\Psi(a))^{\mu} r + \frac{\mathcal{M}\,(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}.\end{aligned}$$
Since $$r \geq 2\left(\frac{1}{\Gamma(\varrho)}\left\{|\delta|+\sum_{i=1}^{m}|\zeta_i|\right\} + \frac{\mathcal{M}(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)} \right)$$ and $$L\leq \frac{\Gamma(\mu+\varrho)}{2\Gamma(\varrho)(\Psi(T)-\Psi(a))^\mu},$$ we have $$\begin{aligned}
\left|(\Psi(t)-\Psi(a))^{1-\varrho}(\mathcal {P}u(t)+\mathcal {Q}v(t))\right| \leq r, ~ t\in \mathcal{J}.\end{aligned}$$
Therefore $$\begin{aligned}
\left\|(\mathcal {P}u+\mathcal {Q}v)\right\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} \leq r.\end{aligned}$$
Further, from definition of the operator $\mathcal{P}$ and $\mathcal{Q}$, one can verify that $$\begin{aligned}
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}(\mathcal {P}u+\mathcal {Q}v)(a)=\delta.\end{aligned}$$ We have proved that, $\mathcal {P}u+\mathcal {Q}v \in \mathcal{B}_r.$\
$\mathit{\rm \textbf{Step ~2 :~}}$ Clearly $\mathcal {P}$ is a contraction with the contraction constant zero.\
$\mathit{\rm\textbf{Step ~3 :~}}$ $\mathcal {Q}$ is compact and continuous.\
The continuity of $\mathcal {Q}$ follows from the continuity of $f$. Next we prove that $\mathcal {Q}$ is uniformly bounded on $\mathcal{B}_r$.
Let any $u\in \mathcal{B}_r$. Then by ($A_2$), for any $t \in \mathcal{J}$, we have $$\begin{aligned}
\left|(\Psi(t)-\Psi(a))^{1-\varrho}\mathcal {Q}u(t)\right|
& = \left|(\Psi(t)-\Psi(a))^{1-\varrho}\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))\right| \\
&\leq \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,u(\sigma))|\,d\sigma\\
&\leq \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,u(\sigma))- f(\sigma,0)|\,d\sigma\\
& \qquad + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,0)|\,d\sigma\\
&\leq \frac{L\,(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|u(\sigma)|\,d\sigma\\
& \qquad + \frac{M\,(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,d\sigma\\
&\leq \frac{L\,\Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{\mu}\, \|u\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}+\frac{\mathcal{M}\,(\Psi(t)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}\\
& \leq \frac{L\, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(T)-\Psi(a))^{\mu}\, r+\frac{\mathcal{M}\, (\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}.
\end{aligned}$$
Therefore $$\left\|\mathcal {Q}u\right\|_{\mathcal{PC}_{1-\varrho; \, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}
\leq \frac{L\, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(T)-\Psi(a))^{\mu}\, r+\frac{\mathcal{M}(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}.$$
This proves $\mathcal {Q}$ is uniformly bounded on $\mathcal{B}_r.$ Next, we show that $\mathcal {Q}\mathcal{B}_r$ is equicontinuous.
Let any $ u \in \mathcal{B}_r$ and $t_1, t_2 \in(t_k, t_{k+1}]$ for some $k, (k=0,1,\cdots,m)$ with $t_1 < t_2$. Then, $$\begin{aligned}
&\left|\mathcal {Q}u(t_2)-\mathcal {Q}u(t_1)\right|\\
&= \left|\left( \left.\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))\right|_{t=t_2}\right) -\left( \left.\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))\right|_{t=t_1}\right) \right|\\
& \leq \frac{1}{\Gamma(\mu)} \int_{a}^{t_2}\mathcal{L}_{\Psi}^{\mu}(t_2 ,\sigma)\,|f(\sigma,u(\sigma))|\,d\sigma\\
& - \frac{1}{\Gamma(\mu)} \int_{a}^{t_1}\mathcal{L}_{\Psi}^{\mu}(t_1 ,\sigma)\,|f(\sigma,u(\sigma))|\,d\sigma\\
& = \frac{1}{\Gamma(\mu)} \int_{a}^{t_2}\mathcal{L}_{\Psi}^{\mu}(t_2 ,\sigma)\,(\Psi(\sigma)-\Psi(a))^{\varrho-1}\left|(\Psi(\sigma)-\Psi(a))^{1-\varrho}f(\sigma,u(\sigma))\right|\, d\sigma\\
&- \frac{1}{\Gamma(\mu)} \int_{a}^{t_1}\mathcal{L}_{\Psi}^{\mu}(t_1 ,\sigma)\,(\Psi(\sigma)-\Psi(a))^{\varrho-1}\left|(\Psi(\sigma)-\Psi(a))^{1-\varrho}f(\sigma,u(\sigma))\right|\, d\sigma\\
&\leq \left\{ \left.\mathbf{I}_{a^+}^{\mu; \, \Psi}(\Psi(t)-\Psi(a))^{\varrho-1}\right|_{t=t_2}-\left.\mathbf{I}_{a^+}^{\mu; \, \Psi}(\Psi(t)-\Psi(a))^{\varrho-1}\right|_{t=t_1} \right \}\times \left\|f\right\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}\\
&= \frac{\Gamma(\varrho)}{\Gamma(\mu+\varrho)}\left\{(\Psi(t_2)-\Psi(a))^{1-\varrho+\mu}-(\Psi(t_1)-\Psi(a))^{1-\varrho+\mu}\right \}\left\|f\right\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}.
\end{aligned}$$ Note that $$\left|\mathcal {Q}u(t_2)-\mathcal {Q}u(t_1)\right| \to 0 \quad \mbox{as} \quad |t_1-t_2| \to 0.$$ This shows that $\mathcal {Q}$ is equicontinuous on $(t_k, t_{k+1}]$. Therefore $\mathcal {Q}$ is relatively compact on $\mathcal{B}_r$. By ${\mathcal{PC}}_{1-\varrho;\, \Psi}$ type Arzela-Ascoli Theorem (Theorem\[pc\]) $\mathcal {Q}$ is compact on $\mathcal{B}_r$. Since all the assumptions of Krasnoselskii’s fixed point theorem (Theorem \[kf\]) are satisfied, the operator equation $$u = \mathcal {P}u+\mathcal {Q}u$$ has fixed point $\tilde{u} \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$, which is the solution of the impulsive $\Psi$-HFDE -.
[**(Uniqueness)**]{} \[unique1\] Assume that the function $f:(a,T]\times \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the conditions $(A_1)-(A_2)$. Then, impulsive $\Psi$–HFDE - has a unique solution in the weighted space ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$.
Consider the set $\mathcal{B}_r$ as defined in the Theorem \[ex\] and define the operator $\mathcal{T}$ on $\mathcal{B}_r$ by $$\mathcal{T} u(t) =
\Omega_{\Psi}^{\varrho}(t,a) \left(\delta+\sum_{a <t_k <t}\zeta_k \right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t, u(t)), \text{ $t \in \mathcal{J} $}.$$
To prove $u=\mathcal{T} u$ has a fixed point, we show that $\mathcal{T}\mathcal{B}_r\subset \mathcal{B}_r$. For that take any $u\in \mathcal{B}_r$. Then, by ($A_2$) for any $t\in \mathcal{J}$, we have $$\begin{aligned}
&|\mathcal{T} u(t)|\\
&= \left|\Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k \right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))\right|\\
&\leq \Omega_{\Psi}^{\varrho}(t,a) \left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right) + \frac{1}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,u(\sigma))|\,d\sigma\\
&\leq \Omega_{\Psi}^{\varrho}(t,a) \left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+ \frac{1}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,u(\sigma))- f(\sigma,0)|\,d\sigma\\
& \qquad + \frac{1}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,0)|\,d\sigma\\
&\leq \Omega_{\Psi}^{\varrho}(t,a) \left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
\frac{L \, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{1-\varrho+\mu} \|u\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}\\
& \qquad + \frac{\mathcal{M} \, (\Psi(t)-\Psi(a))^{\mu}}{\Gamma(\mu+1)} \\
&\leq \Omega_{\Psi}^{\varrho}(t,a) \left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
\frac{L \, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{1-\varrho +\mu} \, r\\
& \qquad + \frac{\mathcal{M}\, (\Psi(t)-\Psi(a))^{\mu}}{\Gamma(\mu+1)}.\end{aligned}$$
Thus, $$\begin{aligned}
|(\Psi(t)-\Psi(a))^{1-\varrho}\, \mathcal{T} u(t)|
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+\sum_{k=1}^{m}|\zeta_k|\right)+
\frac{L \, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(T)-\Psi(a))^{\mu} \, r\\
& \qquad + \frac{\mathcal{M} \,(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)},~ t \in \mathcal{J}.\end{aligned}$$
From the choices of constants $r$ and $L$, it can be easily verified that $$\begin{aligned}
\left\|\mathcal{T} u\right\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} \leq r.\end{aligned}$$ This proves $\mathcal{T} \mathcal{B}_r\subset \mathcal{B}_r.$
Now, we prove that the operator $\mathcal{T}$ is a contraction on $\mathcal{B}_r$. Let any $u,v \in \mathcal{B}_r$. Then by assumption ($A_2$) for any $t\in \mathcal{J}$, $$\begin{aligned}
&\left|(\Psi(t)-\Psi(a))^{1-\varrho}(\mathcal{T}u(t)-\mathcal{T}v(t))\right|\\
& = \left|(\Psi(t)-\Psi(a))^{1-\varrho} \left( \left\{\Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k \right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))\right \}\right.\right.\\
& \qquad- \left. \left.\left\{\Omega_{\Psi}^{\varrho}(t,a)\left(\delta+\sum_{a <t_k <t}\zeta_k\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,v(t))\right\} \right) \right|\\
&= \left|(\Psi(t)-\Psi(a))^{1-\varrho}\left(\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t))-\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,v(t))\right)\right|\\
&\leq \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)}\int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,
\left|f(\sigma,u(\sigma))- f(\sigma,v(\sigma))\right|\,d\sigma\\
&\leq \frac{L\, (\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)}\int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,\left|u(\sigma)- v(\sigma)\right|\,d\sigma\\
&\leq \frac{L\, \Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{\mu} \|u-v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}.\end{aligned}$$
From the choice of constant $L$, it follows that $$\begin{aligned}
\|\mathcal{T}u-\mathcal{T}v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}
&\leq \frac{1}{2} \|u-v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}.\end{aligned}$$
Thus, $\mathcal{T}$ is a contraction and by the Banach contraction principle it has a unique fixed point in $ \mathcal{B}_r \subseteq {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$ which is the unique solution of impulsive $\Psi$-HFDE -.
Nonlocal Impulsive $\Psi$-HFDE
===============================
In this section we examine the existence and uniqueness results for impulsive $\Psi$-HFDE with non local initial conditions given by $$\begin{aligned}
&^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{a^+}u(t)=f(t, u(t)),~t \in \mathcal{J}-\{t_1, t_2,\cdots ,t_m\},\label{51}\\
&\Delta \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(t_k)= \zeta_k \in \mathbb{R}, ~~\, ~k = 1,2,\cdots,m, \label{52}\\
&\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)+ g(u)=\delta \in \mathbb{R} , \label{53} \end{aligned}$$ where $\mu, \nu, \varrho$ and the function $f$ are as given in the problem - and $g:{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right) \to \mathbb{R} $ is a continuous function.
$(\mathbf{Existence})$ Assume that the function $f:(a,T]\times \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the conditions $(A_1)-(A_2)$. Further, assume that $g:{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right) \to \mathbb{R} $ is a continuous function that satisfy:
- $\left|g(u)-g(v)\right|\leq L_g \|u-v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}, ~ u,v \in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right),$ with $0< L_g \leq \dfrac{1}{6}\, \Gamma(\varrho).$
Then, the nonlocal impulsive $\Psi$-HFDE - has at least one solution in ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$.
By applying the Lemma \[fie\], the equivalent fractional integral equation of the nonlocal impulsive $\Psi$-HFDE - is given as follows $$\label{ie}
u(t) =\Omega_{\Psi}^{\varrho}(t,a)\,\left(\delta-g(u)+\sum_{a<t_k<t}\zeta_k\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t)), ~t \in \mathcal{J}.$$
Consider the set $$\mathcal{B}_{r^{*}} = \left\{u\in \mathcal{PC}_{1-\varrho;\, \Psi}\left( \mathcal {J},\,\mathbb{R}\right) : \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)+g(u)=\delta, ~\|u\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} \leq {r^{*}} \right\},$$ where $${r^{*}} \geq 3
\left(\frac{1}{\Gamma(\varrho)}\left\{|\delta|+G+\sum_{k=1}^{m}|\zeta_k|\right\} + \frac{\mathcal{M}}{\Gamma(\mu+1)}\,(\Psi(T)-\Psi(a))^{1-\varrho+\mu} \right),$$ $G= |g(0)|$ and $ \mathcal{M} = \sup_{\sigma\in \mathcal{J}} |f(\sigma,0)|$.
Define operator $\mathcal{R}$ and $\mathcal{Q} ^*$ on $\mathcal{B}_{r^{*}}$ by $$\begin{aligned}
&\mathcal{R}u(t)=\Omega_{\Psi}^{\varrho}(t,a)\,\left(\delta-g(u)+\sum_{a<t_k<t}\zeta_k\right) ,\, t \in \mathcal{J},\\
&\mathcal{Q}^*u (t) = \mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,u(t)) ,\, t \in \mathcal{J}.\end{aligned}$$
Then the fractional integral equation is equivalent to the operator equation $$\begin{aligned}
\label{ope}
u=\mathcal{R}u + \mathcal{Q}^*u,\, u\in {\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right). \end{aligned}$$
We apply the Krasnoselskii’s fixed point theorem (Theorem \[kf\]) to prove that the operator equation has fixed point. Firstly, we show that $\mathcal{R}u+\mathcal{Q}^*v \in \mathcal{B}_{r^{*}}$ for any $u,v\in \mathcal{B}_{r^{*}}$. By assumption ($A_2$) and ($A_3$), for any $u,v \in \mathcal{B}_{r^{*}}$ and $t\in \mathcal{J}$, $$\begin{aligned}
&\left|(\Psi(t)-\Psi(a))^{1-\varrho}(\mathcal {R}u(t)+\mathcal {Q}^*v(t))\right|\\
& = \left|(\Psi(t)-\Psi(a))^{1-\varrho}\left\{\Omega_{\Psi}^{\varrho}(t,a)\,\left(\delta- g(u)+\sum_{a<t_k<t}\zeta_k\right)+\mathbf{I}_{a^+}^{\mu; \, \Psi}f(t,v(t))\right \}\right|\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+|g(u)|+\sum_{k=1}^{m}|\zeta_k|\right) + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,v(\sigma))|\,d\sigma\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+|g(u)-g(0)|+|g(0)|+\sum_{k=1}^{m}|\zeta_k|\right)\\
&\qquad + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,v(\sigma))- f(\sigma,0)|\,d\sigma\\
&\qquad + \frac{(\Psi(t)-\Psi(a))^{1-\varrho}}{\Gamma(\mu)} \int_{a}^{t}\mathcal{L}_{\Psi}^{\mu}(t,\sigma)\,|f(\sigma,0)|\,d\sigma\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+L_g \|u\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}+ G +\sum_{k=1}^{m}|\zeta_k|\right) \\
&\qquad + \frac{L \,\Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(t)-\Psi(a))^{\mu} \|v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}+ \frac{\mathcal{M} \,(\Psi(t)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}\\
&\leq \frac{1}{\Gamma(\varrho)}\left(|\delta|+ G +\sum_{k=1}^{m}|\zeta_k|\right) + \frac{L_g}{\Gamma(\varrho)}r^{*}\\
&\qquad +\frac{L\Gamma(\varrho)}{\Gamma(\mu+\varrho)}(\Psi(T)-\Psi(a))^{\mu}r^{*}+\frac{\mathcal{M}(\Psi(T)-\Psi(a))^{1-\varrho+\mu}}{\Gamma(\mu+1)}.\end{aligned}$$
From the choice of $r^{*}$, $L$ and $L_g$, from the above inequality, we obtain $$\begin{aligned}
\left\|(\mathcal {R}u+\mathcal {Q}^*v)\right\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)} \leq r^{*}.\end{aligned}$$
Further, one can verify that $$\begin{aligned}
\mathbf{I}_{a^+}^{1-\varrho; \, \Psi}(\mathcal {R}u+\mathcal {Q}^*v)(a) + g(u)=\delta .\end{aligned}$$ This shows that $\mathcal{R}u+\mathcal{Q}^*v \in \mathcal{B}_{r^{*}}.$
Next, we prove that $\mathcal{R}$ is a contraction mapping. Let any $u,v \in \mathcal{B}_{r^{*}}$ and $t \in \mathcal{J}$.
Consider $$\begin{aligned}
&\left|(\Psi(t)-\Psi(a))^{1-\varrho}(\mathcal {R}u(t)-\mathcal {R}v(t))\right|\\
&=\left|(\Psi(t)-\Psi(a))^{1-\varrho}\left\{\Omega_{\Psi}^{\varrho}(t,a)\,\left(\delta- g(u)+\sum_{a<t_k<t}\zeta_k\right)\right.\right.\\
&\qquad \left.\left. -\, \Omega_{\Psi}^{\varrho}(t,a)\,\left(\delta-g(v)+\sum_{a<t_k<t}\zeta_k\right)\right \} \right|\\
&=\frac{1}{\Gamma(\varrho)}\left| g(u)-g(v)\right|\\
&\leq \frac{L_g}{\Gamma(\varrho)}\, \|u-v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}.\end{aligned}$$
From the choice of $L_g$, we obtain $$\begin{aligned}
\|Ru-Rv\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}
\leq \frac{1}{6} \, \|u-v\|_{{\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)}.\end{aligned}$$
This shows that $\mathcal {R}$ is a contraction. The operator $\mathcal {Q}^*$ is compact and continuous as proved in the Theorem \[ex\]. Hence by ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$-type Arzela-Ascoli Theorem \[pc\] $\mathcal {Q}^*$ is compact on $\mathcal{B}_{r^{*}}$. Further, as discussed in the proof of Theorem \[ex\] the non local impulsive $\Psi$-HFDE - has at least one solution in ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$.
**(Uniqueness)** \[unique2\] Assume that the function $f:(a,T]\times \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the conditions $(A_1)-(A_3)$. Then, non local impulsive $\Psi$–HFDE - has a unique solution in the weighted space ${\mathcal{PC}}_{1-\varrho;\, \Psi}\left( \mathcal{J},\,\mathbb{R}\right)$.
The proof can be completed following the same steps as in the proof the Theorem \[unique1\].
Applications
============
By taking $$\Psi(t)=t ~\mbox{and} ~~\nu \to 1.$$ the impulsive $\Psi$-HFDE - reduces to Caputo impulsive FDE of the form: $$\begin{aligned}
& ^CD^{\mu}_{0^+}u(t)=f(t, u(t)),~t \in \mathcal{J}=[a,T]-\{t_1, t_2,\cdots ,t_m\},\label{ap11}\\
&\Delta u(t_k)= \zeta_k \in \mathbb{R}, ~k = 1,2,\cdots,m, \label{ap12}\\
& u(a)=\delta \in \mathbb{R}, \label{ap13}\end{aligned}$$ and we have the following existence and uniqueness theorems for Caputo impulsive FDE - as an applications of the Theorem \[ex\] and Theorem \[unique1\].
\[excaputo\] Assume that the function $f\in C(\mathcal{J}, \, \mathbb{R})$ satisfies the Lipschitz condition $$|f(t,u)-f(t,v)|\leq L |u-v|, ~ t\in \mathcal{J}, ~u,v \in \mathbb{R}$$ with $0<L\leq \dfrac{\Gamma(\mu+1)}{2(T-a)^\mu}$. Then, the Caputo impulsive FDE - has at least one solution in the space ${\mathcal{PC}}\left( \mathcal{J},\,\mathbb{R}\right)$.
\[uniquecaputo\] Under the suppositions of the Theorem \[excaputo\] the impulsive Caputo FDE - has a unique solution in the space ${\mathcal{PC}}\left( \mathcal{J},\,\mathbb{R}\right)$.
Examples
========
In this section, we give examples to illustrate the utility of the results we obtained.
Consider, the impulsive $\Psi$-HFDE $$\begin{aligned}
&^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{0^+}u(t)=\frac{9}{5\Gamma(\frac{2}{3})}(\Psi(t)-\Psi(0))^{\frac{5}{3}}-\frac{(\Psi(t)-\Psi(0))^4}{16}+\frac{1}{16}u^2, t\in [0,1]-\left\{\frac{1}{2}\right\},\label{p1}\\
&\mathbf{I}_{0^+}^{1-\varrho; \, \Psi}u(0)=0, \label{p2} \\
&\Delta \mathbf{I}_{0^+}^{1-\varrho; \, \Psi}u\left( \frac{1}{2}\right) = \sigma\in \mathbb{R},\label{p3} \end{aligned}$$ $0<\mu<1,~0\leq\nu\leq 1, ~\varrho=\mu+\nu-\mu\nu$ and $\Psi:[0,1] \to \mathbb{R}$ is as defined in preliminaries.
Define the function $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ by $$f(t,u)=\frac{9}{5\Gamma(\frac{2}{3})}(\Psi(t)-\Psi(0))^{\frac{5}{3}}-\frac{(\Psi(t)-\Psi(0))^4}{16}+\frac{1}{16}u^2.$$
Clearly, $$|f(t,u)-f(t,v)|\leq\frac{1}{8}|u-v|, u, v \in \mathbb{R}, ~ t \in [0,1].$$ Thus $f$ satisfies the Lipschitz condition with the constant $L=\frac{1}{8}.$ If the function $\Psi$ satisfies the condition $$\begin{aligned}
\label{L}
L\leq \frac{\Gamma(\mu+\varrho)}{2\Gamma(\varrho)(\Psi(1)-\Psi(0))^\mu}\end{aligned}$$ then problem - has unique solution.
For instance, consider the particular case of the problem -. By taking $$\Psi(t)=t, ~\mu=\frac{1}{3} ~\mbox{and} ~~\nu \to 1.$$
Then the problem - reduces to impulsive FDE involving Caputo fractional derivative operator of the form: $$\begin{aligned}
&^CD^{\frac{1}{3}}_{0^+}u(t)=\frac{9}{5\Gamma(\frac{2}{3})}t^{\frac{5}{3}}-\frac{t^4}{16}+\frac{1}{16}u^2, t\in [0,1]-\left\{\frac{1}{2}\right\} \label{p4}\\
&\Delta u\left( \frac{1}{2}\right) = 0\label{p5}\\
&u(0)=0. \label{p6}\end{aligned}$$
Note that $$\begin{aligned}
\frac{\Gamma(\mu+\varrho)}{2\Gamma(\varrho)(\Psi(T)-\Psi(a))^\mu}= \frac{1}{2} \, \Gamma \left( \frac{4}{3}\right) \approx 0.445.\end{aligned}$$
Since $L =\frac{1}{8}$, the condition is satisfied. Using the Theorem \[ex\] with $\Psi(t)=t,~a=0, ~T=1, ~\mu=\frac{1}{3} ~\mbox{and} ~~\nu \to 1$ the problem - has a solution on $[0,1]$.
By direct substitution one can verify that $ u(t)=t^2$ is the solution of the problem - .
Consider an impulsive $\Psi$-HFDE $$\begin{aligned}
&^H \mathbf{D}^{\mu, \, \nu; \, \Psi}_{0^+}u(t)=\frac{\sin^4(\Psi(t)-\Psi(0))}{((\Psi(t)-\Psi(0))+3)^3} \frac{|u(t)|}{1+|u(t)|}, t\in [0,1]-\left\{\frac{1}{3}\right\},\label{p7}\\
&\Delta \mathbf{I}_{0^+}^{1-\varrho; \, \Psi}u\left(\frac{1}{3}\right)= \sigma ,\label{p8} \\
&\mathbf{I}_{0^+}^{1-\varrho; \, \Psi}u(0)=\delta,\label{p9}\end{aligned}$$ where $0<\mu<1,~0\leq\nu\leq 1, ~\varrho=\mu+\nu-\mu\nu$.
Define the function $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ by $$f(t,u)= \frac{\sin^4(\Psi(t)-\Psi(0))}{((\Psi(t)-\Psi(0))+3)^3} \frac{|u|}{1+|u|}.$$
Let $u,v \in \mathbb{R} ~\mbox{and} ~~t\in [0,1]$. Then, $$\begin{aligned}
|f(t,u)- f(t,v)|&= \left|\frac{\sin^4(\Psi(t)-\Psi(0))}{((\Psi(t)-\Psi(0))+3)^3} \frac{|u|}{1+|u|}-\frac{\sin^4(\Psi(t)-\Psi(0))}{((\Psi(t)-\Psi(0))+3)^3} \frac{|v|}{1+|v|}\right|\\
& \leq \frac{1}{\left( (\Psi(t)-\Psi(0))+3\right) ^3} \left| \frac{|u|}{1+|u|}- \frac{|v|}{1+|v|} \right|\\
&\leq \frac{1}{\left( (\Psi(1)-\Psi(0))+3\right) ^3}|u-v|.\end{aligned}$$
This proves $f$ is Lipschitz function with the constant $$L=\frac{1}{\left( (\Psi(1)-\Psi(0))+3\right) ^3}.$$
By Theorem \[ex\] the problem - has a solution if $$\frac{1}{\left( (\Psi(1)-\Psi(0))+3\right) ^3}\leq \frac{\Gamma(\mu+\varrho)}{2\Gamma(\varrho)(\Psi(1)-\Psi(0))^\mu}.$$
Concluding remarks {#concluding-remarks .unnumbered}
==================
We close the present paper with the destinations we accomplished. We investigated the existence and uniqueness of solutions of nonlinear $\Psi$-HFDE and of also their respective extension to nonlocal case by means of strong analysis results. Some examples were illustrated in order to elucidate the results obtained. It is noted that since the $\Psi$-Hilfer fractional derivative is global and contains a wide class of fractional derivatives, the properties investigated herein are also valid for their respective particular cases.
Here we have not investigated the continuous dependence on the various data and Ulam-Hyers stabilities of solution of (\[e11\])-(\[e13\]), which is the point of our next investigation and will be published a future work.
Now, if we consider $\Psi(t)=t$ in the problem (\[e11\])-(\[e13\]) with $A: D(A)\subset X \rightarrow X$ generator of $C_{0}$-semigroup ($\mathbb{P}_{t\geq 0}$) on a Banach space $X$, we have the following impulsive $\Psi$-HFDE with initial condition $$\begin{aligned}
& ^H \mathbf{D}^{\mu,\, \nu}_{a^+}u(t)=Au(t)+f(t, u(t)),~t \in \mathcal{J}=[a,T]-\{t_1, t_2,\cdots ,t_m\}, \label{r1}\\
&\Delta \mathbf{I}_{a^+}^{1-\varrho}u(t_k)= \zeta_k \in \mathbb{R}, ~k = 1,2,\cdots,m, \\
& \mathbf{I}_{a^+}^{1-\varrho; \, \Psi}u(a)=\delta \in \mathbb{R}\label{r3}, \end{aligned}$$ with the same conditions of problem (\[e11\])-(\[e13\]). The next step of the research is to analyze the problem (\[r1\])-(\[r3\]). But the question may arise “ Why not get the existence and uniqueness of mild solutions to the problem (\[r1\])-(\[r3\]), with a formulation in the sense?". The reason for non-investigation with the $\Psi$-Hilfer fractional derivative comes from the fact of the non-existence of an integral transform, in particular, of Laplace with respect to another function, since it is an important condition in the investigation of the mild solution. Research in this sense has been developed and, in the near future, results can be published.
Acknowledgment {#acknowledgment .unnumbered}
==============
The third author of this paper is financially supported by the PNPD-CAPES scholarship of the Pos-Graduate Program in Applied Mathematics IMECC-Unicamp.
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---
abstract: 'To every product of $2\times2$ matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of [*random*]{} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}\left( 2, {\mathbb R} \right)$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.'
address:
- 'Laboratoire de Physique Théorique et Modèles Statistiques, Bât. 100, Université Paris-Sud, UMR 8626 du CNRS, F-91405 Orsay Cedex, France ; Université Pierre et Marie Curie – Paris 6, 4 place Jussieu, F-75552 Paris Cedex, France'
- 'Laboratoire de Physique Théorique et Modèles Statistiques, Bât. 100, Université Paris-Sud, UMR 8626 du CNRS, F-91405 Orsay Cedex, France ; Laboratoire de Physique des Solides, Bât. 510, Universite Paris-Sud, UMR 8502 du CNRS, F-91405 Orsay Cedex, France'
- 'School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom'
author:
- Alain Comtet
- Christophe Texier
- Yves Tourigny
date: 'June 14, 2010'
title: Products of random matrices and generalised quantum point scatterers
---
=1
[^1]
Introduction {#introductionSection}
============
Products of random $2 \times 2$ matrices arise in many physical contexts: in the study of random spin chains, or when calculating the distribution of the natural frequencies of a classical random spring chain, or more generally when considering the propagation of a wave in a one-dimensional disordered medium [@BL; @LGP; @Lu]. It is often the case that, in the presence of disorder (i.e. randomness), the waves become sharply localised in space. This physical phenomenon is known as [*Anderson localisation*]{}; one of its mathematical manifestations is the exponential growth of the product of random matrices.
The rate of growth is called the [*Lyapunov exponent*]{}; it often has a physical interpretation in terms of the exponential decay of the transmission probability as the size of the disordered region grows. One method for calculating the Lyapunov exponent is based on a general theory developed by Furstenberg and others [@BL; @CaLa; @Fu]. This method requires the explicit knowledge of a certain measure on the projective space, invariant under the action of the matrices in the product. Examples of products of random matrices for which this invariant measure can be obtained in analytical form are, however, very few; see for instance [@BL; @ChLe; @Ish73; @MTW] and the references therein.
The calculation of the Lyapunov exponent need not always make use of this invariant measure. There are alternative approaches; see for instance [@Lu; @Ni; @Pol10]. Nevertheless, the problem of determining the invariant measure is interesting in itself, and the present paper will focus on the presentation of new explicit examples from corresponding examples of exactly solvable models of one-dimensional disordered systems with [*point scatterers*]{} [@ADK; @AGHH; @Se]. In our context, the phrase “exactly solvable” means that the calculation of the Lyapunov exponent associated with the disordered system is reduced to a problem of quadrature. Some of the models were solved by Nieuwenhuizen [@Ni] (without the use of the invariant measure); some of them are, apparently, new. Although the work reported here is, for the most part, mathematically driven, these new models are of independent physical interest. To the best of our knowledge, all the explicit formulae for the invariant measures constitute new results. In the remainder of this introductory section, we review some relevant concepts and some known facts, summarise our main results, and give a sketch of our approach.
Products of random matrices {#productSubsection}
---------------------------
Let $$A_1,\, A_2,\, A_3,\,\cdots$$ denote independent, identically-distributed $2\times2$ matrices with unit determinant, let $\mu$ be their common distribution, and consider the product $$\Pi_n := A_{n} A_{n-1} \cdots A_1\,.
\label{productOfMatrices}$$ The number $$\gamma_\mu
:= \lim_{n \rightarrow \infty}
\frac{{\mathbb E} \left ( \ln \left | \Pi_n \right | \right )}{n}
\label{lyapunovExponent}$$ where $|\cdot|$ denotes the norm on matrices induced by the euclidean norm on vectors, also denoted $|\cdot|$, is called the [*Lyapunov exponent*]{} of the product.
The product grows if the angle between the columns decreases or, equivalently, if the columns tend to align along some common direction. In precise mathematical terms, a [*direction*]{} in ${\mathbb R}^d$ is a straight line through the origin, and the set of all directions is, by definition, the projective space $P \left ( {\mathbb R}^d \right )$. The case $d=2$ is particularly simple: any direction $$\left \{ \lambda \begin{pmatrix}
x \\
y
\end{pmatrix} \,: \;\lambda \in {\mathbb R} \right \} \,,$$ is characterised by the reciprocal, say $$z = \frac{x}{y} \in \overline{\mathbb R} := {\mathbb R} \cup \{\infty\}\,,$$ of its slope. So we can identify $P\left( {\mathbb R}^2 \right)$ with $\overline{\mathbb R}$. The calculation of the Lyapunov exponent is often based on the formula [@BL; @CaLa; @Fu]: $$\gamma_\mu =
\int_{\overline{\mathbb R}} \nu ( \d z )
\int_{\text{SL}\left(2,{\mathbb R} \right)}\mu (\d A) \,
\ln \frac{\left | A \begin{pmatrix} z \\ 1 \end{pmatrix} \right |}{ \left |
\begin{pmatrix} z \\ 1 \end{pmatrix} \right |} \, .
\label{furstenbergFormula}$$ In this expression, $\mu$ is the [*known*]{} common distribution of the matrices $A_n$ in the product, whereas $\nu$ is the— a priori [*unknown*]{}—probability measure on the projective line which is invariant under the action of matrices drawn from $\mu$. Here, invariance means that if $$A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$ is a $\mu$-distributed random matrix and $z$ is a $\nu$-distributed random direction, then the direction $${\mathscr A} (z) := \frac{a z + b}{c z + d}
\label{action}$$ —of the vector obtained after $A$ has multiplied a vector of direction $z$— is also $\nu$-distributed. In the particular case where $\nu$ has a density, i.e. $$\nu (\d z) = f(z) \,\d z \,,$$ it may be shown that $$f(z) = \int_{\text{SL} \left ( 2, {\mathbb R} \right )} \mu (\d A )\,
\left ( f \circ {\mathscr A}^{-1} \right ) (z)\,
\frac{\d {\mathscr A}^{-1}}{\d z} (z) \,.
\label{integralEquation}$$ However, there is no systematic method for solving this integral equation.
The particular products of random matrices considered {#mainResultsSubsection}
-----------------------------------------------------
To describe them, let us first remark that every $A\in\text{SL}\left( 2, {\mathbb R} \right)$ has a unique Iwasawa decomposition $$A =
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\begin{pmatrix}
\e^{w} & 0 \\
0 & \e^{-w}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}
\label{gramSchmidtDecomposition}$$ for some $\theta$, $u,\, w \in {\mathbb R}$. This follows easily by applying the familiar Gram–Schmidt algorithm to the columns of $A$. The three parameters in this decomposition have simple geometrical meanings: $-\theta$ is the angle that the first column of $A$ makes with the horizontal axis, $\e^{w}$ is its magnitude, and $u$ is related to the angle between the columns; in particular, $u=0$ if and only if the columns are orthogonal.
Now, suppose that these three parameters are independent random variables. We use the notation $$v \sim \text{\tt Exp}(r)$$ to indicate that $v$ is a random variable with an exponential distribution of parameter $r$, i.e. its density is given by $$r\, \e^{-r v }\, {\mathbf 1}_{(0,\infty)}(v)
\:,$$ where for every set $A\subset{\mathbb R}$, $${\mathbf 1}_{A}(x) =
\begin{cases}
1 & \mbox{ for } x\in A \\
0 & \mbox{ otherwise}
\end{cases}
\:.$$ Also, $\delta_x$ will denote the discrete probability distribution on ${\mathbb R}$ with all the mass at $x$. We shall provide an explicit formula for the $\mu$-invariant measure of the product $\Pi_n$ when the matrices are independent draws from the distribution $\mu$ of $A$ corresponding to either $$\theta \sim \text{\tt Exp} (p)\,,\;
\pm u \sim \text{\tt Exp} (q)\,,\;
w \sim \delta_0\,.
\label{frischLloydCase}$$ or $$\theta \sim \text{\tt Exp} (p)\,, \;
u \sim \delta_0\,,\:
\pm w \sim \text{\tt Exp} (q)\,.
\label{susyCase}$$
We shall also look at other closely related products: for instance, products involving matrices of the form $$A = \begin{pmatrix}
\cosh \theta & \sinh \theta \\
\sinh \theta & \cosh \theta
\end{pmatrix}
\begin{pmatrix}
\e^{w} & 0 \\
0 & \e^{-w}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}$$ and we shall exhibit invariant measures for such cases too.
The Schrödinger equation with a random potential {#schroedingerSubsection}
------------------------------------------------
Our approach to computing the invariant measure will not make explicit use of the integral equation (\[integralEquation\]). Instead, we shall exploit the fact that these products arise when solving the Schrödinger equation (in units such that $\hbar=2m=1$) $$-\psi''(x) + V(x) \,\psi (x) = E \psi(x)
\label{schroedingerEquation}$$ for a given energy $E$ and a potential function $V$ that vanishes everywhere except on a countable set of points $\{x_j\}$. Physically speaking, one can think of $\psi$ as the wave function of a quantum particle in a crystal with impurities; the effect of the impurity located at $x_j$ is modelled by the boundary condition $$\begin{pmatrix}
\psi'(x_j+) \\
\psi (x_j+)
\end{pmatrix}
= B_j
\begin{pmatrix}
\psi'(x_j-) \\
\psi (x_j-)
\end{pmatrix}$$ where $B_j \in \text{SL}(2,{\mathbb R})$. The potential $V$ is therefore a sum of simpler potentials, one for each pair $(x_j,B_j)$, known variously as [ *point scatterers*]{}, [*generalised contact scatterers*]{} or [ *pointlike scatterers*]{} [@ADK; @AGHH; @CheShi04; @CheHug93; @Ex; @Se]. The case (\[frischLloydCase\]) corresponds to the disordered version of the familiar Kronig–Penney model [@KP] considered by Frisch & Lloyd [@FL] and Kotani [@Ko]. The case (\[susyCase\]) corresponds to a “supersymmetric version” of the same model, in which the Schrödinger operator factorises as $$\begin{gathered}
-\frac{\d^2}{\d x^2} + V(x)
= -\frac{\d^2}{\d x^2} + W(x)^2 - W'(x)
\\
= \left [-\frac{\d}{\d x} + W(x) \right ]
\left [ \frac{\d}{\d x} + W(x) \right ]
\label{eq:Hsusy}\end{gathered}$$ and the [*superpotential*]{} $W$ is of the Kronig–Penney type. Such a supersymmetric Hamiltonian is related to the square of a Dirac operator with a random mass $W$— a model that is of independent interest in many contexts of condensed matter physics [@BouComGeoLeD90; @ComTex98; @CKS; @GurCha03; @TexHag09].
The strategy for calculating $\nu$ is based on the observation that it is also the stationary distribution of a certain Markov process $\{ z(x) \}$, where $$z := \frac{\psi'}{\psi}$$ is the Riccati variable associated with the Schrödinger equation. In the particular case where $$x_{j+1} - x_j \sim \text{Exp}(p)$$ and the $B_j$ are independent and identically distributed random variables in $\text{SL} (2,{\mathbb R})$, one can, following Frisch & Lloyd [@FL], show that the density of the stationary distribution satisfies a certain integro-differential equation. The cases (\[frischLloydCase\]) and (\[susyCase\]) share a special feature: the distribution of the $B_j$ is such that the integro-differential equation may be reduced to a [*differential*]{} equation. Furthermore, this differential equation is simple enough to admit an exact solution in terms of elementary functions.
The idea of using the Riccati variable to study disordered systems goes back to Frisch & Lloyd [@FL]. The well-known “phase formalism” introduced in [@AntPasSly81; @LGP] is another version of the same idea. The trick that allows one to express the equation for the stationary distribution of the Riccati variable in a purely differential form is borrowed from Nieuwenhuizen’s work [@Ni] on the particular case (\[frischLloydCase\]), in which the Dyson–Schmidt method is used to compute the Lyapunov exponent directly from a so-called characteristic function. The same trick has been used by others in various contexts [@CPY; @GP; @MTW]. The key fact is that the density of the exponential distribution satisfies a linear differential equation with constant coefficients. Our results on products of matrices therefore admit a number of extensions; for instance when $\pm v$ (or $\pm w$) has, say, a gamma or a Laplace (i.e. piecewise exponential) distribution. One difficulty that arises with these distributions is that the differential equation for the invariant density is then of second or higher order. This makes it harder to identify the relevant solution; furthermore, this solution is seldom expressible in terms of elementary functions. Without aiming at an exhaustive treatment, we shall have occasion to illustrate some of these technical difficulties.
Outline of the paper {#outlineSubsection}
--------------------
The remainder of the paper is as follows: in §\[pointSection\], we review the concept of point scatterer. The Frisch–Lloyd equation for the stationary density of the Riccati variable is derived in §\[generalisedKronigPenneySection\]. In §\[invariantSection\], we study particular choices of random point scatterers for which the Frisch–Lloyd equation can be reduced to a purely differential form. We can solve this equation in some cases and these results are then translated in terms of invariant measures for products of random matrices. Some possible extensions of our results are discussed in §\[extensionSection\]. We end the paper with a few concluding remarks in §\[sec:Conclusion\].
Point scatterers {#pointSection}
================
Let $u \in {\mathbb R}$ and let $\delta$ denote the Dirac delta. The Schrödinger equation with the potential $$V(x) = u \,\delta(x)$$ can be expressed in the equivalent form $$-\psi'' = E \psi\,, \quad x \ne 0\,,
\label{schroedingerEquationWithPointInteraction}$$ and $$\psi(0+) = \psi(0-)\,, \quad \psi'(0+) = \psi'(0-) + u\, \psi(0-)\,.
\label{deltaInteraction}$$ This familiar “delta scatterer” is a convenient idealisation for a short-range, highly localised potential.
A (mathematically) natural generalisation of this scatterer is obtained when the boundary condition (\[deltaInteraction\]) is replaced by $$\begin{pmatrix}
\psi'(0+) \\
\psi (0+)
\end{pmatrix}
= B
\begin{pmatrix}
\psi'(0-) \\
\psi (0-)
\end{pmatrix}
\label{pointInteraction}$$ where $B$ is some $2\times2$ matrix. We shall refer to $B$ as the “boundary matrix”. In order to ascertain what boundary matrices yield a Schrödinger operator with a self-adjoint extension, we start with the observation that the probability current associated with the wavefunction is proportional to $$\begin{pmatrix}
\, \overline{\psi'(x)} & \overline{\psi(x)} \,
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
\psi'(x) \\
\psi (x)
\end{pmatrix}$$ where the bar denotes complex conjugation. The requirement that the probability current should be the same on both sides of the scatterer translates into the following condition on $B$ [@CheShi04]: $$B^\dagger
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
B
=
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}$$ where the dagger denotes hermitian transposition. Equivalently, $$b_{11}\overline{b_{22}}-b_{21}\overline{b_{12}}=1\;\;\text{and}\;\;
\text{Im}(b_{11}\overline{b_{21}})=\text{Im}(b_{22}\overline{b_{12}})=0\,.$$ It is easily seen that this forces [@ADK; @Se] $$\e^{-\i \chi} B \in \text{SL} \left (2, {\mathbb R} \right )$$ for some real number $\chi$. As discussed in Appendix \[app:ScatteringTransfer\], for the purposes of this paper there is no loss of generality in setting $\chi = 0$ and restricting our attention to the case of [*real*]{} boundary matrices.
We write $$V (x) = \sigma_B (x)
\label{pointInteractionPotential}$$ for the potential with these properties, and call it a [*point scatterer*]{} (at the origin) or, as it is also known, a generalised contact scatterer or pointlike scatterer [@ADK; @AGHH; @Ex; @Se]. We remark that the Riccati variable $z = \psi'/\psi$ of the Schrödinger equation with this potential satisfies $$z' = - \left ( E+z^2 \right )\,, \quad x \ne 0\,,
\label{riccatiEquationWithPointInteraction}$$ and $$z ( 0+) = {\mathscr B} \left ( z(0-) \right )
\label{riccatiBoundaryConditionAtAnInteraction}$$ where ${\mathscr B}$ is the linear fractional transformation associated with the matrix $B$: $${\mathscr B}(z) = \frac{b_{11} z + b_{12}}{b_{21} z + b_{22}}\,.
\label{linearFractionalTransformation}$$ The fact that $B\in\text{SL}(2,{\mathbb R})$ ensures that ${\mathscr B}$ is invertible.
In order to gain some insight into the possible physical significance of the boundary matrix $B$, we set $E=k^2$, $k > 0$, and look for solutions of Equations (\[schroedingerEquationWithPointInteraction\]) and (\[pointInteraction\]) of the form $$\psi(x) =
\begin{cases}
a^\text{in}_- \, \e^{\i kx} + a^\text{out}_- \, \e^{-\i kx} &
\text{for $x<0$} \\
a^\text{out}_+ \, \e^{\i kx} + a^\text{in}_+ \, \e^{-\i kx} &
\text{for $x>0$}
\end{cases}\,.
\label{eq:ScatteringState}$$ By definition, the scattering matrix $S$ relates the incoming amplitudes to the outgoing amplitudes via $$\begin{pmatrix}
a^\text{out}_- \\
a^\text{out}_+
\end{pmatrix}
= S \begin{pmatrix}
a^\text{in}_- \\
a^\text{in}_+
\end{pmatrix}\,.
\label{eq:ScatteringMatrix}$$ Hence $$\begin{gathered}
\notag
S = \frac{1}{b_{21} k^2 + \i k (b_{11}+b_{22}) - b_{12}} \\
\times \begin{pmatrix}
b_{21} k^2 - \i k (b_{22}-b_{11})+b_{12} & 2 \i k\\
2 \i k ( b_{11} b_{22}-b_{12} b_{21}) & b_{21} k^2 + \i k (b_{22}-b_{11})+b_{12}
\end{pmatrix}\,.\end{gathered}$$ The relationship between boundary and scattering matrices is discussed at greater length in Appendix \[app:ScatteringTransfer\].
For the delta scatterer defined by (\[deltaInteraction\]), $$B = \begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}\,.$$ The wave function is continuous at the origin, but its derivative experiences a jump proportional to the value of the wave function there. We have $$S = \frac{1}{2 \i k -u} \begin{pmatrix}
u & 2 \i k \\
2 \i k & u
\end{pmatrix}
\quad \text{and} \quad
{\mathscr B}(z) = z+u\,.$$ The fact that $$\lim_{u\rightarrow\pm\infty}S=-I\,,$$ where $I$ is the identity matrix, indicates that the limiting case of an infinitely large “impurity strength” $u$ corresponds to imposing a Dirichlet boundary condition at the scatterer’s position. \[deltaExample\]
The “delta–prime” scatterer (see for instance [@AGHH; @Se]) is defined by $$B = \begin{pmatrix}
1 & 0\\
v & 1
\end{pmatrix}$$ where $v\in{\mathbb R}$. Now it is the derivative of the wave function that is continuous at the origin, and the wave function that jumps: $$\psi(0+)-\psi(0-)=v\,\psi'(0)\,.$$ We emphasise that, in spite of its (widely used) name, the delta-prime scatterer [*does not*]{} correspond to using the distributional derivative $\delta'$ as a potential [@ADK].
We have $$S = \frac{1}{2 \i + v k} \begin{pmatrix}
v k & 2 \i \\
2 \i & v k
\end{pmatrix}
\quad \text{and} \quad
{\mathscr B}(z) = \frac{z}{v z+ 1}\,.$$ The fact that $$\lim_{v\rightarrow\pm\infty}S=+I$$ indicates that a Neumann boundary condition is obtained in the limit of infinite strength $v$.
The question of the possible physical significance of the delta-prime scatterer was considered by Cheon & Shigehara [@CS], who showed that it can in principle be “realised” by taking an appropriate limit of three neighbouring delta scatterers. \[deltaPrimeExample\]
Let $w \in {\mathbb R}$ and $$B = \begin{pmatrix}
\e^{w} & 0\\
0 & \e^{-w}
\end{pmatrix}\,.
\label{eq:matrixBsusy}$$ In this case, the scatterer produces a discontinuity in both the wave function and its derivative. As pointed out in [@CheHug93], the Schrödinger equation (\[schroedingerEquationWithPointInteraction\]) can be recast as the first-order system $$\begin{aligned}
- \psi' - W \psi &= k \phi \\
\phi' - W \phi &= k \psi\end{aligned}$$ with $$W(x) = w \,\delta(x)\,.$$ The meaning of these equations becomes clear if we introduce an integrating factor: $$\begin{aligned}
- \frac{\d}{\d x} \left [ \exp \left ( \int_{-\infty}^x W(y)\,\d y \right ) \psi \right ] &= k \exp \left ( \int_{-\infty}^x W(y)\,\d y \right ) \phi \\
\frac{\d}{\d x} \left [ \exp \left ( -\int_{-\infty}^x W(y)\,\d y \right ) \phi \right ] &= k \exp \left ( -\int_{-\infty}^x W(y)\,\d y \right ) \psi\,.\end{aligned}$$ We call this scatterer the [*supersymmetric scatterer*]{}. We have $$S = \begin{pmatrix}
\tanh w & \text{sech} \,w \\
\text{sech} \,w & -\tanh w
\end{pmatrix}
\quad \text{and} \quad
{\mathscr B}(z) = \e^{2 w} z\,.$$ Hence the scattering is independent of the wave number $k$— a property consistent with the observation, made in Albeverio [*et al*]{}[@ADK], that diagonal matrices (are the only matrices in $\text{SL}(2,{\mathbb R})$ that) yield boundary conditions invariant under the scaling $$\psi (x) \mapsto \sqrt{\lambda} \,\psi (\lambda x)\,, \;\; \lambda > 0\,.$$ However, in contrast with the previous examples, the scattering is asymmetric, i.e. not invariant under the transformation $x\mapsto-x$. The limit of infinite strength $w$ has a clear interpretation: it corresponds to a Neumann boundary condition on the left of the barrier, and to a Dirichlet condition on the right. \[supersymmetricExample\]
Let $$B = \begin{pmatrix}
\e^{w} & 0 \\
0 & \e^{-w}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}
\,.$$ This point scatterer can be thought of as two neighbouring scatterers— a supersymmetric scatterer of strength $w$ on the right, and a delta scatterer of strength $u$ on the left— in the limit as the distance $\varepsilon$ separating them tends to $0$; see Figure \[doubleImpurityFigure\]. For want of a better name, we shall refer to it as the [*double impurity*]{}.
(0,0) (-100,20)[(1,0)[220]{}]{} (-55,20) (-57,8)[$0$]{} (-64,54)[$u$]{} (75,20) (73,8)[$\varepsilon$]{} (64,54)[$w$]{} (-55,22)[(0,1)[50]{}]{} (-56,73) (75,20)[(0,1)[50]{}]{} (74,71)
\[doubleImpurityExample\]
We have $$S = \frac{1}{2 \i k \cosh w - u \e^{w}} \begin{pmatrix}
2 \i k \sinh w + u \e^{w} & 2 \i k \\
2 \i k & -2 \i k \sinh w + u \e^{w}
\end{pmatrix}$$ and $${\mathscr B} (z) = \e^{2 w} (z+u)\,.$$
This particular scatterer is interesting for the following reason: the Iwasawa decomposition (\[gramSchmidtDecomposition\]) implies that [*any*]{} point scatterer for a real boundary matrix can be thought of as a double impurity “up to a rotation”. For example, the boundary matrix for the delta-prime scatterer may be decomposed as $$\begin{pmatrix}
1 & 0 \\
v & 1
\end{pmatrix}
= \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\begin{pmatrix}
\e^{w} & 0 \\
0 & \e^{-w}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}$$ with $$\theta= \arctan v, \;\; w=\frac{1}{2} \ln(1+v^2)\;\; \text{and}\;\;
u=\frac{v}{1+v^2}\,.$$ We shall return to this point in the next section.
A generalised Kronig–Penney model with disorder {#generalisedKronigPenneySection}
===============================================
In this section, we elaborate the correspondence between disordered systems with point scatterers and products of random matrices. Then, for a particular type of disorder, we show how, following Frisch & Lloyd [@FL], one can derive a useful equation for the stationary density of the Riccati variable associated with the system.
The generalised Kronig–Penney model {#generalisedSubsection}
-----------------------------------
Given a sequence $$\{ B_j \} \subset \text{SL}(2,{\mathbb R})$$ and an increasing sequence $\{ x_j \}$ of non-negative numbers, we call the equation (\[schroedingerEquation\]) with the potential $$V(x) = \sum_{j=1}^\infty \sigma_{B_j} (x-x_j)
\label{generalisedKronigPenneyPotential}$$ a [*generalised Kronig–Penney model*]{}. The notation $\sigma_B(x)$ was defined in Equation .
Let us consider first the case where the energy is positive, i.e. $E = k^2$, $k >0$. In principle, one could dispense with the parameter $k$ and set it to unity by rescaling $x$ but, as we shall see later in §\[halperinSubsection\], there is some advantage in making the dependence on the energy explicit. For $x_j < x < x_{j+1}$, the solution is given by $$\begin{gathered}
\begin{pmatrix}
\psi' (x) \\
\psi (x)
\end{pmatrix}
=
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cos \left ( k [x-x_j] \right ) & -\sin \left ( k [x-x_j] \right ) \\
\sin \left ( k [x-x_j] \right ) & \cos \left ( k [x-x_j] \right )
\end{pmatrix} \\
\notag \times
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix} B_j
\begin{pmatrix}
\psi'( x_j- ) \\
\psi (x_j- )
\end{pmatrix}
\label{solutionOftheGeneralisedKronigPenneyModel}\,.\end{gathered}$$ By recurrence, we obtain the solution for every $x > 0$ in terms of a product of matrices. In particular, $$\begin{pmatrix}
\psi' (x_{n+1}-) \\
\psi (x_{n+1}-)
\end{pmatrix}
= A_{n} A_{n-1} \cdots A_{1}
\begin{pmatrix}
\psi' (x_1-) \\
\psi (x_1-)
\end{pmatrix}
\label{generalisedKronigPenneySolution}$$ where $$A_j = \begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cos \left ( k \theta_j \right ) & -\sin \left ( k \theta_j \right ) \\
\sin \left ( k \theta_j \right ) & \cos \left ( k \theta_j \right )
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix} B_j
\label{generalisedKronigPenneyMatrixForPositiveEnergy}$$ and $$\theta_j := x_{j+1}-x_j\,.$$ Thus, for instance, we see that a product of matrices of the form (\[gramSchmidtDecomposition\]) corresponds to a generalised Kronig–Penney model of unit energy in which the $\sigma_{B_j}$ are double impurities. It is worth emphasising this point: between impurities, the Schrödinger operator itself produces the “rotation part” of the matrices in the product. Therefore, in order to associate a quantum model to the most general product of matrices, it is sufficient to use a potential made up of (suitably spaced) double impurities.
The case of negative energy, i.e. $E =-k^2$, $k > 0$, is also of mathematical interest. Then Equation (\[generalisedKronigPenneySolution\]) holds with $$A_j = \begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cosh \left ( k \theta_j \right ) & \sinh \left ( k \theta_j \right ) \\
\sinh \left ( k \theta_j \right ) & \cosh \left ( k \theta_j \right )
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}B_j \,.
\label{generalisedKronigPenneyMatrixForNegativeEnergy}$$
The generalised Frisch–Lloyd equation {#frischLloydEquationSubsection}
-------------------------------------
It is physically reasonable to assume that the scatterers are randomly, independently and uniformly distributed. We denote by $p$ the mean density of impurities. If we label the scatterers in order of increasing position along the positive semi-axis, so that $x_j$ denotes the position of the $j$th impurity, then $$0 < x_1<x_2<x_3<\cdots$$ and the spacings between consecutive scatterers are independent and have the same exponential distribution, i.e. $$\theta_j \sim \text{\tt Exp}(p)\,, \quad p > 0\,.
\label{exponentiallyDistributedTheta}$$ For this distribution of the $\theta_j$, $$n(x) := \# \left \{ x_j :\; x_j < x \right \}$$ is the familiar Poisson process.
We shall be interested in the statistical behaviour of the Riccati variable $$z(x) = \frac{\psi'(x)}{\psi(x)}\,.
\label{definitionRiccatiEquation}$$ Its evolution is governed by $$z' = - \left ( z^2 + E \right )\,, \quad x \notin \{ x_j \}\,,
\label{freeRiccatiEquation}$$ and $$z(x_j+) = {\mathscr B}_j \left ( z(x_j-) \right )\,,
\quad j \in {\mathbb N}\,.
\label{riccatiJump}$$ The “lack of memory” property of the exponential distribution implies that the process $\{ z(x) \}$ thus defined is Markov.
It should be clear from §\[productSubsection\] and the previous subsection that, if we set $k=1$, then the invariant measure $\nu$ associated with the product (\[generalisedKronigPenneySolution\]) is precisely the stationary distribution of the Riccati variable. So we shall look for particular cases where this stationary distribution may be obtained in analytical form.
To simplify matters, we also suppose in the first instance that the $B_j$ are all the same, deterministic, and we drop the subscript.
Let $f(z;x)$ be the density of the distribution of the Riccati variable. Let $h>0$ and let $d z$ denote an interval of infinitesimal length $\d z$ centered on the number $z$. Then $$\begin{gathered}
\notag
f(z;x+h) \,\d z = {\mathbb P} \left ( z(x+h) \in dz \right ) \\
= \sum_{\ell=0}^\infty
{\mathbb P} \left ( z(x+h) \in dz \,\Bigl | \,n(x+h)-n(x) = \ell
\right )
{\mathbb P} \left ( n(x+h)-n(x) = \ell \right ) \,.\end{gathered}$$ It is well-known (see [@Fe]) that, with an error of order $o(h)$ as $h \rightarrow 0+$, $$\notag
{\mathbb P} \left ( n(x+h) - n(x) = \ell \right ) = \begin{cases}
1-p \,h & \text{if $\ell=0$} \\
p \,h & \text{if $\ell = 1$} \\
0 & \text{if $\ell > 1$}
\end{cases}$$ and so $$\begin{gathered}
f(z;x+h) \,\d z =
{\mathbb P} \left ( z(x+h) \in dz \,\Bigl | \,n(x+h)-n(x) = 0 \right)
(1- p h) \\
+ {\mathbb P} \left ( z(x+h) \in dz \,\Bigl | \,n(x+h)-n(x) = 1 \right )
p h + \d z \,o (h) \quad \text{as $h \rightarrow 0+$}\,.
\label{sumOfProbabilities}\end{gathered}$$ The condition $n(x+h)-n(x)=0$ means that no $x_j$ lies in $(x,x+h)$, and so implies that the Riccati variable is governed solely by the differential equation (\[freeRiccatiEquation\]) in this interval. Therefore, the first conditional probability on the right-hand side of (\[sumOfProbabilities\]) equals $$f \left ( z+[z^2+E]h ; x \right )
\left [ 1 + 2 z h + o(h) \right ] \d z \quad \text{as $h \rightarrow 0+$}\,.$$ The condition $n(x+h)-n(x)=1$ means that exactly one of the $x_j$ lies in $(x,x+h)$, and so the Riccati variable experiences a jump defined by Equation (\[riccatiJump\]) in this interval. A simple calculation then yields, for the second conditional probability on the right-hand side of (\[sumOfProbabilities\]), $$\begin{gathered}
\notag
{\mathbb P} \left ( z(x+h) \in dz \,\Bigl | \,n(x+h)-n(x) = 1 \right)
= {\mathbb P} \left ( {\mathscr B} \left (z(x_j-) \right ) \in dz \right )
+ \d z \,O(h) \\
= f \left( {\mathscr B}^{-1}(z) ; x \right)
\frac{\d {\mathscr B}^{-1}}{\d z} (z) \,\d z \,\left [ 1 + O(h)
\right ]
\quad \text{as $h \rightarrow 0+$}\,.\end{gathered}$$ After reporting these results in Equation (\[sumOfProbabilities\]) and taking the limit as $h \rightarrow 0+$, we obtain a generalisation of the equation (6.69) in [@LGP], §6.7: $$\notag
\frac{\partial f}{\partial x} (z;x) =
\frac{\partial}{\partial z} \left [ (z^2+E) \,f(z;x) \right ] + p
\left [ f \left ( {\mathscr B}^{-1}(z) ; x \right )
\frac{\d {\mathscr B}^{-1}}{\d z} (z) - f(z;x) \right ]\,.$$ The stationary distribution, denoted again $f=f(z)$, therefore satisfies $$\frac{\d}{\d z} \left [ (z^2+E) \,f(z) \right ] + p
\left [ f \left ( {\mathscr B}^{-1}(z) \right ) \frac{\d {\mathscr B}^{-1}}{\d z} (z) - f(z) \right ]= 0\,.$$
More generally, if we permit the $B_j$ to be independent random variables with a common distribution denoted $\kappa$, then it is straightforward to derive the equation $$\frac{\d}{\d z} \left [ (z^2 + E) \,f(z) \right ]
+ p \int_{\text{SL}(2, \mathbb R)} \kappa ( \d B)\,
\left [ f \left ( {\mathscr B}^{-1} (z) \right )
\frac{\d {\mathscr B}^{-1}}{\d z}(z) - f(z) \right ]=0\,.
\label{frischLloydEquationInDifferentialForm}$$ By integrating with respect to $z$, we obtain $$(z^2 + E)\,f(z) + p \int_{\text{SL}(2, \mathbb R)} \kappa ( \d B)
\int_{z}^{{\mathscr B}^{-1} (z)}\d t\, f(t) =N\,.
\label{frischLloydEquation}$$ The constant of integration $N$ in this equation depends on $E$; as will be explained shortly, it represents the [*integrated density of states*]{} per unit length of the Schrödinger Hamiltonian for the potential [@FL; @Ko; @LGP].
We shall refer to Equation (\[frischLloydEquationInDifferentialForm\]), or to its integrated version (\[frischLloydEquation\]), as the (generalised) [*Frisch–Lloyd equation*]{}. In the following sections, we shall consider again the particular point scatterers described in §\[pointSection\], and exhibit choices of the measure $\kappa$ for which this equation can be converted to a differential equation.
The qualitative behaviour of the Riccati variable {#qualitativeSubsection}
-------------------------------------------------
It is instructive to think of the Riccati equation (\[freeRiccatiEquation\]) in the absence of scatterers as an autonomous system describing the motion of a fictitious “particle” constrained to roll along the “potential” curve $$U(z) = E z + \frac{z^3}{3}$$ in such a way that its “velocity” at “time” $x$ and “position” $z$ is given by the slope $-U'(z)$; see Figure \[fig:pot\_U\]. We may regard the occurence of the jumps in Equation (\[riccatiJump\]) as a perturbation of this autonomous system, and the intensity $p$ of the Poisson process as the perturbation parameter.
![The “potential” $U(z)$ associated with the unperturbed Riccati equation $z' = -U'(z) = - (z^2+E)$.[]{data-label="fig:pot_U"}](pot_U)
Let us consider first the unperturbed system (i.e. $p=0$). For $E>0$, the system has no equilibrium point: the particle rolls down to $-\infty$, and re-appears immediately at $+\infty$, reflecting the fact that the solution $\psi$ of the corresponding Schrödinger equation has a zero at the “time” $x$ when the particle escapes to infinity. This behaviour of the Riccati variable indicates that every $E>0$ belongs to the spectrum of the Schrödinger operator.
Equation gives the “velocity” of the fictitious particle as a function of its position. Hence the “time” taken to go from $+\infty$ to $-\infty$ is $$-\int_{+\infty}^{-\infty}\frac{\d{z}}{z^2+k^2}=\frac{\pi}{k}\,.$$ On the other hand, the solution of the Frisch-Lloyd equation for $E>0$ and $p=0$ is the Cauchy density $$f(z) = \frac{N}{z^2+k^2} \;\;\text{with}\;\; N = \frac{k}{\pi}\,.
\label{cauchyDensity}$$ Therefore the normalisation constant $N$ may be interpreted as the reciprocal of the “time” that the particle takes to run through ${\mathbb R}$. Another equivalent interpretation is as follows: recall that, when the particle escapes to $-\infty$, it is immediately re-injected at $+\infty$ to commence a new journey through ${\mathbb R}$. $N$ may therefore also be viewed as the [*current*]{} of the fictitious particle [@LGP], and the Rice formula $$\lim_{z\to\pm\infty}z^2f(z)=N$$ can be understood as expressing a relation between the stationary distribution and a current of probability. This current equals the number of infinitudes of $z(x)$— i.e. the number of nodes of the wavefunction $\psi(x)$— per unit length. By the familiar oscillation theorem of Sturm–Liouville theory, it is therefore the same as the integrated density of states per unit length of the corresponding Schrödinger Hamiltonian.
By contrast, in the case $E=-k^2 < 0$, $k>0$, the unperturbed system has an unstable equilibrium point at $-k$, and a stable equilibrium point at $k$. Unless the particle starts from a position on the left of the unstable equilibrium, it must tend asymptotically to the stable equilibrium point. The fact that the particle cannot reach infinity more than once indicates that the spectrum lies entirely in ${\mathbb R}_+$. The solution of the Frisch–Lloyd equation is $$ f(z) = \delta(z-k)\,.$$
Let us now consider how the occurence of jumps can affect the system. For $E>0$, the jumps defined by Equation (\[riccatiJump\]), as long as they are finite and infrequent (i.e. $p$ is small), cannot prevent the particle from visiting $-\infty$ repeatedly; the system should therefore behave in much the same way as in the unperturbed case, and we expect the density $f$ to be Cauchy-like. In particular, the interpretation of the normalisation constant in terms of a probability current remains valid because, for $z$ large enough, the deterministic part of the evolution of the Riccati variable dominates the stochastic part (\[riccatiJump\]). The situation for $E<0$ is more complicated. Roughly speaking, [*positive*]{} jumps, i.e. discontinuous increases of $z$, enable the particle to make excursions to the right of the stable equilibrium point $z=k$, but the particle can never overcome the infinite barrier and so it rolls back down towards $k$. On the other hand, [*negative*]{} jumps, i.e. discontinuous decreases of $z$, enable the particle to make excursions to the left of $k$. If the jump is large enough, the particle can overcome the potential barrier at $-k$ and escape to $-\infty$, raising the possibility that part of the spectrum of the Schrödinger operator lies in ${\mathbb R}_-$. For small $p$, we expect the density $f$ to be large in the neighbourhood of $z=k$.
We shall return to this useful particle analogy in later sections when we examine the detailed behaviour of the Riccati variable for specific random point scatterers.
The reduced Lyapunov exponent {#lyapunovSubsection}
-----------------------------
Knowing the density of the invariant measure, the calculation of the Lyapunov exponent reduces, thanks to Formula (\[furstenbergFormula\]), to the evaluation of a multiple integral. We will show in this subsection that, if $A$ is of the form (\[generalisedKronigPenneyMatrixForPositiveEnergy\]) and the boundary matrix $B$ is [*triangular*]{}, then this formula may be greatly simplified. As pointed out in §\[generalisedSubsection\], there is no loss of generality in setting $E=k^2=1$ since the parameter $k$ may be re-introduced subsequently by rescaling. This has the advantage of making the calculation simpler.
For definiteness, let us begin with the upper triangular case, i.e. $b_{21} = 0$. Then the density of the invariant measure satisfies the Frisch–Lloyd equation (\[frischLloydEquationInDifferentialForm\]), again with $E=1$, and we have $$\begin{gathered}
\notag
\left | A \begin{pmatrix}
z \\ 1
\end{pmatrix}
\right |^2
= \left | B \begin{pmatrix}
z \\ 1
\end{pmatrix}
\right |^2 =
\left ( b_{11} z + b_{12} \right )^2 + \left ( b_{21} z + b_{22} \right )^2 \\
= \left( b_{21} z + b_{22} \right)^2 \left[ 1 + {\mathscr B}(z)^2 \right]
= b_{22}^2 \left [ 1 + {\mathscr B}(z)^2 \right]\,.\end{gathered}$$ Hence $$\begin{gathered}
\notag
\gamma_\mu
= \frac{1}{2} \int_{\mathbb R} \d z \int_{\text{SL} (2,\mathbb R )}\kappa (\d B)\,
\ln \frac{\left | A \begin{pmatrix}
z \\ 1
\end{pmatrix}
\right |^2}{\left | \begin{pmatrix}
z \\ 1
\end{pmatrix} \right |^2} \,f(z) \\
= \frac{1}{2} \int_{\mathbb R} \d z \int_{\text{SL} (2,\mathbb R )}\kappa (\d B)\,
\ln \frac{b_{22}^2 \left [ 1 + {\mathscr B}(z)^2 \right ]}{1+z^2} \,f(z) \end{gathered}$$ and, after some re-arrangement, $$\begin{gathered}
\gamma_\mu = {\mathbb E} \left ( \ln | b_{22} | \right ) + \frac{1}{2} \int_{\mathbb R} \d z \int_{\text{SL}(2,\mathbb R)}\kappa (\d B)\, \ln \left [ 1 + {\mathscr B}(z)^2 \right ] \,f(z) \\
- \frac{1}{2} \int_{\mathbb R} \d z \int_{\text{SL}(2,\mathbb R)}\kappa (\d B)\, \ln ( 1+z^2 )\,f(z)\,.
\label{lyapunovEquality}\end{gathered}$$ Consider the second term on the right-hand side of the last equality: by changing the order of integration, and making the substitution $y = {\mathscr B}(z)$ in the inner integral, we obtain $$\begin{gathered}
\notag
\frac{1}{2} \int_{\mathbb R} \d z \int_{\text{SL}(2,\mathbb R)}
\kappa(\d B)\,
\ln \left [ 1 + {\mathscr B}(z)^2 \right ] \, f(z) \\
= \frac{1}{2} \int_{\text{SL}(2,\mathbb R)} \kappa (\d B)\,
\int_{\mathbb R}\d z\, \ln \left [ 1 + {\mathscr B}(z)^2 \right ] f(z) \\
= \frac{1}{2} \int_{\text{SL}(2,\mathbb R)} \kappa (\d B)\,
\int_{\mathbb R}\d y \, \ln \left [ 1 + y^2 \right ]
f \left ( {\mathscr B}^{-1}(y) \right )
\frac{\d {\mathscr B}^{-1}}{\d y} (y) \,.\end{gathered}$$ Next, we use the letter $z$ instead of $y$, and change the order of integration again: Equation (\[lyapunovEquality\]) becomes $$\begin{gathered}
\notag
\gamma_\mu = {\mathbb E} \left ( \ln | b_{22} | \right ) \\
+ \frac{1}{2} \int_{\mathbb R} \, \d z
\int_{\text{SL}(2,\mathbb R)}\kappa (\d B)\,
\left[ f \left( {\mathscr B}^{-1}(z) \right)\,
\frac{\d {\mathscr B}^{-1}}{\d z}(z) - f(z)
\right] \,
\ln (1+z^2)\,. \end{gathered}$$ Finally, by making use of Equation (\[frischLloydEquationInDifferentialForm\]), and then integrating by parts, we arrive at the following formula: $$\gamma_\mu = \frac{1}{p} \, \gamma
\label{simplifiedLyapunovFormula}$$ where $$\gamma := p\,{\mathbb E} \left ( \ln | b_{22} | \right )
+ \dashint_{-\infty}^{\infty}\d z\, z f(z)\,, \quad b_{21}=0\,.
\label{reducedLyapunovForUpperTriangularB}$$ This formula remains unchanged after restoring $k$ by rescaling. A similar calculation may be carried out if, instead, $B$ is lower triangular. Equation (\[simplifiedLyapunovFormula\]) then holds with $$\gamma := p\,{\mathbb E} \left ( \ln | b_{11} | \right )
- \dashint_{-\infty}^{\infty}\d z\, \frac{E}{z} f(z)\,, \quad b_{12} =0\,.
\label{reducedLyapunovForLowerTriangularB}$$ The integrals in these expressions are Cauchy principal value integrals.
We shall henceforth refer to $\gamma$ as the [*reduced Lyapunov exponent*]{}. Although our derivation of the relation between $\gamma_\mu$ and $\gamma$ assumed that $E>0$, we conjecture, on the basis of the numerical evidence obtained in all the examples we considered, that it holds also when $E<0$.
Such simplified formulae for the Lyapunov exponent are well-known in the physics literature [@LGP]. The reduced Lyapunov exponent is the rate of growth of the solution of the Schrödinger equation: $$\gamma = \lim_{x \rightarrow \infty} \frac{1}{x} \ln \sqrt{\psi(x)^2 + \left [ \psi'(x) \right ]^2 }\,.$$ Alternatively, using the stationarity of the process $\{z(x)\}$, $$\notag
\gamma
= \lim_{x \rightarrow \infty} \frac{1}{x} \ln \left | \psi(x)\right | + \lim_{x \rightarrow \infty} \frac{1}{x} \ln \sqrt{1 + z^2(x) }
= \lim_{x \rightarrow \infty} \frac{1}{x} \ln \left | \psi(x)\right |$$ and $$\notag
\gamma
= \lim_{x \rightarrow \infty} \frac{1}{x} \ln \left | \psi'(x)\right | + \lim_{x \rightarrow \infty} \frac{1}{x} \ln \sqrt{1/z^2(x) + 1 }
= \lim_{x \rightarrow \infty} \frac{1}{x} \ln \left | \psi'(x)\right |\,.$$ $\gamma$ also provides a reasonable definition of (the reciprocal of) the [*localisation length*]{} of the system.
The presence of the expectation term on the right-hand side of Equation may, at first sight, surprise readers familiar with the case of delta scatterers, but its occurence in our more general context is easily explained as follows: between consecutive scatterers, the wave function is continuous and so, for $x_n < x < x_{n+1}$, we can write $$\begin{gathered}
\ln |\psi(x) |
=\ln |\psi(x_{n}+) |+\int_{x_{n}}^x \d y \,\frac{\d}{\d y}\ln | \psi(y) | \\
=\ln| \psi(x_{n}+) | + \int_{x_{n}}^x \d y\,z(y)
\,.
\label{logOfPsi}\end{gathered}$$ Let us denote by $b_{ij}^{(n)}$ the entry of $B_n$ in the $i$th row and $j$th column. If $B$ is upper triangular, we have, at $x_n$, $$\psi(x_n+) = b_{22}^{(n)} \,\psi(x_n-)$$ and so the wavefunction is discontinuous there unless $b_{22}^{(n)} = 1$. Reporting this in (\[logOfPsi\]) and iterating, we obtain $$\ln|\psi(x)|
= \ln|\psi(0)| + \sum_{j=1}^{n(x)} \ln \left | b_{22}^{(j)} \right |
+ \int_0^x \d y\,z(y)$$ where $n(x)$, as defined in §\[frischLloydEquationSubsection\], is the number of point scatterers in the interval $[0,x]$. Using $${\mathbb E}(n(x))=px$$ and the ergodicity of the Riccati variable, we recover by this other route the equation obtained earlier. It is now clear that the expectation term arises from the possible discontinuities of the wave function at the scatterers. To give two examples: for the delta scatterer, the wave function is continuous everywhere, $b_{22} = 1$ and so Equation (\[reducedLyapunovForUpperTriangularB\]) is just the familiar formula in [@LGP]. For the supersymmetric scatterer, however, the wavefunction has discontinuites, $$b_{22} = \e^{-w}$$ and so, as noted in [@TexHag09], the formula for the reduced Lyapunov exponent must include the additional term $${\mathbb E}\left (\ln|b_{22}| \right )=-{\mathbb E}(w)\,.$$
If $B$ is lower triangular instead, it is more natural to work with $\psi'$: for $x_n < x < x_{n+1}$, we have $$\begin{gathered}
\notag
\ln |\psi'(x) |
=\ln |\psi'(x_{n}+) |+\int_{x_{n}}^x \d y \,\frac{\d}{\d y}\ln | \psi'(y) | \\
=\ln| \psi'(x_{n}+) | + \int_{x_n}^ x \frac{\psi''(y)}{\psi'(y)}\,\d y = \ln| \psi'(x_{n}+) | -k^2 \int_{x_n}^ x \frac{\psi(y)}{\psi'(y)}\,\d y \\
= \ln| \psi'(x_{n}+) | - k^2 \int_{x_{n}}^x \d y\,\frac{1}{z(y)}
\,.\end{gathered}$$ Using the lower triangularity of $B_n$, we obtain, at $x_n$, $$\psi'(x_n+) = b_{11}^{(n)} \,\psi'(x_n-)\,.$$ By repeating our earlier argument, we recover Equation (\[reducedLyapunovForLowerTriangularB\]).
Halperin’s trick and the energy parameter {#halperinSubsection}
-----------------------------------------
For the particular case of delta scatterers, Halperin [@Ha] devised an ingenious method that, at least in some cases, by-passes the need for quadrature and yields analytical expressions for the reduced Lyapunov exponent. Let us give a brief outline of Halperin’s trick and discuss some of its consequences.
Halperin works with the Fourier transform of the invariant density: $$F (x) := \int_{\mathbb R} f(z) \,\e^{-\i x z } \,\d z\,.
\label{fourierTransform}$$
For the delta scatterer, the Frisch–Lloyd equation (\[frischLloydEquation\]) in Fourier space is then $$F''(x) - E \,F(x)
-p \,\frac{{\mathbb E} \left ( \e^{-\i x u} \right ) -1}{\i x} \,
F(x)
= - 2 \pi \, N \,\delta(x)\,.
\label{EquationForDeltaScatterers}$$
Let $\varepsilon>0$ and integrate over the interval $(-\varepsilon,\varepsilon)$. Using the fact that $$F' (-\varepsilon) = -\overline{F' (\varepsilon)}$$ and letting $\varepsilon \rightarrow 0+$, we obtain $$N = -\frac{1}{\pi} \text{Re} \left [ F'(0+) \right ]
\:.$$ Furthermore, since in this case $b_{11}=1$ and $b_{21}=0$, Equation (\[reducedLyapunovForUpperTriangularB\]) leads to $$\gamma =
\dashint_{-\infty}^{\infty} z\, f(z) \,\d z
= - \text{Im} \left[ F'(0+) \right]\,.$$
These two formulae may be combined neatly by introducing the so-called [*characteristic function*]{} $\Omega$ associated with the system [@Lu; @Nie82]: $$\Omega (E) := \gamma(E) - \i \pi N(E) \,.
\label{characteristicFunction}$$ Then Halperin observes that $$\Omega (E) = \i \frac{F'(0+)}{F(0+)}
\label{halperinFormula}$$ where, with a slight abuse of notation, $F$ is now the particular solution of the [*homogeneous version*]{} of Equation (\[EquationForDeltaScatterers\]) satisfying the condition $$\lim_{x \rightarrow +\infty} F(x) = 0\,.$$ Thus the problem of evaluating $\gamma$ and $N$ has been reduced to that of finding the recessive solution of a linear homogeneous differential equation.
Equation (\[halperinFormula\]) expresses a relationship between the density of states and the Lyapunov exponent— a relationship made more explicit in the [*Herbert–Jones–Thouless formula*]{} [@HJ; @Th] well-known in the theory of quantum disordered systems. A further consequence of the same equation is that, if the recessive solution $F$ depends analytically on the energy parameter, so does the characteristic function. $\Omega$ should thus have an analytic continuation everywhere in the complex plane, save on the cut where the essential spectrum of the Schrödinger Hamiltonian lies.
More generally, for an arbitrary scatterer, the Fourier transform $F$ of the invariant density satisfies the equation $$\begin{gathered}
F''(x) - E \,F(x)
- p \int_{\text{SL} (2,\mathbb R)} \kappa (\d B) \int_{\mathbb R} \d z\,\frac{\e^{-\i x {\mathscr B}(z)}-\e^{-\i x z}}{\i x} \, f(z)
= 0\,.
\label{frischLloydEquationInFourierSpace}\end{gathered}$$ We shall not make explicit use of this equation in what follows: instead, we shall obtain closed formulae for the characteristic function by making use of analytic continuation. This is one important benefit of having retained the energy parameter.
Random continued fractions {#continuedFractionSubsection}
--------------------------
There is a close correspondence between products of $2 \times 2$ matrices and continued fractions. Let $z_0$ be an arbitrary starting value, recall the definition (\[action\]) and set $$z_{n} := {\mathscr A}_{n-1} \circ \cdots \circ {\mathscr A}_0 (z_0)\,.
\label{forwardIteration}$$ Then the sequence $\{ z_n \}_{n \in {\mathbb N}}$ is a Markov chain on the projective line, and $\nu$ is $\mu$-invariant if and only if it is a stationary distribution of this Markov chain. Now, reverse the order of the matrices in the product, set $\zeta_0 = z_0$ and $$\zeta_{n} := {\mathscr A}_0 \circ \cdots \circ {\mathscr A}_{n-1} (\zeta_0)\,.
\label{backwardIteration}$$ Although, for every $n$, $z_n$ and $\zeta_n$ have the same distribution, the large-$n$ behaviour of a typical realisation of the sequence $\{z_n\}_{n \in {\mathbb N}}$ differs greatly from that of a typical realisation of the sequence $\{\zeta_n\}_{n \in {\mathbb N}}$ [@Fu].
$\{\zeta_n\}_{n \in {\mathbb N}}$ converges to a (random) limit, say $\zeta$. Write $$A_n := \begin{pmatrix}
a_n & b_n \\
c_n & d_n
\end{pmatrix}\,.$$ Then $${\mathscr A}_n (\zeta) = a_n/c_n - \frac{1/c_n^2}{d_n/c_n + \zeta}
\label{continuedFractionForm}$$ and so $$\zeta := a_0/c_0 - \cfrac{1/c_0^2}{d_0/c_0 + a_1/c_1 - \cfrac{1/c_1^2}{d_1/c_1 + a_2/c_2 - \cfrac{1/c_2^2}{d_2/c_2 + \cdots}}}\,.
\label{continuedFraction}$$ It is immediately clear that, if $A$ is $\mu$-distributed, then $${\mathscr A} (\zeta) = a/c - \frac{1/c^2}{d/c + \zeta}$$ has the same distribution as $\zeta$. Hence the distribution of $\zeta$ is $\mu$-invariant. Furthermore, if $\zeta$ is independent of $z_0$, then there can be only one $\mu$-invariant measure. So $f$ is also the density of the infinite random continued fraction $\zeta$.
By contrast, $\{z_n\}_{n \in {\mathbb N}}$ behaves ergodically. Therefore the density $f$ of the invariant measure $\nu$ should be well approximated by a histogram of the $z_n$. We have used this to verify the correctness of our results.
Some explicit invariant measures {#invariantSection}
================================
Delta scatterers {#deltaSubsection}
----------------
In this section, we obtain invariant measures for products where the matrices are of the form $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cos (k\theta) & -\sin (k\theta) \\
\sin (k\theta) & \cos (k\theta )
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}
\label{deltaInteractionForPositiveEnergy}$$ or $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cosh (k \theta) & \sinh (k\theta) \\
\sinh (k \theta) & \cosh (k\theta)
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}
\label{deltaInteractionForNegativeEnergy}$$ where $\theta \sim \text{Exp}(p)$ and $u$ is a random variable, independent of $\theta$, whose density we denote by $\varrho : {\mathbb R} \rightarrow {\mathbb R}_+$. These products are associated with the generalised Kronig–Penney model for $E = k^2 >0$ and $E = -k^2$ respectively, in the case where (see Example \[deltaExample\]) $$B = \begin{pmatrix}
1 & u \\
0 & 1
\end{pmatrix}\,.$$
The corresponding Frisch–Lloyd equation (\[frischLloydEquation\]) is $$(z^2 + E)\,f(z) + p \int_{\mathbb R} \d u \int_{z}^{z-u}\d t \,
f(t)\,\varrho ( u) =N\,.$$ We change the order of integration; the equation becomes $$N = (z^2 + E)\,f(z) + p \int_{\mathbb R} \d t\, K (z-t)\,f(t)
\label{frischLloydForDelta}$$ where $$K(x) = \begin{cases}
- \int_x^\infty \varrho(u) \,\d u & \text{if $x > 0$} \\
\int_{-\infty}^x \varrho(u)\,\d u & \text{if $ x<0$}
\end{cases}\,.
\label{deltaKernel}$$
Suppose that $$\pm u \sim \text{\tt Exp}(q)\,.
\label{exponentialDistributionForTheDeltaInteraction}$$ We shall show that, in this case, the Frisch–Lloyd equation reduces to a first-order differential equation. For the sake of clarity, consider first the case $u \sim \text{Exp} (q)$. For this choice of distribution, $$K(x) = \begin{cases}
- \e^{-q x} & \text{if $x>0$} \\
0 & \text{if $x<0$}
\end{cases}\,.$$ So we have $$K'(x) = - q K(x)\,, \;\; K(0+) = -1\,,$$ and equation (\[frischLloydForDelta\]) is $$N = (z^2+E)\, f(z) + p \int_{-\infty}^z\d t \, K(z-t)\,f(t)\,.$$ Differentiate this equation with respect to $z$: $$\begin{gathered}
\notag
0 = \frac{\d}{\d z} \left [ (z^2+E)\,f(z) \right ] + p K(0+) f(z) + p
\int_{-\infty}^z\d t \, K'(z-t)\,f(t) \\
= \frac{\d}{\d z} \left [ (z^2+E)\,f(z) \right ] - p f(z) - q p
\int_{-\infty}^z\d t \, K(z-t)\,f(t) \\
= \frac{\d}{\d z} \left [ (z^2+E)\,f(z) \right ] - p f(z) - q \left [ N - (z^2+E)\,f(z) \right ]\,.\end{gathered}$$ This is the required differential equation. The case $-u \sim \text{Exp}(q)$ is analogous, and so we find, for the general case (\[exponentialDistributionForTheDeltaInteraction\]), $$\frac{\d}{\d z} \left [ (z^2 + E) \,f(z) \right ]
- p \,f(z) \pm q \left [ (z^2 + E) \,f(z) \right ] = \pm q N\,.
\label{deltaDifferentialEquationForExponential}$$
We seek the particular solution that satisfies the normalisation condition $$\int_{\mathbb R} f(z) \,\d z = 1\,.
\label{normalisationCondition}$$ This condition fixes the constant of integration $N$, and hence provides an expression for the integrated density of states for the Schrödinger Hamiltonian.
### Product of the form (\[deltaInteractionForPositiveEnergy\])
For $E = k^2 > 0$, this leads to $$\begin{gathered}
f(z) =
\frac{\pm q N}{z^2+k^2}
\exp \left[ \mp q z + \frac{p}{k} \arctan \frac{z}{k} \right] \\
\times \int_{\mp \infty}^z
\exp \left[ \pm q t - \frac{p}{k} \arctan \frac{t}{k} \right]\,\d t\,.
\label{nuForDeltaInteractionWithExponentialStrengthandPositiveE}\end{gathered}$$
The density of the Riccati variable is plotted in Figure \[fig:fdelta\_pe\] for (a) positive $u_n$ and (b) negative $u_n$. The continuous black curves correspond to a low density of scatterers (small $p$, compared to $k$ and $1/q$) and are reminiscent of the Cauchy law obtained in the absence of scatterers. Recall that the effect of the $n$th scatterer on the Riccati variable is described by the equation $$z(x_n+)=z(x_n-)+u_n\,.$$ When the $u_n$ are positive, any increase in the concentration of the scatterers produces a decrease in the current and so the distribution is pushed to the right; see the blue dashed curve in Figure \[fig:fdelta\_pe\] (a). On the other hand, when the $u_n$ are negative, any increase in the concentration of the scatterers leads to an increase in the current of the Riccati variable and so spreads the distribution; see the blue dashed curve in Figure \[fig:fdelta\_pe\] (b).
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fdelta\_pe\]
### Product of the form (\[deltaInteractionForNegativeEnergy\])
For $E = -k^2 < 0$ and $u \sim \text{Exp} (q)$, one must take $N=0$ to obtain a normalisable solution— a reflection of the fact that the essential spectrum of the Schrödinger Hamiltonian is ${\mathbb R}_+$. Then $$f(z) = C^{-1}
\frac{\e^{-q z}}{z^2-k^2}
\left ( \frac{z-k}{z+k} \right )^{\frac{p}{2k}} {\mathbf 1}_{(k,\infty)}(z)
\label{nuForFLandNegativeEplus}$$ where $C$ is the normalisation constant.
By contrast, in the case $-u \sim \text{\tt Exp}(q)$, one finds $$f(z) = \frac{q N}{z^2-k^2} \e^{q z}
\left | \frac{z-k}{z+k} \right |^{\frac{p}{2k}} \int_z^{c(z)} \e^{-q t} \left | \frac{t+k}{t-k} \right |^{\frac{p}{2k}}\,\d t
\label{nuForFLandNegativeEminus}$$ where $$c(z) = \begin{cases}
\infty & \text{if $z > k$} \\
-k & \text{if $z < k$}
\end{cases}
\:.$$
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fdelta\_ne\]
The invariant density $f$ is plotted in Figure \[fig:fdelta\_ne\] for the cases (a) $u \sim {\tt Exp}(q)$ and (b) $-u \sim {\tt Exp}(q)$ respectively. The shape of the distribution can again be explained by using the qualitative picture of §\[qualitativeSubsection\]. For positive $u_n$, the sharp peak obtained for a small concentration $p$ of scatterers (black continuous line) reflects the trapping of the process $\{ z(x)\}$ by the potential well at $z=k$; recall Figure \[fig:pot\_U\]. When the concentration of scatterers is increased, the Riccati variable experiences positive jumps more frequently and so the distribution spreads to the right (blue dashed curve). For negative $u_n$, the jumps can take arbitrary negative values. This enables the “particle” to overcome the barrier at $z=-k$, and so we have a non-zero current $N$ (i.e. a non-zero density of states). This effect is enhanced as the density of the scatterers is increased (blue dashed curve).
### Calculation of the characteristic function
Using the invariant measure, it is trivial to express the integrated density of states and the Lyapunov exponent in integral form. Such integral expressions are particularly useful when studying the asymptotics of $N$ and $\gamma$ in various limits. Even so, it is worth seeking analytical expressions (in terms of special functions) for these quantities, as they sometimes reveal unexpected connections to other problems.
Recalling the discussion in §\[halperinSubsection\], we begin with a straightforward application of Halperin’s trick. For $\pm u \sim {\tt Exp}(q)$, we have $${\mathbb E} \left ( \e^{-\i u x } \right ) = \frac{1}{1 \pm \i x/q}$$ and so the homogeneous version of Equation (\[EquationForDeltaScatterers\]) is $$\notag
F''(x) + \left [ -E + \frac{p}{\pm q + \i x} \right ] \,F(x) = 0\,.$$ The recessive solution is $$F(x) = \frac{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}}\left ( 2 \sqrt{-E} \left [ \i x \pm q \right ] \right )}{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}}\left ( \pm 2 \sqrt{-E} q \right )}$$ where $W_{a,b}$ is the Whittaker function [@AS; @GR]. We deduce that, for $E$ outside the essential spectrum of the Schrödinger Hamiltonian, $$\Omega (E) := \gamma (E) -\i \pi N (E)
= - 2 \sqrt{-E} \,
\frac{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}}'\left(\pm2 \sqrt{-E} q \right)}
{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}}\left(\pm2 \sqrt{-E} q \right)}\,.
\label{characteristicFunctionForDeltaScatterer}$$ This formula for the characteristic function was discovered by Nieuwenhuizen [@Ni]. In particular, for $k$ real, $$\gamma (k^2) - \i \pi N (k^2) = \Omega (k^2+\i 0+)
= 2 \i k \frac{W_{\frac{-\i p}{2 k},\frac{1}{2}} ' \left ( \mp 2 \i k q \right )}{W_{\frac{-\i p}{2 k},\frac{1}{2}}\left ( \mp 2 \i k q \right )}$$ and $$\gamma (-k^2) - \i \pi N (-k^2) = \Omega (-k^2+\i 0+)
= - 2 k \frac{W_{\frac{-p}{2 k},\frac{1}{2}} ' \left ( \pm 2 k q \right )}{W_{\frac{- p}{2 k},\frac{1}{2}}\left ( \pm 2 k q \right )}\,.$$
In the case $u \sim {\tt Exp}(q)$ there is an alternative derivation of this formula which does not require the solution of a differential equation: start with the explicit form of the invariant density $f$ for $E=-k^2<0$, given by Equation (\[nuForFLandNegativeEplus\]). By using Formula 3 in [@GR], §3.384, we obtain the following expression for the normalisation constant: $$C := \int_k^{\infty} \frac{\e^{-q z}}{z^2-k^2}
\left ( \frac{z-k}{z+k} \right )^{\frac{p}{2k}} \,\d z = \frac{1}{2 k} \Gamma \left ( \frac{p}{2 k} \right )\,
W_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 k q \right )\,.$$ The reduced Lyapunov exponent $\gamma$ may then be obtained easily by noticing that differentiation with respect to the parameter $q$ yields an additional factor of $z$ in the integrand. Hence, for $k$ real, we find $$\gamma (-k^2) = \int_{k}^{\infty}\d z\, z \, f(z)
= -\frac{\partial}{\partial q}\ln C
= -2 k\, \frac{W'_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 k q \right )}
{W_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 k q \right )}
\,.$$ Since $N=0$ for $E<0$, this yields $$\Omega(-k^2) = -2 k\, \frac{W'_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 k q \right )}
{W_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 k q \right )}\,.
\label{eq:LyapunovFrischLloydNegativeEnergy}$$ Now, the half-line $E<0$ lies outside the essential spectrum of the Schrödinger Hamiltonian because $N=0$ along it. Hence $\Omega$ is analytic along this half-line, and we see that the “$+$ case” of our earlier Equation (\[characteristicFunctionForDeltaScatterer\]) is simply the analytic continuation of Equation (\[eq:LyapunovFrischLloydNegativeEnergy\]). In particular, the formula in the case $E = k^2>0$ may be deduced from the formula in the case $E = -k^2 < 0$ by applying the simple substitution $$k \mapsto - \i k\,.$$
Delta–prime scatterers {#deltaPrimeSubsection}
----------------------
Products of matrices of the form $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cos (k\theta) & -\sin (k\theta) \\
\sin (k\theta) & \cos (k \theta)
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
v & 1
\end{pmatrix}
\label{deltaPrimeInteractionForPositiveEnergy}$$ or $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cosh (k\theta) & \sinh (k\theta) \\
\sinh (k \theta) & \cosh (k\theta)
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
v & 1
\end{pmatrix}
\label{deltaPrimeInteractionForNegativeEnergy}$$ where $\theta \sim \text{Exp}(p)$ and $v$ is a random variable independent of $\theta$, are associated with the delta-prime scatterer (see Example \[deltaPrimeExample\]) $$B = \begin{pmatrix}
1 & 0 \\
v & 1
\end{pmatrix}\,.$$
The Frisch–Lloyd equation (\[frischLloydEquation\]) for this scatterer is $$(z^2 + E)\,f(z)
+ p \int_{\mathbb R} \d v \int_{z}^{\frac{z}{1-vz}}\d t\, f(t)\,\varrho(v) =N
\label{frischLloydForDeltaPrime}$$ where $\varrho$ is the density of $v$. The calculation of the invariant measure in this case can be reduced to the calculation of the invariant measure for some Kronig–Penney model with delta scatterers. For instance, in the negative energy case (\[deltaPrimeInteractionForNegativeEnergy\]) with $k=1$, we have $$\begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}
A
\begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}
=
\begin{pmatrix}
\cosh \theta & \sinh \theta \\
\sinh \theta & \cosh \theta
\end{pmatrix}
\begin{pmatrix}
1 & v \\
0 & 1
\end{pmatrix}
\,.$$ The similarity transformation of the matrix $A$ on the left corresponds to the transformation $z \mapsto 1/z$ of the Riccati variable. So the invariant densities for the delta and the delta-prime cases are in a reciprocal relationship. Accordingly, replace $z$ by $1/z$ in Equation (\[frischLloydForDeltaPrime\]) and set $$g(z) = \frac{1}{z^2} f(1/z)\,.$$ Then $$\begin{gathered}
N = \left ( 1+Ez^2 \right ) g(z)
- p \int_{\mathbb R} \d v \int_{z}^{z-v}\d t \, g(t) \,\varrho(v) \\
= \left ( 1+Ez^2 \right ) g(z) - p \int_{\mathbb R}\d t \, K(z-t) \,g(t)
\label{frischLloydForDeltaPrime2}\end{gathered}$$ where $K$ is the kernel defined by Equation (\[deltaKernel\]). This equation for $g$ is essentially the same as Equation (\[frischLloydForDelta\]) save for the sign of $p$ and the dependence on the energy. For the case $$\pm v \sim {\tt Exp} (q)$$ this equation can, by using the same tricks as before, be converted into a differential equation which is easy to solve.
### Product of the form (\[deltaPrimeInteractionForPositiveEnergy\])
For $E = k^2 > 0$, the upshot is $$\begin{gathered}
f(z) =
\frac{\pm q N}{z^2+k^2}
\exp \left[ \mp \frac{q}{z} - \frac{p}{k} \arctan \frac{k}{z} \right] \\
\times \int_{\mp \infty}^{1/z}
\exp \left[ \pm q t + \frac{p}{k} \arctan (k t) \right]\,\d t\,.
\label{nuForDeltaPrimeInteractionWithExponentialStrengthandPositiveE}\end{gathered}$$
Plots of the distribution are shown in Figure \[fig:fdeltaprime\_pe\] for (a) positive $v_n$ and (b) negative $v_n$. These plots differ somewhat from those obtained in the case of delta scatterers, and we can use the particle analogy of §\[qualitativeSubsection\] to explain the differences. The jump of the particle associated with the $n$th delta-prime scatterer is given implicitly by $$\frac{1}{z(x_n+)} = \frac{1}{z(x_n-)} + v_n\,.
\label{ricattiJumpForDeltaPrime}$$ The strongly asymmetric distribution obtained for negative $v_n$ (Part (b) of Figure \[fig:fdeltaprime\_pe\]) can be explained as follows: starting from $+\infty$, the particle experiences its first jump at “time” $x_1$, and its value after the jump is approximately $1/v_1 < 0$. In fact, for $z$ negative and small in modulus, the invariant density resembles very closely that of $1/v_1$, i.e. $$f(z) \sim \frac{c}{z^2}\e^{q/z} \;\; \text{as $z \rightarrow 0-$}\,.$$ Thereafter, the particle proceeds towards $-\infty$. In particular, if $p$ is large, then the expected value of $x_1$ is small, and the particle spends hardly any time on the positive semi-axis.
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fdeltaprime\_pe\]
[*Product of the form (\[deltaPrimeInteractionForNegativeEnergy\]):*]{} for $E = -k^2 < 0$ and $v \sim \text{\tt Exp}(q)$, we find $$f(z) = C^{-1} \, \frac{\e^{-q/z}}{k^2-z^2}
\left( \frac{k-z}{k+z} \right)^{\frac{p}{2 k}} \,{\mathbf 1}_{(0,k)}(z)
\label{densityForDeltaPrimeEnegativeQpositive}$$ where $$C = \frac{1}{2 k}
\Gamma\left( \frac{p}{2 k} \right)
W_{-\frac{p}{2 k},\frac{1}{2}} \left( 2q/k \right)
\,.
\label{deltaPrimeNormalisationConstant}$$
When $E=-k^2<0$ and $-v \sim \text{\tt Exp}(q)$, we obtain $$f(z) = \frac{q N}{z^2-k^2} \e^{q / z}
\left| \frac{z-k}{z+k} \right|^{\frac{p}{2k}}
\int_{c(z)}^{1/z} \e^{-q t} \left| \frac{1+k t}{1-k t}\right|^{\frac{p}{2k}}
\,\d t
\label{densityForDeltaPrimeEnegativeQnegative}$$ where $$c(z) = \begin{cases}
\infty & \text{if $0 < z < 1/k$} \\
-k & \text{otherwise}
\end{cases}
\,.$$
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fdeltaprime\_ne\]
Again, we can try to understand the qualitative features of the density function $f$ for $E<0$ by invoking the particle analogy of §\[qualitativeSubsection\]. In view of Equation (\[ricattiJumpForDeltaPrime\]), when $v_n \sim {\tt Exp}(q)$ and $z(x_n-)>0$, the value of the Riccati variable decreases but can never become negative. So the particle, once it passes to the left of the equilibrium point at $z=k$, must remain trapped there. This explains why the density is supported on $(0,k)$; see Figure \[fig:fdeltaprime\_ne\] (a). By contrast, when $-v_n \sim {\tt Exp}(q)$, the jumps are unrestricted; the “particle” can escape over the potential barrier at $z=-k$ infinitely often, leading to a non-zero current and a density $f$ spread over ${\mathbb R}$. This is shown in Figure \[fig:fdeltaprime\_ne\] (b).
### Calculation of the characteristic function {#deltaPrimeCharacteristicSubsection}
We begin with the case $v \sim {\tt Exp}(q)$ and $E= -k^2 < 0$.
The invariant density is then given by Equation (\[densityForDeltaPrimeEnegativeQpositive\]). Using Formula (\[reducedLyapunovForLowerTriangularB\]) for the reduced Lyapunov exponent and the expression (\[deltaPrimeNormalisationConstant\]) for the normalisation constant $C$, we find $$\gamma(-k^2) = -k^2 \frac{\partial}{\partial q} \ln C = - 2 k \frac{W_{\frac{-p}{2 k},\frac{1}{2}}'\left ( 2 q/k \right )}{W_{\frac{-p}{2 k},\frac{1}{2}} \left ( 2 q/k \right )}\,.$$ Since $N(-k^2)=0$ in this case, analytic continuation yields $$\Omega (E) = - 2 \sqrt{-E} \,\frac{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}}'\left ( 2 q/\sqrt{-E} \right )}{W_{\frac{-p}{2 \sqrt{-E}},\frac{1}{2}} \left ( 2 q/\sqrt{-E} \right )}\,.
\label{characteristicFunctionForDeltaPrime}$$
The characteristic function in the case $-v \sim {\tt Exp}(q)$ is the same, except that $q$ becomes $-q$. In particular, for $E = k^2 >0$ and $\pm v \sim {\tt Exp}(q)$, we obtain $$\gamma (k^2) - \i \pi N (k^2) = \Omega (k^2+\i0+) = 2 \i k \frac{W_{\frac{-\i p}{2 k},\frac{1}{2}}'\left ( \pm 2 \i q/k \right )}{W_{\frac{-\i p}{2 k},\frac{1}{2}} \left ( \pm 2 \i q/k \right )}\,.$$
An alternative derivation of these results could use the correspondence between the delta and delta-prime cases alluded to earlier.
Supersymmetric scatterers {#susySubsection}
-------------------------
We now consider products where the matrices are of the form $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cos (k\theta) & -\sin (k\theta) \\
\sin (k\theta) & \cos (k \theta)
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
\e^w & 0 \\
0 & \e^{-w}
\end{pmatrix}
\label{susyInteractionForPositiveEnergy}$$ or $$A =
\begin{pmatrix}
\sqrt{k} & 0 \\
0 & \frac{1}{\sqrt{k}}
\end{pmatrix}
\begin{pmatrix}
\cosh (k\theta) & \sinh (k\theta) \\
\sinh (k\theta) & \cosh (k \theta)
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{k}} & 0 \\
0 & \sqrt{k}
\end{pmatrix}
\begin{pmatrix}
\e^w & 0 \\
0 & \e^{-w}
\end{pmatrix}
\label{susyInteractionForNegativeEnergy}$$ where $\theta \sim \text{Exp}(p)$ and $w$ is a random variable independent of $\theta$. These products arise in the solution of the generalised Kronig–Penney model with the supersymmetric interaction of Example \[supersymmetricExample\], i.e. $$B = \begin{pmatrix}
\e^{w} & 0 \\
0 & \e^{-w}
\end{pmatrix}\,.$$
Let $\varrho$ denote the density of $w$. The Frisch–Lloyd equation (\[frischLloydEquation\]) is $$(z^2 + E)\,f(z)
+ p \int_{\mathbb R} \d w \int_{z}^{z \e^{-2w}}\d t \, f(t)\,\varrho(w) =N\,.$$ After changing the order of integration, this becomes $$N = (z^2 + E)\,f(z)
+ p \int_{0}^\infty K\left( \frac{1}{2} \ln \frac{z}{t} \right)\,
f(t) \,\d t
\label{frischLloydForSusy}$$ where $K$ is the kernel defined by Equation (\[deltaKernel\]).
Let $$\pm w \sim \text{\tt Exp}(q)\,.$$ Then the kernel is supported on ${\mathbb R}_{\pm}$ and satisfies the differential equation $$K'(x) = \mp q K(x)\,, \;\; K(0\pm) = \mp 1\,.$$ We deduce $$\frac{\d}{\d z} \left [ (z^2 + E) \,f(z) \right ]
- p \,f(z) \pm q \frac{ z^2 + E}{2z} \,f(z) = \pm \frac{q}{2z}\, N
\label{susyDifferentialEquation}$$ where $N$ is the integrated density of states.
Before going on to solve this equation, let us make a general remark: in the supersymmetric case, if one knows the invariant density, say $f_+$, for a certain distribution of the strength $w$, then one can easily deduce the invariant density, say $f_-$, when the sign of the strength is reversed. For instance, in the case $E=1$, the relationship between $f_-$ and $f_+$ is simply $$f_-(z) = \frac{1}{z^2} f_+ \left ( - \frac{1}{z} \right )\,.$$ This relationship can be deduced directly from the form of the matrices in the product (\[susyInteractionForPositiveEnergy\]). It is also connected with the fact that changing the sign of the superpotential $W$ in Example \[supersymmetricExample\] corresponds to swapping the functions $\phi$ and $\psi$— a manifestation of the so-called supersymmetry of the Hamiltonian.
### Product of the form (\[susyInteractionForPositiveEnergy\]).
For $E=k^2 >0$ and $\pm w \sim {\tt Exp}(q)$, we have $$\begin{gathered}
f(z) = N \frac{\mp \frac{q}{2} |z|^{ \mp \frac{q}{2}}}{z^2+k^2}
\exp \left[ \frac{p}{k} \arctan \frac{z}{k} \right] \\
\times \int_z^{c_{\pm}(z)} | t |^{\pm \frac{q}{2}}
\exp \left[ -\frac{p}{k} \arctan \frac{t}{k} \right] \frac{\d t}{t}
\label{susyExponentialDensityForEpositive}\end{gathered}$$ where $$c_+(z) = 0 \;\;\text{and}\;\; c_{-}(z) =
\begin{cases}
\infty & \text{if $z>0$} \\
-\infty & \text{if $z<0$}
\end{cases}
\,.$$
For the supersymmetric scatterer, $${\mathscr B}(z) = \e^{2 w} z\,.
\label{eq:Bsusy}$$ Hence, for $w \sim {\tt Exp}(q)$, the jumps increase the Riccati variable if it is already positive, and decrease it otherwise. Furthermore there is no bound on the magnitude of the jumps. It follows that the effect of increasing the density $p$ of the scatterers is to decrease the density $f$ on ${\mathbb R}_-$, and to increase it on ${\mathbb R}_+$. This is in agreement with the plots shown in Figure \[fig:fsusy\_pe\] (a).
For $-w \sim {\tt Exp}(q)$, we observe the opposite effect: as shown in Figure \[fig:fsusy\_pe\] (b), for increasing $p$, the density $f$ is lowered on ${\mathbb R}_+$ and raised on ${\mathbb R}_-$. The asymmetry of the plots for a negative strength $w$ is readily explained by using the particle analogy: starting at $+\infty$, the particle rolls down the potential, spurred along by the impurities, and quickly reaches the origin. Once the particle crosses over to the left, the impurities work against the downward force and tend to push the particle back towards the origin.
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fsusy\_pe\]
### Product of the form (\[susyInteractionForNegativeEnergy\]).
For $E = - k^2 <0$ and $w \sim {\tt Exp}(q)$, we must take $N=0$ in Equation (\[susyDifferentialEquation\]) to obtain a normalisable solution. This is consistent with the well-known fact that the spectrum of a supersymmetric Schrödinger Hamiltonian must be contained in $\mathbb{R}_+$. Hence $$f(z) = C_+^{-1} \frac{z^{-\frac{q}{2}}}{z^2-k^2}
\left ( \frac{z-k}{z+k} \right )^{\frac{p}{2 k}} \,{\mathbf 1}_{(k,\infty)}(z)\,.
\label{susyExponentialDensityForEnegativePlus}$$ For $-w \sim {\tt Exp}(q)$, the solution is, instead, $$f(z) = C_-^{-1} \frac{z^{\frac{q}{2}}}{k^2-z^2}
\left ( \frac{k-z}{k+z} \right )^{\frac{p}{2 k}} \,{\mathbf 1}_{(0,k)}(z)\,.
\label{susyExponentialDensityForEnegativeMinus}$$ By Formula 8 in [@GR], §3.197, $$C_{\pm} = k^{\mp \frac{q}{2}-1} \,
{\tt B}\!\left( \frac{q}{2}+1,\frac{p}{2k} \right) \,
{_2}F_1\!\left(
\frac{p}{2k}+1,\frac{q}{2}+1;\frac{p}{2k}+\frac{q}{2}+1;-1
\right)
\label{susyNormalisationConstant}$$ where ${\tt B}$ is the beta function and ${_2}F_1$ is Gauss’s hypergeometric function.
 
(0,0) (-85,80)[(a)]{} (95,80)[(b)]{}
\[fig:fsusy\_ne\]
Plots of the invariant density are shown in Figure \[fig:fsusy\_ne\]. As before, the particle analogy helps to explain their qualitative features: in view of Equation , when $w \sim {\tt Exp}(q)$, the “particle” must eventually end up to the right of the equilibrium point $z=k$; see Figure \[fig:fsusy\_ne\] (b). By contrast, when $-w \in {\tt Exp}(q)$ and $z>0$, the Riccati variable remains positive but its value decreases at every jump. Hence, in this case, the support of the invariant density is $(0,k)$; see Figure \[fig:fsusy\_ne\] (b).
### Calculation of the characteristic function.
The essential spectrum— and hence also the characteristic function— is invariant under a change of sign of the strength $w$. So we need only consider the case $w \sim {\tt Exp}(q)$. For $E = -k^2$, we find, by using Equation (\[susyExponentialDensityForEnegativePlus\]), $$\notag
\dashint_{-\infty}^{\infty} \d z \, z\, f(z) = C_+^{-1} \int_k^{\infty} \d z \,
\frac{z^{-\frac{q}{2}+1}}{z^2-k^2}
\left ( \frac{z-k}{z+k} \right )^{\frac{p}{2 k}} \,.$$ The normalisation constant $C_+$ is given explicitly by Formula (\[susyNormalisationConstant\]), and a similar formula is available for the definite integral; it suffices to replace $q/2$ by $q/2-1$. Using and the fact that $N(-k^2)=0$, the result is $$\notag
\Omega (-k^2) = -\frac p q +
k \frac{{\tt B} \left ( \frac{p}{2 k}, \frac{q}{2} \right ) {_2} F_{1} \left ( \frac{p}{2k}+1, \frac{q}{2}; \frac{p}{2k}+\frac{q}{2};-1 \right ) }{{\tt B} \left ( \frac{p}{2 k}, \frac{q}{2} +1 \right ) {_2} F_{1} \left ( \frac{p}{2k}+1, \frac{q}{2}+1; \frac{p}{2k}+\frac{q}{2}+1;-1 \right )}\,.$$ This formula extends to other values of the energy by analytic continuation; it suffices to replace $k$ by $\sqrt{-E}$. In particular, for $E = k^2 > 0$, the characteristic function is obtained by replacing $k$ by $-\i k$. The density of states and the Lyapunov exponent may then be deduced from the formulae (see Equation ) $$N = -\frac{1}{\pi} \,\text{Im} \,\Omega
\;\;\text{and}\;\;
\gamma = \text{Re} \,\Omega\,.$$
Extensions {#extensionSection}
==========
In this final section, we consider possible extensions of our results: (1) to another scatterer; (2) to another distribution of the strength of the scatterers and (3) to another distribution of the spacing between consecutive scatterers.
Double impurities {#doubleImpuritySubsection}
-----------------
The decomposition formula (\[gramSchmidtDecomposition\]) gives a formal correspondence between products of $2 \times 2$ matrices and generalised Kronig–Penney models of unit energy where the point scatterers are double impurities. In the particular case $$\theta \sim {\tt Exp} (p)\,,$$ the density of the invariant measure solves the Frisch–Lloyd equation $$N = (z^2+1) f(z) + p \int_{\mathbb R} \int_{\mathbb R} \d u\, \d w \, \varrho(u,w)\,\int_z^{z \e^{-2 w}-u} f(y)\,\d y \,.
\label{frischLloydForDoubleImpurity}$$
We have already considered the cases where $w$ vanishes almost surely (the delta scatterer) or $u$ vanishes almost surely (the supersymmetric scatterer). The purpose of this subsection is to consider the truly multivariate case where $u$ and $w$ are independent and $$u \sim {\tt Exp} (q_d)\,,\;\; w \sim {\tt Exp}(q_s)\,.$$ We shall show that the corresponding invariant density $f$ solves the differential equation $$\frac{\d}{\d z} \left [ 2 z \left ( \varphi' - p \,f \right ) \right ] + \left ( q_s + 2 q_d z \right ) \left ( \varphi' - p\, f \right )
+ q_d q_s \varphi = q_d q_s N
\label{doubleImpurityDifferentialEquation}$$ where $$\varphi := (z^2+1) f(z)$$ and $N$ is independent of $z$. To derive this equation from (\[frischLloydForDoubleImpurity\]), we shall consider the cases $z \le 0$ and $z>0$ separately.
Consider the latter case; we write $$\varrho (u,w) = \varrho_d (u) \,\varrho_s(w)\,.$$ By changing the order of integration, we find $$\begin{gathered}
\notag
\int_0^\infty \d w \,\varrho_s(w)\, \int_z^{z \e^{-2 w}-u} f(y) \,\d y \\
=
-\int_{z-u}^{z} f(y)\,\d y + \int_{-u}^{z-u} K_s \left ( \frac{1}{2} \ln \frac{z}{y+u} \right ) f(y)\,\d y\,,\end{gathered}$$ where $$K_s (x) := \begin{cases}
-\e^{-q_s x} & \text{if $x \ge 0$} \\
0 & \text{if $x < 0$}
\end{cases}\,.$$ So Equation (\[frischLloydForDoubleImpurity\]) becomes $$\begin{gathered}
\notag
N = \varphi(z) - p \int_0^{\infty} \d u \,\varrho_d(u) \int_{z-u}^{z} f(y) \,\d y \\ +
p \int_0^{\infty} \d u \,\varrho_d(u) \int_{-u}^{z-u} K_s \left ( \frac{1}{2} \ln \frac{z}{y+u} \right ) f(y)\,\d y \,,\end{gathered}$$ and, by using integration by parts for the first integral on the right-hand side, we find $$\begin{gathered}
N = \varphi(z) - p \int_0^\infty \e^{-q_d u} f(z-u) \,\d u \\
+ p \int_0^{\infty} \d u \,\varrho_d(u) \int_{-u}^{z-u} K_s \left ( \frac{1}{2} \ln \frac{z}{y+u} \right ) f(y)\,\d y \,.
$$ Next, differentiate this equation with respect to $z$. By exploiting the identities $$K_s' = -q_s K_s\,,\;\; K_s(0+) = 1$$ and $$\frac{\d}{\d u} \e^{-q_d u} = -q_d \,\e^{-q_d u}\,,$$ we deduce $$2 z \left [ \varphi'(z) - p \,f(z) \right ] + q_s \left [ \varphi(z)-N\right ] = q_s p \int_0^\infty \e^{-q_d u} \,f(z-u)\,\d u \,.
$$ The integral term may be eliminated by differentiating once more with respect to $z$, and we obtain eventually Equation (\[doubleImpurityDifferentialEquation\]).
The same equation is obtained if, instead, $z<0$. It is a trivial exercise to adapt these arguments to cater for cases where one or both of $u$ and $w$ is always negative. We do not know how to express the solution of this second-order linear differential equation in terms of known functions, except in the limiting cases $$q_d \;\text{fixed}\,,\;q_s \rightarrow \infty\,,$$ and $$q_d \rightarrow \infty\,,\; q_s \; \text{fixed}\,,$$ that have already been considered in §\[deltaSubsection\] and §\[susySubsection\] respectively.
Delta scatterers with a gamma distribution {#gammaSubsection}
------------------------------------------
The equation (\[frischLloydForDelta\]) with the kernel (\[deltaKernel\]) can be reduced to a purely differential form whenever $\varrho$ solves a linear differential equation with piecewise constant coefficients. For instance, suppose that $$\pm u \sim {\tt Gamma}(2,1/q)\,,$$ i.e. $$\varrho (u) =
\pm q^2 u \e^{\mp q u} {\mathbf 1}_{{\mathbb R}_{\pm}}(u)\,.$$ Then $$\left ( \frac{\d}{\d u} \pm q \right )^2 \varrho = 0\,.$$ Using the same trick as before, we obtain the following differential equation for $\varphi := (z^2+E) f$: $$\left ( \frac{\d}{\d z} \pm q \right )^2 \varphi - p \left ( \frac{\d}{\d z} \pm 2 q \right ) \frac{\varphi}{z^2+E} = q^2 N\,.
\label{frischLloydGammaEquation}$$ Suppose that $E = k^2 >0$ and use the ansatz $$\varphi (z) := \exp \left [ \mp q z + \frac{p}{k} \arctan \frac{z}{k} \right ] h (z)\,.$$ Then $$(z^2+k^2) \,h '' + p \,h' \mp p \,q \,h =0\,.
\label{gammaEquation}$$ This equation may be solved in terms of hypergeometric series and, by imposing suitable auxiliary conditions, we can find a particular solution $h$ that is positive. The method of variation of constants then yields $$\begin{gathered}
f(z) = \frac{q^2 N}{z^2+k^2} \exp \left [ \mp q z + \frac{p}{k} \arctan \frac{z}{k} \right ] h(z) \\
\times \int_{\mp \infty}^z
\exp \left [\pm q t - \frac{p}{k} \arctan \frac{t}{k} \right ] H(t) \,\d t\,,
\label{densityForTheFrischLloydGammaCaseAndEpositive}\end{gathered}$$ where $$H(z) := \frac{\e^{\mp q z}}{h^2(z)} \int_{\mp \infty}^z \e^{\pm q t} h(t)\,\d t\,.$$
We now return to the calculation of the function $h(z)$ appearing in this formula. The general solution of Equation (\[gammaEquation\]) takes a remarkably simple form when $$p \,q = j(j-1)\,,\;\; j \in {\mathbb N}\,.$$ Indeed, substitute $$h(z) := \sum_{i=0}^\infty \frac{a_i}{i!} z^i\,,\;\; a_0 = 1\,,
\label{gammaSeries}$$ into Equation (\[gammaEquation\]). This yields a recurrence relation for the $a_i$: $$k^2 a_{i+2} = -p \,a_{i+1} + \left [ p\,q - i (i-1) \right ] a_{i-1} = 0\,,\;\; i=0,\,1,\, \ldots\,.$$ By choosing $a_1$ so that $a_{j+1}=0$, the infinite series reduces to a polynomial, say $P_{j}$: $$P_2(z) = 1+2\,{\frac {pz}{{p}^{2}+2\,{k}^{2}}}+2\,{\frac {{z}^{2}}{{p}^{2}+2\,{
k}^{2}}}
\,.$$ $$P_3(z) = 1+6\,{\frac { \left( {p}^{2}+4\,{k}^{2} \right) z}{p \left( {p}^{2}+10
\,{k}^{2} \right) }}+18\,{\frac {{z}^{2}}{{p}^{2}+10\,{k}^{2}}}+24\,{
\frac {{z}^{3}}{p \left( {p}^{2}+10\,{k}^{2} \right) }}
\,.$$ $$\begin{gathered}
\notag
P_4(z) = 1+12\,{\frac {p \left( {p}^{2}+16\,{k}^{2} \right) z}{{p}^{4}+28\,{p}^
{2}{k}^{2}+72\,{k}^{4}}}+72\,{\frac { \left( {p}^{2}+6\,{k}^{2}
\right) {z}^{2}}{{p}^{4}+28\,{p}^{2}{k}^{2}+72\,{k}^{4}}}\\
+240\,{
\frac {p{z}^{3}}{{p}^{4}+28\,{p}^{2}{k}^{2}+72\,{k}^{4}}}+360\,{\frac
{{z}^{4}}{{p}^{4}+28\,{p}^{2}{k}^{2}+72\,{k}^{4}}}
\,.\end{gathered}$$
Another solution may be found by setting $$h(z) := (z^2+k^2) \exp \left [ -\frac{p}{k} \arctan \frac{z}{k} \right ] \sum_{i=0}^\infty \frac{b_i}{i!} \,z^i\,,\;\; b_0 = 1\,.$$ Then $$k^2 b_{i+2} = p \,b_{i+1} + \left [ p\,q - (i+2)(i+1) \right ] b_{i} = 0\,, \;\; i=0,\,1,\,\ldots\,.$$ By choosing $b_1$ so that $b_{j-1} = 0$, this series reduces to another polynomial, say $Q_{j}$: $$Q_2 (z) = 1\,.$$ $$Q_3 (z) = 1- \frac{4}{p} z\,.$$ $$Q_4 (z) = 1-10\,{\frac {pz}{6\,{k}^{2}+{p}^{2}}}+30\,{\frac {{z}^{2}}{6\,{k}^{2}
+{p}^{2}}}\,.$$ Hence the general solution of Equation (\[gammaEquation\]) is $$h(z) = c_1 P_j (z) + c_2 Q_j(z) (z^2+k^2) \exp \left [ -\frac{p}{k} \arctan \frac{z}{k} \right ]\,.
\label{generalSolution}$$
Even with such detailed knowledge, it is not straightforward to identify the particular solution $h$ that yields the density. We end with the remark that the characteristic function may, nevertheless, be constructed by using Halperin’s trick: in this case, the homogeneous version of Equation (\[EquationForDeltaScatterers\]) is $$F''(x) + \left \{ -E \pm \frac{p/q}{(1 \pm \i x/q)^2} \left [ 2 \pm \i x/q \right ] \right \} F(x) = 0\,.$$ The solutions are expressible in terms of Whittaker functions; in particular, for $k$ real, $$\Omega (k^2) = \gamma (k^2) - \i \pi N (k^2) = 2\i k \,
\frac{W_{-\i\frac{p}{2k},\frac{\sqrt{1 \pm 4pq}}{2}}' \left ( \mp 2 \i k q \right
)}{W_{-\i\frac{p}{2k},\frac{\sqrt{1 \pm 4pq}}{2}} \left ( \mp 2 \i k q \right )}\,.$$ This result was originally found by Nieuwenhuizen [@Ni].
An alternative derivation of the Frisch–Lloyd equation {#alternativeSubsection}
------------------------------------------------------
In deriving Equation (\[frischLloydEquation\]), we made explicit use of the fact that, when the spacing $\theta_j := x_{j+1}-x_j$ is exponentially distributed, $$n(x) := \# \left \{ x_j :\; x_j < x \right \}$$ is a Poisson process. In this subsection, we outline an alternative derivation of the Frisch–Lloyd equation which generalises to other distributions of the $\theta_j$.
There is no real loss of generality in setting $E=1$. We use the decomposition $$A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}
\, B\,.$$ Then $${\mathscr A}^{-1} = {\mathscr B}^{-1} \circ {\mathscr R}_{-\theta}$$ where $${\mathscr R}_{\theta} (z) := \frac{z \cos \theta-\sin \theta}{z \sin \theta+\cos \theta}\,.$$ Denote by $\varrho$ the density of the random variable $\theta$ and by $\kappa$ the distribution of $B$. Equation (\[integralEquation\]) for the invariant density $f$ then becomes $$\begin{gathered}
\notag
f(z) = \int_{\mathbb R} \d \theta\, \varrho (\theta) \int_{\text{SL} \left ( 2,{\mathbb R} \right )} \kappa ( \d B )
\left [ f \circ {\mathscr B}^{-1} \circ {\mathscr R}_{-\theta} \right ] (z) \frac{\d}{\d z} \left [ {\mathscr B}^{-1} \circ {\mathscr R}_{-\theta} \right ] (z) \\
= \int_{\mathbb R} \d \theta\, \varrho (\theta) \int_{\text{SL} \left ( 2,{\mathbb R} \right )} \kappa ( \d B )
\left [ f \circ {\mathscr B}^{-1} \right ] (w) \frac{\d {\mathscr B}^{-1}}{\d z} (w) \,\frac{\partial w}{\partial z}
$$ where $$w := {\mathscr R}_{-\theta} (z) = \frac{z \cos \theta+ \sin \theta}{-z \sin \theta + \cos \theta}\,.$$ The same equation can also be written in the more compact form $$f(z) = \int_{\mathbb R} \d \theta\, \varrho (\theta) \frac{\partial}{\partial z} \int_{\text{SL} \left ( 2,{\mathbb R} \right )}
\kappa (\d B) \int_0^{{\mathscr B}^{-1}(w)} \d t f(t)\,.
\label{alternativeIntegralEquation}$$ Now, $$\frac{\partial w}{\partial \theta} = (1+z^2) \frac{\partial w}{\partial z} \,.$$ Hence, if we multliply Equation (\[alternativeIntegralEquation\]) by $1+z^2$, we obtain $$(1+z^2) f(z) = \int_{\mathbb R} \d \theta\, \varrho (\theta) \frac{\partial}{\partial \theta} \int_{\text{SL} \left ( 2,{\mathbb R} \right )}
\kappa (\d B) \int_0^{{\mathscr B}^{-1}(w)} \d t f(t)\,.$$ Next, differentiate with respect to $z$: $$\begin{gathered}
\notag
\frac{\d}{\d z} \left [ (1+z^2) f(z) \right ] = \int_{\mathbb R} \d \theta\, \varrho (\theta) \frac{\partial^2}{\partial z \partial \theta} \int_{\text{SL} \left ( 2,{\mathbb R} \right )}
\kappa (\d B) \int_0^{{\mathscr B}^{-1}(w)} \d t f(t) \\
= \int_{\mathbb R} \d \theta\, \varrho (\theta) \frac{\partial^2}{\partial \theta \partial z} \int_{\text{SL} \left ( 2,{\mathbb R} \right )}
\kappa (\d B) \int_0^{{\mathscr B}^{-1}(w)} \d t f(t)\,.\end{gathered}$$ We may then use integration by parts for the outer integral; in the particular case $$\varrho (\theta) = p \e^{-p \theta} {\mathbf 1}_{[0,\infty)}$$ the Frisch–Lloyd equation (\[frischLloydEquation\]) follows easily after invoking (\[alternativeIntegralEquation\]) once more.
We can use the same trick whenever the density of $\theta$ satisfies a linear differential equation with constant coefficients. For instance, in the case $$\theta \sim {\tt Gamma} (2, p)$$ it may be shown that $$\frac{\d}{\d z} \left [ (1+z^2) \varphi'(z) \right ] -2 p \varphi'(z) + p^2 f(z) =
p^2 \int_{\text{SL} \left (2,\mathbb R \right )} \kappa (\d B) \left [ f \circ {\mathscr B}^{-1} \right ](z) \frac{\d {\mathscr B}^{-1}}{\d z} (z)$$ where $$\varphi (z) = (1+z^2) f(z)\,.$$
Conclusion {#sec:Conclusion}
==========
In this article we have studied the invariant measure of products of random matrices in $\text{SL}\left(2,{\mathbb R}\right)$. This study relied on the correspondence between such products and a certain class of random Schrödinger equations in which the potential consists of point scatterers. We have considered several instances of this correspondence: delta, delta-prime and supersymmetric scatterers. By generalising the approach developed by Frisch & Lloyd for delta scatterers, we have obtained an integral equation for the invariant density of a Riccati variable; this density yields the invariant measure of the product of random matrices. For the three cases of point scatterers we have obtained explicit formulae for the invariant measures. These are the main new results of this paper.
The integrated density of states and the Lyapunov exponent of these models were also calculated. Two approaches were used for this purpose: the first is “Halperin’s trick” and is specific to the case of delta scatterers (cf. section \[halperinSubsection\]); the second uses analytic continuation of the characteristic function and depends on the explicit knowledge of the invariant measure in some interval of the energy outside the spectrum. By the first of these methods we have recovered the results of Nieuwenhuizen in the case of delta scatterers. By the second method we have found new explicit formulae for the integrated density of states and for the Lyapunov exponent in the cases of delta-prime and of supersymmetric scatterers.
All these analytical results were obtained when the spacing between consecutive scatterers, as well as the impurity strength, have exponential distributions. Possible extensions to the gamma distribution were also discussed.
A more complicated type of scatterer, combining the delta and the supersymmetric scatterers, has also been examined. We called this scatterer the “double impurity”; it is interesting because every product of matrices in $\text{SL}\left(2,{\mathbb R}\right)$ may in principle be studied by considering a Schrödinger problem whose potential consists of double impurities. Although we succeeded in deriving a differential equation for the invariant measure associated with a particular distribution of such scatterers, we were unable to express its solution in terms of known functions.
In this paper we have played down the physical aspects of the models. Apart from the inverse localisation length and the density of states, there are other physical quantities that bear some relation to the Riccati variable and whose statistical properties are of interest. Let us mention three of them: the most obvious is the phase of the reflexion coefficient on the disordered region; for a semi-infinite disordered region, its distribution is trivially related to the invariant density of the Riccati variable [@BarLuc90; @GolNosSch94; @Lu]. Another quantity is the Wigner time delay (the derivative of the phase shift with respect to the energy); it has been considered in the contexts of the Schrödinger [@JayVijKum89; @TexCom99] and Dirac [@SteCheFabGog99] equations. A third quantity is the transmission coefficient (i.e. conductance) [@AntPasSly81; @DeyLisAlt01; @SchTit03]. The study of the distributions of the Wigner time delay and of the transmission coefficient is mathematically more challenging because it requires the analysis of some joint distributions; for this reason it has been confined so far to limiting cases.
Some of the physical aspects arising from our results will form the basis of future work.
Scattering, transfer and boundary matrices {#app:ScatteringTransfer}
==========================================
We discuss in this section the relationship between the scattering matrix $S$, defined by , and the boundary matrix $B$, defined by . Here we need not assume that the scatterer is necessarily pointlike; the scattering matrix and the corresponding boundary matrix could equally well describe the effect of a potential supported on an interval.
We first write the scattering matrix in terms of transmission and reflexion probability amplitudes $t$, $t'$ and $r$, $r'$ : $$S =
\begin{pmatrix} r & t' \\ t & r' \end{pmatrix}
\:.$$ Current conservation implies $$|a^\text{out}_+|^2+|a^\text{out}_-|^2=|a^\text{in}_-|^2+|a^\text{in}_+|^2$$ and so forces the scattering matrix to be unitary, i.e. $S \in U(2)$. The constraints on the coefficients, namely $$|r|^2+|t|^2=|r'|^2+|t'|^2=|r'|^2+|t|^2=|r|^2+|t'|^2=1\,,$$ $${r'}/{t'}=-\overline{{r}/{t}} \;\;\text{and}\;\; {r}/{t'}=-\overline{{r'}/{t}}\,,$$ are conveniently built into the following parametrisation, which also illustrates the factorisation $U(2)=U(1)\times{}SU(2)$: $$S = \i\e^{\i\theta}
\begin{pmatrix}
\e^{\i\varphi} \sqrt{1-\tau} & -\i\e^{-\i\chi} \sqrt{\tau} \\[0.1cm]
-\i\e^{\i\chi} \sqrt{\tau} & \e^{-\i\varphi} \sqrt{1-\tau}
\end{pmatrix}
\:.
\label{eq:ParametrisationS}$$ This representation of the scattering matrix is interesting because the four real parameters have a clear physical interpretation: $\tau\in[0,1]$ is the probability of transmission through the scatterer; $\theta$ is the global phase of the matrix, i.e. $$\det{}S=-\e^{2\i\theta}\,.$$ It is sometimes referred to as the “Friedel phase” since it is the phase appearing in the Krein–Friedel sum rule relating the local density of states of the scattering region to a scattering property. The phase $\varphi$ is a measure of the left-right asymmetry ($\varphi=0$ or $\pi$ corresponds to a scattering invariant under $x\to-x$). Finally the phase $\chi$ is of magnetic origin, since time reversal corresponds to transposition of the scattering matrix.
Next, we introduce the transfer matrix $T$ relating left and right amplitudes: $$\begin{pmatrix}
a^\text{out}_+ \\[0.1cm] a^\text{in}_+
\end{pmatrix}
=
T
\begin{pmatrix}
a^\text{in}_- \\[0.1cm] a^\text{out}_-
\end{pmatrix}
\hspace{0.5cm}\mbox{where}\hspace{0.5cm}
T = \begin{pmatrix}
\overline{1/t} & -\overline{r/t} \\[0.1cm] -r/t' & 1/t'
\end{pmatrix}
\:.$$ This matrix is useful when considering the cumulative effect of many scatterers because it follows a simple composition law. Again, current conservation implies that the transfer matrix is unitary: $$\left | a_+^{\text{out}} \right |^2-\left | a_+^{\text{in}} \right |^2 = \left | a_-^{\text{in}} \right |^2-\left | a_-^{\text{out}} \right |^2\,.$$ In other words, $T\in{}U(1,1)$ (note that $\det{}T=t/t'=\e^{2\i\chi}$).
The boundary matrix is also a “transfer matrix” in the sense that it connects properties of the wavefunction on both sides of the scatterer. The relation between $T$ and $B$ is easily found: from $$\begin{pmatrix}
\psi'(0\pm) \\ \psi(0\pm)
\end{pmatrix}
=
\begin{pmatrix} \i k & -\i k \\ 1 & 1 \end{pmatrix}
\begin{pmatrix}
a_{\pm}^{\substack{\text{out} \\ \text{in}}} \\[0.1cm] a_{\pm}^{\substack{\text{in} \\ \text{out}}}
\end{pmatrix}$$ we deduce $$B=U\,T\,U^\dagger
\hspace{0.5cm}\mbox{where}\hspace{0.5cm}
U = \frac{\e^{-\i\pi/4}}{\sqrt{2k}}
\begin{pmatrix} \i k & -\i k \\ 1 & 1 \end{pmatrix}
\:.$$ Then, using the parametrisation , we arrive at the following alternative form of Equation : $$\label{eq:2}
B = \frac{\e^{\i\chi}}{\sqrt{\tau}}
\begin{pmatrix}
\cos\theta - \sin\varphi\, \sqrt{1-\tau}
& -k \left [ \sin\theta + \cos\varphi\, \sqrt{1-\tau} \right ] \\[0.1cm]
\frac1k \left [ \sin\theta - \cos\varphi\, \sqrt{1-\tau} \right ]
& \cos\theta + \sin\varphi\, \sqrt{1-\tau}
\end{pmatrix}$$ In particular, this expression shows clearly that $$\e^{-\i\chi}B\in\mathrm{SL}(2,{\mathbb R})\,.$$ In one dimension, if a magnetic field is present, it may always be removed by a gauge transformation. Furthermore, setting the magnetic phase in the exponential factor to zero does not affect the spectrum of the Schrödinger operator. Hence there is no loss of generality in restricting our attention to the case $B\in\mathrm{SL}(2,{\mathbb R})$.
We end this appendix with some examples of scatterers, expressed in terms of the parameters $\chi$, $\tau$, $\theta$ and $\varphi$.
For $\tau=1$, $\varphi=0$ and $\chi=0$, $B$ is the matrix describing a rotation of angle $\theta=k\ell$. In this case, the “scattering” is equivalent to free propagation through an interval of length $\ell$.
The scattering matrix for the delta impurity may be written as $$S = \e^{\i\theta}
\begin{pmatrix}
\i\sin\theta & \cos\theta \\ \cos\theta & \i\sin\theta
\end{pmatrix}$$ where $$\theta=-\arctan\frac{u}{2k} \in ( -\pi/2,\,\pi/2)\,.$$ The other parameters are given by $\chi =0$, $$\varphi = \begin{cases}
0 & \text{if $u<0$} \\
\pi & \text{if $u>0$}
\end{cases}$$ and $$\tau=\left [ 1+ \left ( \frac{u}{2k} \right )^2 \right ]^{-1}\,.$$
For the delta-prime scatterer, the scattering matrix $S$ has the same form as in the previous example, but this time with $$\theta=\arctan\frac{vk}{2} \in ( -\pi/2,\,\pi/2)\,,$$ $$\chi = 0\,,$$ $$\varphi = \begin{cases}
0 & \text{if $v<0$} \\
\pi & \text{if $v>0$}
\end{cases}\,,$$ and $$\tau=\left [1+\left ( \frac{vk}{2}\right )^2 \right ]^{-1}\,.$$
The supersymmetric scatterer corresponds to taking $\chi=\theta=0$, $\varphi=-\pi/2$ and $\tau=\text{sech}^2 w$.
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DOI : 10.1007/s10955-010-0005-x. The final publication is available at www.springerlink.com
|
---
abstract: 'Conventional topological insulators and superconductors have topologically protected nodal points on their boundaries, and the recent interests in nodal-line semimetals only concerned bulk band structures. Here, we present a novel four-dimensional topological insulator protected by an anti-unitary reflection symmetry, whose boundary band has a single $PT$-symmetric nodal line with double topological charges. Inspired by the recent experimental realization of the four-dimensional quantum Hall effect, we also propose a cold-atom system which realizes the novel topological insulator with tunable parameters as extra dimensions.'
author:
- 'L. B. Shao'
- 'Y. X. Zhao'
bibliography:
- '4D-TI-Ref.bib'
title: 'Four-Dimensional Topological Insulators with Nodal-Line Boundary States'
---
*Introduction* Topological insulators (TIs) and superconductors have been one of the most developed and prosperous fields in condensed matter physics during the past decade [@Kane-RMP; @XLQi-RMP]. We now understand that these topological phases are protected by certain symmetries, and a complete classification has been made for ten Altland-Zirnbauer (AZ) symmetry classes [@AZ-classes] and for all dimensions, even beyond the physical limit of three dimensions [@Schnyder-classification; @Kitaev-Classification; @Classification-RMP]. Historically the four-dimensional (4D) time-reversal ($T$) invariant TI played a key theoretical role in exploring topological phases [@Zhang-4D-TI], because its dimension reductions give rise to TIs in three and two dimensions [@Qi-PRB-2008]. However, it has only become promising until recently to experimentally explore the topological phases in dimensions beyond three [@4D-Hall-ColdAtom-Theory; @4D-Hall-Cold-Atom-Exper; @4D-Hall-Photonic]. Such possibilities reside in two facts. First, the topological invariants are defined in a band theory, and therefore can be well applied to artificial periodic systems other than electronic crystals, such as photonic [@Haldane-photonic-crystal] and phononic [@Prodan-Phonon; @Kane-Phonon] crystals, and cold atoms in optical lattices . Second, extra dimensions parametrized by momenta can be regarded as highly tunable parameters of these artificial systems, namely, that extra dimensions can be, in a sense, synthesized [@Thouless-Pump; @4D-Hall-ColdAtom-Theory; @4D-Hall-Cold-Atom-Exper; @4D-Hall-Photonic].
Another main focus of topological phases is the topological (semi)metals and nodal superconductors [@Classification-RMP; @Weyl-Dirac-Semimetal], which mainly concerns the topological charges of band crossings [@Volovik-book; @ZhaoWang-Classification]. While Weyl semimetals are of fundamental interest with band crossings occuring at discrete Weyl points in the Brillouin zone (BZ) [@Volovik-book; @NN-NoGo; @XGWan-WSM], nodal-line semimetals, which have the band crossings forming lines in the BZ, are recently a hot topic [@YuRui-Nodal-Line; @Kim-PT-DNLSM; @FangChen-Nodal-Line; @DWZ-ColdAtom-PT; @Rui-Zhao-Schnyder-DNL-Trans]. Belonging to the spacetime-inversion ($PT$) symmetric classification of topological gapless phases [@Zhao-Schnyder-Wang-PT], a nodal line can appear in generic locations in the 3D BZ with topological stability, provided $PT$ is preserved with $(PT)^2=1$ [@Note-Mirror-NL]. One of the significant features of TIs is that gapless modes can appear on boundaries, corresponding to nontrivial bulk topological invariants. Conventionally, the boundary gapless modes are located at isolated band-crossing points in the boundary BZ, which includes all topological phases in the periodic classification table of ten AZ symmetry classes [@Schnyder-classification; @Kitaev-Classification]. For instance, Weyl points of the same chirality appear on the boundary of the aforementioned 4D TI [@Zhang-4D-TI; @Qi-PRB-2008]. In this Letter, we present a 4D $\mathbb{Z}_2$ TI protected by an anisotropic two-fold anti-unitary spatial symmetry $RT$ with $(RT)^2=1$, where $R$ inverses the three dimensions and preserves the fourth. In contrast to boundary nodal points for conventional TIs, the crystalline TI generically has odd number of nodal-lines on the 3D boundary normal to the fourth dimension. Notably, the nodal lines have both 1D and 2D topological charges as detailed below, essentially different from ordinary nodal lines with only the 1D topological charge. We formulate the bulk $\mathbb{Z}_2$ topological invariant as the second Chern number over the half BZ, $T_{1/2}^4$, subtracted by the Chern-Simons integrals over the two $RT$-invariant boundaries of $T_{1/2}^4$. Analogous to the fact that the 4D $T$-invariant TI generalizes the 2D Chern insulator [@Zhang-4D-TI; @Qi-PRB-2008], the proposed 4D crystalline TI can be regarded as a 4D generalization of the well-known 2D $T$-invariant TI [@Fu-Kane-Pump]. Moreover, we discuss the general principles for realizing the topological insulator by artificial systems, and propose a cold-atom realization.
*Nodal line of double topological charges–* Let us begin by introducing the $PT$ symmetry. The spatial inversion $P$ is a unitary symmetry, which maps $\mathbf{x}$ to $-\mathbf{x}$, therefore $\mathbf{k}$ to $-\mathbf{k}$ in momentum space, while time-reversal $T$ as an anti-unitary symmetry maps $\mathbf{k}$ to $-\mathbf{k}$ as well. Accordingly, the combination $PT$ is an anti-unitary operator that leaves any $\mathbf{k}$ invariant. For spinless fermions, it satisfies $(PT)^2=1$. The nodal-line model we would like to emerge on the boundary of a TI is $$\mathcal{H}_{NL}(\mathbf{k})=\sum_{i=1}^3k_i\gamma_i+im_z\gamma_3\gamma_4. \label{Nodal-Line-Model}$$ The $4\times 4$ hermitian matrices $\gamma_\mu$ with $\mu=1,2\cdots,5$ satisfy the Clifford algebra $\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu\nu}$, among which $\gamma_i$ with $i=1,2,3$ are real, while $\gamma_4$ and $\gamma_5$ are purely imaginary. Thus, if $PT$ symmetry is represented as $\mathcal{PT}=\hat{ \mathcal{K}}$ with $\hat{ \mathcal{K}}$ being the complex conjugate, the model of Eq. as a real Hamiltonian is clearly $PT$ invariant. If $m_z=0$, the Hamiltonian of Eq. presents the real Dirac point, which can be regarded as a real generalization of the well-known Weyl points, namely the real counterpart of monopoles in the $PT$ symmetric real band theory [@FangChen-Nodal-Line; @Zhao-PT-Dirac]. But its monopole charge is characterized by a $\mathbb{Z}_2$ invariant $\nu^C_{\mathbb{R}}$, which is defined on a sphere $S^2$ surrounding the crossing point in the real band theory, in contrast to the $\mathbb{Z}$-valued Chern number in the complex band theory [@Zhao-PT-Dirac]. The second term of Eq. is a “partial” mass term, which anti-commutes with $\gamma_3$ while commutes with $\gamma_{1,2}$. Hence, when the second term of Eq. is turned on, the four-fold degeneracy at the momentum-space origin, though, is lifted, there still exist two-fold band crossing points forming a nodal circle of $k_x^2+k_y^2=m_z^2$ in the plane with $k_z=0$ [@FangChen-Nodal-Line]. The fact that the spectrum cannot be fully gapped by the $PT$-symmetric term actually reflects the nontrivial topological charge $\nu_{\mathbb{R}}^C$ of the real Dirac model. It is remarkable that, while the topological charge $\nu^C_{\mathbb{R}}$ on an $S^2$ enclosing the whole nodal circle is inherited, the nodal line also has nontrivial Berry phase $\nu^B_{\mathbb{R}}$ along any loop $S^1$ surrounding it, which is quantized by $PT$ symmetry, and therefore a $\mathbb{Z}_2$ topological invariant. Thus, the Hamiltonian of Eq. describes a $PT$-symmetric nodal-line semimetal of double topological charges.
![The nodal line with double topological charges in the boundary Brillouin zone. $S^2$ encloses the nodal line as a whole, while $S^1$ surrounds the nodal line locally in momentum space. \[Fig-nodal-line\]](Nodal-Line.pdf)
*The model–* We now proceed to construct a minimal model of 4D TIs, such that the nodal line of Eq. can correspond to its low-energy boundary theory. If there is no symmetry \[except $U(1)$\], the only TIs in four dimensions are second Chern insulators, for which Weyl fermions with given chirality exist on the boundary. Hence, certain symmetry is required to realize a boundary nodal line. On the other hand, we expect the bulk symmetry projected on the $3$D boundary to be $PT$ symmetry, since the nodal line is protected by $PT$ symmetry in three dimensions. Considering this condition, the symmetry we propose is an anti-unitary reflection symmetry $RT$. Here, $T$ is time-reversal symmetry, and $R$ is a unitary spatial symmetry operating as $$R: ~(w,\mathbf{x}) \mapsto (w,-\mathbf{x}),~~ (k_w, \mathbf{k})\mapsto (k_w, -\mathbf{k}).$$ in real and momentum spaces, respectively, where $\mathbf{x}$ is the three Cartesian coordinates for boundary, and $w$ is the coordinate perpendicularly towards the bulk. Combined with time reversal, $RT$ is anti-unitary with $\{RT,i\}=0$, and operates in momentum space as $$RT:~(-k_w, \mathbf{k})\mapsto (-k_w, \mathbf{k}),$$ recalling that time reversal inverses all momentum components. In general, $RT$ can be represented in momentum space as $\mathcal{RT}=U_{RT}\hat{ \mathcal{K}}\hat{I}_w$, where $\hat{I}_w$ is the inversion of $k_w$, and $U_{RT}$ is a unitary operator. We further requires that $(\mathcal{RT})^2=1$, or equivalently $U_{RT}^T=U_{RT}$. Explicitly, $PT$ symmetry constrains the momentum-space Hamiltonian as $$U_{RT}\mathcal{H}^*(\mathbf{k},k_w)U^\dagger_{RT}=\mathcal{H}(\mathbf{k},-k_w). \label{RT-H}$$ It is clear that $\mathcal{RT}$ with $(\mathcal{RT})^2=1$ can always be converted to be $\mathcal{RT}=\hat{\mathcal{K}}\hat{I}_w$ by a unitary transformation. Therefore, we assume $\mathcal{RT}=\hat{\mathcal{K}}\hat{I}_w$ in the model construction.
The minimal model we shall construct has eight bands, which can be inferred from the fact that the doubly charged nodal line has four bands. It is conventional for TIs that the bulk band number is a double of that of the boundary, for instance the $4$D Chern insulator has four bands with the boundary Weyl points being two-band crossings. Hence, we shall use the Clifford algebra $Cl_6$, which gives rise to a basis for $8\times 8$ matrices, as building blocks. In $8\times 8$ matrices, there are at most seven mutually anti-commuting hermitian $8\times 8$ ones, $\Gamma_a$ , namely $\{\Gamma_a,\Gamma_b\}=2\delta_{ab}$, and $\Gamma_a^\dagger=\Gamma_a$, with $a,b=0,1,\cdots,6$. Among them, following the standard convention, the first four, $\Gamma_a$ with $a=0,1,2,3$, are real, and remaining three are purely imaginary. For convenience, let $\mathbf{\Gamma}=(\Gamma_1,\Gamma_2,\Gamma_3)$, and the components be $\Gamma_i$ with $i=1,2,3$. An explicit representation of $\Gamma_a$ convenient for our purpose can be found in the Supplemental Material (SM) [@Supp].
A $RT$-symmetric massive Dirac Hamiltonian may be given by $
\mathcal{H}^{C}_0=\mathbf{k}\cdot\mathbf{\Gamma}+k_w\Gamma_4+m\Gamma_0
$ in the continuous form. To compactify momentum space, one may replace $m$ by $m-\lambda (k^2+k^2_w)$ to get the modified Dirac model. This is so far the standard procedure to construct a TI by Dirac matrices, which however is not sufficient in our case of $Cl_6$ in four dimensions. There are other terms including $i\Gamma_i\Gamma_{5, 6}$ with $i=1,2,3$, which are real and therefore $RT$-symmetric, but still hermitian. Such terms, $i\Gamma_i\Gamma_{5, 6}$, are “partial mass” terms, for $i\Gamma_i\Gamma_{5, 6}$ anti-commutes with $k_i\Gamma_i$, while commutes with the rest terms in the Hamiltonian, and turn out to be essentially important for boundary in-gap modes. For simplicity, adding only $i\Gamma_3\Gamma_{5}$ into the continuous Dirac model, we can readily obtain the lattice Dirac model, $$\begin{gathered}
\mathcal{H} =\sum_{a=1}^{4}\sin k_i\Gamma_i+[m-(\sum_{a=1}^4\cos k_i)]\Gamma_0+it\Gamma_3\Gamma_5, \label{The-bulk-model}\end{gathered}$$ which is clearly $RT$ invariant with $\mathcal{RT}=\hat{ \mathcal{K}}\hat{I}_w$, and serves as a minimal model for the novel 4D TI with nodal-line boundary.
*Nodal-line boundary states* We now open a 3D boundary for the Hamiltonian, Eq. , perpendicular to the $w$-direction, and discuss the in-gap boundary states of the semi-infinite system with Dirichlet boundary condition. Considering that translational symmetry is violated by the boundary in the $w$-direction, while is still preserved for the other dimensions, we introduce the ansatz $|\psi_\mathbf{k}\rangle=|\xi_{\mathbf{k}}\rangle\otimes\sum_{i=0}^{\infty}\lambda^i|i\rangle$ with $|\lambda|<1$ for boundary eigenstates, where $i$ labels the $i$th site along the $w$-direction. Solving the Schrödinger equation for the semi-infinite system, we find that the boundary states occur in the 4D subspace corresponding to the positive eigenvalue of $i\Gamma_0\Gamma_4$ or equivalently the image of the projector $P=\frac{1}{2}(1+i\Gamma_0\Gamma_4)$, and $\lambda=m-\sum_{i=1}^3\cos k_i$ (for derivation details, see the SM) [@Supp]. It is noteworthy that $i\Gamma_0\Gamma_4$ anti-commutes with $\Gamma_0$ and $\Gamma_4$, while commutes with the other $\Gamma$’s, as well as $RT$ operator $\mathcal{RT}$. Hence, the boundary low-energy effective theory is given by $\mathcal{H}_{eff}(\mathbf{k})=P\mathcal{H}P$, for the momentum $\mathbf{k}$ satisfying $|\lambda|=|m-\sum_{i=1}^3\cos k_i|<1$. Particularly, we can focus on the eight high symmetry momenta $\mathbf{K^\alpha}$ with $K_i^\alpha=0$ or $\pi$ and $\alpha=1,\cdots,8$. If $|\lambda|=|m-\sum_{i=1}^3\cos K^\alpha_i|<1$, the coarse-grained theory around $\mathbf{K}^\alpha$ is explicitly given by $$\mathcal{H}^\alpha_{eff}(\mathbf{q})=\sum_{i=1}^3\eta^{\alpha,i}q_i\gamma_i+it\gamma_3\gamma_4, \label{Boundary-theory}$$ where $\mathbf{k}=\mathbf{K}^\alpha+\mathbf{q}$, and $\eta^{\alpha,i}$ are signs equal to $-\cos(K^\alpha_i)$. Here, we have introduced $\gamma$’s for $Cl_4$, which are specified as $\gamma_i=-P\Gamma_iP$ with $i=1,2,3$, $\gamma_4=P\Gamma_5P$ and $\gamma_5=P\Gamma_6P$, noting that $\Gamma_0$ and $\Gamma_4$ vanishes under the projection. Recall that $RT$ symmetry is projected to $PT$ in the boundary, but since $\mathcal{RT}$ commutes with the projector $P$, the low-energy effective theory of Eq. is clearly $PT$ invariant.
It is found that if $|m|>4$, there is no boundary state because $|\lambda|<1$ cannot be satisfied for any $\mathbf{k}$. If $2<m<4$ ($-4<m<-2$), the low-energy boundary effective theory of Eq. arises in the neighborhood of $\mathbf{K}=(0,0,0)$ $[(\pi,\pi,\pi)]$. If $0<m<2$ ($-2<m<0$), Eq. arises at three momenta, $\mathbf{K}=(\pi,0,0), (0,\pi,0), (0,0,\pi)$ \[$(\pi,\pi,0), (\pi,0,\pi), (0,\pi,\pi)$\]. However, in general a pair of Eq. ’s can be gapped by $RT$-invariant terms, leaving only a single gapless low-energy theory for the last case. Thus, we find that the boundary effective theory of Eq. in the topological phase corresponds exactly to the nodal-line model of Eq. , as we claimed.
![Two coordinate charts $X$ and $Y$ covering the half Brillouin zone. $Y$ covers the whole half BZ, but has a small singular region, which is covered by the regular chart $X$. The boundary of $X$ consists of two 3D tori, $\partial X=\tau_1'-\tau_2'$, and similarly for $Y$, $\partial Y=\tau_1-\tau_2$. Periodicity is assumed for $k_i$ with $i=1,2,3$. $Y_1$ and $Y_2$ are subspaces of $Y$, and can complement $X$ in the half Brillouin zone. \[T4-vortex\]](T4-Vortex.pdf)
*The bulk topological invariant–* In momentum space $RT$ symmetry relates $(\mathbf{k},k_w)$ to $(\mathbf{k},-k_w)$. According to Eq. , we can require, in a neighborhood of $(\mathbf{k},-k_w)$ and its mirror image, that $$U_{RT}|\alpha,\mathbf{k},k_w\rangle^*=|\alpha,\mathbf{k},-k_w\rangle, \label{State-basis-RT}$$ where $|\alpha,\mathbf{k},k_w\rangle$ are valence eigenstates of $\mathcal{H}(\mathbf{k},k_w)$ with $\alpha$ labeling valence bands. From Eq. , it is sufficient to consider only half of the BZ, $T^4_{1/2}$, as illustrated in Fig. \[T4-vortex\]. Particularly in two mirror-symmetric 3D tori, $\tau_1$ and $\tau_2$, namely the boundary of $T^4_{1/2}$ with $k_w=0$ and $\pi$, equation puts constraints on eigenstates pointwisely as a boundary condition. In the nontrivial topological phase, it is impossible to find a complete basis of valence bands, which are globally well-defined in $T^4_{1/2}$, and satisfy the boundary condition of Eq. as well. In other words, if $|\alpha,\mathbf{k},k_w\rangle_Y$, with $Y$ being the coordinate chart covering the whole $T^4_{1/2}$, are periodic for $k_i$ with $i=1,2,3$, and satisfy Eq. in $\tau_1$ and $\tau_2$, there must exist singular points in the bulk of $T^4_{1/2}$ for $|\alpha,\mathbf{k},k_w\rangle_Y$ which cannot be eliminated without violating the boundary condition given by Eq. , while in the trivial phase, we can smooth out all singular points. To characterize this, we first choose another coordinate chart $X$ in the bulk of $T^4_{1/2}$ (Fig. \[T4-vortex\]) covering all singular points of $Y$, and assume another basis of valence bands $|\alpha,\mathbf{k},k_w\rangle_X$ without singular point in $X$, which is always possible since Eq. puts no constraint in the bulk of $T^4_{1/2}$. Then, we can derive the transition function $t_{XY}$ from $X$ to $Y$ in the boundary of $X$, $\partial X=\tau_1'-\tau_2'$. As $t_{XY}\in U(N)$ with $N$ the valence-band number, and $\pi_3[U(N)]=\mathbb{Z}$, the transition function may has a nontrivial winding number $\mathcal{N}[t_{XY}]$. The winding number can be calculated by the Chern-Simons terms of the Berry connection on two charts restricted on $\partial X$, $\mathcal{A}_{\alpha\beta,i}^{X/Y}={}_{X/Y}\langle \alpha,k|\partial_{k_i}|\beta,k\rangle_{X/Y}$ as $\mathcal{N}=\int_{\partial X}Q_3(\mathcal{A}^X)-\int_{\partial X}Q_3(\mathcal{A}^Y)$, where $Q_3(\mathcal{A})=-\epsilon^{\mu\nu\lambda}\mathrm{tr}(\mathcal{A}_\mu\partial_\nu \mathcal{A}_\lambda+\frac{2}{3}\mathcal{A}_\mu\mathcal{A}_\nu\mathcal{A}_\lambda)/(8\pi^2) d^3k$. It can be shown that (see the SM for detailed derivations [@Supp]) $$\mathcal{N}=\int_{T^4_{1/2}} \mathrm{ch}_2(\mathcal{F})-\int_{\tau_1}Q_3(\mathcal{A})+\int_{\tau_2} Q_3(\mathcal{A})\mod 2,\label{Invariant}$$ where the gauge-invariant second Chern character is given by $\mathrm{ch}_2(\mathcal{F})=-\epsilon^{\mu\nu\lambda\sigma}\mathrm{tr}\mathcal{F}_{\mu\nu}\mathcal{F}_{\lambda\sigma}/(32\pi^2)d^4k$, and $\mathcal{A}$ can be chosen as $\mathcal{A}^Y$ or any basis for valence bands satisfying the boundary condition given by Eq. . The formula is a $\mathbb{Z}_2$ invariant, because a gauge transformation in either one of the Chern-Simons terms in Eq. can change $\mathcal{N}$ by an even integer, noticing that $\pi_3[O(N)]=2\mathbb{Z}$ in the eight-fold periodic homotopy groups of classifying spaces in real K theory [@Karoubi-book; @Zhao-Schnyder-Wang-PT]. Detailed discussions on the topological invariant of the minimal model, Eq. , can be found in the SM [@Supp]. It noteworthy that the topological invariant of Eq. may be regarded as a 4D generalization of the well-known topological invariant for 2D topological insulators with a $\mathbb{Z}_2$ pump interpretation [@Fu-Kane-Pump], where the first Chern character is replaced by the second, and accordingly the Berry phases are replaced by the Chern-Simons terms.
*Cold atom realization* To simulate a 4D system by a physical system with dimension $\le 3$, certain momenta have to be simulated by tunable parameters of the physical system. For the particular Hamiltonian of Eq. , merely the $w$-dimension has to be faithfully simulated in real space, so that a boundary can be opened for exploring the bulk-boundary correspondence. Hence, we simulate the 4D model by a 1D lattice, and identify the three momenta $\mathbf{k}$ with highly tunable parameters of the artificial system. Then, the band structure of the in-gap boundary states can be mapped out by tuning the parameters accordingly. Based on these general considerations of simulating a high-dimensional system, the model of Eq. , in principle, can be realized by artificial systems, such as photonic crystals, mechanical systems, and cold atoms. We now present a simulation by cold atoms trapped in a quasi 1D optical lattice as illustrated in Fig. \[simulation\]. To realize the eight bands of Eq. , cold atoms with pseudo spin are arranged to hop on the quasi 1D tetragonal optical lattice made by four parallel chains. Observing that each term of Eq. is a tensor product of three Pauli matrices, we assign them to the left-right, up-down and quasi-spin spaces, respectively.
![The quasi 1D tetragonal optical lattice. There are four sites positioned as a square in each unit cell, which are labeled as $|ur\rangle$, $|dr\rangle$, $|ul\rangle$ and $|dl\rangle$ according to their positions, respectively. All next-nearest hoppings are marked with arrows. Specifically, only green arrows involve spin-orbital interactions, and the phase difference of red and blue arrows is $\pi$. \[simulation\]](Lattice-1D.pdf)
In such a setup, the model can be well realized by the cutting-edge techniques of ultracold atom experiment. Particularly, all next-nearest-neighbor hoppings marked by colored arrows in Fig. \[simulation\] can be realized by photon-assisted tunnelings [@2015Zhang]. The spin-orbit couplings occur merely in an inner-cell term of Eq. (i.e., the green hoppings in Fig. \[simulation\]), which can be induced by a two-photon process when a pair of incident Raman lasers, counter-propagating along the green lines in Fig. \[simulation\], are resonant with the field-split internal levels [@2015Zhang]. Moreover, the phases of the hoppings can be created by synthetic gauge fields [@Zhu2006; @2011Atala; @Miy2013]. The technical details and the realization of all terms of Eq. are fully addressed in the SM [@Supp]. *Conclusion and Discussions* In conclusion, we have presented a 4D crystalline topological phases with nodal-line boundary states, and formulated its topological invariant. Furthermore, a minimal model has been constructed, and a cold-atom realization has been proposed. Finally, we discuss two aspects of the topological phase. First, the 2D topological charge of the nodal line is essential for the bulk boundary correspondence, not the 1D topological charge. The 2D topological charge implies that the boundary band structure with odd number of nodal lines cannot be realized by an independent 3D system, and therefore has to be connected to a higher-dimensional bulk, which follows from the index theorem of the bulk boundary correspondence [@Zhao-Index]. Indeed, any 3D $PT$ symmetric semimetal satisfies the Nielsen-Ninomiya no-go theorem, namely always has even number of such nodal lines [@NN-NoGo; @Zhao-PT-Dirac]. Second, the topological phase is related to time-reversal symmetry with $T^2=1$, so that $(RT)^2=1$ is satisfied, provided $[R,T]=0$. Although the requirement of $T^2=1$ is violated by electron systems with spin-orbital couplings (with $T^2=-1$), it is natural to artificial systems, such as photonic and phononic crystals, and so is to cold atoms with pseudo spin degrees of freedom.
*Acknowledgments–* We thank S. L. Zhu for discussions on the cold-atom realization. This work is supported by the startup funding of Nanjing University, China.
**Supplemental Materials: “Four-Dimensional Topological Insulators with Nodal-Line Boundary States"**
Clifford Algebras
=================
For the Clifford algebra $C_{2n}$, there are $2n$ generators, namely $2^n\times 2^n$ Dirac gamma matrices $\gamma_{a}$ with $a=1,2,\cdots,2n$, satisfying $$\{\gamma^a,\gamma^b\}=2\delta^{ab}.$$ It is clear that there is the $(2n+1)$th gamma matrix anti-commuting with all $\gamma_a$ with $a=1,\cdots,2n$, which is $$\gamma^{2n+1}=\pm i^n\gamma^1\gamma^2\cdots\gamma^{2n}.$$ Note that all gamma matrices are hermitian, $\gamma_a^\dagger=\gamma_a$, and $\gamma_a^2=1$, with $a=1,2,\cdots,2n+1$. In this paper, $Cl_4$ and $ Cl_6 $ appear as building blocks of the models. Since the representation of the Clifford algebra by $2^n\times 2^n$ matrices is unique up to unitary transformations, we adopt the following convention for gamma matrices for our convenience. For $Cl_6$, the $8\times 8$ gamma matrices are given by $$\begin{split}
&\Gamma_0=\sigma_1\otimes \sigma_0\otimes \sigma_0,~~~
\Gamma_1=\sigma_3\otimes\sigma_1\otimes\sigma_1,~~~ \Gamma_2=\sigma_3\otimes\sigma_1\otimes\sigma_3,~~~ \Gamma_3=\sigma_3\otimes\sigma_3\otimes \sigma_0,~~~\\
&\Gamma_4=\sigma_2\otimes \sigma_0\otimes \sigma_0,~~~ \Gamma_5=\sigma_3\otimes\sigma_1\otimes\sigma_2,~~~ \Gamma_6=\sigma_3\otimes\sigma_2\otimes \sigma_0,
\end{split}$$ where the first four $\Gamma_\mu$ with $\mu=0,1,\cdots,3$ are real, while the last three are purely imaginary. Then, we apply the projector $P=\frac{1}{2}(1+i\Gamma_0\Gamma_4)$ to get $4\times 4$ gamma matrices for $Cl_4$, $$\begin{split}
&\gamma_1=\sigma_1\otimes\sigma_1,~~~ \gamma_2=\sigma_1\otimes\sigma_3,~~~ \gamma_3=\sigma_3\otimes \sigma_0,~~~\gamma_4=\sigma_1\otimes\sigma_2,~~~ \gamma_5=\sigma_2\otimes \sigma_0,
\end{split}$$ where $\gamma_{i}=-P\Gamma_{i}P$ with $i=1,2,3$, $\gamma_{4}=-P\Gamma_{5}P$, and $\gamma_{5}=-P\Gamma_{6}P$. Note that $P\Gamma_{0,4}P=0$.
The boundary effective theory
=============================
To open a boundary perpendicular to the $w$-direction, we first apply the inverse Fourier transform for $k_w$ to get the first quantized Hamiltonian with the $w$-dimension in real space, $$\H=\sum_{i=1}^3\sin k_i\Gamma_i+\frac{1}{2i}(S-S^\dagger)\Gamma_4+\left[m-\sum_{i=1}^3\cos k_i-\frac{1}{2}(S+S^\dagger)\right]\Gamma_0+it\Gamma_3\Gamma_5$$ where $S$ is the translation operator along the $w$-direction, and $S^\dagger$ is that along the $-w$-direction. So, $S|i\rangle=|i+1\rangle$ and $S^\dagger|i\rangle=|i-1\rangle$ with integer $i$ numbering lattice site of the $w$-dimension, and accordingly the matrices are $$S=\begin{pmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \scalebox{-1}[1]{$\ddots$}\\
\ddots&0 & 0 & 0 & 0 &\cdots \\
\cdots&1 & 0 & 0 & 0 &\cdots\\
\cdots&0 & 1 & 0 & 0 &\cdots\\
\cdots&0 & 0 & 1 & 0 & \cdots\\
\scalebox{-1}[1]{$\ddots$} & \vdots & \vdots & \vdots & \ddots & \ddots
\end{pmatrix},\quad S^\dagger=\begin{pmatrix}
\ddots & \ddots & \vdots & \vdots & \vdots & \scalebox{-1}[1]{$\ddots$}\\
\cdots&0 & 1 & 0 & 0 &\cdots \\
\cdots&0 & 0 & 1 & 0 &\cdots\\
\cdots&0 & 0 & 0 & 1 &\cdots\\
\cdots&0 & 0 & 0 & 0 & \ddots\\
\scalebox{-1}[1]{$\ddots$} & \vdots & \vdots & \vdots & \vdots & \ddots
\end{pmatrix}.$$ If the boundary is opened with the system being on the non-negative part of the $w$-axis, the corresponding Hamiltonian is given by $$\widehat{\H}=\sum_{i=1}^3\sin k_i\Gamma_i+\frac{1}{2i}(\widehat{S}-\widehat{S}^\dagger)\Gamma_4+\left[m-\sum_{i=1}^3\cos k_i-\frac{1}{2}(\widehat{S}+\widehat{S}^\dagger)\right]\Gamma_0+it\Gamma_3\Gamma_5, \label{Half-H}$$ where still $\widehat{S}|i\rangle=|i+1\rangle$ with $i=0,1,2,\cdots$, but $\widehat{S}^\dagger|0\rangle=0$ and $\widehat{S}^\dagger|i\rangle=|i-1\rangle$ with $i\ge 1$. Explicitly, $$\widehat{S}=\begin{pmatrix}
0 & 0 & 0 & 0 &\cdots \\
1 & 0 & 0 & 0 &\cdots\\
0 & 1 & 0 & 0 &\cdots\\
0 & 0 & 1 & 0 & \cdots\\
\vdots & \vdots & \vdots & \ddots & \ddots
\end{pmatrix},\quad
\widehat{S}^\dagger=\begin{pmatrix}
0 & 1 & 0 & 0 &\cdots \\
0 & 0 & 1 & 0 &\cdots\\
0 & 0 & 0 & 1 &\cdots\\
0 & 0 & 0 & 0 & \ddots\\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{pmatrix}.$$ We now solve the Schrödinger equation for Eq. by substituting the ansatz $|\psi_\mathbf{k}\rangle=|\xi_{\mathbf{k}}\rangle\otimes\sum_{i=0}^{\infty}\lambda^i|i\rangle$ with $|\lambda|<1$ for boundary eigenstates. Inside the bulk with $|i\ge 1\rangle$, the Schrödinger equation gives $$\begin{aligned}
\left[\sum_{i}\sin k_i\Gamma_{i} +\frac{1}{2i}(\lambda-\lambda^{-1})\Gamma_4 +\left(m-\sum_{i}\cos k_i-\frac{1}{2}(\lambda+\lambda^{-1})\right) \Gamma_0+it\Gamma_3\Gamma_5 \right]\ket{\xi}=\mathcal{E}\ket{\xi}\label{bulk},\end{aligned}$$ while at $|i=0\rangle$ $$\begin{aligned}
\left[\sum_{i}\sin k_i\Gamma_{i} +\frac{1}{2i}\lambda\Gamma_4 +\left(m-\sum_{i}\cos k_i-\frac{1}{2}\lambda\right) \Gamma_0+it\Gamma_3\Gamma_5 \right]\ket{\xi}=\mathcal{E}\ket{\xi}.\label{boundary}\end{aligned}$$ The difference of Eqs. and gives $$i\Gamma_0\Gamma_4\ket{\xi}=\ket{\xi}. \label{Sub-space-condition}$$ This equation implies the boundary states occur merely in the 4D subspace with positive eigenvalue $1$ of $i\Gamma_0\Gamma_4$, which corresponds to the projector $$P=\frac{1}{2}(1+i\Gamma_0\Gamma_4).$$ Applying the projector to Eq. , we have $$\left(\sum_{i}\sin k_i\Gamma_{i}+it\Gamma_3\Gamma_5 \right)\ket{\xi}=\mathcal{E}\ket{\xi}, \label{Boundary-equation}$$ The difference of Eqs. and , together with Eq. , gives $$\lambda=m-\sum_{i}\cos k_i. \label{lambda}$$ The effective Hamiltonian for boundary states is just $$\mathcal{H}_{eff}(\mathbf{k})=P\mathcal{H}(\k)P$$ for regions in $\mathbf{k}$ space satisfying Eq. .
The Topological Invariant
=========================
In this section, we first give a detailed derivation of the topological invariant in the main text, and then explicitly calculate the topological invariant of the Dirac model.
The formula
-----------
For any region with a well-defined Berry connection, the second Chern character is the total derivative of the Chern-Simons form, and therefore we can apply Stokes’ theorem to obtain the following identities, $$\begin{aligned}
\int_X \mathrm{ch}_2(\F)&=&\int_{\partial X}Q_3(\A^X),\\ \int_{Y_1}\mathrm{ch}_2(\F)&=&\int_{\tau_1}Q_3(\A^Y)-\int_{\tau_1'}Q_3(\A^Y),\\
\int_{Y_2} \mathrm{ch}_2(\F)&=&\int_{\tau'_2}Q_3(\A^Y)-\int_{\tau_2}Q_3(\A^Y).\end{aligned}$$
With these identities, we proceed that $$\begin{split}
\mathcal{N} &=\int_{\partial X}Q_3(\A^X)-\int_{\partial_X}Q_3(\A^Y)\\
&=\int_X \mathrm{ch}_2(\F)-\int_{\tau_1'}Q_3(\A^Y)+\int_{\tau_2'}Q_3(\A^Y)\\
&=\int_X\mathrm{ch}_2(\F)+\int_{Y_1+Y_2}\mathrm{ch}_2(\F)-\int_{\tau_1}Q_3(\A^Y)+\int_{\tau_2}Q_3(\A^Y)\\
&=\int_{T^4_{1/2}}\mathrm{ch}_2(\F)-\int_{\tau_1}Q_3(\A^Y)+\int_{\tau_2}Q_3(\A^Y)\\
&=\int_{T^4_{1/2}}\mathrm{ch}_2(\F)-\int_{\tau_1}Q_3(\A)+\int_{\tau_2}Q_3(\A)\mod 2.
\end{split}$$ In the last equality, the Chern-Simons forms can be derived from any valence basis satisfying the boundary condition on $\tau_1$ and $\tau_2$ imposed by $RT$ symmetry, and we have used the fact that a gauge transformation preserving the boundary condition can change either Chern-Simons term by an even number, namely, that only the expression in the last line is gauge invariant.
The second Chern number
-----------------------
Since the topological invariant is preserved under adiabatic deformations of the Hamiltonian without closing the energy gap, let us set $t=0$ throughout this section for technical simplification. We first calculate the second Chern number in the half BZ, $T^4_{1/2}$. For this purpose, we need to derive a complete set of globally well-defined wave functions in the BZ for valence bands, of which the existence is clear because $RT$ symmetry makes the Chern number in the whole BZ vanishing. Introducing $$\begin{aligned}
R_0(k) &=& m-\sum_{i=1}^3\cos k_i-\cos k_w,\\
R_i(k)&=&\sin k_i,\quad i=1,2,3,\\
R_4(k) &=& \sin k_w,\end{aligned}$$ we write the Hamiltonian in the concise form, $$\mathcal{H}(k)=\sum_{a=0}^4R_{a}(k)\Gamma^a.$$ Applying the unitary transformation, $$U=e^{\frac{\pi}{4}\Gamma_0\Gamma_4} e^{\frac{\pi i}{4}\Gamma_{0}}e^{-\frac{\pi}{4}\Gamma_4\Gamma_6},$$ we have $$\tilde{\Gamma}_\mu=U\Gamma_\mu U^\dagger$$
$$\tilde{\Gamma}_0=\begin{pmatrix}
0 & i1_4\\
-i 1_4 & 0
\end{pmatrix},\quad
\tilde{\Gamma}_i=\begin{pmatrix}
0 & \tilde{\gamma}_i\\
\tilde{\gamma}_i & 0
\end{pmatrix},\quad \tilde{\Gamma}_4=\begin{pmatrix}
0 & \tilde{\gamma}_4\\
\tilde{\gamma}_4 & 0
\end{pmatrix},$$
where $$\tilde{\gamma}_1=\sigma_1\otimes\sigma_1,\quad \tilde{\gamma}_2=\sigma_1\otimes\sigma_3,\quad \tilde{\gamma}_3=\sigma_3\otimes\sigma_0,\quad \tilde{\gamma}_4=\sigma_2\otimes\sigma_0,$$ and thereby $$U\mathcal{H}U^\dagger=\begin{pmatrix}
0 & Q^\dagger\\
Q & 0
\end{pmatrix}$$ with $$Q=-i R_0+R_j\tilde{\gamma}^j.$$ We can renormalize $Q$ to be a unitary matrix, $$\hat{Q}=-i \hat{R}_0+\hat{R}_j\tilde{\gamma}^j,$$ where $\hat{R}_\mu=R_\mu/R$ with $R=\sqrt{\sum_\mu R_\mu^2}$ and $\mu=0,1,2,3$. It is noteworthy that $$\hat{Q}^T(\k,k_w)=\hat{Q}(\k,-k_w),$$ as required by $RT$ symmetry. Accordingly, the valence eigenstates are given by $$\begin{split}
|\alpha,k\rangle&=\frac{1}{\sqrt{2}}U^\dagger\begin{pmatrix}
-\chi_\alpha(k)\\
\hat{Q}(k)\chi_{\alpha}(k)
\end{pmatrix}\end{split}\label{Wave-functions}$$ where $\chi_\alpha(k)$ is an orthonormal basis of the 4D Hilbert space for valence bands at each $k$, namely $$\chi^\dagger_\alpha(k)\chi_\beta(k)=\delta_{\alpha\beta}.$$ For convenience, we choose $$\chi_1=\begin{pmatrix}
1\\
0\\
0\\
0
\end{pmatrix},\quad
\chi_2=\begin{pmatrix}
0\\
1\\
0\\
0
\end{pmatrix}\quad
\chi_3=\begin{pmatrix}
0\\
0\\
1\\
0
\end{pmatrix},\quad
\chi_4=\begin{pmatrix}
0\\
0\\
0\\
1
\end{pmatrix}. \label{Chi-vectors}$$
The globally well-defined wave functions of Eq. with Eq. for valence bands give the concise expression for the Berry connection, $$\A^\mu=\frac{1}{2}\hat{Q}^\dagger \partial_\mu \hat{Q}. \label{Berry-connection}$$ Substituting Eq. into the formula for the Berry curvature, $$\F_{\mu\nu}=\partial_\mu\A_{\nu}-\partial_\nu\A_{\mu}+[\A_\mu,\A_\nu]$$ we find $$\F_{\mu\nu}=\frac{1}{4}(\partial_\mu \hat{Q}^\dagger \partial_\nu\hat{Q}-\partial_\nu \hat{Q}^\dagger \partial_\mu\hat{Q}),$$ and thereby $$\mathrm{ch}_2=-\frac{1}{128\pi^2}\epsilon^{\mu\nu\lambda\sigma}\mathrm{tr}\partial_\mu \hat{Q}^\dagger \partial_\nu\hat{Q}\partial_\lambda \hat{Q}^\dagger \partial_\sigma\hat{Q},$$ which turns out to be vanished after tedious derivations. Thus, the topological invariant for this model is determined merely by the boundary Chern-Simons terms, derived from valence wave functions with the boundary condition given by $RT$ symmetry.
The representation of $RT$ symmetry
-----------------------------------
We now discuss how the $RT$ operator $\mathcal{RT}=U_{RT}\hat{\mathcal{K}}\hat{I}_w$ is represented in the valence band structure. First, we have the identities, $$U_{RT} U_{RT}^\dagger=1, \quad U_{RT}=U_{RT}^T.$$ The symmetry operator $\mathcal{RT}=U_{RT}\hat{\mathcal{K}}\hat{I}_w$ gives the constraint on the Hamiltonian, $$U_{RT} \mathcal{H}^*(\mathbf{k},k_w)U^\dagger_{RT}=\mathcal{H}(\mathbf{k},-k_w),$$ which implies $RT$ symmetry is represented by the wave functions as
$$U_{RT}|\alpha,\mathbf{k},k_w\rangle^*=|\beta,\mathbf{k},-k_w\rangle \mathcal{U}_{\beta\alpha}(\mathbf{k},-k_w).$$
$$\mathcal{U}_{\alpha\beta}(\mathbf{k},k_w)\mathcal{U}_{\beta\gamma}^*(\mathbf{k},-k_w)=\delta_{\alpha\gamma}, \quad \mathcal{U}^T_{\alpha\beta}(\mathbf{k},k_w)=\mathcal{U}_{\alpha\beta}(\mathbf{k},-k_w).$$
The topological obstruction in the topological phase can be essentially stated as that there exist no globally well-defined valence basis that can simultaneously “diagonalize” $\mathcal{H}(k)$ and $\mathcal{RT}$ in the whole BZ. In other words, if we require that $$\mathcal{U}(\k,0)=U_{RT}=\mathcal{U}(\k,\pi), \label{Boundary-condition}$$ which is just the boundary condition of Eq. (7) in the main text, the valence basis must be singular at some points in the bulk of the half BZ, $T^4_{1/2}$. On the other hand, if the basis is well-defined in the whole BZ, the boundary condition of Eq. must be violated.
In particular, the wave functions of Eq. are clearly well defined in the whole BZ, provided $\chi_\alpha$ are well-defined periodic functions, and accordingly give $$\mathcal{U}_{\alpha\beta}(\k,k_w)=i\chi^\dagger_\alpha(\k,k_w)\hat{Q}^\dagger(\k,k_w)\chi^*_\beta(\k,-k_w).$$ In this case, $U_{RT}$ is just the identity matrix, and one can verify that there are no $\chi_\alpha(k)$ that can transform $\hat{Q}^\dagger$ into identity matrix in the two boundaries with $k_w=0$ and $\pi$ at the same time in the topological phases.
The boundary Chern-Simons terms
-------------------------------
The Hamiltonian restricted on the two boundaries $\tau_1$ and $\tau_2$ is $$\H(\mathbf{k},\pi/0)=\sum_{i=1}^3\sin k_i\Gamma_{i}+\left (m\pm 1-\sum_{i=1}^{3}\cos k_i\right)\Gamma_4,$$ In principle, there exists a global basis in $\tau_1$ ($\tau_2$) with the boundary condition for valence bands, $|\alpha,\k\rangle$ with $\alpha=1,2,3,4$, for $\H(\k,\pi)$ ($\H(\k,0)$), because there is no 3D topological insulators protected by $PT$ symmetry with $(PT)^2=1$. Note that $RT$ restricted on either $\tau_1$ or $\tau_2$ is just $PT$. However, to derive the global wave function is quite technically involved.
To warm up, we diagonalize the 1D Hamiltonian, $$\H(k)=\cos k~\sigma_3\otimes \sigma_0+\sin k~\sigma_1\otimes\sigma_1.$$ It is easy to obtain the two sets of valence eigenstates, $$\psi_1(k)=\begin{pmatrix}
\sin k \\0\\0\\-1-\cos k
\end{pmatrix},\quad \psi_2(k)=\begin{pmatrix}
0\\ \sin k\\-1-\cos k\\0
\end{pmatrix},\quad k\in[-\frac{\pi}{2},\frac{\pi}{2}]$$ and $$\tilde{\psi}_1(k)=\begin{pmatrix}
1-\cos k \\0\\0\\-\sin k
\end{pmatrix},\quad \tilde{\psi}_2(k)=\begin{pmatrix}
0\\ 1-\cos k\\-\sin k\\0
\end{pmatrix},\quad k\in[\frac{\pi}{2},\frac{3\pi}{2}],$$ where $\psi_{1,2}$ and $\tilde{\psi}_{1,2}$ have the singular points $k=\pi$ and $k=0$, respectively. To achieve a global basis, we shall glue them at $k=-\frac{\pi}{2}$ and $\frac{\pi}{2}$ periodically and continuously. Through they are equal at $k=\pi/2$, they are not smoothly connected at ${3\pi}/{2}$ and $-\pi/2$. The following recombination realizes the continuity, $$\begin{aligned}
(\psi_1(k)+\psi_2(k))/\sqrt{2},\quad (\psi_1(k)-\psi_2(k))/\sqrt{2},\quad && k\in[-\pi/2,\pi/2]\\
\cos(k/2)\tilde{\psi}_1(k)+\sin(k/2)\tilde{\psi}_2(k),\quad \sin(k/2)\tilde{\psi}_1(k)-\cos(k/2)\tilde{\psi}_2(k),\quad && k\in[-\pi/2,\pi/2]\end{aligned}$$ After further simplification, we arrive at the globally well-defined valence orthonormal basis, $$|1,k\rangle=\frac{1}{2}\begin{pmatrix}
\sin k\\ 1-\cos k \\ -\sin k\\-1-\cos k
\end{pmatrix},\quad |2,k\rangle=\frac{1}{2}\begin{pmatrix}
1-\cos k\\ -\sin k \\ 1+\cos k\\-\sin k
\end{pmatrix}$$ which can also be derived as $$|1,k\rangle=\frac{1}{2}(\psi_1+\tilde{\psi}_2),\quad
|2,k\rangle=\frac{1}{2}(\tilde{\psi}_1-\psi_2).$$
With the insight from the above simple model, we now return to solving the Hamiltonian. First, we flatten the Hamiltonian through dividing it by $d=R(\k,\pi/0)$, $$\begin{split}
\tilde{\H} &=\H(\k,\pi/0)/d\\
&=\hat{d}_0\Gamma_0+\hat{d}_i\Gamma^i\\
&=\begin{pmatrix}
\hat{d}_0 & \hat{\Delta}\\
\hat{\Delta} & -\hat{d}_0
\end{pmatrix},
\end{split}$$ where $\sum_{\mu=0}^3 \d_\mu^2=1$, and $\hat{\Delta}$ is a real symmetric matrix, $$\hat{\Delta}=\begin{pmatrix}
\hat{d}_3 & 0 & \hat{d}_2 & \hat{d}_1\\
0 & \hat{d}_3 & \hat{d}_1 & -\hat{d}_2\\
\hat{d}_2 & \hat{d}_1 & -\hat{d}_3 & 0\\
\hat{d}_1 & -\hat{d}_2 & 0 & -\hat{d}_3
\end{pmatrix}.$$ Then, $$\psi_1(\xi_1)=\begin{pmatrix}
\hat{\Delta}\xi_1\\
-(1+\hat{d}_0)\xi_1
\end{pmatrix},\quad
\psi_2(\xi_2)=\begin{pmatrix}
(1-\hat{d}_0)\xi_2\\
-\hat{\Delta}\xi_2
\end{pmatrix}$$ are two forms of eigenstates. $\psi_{1,2}$ is singular if and only if $\hat{d}_0=\mp 1$, and therefore $\psi_{1}$ and $\psi_{2}$ cannot both be singular at the same $\k$. To construct globally well-defined wave functions, we combine them as $$\psi_+=\psi_1+\psi_2=\begin{pmatrix}
\hat{\Delta}\xi_1+(1-\hat{d}_0)\xi_2\\
-\hat{\Delta}\xi_2-(1+\hat{d}_0)\xi_1
\end{pmatrix},$$ and find $$\psi_+^\dagger \psi_+=4+2(\xi^\dagger_1\hat{\Delta}\xi_2+\xi^\dagger_2\hat{\Delta}\xi_1). \label{psi-plus}$$ The norm of $\psi_+$ is well defined if the matrix elements of $\hat{\Delta}$ in the above equation is zero. We observe that if $\xi_1=\chi_i$ and $\xi_2=\chi_j$ in Eq. , $\xi^\dagger_1\hat{\Delta}\xi_2$ is just the $(i,j)$-entry of $\Delta$. Since $\hat{\Delta}$ exactly has four zero entries for $\hat{\Delta}$, we can certainly construct a complete set of global wave functions, which are given by $$\begin{split}
|1,\k\rangle &=\frac{1}{2}[\psi_1(\chi_2)-\psi_2(\chi_1)],\\
|2,\k\rangle &=\frac{1}{2}[\psi_1(\chi_1)+\psi_2(\chi_2)],\\
|3,\k\rangle &=\frac{1}{2}[-\psi_1(\chi_3)+\psi_2(\chi_4)],\\
|4,\k\rangle &=\frac{1}{2}[-\psi_1(\chi_4)-\psi_2(\chi_3)].
\end{split}$$ The signs in the combinations above have been carefully chosen, such that the four wave functions are orthogonal. Explicitly, the orthonormal wave functions are $$|1,\k\rangle=\frac{1}{2}\begin{pmatrix}
-1+\hat{d}_0\\
\hat{d}_3\\
\hat{d}_1\\
-\hat{d}_2\\
\hat{d}_3\\
-1-\hat{d}_0\\
\hat{d}_2\\
\hat{d}_1
\end{pmatrix},\quad
|2,\k\rangle=\frac{1}{2}\begin{pmatrix}
\hat{d}_3\\
1-\hat{d}_0\\
\hat{d}_2\\
\hat{d}_1\\
-1-\hat{d}_0\\
-\hat{d}_3\\
-\hat{d}_1\\
\hat{d}_2
\end{pmatrix},\quad
|3,\k\rangle=\frac{1}{2}\begin{pmatrix}
-\hat{d}_2\\
-\hat{d}_1\\
\hat{d}_3\\
1-\hat{d}_0\\
-\hat{d}_1\\
\hat{d}_2\\
1+\hat{d}_0\\
\hat{d}_3
\end{pmatrix},\quad
|4,\k\rangle=\frac{1}{2}\begin{pmatrix}
-\hat{d}_1\\
\hat{d}_2\\
-1+\hat{d}_0\\
\hat{d}_3\\
\hat{d}_2\\
\hat{d}_1\\
-\hat{d}_3\\
1+\hat{d}_0
\end{pmatrix}.$$
Actually the wave functions are carefully ordered, and hence the Berry connection can be written in the concise form, $$\mathcal{A}_{\alpha\beta,j}=\frac{1}{2}(\d_\alpha\partial_j\d_\beta-\d_\beta\partial_j \d_\alpha-\epsilon_{\alpha\beta\gamma\delta}\d_\gamma\partial_j\d_{\delta}).$$ Substituting the above expression into the formula of the Chern-Simons form, we find $$\begin{split}
Q_3(\A)&=-\frac{1}{8\pi^2}\epsilon^{ijk}\mathrm{tr}(\A_i\partial_j\A_k+\frac{2}{3}\A_i\A_j\A_k)\\
&=-\frac{1}{12\pi^2}\epsilon^{ijk}\epsilon^{\alpha\beta\gamma\delta}\d_\alpha\partial_i \d_\beta \partial_j \d_\gamma \partial_k \d_\delta,
\end{split}$$ which is just the winding number of the unit vector valued in the unit sphere $S^3$ over the 3D BZ.
The topological invariant
--------------------------
Since the second Chern number is zero, the topological invariant is given by the boundary Chern-Simons terms, more clearly the difference of winding numbers $\mathcal{N}^\pi_W$ and $\mathcal{N}^0_W$ of $\hat{d}_\mu$’s in the two boundaries, $$\mathcal{N}=\mathcal{N}^\pi_W-\mathcal{N}^0_W \mod 2.$$ In the particular model, the winding numbers are determined by $$\tilde{m}_\pi=m+1,\quad \tilde{m}_0=m-1,$$ for the two boundaries, respectively. The results for typical masses in gapped regions of $m$ are summarized in the Tab. \[Topological-Invariant\].
$m$ $\tilde{m}_\pi$ $\tilde{m}_0$ $\mathcal{N}=\mathcal{N}^\pi_W-\mathcal{N}^0_W$
----- ----------------- --------------- -------------------------------------------------
5 6 4 0-0=0
3 4 2 0-1=-1
1 2 0 1+2=3
-1 0 -2 -2+1=-1
-3 -2 -4 -1-0=-1
-5 -4 -6 0-0=0
: The topological invariant for different $m$’s.\[Topological-Invariant\]
We find that all gapped regions with $|m|<5$ correspond to the topologically nontrivial phase.
Simulation with Ultracold Atoms
===============================
For the simplicity of the simulation, we choose another set of Clifford algebra generators $$\begin{split}
&\Gamma_0'=\sigma_3\otimes\sigma_0\otimes\sigma_0,\ \Gamma_1'=\sigma_1\otimes\sigma_1\otimes\sigma_1,\ \Gamma_2'=\sigma_1\otimes\sigma_1\otimes\sigma_3,\ \Gamma_3'=\sigma_1\otimes\sigma_3\otimes\sigma_0,\\
&\Gamma_4'=\sigma_2\otimes\sigma_0\otimes\sigma_0,\ \Gamma_5'=\sigma_1\otimes\sigma_2\otimes\sigma_0,\ \Gamma_6'=\sigma_1\otimes\sigma_1\otimes\sigma_2,
\end{split}$$ which is related to the original ones by a unitary transformation that does not change the symmetry operator $\mathcal{RT}=\hat{\mathcal{K}}$. According, the Hamiltonian is transformed to be $$\label{hh1}
\begin{split}
\H'(\k,k_{w})=&\sin k_w\Gamma_4'-\cos k_w\Gamma_0'+(m-\sum_{i=1}^{3}\cos k_i)\Gamma_0'\\
&+\sin k_1\Gamma_1'+\sin k_2\Gamma_2'+\sin k_3\Gamma_3'\\
&+t_0 \left(i\Gamma_3'\Gamma_5'\right),
\end{split}$$ which has been divided into different parts for simulation.
![(a) The quasi 1D tetragonal optical lattice. The four parallel chains are denoted as $ A,B,C,D $. (b) The unit cell. (c) The internal ground levels of $ F=1 $ manifold split by an external magnetic field. \[simulation\]](simulation.pdf)
We consider ultra-cold atoms with internal pseudospin moving in two ladders of optical lattice consisting of four chains denoted as $ A,B,C,D $ in Fig. \[simulation\](a) to simulate the Hamiltonian in Eq. . The first two Pauli matrices of $ \Gamma' $ matrices act in the spaces spanned by $ \{\ket{l}, \ket{r}\} $ and $ \{\ket{u}, \ket{d}\} $, respectively. Then, $ \ket{\psi_{A}}=\ket{l}\otimes\ket{u} $, $ \ket{\psi_{B}}=\ket{l}\otimes\ket{d} $, $ \ket{\psi_{C}}=\ket{r}\otimes\ket{u} $, $ \ket{\psi_{D}}=\ket{r}\otimes\ket{d} $ in Fig. \[simulation\](b).
The internal pseudospin is simulated by two levels $ \ket{F=1,F_{z}=-1} $ and $\ket{F=1,F_{z}=0}$ among the three levels with $ {F=1}$, which are slightly split by an external magnetic field, as shown in Fig. \[simulation\](c). We set the amplitude of inter-chain nearest-neighbor (NN) hoppings as unit, and all next-nearest-neighbor (NNN) hoppings are introduced by the photon-assisted tunneling. When two counter-propagating Raman lasers along $ \mathbf{r} $ are resonant with the internal levels, the two-photon process can lead to the effective spin-orbital interaction, $$H_{SO}=\left(\begin{array}{cc}
\frac{\hat{p}^{2}}{2M}+\delta & \Omega e^{2ik_{0}r} \\ \Omega e^{-2ik_{0}r} & \frac{\hat{p}^{2}}{2M}-\delta
\end{array}\right)\label{hso}$$ in the subspace spanned by $ \{\ket{F=1,F_z=0}, \ket{F=1,F_z=-1}\} $. Here, $ \hat{p} $ is the momentum operator along $\mathbf{r}$, $\delta$ is the energy splitting by the Zeeman effect, $ k_{0} $ is the wave vector of the lasers along $\mathbf{r}$ and $ \Omega $ is the Rabbi frequency. To obtain a tight-binding model, the hopping coefficient $ T_{RR'} $ between sites $ \textbf{R} $ and $ \textbf{R}' $ can be derived as $$\begin{aligned}
T_{RR'}=\bra{W_{\textbf{R}}}H_{SO}\ket{W_{\textbf{R}'}}, \label{tunnel}\end{aligned}$$ where $ \ket{W_{\textbf{R}}} $ are the Wannier wave functions of the optical lattice. The general form of $ T_{\textbf{R}\textbf{R}'} $ is $ t_0\sigma_0+t_1\sigma_1+t_2\sigma_2+t_3\sigma_3 $, with all $ t_i $ highly tunable. In addition, we note that all pseudospin-independent phases of hoppings can be tuned by the usual synthetic gauge field in cold-atom experiments.
![The simulation of all terms in Eq. . The light-blue and red arrows represent pseudospin-independent hoppings, and the former has an additional phase $ \pi $. The hoppings depicted by the green arrows involve pseudospin-orbital couplings. \[hoppings\]](hoppings.pdf)
$ \sin k_{w}\Gamma_{4}' $
-------------------------
This term can be translated to the tight-binding form as $$\begin{split}
&\sin k_{w}\Gamma_{6}=\sin k_{w} \sigma_{2}\otimes \sigma_{0}\otimes\sigma_{0}\\
\Rightarrow&\frac{1}{2}\sum_{n}\sum_{s=\uparrow,\downarrow}\Big[\left(f_{A,n,s}^{\dagger} f_{C,n+1,s}-f_{A,n+1,s}^{\dagger} f_{C,n,s}\right) + \left(f_{B,n,s}^{\dagger} f_{D,n+1,s}-f_{B,n+1,s}^{\dagger} f_{D,n,s}\right) \Big] +\mathrm{H.c.},
\end{split}$$ where the first term corresponds to the NNN hoppings between two up-chains and the second term is the NNN hopping between two down-chains. Here and below, $ f_{i,n,s} $ with $ i=A,B,C,D $ is the annihilation operator of site $ n $ of the $ i $-chain with pseudospin $ s $. It is depicted in Fig. \[hoppings\](a), where the light-blue arrows represent pseudospin-independent hoppings with an additional phase $ \pi $, and the red arrows represent pseudospin-independent hopping without additional phase.
$ -\cos k_{w}\Gamma_{0}' $
--------------------------
This term only contains the NN inter-chain hoppings. There is an phase $ \pi $ for hoppings between two left-chains $ A,B $, and no phase for those between two right chains, $C$ and $D$. We then have $$\begin{split}
&-\cos k_{w}\Gamma_{4}=-\cos k_{w} \sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\\
\Rightarrow& -\frac{1}{2}\sum_{n}\sum_{s=\uparrow,\downarrow}\left(f_{A,n+1,s}^{\dagger} f_{A,n,s} +f_{B,n+1,s}^{\dagger} f_{B,n,s}\right) +\frac{1}{2}\sum_{n,s}\left(f_{C,n+1,s}^{\dagger} f_{C,n,s} +f_{D,n+1,s}^{\dagger} f_{D,n,s}\right)+\mathrm{H.c.},
\end{split}$$ which is illustrated in Fig. \[hoppings\](b).
$ \left(m-\sum_{j=1}^{3}\cos k_{j}\right)\Gamma_0' $
----------------------------------------------------
We regard $ m_{0}=m-\sum_{j=1}^{3}\cos k_{j} $ as an internal parameter of the cold-atom system. Then, $$\begin{split}
&\left(m-\sum_{j=1}^{3}\cos k_{j}\right)\Gamma_{4}=m_{0}\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\\
\Rightarrow & m_{0}\sum_{n}\sum_{s=\uparrow,\downarrow}~\left(f_{A,n,s}^{\dagger} f_{A,n,s}+f_{B,n,s}^{\dagger} f_{B,n,s}\right)-\left(f_{C,n,s}^{\dagger} f_{C,n,s}+f_{D,n,s}^{\dagger} f_{D,n,s}\right),
\end{split}$$ which means the on-site energy of a cold atom at two left-chains $ A,B $ is $ m_0 $, and that at two right-chains $ C,D $ is $ -m_0 $, as illustrated in Fig. \[hoppings\](c). This term can be readily simulated by applying a tilted $ W $-potential.
$ \sum_{j=1,2} \sin k_{j}\Gamma_{j}'$
-------------------------------------
Defining $ t_{j}=\sin k_{j} $, for $ j=1,2 $, this term gives the NNN pseudospin-dependent hoppings by $$\begin{split}
&\sum_{j=1,2} \sin k_{j}\Gamma_{j}'=\sigma_{1}\otimes\sigma_{1}\otimes\left(t_{1}\sigma_{1}+t_{2}\sigma_{3}\right)\\
\Rightarrow& t_1\sum_{n}\Big[\left(f_{A,n,\uparrow}^{\dagger}f_{D,n,\downarrow}+f_{A,n,\downarrow}^{\dagger}f_{D,n,\uparrow}\right)+\left(f_{B,n,\uparrow}^{\dagger}f_{C,n,\downarrow}+f_{B,n,\downarrow}^{\dagger}f_{C,n,\uparrow}\right)\Big]+\mathrm{H.c.}\\
&+t_2\sum_{n}\Big[\left(f_{A,n,\uparrow}^{\dagger}f_{D,n,\uparrow}-f_{A,n,\downarrow}^{\dagger}f_{D,n,\downarrow}\right)+\left(f_{B,n,\uparrow}^{\dagger}f_{C,n,\uparrow}-f_{B,n,\downarrow}^{\dagger}f_{C,n,\downarrow}\right)\Big]+\mathrm{H.c.},
\end{split}$$ as illustrated in Fig. \[hoppings\](d). The pseudospin is flipped during the hoppings of term $ t_{1}\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1} $. For term $ t_{2}\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{3} $, an additional phase $ \pi $ is acquired for pseudospin $ \ket{\downarrow} $ during the hoppings. Here, the photon-assisted tunneling is applied to realize the hoppings between chain $ A $ ($ B $) and chain $ D $ ($ C $), according to Eqs. and .
$ \sin k_{3}\Gamma_{3}' $
-------------------------
Let $ t_{3}=\sin k_{3} $, we have $$\begin{split}
&\sin k_{3}\Gamma_{3}=t_{3} \sigma_{1} \otimes \sigma_{3} \otimes\sigma_{0}\\
\Rightarrow& t_3\sum_{n}\sum_{s=\uparrow,\downarrow}\left(f_{A,n,s}^{\dagger}f_{C,n,s}-f_{B,n,s}^{\dagger}f_{D,n,s}\right)+\mathrm{H.c.},
\end{split}$$ which represents NN hopping between two up (down) chains in Fig. \[hoppings\](e). An additional phase $ \pi $ is required for the hopping between two down chains $ B,D $. This $ \pi $ phase can be induced by rotating the optical lattice along $ w $ direction to obtain an effective gauge field by the Coriolis effect.
Partial Mass Term $ H_{1}=it\Gamma_{3}'\Gamma_{5}' $
----------------------------------------------------
The tight-binding model for this term is $$\begin{aligned}
&it\Gamma_{3}\Gamma_{7}=t \sigma_{0}\otimes\sigma_{1}\otimes \sigma_{0}\nonumber\\
\Rightarrow& t\sum_{n}\sum_{s=\uparrow,\downarrow}\left(f_{A,n,s}^{\dagger}f_{B,n,s}+f_{D,n,s}^{\dagger}f_{C,n,s}\right)+H.c.\end{aligned}$$ which are just NN hoppings shown by red arrows in Fig. \[hoppings\](f) between two left (right) chains.
|
---
abstract: |
Based on a method that produces the solutions to the Schrödinger equations of partner potentials, we give two conditionally exactly solvable partner potentials of exponential type defined on the half line. These potentials are multiplicative shape invariant and each of their linearly independent solution includes a sum of two hypergeometric functions. Furthermore we calculate the scattering amplitudes and study some of their properties.
KEYWORDS: Exactly solvable potentials; Shape invariance; Hypergeometric function; Scattering amplitude.
PACS: 03.65.Ge, 03.65.Nk, 03.65.Ca, 02.90.+p
author:
- |
A. López-Ortega\
Departamento de Física.\
Escuela Superior de Física y Matemáticas.\
Instituto Politécnico Nacional.\
Unidad Profesional Adolfo López Mateos. Edificio 9.\
México, D. F., México.\
C. P. 07738\
email: [email protected]
title: New conditionally exactly solvable potentials of exponential type
---
Introduction {#s: Introduction}
============
We know that in physics the exactly solvable problems are useful in the analysis of physical systems since they allow us to study in detail their properties or they are suitable approximations to more complex systems. In non relativistic quantum mechanics the potentials for which we can solve exactly the Schrödinger equation are used in the analysis of several phenomena. Therefore the search of new solvable potentials and the study of their properties is thoroughly investigated [@Cooper-book]–[@Bagrov]. At present time there are several methods to find exact solutions to the Schrödinger equation. We know the factorization method [@Schrodinger]–[@Infeld-Hull], the methods based on supersymmetric quantum mechanics [@Cooper-book]–[@Bagchi-book], [@Witten-susy]–[@Dutt-ajp-1988], on the point canonical transformations [@Bhattacharjie], and on the Darboux transformations [@Darboux], [@Bagrov].
Recently in Ref. [@ALO-2014-I] it is shown that for $x \in (0, +\infty)$ we can solve exactly the Schrödinger equations of the partner potentials $$\label{e: CES potentials previous}
V_\pm^I (x) = \frac{m^2}{x} \pm \frac{m}{2} \frac{1}{x^{3/2}} ,$$ where $m$ is a constant. We also find that these potentials are multiplicative shape invariant and each linearly independent solution includes the sum of two confluent hypergeometric functions.
For the inverse square root potential $$\label{e: square root potential}
V_{SR} = \frac{\tilde{V}_0}{x^{1/2}},$$ where $\tilde{V}_0$ is a constant, in Ref. [@Ishkhanyan-1] Ishkhanyan finds that the Schrödinger equation is exactly solvable. Furthermore in Ref. [@Ishkhanyan-PLA] it is shown that another exactly solvable potential is the Lambert $W$-function step potential $$\label{e: lambert potential}
V_W = \frac{\tilde{V}_0}{1 + W(\textrm{e}^{-u/\sigma})},$$ where $u \in (-\infty,+\infty)$, $W$ is the Lambert function, and $\sigma$ is a constant. We notice that each linearly independent solution found in Refs. [@Ishkhanyan-1], [@Ishkhanyan-PLA] includes the sum of two confluent hypergeometric functions with non-constant coefficients as the exact solutions previously studied in Ref. [@ALO-2014-I]. Furthermore, in Ref. [@Ishkhanyan-2] it is shown that for the sum of the potentials (\[e: CES potentials previous\]) and (\[e: square root potential\]) the Schrödinger equation can be exactly solved and the linearly independent solutions include a sum of two confluent hypergeometric functions.
Based on the method of Ref. [@ALO-2014-I], for the partner potentials $$\begin{aligned}
\label{e: ces step potentials}
V^{II}_\pm = m^2 \frac{ e^u}{e^u + 1} \mp \frac{m}{2} \frac{e^{u/2}}{(e^u + 1)^{3/2}} ,\end{aligned}$$ where $m$ is a constant (as for the potentials (\[e: CES potentials previous\])), in Ref. [@ALO-2015-arxiv] we showed that the Schrödinger equation is exactly solvable and each linearly independent solution involves a sum of two hypergeometric functions with non-constant coefficients. Thus the results of Ref. [@ALO-2015-arxiv] complement those of Refs. [@ALO-2014-I]–[@Ishkhanyan-2]. Recently these results are extended in Ref. [@Ishkhanyan-last] where an extensive study is carried out of the potentials whose solutions involve Heun functions and a list of known potentials with linearly independent solutions expanded as a sum of (confluent) hypergeometric functions is given.
Our purpose in this work is to extend the results on the potentials (\[e: CES potentials previous\]) and (\[e: ces step potentials\]). Here we study the properties of two partner potentials defined on the half line and possessing the property that each of their linearly independent solutions includes two hypergeometric functions as those of Ref. [@ALO-2015-arxiv] and in contrast to the potentials of Refs. [@ALO-2014-I]–[@Ishkhanyan-2] whose linearly independent solutions include two confluent hypergeometric functions. The expressions of the potentials that we study are $$\begin{aligned}
\label{e: potentials ces}
V_\pm (x, m) &=& \frac{m^2 }{e^x - 1} \pm \frac{m}{2} \frac{e^{x}}{(e^x - 1)^{3/2}} ,\end{aligned}$$ where, as in Refs. [@ALO-2014-I], [@ALO-2015-arxiv], $m$ is a constant. As far as we know the potentials that we present in this work are first studied. Also we notice that the potentials (\[e: potentials ces\]) are one example of the potentials written in explicit form and whose linearly independent solutions include a sum of two hypergeometric functions with non-constant coefficients (as we show below). Previous potentials with this property appear in Ref. [@ALO-2015-arxiv] or are given in implicit form [@Cooper:1986tz].
The potentials (\[e: potentials ces\]) are algebraic modifications of the Hulthen potential [@Gangopadhyayabook] $$\label{e: Hulthen potential}
V_H = \frac{Q}{e^x-1},$$ where $Q$ is a constant. We notice that the Hulthen potential near $x=0$ behaves as $1/x$ and decays exponentially as $x \to + \infty$. It is convenient to notice that we can not obtain the Hulthen potential as a limit of the potentials (\[e: potentials ces\]). From the shape of the potentials (\[e: potentials ces\]) we think that they can be useful to study scattering or tunneling phenomena (see Figs. 2-5) [@Flugge]. For some values of the parameters, in an interval, one of our potentials reminds us the shape of the effective potentials that govern the propagation of the Dirac field in a Schwarzschild black hole [@Chandra-book], [@Cho-Dirac-qnms] and therefore they can be used as a model to understand its dynamics in this spacetime. Furthermore we do not find the potentials (\[e: potentials ces\]) in the list of Ref. [@Ishkhanyan-last] that enumerates the known examples of potentials with linearly independent solutions involving a sum of (confluent) hypergeometric functions (see Table 3 of Ref. [@Ishkhanyan-last]). Thus we believe that these potentials are first studied in this work.
We think that the partner potentials (\[e: potentials ces\]) may be useful in supersymmetric quantum mechanics as a basis to generate new exactly solvable potentials of the Schrodinger equation [@Cooper-book]–[@Bagchi-book], [@Fernandez], [@Cooper:1994eh]. Also the mathematical form of the exact solutions is not common (see the expressions (\[e: Zeta 1\]), (\[e: Zeta 2\]), (\[e: solutions v variable\])) and they can be used as a model to search new exact solutions of the Schrödinger equation, since exact solutions of this mathematical form appear previously in Refs. [@ALO-2014-I]–[@ALO-2015-arxiv].
For several potentials we can find exact solutions to their Schrödinger equations in terms of special functions only when the parameters of the potentials satisfy some restrictions [@Bagchi-book], [@Ishkhanyan-2], [@SouzaDutra]–[@Roychoudhury]. These potentials are known as conditionally exactly solvable potentials (CES potentials in what follows). For the partner potentials (\[e: potentials ces\]) that we study in this work we show that their parameters satisfy an algebraic constriction and therefore they are CES in the sense of Ref. [@Ishkhanyan-2], that is, we call a potential as CES when its parameters can not be varied independently, that is, they satisfy a constriction [@Ishkhanyan-2]. Notice that this definition of CES potential does not impose that some parameter takes a fixed value [@Ishkhanyan-2].
We organize this paper as follows. In Sect. \[s: Method\] we study the properties of the partner potentials (\[e: potentials ces\]) that we analyze in this work. Using the method of Ref. [@ALO-2014-I] we solve exactly the Schrödinger equations of the studied potentials. We also expound some facts on these partner potentials and verify the solutions that we previously found. In Sect. \[s: scattering amplitude\] we calculate the scattering amplitudes of the potentials (\[e: potentials ces\]). We study some additional characteristics of the potentials that we analyze in this paper in Sect. \[s: Discussion\]. Finally, for the method used in this work, in Appendix we verify that it produces the solutions to the Schrödinger equations of partner potentials.
Solution method {#s: Method}
===============
In a similar way to Refs. [@ALO-2014-I], [@ALO-2015-arxiv] here we show that in the interval $x \in (0,+\infty)$, for the partner potentials (\[e: potentials ces\]) we can solve exactly their Schrödinger equations in terms of hypergeometric functions. For these partner potentials the superpotential $W$ is equal to[^1] $$\label{e: superpotential ces}
W (x,m) = - \frac{ m}{\sqrt{e^x - 1}} .$$ We notice that in Refs. [@Cooper-book]–[@Bagchi-book], [@Ishkhanyan-last], [@Khare-scattering]–[@Levai-search] that enumerate the solvable potentials already known, a search for the partner potentials (\[e: potentials ces\]) shows that they have not been previously discussed.
\[figure1\]
Since for the potentials (\[e: potentials ces\]) the constants multiplying to the factors $1/(e^x -1 )$ ($m^2$) and $ e^{x} /(e^x - 1)^{3/2}$ ($ \pm m/2$) fulfill the expression $- m^2 / 4 + (\pm m /2 )^2 = 0$ these are CES potentials, as those previously studied in Refs. [@ALO-2014-I], [@Ishkhanyan-2], [@ALO-2015-arxiv], [@SouzaDutra]–[@Roychoudhury]. It is convenient to notice that we classify the partner potentials (\[e: potentials ces\]) as CES since its parameters can not be varied independently [@Ishkhanyan-2]. Some previously found CES potentials are [@Bagchi-book], [@Dutt-CES] $$\begin{aligned}
\label{e: previous ces Bagchi}
\hat{V}_1 (x) &=& \frac{\hat{a}_1}{1 + e^{-2 u}} - \frac{\hat{b}_1}{(1 + e^{-2 u})^{1/2}} - \frac{3}{4(1 + e^{-2 u})^2} , \\
\hat{V}_2 (x) &=& \frac{\hat{a}_2}{1 + e^{-2 u}} - \frac{\hat{b}_2 e^{-u} }{ (1 + e^{-2 u})^{1/2} } - \frac{3}{4 (1 + e^{-2 u})^2 }, \nonumber\end{aligned}$$ where the constants $\hat{a}_1$ and $\hat{b}_1$ ($\hat{a}_2$ and $\hat{b}_2$) satisfy some constraints [@Bagchi-book], [@Dutt-CES]. We point out that the CES potentials (\[e: previous ces Bagchi\]) remind us to our potentials (\[e: potentials ces\]), but notice that we can not get these as a limit of the CES potentials (\[e: previous ces Bagchi\]) of Refs. [@Bagchi-book], [@Dutt-CES]. Moreover the interval where they are defined is different for the CES potentials (\[e: potentials ces\]) and (\[e: previous ces Bagchi\]).
![Plots of the potential $V_+$ (solid line) and $V_-$ (dashed line) for $m=1$.[]{data-label="figure3"}](fig2-II-m2.eps){width="\linewidth"}
![Plots of the potential $V_+$ (solid line) and $V_-$ (dashed line) for $m=1$.[]{data-label="figure3"}](fig3-II-m1.eps){width="\linewidth"}
In what follows we assume that $m > 0$, since for $m < 0$ we get the same results with the potentials $V_+$ and $V_-$ interchanged. Since for $x > 0$ it is true that $1/\sqrt{e^x - 1} > 0 $, we note that the superpotential (\[e: superpotential ces\]) does not cross the $x$-axis and therefore the supersymmetry is broken [@Cooper-book], [@Gangopadhyayabook]. We also find $$\label{e: limits superpotential}
W_+ = \lim_{x \to + \infty} W = 0^-, \qquad \qquad \lim_{x \to 0^+} W = - \infty,$$ where $0^+$ ($0^-$) means that the quantity goes to zero taking positive (negative) values. Notice that as $x \to + \infty$ the superpotential $W$ decays exponentially, whereas near $x = 0$ it behaves as $1/\sqrt{x}$. Since for $m > 0$ the derivative of $W$ satisfies ${\textrm{d}}W /{\textrm{d}}x > 0 $, we obtain that the superpotential is an increasing function for $x \in (0, + \infty)$. To illustrate these facts we plot the superpotential (\[e: superpotential ces\]) in Fig. 1.
For the potentials $V_{\pm}$ we get the following limits $$\begin{aligned}
\label{e: limits potentials}
&& \lim_{x \to + \infty} V_+ = 0^+, \qquad \qquad \lim_{x \to 0^+} V_+ = + \infty ,\nonumber \\
&& \lim_{x \to + \infty} V_- = 0^-, \qquad \qquad \lim_{x \to 0^+} V_- = - \infty .\end{aligned}$$ Furthermore these potentials decay exponentially to zero as $x \to + \infty$ (as the Hulthen potential (\[e: Hulthen potential\])) and near $x = 0$ they diverge as $1/ x^{3/2}$ (in a different way than the Hulthen potential (\[e: Hulthen potential\])). We point out that near $x = 0$ the potential $V_+$ diverges to $+ \infty$, whereas the potential $V_-$ diverges to $- \infty$. The potential $V_+$ does not cross the $x$-axis and it is strictly positive, but for $m > 1$ the potential $V_-$ crosses the $x$-axis at the two points $s_{\pm} = 2 m^2 \pm 2m \sqrt{m^2-1},$ where $s = e^x $. Notice that $s_+ > s_- > 0$. For $m < 1$ the potential $V_-$ does not cross the $x$-axis and it is strictly negative.
![Plots of the potential $V_+$ (solid line) and $V_-$ (dashed line) for $m=1/2$.[]{data-label="figure5"}](fig4-II-msqrt.eps){width="\linewidth"}
![Plots of the potential $V_+$ (solid line) and $V_-$ (dashed line) for $m=1/2$.[]{data-label="figure5"}](fig5-II-mp5.eps){width="\linewidth"}
Owing to the derivative of the potential $V_+$ satisfies ${\textrm{d}}V_+ / {\textrm{d}}x < 0 $, for $x \in (0,+\infty)$ we obtain that the potential $V_+$ decreases in this interval. For the potential $V_-$ we find that its derivative $$\frac{{\textrm{d}}V_-}{{\textrm{d}}x} = - \frac{e^x}{(e^x-1)^2}\left(m^2 - \frac{m}{2} \frac{1 + e^x/2}{(e^x-1)^{1/2}} \right) ,$$ has critical points at $s_{1,2} = 8 m^2 - 2 \pm 4 m \sqrt{4m^2-3}$. Hence for $m > \sqrt{3}/2$ the potential $V_-$ has two real critical points, whereas for $m < \sqrt{3}/2$ it does not have real critical points. We note that $s_1 > s_2 >0$, and we also point out that the critical point $s_2$ is a maximum and $s_1$ is a minimum. Furthermore we notice that for $m > 1$ the quantities $s_{\pm}$ and $s_{1,2} $ satisfy $s_1 > s_+ > s_2 > s_- $, that is, the maximum of the potential $V_-$ is located between the points $s_\pm$ where $V_-$ intersects the $x$ axis and its minimum has a coordinate greater than the intersections of the potential $V_-$ with the $x$ axis. We illustrate these facts in Figs. 2–5.
In what follows, using the method of Ref. [@ALO-2014-I], (see also Ref. [@ALO-2015-arxiv]) we solve exactly the Schrödinger equations of the partner potentials (\[e: potentials ces\]). With this objective we write these equations as $$\begin{aligned}
\label{e: Schrodinger equations}
\frac{{\textrm{d}}^{2} Z_-}{{\textrm{d}}x^{2}} + \omega^{2} Z_- = \left( W^2 - \frac{{\textrm{d}}W}{{\textrm{d}}x} \right) Z_- , \\
\frac{{\textrm{d}}^{2} Z_+}{{\textrm{d}}x^{2}} + \omega^{2} Z_+ = \left( W^2 + \frac{{\textrm{d}}W}{{\textrm{d}}x} \right) Z_+ , \nonumber \end{aligned}$$ and as in Ref. [@ALO-2014-I], to simplify the equations that follow, we denote the energy $E$ as $\omega^2$. In Ref. [@ALO-2014-I] it is shown that for $\omega \neq 0$ the Schrödinger equations (\[e: Schrodinger equations\]) can be written as $$\begin{aligned}
\left( \frac{{\textrm{d}}}{{\textrm{d}}x } - W \right) \frac{1}{i \omega} \left( \frac{{\textrm{d}}}{{\textrm{d}}x } + W \right) Z_- = i \omega Z_-, \\
\left( \frac{{\textrm{d}}}{{\textrm{d}}x } + W \right) \frac{1}{i \omega} \left( \frac{{\textrm{d}}}{{\textrm{d}}x } - W \right) Z_+ = i \omega Z_+, \nonumber \end{aligned}$$ from which we obtain that the functions $Z_+$ and $Z_-$ satisfy the coupled system $$\begin{aligned}
\label{e: Z coupled}
\left( \frac{{\textrm{d}}}{{\textrm{d}}x } + W \right) Z_- = i \omega Z_+, \qquad \left( \frac{{\textrm{d}}}{{\textrm{d}}x } - W \right) Z_+ = i \omega Z_- .\end{aligned}$$
Defining $Z_{\pm} = R_1 \pm R_2, $ we get that Eqs. (\[e: Z coupled\]) transform into the coupled system $$\begin{aligned}
\label{e: equations R}
\frac{{\textrm{d}}R_{1} }{{\textrm{d}}x} - i\omega R_{1} = W R_{2}, \qquad \qquad \frac{{\textrm{d}}R_{2} }{{\textrm{d}}x} + i\omega R_{2} = W R_{1} ,\end{aligned}$$ (see Eqs. (11) of Ref. [@ALO-2014-I]). In Appendix we show that the solutions of these coupled equations produce the solutions to the Schrödinger equations of partner potentials.
As in Ref. [@ALO-2014-I] we take $R_1 = e^{ - i \pi /4} \tilde{R}_1 ,$ $R_2 = e^{i \pi /4} \tilde{R}_2 ,$ and defining the variable $z$ by[^2] $$\label{e: z definition}
z = e^{-x},$$ we find that the coupled system of differential equations (\[e: equations R\]) transforms into $$\begin{aligned}
\label{e: R tilde coupled}
z \frac{{\textrm{d}}\tilde{R}_{1} }{{\textrm{d}}z} + i \omega \tilde{R}_{1} &=& i m \frac{ z^{1/2}}{(1-z)^{1/2}} \tilde{R}_2 , \nonumber \\
z \frac{{\textrm{d}}\tilde{R}_{2} }{{\textrm{d}}z} - i \omega \tilde{R}_{2} &=& - i m \frac{ z^{1/2} }{(1-z)^{1/2}} \tilde{R}_1 .\end{aligned}$$ From this coupled system we obtain that the functions $\tilde{R}_{1}$ and $\tilde{R}_{2}$ must be solutions of the decoupled differential equations $$\begin{aligned}
\label{e: radial equations tilde}
\frac{{\textrm{d}}^{2} \tilde{R}_k }{{\textrm{d}}z^{2}} + \left( \frac{1/2}{z} - \frac{1/2}{1-z} \right) \frac{{\textrm{d}}\tilde{R}_k }{{\textrm{d}}z} + \frac{\omega^2- i \omega \epsilon /2 }{ z^{2} } \tilde{R}_k - \frac{m^{2} + i \omega \epsilon /2 }{z (1-z )} \tilde{R}_k =0 ,\end{aligned}$$ where $k=1,2$, and $\epsilon = 1$ ($\epsilon = -1$) for $\tilde{R}_{1}$ ($\tilde{R}_{2}$).
If the functions $\tilde{R}_{1}$ and $\tilde{R}_{2}$ take the form $\tilde{R}_k = z^{A_k} \bar{R}_k ,$ with the quantities $A_k$ being solutions of the algebraic equations $$\label{e: A equation}
A_k^2 - \frac{A_k}{2} - \frac{i \omega \epsilon}{2} + \omega^2 = 0,$$ we find that the functions $\bar{R}_k$ satisfy the differential equations $$\label{e: equations bar R}
\frac{{\textrm{d}}^{2} \bar{R}_k }{{\textrm{d}}z^{2}} + \left( \frac{2 A_k + 1/2}{z} - \frac{1/2}{1-z} \right) \frac{{\textrm{d}}\bar{R}_k }{{\textrm{d}}z} - \frac{m^2 + i \omega \epsilon /2 + A_k/2 }{z(1-z)} \bar{R}_k = 0.$$ These equations are of hypergeometric type [@Abramowitz-book]–[@NIST-book] $$\label{e: hypergeometric equation}
z(1-z)\frac{{\textrm{d}}^2 F}{{\textrm{d}}z^2} + (c -(a+b+1)z)\frac{{\textrm{d}}F}{{\textrm{d}}z} - a b F = 0 ,$$ with the parameters $a_k$, $b_k$, $c_k$ equal to $$\begin{aligned}
\label{e: a b c hypergeometric}
a_k = A_k + i(m^2 + \omega^2 )^{1/2}, \,\,\,\,
b_k = A_k - i(m^2 + \omega^2 )^{1/2}, \,\,\,\,
c_k = 2 A_k + 1/2. \end{aligned}$$
If the parameters $c_k$ are not integers,[^3] then the functions $\tilde{R}_{k}$ are $$\begin{aligned}
\label{e: solutions R tilde}
\tilde{R}_k &=& z^{A_k} \left[ G_k \, {}_{2}F_{1} (a_k,b_k;c_k;z) \right. \nonumber \\
& & \left. + H_k \, z^{1-c_k} {}_{2}F_{1}(a_k-c_k+1,b_k-c_k+1;2-c_k;z) \right] ,\end{aligned}$$ where ${{}_{2}F_{1}}(a,b;c;z)$ denotes the hypergeometric function [@Abramowitz-book]–[@NIST-book], and the quantities $G_k$, $H_k$ are constants.
In a straightforward way we find that Eqs. (\[e: equations R\]) impose conditions on the constants $G_k$ and $H_k$. To discuss this fact we take the quantities $A_1$ and $A_2$ as $A_1 = i \omega + 1/2 = A_2 + 1/2$. Therefore the constants $a_k$, $b_k$, $c_k$ are equal to $$\begin{aligned}
\label{e: a b c hypergeometric defined}
a_1 &=& a_2 + 1/2 = 1/2 + i \omega + i(m^2 + \omega^2 )^{1/2}, \nonumber \\
b_1 &=& b_2 + 1/2= 1/2 + i \omega - i(m^2 + \omega^2 )^{1/2}, \\
c_1 &=& c_2 + 1 = 2 i \omega + 3/2. \nonumber\end{aligned}$$
From Eqs. (\[e: equations R\]) and the contiguous relations of the hypergeometric function [@Lebedev] we obtain:
a\) If we choose the function $\tilde{R}_{1}$ as $$\label{e: R one tilde first}
\tilde{R}_1 = G_1 z^{A_1} {}_{2}F_{1}(a_1,b_1;c_1;z),$$ then from Eqs. (\[e: equations R\]) we get that the function $\tilde{R}_2$ must be equal to $$\label{e: R two tilde first}
\tilde{R}_2 = G_1 \frac{c_1 - 1}{im} z^{A_2} {}_{2}F_{1}(a_2,b_2;c_2;z),$$ and the constants $G_1$ and $G_2$ are related by $G_2 = G_1 (c_1 - 1)/ (im ) $.
b\) If we select the function $\tilde{R}_{1}$ in the form $$\tilde{R}_1 = H_1 z^{A_1 + 1-c_1} {}_{2}F_{1}(a_1-c_1+1,b_1-c_1+1;2-c_1;z) ,$$ then from Eqs. (\[e: equations R\]) we obtain that the function $\tilde{R}_2$ must be equal to $$\begin{aligned}
\tilde{R}_2 &=& H_1 \frac{ (a_1-c_1+1)(b_1-c_1+1)}{im(2-c_1)} z^{A_2 + 1-c_2} \nonumber \\
& & \times {}_{2}F_{1}(a_2-c_2+1,b_2-c_2+1;2-c_2;z) ,\end{aligned}$$ and the constants $H_1$ and $H_2$ satisfy $H_2 = H_1 (a_1-c_1+1)(b_1-c_1+1)/(im(2-c_1))$. Hence we find that Eqs. (\[e: equations R\]) impose the previous constrictions on the constants $G_k$ and $H_k$.
Considering the previous definitions we get that as function of $\tilde{R}_1$ and $\tilde{R}_2$ the solutions $Z_{\pm}$ take the form $Z_\pm = e^{-i \pi /4} (\tilde{R}_1 \pm i \tilde{R}_2) $. Thus from our results we get that the linearly independent solutions to the Schrödinger equations of the potentials $V_\pm$ are $$\label{e: Zeta 1}
Z_\pm^I = G_1 \textrm{e}^{-i \pi /4} \left( z^{A_1} {{}_{2}F_{1}}(a_1,b_1;c_1;z) \pm \frac{c_1-1}{m} z^{A_2} {{}_{2}F_{1}}(a_2,b_2;c_2;z) \right),$$ and $$\begin{aligned}
\label{e: Zeta 2}
Z_\pm^{II} &=& H_1 \textrm{e}^{-i \pi /4} \left( z^{A_1+1-c_1} {{}_{2}F_{1}}(a_1-c_1+1,b_1-c_1+1;2-c_1;z) \right. \\
&& \pm z^{A_2+1-c_2} \frac{(a_1-c_1+1)(b_1-c_1+1)}{(2-c_1) m} \nonumber \\
& &\times
\left. {{}_{2}F_{1}}(a_2-c_2+1,b_2-c_2+1;2-c_2;z) \right). \nonumber\end{aligned}$$
Using that for the linearly independent solutions to the hypergeometric differential equation (\[e: hypergeometric equation\]) its Wronskian is [@NIST-book] $$\tilde{W}_z [ {{}_{2}F_{1}}(a,b;c;z), z^{1-c} {{}_{2}F_{1}}(a-c+1, b-c+1; 2-c;z) ] = \frac{(1-c)}{ z^{c} (1-z)^{a+b+1-c}},$$ in a straightforward way we find that the Wronskian of the solutions $Z_\pm^I$ and $Z_\pm^{II}$ is equal to (for $G_1 = H_1 = 1$) $$\mathfrak{W}_x [Z_\pm^I,Z_\pm^{II}] = \pm \frac{2 \omega (c_1-1)}{m} .$$
Furthermore, taking into account Eqs. (\[e: equations bar R\]), we get that the functions $\tilde{R}_{k}$ satisfy $$\begin{aligned}
\label{e: second derivative tilde R}
\frac{{\textrm{d}}}{{\textrm{d}}z} \left( z \frac{{\textrm{d}}\tilde{R}_k}{{\textrm{d}}z} \right) &=& \frac{1/2}{1-z} \frac{{\textrm{d}}\tilde{R}_k }{{\textrm{d}}z} - \frac{A_k}{2(1-z)} \tilde{R}_k \nonumber \\
& +& \frac{A_k^2 - A_k/2}{z} \tilde{R}_k + \frac{m^2 + A_k/2 + i \epsilon \omega/2}{1-z} \tilde{R}_k.\end{aligned}$$ From these equations we get that the functions $Z_\pm$ fulfill $$\label{e: Schrodinger equation z variable}
\frac{{\textrm{d}}}{{\textrm{d}}z} \left( z \frac{{\textrm{d}}Z_\pm}{{\textrm{d}}z} \right) + \left( \frac{\omega^2}{z} - \frac{m^2}{1-z} \mp \frac{m}{2} \frac{1}{z^{1/2}(1-z)^{3/2}} \right) Z_\pm = 0,$$ that are the Schrödinger equations (\[e: Schrodinger equations\]) in the variable $z$ defined in the expression (\[e: z definition\]). Hence, if the functions $\bar{R}_k$ are solutions of Eqs. (\[e: equations bar R\]), then the functions $Z_{\pm}$ solve the Schrödinger equations (\[e: Schrodinger equations\]) with the potentials (\[e: potentials ces\]).
Scattering amplitude {#s: scattering amplitude}
====================
In what follows we determine the scattering amplitudes for the potentials $V_\pm$. To calculate the scattering amplitude it is convenient to use the variable $v=1-z$ to write the solutions of the Schrödinger equations.[^4] We find that in this variable, the linearly independent solutions of the Schrödinger equations for the potentials (\[e: potentials ces\]) take the form $$\begin{aligned}
\label{e: solutions v variable}
\tilde{Z}_\pm^I &=& \tilde{C}_1 \left( (1-v)^{B_1} {{}_{2}F_{1}}(\alpha_1,\beta_1;\gamma_1;v) \right. \nonumber \\
&-& \left. \frac{m}{\gamma_1} (1-v)^{B_2} v^{1-\gamma_2} {{}_{2}F_{1}}(\alpha_2-\gamma_2+1,\beta_2-\gamma_2+1;2-\gamma_2;v) \right) ,\nonumber \\
\tilde{Z}_\pm^{II} &=& \tilde{C}_2 \left( (1-v)^{B_1} v^{1-\gamma_1} {{}_{2}F_{1}}(\alpha_1-\gamma_1+1,\beta_1-\gamma_1+1;2-\gamma_1;v) \nonumber \right. \\
&-& \left. \frac{\gamma_1}{m} (1-v)^{B_2} {{}_{2}F_{1}}(\alpha_2,\beta_2;\gamma_2;v) \right) ,\end{aligned}$$ where $\tilde{C}_1$, $\tilde{C}_2$ are constants and $$\begin{aligned}
\label{e: alpha beta gamma hypergeometric}
B_1 &=& 1/2 + i \omega, \,\,\,\,\,\,\,\,\,\, \quad B_2 = i \omega , \nonumber \\
\alpha_k &=& B_k + i(m^2 + \omega^2 )^{1/2}, \,\,\,\,
\beta_k = B_k - i(m^2 + \omega^2 )^{1/2}, \,\,\,\,
\gamma_k = 1/2. \end{aligned}$$
In what follows we study in detail the potential $V_+$ since a similar calculation produces the result for the potential $V_-$. We see that near $x=0$ ($v=0$) the solutions $\tilde{Z}_\pm$ behave as $$\tilde{Z}_\pm^I \approx 1 - \frac{m}{\gamma_1} v^{1/2} , \qquad \qquad \tilde{Z}_\pm^{II} \approx - \frac{\gamma_1}{m} + v^{1/2} .$$ Since we like to impose as boundary condition that the solution is equal to zero at $x=0$ it is convenient to define the new solutions $$Y_+^I = \tilde{Z}_\pm^I - \frac{m}{\gamma_1} \tilde{Z}_\pm^{II} , \qquad \qquad Y_+^{II} = \tilde{Z}_\pm^I + \frac{m}{\gamma_1} \tilde{Z}_\pm^{II},$$ that near $x=0$ behave as $$Y_+^I \approx 2 - \frac{2 m}{\gamma_1} v^{1/2} , \qquad \qquad Y_+^{II} \approx 0.$$ Therefore to satisfy the boundary condition we choose the solution $Y_+^{II}$ that is equal to zero at $x=0$.
Taking into account the Kummer property of the hypergeometric function [@Abramowitz-book]–[@NIST-book] $$\begin{aligned}
\label{e: Kummer property v 1-v}
& &{}_2F_1(a,b;c;v) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c - b)} {}_2 F_1 (a,b;a +b +1-c;1-v) \\
&+& \frac{\Gamma(c) \Gamma( a + b - c)}{\Gamma(a) \Gamma(b)} (1-v)^{c-a-b} {}_2F_1(c-a, c-b; c + 1 -a-b; 1 -v) , \nonumber\end{aligned}$$ we obtain that as $x \to \infty$ ($v \to 1$) the solution $Y_+^{II}$ behaves as $$\begin{aligned}
Y_+^{II} &\approx& \frac{\Gamma(1/2 + 2 i \omega) 2^{8 i \omega}}{\Gamma(1/2 - 2 i \omega)} \frac{ \Gamma(-2 \alpha_2)\Gamma(-2 \beta_2) }{\Gamma(2 \alpha_2)\Gamma(2 \beta_2)} \\
&\times& \frac{m \Gamma( \alpha_2) \Gamma( \beta_2) +\Gamma( 1/2+\alpha_2) \Gamma(1/2+ \beta_2)}{m \Gamma(- \alpha_2) \Gamma(- \beta_2) +\Gamma( 1/2 -\alpha_2) \Gamma(1/2 - \beta_2)} \textrm{e}^{i \omega x} - \textrm{e}^{- i \omega x} , \nonumber \end{aligned}$$ that is, the scattering amplitude of the potential $V_+$ is equal to [@Cooper-book], [@Flugge] $$\begin{aligned}
S_+ &=& \frac{\Gamma(1/2 + 2 i \omega) 2^{8 i \omega}}{\Gamma(1/2 - 2 i \omega)} \frac{ \Gamma(-2 \alpha_2)\Gamma(-2 \beta_2) }{\Gamma(2 \alpha_2)\Gamma(2 \beta_2)} \nonumber \\
&\times& \frac{m \Gamma( \alpha_2) \Gamma( \beta_2) +\Gamma( 1/2+\alpha_2) \Gamma(1/2+ \beta_2)}{m \Gamma(- \alpha_2) \Gamma(- \beta_2) +\Gamma( 1/2 -\alpha_2) \Gamma(1/2 - \beta_2)} .\end{aligned}$$ Notice that the previous scattering amplitude satisfies $S_+ S_+^* = 1 $. A similar result is valid for the scattering amplitude of the potential $V_-$.
Discussion {#s: Discussion}
==========
Here we show that each of the exact solutions (\[e: Zeta 1\]) and (\[e: Zeta 2\]) of the Schrödinger equations for the partner potentials (\[e: potentials ces\]) includes a sum with non-constant coefficients of two hypergeometric functions (see also the exact solutions (\[e: solutions v variable\])). We note that this form of the solutions is not common in the previous references [@Cooper-book]–[@Bagchi-book], [@Cooper:1994eh]. The potentials that we study in this work may be suitable to analyze scattering and tunneling phenomena and they can be taken as a basis to search new exactly solvable potentials, since the mathematical form of their solutions is not widely explored.
To finish this work we notice the following facts on the potentials (\[e: potentials ces\]).
- Considering that $ e^{x} = e^{x/2} / e^{-x/2}$ and employing hyperbolic functions we obtain that the superpotential (\[e: superpotential ces\]) and the potentials (\[e: potentials ces\]) take the form $$\begin{aligned}
W &=& - \mu (\coth(x/2)-1)^{1/2}, \nonumber \\
V_\pm &=& \mu^2 (\coth(x/2) -1) \pm \frac{\mu}{4} \frac{(1 + \coth(x/2) )^{1/2}}{\sinh(x/2)} ,\end{aligned}$$ with $\mu = m / \sqrt{2}$.
- We notice that near $x = 0$, the potentials $V_\pm$ behave as (preserving the leading and subleading terms) $$\label{e: behavior near zero}
\frac{m^2}{x} \pm \frac{m}{2} \frac{1}{x^{3/2}} ,$$ that are the potentials (\[e: CES potentials previous\]) previously studied in Ref. [@ALO-2014-I], that is, near $x=0$ our potentials $V_\pm$ yield the behavior analyzed in Ref. [@ALO-2014-I]. In contrast to the potentials of Ref. [@ALO-2014-I], the potentials $V_\pm$ decay exponentially as $x \to + \infty$ (the potentials (\[e: CES potentials previous\]) decay as $1/x$ as $x \to + \infty$). Thus we can consider to the potentials (\[e: potentials ces\]) as a generalization of the potentials (\[e: CES potentials previous\]).
- The partner potentials are shape invariant if they satisfy $V_+(x,\alpha_0) = V_-(x,\alpha_1) + R (\alpha_0) $ [@Gendenshtein], where the parameters $\alpha_0$, $\alpha_1$ are independent of the coordinate $x$, with $\alpha_1 = f(\alpha_0)$, and $R (\alpha_0)$ is also a function of $\alpha_0$. From the expressions (\[e: potentials ces\]) we notice that the potentials $V_\pm$ fulfill $V_-(x,-m) = V_+(x,m),$ and therefore they are multiplicative shape invariant, since $\alpha_0 = m$, $\alpha_1 = - \alpha_0 = q \alpha_0$ with $q=-1$ and $R (\alpha_0) = 0$. For several multiplicative shape invariant potentials that are already found [@Cooper-book], [@Cooper:1994eh], we know them in series form, but we get the potentials (\[e: potentials ces\]) in closed form, in a similar way to the multiplicative shape invariant potentials of Refs. [@ALO-2014-I], [@ALO-2015-arxiv].
Recently in Ref. [@Aleixo-Balantekin] is studied the concept of shape invariance with reflection transformations. The analyzed transformations include reflections of the coordinates and translations of the parameters. For the potentials (\[e: potentials ces\]) the formula $\alpha_1 = - \alpha_0 $ remind us a reflection, but for the parameters of the potential and it is different from the mathematical operations considered in Ref. [@Aleixo-Balantekin].
- As previously noted, the superpotential $W$ does not cross the $x$-axis and therefore the supersymmetry is broken [@Cooper-book], [@Gangopadhyayabook]. Thus the functions $\psi^{\mp}_0$ that are solutions of the differential equations $$\left(\frac{{\textrm{d}}}{{\textrm{d}}x} + W \right) \psi_0^- = 0, \qquad \left(- \frac{{\textrm{d}}}{{\textrm{d}}x} + W \right) \psi_0^+ = 0,$$ are equal to $$\psi^{\mp}_0 = \exp \left( \mp \int W(x^{\prime}) {\textrm{d}}x^{\prime} \right) = \left[ \sqrt{1-\textrm{e}^{-x}} + i \textrm{e}^{-x/2} \right]^{\pm 2 i m} ,$$ and they are not normalizable.
Taking into account the formulas (15.4.11) and (15.4.15) of Ref. [@NIST-book], we obtain that the functions $\psi^{\mp}_0$ can be written in the form $$\label{e: psi 0 1 hypergeometric}
\psi^{\mp}_0 = {{}_{2}F_{1}}(\mp im, \pm im; 1/2;z) \mp 2 m z^{1/2} {{}_{2}F_{1}}(1/2 \mp im, 1/2 \pm i m; 3/2; z) .$$ From the values of the parameters $A_k$ and $a_k$, $b_k$, and $c_k$ with $\omega = 0$ we find that the functions $\tilde{R}_k$ of the expressions (\[e: R one tilde first\]) and (\[e: R two tilde first\]) simplify to $$\begin{aligned}
\label{e: tilde for omega 0}
\tilde{R}_1 &=& z^{1/2} {{}_{2}F_{1}}(1/2 + im, 1/2 - i m; 3/2; z) , \nonumber \\
\tilde{R}_2 &=& \frac{1}{2mi} {{}_{2}F_{1}}(im, - im; 1/2;z).\end{aligned}$$ Thus from the expressions (\[e: psi 0 1 hypergeometric\]) and (\[e: tilde for omega 0\]) we obtain $$\psi^{\mp}_0 = \mp 2 m (\tilde{R}_1 \mp i \tilde{R}_2) = \mp 2 m \textrm{e}^{i \pi /4} Z_\mp.$$ Hence the functions $\psi^{\mp}_0$ are proportional to $Z_\mp$, as we expect from the previous analysis.
- Finally we notice that for the potentials (\[e: potentials ces\]) each linearly independent solution of the Schrödinger equations (\[e: Schrodinger equations\]) includes a sum with non-constant coefficients of two hypergeometric functions (see the expressions (\[e: Zeta 1\]), (\[e: Zeta 2\]), and (\[e: solutions v variable\])), and we have not been able to simplify this sum to a single hypergeometric function, but this fact must be studied carefully. Therefore, as those of Refs. [@ALO-2014-I]–[@ALO-2015-arxiv], the potentials (\[e: potentials ces\]) are examples of the potentials analyzed, but not given in explicit form in Ref. [@Cooper:1986tz] whose linearly independent solutions include a sum of (confluent) hypergeometric functions.
From the results of Ref. [@Ishkhanyan-last] we think that the solutions of the Schrödinger equations for the potentials (\[e: potentials ces\]) also can be expanded in terms of Heun functions. As far as we can see the advantage of writing the solutions (\[e: Zeta 1\]) and (\[e: Zeta 2\]) (see also (\[e: solutions v variable\])) as a sum of hypergeometric functions (instead of Heun functions) is that we can use the well developed techniques involving (confluent) hypergeometric functions (as Kummer’s property (\[e: Kummer property v 1-v\])) in the study of the characteristics for the potentials (\[e: potentials ces\]), as illustrated in Sect. \[s: scattering amplitude\] (and in Refs. [@Ishkhanyan-PLA], [@ALO-2015-arxiv]). Thus we think that it is convenient to search and study the potentials whose linearly independent solutions have this property.
To generalize the results of Refs. [@ALO-2014-I]–[@ALO-2015-arxiv] and this paper, a problem to analyze in detail is the search of potentials with the property that each linearly independent solution includes a sum of three or more (confluent) hypergeometric functions.
Appendix {#Appendix}
========
In this Appendix we show that the solutions to the system of coupled equations (\[e: equations R\]) produce the solutions to the Schrödinger equations of the partner potentials $$\label{e: partner potentials}
\hat{V}_\pm = W^2 \pm \frac{{\textrm{d}}W}{{\textrm{d}}x}.$$ First, from Eqs. (\[e: Schrodinger equations\]) and the definition of $Z_\mp$ we notice that the Schrödinger equations for the partner potentials can be written in the form $$\frac{{\textrm{d}}}{{\textrm{d}}x} \frac{{\textrm{d}}}{{\textrm{d}}x} (R_1 \pm R_2) + \omega^2 (R_1 \pm R_2) - \left(W^2 \pm \frac{{\textrm{d}}W}{{\textrm{d}}x} \right) (R_1 \pm R_2) = 0.$$ Using Eqs. (\[e: equations R\]) we get that the left hand sides of the previous equations transform into $$\begin{aligned}
\label{e: equations appendix second}
\frac{{\textrm{d}}}{{\textrm{d}}x} \left( i \omega R_1 \mp i \omega R_2 + W R_2 \pm W R_1 \right) &+& \omega^2 (R_1 \pm R_2) \\
&-& \left(W^2 \pm \frac{{\textrm{d}}W}{{\textrm{d}}x} \right) (R_1 \pm R_2) . \nonumber\end{aligned}$$
Expanding the first four factors of the previous expressions and considering Eqs. (\[e: equations R\]) we find that the formulas (\[e: equations appendix second\]) become $$\begin{aligned}
&-& \omega^2 R_1 \mp \omega^2 R_2 + \frac{{\textrm{d}}W}{{\textrm{d}}x} (R_2 \pm R_1) + W^2 R_1 \pm W^2 R_2 \nonumber \\
&+& \omega^2 (R_1 \pm R_2) - \left(W^2 \pm \frac{{\textrm{d}}W}{{\textrm{d}}x} \right) (R_1 \pm R_2) .\end{aligned}$$ Simplifying the previous expressions we obtain $$\frac{{\textrm{d}}W}{{\textrm{d}}x} (R_2 \pm R_1) - \frac{{\textrm{d}}W}{{\textrm{d}}x} (R_2 \pm R_1) = 0.$$ Therefore from the solutions of the coupled system (\[e: equations R\]) we obtain the solutions to the Schrödinger equations of the partner potentials (\[e: partner potentials\]).
Acknowledgments
===============
I thank the support by CONACYT México, SNI México, EDI-IPN, COFAA-IPN, and Research Project IPN SIP-20160074.
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[^1]: For the superpotential $\hat{W} = - m (\mathcal{A} e^x - \mathcal{B})^{-1/2}$ (with the constants $\mathcal{A} > \mathcal{B} > 0$), by making the change of variable $y = x + \ln \left(\mathcal{A}/\mathcal{B} \right)$ and redefining the constant $m$ by $\hat{m} = m / \sqrt{\mathcal{B}}$, we simplify the Schrödinger equations of their partner potentials to those of the potentials for the superpotential (\[e: superpotential ces\]).
[^2]: Notice that for $x \in (0,+\infty)$ the variable $z$ varies over the range $ 0 < z < 1$.
[^3]: We take the constants $c_k$ different from integral numbers to discard the solutions of the hypergeometric equation (\[e: hypergeometric equation\]) that include logarithmic terms [@Abramowitz-book]–[@NIST-book].
[^4]: We point out that the variable $v$ varies over the range $0 < v < 1$.
|
---
abstract: 'We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.'
address: 'Mathematisches Institut, Bunsenstrasse 3-5, D-37073 Göttingen, Germany'
author:
- Berke Topacogullari
bibliography:
- 'ConvSumsArt.bib'
title: The shifted convolution of divisor functions
---
Introduction
============
The binary additive divisor problem is concerned with sums of the form $$\sum_{n \leq x} d(n) d(n + h), \quad h \geq 1,$$ where $ d(n) $ is the usual divisor function. In the past decades a lot of effort has been made to study this problem and several results have been obtained (see [@Moto_ThBinAddDivProb] for a historical survey).
Here we will go one step further and look at the sums $$D^+(x; h) := \sum_{n \leq x} d_3(n) d(n + h) \quad \text{and} \quad D^-(x; h) := \sum_{n \leq x} d_3(n + h) d(n), \quad h \geq 1,$$ where $ d_3(n) $ is the ternary divisor function. This problem has also been studied by several authors, beginning with Hooley [@Hoo_AsympFormTheNumb]. The first result with a power-saving error term seems to be given by Deshouillers [@Desh_MajMoySomKl], who used spectral methods to attack a smoothed version of this problem, much in the spirit of his earlier joint work with Iwaniec [@DeshIwa_AddDivProb] on the binary additive divisor problem. Naturally, Deshouillers’ result can also be used to treat sums like $ D^{\pm}(x, h) $ with sharp cut-off, although he did not work out the details.
As Friedlander and Iwaniec [@FriedIwan_IncomplKlSumsDivProb] pointed out, another approach was possible as a consequence on their work on the ternary divisor function in arithmetic progressions. Heath-Brown [@HB_DivFuncArithProg] improved their result, and showed that $$\begin{aligned}
\label{eqn: result of HB}
D^-(x; 1) = x P(\log x) + \mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^{\frac{101}{102} + \varepsilon} {\aftergroup\egroup\originalright})\end{aligned}$$ for any $ \varepsilon > 0 $, where $P$ is a polynomial of degree three.
Bykovski[ĭ]{} and Vinogradov [@BykVino_InhomConv] returned to the spectral approach of Deshouillers [@Desh_MajMoySomKl] based on the Kuznetsov formula and stated with an exponent $ \frac89 $ in the error term. Unfortunately, not more than a few brief hints were given to support this claim, and our first result is a detailed proof of the following asymptotic formula, which yields in addition a substantial range of uniformity in the shift parameter $h$.
\[thm: main theorem for d(n)\] We have for $ h \ll x^\frac23 $, $$D^\pm(x; h) = x P_h(\log x) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^{ \frac89 + \varepsilon } {\aftergroup\egroup\originalright})},$$ where $ P_h $ is a polynomial of degree three, and where the implied constants depend only on $\varepsilon$.
Let us also state the analogous result for the smoothed sum. For a smooth function $ w : \mathbb{R} \rightarrow \mathbb{R} $, which is compactly supported in $ {\mathopen{}\mathclose\bgroup\originalleft}[ \frac12, 1 {\aftergroup\egroup\originalright}] $, define $$D_w^\pm(x; h) := \sum_n w{\mathopen{}\mathclose\bgroup\originalleft}( \frac nx {\aftergroup\egroup\originalright}) d_3(n) d(n \pm h).$$ Then we have the following
\[thm: main theorem for smoothed d(n)\] We have for $ h \ll x^\frac23 $, $$D_w^\pm(x; h) = x P_{w, h}(\log x) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^{\frac56 + \frac\theta3 + \varepsilon} {\aftergroup\egroup\originalright})},$$ where $ P_{w, h} $ is a polynomial of degree three, and where the implied constants depend at most on $w$ and $\varepsilon$.
By $\theta$ we denote the bound in the Ramanujan-Petersson conjecture (see section \[subsection: The Kuznetsov trace formula and the Large sieve inequalities\] for a precise definition). With the currently best value for $\theta$ we get an error term which is $ \ll x^\frac{7}{8} $, thus improving the result of Deshouillers [@Desh_MajMoySomKl].
Our method applies as well to the dual sum $$D(N) := \sum_{n = 1}^{N - 1} d_3(n) d(N - n).$$ In contrast to the analogous sum with two binary divisor functions (see [@Moto_ThBinAddDivProb Theorem 2]), the main term in our case is a little bit more complicated. Our result is
\[thm: main theorem for the dual sum of d(n)\] We have for any $ \varepsilon > 0 $, $$D(N) = M(N) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( N^{\frac{11}{12} + \varepsilon} {\aftergroup\egroup\originalright})},$$ where the main term $ M(N) $ has the form $$M(N) = N \sum_{{\substack{ 0 \leq i, j, k, \ell \leq 3 \\ i + j + k + \ell \leq 3 }}} c_{i, j, k, \ell} F^{ (i, j, k, \ell) } (0, 0, 0, 0)$$ with certain constants $ c_{i, j, k, \ell} $ and $$F(\alpha, \beta, \gamma, \delta) := N^\alpha \sum_{d \mid N} \frac{ \chi_1(d) }{ d^{1 - \beta} } \sum_{c \mid d} \chi_2(c) \chi_3{\mathopen{}\mathclose\bgroup\originalleft}( \frac dc {\aftergroup\egroup\originalright}),$$ where the arithmetic functions $ \chi_1 $, $ \chi_2 $ and $ \chi_3 $ are defined by $$\begin{aligned}
\begin{gathered} \label{eqn: definition of chi_1, chi_2 and chi_3}
\chi_1(n) := \prod_{p \mid n} {\mathopen{}\mathclose\bgroup\originalleft}( 1 - \frac1{ p^{3 - \gamma - \beta } - p^{ 1 - \gamma + \delta } - p^{1 - \gamma} + 1 } {\aftergroup\egroup\originalright}), \\
\chi_2(n) := \prod_{p \mid n} {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac1{ p^{2 - \beta - \delta} - p^{-\delta} - 1 } {\aftergroup\egroup\originalright}), \quad \chi_3(n) := \prod_{p \mid n} {\mathopen{}\mathclose\bgroup\originalleft}( 1 - \frac1{ p^{1 - \gamma - \delta} } {\aftergroup\egroup\originalright}).
\end{gathered}
\end{aligned}$$ The implied constant depends only on $\varepsilon$.
In particular, we have as leading term $$D(N) = {\mathopen{}\mathclose\bgroup\originalleft}( 1 + o(1) {\aftergroup\egroup\originalright}) C_0 C(N) N \log^3 N,$$ where the constant is given by $$C_0 := \frac3{\pi^2} \prod_p {\mathopen{}\mathclose\bgroup\originalleft}( 1 - \frac1{ p(p + 1) } {\aftergroup\egroup\originalright}),$$ and where $ C(N) $ is a multiplicative function defined on prime powers by $$C{\mathopen{}\mathclose\bgroup\originalleft}( p^k {\aftergroup\egroup\originalright}) := 1 + {\mathopen{}\mathclose\bgroup\originalleft}( 1 - \frac1{ p^k } {\aftergroup\egroup\originalright}) \frac{ 2 p^2 + 2p - 1}{ p^3 - 2p + 1 } - \frac k{p^k} \frac{p + 1}{ (p^2 + p - 1) }.$$
Let $ \varphi(z) $ be a holomorphic cusp form of weight $ \kappa $ for the modular group $ \operatorname{SL}_2( \mathbb{Z} ) $. Let $ a(n) $ be its normalized Fourier coefficients, so that $ \varphi(z) $ has the Fourier expansion $$\varphi(z) = \sum_{n = 1}^\infty a(n) n^\frac{ \kappa - 1 }2 e(nz).$$ The divisor function and the Fourier coefficients $ a(n) $ share a lot of similarities in their behaviour, so one might expect to get analogous results as in Theorems \[thm: main theorem for d(n)\] and \[thm: main theorem for smoothed d(n)\] for the sums $$A^+(x; h) := \sum_{n \leq x} d_3(n) a(n + h) \quad \text{and} \quad A^-(x; h) := \sum_{n \leq x} d_3(n + h) a(n), \quad h \geq 1,$$ and $$A_w^\pm(x; h) := \sum_n w{\mathopen{}\mathclose\bgroup\originalleft}( \frac nx {\aftergroup\egroup\originalright}) d_3(n) a(n \pm h),$$ with the difference that now we cannot expect a main term to appear anymore. Indeed, Pitt [@Pitt_ShiftConvZetaAutLFunc] and Munshi [@Mun_ShiftConvDivFuncRamTauFunc] already obtained results of this sort. Using our method we will be able to partially improve their results by showing
\[thm: main theorem for a(n)\] We have for $ h \ll x^\frac23 $, $$A^\pm(x; h) \ll x^{ \frac89 + \varepsilon } \quad \text{and} \quad A_w^\pm(x; h) \ll x^{\frac56 + \frac\theta3 + \varepsilon},$$ where the implied constants depend at most on $w$, on the holomorphic cusp form $ \varphi(z) $ and on $\varepsilon$.
Of course the dual sum $$A(N) := \sum_{n = 1}^{N - 1} d_3(n) a(N - n),$$ can be treated as well.
\[thm: main theorem for the dual sum of a(n)\] We have $$A(N) \ll N^{\frac{11}{12} + \varepsilon},$$ where the implied constant depends only on $\varepsilon$.
As in [@BykVino_InhomConv] and [@Desh_MajMoySomKl], our main ingredient is the Kuznetsov trace formula, which enables us to exploit the cancellation between Kloosterman sums. This approach yields much better error terms than by using results from algebraic geometry to bound complicated exponential sums individually, as it is done in the other works [@FriedIwan_IncomplKlSumsDivProb], [@HB_DivFuncArithProg], [@Mun_ShiftConvDivFuncRamTauFunc] and [@Pitt_ShiftConvZetaAutLFunc] on $ D^\pm(x; h) $ and $ A^\pm(x; h) $, which give power-saving error terms.
Prerequisites
=============
Note that $ \varepsilon $ always stands for some positive real number, which can be chosen arbitrarily small. However, it need not be the same on every occurrence, even if it appears in the same equation. To avoid confusion we also want to recall that as usually $ e(q) := e^{2\pi i q} $, and that $$S(a, b; c) := \sum_{{\substack{d {\, (c)} \\ (d, c) = 1 }}} {e{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ ad + b \overline{d} }{c} {\aftergroup\egroup\originalright})} \quad \text{and} \quad c_q(n) := \sum_{{\substack{d {\, (q)} \\ (d, q) = 1 }}} {e{\mathopen{}\mathclose\bgroup\originalleft}( \frac{dn}{q} {\aftergroup\egroup\originalright})},$$ which are the usual notations for Kloosterman sums and Ramanujan sums. For Kloosterman sums we have Weil’s bound, $$|S(a, b; c)| \leq d(c) (a, b, c)^\frac12 c^\frac12,$$ while for Ramanujan sums it is well-known that $$|c_q(n)| \leq (n, q).$$
The Voronoi summation formula and Bessel functions
--------------------------------------------------
Using the well-known Voronoi formula for the divisor function (see [@IwaKow_ANT Chapter 4.5] or [@Jut_LectMethThExpSums Theorem 1.6]) and the identity $$\sum_{{\substack{n = 1 \\ n \equiv b {\, (c)} }}}^\infty d(n) f(n) = \frac1c \sum_{d \mid c} \sum_{{\substack{\ell {\, (d)} \\ (\ell, d) = 1 }}} {e{\mathopen{}\mathclose\bgroup\originalleft}( \frac{-b\ell}{d} {\aftergroup\egroup\originalright})} \sum_{n = 1}^\infty d(n) f(n) {e{\mathopen{}\mathclose\bgroup\originalleft}( \frac{n\ell}{d} {\aftergroup\egroup\originalright})},$$ it is not hard to show the following summation formula for the divisor function in arithmetic progressions:
\[thm: Voronoi summation for d(n) in arithmetic progressions\] Let $b$ and $ c \geq 1 $ be integers. Let $ f : (0, \infty) \rightarrow \mathbb{R} $ be smooth and compactly supported. Then $$\begin{aligned}
\sum_{{\substack{n = 1 \\ n \equiv b {\, (c)} }}}^\infty d(n) f(n) &= \frac1c \int \! \lambda_{b, c}(\xi) f(\xi) \, d\xi \\
&\qquad -\frac{2 \pi}c \sum_{d \mid c} \sum_{n = 1}^\infty d(n) \frac{ S(b, n; d) }d \int \! Y_0{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4 \pi}d \sqrt{n \xi} {\aftergroup\egroup\originalright}) f(\xi) \, d\xi \\
&\qquad +\frac4c \sum_{d \mid c} \sum_{n = 1}^\infty d(n) \frac{ S(b, -n; d) }d \int \! K_0{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4 \pi}d \sqrt{n \xi} {\aftergroup\egroup\originalright}) f(\xi) \, d\xi,
\end{aligned}$$ with $$\lambda_{b, c}(\xi) = \frac{ \varphi_b(c) }c \log\xi + 2\gamma \frac{ \varphi_b(c) }c - 2 \frac{ \varphi_b'(c) }c,$$ where $$\varphi_b(c) := c \sum_{d \mid c} \frac{ c_d(b) }d, \quad \text{and} \quad \varphi_b'(c) := c \sum_{d \mid c} \frac{ c_d(b) \log d }d.$$
In the same way, an analogous formula for Fourier coefficients of holomorphic cusp forms can be obtained by using the corresponding Voronoi formula (see [@Jut_LectMethThExpSums Theorem 1.6]):
\[thm: Voronoi summation for a(n) in arithmetic progressions\] Let $b$ and $ c \geq 1 $ be integers. Let $ f : (0, \infty) \rightarrow \mathbb{R} $ be smooth and compactly supported. Then $$\sum_{{\substack{n = 1 \\ n \equiv b {\, (c)} }}} a(n) f(n) = (-1)^\frac\kappa2 \frac{2 \pi}c \sum_{d \mid c} \sum_{n = 1}^\infty a(n) \frac{ S(b, n; d) }d \int_0^\infty \! J_{\kappa - 1} {\mathopen{}\mathclose\bgroup\originalleft}( 4\pi \frac{ \sqrt{n \xi} }d {\aftergroup\egroup\originalright}) f(\xi) \, d\xi.$$
Here we also want to recall the bounds $$d(n) \ll n^\varepsilon \quad \text{and} \quad a(n) \ll n^\varepsilon,$$ the latter following from the Ramanujan-Petersson conjecture proven by Deligne.
Concerning the Bessel function appearing in Theorems \[thm: Voronoi summation for d(n) in arithmetic progressions\] and \[thm: Voronoi summation for a(n) in arithmetic progressions\], we want to sum up some well-known facts. We know that $$\begin{aligned}
K_0(\xi) \ll | \log\xi | \quad \text{for \( \xi \ll 1 \),} \quad \text{and} \quad K_0 \ll \frac1{ e^\xi \sqrt{\xi} } \quad \text{for \( \xi \gg 1 \),}\end{aligned}$$ and that for $ \mu \geq 1 $, $$K_0^{ (\mu) }(\xi) \ll \frac1{ \xi^\mu } \quad \text{for \( \xi \ll 1 \),} \quad \text{and} \quad K_0^{ (\mu) } \ll \frac1{ e^\xi \sqrt{\xi} } \quad \text{for \( \xi \gg 1 \).}$$ Regarding the other two Bessel functions, we have for $ \nu \geq 0 $ and $ \xi \ll 1 $, $$J_\nu(\xi) \ll \xi^\nu \quad \text{and} \quad J_\nu^{ (\mu) } \ll \xi^{\nu - \mu} \quad \text{for } \mu \geq 0,$$ and for $ \nu \geq 1 $ and $ \xi \ll 1 $, $$Y_0(\xi) \ll | \log\xi |, \quad Y_\nu(\xi) \ll \frac1{ \xi^\nu }, \quad \text{and} \quad Y_0^{ (\mu) } \ll \frac1{ \xi^\mu }, \quad Y_\nu^{ (\mu) } \ll \frac1{ \xi^{\nu + \mu} } \quad \text{for } \mu \geq 1.$$ Finally, for $ \nu \geq 0 $ and $ \xi \gg 1 $, it is known that $$J_\nu^{ (\mu) }(\xi), Y_\nu^{ (\mu) }(\xi) \ll \frac1{ \sqrt{\xi} } \quad \text{for \( \mu \geq 0 \).}$$
From the recurrence relations $$\begin{aligned}
\label{eqn: recurrence relations for Bessel functions}
{\mathopen{}\mathclose\bgroup\originalleft}( \xi^\nu B_\nu(\xi) {\aftergroup\egroup\originalright})' = \xi^\nu B_{\nu - 1}(\xi) \quad \text{and} \quad B_{\nu - 1}(\xi) - B_{\nu + 1}(\xi) = 2 B_\nu'(\xi),\end{aligned}$$ which are true for $ B_\nu(\xi) = J_\nu(\xi) $ or $ B_\nu(\xi) = Y_\nu(\xi) $, we get the identity $$\begin{aligned}
\label{eqn: consequence of the recurrence relations}
\int \! B_0{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4\pi}c \sqrt{h \xi} {\aftergroup\egroup\originalright}) f(\xi) \, d\xi = {\mathopen{}\mathclose\bgroup\originalleft}( \frac{-2c}{ 4\pi \sqrt{h} } {\aftergroup\egroup\originalright})^\nu \int \! \xi^\frac\nu2 B_\nu{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4\pi}c \sqrt{h \xi} {\aftergroup\egroup\originalright}) \frac{\partial^\nu f}{\partial \xi^\nu}(\xi) \, d\xi,\end{aligned}$$ which will be useful later. These Bessel functions oscillate for large values, and to make use of this behaviour we have the following
\[lemma: Bessel functions as oscillating functions\] For any $ \nu \geq 0 $ there are smooth functions $ v_J, v_Y: (0, \infty) \rightarrow \mathbb{C} $ such that $$\begin{aligned}
J_\nu(\xi) &= 2{\operatorname{Re}}{\mathopen{}\mathclose\bgroup\originalleft}( {e}{\mathopen{}\mathclose\bgroup\originalleft}( \frac \xi{2\pi} {\aftergroup\egroup\originalright}) v_J{\mathopen{}\mathclose\bgroup\originalleft}( \frac \xi\pi {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}), \\
Y_\nu(\xi) &= 2{\operatorname{Re}}{\mathopen{}\mathclose\bgroup\originalleft}( {e}{\mathopen{}\mathclose\bgroup\originalleft}( \frac \xi{2\pi} {\aftergroup\egroup\originalright}) v_Y{\mathopen{}\mathclose\bgroup\originalleft}( \frac \xi\pi {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}), \label{eqn: representation of Y_0}
\end{aligned}$$ and such that for any $ \mu \geq 0 $, $$\begin{aligned}
v_J^{ (\mu) }, v_Y^{ (\mu) } \ll \frac1{ \xi^{\mu + \frac12} } \quad \text{for } \xi \gg 1. \label{eqn: estimates for v_J and v_Y}
\end{aligned}$$
We start with the integral representations $$J_0(\xi) = \frac1\pi \int_0^\infty \! \sin{\mathopen{}\mathclose\bgroup\originalleft}( \frac x{2\pi} + \frac{\pi \xi^2}{2x} {\aftergroup\egroup\originalright}) \, \frac{dx}x \quad \text{and} \quad Y_0(\xi) = -\frac1\pi \int_0^\infty \! \cos{\mathopen{}\mathclose\bgroup\originalleft}( \frac x{2\pi} + \frac{\pi \xi^2}{2x} {\aftergroup\egroup\originalright}) \, \frac{dx}x,$$ which can be found in [@GradRyzh_TableIntSerProd 3.871]. Here we will only look at $ Y_\nu(\xi) $, as the proof for $ J_\nu(\xi) $ is almost identical. As in [@DeshIwa_AddDivProb Lemma 4], we use a substitution $$y = \frac{ \sqrt{x} }{2\pi} - \frac{\xi}{ 2 \sqrt{x} }, \quad x = \pi^2 {\mathopen{}\mathclose\bgroup\originalleft}( y + \sqrt{ y^2 + \frac\xi\pi } {\aftergroup\egroup\originalright})^2,$$ so that we can write the integral above as $$Y_0(\xi) = -\frac2\pi \int_{-\infty}^\infty \! \cos{\mathopen{}\mathclose\bgroup\originalleft}( 2\pi {\mathopen{}\mathclose\bgroup\originalleft}( y^2 + \frac\xi{2\pi} {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( y^2 + \frac\xi\pi {\aftergroup\egroup\originalright})^{-\frac12} \, dy.$$ Now writing the cosine function out as a sum of exponential functions, we get for $ Y_0 $ with $$v_Y(\xi) = -\frac2\pi \int_0^\infty \! \frac{ {e}(y^2) }{ \sqrt{ y^2 + \xi } } \, dy.$$ The estimate can be shown by splitting the integral at $1$ and repeatedly using partial integration on the part which goes to $\infty$. The statements for $ Y_\nu(\xi) $ follow from .
The Kuznetsov trace formula and the Large sieve inequalities {#subsection: The Kuznetsov trace formula and the Large sieve inequalities}
------------------------------------------------------------
We follow in great parts the notation used in [@DeshIwan_KlSumsFourCoeffCuspForms]. Let $q$ be some positive integer, which will stay fixed throughout this section, and let $ \Gamma := \Gamma_0(q) $ be the Hecke congruence subgroup of level $q$. For these groups we have the spectral decomposition $$L^2(\Gamma \backslash \mathbb{H}) = \mathbb{C} \oplus L_\text{cusp}^2(\Gamma \backslash \mathbb{H}) \oplus L_\text{Eis}^2(\Gamma \backslash \mathbb{H}),$$ where $ L_\text{cusp}^2( \Gamma \backslash \mathbb{H} ) $ is the space spanned by the cusp forms, and $ L_\text{Eis}^2( \Gamma \backslash \mathbb{H} ) $ is a continuous sum spanned by Eisenstein series.
Let $ u_0 $ be the constant function, and let $ u_j $, $ j \geq 1 $, run over an orthonormal basis of $ L_\text{cusp}^2(\Gamma \backslash \mathbb{H}) $, with the corresponding real eigenvalues $ \lambda_0 < \lambda_1 \leq \lambda_2 \leq \ldots $. We set $ {\kappa_j}^2 = \lambda_j - \frac14 $, where we choose the sign of $ \kappa_j $ so that $ i \kappa_j \geq 0 $ if $ \lambda_j < \frac14 $, and $ \kappa_j \geq 0 $ if $ \lambda_j \geq \frac14 $. Then the Fourier expansions of these functions is given by $$u_j(z) = y^\frac12 \sum_{m \neq 0} \rho_j(m) K_{i \kappa_j}(2 \pi |m| y) e(mx).$$ The Selberg eigenvalue conjecture says that $ \lambda_1 \geq \frac14 $, which would imply that all $ \kappa_j $ are real and non-negative, however this still remains to be proven. The eigenvalues with $ 0 < \lambda_j < \frac14 $ as well as the corresponding values $ \kappa_j $ are called exceptional, and lower bounds for these exceptional $ \lambda_j $ imply upper bounds for the corresponding $ i \kappa_j $. Let $ \theta \in \mathbb{R}_0^+ $ be such that $ i \kappa_j \leq \theta $ for all exceptional $ \kappa_j $ uniformly for all levels $q$; by the work of Kim and Sarnak [@Kim_FunctExtSqGL4SymFouGL2] we know that we can choose $$\begin{aligned}
\label{eqn: Kim-Sarnak-bound}
\theta = \frac7{64}.\end{aligned}$$
For any cusp $ \mathfrak{c} $ of $ \Gamma $ we have the Eisenstein series, defined for $ {\operatorname{Re}}s > 1 $ and $ z \in \mathbb{H} $ by $$E_\mathfrak{c}(z; s) = \sum_{ \tau \in \Gamma_\mathfrak{c} \backslash \Gamma } {\operatorname{Im}}( \sigma_\mathfrak{c}^{-1} \tau z )^s,$$ which can be continued meromorphically to the whole complex plane. The space $ L_\text{Eis}^2( \Gamma \backslash \mathbb{H} ) $ is then the continuous direct sum spanned by the $ E_\mathfrak{c}(z; \frac12 + ir) $, $ r \in \mathbb{R} $, and the Fourier expansion of these Eisenstein series around $ \infty $ is given by $$\begin{aligned}
E_\mathfrak{c}(z; s) = \delta_{ \mathfrak{c} \infty } y^s &+ \pi^\frac12 \frac{ \Gamma{\mathopen{}\mathclose\bgroup\originalleft}( s - \frac12 {\aftergroup\egroup\originalright}) }{ \Gamma(s) } \varphi_{ \mathfrak{c}, 0 }(s) y^{1 - s} \\
&+ 2 y^\frac12 \frac{ \pi^s }{ \Gamma(s) } \sum_{n \neq 0} |n|^{s - \frac12} \varphi_{ \mathfrak{c}, n }(s) K_{s - \frac12}(2 \pi |n| y) e(nx).\end{aligned}$$
Finally, denote by $ \mathfrak{M}_k(\Gamma) $ the space of holomorphic cusp forms of weight $k$ and by $ \theta_k(q) $ its dimension. Let $ f_{j, k} $, $ 1 \leq j \leq \theta_k(q) $, be an orthonormal basis of $ \mathfrak{M}_k(\Gamma) $. Then the Fourier expansion of $ f_{j, k} $ around $\infty$ is given by $$f_{j, k}(z) = \sum_{m = 1}^\infty \psi_{j, k}(m) e(mz).$$ With the whole notation set up, we can now formulate the famous Kuznetsov trace formula (see [@DeshIwan_KlSumsFourCoeffCuspForms Theorem 1]).
\[thm: Kuznetsov trace formula\] Let $ f : (0, \infty) \rightarrow \mathbb{C} $ be smooth with compact support. Let $m$, $n$ be two positive integers. Then $$\begin{aligned}
\sum_{ c \equiv 0 {\, (q)} } \frac{ S(m, n; c) }c &f{\mathopen{}\mathclose\bgroup\originalleft}( 4\pi \frac{ \sqrt{mn} }c {\aftergroup\egroup\originalright}) = \sum_{j = 1}^\infty \frac{ \overline{ \rho_j }(m) \rho_j(n) }{ \cosh(\pi \kappa_j) } \hat f( \kappa_j ) \\
&+ \frac1\pi \sum_\mathfrak{c} \int_{-\infty}^\infty \! {\mathopen{}\mathclose\bgroup\originalleft}( \frac mn {\aftergroup\egroup\originalright})^{-ir} \overline{ \varphi_{ \mathfrak{c}, m } } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \varphi_{\mathfrak{c}, n} {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \hat f(r) \, dr \\
&+ \frac1{2 \pi} \sum_{{\substack{ k \equiv 0 {\, (2)} \\ 1 \leq j \leq \theta_k(q) }}} \frac{ i^k (k - 1)! }{ {\mathopen{}\mathclose\bgroup\originalleft}( 4\pi \sqrt{mn} {\aftergroup\egroup\originalright})^{k - 1} } \overline{ \psi_{j, k} }(m) \psi_{j, k}(n) \tilde f(k - 1),
\intertext{and}
\sum_{ c \equiv 0 {\, (q)} } \frac{ S(m, -n; c) }c &f{\mathopen{}\mathclose\bgroup\originalleft}( 4\pi \frac{ \sqrt{mn} }c {\aftergroup\egroup\originalright}) = \sum_{j = 1}^\infty \frac{ \rho_j(m) \rho_j(n) }{ \cosh(\pi \kappa_j) } \check f( \kappa_j ) \\
&+ \frac1\pi \sum_\mathfrak{c} \int_{-\infty}^\infty \! (mn)^{ir} \varphi_{ \mathfrak{c}, m } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \varphi_{\mathfrak{c}, n} {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \check f(r) \, dr,
\end{aligned}$$ where the Bessel transforms are defined by $$\begin{aligned}
\hat f(r) &= \frac\pi{ \sinh(\pi r) } \int_0^\infty \! \frac{ J_{2ir}(\xi) - J_{-2ir}(\xi) }{2i} f(\xi) \, \frac{d\xi}\xi, \\
\check f(r) &= \frac4\pi \cosh(\pi r) \int_0^\infty \! K_{2ir}(\xi) f(\xi) \, \frac{d\xi}\xi, \\
\tilde f(\ell) &= \int_0^\infty \! J_\ell(\xi) f(\xi) \, \frac{d\xi}\xi.
\end{aligned}$$
To get some first estimates for the appearing Bessel transforms we refer to [@BloHarMich_BurgSubconvBoundTwiLFunc Lemma 2.1]:
\[lemma: estimates for the Bessel transforms\] Let $ f : (0, \infty) \rightarrow \mathbb{C} $ be a smooth and compactly supported function such that $${\operatorname{supp}}f \asymp X \quad \text{and} \quad f^{ (\nu) } \ll \frac1{Y^\nu} \quad \text{for} \quad \nu = 0, 1, 2,$$ for positive $X$ and $Y$ with $ X \gg Y $. Then $$\begin{aligned}
\hat f(ir), \check f(ir) &\ll \frac{ 1 + Y^{-2r} }{1 + Y} & \text{for} \quad 0 \leq r < \frac14, \label{eqn: first estimate for the bessel transforms} \\
\hat f(r), \check f(r), \tilde f(r) &\ll \frac{ 1 + |\log Y| }{1 + Y} & \text{for} \quad r \geq 0, \\
\hat f(r), \check f(r), \tilde f(r) &\ll {\mathopen{}\mathclose\bgroup\originalleft}( \frac XY {\aftergroup\egroup\originalright})^2 {\mathopen{}\mathclose\bgroup\originalleft}( \frac1{ r^\frac52 } + \frac X{ r^3 } {\aftergroup\egroup\originalright}) & \text{for} \quad r \gg \max(X, 1).
\end{aligned}$$
For oscillating functions, we can do better. Assume $ w : (0, \infty) \rightarrow \mathbb{C} $ to be a smooth and compactly supported function such that $${\operatorname{supp}}w \asymp X \quad \text{and} \quad w^{ (\nu) } \ll \frac1{X^\nu} \quad \text{for} \quad \nu \geq 0,$$ and for $ \alpha > 0 $ define $$f(\xi) := {e}{\mathopen{}\mathclose\bgroup\originalleft}( \xi \frac\alpha{2 \pi} {\aftergroup\egroup\originalright}) w(\xi).$$ Then the following two lemmas give bounds for the Bessel transforms of $ f(\xi) $, depending on the sizes of $X$ and $\alpha$.
\[lemma: estimates for the Bessel transforms of oscillating functions for large alpha\] Assume that $$X \ll 1 \quad \text{and} \quad \alpha X \gg 1.$$ Then for $ \nu, \mu \geq 0 $, $$\begin{aligned}
\hat f(ir), \check f(ir) &\ll X^{ -2r + \varepsilon } {\mathopen{}\mathclose\bgroup\originalleft}( X^\mu + \frac1{ (\alpha X)^\nu } {\aftergroup\egroup\originalright}) &\text{for} \quad 0 < r \leq \frac14, \label{eqn: first estimate for the bessel transforms for osciallting functions} \\
\hat f(r), \check f(r), \tilde f(r) &\ll \frac{\alpha^\varepsilon}{\alpha X} {\mathopen{}\mathclose\bgroup\originalleft}( \frac{\alpha X}r {\aftergroup\egroup\originalright})^\nu &\text{for} \quad r > 0. \label{eqn: second estimate for the bessel transforms for osciallting functions}
\end{aligned}$$
We begin with . Using the Taylor series of the $ J_\nu $-Bessel function we can write the Bessel transform $ \hat f(ir) $ as $$\begin{aligned}
\hat f(ir) = \frac\pi2 \sum_{m = 0}^\infty \frac{ (-1)^m }{ 4^m m! } \int_0^\infty \! e{\mathopen{}\mathclose\bgroup\originalleft}( \xi \frac\alpha{2 \pi} {\aftergroup\egroup\originalright}) g(\xi, r, m) w(\xi) \xi^{2m - 1} \, d\xi \label{eqn: Taylor series for hat f}
\end{aligned}$$ with $$g(\xi, r, m) := \frac1{ \sin(\pi r) } {\mathopen{}\mathclose\bgroup\originalleft}( \frac1{ \Gamma(m + 2r + 1) } {\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi2 {\aftergroup\egroup\originalright})^{2r} - \frac1{ \Gamma(m - 2r + 1) } {\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi2 {\aftergroup\egroup\originalright})^{-2r} {\aftergroup\egroup\originalright}).$$ For $ 0 < r \leq \frac12 $, one can check that we have the bound $$\frac{ \partial^\nu g }{ \partial \xi^\nu } (\xi, r, m) \ll \frac{ X^{-2r + \nu + \varepsilon} }{ (m - 1)! }.$$ By splitting the sum in at $ m = \mu $, and using partial integration for the finite part while estimating trivially the rest, we get that $$\hat f(ir) \ll X^{-2r + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( X^{2\mu} + \frac1{ (\alpha X)^\nu } {\aftergroup\egroup\originalright}).$$ The estimate for $ \check f(ir) $ follows in exactly the same way by using the corresponding Taylor series for $ K_{2ir}(\xi) $.
For the proof of we follow [@Jut_ConvFourCoeffCuspForms Lemma 3]. We begin with the following identity (see [@GradRyzh_TableIntSerProd 8.411.11]), $$\begin{aligned}
\label{eqn: identity for J-Bessel function}
\frac{ J_{2ir}(\eta) - J_{-2ir}(\eta) }{ \sinh(\pi r) } = \frac2{\pi i} \int_{-\infty}^\infty \! \cos( \eta \cosh\zeta ) \cos(2c \zeta) \, d\zeta,
\end{aligned}$$ which gives $$\hat f(r) = - \int_{-\infty}^\infty \int \! \cos(\eta \cosh\zeta) \cos(2r \zeta) f(\eta) \, \frac{d\eta}\eta d\zeta =: -(I^+ + I^-)$$ with $$I^\pm := \int_{-\infty}^\infty \int \! {e}{\mathopen{}\mathclose\bgroup\originalleft}( \eta {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ \alpha \pm \cosh \zeta }{2\pi} {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}) \frac{ w(\eta) }\eta \cos(2r \zeta) \, d\eta d\zeta.$$ To bound $ I^+ $ we use partial integration $\mu$-times on the integral over $\eta$ and get $$I^+ \ll \frac{ \alpha^\varepsilon }{ (\alpha X)^\mu }.$$
The treatment of $ I^- $ is a little trickier since the factor $$\gamma(\zeta) := \alpha - \cosh \zeta$$ occuring in the exponent may vanish, so that we have to treat the integral differently depending on whether $ \gamma(\zeta) $ is near $0$ or not. Out of technical reasons, it is easier to use smooth weight functions to split the integral. Set $$Z_1 := {\operatorname{arcosh}}(\alpha - A), \quad Z_2 := {\operatorname{arcosh}}(\alpha + A), \quad \text{with} \quad A := \frac1X.$$ Let $ u_i : \mathbb{R} \rightarrow [0, \infty) $, $ i = 1, 2, 3 $, be suitable weight functions such that $$\begin{aligned}
u_1(\xi) &= 1 \quad \text{for} \quad | \xi | \leq \frac12 Z_1 \quad \text{and} \quad {\operatorname{supp}}u_1 \subseteq [ -Z_1, Z_1 ], \\
u_2(\xi) &= 1 \quad \text{for} \quad | \xi | \geq 2 Z_2 \quad \text{and} \quad {\operatorname{supp}}u_2 \subseteq [ -\infty, -Z_2 ] \cup [ Z_2, \infty ],
\end{aligned}$$ and define $$u_3(\xi) := 1 - u_1(\xi) - u_2(\xi).$$ Note that for all $ i = 1, 2, 3 $, $$u_i^{ (\nu) }(\xi) \ll 1 \quad \text{for} \quad \nu \geq 0.$$ Then we have to consider the integrals $$\begin{aligned}
\label{eqn: splitted I^-}
I_i^- := \int \int \! u_i(\zeta) {e}{\mathopen{}\mathclose\bgroup\originalleft}( \eta \frac{ \gamma(\zeta) }{2\pi} {\aftergroup\egroup\originalright}) \frac{ w(\eta) }\eta \cos(2r \zeta) \, d\eta d\zeta,
\end{aligned}$$ and using partial integration $\mu$-times over $\eta$ we get $$I_1^-, I_2^- \ll \frac A{ \alpha (XA)^\mu } + \frac{ \alpha^\varepsilon }{ (\alpha X)^\mu },$$ whereas bounding $ I_3^- $ directly gives $$I_3^- \ll \frac A\alpha.$$ This already proves for $ \nu = 0 $. The result for $ \nu \geq 1 $ can be shown the same way by partially integrating $\nu$-times over $\zeta$ before estimating the integrals absolutely.
The estimate for $ \check f(r) $ can be shown analogously by using the integral representation $$K_{2ir}(\eta) = \frac1{ \cosh(\pi r) } \int_0^\infty \! \cos(\eta \sinh\zeta) \cos(2r \zeta) \, d\zeta$$ (see [@GradRyzh_TableIntSerProd 8.432.4]). Finally, the proof for $ \tilde f(r) $ also goes along the same lines – in this case we use the identity $$J_\ell(\eta) = \frac1\pi \int_0^\pi \! \cos( \ell \zeta - \eta \sin\zeta ) \, d\zeta,$$ which can be found for instance in [@GradRyzh_TableIntSerProd 8.411.1].
\[lemma: estimates for the Bessel transforms of oscillating functions for alpha near 1\] Assume that $$X \gg 1 \quad \text{and} \quad | \alpha - 1 | \ll \frac{ X^\varepsilon }X.$$ Then for $ \nu \geq 0 $, $$\begin{aligned}
\hat f(ir), \check f(ir) &\ll 1 &\text{for} \quad 0 < r \leq \frac14, \label{eqn: first estimate for alpha near 1} \\
\hat f(r), \tilde f(r) &\ll \frac{X^\varepsilon}{ X^\frac12 } {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ X^\frac12 }r {\aftergroup\egroup\originalright})^\nu &\text{for} \quad r > 0, \label{eqn: second estimate for alpha near 1} \\
\check f(r) &\ll \frac{X^\varepsilon}X {\mathopen{}\mathclose\bgroup\originalleft}( \frac Xr {\aftergroup\egroup\originalright})^\nu &\text{for} \quad r > 0.
\end{aligned}$$
The first bound follows directly from . The proof of the other bounds follows the same path as in Lemma \[lemma: estimates for the Bessel transforms of oscillating functions for large alpha\], so we only want to point out some differences. In the case of $ \hat f(r) $, we again use the identity . For $ I^+ $ we here get the bound $$I^+ \ll \frac1{ X^\mu }.$$ It is again necessary to split $ I^- $, and in order to do so, we choose a suitable weight function $ u_1(\xi) $ which satisfies $$u_1(\xi) = 1 \quad \text{for} \quad |\xi| \geq 2Z, \quad u_1(\xi) = 0 \quad \text{for} \quad |\xi| \leq Z,$$ and $$u_1^{ (\nu) }(\xi) \ll \frac1{ Z^\nu } \asymp \frac1{ A^\frac\nu2 },$$ where $$A := \frac{ X^\varepsilon }X \quad \text{and} \quad Z := {\operatorname{arcosh}}(2A + \alpha).$$ Set $ u_2(\xi) := 1 - u_1(\xi) $. Then $$I^- =: I_1^- + I_2^-$$ in the same way as in , and we get $$I_1^- \ll \frac{ A^\frac12 }{ (XA)^\mu } + \frac1{ X^\mu } \ll \frac{ X^\varepsilon }{ X^\frac12 } \quad \text{and} \quad I_2^- \ll A^\frac12 \ll \frac{ X^\varepsilon }{ X^\frac12 }.$$ This gives for $ \nu = 0 $. By partially integrating over $\zeta$, we get the result for higher $\nu$. Finally, the results for $ \tilde f(r) $ and $ \check f(r) $ can be deduced similarly by using the appropriate integral representations for the occuring Bessel functions.
Another important tool are the large sieve inequalities for Fourier coefficients of cusp forms and Eisenstein series (see [@DeshIwan_KlSumsFourCoeffCuspForms Theorem 2]). For a sequence $ a_n $ of complex numbers define $$\| a_n \|_N := \sqrt{ \sum_{N < n \leq 2N} |a_n|^2 },$$ and furthermore set $$\begin{aligned}
\Sigma_j^{(1)} (N) &:= \frac1{ \sqrt{ \cosh(\pi \kappa_j) } } \sum_{N < n \leq 2N} a_n \rho_j(n), \\
\Sigma_{ \mathfrak{c}, r }^{(2)} (N) &:= \sum_{N < n \leq 2N} a_n n^{ir} \varphi_{ \mathfrak{c}, n } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}), \\
\Sigma_{j, k}^{(3)} (N) &:= i^\frac k2 \sqrt{ \frac{ (k - 1)! }{ (4\pi)^{k - 1} } } \sum_{N < n \leq 2N} a_n n^{ -\frac{k - 1}2 } \psi_{j, k}(n).\end{aligned}$$
Then we have the following
\[thm: large sieve inequalities\] Let $ K \geq 1 $ and $ N \geq \frac12 $ be real numbers, $ a_n $ a sequence of complex numbers and $ \mathfrak{c} $ a cusp of $ \Gamma $. Then $$\begin{aligned}
\sum_{ | \kappa_j | \leq K } {\mathopen{}\mathclose\bgroup\originalleft}| \Sigma_j^{(1)} (N) {\aftergroup\egroup\originalright}|^2 &\ll {\mathopen{}\mathclose\bgroup\originalleft}( K^2 + \frac{ N^{1 + \varepsilon} }q {\aftergroup\egroup\originalright}) \| a_n \|_N^2, \\
\sum_\mathfrak{c} \int_{-K}^K \! {\mathopen{}\mathclose\bgroup\originalleft}| \Sigma_{ \mathfrak{c}, r }^{(2)} (N) {\aftergroup\egroup\originalright}|^2 \, dr &\ll {\mathopen{}\mathclose\bgroup\originalleft}( K^2 + \frac{ N^{1 + \varepsilon} }q {\aftergroup\egroup\originalright}) \| a_n \|_N^2, \\
\sum_{{\substack{2 \leq k \leq K, \, 2 \mid k \\ 1 \leq j \leq \theta_k(q) }}} {\mathopen{}\mathclose\bgroup\originalleft}| \Sigma_{k, j}^{(3)}(N) {\aftergroup\egroup\originalright}|^2 &\ll {\mathopen{}\mathclose\bgroup\originalleft}( K^2 + \frac{ N^{1 + \varepsilon} }q {\aftergroup\egroup\originalright}) \| a_n \|_N^2,
\end{aligned}$$ where the implicit constants depend only on $ \varepsilon $.
When there is no averaging over $n$, the bounds given by the large sieve inequalities are not optimal. So, we also want to mention
\[lemma: estimates for Fourier coefficients\] Let $ K \geq 1 $ and $ n \geq 1 $. Then $$\begin{aligned}
\sum_{ | \kappa_j | \leq K } \frac{ | \rho_j(n) |^2 }{ \cosh(\pi \kappa_j) } &\ll K^2 + (qKn)^\varepsilon (q, n)^\frac12 \frac{ n^\frac12 }q, \label{eqn: first estimate for Fourier coefficients} \\
\sum_\mathfrak{c} \int_{-K}^K \! {\mathopen{}\mathclose\bgroup\originalleft}| \varphi_{ \mathfrak{c}, n } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}|^2 dr &\ll K^2 + (qKn)^\varepsilon (q, n)^\frac12 \frac{ n^\frac12 }q, \label{eqn: second estimate for Fourier coefficients} \\
\sum_{{\substack{2 \leq k \leq K, \, 2 \mid k \\ 1 \leq j \leq \theta_k(q) }}} \frac{ (k - 1)! }{ (4\pi n)^{k - 1} } {\mathopen{}\mathclose\bgroup\originalleft}| \psi_{j, k}(n) {\aftergroup\egroup\originalright}|^2 &\ll K^2 + (qKn)^\varepsilon (q, n)^\frac12 \frac{ n^\frac12 }q, \label{eqn: third estimate for Fourier coefficients}
\end{aligned}$$ where the implicit constants depend only on $ \varepsilon $.
For the full modular group, and are proven in [@Moto_SpecTheRieZetaFunc Lemma 2.4]. Except for some obvious modifications, the proof applies as well to general Hecke congruence subgroups. The proof of is a simpler variant of the proof of [@DeshIwan_KlSumsFourCoeffCuspForms Proposition 4].
Finally, to treat the exceptional eigenvalues we need a result, which we state here as
\[lemma: lemma to treat the exceptional eigenvalues\] Let $ X, q, h \geq 1 $ be such that $ h^\frac12 X \geq q $. Then $$\sum_{ \kappa_j \text{ exc.} } | \rho_j(h) |^2 X^{ 4i \kappa_j } \ll (Xh)^\varepsilon \frac{ h^{2\theta} X^{4\theta} }{ q^{4\theta} } (h, q)^\frac12 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac12 }q {\aftergroup\egroup\originalright}),$$ where the implicit constant depends only on $ \varepsilon $.
We have $$\sum_{ \kappa_j \text{ exc.} } | \rho_j(h) |^2 X^{ 4i \kappa_j } \leq {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ h^\frac12 X }q {\aftergroup\egroup\originalright})^{4\theta} \sum_{ \kappa_j \text{ exc.} } | \rho_j(h) |^2 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac q{ h^\frac12 } {\aftergroup\egroup\originalright})^{ 4i \kappa_j }.$$ To treat the sum on the right hand side we make use of [@IwaKow_ANT (16.58)], which says that $$\sum_{ \kappa_j \text{ exc.} } | \rho_j(h) |^2 Y^{ 4i \kappa_j } \ll (qYh)^\varepsilon (h, q)^\frac12 \frac{ h^\frac12 Y }q \quad \text{for} \quad Y \geq 1,$$ and the result follows.
Proof of Theorems \[thm: main theorem for d(n)\] and \[thm: main theorem for a(n)\] {#section: main proof}
===================================================================================
Our method applies to $ D^\pm(x; h) $ as well as $ A^\pm(x; h) $, and it will pose no further difficulty to treat both cases simultaneously. With this in mind, we let $ \alpha(n) $ be a placeholder for $ d(n) $ or $ a(n) $.
From now on we consider $x$ and $h$ as fixed. Let $ w : \mathbb{R} \rightarrow [0, \infty) $ be a smooth function with compact support in $ {\mathopen{}\mathclose\bgroup\originalleft}[ \frac12, 1 {\aftergroup\egroup\originalright}] $ such that $$w^{ (\nu) } \ll \frac1{ \Omega^\nu } \quad \text{for } \nu \geq 0, \quad \text{and} \quad \int \! \big| w^{ (\nu) }(\xi) \big| d\xi \ll \frac1{ \Omega^{\nu - 1} } \, d\xi \quad \text{for} \quad \nu \geq 1,$$ where $ \Omega := x^{-\omega} $ with $ 0 \leq \omega < \frac16. $ We will look at the sum $$\begin{aligned}
\Phi(w) := \sum_n d_3(n) \alpha(n + h) w{\mathopen{}\mathclose\bgroup\originalleft}( \frac nx {\aftergroup\egroup\originalright}), \quad h \in \mathbb{Z}, h \neq 0,\end{aligned}$$ with the aim of showing that $$\begin{aligned}
\label{eqn: main sum}
\Phi(w) = M(w) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^{\frac56 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( x^\frac\theta3 + x^\frac\omega2 {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac14}{ x^\frac16 } {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright})}\end{aligned}$$ for $ h \ll x^\frac56 $ and any $ \varepsilon > 0 $; recall that $\theta$ was defined at . The main term $ M(w) $ vanishes if $ \alpha = a $, and otherwise has the form $ M(w) = x Q_{w, h}(\log x) $ with a cubic polynomial $ Q_{w, h} $. The choice $ \omega = 0 $ gives Theorem \[thm: main theorem for smoothed d(n)\] and the second bound in Theorem \[thm: main theorem for a(n)\], while the choice $ \omega = \frac19 $ together with a suitable weight function $w$ gives Theorem \[thm: main theorem for d(n)\] and the first bound in Theorem \[thm: main theorem for a(n)\].
We will need a smooth decomposition of the ternary divisor function, for which we will use a similar construction as the one used in [@Meur_BinAddDivProb]. Let $ v : \mathbb{R} \rightarrow [0, \infty) $ be a smooth function such that $${\operatorname{supp}}v \subset [-2, 2], \quad \text{and} \quad v(\xi) = 1 \quad \text{for } \xi \in [-1, 1],$$ and define $$v_1(\xi) := v{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi{ x^\frac13 } {\aftergroup\egroup\originalright}), \quad v_2(\xi) := v{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi{ \sqrt{ \frac xa } } {\aftergroup\egroup\originalright}).$$ If $ abc \leq x $, then obviously $${\mathopen{}\mathclose\bgroup\originalleft}( v_1(a) - 1 {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( v_1(b) - 1 {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( v_1(c) - 1 {\aftergroup\egroup\originalright}) = 0 \quad \text{and} \quad {\mathopen{}\mathclose\bgroup\originalleft}( v_2(b) - 1 {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( v_2(c) - 1 {\aftergroup\egroup\originalright}) = 0,$$ and hence $$d{\mathopen{}\mathclose\bgroup\originalleft}( \frac na {\aftergroup\egroup\originalright}) = \sum_{bc = \frac na} v_2(b) {\mathopen{}\mathclose\bgroup\originalleft}( 2 - v_2(c) {\aftergroup\egroup\originalright})$$ as well as $$\begin{aligned}
d_3(n) &= \sum_{abc = n} {\mathopen{}\mathclose\bgroup\originalleft}( v_1(a) v_1(b) v_1(c) - 3 v_1(a) v_1(b) + 3 v_1(a) {\aftergroup\egroup\originalright}) \notag \\
&= \sum_{abc = n} {\mathopen{}\mathclose\bgroup\originalleft}( v_1(a) v_1(b) v_1(c) - 3 v_1(a) v_1(b) {\aftergroup\egroup\originalright}) + 3 \sum_{a \mid n} d{\mathopen{}\mathclose\bgroup\originalleft}( \frac na {\aftergroup\egroup\originalright}) \notag \\
&= \sum_{abc = n} h(a, b, c) \label{eqn: decomposition of d_3(n)}\end{aligned}$$ with $$h(a, b, c) := v_1(a) v_1(b) v_1(c) - 3 v_1(a) v_1(b) + 3 v_1(a) v_2(b) ( 2 - v_2(c) ).$$ Note that this function is non-zero only when $$a, b \ll c.$$
It will be useful to use a partition of unity on $ (0, \infty) $ constructed as follows. Let $ h_X $ be smooth and compactly supported functions such that $${\operatorname{supp}}h_X \subset {\mathopen{}\mathclose\bgroup\originalleft}[ \frac X2, 2X {\aftergroup\egroup\originalright}], \quad h^{ (\nu) } \ll \frac1{ X^\nu } \quad \text{and} \quad \sum_X h_X = 1,$$ where the last sum runs over powers of $2$. Then we set $$h_{ABC}(a, b, c) := h(a, b, c) h_A(a) h_B(a) h_C(c)$$ and $$\Phi_{ABC}(w) := \sum_{a, b, c} h_{ABC}(a, b, c) \alpha(abc + h) w{\mathopen{}\mathclose\bgroup\originalleft}( \frac{abc}x {\aftergroup\egroup\originalright}),$$ so that $$\Phi(w) = \sum_{A, B, C} \Phi_{ABC}(w).$$ Note that we can bound the derivatives of $ h_{ABC} $ by $$\frac{ \partial^{\nu_1 + \nu_2 + \nu_3} }{ \partial a^{\nu_1} \partial b^{\nu_2} \partial c^{\nu_3} } h_{ABC}(a, b, c) \ll \frac1{ A^{\nu_1} B^{\nu_2} C^{\nu_3} }.$$ Furthermore we can assume $$\begin{aligned}
ABC \asymp x \quad \text{and} \quad A \ll B \ll C,\end{aligned}$$ since otherwise $ \Phi_{ABC}(w) $ is empty, and since our argument is symmetric in $A$ and $B$. This also implies that $$A \ll x^\frac13, \quad AB^2 \ll x \quad \text{and} \quad AB \ll x^\frac23.$$
Use of the Voronoi summation formula
------------------------------------
We have $$\begin{aligned}
\Phi_{ABC}(w) &= \sum_{a, b} \sum_{ m \equiv h {\, (ab)} } \alpha(m) w{\mathopen{}\mathclose\bgroup\originalleft}( \frac{m - h}x {\aftergroup\egroup\originalright}) h_{ABC} {\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{m - h}{ab} {\aftergroup\egroup\originalright}) \\
&= \sum_{a, b} \sum_{ m \equiv h {\, (ab)} } \alpha(m) f(m; a, b),\end{aligned}$$ where we have set $$f(\xi; a, b) := w{\mathopen{}\mathclose\bgroup\originalleft}( \frac{\xi - h}x {\aftergroup\egroup\originalright}) h_{ABC} {\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{\xi - h}{ab} {\aftergroup\egroup\originalright}).$$ Note that $${\operatorname{supp}}f( \,\bullet\, ; a, b ) \asymp x \quad \text{and} \quad \frac{ \partial^{\nu_1 + \nu_2} }{ \partial \xi^{\nu_1} \partial b^{\nu_2} } f(\xi; a, b) \ll \frac1{ (x \Omega)^{\nu_1} B^{\nu_2} }.$$
Now we use Theorem \[thm: Voronoi summation for d(n) in arithmetic progressions\] in the case $ \alpha = d $ to get $$\begin{aligned}
\Phi_{ABC}(w) &= \sum_{a, b} \frac1{ab} \int \! \lambda_{h, ab}(\xi) f(\xi; a, b) \, d\xi \\
&\qquad - 2\pi \sum_{{\substack{a, b, c \\ c \mid ab}}} \frac1{ab} \sum_m d(m) \frac{ S(h, m; c) }c \int_0^\infty \! Y_0{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4\pi}c \sqrt{m \xi} {\aftergroup\egroup\originalright}) f(\xi; a, b) \, d\xi \\
&\qquad + 4 \sum_{{\substack{a, b, c \\ c \mid ab}}} \frac1{ab} \sum_m d(m) \frac{ S(h, -m; c) }c \int_0^\infty \! K_0{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4\pi}c \sqrt{m \xi} {\aftergroup\egroup\originalright}) f(\xi; a, b) \, d\xi,\end{aligned}$$ and Theorem \[thm: Voronoi summation for a(n) in arithmetic progressions\] in the case $ \alpha = a $, which gives $$\Phi_{ABC}(w) = (-1)^\frac\kappa2 2\pi \sum_{{\substack{a, b, c \\ c \mid ab}}} \frac1{ab} \sum_m a(m) \frac{ S(h, m; c) }c \int_0^\infty \! J_{\kappa - 1} {\mathopen{}\mathclose\bgroup\originalleft}( 4\pi \frac{ \sqrt{m\xi} }c {\aftergroup\egroup\originalright}) f(\xi; a, b) \, d\xi.$$ The possible main term will be given by $$M_0(w) := \sum_{A, B, C} \sum_{a, b} \frac1{ab} \int \! \lambda_{h, ab}(\xi) f(\xi; a, b) \, d\xi,$$ which we will compute at the end. First we want to treat the other sums and show that they are small enough.
Here we can restate the outer sum as follows $$\begin{aligned}
\sum_{{\substack{a, b, c \\ c \mid ab}}} (\ldots) = \sum_{{\substack{a, b, c, t \\ ab = ct}}} (\ldots) &= \sum_s \sum_{{\substack{a_1, t_1 \\ (a_1, t_1) = 1 }}} \sum_{{\substack{c, b \\ c t_1 = a_1 b }}} (\ldots) &\text{with \( a_1 := \frac as, t_1 := \frac ts \)} \\
&= \sum_s \sum_{{\substack{a_1, t_1 \\ (a_1, t_1) = 1 }}} \sum_{a_1 \mid c} (\ldots) &\text{with \( b = \frac c{a_1} t_1 \)} \\
&= \sum_s \sum_{a_1, t_1} \sum_{ r \mid (a_1, t_1) } \mu(r) \sum_{a_1 \mid c} (\ldots) \\
&= \sum_{s, r} \sum_{a_2, t_2} \mu(r) \sum_{a_2 r \mid c} (\ldots) &\text{with \( a_2 := \frac{a_1}r, t_2 := \frac{t_1}r \)}.\end{aligned}$$ We set $$\begin{aligned}
\label{eqn: definition of F}
F^\pm(c, m) := \frac{ar}c \int_0^\infty \! B^\pm{\mathopen{}\mathclose\bgroup\originalleft}( \frac{4 \pi}c \sqrt{m \xi} {\aftergroup\egroup\originalright}) f{\mathopen{}\mathclose\bgroup\originalleft}( \xi; ars, \frac{ct}a {\aftergroup\egroup\originalright}) \, d\xi\end{aligned}$$ with $$\begin{aligned}
B^+(\xi) &= Y_0(\xi), \quad & &B^-(\xi) = K_0(\xi), \quad & &\text{if \( \alpha = d \),} \\
B^+(\xi) &= J_{\kappa - 1}(\xi), \quad & &B^-(\xi) = 0, \quad & &\text{if \( \alpha = a \),}\end{aligned}$$ and after renaming $a_2$ and $t_2$, we end up with $$\begin{aligned}
R^\pm_{ABC} &:= \sum_{r, s, t} \frac{ \mu(r) }{r^2 st} \sum_a \sum_m \alpha(m) \sum_{ar \mid c} \frac{ S(h, \pm m; c) }c \frac{ F^\pm(c, m) }a \\
&= \sum_{r, s, t} \frac{ \mu(r) }{r^2 st} \sum_a \frac{ R^\pm_{ABC}(a; r, s, t) }a\end{aligned}$$ where $$R^\pm_{ABC}(a; r, s, t) := \sum_m \alpha(m) \sum_{ar \mid c} \frac{ S(h, \pm m; c) }c F^\pm(c, m),$$ for which we need to find good bounds. Note that the sums over $a$ and $c$ are supported in $$a \asymp \frac A{rs} \quad \text{and} \quad c \asymp \frac{AB}{rst}.$$ The function $ F^\pm(c, m) $ can be bound by $$F^\pm(c, m) \ll x^{1 + \varepsilon} \frac A{sc} \asymp x^{1 + \varepsilon} \frac{rt}B,$$ however, when $ m \gg \frac{ c^2 }x $ we can use to get $$F^+(c, n) \ll \frac1{ x^{\frac\nu2 - \frac34} \Omega^{\nu - 1} } \frac{ c^{\nu - \frac12} }{ n^{\frac\nu2 + \frac14} } \frac As \quad \text{and} \quad F^-(c, n) \ll \frac1{ x^{\frac\nu2 - \frac34} } \frac{ c^{\nu - \frac12} }{ n^{\frac\nu2 + \frac14} } \frac As.$$ We set $$M_0^- := \frac{x^\varepsilon}x {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2 \quad \text{and} \quad M_0^+ := \frac{x^\varepsilon}{x \Omega^2} {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2,$$ and a standard exercise then shows that we can cut the sum over $m$ in $ R_{ABC}^\pm $ at $ M_0^\pm $, so that it is sufficient to look at the sums $$\begin{aligned}
R^\pm_{ABC}(M) := \sum_{M < m \leq 2M} \alpha(m) \sum_{ar \mid c} \frac{ S(h, \pm m; c) }c F^\pm(c, m), \label{eqn: cut off sum}\end{aligned}$$ where we have divided the range of summation over $n$ into dyadic intervalls $ [M, 2M] $ with $ M = \frac{M_0^\pm}{2^k} $, where $k$ runs over positive integers.
Auxiliary estimates
-------------------
We want to use the Kuznetsov formula given in Theorem \[thm: Kuznetsov trace formula\] for the inner sum in . To bring the functions $ F^\pm(c, n) $ into the right shape, we define $$\tilde F^\pm(c, m) := h(m) \frac{arc}{ 4\pi \sqrt{|h|m} } \int_0^\infty \! B^\pm{\mathopen{}\mathclose\bgroup\originalleft}( c \sqrt{ \frac\xi{|h|} } {\aftergroup\egroup\originalright}) f{\mathopen{}\mathclose\bgroup\originalleft}( \xi; ars, \frac{ 4\pi \sqrt{|h|m} }{c} \frac ta {\aftergroup\egroup\originalright}) \, d\xi,$$ where $ h(m) $ is a smooth and compactly supported bump function such that $$h(m) \equiv 1 \quad \text{for} \quad m \in [M, 2M], \quad {\operatorname{supp}}h \asymp M \quad \text{and} \quad h^{ (\nu) }(m) \ll \frac1{ M^\nu }.$$ Then we have $$F^\pm(c, m) = \tilde F^\pm {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ 4\pi \sqrt{|h|m} }c, m {\aftergroup\egroup\originalright}) \quad \text{for} \quad m \in [M, 2M].$$
In order to seperate the variable $m$ we use Fourier inversion. First define $$G_0(\lambda) := x^{1 + \varepsilon} \frac{rt}B \min{\mathopen{}\mathclose\bgroup\originalleft}( M, \frac1\lambda, \frac1{ M \lambda^2 } {\aftergroup\egroup\originalright}),$$ which is just a normalization factor. We have $$\tilde F^\pm(c, m) = \int \! G_0(\lambda) G_\lambda^\pm(c) e(\lambda m) \, d\lambda, \quad G_\lambda^\pm(c) := \frac1{ G_0(\lambda) } \int \! \tilde F^\pm(c, m) e(-\lambda m) \, dm,$$ so that $$R_{ABC}^\pm(M) = \int \! G_0(\lambda) \sum_{M < m \leq 2M} \alpha(m) {e}(\lambda m) \sum_{ar \mid c} \frac{ S(h, \pm m; c) }c G_\lambda^\pm{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ 4\pi \sqrt{|h|m} }c {\aftergroup\egroup\originalright}) \, d\lambda.$$ Before going on, we need some good estimates for the Bessel transforms occuring in the Kuznetsov formula. For convenience set $$W := \sqrt{ |h| M } \frac{rst}{AB} \quad \text{and} \quad Z := \sqrt{xM} \frac{rst}{AB}.$$
\[lemma: bounds for the Bessel transforms of G\] We have for $ M \ll M_0^- $, $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll W^{-2c} \quad & &\text{for} \quad 0 \leq c < \frac14, \label{eqn: First bound for Bessel transforms of G} \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ x^\varepsilon }{1 + c^\frac52} \quad & &\text{for} \quad c \geq 0. \label{eqn: Second bound for Bessel transforms of G}
\end{aligned}$$ If $ M_0^- \ll M \ll M_0^+ $, we have for any $ \nu \geq 0 $, $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll x^{-\nu} \quad & &\text{for} \quad 0 \leq c < \frac14, \label{eqn: Third bound for Bessel transforms of G} \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ x^\varepsilon }{ Z^\frac52 } {\mathopen{}\mathclose\bgroup\originalleft}( \frac Zc {\aftergroup\egroup\originalright})^\nu \quad & &\text{for} \quad c \geq 0. \label{eqn: Fourth bound for Bessel transforms of G}
\end{aligned}$$
Since all occurring integrals can be interchanged, we can look directly at the Bessel transforms of $ \tilde F^\pm(c, m) $ and its first two partial derivatives in $m$. We will confine ourselves with the treatment of $ \tilde F^\pm(c, m) $, since the corresponding estimates for the derivatives can be shown the same way.
First we want to use Lemma \[lemma: estimates for the Bessel transforms\] to prove the first two bounds. Again we can look directly at the function inside the integral over $\xi$, given by $$H_1(c) := c B^\pm{\mathopen{}\mathclose\bgroup\originalleft}( c \sqrt{ \frac\xi{ |h| } } {\aftergroup\egroup\originalright}) f{\mathopen{}\mathclose\bgroup\originalleft}( \xi; ars, \frac{ 4\pi \sqrt{ |h| m } }c \frac ta {\aftergroup\egroup\originalright}),$$ for which we have the bounds $${\operatorname{supp}}H_1(c) \asymp W \quad \text{and} \quad H_1^{ (\nu) }(c) \ll x^\varepsilon W {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ x^\varepsilon }W {\aftergroup\egroup\originalright})^\nu.$$ Hence by the mentioned lemma $$\begin{aligned}
\hat H_1(ic), \check H_1(ic) &\ll W^{1 - 2c} \quad \text{for} \quad 0 \leq c < \frac14, \\
\hat H_1(c), \check H_1(c), \tilde H_1(c) &\ll \frac{ x^\varepsilon W }{ 1 + c^\frac52 } \quad \text{for} \quad c \geq 0,
\end{aligned}$$ from which we get and .
When $ M \gg M_0^- $, oscillation effects come into play. By using Lemma \[lemma: Bessel functions as oscillating functions\] and partially integrating once over $\xi$, we get $$\tilde F^+(c, m) = -h(m) \frac{ar}{ 2\pi i \sqrt{ |h| m } } {\operatorname{Re}}{\mathopen{}\mathclose\bgroup\originalleft}( \int_0^\infty \! {e}{\mathopen{}\mathclose\bgroup\originalleft}( \frac c{2\pi} \sqrt{ \frac\xi{ |h| } } {\aftergroup\egroup\originalright}) \tilde w(c) \, d\xi {\aftergroup\egroup\originalright})$$ with $$\tilde w(c) := \frac\partial{\partial \xi} {\mathopen{}\mathclose\bgroup\originalleft}( \sqrt{ \xi |h| } v_Y{\mathopen{}\mathclose\bgroup\originalleft}( \frac c\pi \sqrt{ \frac\xi{ |h| } } {\aftergroup\egroup\originalright}) f{\mathopen{}\mathclose\bgroup\originalleft}( \xi; ars, \frac{ 4\pi t \sqrt{ |h| m } }{ac} {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}).$$ It is hence enough to look at $$H_2(c) := {e}{\mathopen{}\mathclose\bgroup\originalleft}( \frac c{2\pi} \sqrt{ \frac\xi{ |h| } } {\aftergroup\egroup\originalright}) \tilde w(c),$$ where we have the bounds $${\operatorname{supp}}\tilde w \asymp W, \quad \text{and} \quad \tilde w^{ (\nu) }(\xi) \ll \frac{ W^{1 - \nu} }{ Z^\frac32 } C(\xi) \quad \text{with} \quad C(\xi) := 1 + {\mathopen{}\mathclose\bgroup\originalleft}| w'{\mathopen{}\mathclose\bgroup\originalleft}( \frac{\xi - h}x {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}|.$$ We use Lemma \[lemma: estimates for the Bessel transforms of oscillating functions for large alpha\] with $ \alpha = \frac1{2\pi} \sqrt{ \frac\xi{ |h| } } $ and $ X = W $, which is possible since $$W \ll x^{\omega + \varepsilon} \sqrt{ \frac{ |h| }x } \ll \frac1{ x^\varepsilon } \quad \text{and} \quad \alpha W \asymp Z \gg x^\varepsilon,$$ and so we get $$\begin{aligned}
\hat H_2(ic), \check H_2(ic) &\ll x^{-\nu} \quad \text{for} \quad 0 \leq c < \frac14, \\
\hat H_2(c), \check H_2(c), \tilde H_2(c) &\ll x^\varepsilon C(\xi) \frac W{ Z^\frac52 } {\mathopen{}\mathclose\bgroup\originalleft}( \frac Zc {\aftergroup\egroup\originalright})^\nu \quad \text{for} \quad c \geq 0,
\end{aligned}$$ which then give and .
Use of the Kuznetsov trace formula
----------------------------------
Now we are ready to apply the Kuznetsov trace formula. We will only look at $ R_{ABC}^+(M) $ and we will assume that $ h \geq 1 $, since all other cases can be treated in very similar ways. Here we use Theorem \[thm: Kuznetsov trace formula\] on the inner sum, $$\begin{aligned}
\sum_{ar \mid c} \frac{ S(h, m; c) }c &G_\lambda^+{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ 4\pi \sqrt{hm} }c {\aftergroup\egroup\originalright}) = \sum_{j = 1}^\infty \frac{ \overline{\rho_j}(h) \rho_j(m) }{ \cosh(\pi \kappa_j) } \hat G_\lambda^+(\kappa_j) \\
&+ \frac1\pi \sum_\mathfrak{c} \int_{-\infty}^\infty \! m^{ir} \overline{ \varphi_{ \mathfrak{c}, h } } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \varphi_{ \mathfrak{c}, m } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) \hat G_\lambda^+(r) \, dr \\
&+ \frac1{2\pi} \sum_{{\substack{k \equiv 0 {\, (2)} \\ 1 \leq j \leq \theta_k(ar) }}} \frac{ i^k (k - 1)! }{ ( 4\pi \sqrt{m} )^{k - 1} } \overline{ \psi_{j, k} }(h) \psi_{j, k}(m) \tilde G_\lambda^+(k - 1).\end{aligned}$$ Hence we can write our sum as $$R_{ABC}^+(M) = \int \! G_0(\lambda) {\mathopen{}\mathclose\bgroup\originalleft}( \Xi_\text{exc.}(M) + \Xi_1(M) + \frac1\pi \Xi_2(M) + \frac1{2\pi} \Xi_3(M) {\aftergroup\egroup\originalright}) \, d\lambda,$$ where $$\begin{aligned}
\Xi_\text{exc.}(M) &= \sum_{ \kappa_j \text{ exc.} } \hat G_\lambda^+(\kappa_j) {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ \overline{\rho_j}(h) }{ \sqrt{ \cosh(\pi \kappa_j) } } {\aftergroup\egroup\originalright}) \Sigma_j^{( \text{exc.} )}(M), \\
\Xi_1(M) &= \sum_{\kappa_j \geq 0} \hat G_\lambda^+(\kappa_j) {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ \overline{\rho_j}(h) }{ \sqrt{ \cosh(\pi \kappa_j) } } {\aftergroup\egroup\originalright}) \Sigma_j^{(1)}(M), \\
\Xi_2(M) &= \sum_\mathfrak{c} \int \! \hat G_\lambda^+(r) {\mathopen{}\mathclose\bgroup\originalleft}( \overline{ \varphi_{ \mathfrak{c}, h } }{\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}) \Sigma_{ \mathfrak{c},r }^{(2)}(M) \, dr, \\
\Xi_3(M) &= \sum_{{\substack{k \equiv 0 {\, (2)} \\ 1 \leq j \leq \theta_k(ar) }}} \tilde G_\lambda^+(k - 1) {\mathopen{}\mathclose\bgroup\originalleft}( i^\frac k2 \sqrt{ \frac{ (k - 1)! }{ (4\pi)^{k - 1} } } \overline{ \psi_{j, k} }(h) {\aftergroup\egroup\originalright}) \Sigma_{j, k}^{(3)}(M),
\intertext{and}
\Sigma_j^{( \text{exc.} )}(M) := \Sigma_j^{(1)} (M) &:= \frac1{ \sqrt{ \cosh(\pi \kappa_j) } } \sum_{M < m \leq 2M} \alpha(m) {e}(\lambda m) \rho_j(m), \\
\Sigma_{ \mathfrak{c}, r }^{(2)} (M) &:= \sum_{M < m \leq 2M} \alpha(m) {e}(\lambda m) m^{ir} \varphi_{ \mathfrak{c}, m } {\mathopen{}\mathclose\bgroup\originalleft}( \frac12 + ir {\aftergroup\egroup\originalright}), \\
\Sigma_{j, k}^{(3)} (M) &:= i^\frac k2 \sqrt{ \frac{ (k - 1)! }{ (4\pi)^{k - 1} } } \sum_{M < m \leq 2M} \alpha(m) {e}(\lambda m) m^{ -\frac{k - 1}2 } \psi_{j, k}(m).\end{aligned}$$
$ \Xi_\text{exc.}(M) $ needs a special treatment, which we will do in the following section. First, we want to look at the other summands, and here we will restrict ourselves to $ \Xi_1 $, since the treatment of the other sums can be done along the same lines.
First assume $ M \ll M_0^- $. We divide $ \Xi_1(M) $ into two parts: $$\Xi_1(M) = \sum_{\kappa_j \leq 1} (\ldots) + \sum_{1 < \kappa_j} (\ldots) =: \Xi_{ 1 \text{a} }(M) + \Xi_{ 1 \text{b} }(M).$$ For $ \Xi_{ 1 \text{a} }(M) $ we get using , Cauchy-Schwarz, Theorem \[thm: large sieve inequalities\] and Lemma \[lemma: estimates for Fourier coefficients\], $$\begin{aligned}
\Xi_{ 1 \text{a} }(M) &\ll \max_{0 \leq \kappa_j \leq 1} {\mathopen{}\mathclose\bgroup\originalleft}| \hat G_\lambda^+(\kappa_j) {\aftergroup\egroup\originalright}| \sum_{\kappa_j \leq 1} \frac{ | \rho_j(h) | }{ \sqrt{ \cosh(\pi \kappa_j) } } {\mathopen{}\mathclose\bgroup\originalleft}| \Sigma_j^{(1)}(M) {\aftergroup\egroup\originalright}| \\
&\ll x^\varepsilon {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ (ar, h)^\frac12 h^\frac12 }{ar} {\aftergroup\egroup\originalright})^\frac12 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac M{ar} {\aftergroup\egroup\originalright})^\frac12 M^\frac12 \\
&\ll x^\varepsilon \frac1{rt} {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{ x^\frac12 } + \frac{ A^\frac32 B^2 }x {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ (ar, h)^\frac14 h^\frac14 }{ A^\frac12 } {\aftergroup\egroup\originalright})\end{aligned}$$ so that $$\int \! G_0(\lambda) \Xi_{ 1 \text{a} }(M) \, d\lambda \ll x^{\frac56 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ (ar, h)^\frac14 h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}).$$ We split up the remainig sums into dyadic segments $$\Xi_1(M, K) := \sum_{K < \kappa_j \leq 2K} \hat G_\lambda^+(\kappa_j) \frac{ \overline{\rho_j}(h) }{ \sqrt{ \cosh(\pi \kappa_j) } } \Sigma_j^{(1)}(M),$$ and in the same way as above we get $$\Xi_1(M, K) \ll x^\varepsilon \frac1{rt} {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{ x^\frac12 } \frac1{ K^\frac12 } + \frac{ A^\frac32 B^2 }x \frac1{ K^\frac32 } {\aftergroup\egroup\originalright}) {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac1K \frac{ (ar, h)^\frac14 h^\frac14 }{ A^\frac12 } {\aftergroup\egroup\originalright}),$$ which then gives $$\int \! G_0(\lambda) \Xi_{ 1 \text{b} }(M) \, d\lambda \ll x^{\frac56 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ (ar, h)^\frac14 h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}).$$
The case $ M \gg M_0^- $ is handled the same way: We again divide $ \Xi_1(M) $ into two parts $$\Xi_1(M) = \sum_{\kappa_j \leq Z} (\ldots) + \sum_{Z < \kappa_j} (\ldots),$$ and this time we use the bound , which gives $$\int \! G_0(\lambda) \Xi_1(M) \, d\lambda \ll x^{\frac56 + \frac\omega2 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ (ar, h)^\frac14 h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}).$$
The same bounds apply for $ \Xi_2(M) $ and $ \Xi_3(M) $, so that we end up with $$\begin{aligned}
\label{eqn: intermediate result for R(M)}
R_{ABC}^+(M) = \int \! G_0(\lambda) \Xi_\text{exc.}(M) \, d\lambda + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^{\frac56 + \frac\omega2 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( 1 + (ar, h)^\frac14 \frac{ h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright})}.\end{aligned}$$
Treatment of the exceptional eigenvalues
----------------------------------------
For $ M \gg M_0^- $, the exceptional eigenvalues pose no problem at all, since the Bessel transforms $ \hat G_\lambda^+(\kappa_j) $ are very small, as can be seen at . So, $ \Xi_\text{exc.}(M) $ certainly does not exceed the size of the error term in .
For $ M \ll M_0^- $, this is a totally different story. If we would bound $ \Xi_\text{exc.}(M) $ the same way as in the section above using , we would end up with $$\begin{aligned}
\label{eqn: first bound for Xi_exc}
\int \! G_0(\lambda) \Xi_\text{exc.}(M) \, d\lambda \ll x^{\frac56 + \theta + \varepsilon} \frac1{ h^\theta } {\mathopen{}\mathclose\bgroup\originalleft}( 1 + (ar, h)^\frac14 \frac{ h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}).\end{aligned}$$ With the currently best value for $\theta$, this would weaken our result considerably. However, we can reduce the effect of the exceptional eigenvalues by exploiting the fact that these eigenvalues appear infrequently. Cauchy-Schwarz and give $$\begin{aligned}
\Xi_\text{exc.}(M) \ll {\mathopen{}\mathclose\bgroup\originalleft}( \sum_{ \kappa_j \text{ exc.} } {\mathopen{}\mathclose\bgroup\originalleft}( \frac1{ \sqrt{hM} } \frac{AB}{rst} {\aftergroup\egroup\originalright})^{4i \kappa_j} | \rho_j(h) |^2 {\aftergroup\egroup\originalright})^\frac12 {\mathopen{}\mathclose\bgroup\originalleft}( \sum_{ \kappa_j \text{ exc.} } {\mathopen{}\mathclose\bgroup\originalleft}| \Sigma_j^\text{exc.} (M) {\aftergroup\egroup\originalright}|^2 {\aftergroup\egroup\originalright})^\frac12.\end{aligned}$$ The second factor be can treated with the large sieve inequalities. Because of $$h^\frac12 \frac1{ \sqrt{hM} } \frac{AB}{rst} \gg x^{\frac12 - \varepsilon} \gg ar,$$ we can use Lemma \[lemma: lemma to treat the exceptional eigenvalues\] to bound the first factor. So, $$\begin{aligned}
\Xi_\text{exc.}(M) &\ll x^\varepsilon {\mathopen{}\mathclose\bgroup\originalleft}( \frac1{ \sqrt{M} } \frac{AB}{rst} \frac1{ar} {\aftergroup\egroup\originalright})^{2\theta} (ar, h)^\frac14 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac12 }{ar} {\aftergroup\egroup\originalright})^\frac12 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac M{ar} {\aftergroup\egroup\originalright})^\frac12 M^\frac12 \\
&\ll x^\varepsilon \frac1{rt} \frac{ x^\theta }{ A^{2\theta} } {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{ x^\frac12 } + \frac{A^\frac32 B^2}x {\aftergroup\egroup\originalright}) (ar, h)^\frac14 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac14 }{ A^\frac12 } {\aftergroup\egroup\originalright}),\end{aligned}$$ and hence $$\begin{aligned}
\int \! G_0(\lambda) \Xi_\text{exc.}(M) \, d\lambda &\ll x^\varepsilon \frac{ x^\theta }{ A^{2\theta} } {\mathopen{}\mathclose\bgroup\originalleft}( x^\frac12 A + A^\frac32 B {\aftergroup\egroup\originalright}) (ar, h)^\frac14 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac14 }{ A^\frac12 } {\aftergroup\egroup\originalright}) \\
&\ll x^{ \frac56 + \frac\theta3 + \varepsilon} (ar, h)^\frac14 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}),\end{aligned}$$ which is a substantial improvement to .
Eventually we get $$R_{ABC}^\pm(N) \ll x^{ \frac56 + \varepsilon} {\mathopen{}\mathclose\bgroup\originalleft}( x^\frac\omega2 + x^\frac\theta3 {\aftergroup\egroup\originalright}) (ar, h)^\frac14 {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac{ h^\frac14 }{ x^\frac16 } {\aftergroup\egroup\originalright}),$$ which as a consequence gives the error term in .
The main term {#subsection: The main terms}
-------------
To finish the proof of , we have to evaluate the main term, which occurs in the case $ \alpha(n) = d(n) $, and which is given by $$\begin{aligned}
M_0(w) &= \sum_{a, b} \frac1{ab} \int \! \lambda_{h, ab}(\xi + h) w{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi x {\aftergroup\egroup\originalright}) h_{ABC} {\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac\xi{ab} {\aftergroup\egroup\originalright}) \, d\xi \\
&= x \int \! w(\xi) \sum_{a, b} \frac{ \lambda_{h, ab}(x \xi) }{ab} h{\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{x \xi}{ab} {\aftergroup\egroup\originalright}) \, d\xi + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( x^\varepsilon h {\aftergroup\egroup\originalright})},\end{aligned}$$ so effectively we are concerned with $$M_1(\xi) := \sum_{a, b} \frac{ \lambda_{h, ab}(x \xi) }{ab} H_1(a, b; \xi), \quad \text{where} \quad H_1(a, b; \xi) := h{\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{x \xi}{ab} {\aftergroup\egroup\originalright}).$$
Using Mellin inversion this sum can be written as $$M_1(\xi) = \frac1{2\pi i} \sum_a \frac1a \int_{ (\sigma_1) } \! \hat H_1(a, s; \xi) \sum_{b = 1}^\infty \frac{ \lambda_{h, ab}(x \xi) }{ b^{1 + s} } \, ds, \quad \sigma_1 > 0,$$ where the Mellin transform of $ H_1(a, b; \xi) $ is given by $$\hat H_1(a, s; \xi) := \int_0^\infty \! H_1(a, b; \xi) b^{s - 1} \, db, \quad {\operatorname{Re}}(s) > 0.$$ A routine calculation then shows that for $ {\operatorname{Re}}(s) > 0 $, $$\sum_{b = 1}^\infty \frac{ \lambda_{h, ab}(x \xi) }{ b^{1 + s} } \, ds = \zeta(1 + s) \sum_{d = 1}^\infty \frac{ c_d(h) ( \log(x \xi) + 2\gamma - 2 \log d ) (a, d)^{1 + s} }{ d^{2 + s} },$$ so that it is sufficient to look at $$\begin{aligned}
\label{eqn: first Mellin transformed main term}
\tilde M_1(\xi, d) := \frac1{2\pi i} \sum_a \frac{ (a, d) }a \int_{ (\sigma_1) } \! \hat H_1(a, s; \xi) \zeta(1 + s) \frac{ (a, d)^s }{ d^s} \, ds.\end{aligned}$$
Here we want to use the residue theorem. $ \hat H_1(a, s; \xi) $ can be continued meromorphically to the whole complex plane with a simple pole at $ s = 0 $, and its Laurent series is given by $$\begin{aligned}
\hat H_1(a, s; \xi) = 3 v_1(a) \frac1s &+ 3 v_1(a) {\mathopen{}\mathclose\bgroup\originalleft}( \log\frac{x \xi}a + C(a) {\aftergroup\egroup\originalright}) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}(s{\aftergroup\egroup\originalright})},\end{aligned}$$ where $$C(a) := \int_0^\infty \! v_1'(b) \log b \, db + \frac13 \int_0^\infty \! v_1'(b) v_1{\mathopen{}\mathclose\bgroup\originalleft}( \frac{x \xi}{ab} {\aftergroup\egroup\originalright}) \log\frac{x \xi}{a b^2} \, db.$$ We also have that, $$\hat H_1(a, s; \xi) \ll \frac1{ |s| |s + 1| } x^{ \frac13 {\operatorname{Re}}(s) }.$$ Now we shift the line of integration in to $ {\operatorname{Re}}(s) = -1 + \varepsilon $ and the residue theorem gives $$\tilde M_1(\xi, d) = 3 M_{ 2 \text{a} }(\xi, d) + 3 M_{ 2 \text{b} }(\xi, d) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ d^{1 - \varepsilon} }{ x^{\frac13 - \varepsilon} }{\aftergroup\egroup\originalright})},$$ where $$\begin{aligned}
M_{ 2 \text{a} }(\xi, d) &:= \sum_a \frac{ (a, d) }a \log\frac{ (a, d) }a H_{ 2 \text{a} }(a), \quad M_{ 2 \text{b} }(\xi, d) := \sum_a \frac{ (a, d) }a H_{ 2 \text{b} }(a, \xi), \\
\intertext{and}
H_{ 2 \text{a} }(a) &:= v_1(a), \quad H_{ 2 \text{b} }(a; \xi) := v_1(a) {\mathopen{}\mathclose\bgroup\originalleft}( \log\frac{x \xi}d + \gamma + C_1(a) {\aftergroup\egroup\originalright}).\end{aligned}$$
The evaluation of these two sums can be done the same way as above using Mellin inversion and the residue theorem. The appearing Dirichlet series can be continued meromorphically via $$\begin{aligned}
\sum_a \frac{ (a, d) }{ a^{1 + s} } \log\frac{ (a, d) }a &= \sum_{r \mid d} \frac{ \mu(r) }r \sigma_s{\mathopen{}\mathclose\bgroup\originalleft}( \frac dr {\aftergroup\egroup\originalright}) \frac{ ( \zeta'(1 + s) - \zeta(1 + s) \log r ) }{d^s}, \\
\sum_a \frac{ (a, d) }{ a^{1 + s} } &= \frac{ \zeta(1 + s) }{d^s} \sum_{r \mid d} \frac{ \mu(r) }r \sigma_s{\mathopen{}\mathclose\bgroup\originalleft}( \frac dr {\aftergroup\egroup\originalright}),\end{aligned}$$ which are identites for $ {\operatorname{Re}}(s) > 0 $. Furthermore, the Mellin transforms $ \hat H_{ 2 \text{a} }(s) $ and $ \hat H_{ 2 \text{b} }(s; \xi) $ too have a meromorphic continuation to the whole complex plane, both with a simple pole at $ s = 0 $, and with Laurent series of the form $$\begin{aligned}
\hat H_{ 2 \text{a} }(s) &= \frac1s + P_{ 1 \text{a} }(\log x) + s P_{ 2 \text{a} }(\log x) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}(s^2{\aftergroup\egroup\originalright})}, \\
\hat H_{ 2 \text{b} }(s) &= \frac1s P_{ 1 \text{b} }(\log x, \log \xi) + P_{ 2 \text{b} }(\log x, \log \xi) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}(s{\aftergroup\egroup\originalright})},\end{aligned}$$ where $ P_{ 1 \text{a} } $ and $ P_{ 1 \text{b} } $ are linear polynomials, and $ P_{ 2 \text{a} } $ and $ P_{ 2 \text{b} } $ quadratic ones (which may depend on $d$ and $v$). We also have the bounds $$\hat H_{ 2 \text{a} }(s), \hat H_{ 2 \text{b} }(s; \xi) \ll \frac1{ |s| |s + 1| } x^{ \frac13 {\operatorname{Re}}(s) + \varepsilon }.$$ Now applying the residue theorem the same way as before we get $$\tilde M_1(\xi, d) = P(\log x, \log\xi) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ d^{1 - \varepsilon} }{ x^{\frac13 - \varepsilon} } {\aftergroup\egroup\originalright})},$$ where $P$ is a quadratic polynomial depending only on $d$, which as a consequence then gives .
Proof of Theorems \[thm: main theorem for the dual sum of d(n)\] and \[thm: main theorem for the dual sum of a(n)\]
===================================================================================================================
Now we are interested in the sums $$\sum_{n = 1}^{N - 1} d_3(n) d(N - n) \quad \text{and} \quad \sum_{n = 1}^{N - 1} d_3(n) a(N - n),$$ and as before we can consider both sums simultaneously, so that we will stick to the convention that $ \alpha(n) $ is a placeholder for $ d(n) $ or $ a(n) $. We first construct a smooth decomposition of the unit interval in a form suiting our needs. There exist smooth and compactly supported functions $ \tilde u_i : \mathbb{R} \rightarrow [0, \infty) $, $ i \geq 1 $, such that $${\operatorname{supp}}\tilde u_i \subset {\mathopen{}\mathclose\bgroup\originalleft}[ \frac1{ 2^{i + 2} }, \frac1{ 2^i } {\aftergroup\egroup\originalright}], \quad \text{and} \quad \sum_{i = 1}^\infty \tilde u_i(\xi) = 1 \quad \text{for} \quad \xi \in {\mathopen{}\mathclose\bgroup\originalleft}(0, 1/4 {\aftergroup\egroup\originalright}].$$ For $ i \geq 1 $ we then define $$u_i(\xi) := \tilde u_i{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi{N - 1} {\aftergroup\egroup\originalright}), \quad u_{-i}(\xi) := u_i{\mathopen{}\mathclose\bgroup\originalleft}(N - 1 - \xi {\aftergroup\egroup\originalright}) \quad \text{and} \quad u_0(\xi) := 1 - u_1(\xi) - u_{-1}(\xi),$$ so that by construction $$\sum_{ i \in \mathbb{Z} } u_i(\xi) = 1 \quad \text{for} \quad \xi \in (0, N - 1).$$
We have $$\sum_{n = 1}^{N - 1} d_3(n) \alpha(N - n) = \sum_{ i \in \mathbb{Z} } \sum_n u_i(n) d_3(n) \alpha(N - n),$$ hence it is enough to look at the sums $$\Psi_i(N) := \sum_n u_i(n) d_3(n) \alpha(N - n).$$ The evaluation of these sums follows the same path as in section \[section: main proof\], we will therefore use in large parts the same notation and omit many details.
For the sake of easier notation, we will leave out the $i$-subscript from now on. So $ u(\xi) := u_i(\xi) $, and we have $${\operatorname{supp}}u(\xi) \subseteq {\mathopen{}\mathclose\bgroup\originalleft}[ \frac x2, 2x {\aftergroup\egroup\originalright}] \quad \text{for} \quad i \geq 0, \quad {\operatorname{supp}}u(\xi) \subseteq {\mathopen{}\mathclose\bgroup\originalleft}[ N - 2x, N - \frac x2 {\aftergroup\egroup\originalright}] \quad \text{for} \quad i < 0,$$ with $$x := \frac N{ 2^{|i| + 1} }.$$ A first trivial bound is then given by $$\Psi(N) := \Psi_i(N) \ll N^\varepsilon x.$$ The decomposition we use for $ d_3(n) $ is the same as in , but with a different normalization, namely $$v_1(\xi) := v{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi{ (N - 1)^\frac13} {\aftergroup\egroup\originalright}), \quad v_2(\xi) := v{\mathopen{}\mathclose\bgroup\originalleft}( \frac\xi{ \sqrt{ \frac{N - 1}a } } {\aftergroup\egroup\originalright}).$$
It is enough to look at $$\Psi_{ABC} := \sum_{a, b, c} h_{ABC}(a, b, c) \alpha(N - abc) u(abc) = \sum_{a, b} \sum_{ m \equiv N (ab) } \alpha(m) f(m; a, b)$$ with $$f(m; a, b) := h_{ABC}{\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{N - m}{ab} {\aftergroup\egroup\originalright}) u(N - m).$$ After using the Voronoi formula and reordering the sums, we get as a possible main term $$M_0(N) := \sum_{a, b} \frac1{ab} \int \! \lambda_{N, ab}(\xi) f(\xi; a, b) \, d\xi,$$ and as error terms we eventually have to deal with $$R_{ABC}^\pm(M) := \sum_{M < m \leq 2M} \alpha(m) \sum_{ar \mid c} \frac{ S(N, \pm m; c) }c F^\pm(c, m),$$ where $ F^\pm(c, m) $ is defined the same way as in . We only need to look at the $ R_{ABC}^\pm(M) $ with $ M \leq M_0^\pm $, given by $$\begin{aligned}
&M_0^+ := \frac{ N^{1 + \varepsilon} }{x^2} {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2, \quad M_0^- := \frac{ N^\varepsilon }N {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2 \quad \text{for} \quad i \geq 0, \\
\intertext{and}
&M_0^+ := M_0^- := \frac{ N^\varepsilon }x {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2 \quad \text{for} \quad i < 0,\end{aligned}$$ since otherwise $ R_{ABC}^\pm(M) $ is small.
We bring again everything into the right shape for the use of the Kuznetsov formula by setting $$\tilde F^\pm(c, m) := F^\pm{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ 4\pi \sqrt{Nm} }c, m {\aftergroup\egroup\originalright})$$ and using Poisson inversion so separate the variable $m$, so that $$R_{ABC}^\pm(M) = \int \! G_0(\lambda) \sum_{M < m \leq 2M} \alpha(m) {e}(\lambda m) \sum_{ar \mid c} \frac{ S(N, \pm m; c) }c G_\lambda^\pm{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ 4\pi \sqrt{Nm} }c {\aftergroup\egroup\originalright}) \, d\lambda,$$ where $$G_0(\lambda) := N^\varepsilon x \frac{rt}B \min{\mathopen{}\mathclose\bgroup\originalleft}( M, \frac1\lambda, \frac1{ M \lambda^2 } {\aftergroup\egroup\originalright}).$$ Set $$W := \sqrt{NM} \frac{rst}{AB}.$$ When bounding the Bessel transforms, we have to distinguish between the cases $ i \geq 0 $ and $ i < 0 $.
The case $ i \geq 0 $
---------------------
In this case, we have the following bounds when $ M \ll M_0^- $, $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll N^\varepsilon W^{-2c} \quad & &\text{for} \quad 0 \leq c < \frac14, \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }{1 + c^\frac52} \quad & &\text{for} \quad c \geq 0,\end{aligned}$$ while for $ M_0^- \ll M \ll M_0^+ $ we have $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll \frac{ N^\varepsilon }W \quad & &\text{for} \quad 0 \leq c < \frac14, \\
\hat G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }W {\mathopen{}\mathclose\bgroup\originalleft}( \frac{ W^\frac12 }c {\aftergroup\egroup\originalright})^\nu \quad & &\text{for} \quad c \geq 0, \\
\check G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }{ W^\frac32 } {\mathopen{}\mathclose\bgroup\originalleft}( \frac Wc {\aftergroup\egroup\originalright})^\nu \quad & &\text{for} \quad c \geq 0.\end{aligned}$$ All these bounds can be derived the same way as in Lemma \[lemma: bounds for the Bessel transforms of G\]. There are two slight differences, though: Applying partial integration once over $\xi$ is useless here. Furthermore, instead of Lemma \[lemma: estimates for the Bessel transforms of oscillating functions for large alpha\] we need to use Lemma \[lemma: estimates for the Bessel transforms of oscillating functions for alpha near 1\].
Now applying the Kuznetsov formula and the large sieve inequalities, we get that $$R_{ABC}^+(M) \ll N^{ \frac{11}{12} + \varepsilon } \quad \text{and} \quad R_{ABC}^-(M) \ll N^{ \frac{11}{12} + \varepsilon } + \frac{ N^{\frac43 + \varepsilon} }{ x^\frac12}.$$ In contrast to section \[section: main proof\], the exceptional eigenvalues cause no problem at all.
The case $ i < 0 $
------------------
The bounds for the Bessel transforms for $ M \ll \frac{ N^\varepsilon}N {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2 $ are given by $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll N^\varepsilon W^{-2c} \quad & &\text{for} \quad 0 \leq c < \frac14, \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }{1 + c^\frac52} \quad & &\text{for} \quad c \geq 0,\end{aligned}$$ and for $ \frac{ N^\varepsilon}N {\mathopen{}\mathclose\bgroup\originalleft}( \frac{AB}{rst} {\aftergroup\egroup\originalright})^2 \ll M \ll M_0^\pm $ by $$\begin{aligned}
\hat G_\lambda^\pm(ic), \check G_\lambda^\pm(ic) &\ll \frac{ N^\varepsilon }W \quad & &\text{for} \quad 0 \leq c < \frac14, \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }W \quad & &\text{for} \quad c \geq 0, \\
\hat G_\lambda^\pm(c), \check G_\lambda^\pm(c), \tilde G_\lambda^\pm(c) &\ll \frac{ N^\varepsilon }{ c^\frac52 } {\mathopen{}\mathclose\bgroup\originalleft}( 1 + \frac W{ c^\frac12 } {\aftergroup\egroup\originalright}) \quad & &\text{for} \quad c \gg W.\end{aligned}$$ Another use of the Kuznetsov formula gives $$R_{ABC}^\pm(M) \ll N^{ \frac{11}{12} + \varepsilon }.$$ So, altogether we have for all $ i \in \mathbb{Z} $, $$R_{ABC}^\pm(M) \ll N^{ \frac{11}{12} + \varepsilon } + \frac{ N^{\frac43 + \varepsilon} }{ x^\frac12}.$$ We use this bound for $ x \gg N^\frac89 $ and otherwise bound trivially, to get the error terms claimed in Theorems \[thm: main theorem for the dual sum of d(n)\] and \[thm: main theorem for the dual sum of a(n)\].
The main term {#the-main-term}
-------------
To finish the proof, we have to evaluate the main term, which occurs in the case $ \alpha = d $ and which is given by $$\begin{aligned}
M_0(N) &= \sum_{a, b} \frac1{ab} \int_1^{N - 1} \! \lambda_{N, ab}(\xi) h{\mathopen{}\mathclose\bgroup\originalleft}(a, b, \frac{N - \xi}{ab} {\aftergroup\egroup\originalright}) \, d\xi \\
&= N \int_0^1 \! \sum_{a, b} \frac{ \lambda_{N, ab}( N(1 - \xi) ) }{ab} h{\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{ (N - 1)\xi }{ab} {\aftergroup\egroup\originalright}) \, d\xi + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( N^\varepsilon {\aftergroup\egroup\originalright})}.\end{aligned}$$ This, too, can be done the same way as in section \[subsection: The main terms\], so we will just state some intermediate results. It is enough to look at $$M_1(\xi) = \sum_{a, b} \frac{ \lambda_{N, ab}( N(1 - \xi) ) }{ab} h{\mathopen{}\mathclose\bgroup\originalleft}( a, b, \frac{ (N - 1)\xi }{ab} {\aftergroup\egroup\originalright}) \, d\xi,$$ and this sum can be evaluated by using Mellin inversion and the residue theorem, so that we get $$\begin{aligned}
M_1(\xi) &= 3 \sum_{d = 1}^\infty \frac{ c_d(N) {\mathopen{}\mathclose\bgroup\originalleft}( \log( N (1 - \xi) ) + 2\gamma - 2 \log d {\aftergroup\egroup\originalright}) }{d^2} {\mathopen{}\mathclose\bgroup\originalleft}( M_{ 2 \text{a} }(\xi, d) + M_{ 2 \text{b} }(\xi, d) {\aftergroup\egroup\originalright}) \\
&\phantom{ ={} } + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( \frac1{ N^{\frac13 - \varepsilon} \xi^{1 - \varepsilon} } {\aftergroup\egroup\originalright})},\end{aligned}$$ with $$\begin{aligned}
M_{ 2 \text{a} }(d) &= \sum_a \frac{ (a, d) }a \log{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ (a, d) }a {\aftergroup\egroup\originalright}) v_1(a), \\
M_{ 2 \text{b} }(\xi, d) &= \sum_a \frac{ (a, d) }a v_1(a) {\mathopen{}\mathclose\bgroup\originalleft}( \log{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ N\xi }d {\aftergroup\egroup\originalright}) + \gamma + C(a) {\aftergroup\egroup\originalright}),\end{aligned}$$ and $$C(a) = \int_0^\infty \! v_1'(b) \log b \, db + \frac13 \int_0^\infty \! v_1'(b) v_1{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ (N - 1)\xi }{ab} {\aftergroup\egroup\originalright}) \log{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ (N - 1)\xi }{ ab^2 } {\aftergroup\egroup\originalright}) \, db .$$
The evaluation of $ M_{ 2 \text{a} }(d) $ and $ M_{ 2 \text{b} }(\xi, d) $ follows the usual pattern, and as result we get $$M_{ 2 \text{a} }(d) + M_{ 2 \text{b} }(\xi, d) = \sum_{r \mid d} \frac{ \mu(r) }r \sum_{ m \mid \frac dr } P_2(\log N, \log d, \log r, \log m) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( \frac{ d^{1 - \varepsilon} }{ N^{\frac13 - \varepsilon} \xi^{1 - \varepsilon} } {\aftergroup\egroup\originalright})},$$ where $ P_2 $ is a quadratic polynomial (which depends on $\xi$). From this we see that $ M_0(N) $ has the form $$M_0(N) = N \sum_{d = 1}^\infty \frac{ c_d(N) }{d^2} \sum_{r \mid d} \frac{ \mu(r) }r \sum_{ m \mid \frac dr } P_3(\log N, \log d, \log r, \log m) + {\mathcal{O}{\mathopen{}\mathclose\bgroup\originalleft}( N^{\frac23 + \varepsilon} {\aftergroup\egroup\originalright})},$$ with a cubic polynomial $ P_3 $.
We want to reshape this result a little bit. Set $$G(\alpha, \beta, \gamma, \delta) := N^\alpha \sum_{d = 1}^\infty \frac{ c_d(N) }{ d^{2 - \beta} } \sum_{r \mid d} \frac{ \mu(r) }{ r^{1 - \gamma} } \sigma_\delta{\mathopen{}\mathclose\bgroup\originalleft}( \frac dr {\aftergroup\egroup\originalright}),$$ so that the main term can be stated in terms of the partial derivatives of $G$ up to third order evaluated at $ (0, 0, 0, 0) $. A lengthy but elementary calculation shows that $$\begin{aligned}
G(\alpha, \beta, \gamma, \delta) &= N^\alpha \sum_{d \mid N} \sum_{{\substack{c \mid d \\ b \mid c}}} \mu{\mathopen{}\mathclose\bgroup\originalleft}( \frac dc {\aftergroup\egroup\originalright}) \frac{ c^{1 - \gamma + \delta} }{ d^{2 - \gamma - \beta} b^\delta } \sum_{ (r, d) = 1 } \sum_{{\substack{ (s, br) = 1 \\ (d, rs) = 1 }}} \frac{ \mu^2(r) \mu(s) \mu(d) }{ r^{3 - \gamma - \delta} s^{2 - \beta - \delta } d^{2 - \beta} } \\
&= C(\beta, \gamma, \delta) N^\alpha \sum_{d \mid N} \frac{ \chi_1(d) }{ d^{1 - \beta} } \sum_{c \mid d} \chi_2{\mathopen{}\mathclose\bgroup\originalleft}( \frac dc {\aftergroup\egroup\originalright}) \chi_3(c),\end{aligned}$$ with $$C(\beta, \gamma, \delta) := \frac1{ \zeta(2 - \delta) } \prod_p {\mathopen{}\mathclose\bgroup\originalleft}( 1 - \frac{ p^{1 - \gamma + \delta} - 1 }{ p^{1 - \gamma} ( p^{2 - \beta} - 1 ) } {\aftergroup\egroup\originalright})$$ and $ \chi_1 $, $ \chi_2 $, and $ \chi_3 $ defined as in . This eventually gives Theorem \[thm: main theorem for the dual sum of d(n)\].
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EMPG–13–14
2.0cm
[**Lie 2-algebra models**]{} 1.5cm [Patricia Ritter$^a$ and Christian Sämann$^b$]{}
\
[*${}^b$ Maxwell Institute for Mathematical Sciences\
Department of Mathematics, Heriot-Watt University\
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.*]{}\
[Email: [[email protected] , [email protected]]{}]{}
1.0cm
[**Abstract**]{}
> In this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of ${\mathbbm{R}}^3$, $S^3$ and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized ${\mathbbm{R}}^3$, we obtain higher BF-theory on this quantized space.
Introduction and motivation
===========================
One of the fundamental problems in theoretical physics today is the construction of theories that are formulated without reference to any specific space-time geometry. In such background independent models, space-time is expected to emerge from the dynamics of the theory, for example as vacuum configurations. A good example of such a theory is the IKKT matrix model [@Ishibashi:1996xs], which was conjectured to provide a non-perturbative and background independent formulation of superstring theory. This model arises as a finite regularization of the type IIB superstring in Schild gauge. It is a zero-dimensional theory, in which fields take values in a (matrix) Lie algebra.
It has become more and more evident that many of the algebraic structures underlying string and M-theory are not Lie algebras but rather extensions of Lie algebras which are known as strong homotopy Lie algebras or $L_\infty$-algebras. In particular, regularizations of the membrane action yield models with fields taking values in truncated, 2-term $L_\infty$-algebras. It is therefore desirable to study generalized IKKT-like models, in which fields can take values in strong homotopy Lie algebras. The purpose of this paper is to initiate such a study.
To keep our models manageable, we will restrict ourselves to the 2-term strong homotopy Lie algebras, which are categorically equivalent to semistrict Lie 2-algebras[^1]. These algebras feature prominently in higher gauge theories which seem to underlie M-brane models, and a subclass of these form the gauge structure of the recently popular M2-brane models [@Bagger:2007jr; @Gustavsson:2007vu; @Aharony:2008ug]. This is to be seen in analogy to the conventional Lie algebras of the IKKT model underlying the gauge theories arising in the effective description of D-brane configurations in string theory.
This paper is structured as follows. In the remainder of this section, we will give a more detailed motivation for studying Lie 2-algebra models. We then review relevant definitions of Lie 2-algebras and discuss various notions of inner products on them in section 2. Section 3 makes contact with the quantization of 2-plectic manifolds. Homogeneous and inhomogeneous Lie 2-algebra models are then discussed in section 4 and section 5, respectively. We present our conclusions in section 6. Two appendices summarize useful definitions and review the gauge symmetry of semistrict higher gauge theory for the reader’s convenience.
Background independence and the IKKT model {#ssec:background_IKKT}
------------------------------------------
As stated above, it is an important goal to construct and study background independent theories to replace our mostly background dependent formulations of string theory. A straightforward method for eliminating the background geometry from any field theory is to dimensionally reduce it to a point. If the fields in the original theory took values in a Lie algebra and its adjoint representation, one is left with a matrix model.
Matrix models have indeed contributed greatly to the understanding of non-perturbative phenomena in string theory. This started with the Hermitian matrix models describing $c<1$ string theory [@DiFrancesco:1993nw] and continued with the success of the IKKT matrix model [@Ishibashi:1996xs], see also [@Aoki:1998bq].
The IKKT model is obtained by regularizing the Green-Schwarz action of the type IIB superstring in Schild gauge, $$S=\int {\mathrm{d}}^2\sigma \sqrt{g} \alpha \left(\tfrac{1}{4}\{X_\mu,X_\nu\}^2-\tfrac{{\mathrm{i}}}{2}{{\bar{\psi}}}\Gamma^\mu\{X_\mu,\psi\}\right)+\beta\sqrt{g}~.$$ In this regularization, the worldsheet fields $X_\mu$ and $\psi$ are replaced by hermitian matrices $A_\mu$ and $\psi$, while the integral becomes a trace and the Poisson bracket $\{{-},{-}\}$ is turned into the commutator $-{\mathrm{i}}[{-},{-}]$. Note that this process is standard in noncommutative field theory and the result is the following: $$\label{eq:action_IKKT}
S_{\rm IKKT}=\alpha{\,\mathrm{tr}\,}\left(-\tfrac{1}{4}\,[A_\mu,A_\nu]^2-\tfrac{1}{2}\,{{\bar{\psi}}}\Gamma^\mu[A_\mu,\psi]+\beta {\mathbbm{1}}\right)~.$$ Alternatively, one can obtain the IKKT model by dimensionally reducing maximally supersymmetric Yang-Mills theory in ten dimensions to a point. The fields $A_\mu$ and $\psi$ here take values in the gauge algebra of the ten-dimensional theory.
As equations of motion of the action , we have $$\label{eq:eom_IKKT}
\begin{aligned}
{}[A_\mu,[A^\mu,A^\nu]]-\tfrac{{\mathrm{i}}}{2}\Gamma^\nu_{\alpha\beta}\{\psi^\beta,{{\bar{\psi}}}^\alpha\}&=0~,\\
\Gamma^\mu_{\alpha\beta}[A_\mu,\psi^\beta]&=0~.
\end{aligned}$$ Amongst the solutions to these equations are matrices $A_m$, $m=1,\ldots,2d$, that we can identify with the generators $\hat{x}^m$ of the Heisenberg algebra $[\hat{x}^m,\hat{x}^n]={\mathrm{i}}\theta^{mn}{\mathbbm{1}}$. The generators $\hat{x}^m$ are the coordinate functions on the Moyal space ${\mathbbm{R}}^{2d}_\theta$, and this is the most prominent example of a geometry emerging as the vacuum configuration of the IKKT model. Expanding the action around this background solution as $A_m=\hat{x}_m+\hat{A}_m$, we obtain Yang-Mills theory on noncommutative ${\mathbbm{R}}^{2d}_\theta$ [@Aoki:1999vr]. The action therefore simultaneously provides the background and the dynamics of the theory.
More general noncommutative geometries are obtained as vacuum solutions of deformations of the IKKT model. A particularly interesting class of deformations comprise mass-terms as well as a cubic potential term, $$\label{eq:action_IKKT_def}
S_{\rm def}=S_{\rm IKKT}+{\,\mathrm{tr}\,}\left( -\tfrac{1}{2}\sum_\mu m^2_{1,\mu}A_\mu A_\mu+\tfrac{{\mathrm{i}}}{2}m_2 {{\bar{\psi}}}\psi+c_{\mu\nu\kappa}A^\mu A^\nu A^\kappa\right)~,$$ where $c_{\mu\nu\kappa}$ is some background tensor field, cf. [@Berenstein:2002jq]. This action has classical configurations corresponding to fuzzy spheres and the space ${\mathbbm{R}}^3_\lambda$, which is a discrete foliation of ${\mathbbm{R}}^3$ by fuzzy spheres, as well as noncommutative Hpp waves, see [@DeBellis:2010sy] and references therein.
Finally, note that in a very similar manner in which a background expansion of the IKKT model yields Yang-Mills theories on noncommutative spaces, one can also obtain models of gravity, see e.g. [@Steinacker:1003.4134].
Lie $n$-algebras in string theory
---------------------------------
Lie $2$-algebras arise in the categorification of the notion of a Lie algebra. In this process, the vector space underlying the Lie algebra is replaced by a category. Furthermore, the standard structural equations of a Lie algebra, which state that the Lie bracket is antisymmetric and satisfies a Jacobi identity, are lifted in a controlled way and hold only up to an isomorphism in this category. Lie $n$-algebras arise analogously by $n$-fold, iterative categorification of Lie algebras. In the semistrict case, which is the one we will consider exclusively in this paper, Lie $n$-algebras are equivalent to truncated $n$-term strong homotopy Lie or $L_\infty$-algebras, which are also known as $L_n$-algebras.
Strong homotopy Lie algebras and in particular their truncated versions appear in a variety of contexts related to string theory, for example:
- Strong homotopy Lie algebras arise in string field theory, cf. [@Zwiebach:1992ie; @Kajiura:0410291], as well as in Kontsevich’s deformation quantization.
- Lie 2-algebras appear in topological open M2-brane actions in the form of Courant Lie 2-algebroids [@Hofman:2002rv].
- Special Lie 2-algebras, which are known as differential crossed modules, form the gauge structure of the recently popular M2-brane models [@Bagger:2007jr; @Gustavsson:2007vu; @Aharony:2008ug] as shown in [@Palmer:2012ya].
- The full M2-brane action is coupled to the $C$-field of supergravity and is thus expected to be related to parallel transport of two-dimensional objects, which has an underlying Lie 3-algebra [@Sati:0801.3480].
- Interactions of M5-branes are mediated by M2-branes ending on them and their boundaries are one-dimensional objects known as self-dual strings. It is natural to assume that an effective description of M5-branes yields a higher gauge theory describing the parallel transport of these self-dual strings. The gauge structure of such a higher gauge theory is described by a Lie 2-algebra, cf. [@Baez:2010ya].
- Equations of motion of interacting non-abelian superconformal field theories in six dimensions have been derived using twistor spaces in [@Saemann:2012uq; @Saemann:2013pca]. These constructions again make use of the framework of higher gauge theory, employing Lie 2- and 3-algebras.
Our goals in this paper
-----------------------
We saw above that the Lie algebras describing gauge symmetries in effective descriptions of D-branes within string theory are replaced by Lie 2-algebras in M-theory. It is therefore natural to suspect that a potential non-perturbative description of M-theory along the lines of the IKKT model may be based on Lie 2-algebras.
In this paper, we perform an initial study of zero-dimensional field theories in which the fields take values in a Lie 2-algebra. We discuss the mathematical notions required in the description of Lie 2-algebra models, put them into context and test how far the analogies with the IKKT model reach.
Throughout this paper, we will distinguish two types of models. First, there are [*homogeneous Lie 2-algebra models*]{}, in which the fields $\{X^a\}$ take values in the direct sum of the two vector spaces $V$ and $W$ that underlie a Lie 2-algebra. In the [*inhomogeneous models*]{}, we have two types of fields $\{X^a\}$ and $\{Y^i\}$, where the $X^a$ take values in $V$ while the $Y^i$ take values in $W$. Note that homogeneous models form a subset of the inhomogeneous models.
Since semistrict Lie 2-algebras contain ordinary Lie algebras, homogeneous Lie 2-algebra models will trivially contain the IKKT model as a special case. Moreover, Lie 2-algebras contain the 3-algebras appearing in M2-brane models, and we therefore also expect our Lie 2-algebra models to contain the 3-algebra models discussed previously in [@Sato:2009mf; @Lee:2009ue; @Sato:2009tr; @Sato:2010ca], [@Furuuchi:2009ax] and [@DeBellis:2010sy].
In [@Sato:2009mf], the author followed the logic of the IKKT model, starting from a Schild-type action of the M2-brane [@Park:2008qe], $$S=T_{\rm M2} \int {\mathrm{d}}^3\sigma \{X^M,X^N,X^K\}^2~,~~~M,N,K=0,\ldots,10~.$$ He then suggested to regularize this action by replacing the Nambu-bracket by that of a 3-Lie algebra. Note that it has often been suggested that, at quantum level, Nambu-Poisson structures should turn into 3-Lie algebras, see [@DeBellis:2010pf] and references therein. To a certain extent, one can even make the resulting action supersymmetric[^2], and the result is [@Furuuchi:2009ax; @Sato:2009tr] $$\label{eq:3LA_action}
S_{\rm 3LA}=\langle [X^M,X^N,X^K],[X^M,X^N,X^K]\rangle+\langle \bar{\Psi},\Gamma_{MN} [X^M,X^N,\Psi]\rangle~,$$ where $\Psi$ is a Majorana spinor of ${\mathsf{SO}}(1,10)$. A very similar model has been studied in [@Chu:2011yd] as a matrix model for the description of multiple M5-branes.
Alternatively, one can obtain a zero-dimensional action with fields living in a 3-Lie algebra by dimensionally reducing the M2-brane models to a point. The case of the BLG-model was discussed in [@DeBellis:2010sy], where various solutions have been interpreted as quantized Nambu-Poisson manifolds. Compared to , there are additional scalar fields present, living in the inner derivations of the underlying 3-Lie algebra that arise from the dimensional reduction of the Chern-Simons part and the covariant couplings to the matter fields. While there is now a dichotomy of fields compared to , the resulting action is invariant under 16 supercharges. Moreover, applying a dimensionally reduced form of the Higgs mechanism proposed in [@Mukhi:2008ux], this action reduces to in the strong coupling limit as shown in [@DeBellis:2010sy].
An important feature of the IKKT model is that familiar examples of quantized symplectic manifolds arise as solutions of the classical equations of motion. Correspondingly, we expect that “higher quantized” manifolds arise as solutions of our Lie 2-algebra models. There are essentially two approaches in the literature of how to extend geometric quantization to a higher setting. First, we can focus on the Poisson structure and generalize this structure to a Nambu-Poisson structure. The geometric quantization of Nambu-Poisson manifolds, however, is problematic and the answers obtained in this context are not very satisfying, see [@DeBellis:2010pf] and references therein. The second approach focuses on extending the symplectic structure to a 2-plectic one, which yields a Lie 2-algebra of Hamiltonian 1-forms on the manifold. This is by now a fairly standard construction in multisymplectic geometry [@Cantrijn:1999aa; @Baez:2008bu]. Quantizing the 2-plectic manifold amounts here to quantizing the Lie 2-algebra of Hamiltonian 1-forms. While more appealing than the first one, this approach has its own shortcomings, and a more detailed discussion is found in section \[ssec:2-plectic\_quantization\]. Here it is important to note that this point of view is clearly very suitable for our purposes, and we expect that quantized versions of Lie 2-algebras of Hamiltonian 1-forms yield solutions to the classical equations of motion of our Lie 2-algebra models.
From this perspective, our Lie 2-algebra models are a good testing ground for the extension of the notion of a space. In noncommutative geometry, the first step in such an extension is made by replacing the commutative product in the algebra of functions by a noncommutative one. The next step is to generalize this to a nonassociative product, which requires the use of 2-term $L_\infty$- and $A_\infty$-algebras. Ultimately, the notion of a commutative algebra of functions on a manifold should be generalized to that of a certain type of operad or an even more general mathematical structure.
Lie 2-algebras
==============
Lie 2-algebras are categorified versions of Lie algebras. While categorification is not a unique or straightforward recipe, the procedure is roughly the following: most mathematical notions are based on spaces endowed with extra structure satisfying certain basic equations. To categorify such a notion, replace the spaces with categories and endow them with extra structure given by functors that satisfy the basic equations up to an isomorphism. The isomorphisms, in turn, have to satisfy reasonable coherence equations. In the case of Lie algebras, one thus obtains the [*weak Lie 2-algebras*]{} [@Roytenberg:0712.3461]: the linear space underlying the Lie algebra gets replaced by a linear category. We demand that we have a Lie bracket functor on this category, but it is antisymmetric and satisfies the Jacobi identity only up to isomorphisms. These isomorphisms are called the alternator and the Jacobiator, respectively.
Demanding that the alternator is trivial, which implies that the categorified Lie bracket is antisymmetric, one obtains the so-called [*semistrict Lie 2-algebras*]{}. It is these that we will be considering in this paper. They are particularly nice to work with, as they are categorically equivalent to 2-term $L_\infty$-algebras, cf. [@Baez:2003aa].
One can go one step further and demand that the Jacobi identity is satisfied, too. In this case, one ends up with [*strict Lie 2-algebras*]{}, which can be identified with differential crossed modules [@Baez:2002jn]. Although most of the structural generalizations of categories have been lost at this point, strict Lie 2-algebras are still interesting. For example, they underlie the definition of non-abelian gerbes, see e.g. [@Baez:2010ya]. Moreover, when endowed with a metric, they contain all the 3-algebras that have appeared recently in M2-brane models [@Palmer:2012ya].
Semistrict Lie 2-algebras
-------------------------
As stated above, semistrict Lie 2-algebras are categorically equivalent to 2-term $L_\infty$-algebras, and we can relatively easily specify their structure in terms of vector spaces. The general definition of an $L_\infty$-algebra is recalled for the reader’s convenience in appendix \[app:A\].
A [*2-term $L_\infty$-algebra*]{} is given by a two-term complex of real[^3] vector spaces, $$V~\xrightarrow{~\mu_1~}~W~\xrightarrow{~\mu_1~}~0~,$$ where gradings $-1$ and $0$ are assigned to elements of $V$ and $W$, respectively. This complex is equipped with unary, binary and ternary totally graded antisymmetric and multilinear “products” $\mu_1$, $\mu_2$ and $\mu_3$ satisfying the following higher homotopy relations:
\[eq:homotopy\_relations\] $$\begin{aligned}
\mu_1(w)&=0~,~~~\mu_2(v_1,v_2)=0~,\\
\mu_1(\mu_2(w,v))&=\mu_2(w,\mu_1(v))~,~~~\mu_2(\mu_1(v_1),v_2)=\mu_2(v_1,\mu_1(v_2))~,\\
\mu_3(v_1,v_2,v_3)&=\mu_3(v_1,v_2,w)=\mu_3(v_1,w_1,w_2)=0~,\\
\mu_1(\mu_3(w_1,w_2,w_3))&=-\mu_2(\mu_2(w_1,w_2),w_3)-\mu_2(\mu_2(w_3,w_1),w_2)-\mu_2(\mu_2(w_2,w_3),w_1)~,\\
\mu_3(\mu_1(v),w_1,w_2)&=-\mu_2(\mu_2(w_1,w_2),v)-\mu_2(\mu_2(v,w_1),w_2)-\mu_2(\mu_2(w_2,v),w_1)
\end{aligned}$$ and $$\begin{aligned}
\mu_2(\mu_3(w_1,&w_2,w_3),w_4)-\mu_2(\mu_3(w_4,w_1,w_2),w_3)+\mu_2(\mu_3(w_3,w_4,w_1),w_2)\\
& -\mu_2(\mu_3(w_2,w_3,w_4),w_1)=\\
&\mu_3(\mu_2(w_1,w_2),w_3,w_4)-\mu_3(\mu_2(w_2,w_3),w_4,w_1)+\mu_3(\mu_2(w_3,w_4),w_1,w_2)\\
&-\mu_3(\mu_2(w_4,w_1),w_2,w_3)
-\mu_3(\mu_2(w_1,w_3),w_2,w_4)-\mu_3(\mu_2(w_2,w_4),w_1,w_3)~,
\end{aligned}$$
where $v,v_i\in V$ and $w,w_i\in W$.
Besides the above product, we also introduce the product $\kappa_2:V\times V\rightarrow V$ with $$\kappa_2(v_1,v_2):=\mu_2(\mu_1(v_1),v_2)=-\mu_2(\mu_1(v_2),v_1)=-\kappa_2(v_2,v_1)~.$$
A simple example of a semistrict Lie 2-algebra is the following one [@Baez:2003aa], which we will denote by $({\mathfrak{g}},V,\rho,c)$: as two-term complex, we take $V\rightarrow {\mathfrak{g}}$, where ${\mathfrak{g}}$ is a finite-dimensional real Lie algebra and $V$ is a vector space carrying a representation $\rho$ of ${\mathfrak{g}}$. The non-vanishing products are given by $$\mu_2(g_1,g_2):=[g_1,g_2]~,~~~\mu_2(g,v)=-\mu_2(v,g):=\rho(g)v~,~~~\mu_3(g_1,g_2,g_3)=c(g_1,g_2,g_3)~,$$ where $g\in {\mathfrak{g}}$, $v\in V$ and $c\in H^3({\mathfrak{g}},V)$. Since $\mu_1$ is trivial, isomorphic objects in the category corresponding to this Lie 2-algebra are identical. Such Lie 2-algebras are called [*skeletal*]{}.
Any semistrict Lie 2-algebra is in fact categorically equivalent to a skeletal one, and all skeletal semistrict Lie 2-algebras are equivalent to one of the form $({\mathfrak{g}},V,\rho,c)$ [@Baez:2003aa Thm. 55]. This fact can be used to classify Lie 2-algebras.
If $V={\mathbbm{R}}$ then an interesting example of a Lie-algebra cocycle is given by $c(g_1,g_2,g_3)=k\langle g_1,[g_2,g_3]\rangle$, where $\langle{-},{-}\rangle$ is the Killing form and $k\in {\mathbbm{R}}$. The resulting semistrict Lie 2-algebra is also called the string Lie 2-algebra of ${\mathfrak{g}}$.
Other examples are given by the Lie 2-algebra of Hamiltonian 1-forms on 2-plectic manifolds, and we describe these in detail in section \[ssec:2Plectic\_Manifolds\].
If the Jacobiator $\mu_3$ in a semistrict Lie 2-algebra vanishes, we arrive at a [*strict Lie 2-algebra*]{} or, equivalently, a [*differential crossed module*]{}: both $\mu_2$ and $\kappa_2$ now satisfy the Jacobi identity, and we have a two-term complex of Lie algebras $V\xrightarrow{\mu_1}W$ with an action ${\vartriangleright}W\times V\rightarrow V: w{\vartriangleright}v:=\mu_2(w,v)$ satisfying $$\mu_1(w{\vartriangleright}v)=[w,\mu_1(v)]{{\qquad\mbox{and}\qquad}}\mu_1(v_1){\vartriangleright}v_2=[v_1,v_2]~,$$ for all $v\in V$ and $w\in W$, where the commutators are identified with $\mu_2$ on $W$ and $\kappa_2$ on $V$.
The simplest examples of strict Lie 2-algebras are the gauge algebras of ${\mathfrak{u}}(1)$-bundles and ${\mathfrak{u}}(1)$-gerbes: the Lie algebra ${\mathfrak{u}}(1)$ can be regarded as a Lie 2-algebra[^4] $(*\xrightarrow{\mu_1} {\mathfrak{u}}(1),{\vartriangleright})$, where $\mu_1(*)=0\in {\mathfrak{u}}(1)$ and ${\vartriangleright}$ is trivial. The gauge algebra of a ${\mathfrak{u}}(1)$-gerbe is the Lie 2-algebra ${\mathsf{b}}{\mathfrak{u}}(1)=({\mathfrak{u}}(1)\xrightarrow{\mu_1} *,{\vartriangleright})$, where $\mu_1$ and ${\vartriangleright}$ are trivial. A non-abelian example is the derivation Lie 2-algebra $\mathsf{Der}({\mathfrak{g}})$ of a Lie algebra ${\mathfrak{g}}$, $({\mathfrak{g}}\xrightarrow{\rm ad}\mathsf{der}({\mathfrak{g}}),{\vartriangleright})$, where $\mathsf{der}({\mathfrak{g}})$ are the derivations of the Lie algebra ${\mathfrak{g}}$, ${\mathrm{ad}}$ is the embedding of ${\mathfrak{g}}$ as inner derivations via the adjoint map, and ${\vartriangleright}$ is the natural action of derivations of ${\mathfrak{g}}$ onto ${\mathfrak{g}}$.
Lie 2-algebra homomorphisms {#ssec:Lie_2_algebra_homomorphism}
---------------------------
To analyze symmetries in our models, we will require the notion of a homomorphism between Lie 2-algebras. Such a homomorphism should preserve both the vector space structure as well as the higher products. However, as we are working in a categorified setting, we will require the higher products to be preserved only up to an isomorphism. The appropriate definition for Lie 2-algebras has been developed in [@Baez:2003aa Def. 23]. Translated to the equivalent 2-term $L_\infty$-algebras, we have the following definition [@Baez:2003aa Def. 34].
An $L_\infty$-homomorphisms $\Psi:\,L\rightarrow L'$ between two 2-term $L_\infty$-algebras $L=V\rightarrow W$ and $L'=V'\rightarrow W'$ is defined as a set of maps $$\label{eq:55}
\Psi_{-1}:\, V\rightarrow V'~,~~~\Psi_0:\, W\rightarrow W'~,~~~\Psi_2:\, W\times W\rightarrow V'~,$$ where $\Psi_{-1}$ and $\Psi_0$ form a linear chain map and $\Psi_2$ is a skew-symmetric bilinear map preserving the higher product structure. That is, for $w,\,w_i\in W$ and $v,\,v_i\in V$, the following hold: $$\label{eq:Lie_2_algebra_homomorphism_conditions}
\begin{array}{rcl}
\Psi_0\left(\mu_2(w_1,w_2)\right)&=&\mu_2\left(\Psi_0(w_1),\Psi_0(w_2)\right)+\mu_{1}(\Psi_2(w_1,w_2))~,\\
\Psi_{-1}\left(\mu_2(w,v)\right)&=&\mu_2(\Psi_0(w),\Psi_{-1}(v))+\Psi_2(w,\mu_{1}(v))~,\\
\mu_3(\Psi_0(w_1),\Psi_0(w_2),\Psi_0(w_3))&=&\Psi_{-1}\left(\mu_3(w_1,w_2,w_3)\right)-\left[
\Psi_2(w_1,\mu_2(w_2,w_3))\right. \\
\ & \ &\left.+\mu_2\left(\Psi_0(w_1),\Psi_2(w_2,w_3)\right)+\text{cyclic}\,(w_1,w_2,w_3)\right].
\end{array}$$ Two homomorphisms $\Psi:\,L\rightarrow L'$ and $\Phi:\,L'\rightarrow L''$ can be combined via the composition rules
\[composition\] $$\begin{aligned}
\label{eq:56}
(\Psi\circ\Phi)_0(w)&=\Psi_0\Phi_0(w)~,\\
(\Psi\circ\Phi)_{-1}(v)&=\Psi_{-1}\Phi_{-1}(v)~,\\
(\Psi\circ\Phi)_2(w_1,w_2)&=\Psi_{-1}\Phi_2(w_1,w_2)+\Psi_2\left(\Phi_0(w_1),\Phi_0(w_2)\right)~.\end{aligned}$$
The identity automorphism $\text{Id}_L:\,L\rightarrow L$ is given by the maps $$\label{eq:58}
(\text{id}_L)_0(w)=w~,~~~(\text{id}_L)_{-1}=v~,~~~(\text{id}_L)_2(w_1,w_2)=0~.$$ The inverse to an automorphism $\Phi:L\rightarrow L$ under the composition $\circ$ given in is indicated by $\Phi^{-1\circ}: L\rightarrow L$. It satisfies $\Phi^{-1\circ}\circ\Phi=\Phi\circ\Phi^{-1\circ}=\text{id}_L$, and is made of three maps given by
$$\begin{aligned}
\label{eq:59}
(\Phi^{-1\circ})_0(w)&=(\Phi_0)^{-1}(w)~,\\
(\Phi^{-1\circ})_{-1}(v)&=(\Phi_{-1})^{-1}(v)~,\\
(\Phi^{-1\circ})_2(w_1,w_2)&=-(\Phi_{-1})^{-1}\left(\Phi_2\left((\Phi_0)^{-1}(w_1),(\Phi_0)^{-1}(w_2)\right)\right)~.\end{aligned}$$
For more details on morphisms between semistrict Lie 2-algebras see for instance [@Baez:2003aa; @Zucchini:2011aa].
Inner products on semistrict Lie 2-algebras {#ssec:inner_products}
-------------------------------------------
Let us now discuss the notion of an inner product on semistrict Lie 2-algebras, which we will need to write down action functionals. Naturally, an inner product on a semistrict Lie 2-algebra should originate from an inner product on its underlying Baez-Crans 2-vector space. Moreover, it should be compatible with certain actions of Lie 2-algebra homomorphisms. And finally, as we want to be able to reproduce dimensionally reduced M2-brane models, we allow for indefinite scalar products, cf. appendix \[app:A\].
Unfortunately, there are at least three different notions of inner product that satisfy these properties. First, there is a scalar product on $L_\infty$-algebras[^5] that was used in [@Zwiebach:1992ie] and [@Kontsevich:1992aa], see also [@Igusa:2003yg] and [@0821843621]. Given an $L_\infty$-algebra $L=\oplus_i L_i$, a [*(cyclic) scalar product*]{} $\langle {-},{-}\rangle_\infty$ on $L$ is a non-degenerate, even, bilinear form that is compatible with all the homotopy products $\mu_n$, $n\in{\mathbbm{N}}^*$. Explicitly, we have $$\begin{aligned}
\langle x_1,x_2\rangle_\infty&=(-1)^{{\tilde{x}}_1+{\tilde{x}}_2}\langle x_2,x_1\rangle_\infty~.\\
\langle \mu_n(x_1,\ldots,x_n),x_0\rangle_\infty&=(-1)^{n+{\tilde{x}}_0({\tilde{x}}_1+\cdots+{\tilde{x}}_n)}\langle \mu_n(x_0,\ldots,x_{n-1}),x_n\rangle_\infty~,
\end{aligned}$$ $x_i\in L$. Adapted to 2-term $L_\infty$-algebras, it follows that a [*cyclic scalar product*]{} on a semistrict Lie 2-algebra $V\longrightarrow W$ is a scalar product $\langle {-},{-}\rangle_\infty$ on $V\oplus W$, which satisfies the following conditions.
It is even symmetric, that is: $$\langle v_1,v_2\rangle_\infty=\langle v_2,v_1\rangle_\infty~,~~~\langle w_1,w_2\rangle_\infty=\langle w_2,w_1\rangle_\infty~,~~~\langle v,w\rangle_\infty =\langle w,v\rangle_\infty=0~.$$
It is cyclically graded symmetric with respect to $\mu_2$ and cyclically graded antisymmetric with respect to $\mu_1$ and $\mu_3$, which implies $$\label{eq:too_strong}
\langle \mu_1(v),w\rangle_\infty=\langle \mu_2(v_1,w),v_2\rangle_\infty=\langle \mu_3(w_1,w_2,w_3),v\rangle_\infty=0~.$$
We thus see that this kind of inner product is very restrictive.
Another metric $\langle {-},{-}\rangle_{\rm red}$ was introduced on [*reduced*]{} semistrict Lie 2-algebras, where $\mu_1$ is injective [@Zucchini:2011aa]. In this case, $V$ can be regarded as a subspace of $W$ and the domain and range of all products collapse to $W$. One can then impose the following invariance conditions $$\begin{aligned}
\langle\mu_2(w_1,w_2),w_3\rangle_{\rm red}+\langle w_2,\mu_2(w_1,w_3)\rangle_{\rm red}=0~,\\
\langle\mu_1(\mu_3(w_1,w_2,w_3)),w_4\rangle_{\rm red}+\langle w_3,\mu_1(\mu_3(w_1,w_2,w_4))\rangle_{\rm red}&=0~.
\end{aligned}$$ While the latter equation is very reminiscent of the fundamental identity for 3-Lie algebras, cf. appendix \[app:A\], focusing on reduced semistrict Lie 2-algebras is a severe restriction. In particular, it excludes the semistrict (and strict) Lie 2-algebra ${\mathsf{b}}{\mathfrak{u}}(1)={\mathfrak{u}}(1)\rightarrow *$, which is the gauge 2-algebra of an abelian gerbe. Moreover, it will collide with the semistrict Lie 2-algebra structures obtained on 2-plectic manifolds in section \[ssec:2Plectic\_Manifolds\].
The final metric we want to consider arises from extending the definition on strict Lie 2-algebras to the semistrict case, cf. e.g. [@Baez:2002jn; @Martins:2010ry; @Palmer:2012ya]. On a semistrict Lie 2-algebra $V\oplus W$, an inner product is an even and graded symmetric bilinear map $\langle{-},{-}\rangle_0$ such that $$\label{eq:15}
\begin{aligned}
\langle v_1,v_2\rangle_0=\langle v_2,v_1\rangle_0~,~~~\langle w_1,w_2\rangle_0=\langle w_2,w_1\rangle_0~,~~~&\langle v,w\rangle_0=\langle w,v\rangle_0=0~,\\
\langle \mu_2(w_1,x_1),x_2\rangle_0+\langle x_1,\mu_2(w_1,x_2)\rangle_0&=0
\end{aligned}$$ for all $v_i\in V$, $w_i\in W$ and $x_i\in V\oplus W$. We will call this inner product the [*minimally invariant*]{} inner product. Note that demanding $\langle \mu_2(x_3,x_1),x_2\rangle_0+\langle
x_1,\mu_2(x_3,x_2)\rangle_0=0$ in general is too restrictive, as this would imply that $\mu_2(v_1,w_1)=0$ due to $\langle
\mu_2(v_1,w_1),v_2\rangle_0+\langle w_1,\mu_2(v_1,v_2)\rangle_0=0$. Note furthermore that the above relations automatically imply that $$\langle \kappa_2(x_1,x_2),x_3\rangle_0+\langle x_2,\kappa_2(x_1,x_3)\rangle_0=0~.$$
Besides matching the natural definition of an inner product on differential crossed modules, this definition includes also natural inner products on the semistrict Lie 2-algebras $({\mathfrak{g}},V,\rho,c)$ if ${\mathfrak{g}}$ is the Lie algebra of metric preserving transformations on $V$. And finally, it will turn out to match the natural metrics on semistrict Lie 2-algebras arising from 2-plectic manifolds.
Transposed products
-------------------
To facilitate computations with metric semistrict Lie 2-algebras, it is useful to introduce “transposed products” $\mu_n^*$ for each $\mu_n$. These products are defined by regarding the products as operators acting on the element in the last slot and taking the dual: $$\begin{aligned}
\label{eq:1}
\langle \mu_1(y_1),y_2\rangle&=:\langle y_1,\mu_1^*(y_2)\rangle~,\\
\langle \mu_2(x_1,y_1),y_2\rangle&=:\langle y_1,\mu_2^*(x_1,y_2)\rangle~,\\
\langle \kappa_2(x_1,y_1),y_2\rangle&=:\langle y_1,\kappa_2^*(x_1,y_2)\rangle~,\\
\langle \mu_3(x_1,x_2,y_1),y_2\rangle&=:\langle y_1,\mu_3^*(x_1,x_2,y_2)\rangle
\end{aligned}$$ for all $x_i,y_i\in V\oplus W$. The product $\mu_1^*$ had already been introduced in [@Baez:2002jn] and used extensively in [@Palmer:2012ya]. Note that $\mu_3^*$ is not antisymmetric.
Let us examine the transposed products for each of the inner products in more detail. First, in the case of $\infty$-metrics $\langle {-},{-}\rangle_\infty$, the only non-vanishing transposed product is $$\mu_2^*(w_1,w_2)=-\mu_2(w_1,w_2)~.$$
In the case of metric $\langle {-},{-}\rangle_{\rm red}$, we have $$\mu_2(w_1,w_2)=-\mu_2^*(w_1,w_2){{\qquad\mbox{and}\qquad}}\kappa^*_2(w_1,w_2)=-\kappa_2(w_1,w_2)$$ for all $w_1,w_2\in W$. These are the only two transposed products that are needed, since here one really only has to deal with the metric on $W$.
For the metric $\langle {-},{-}\rangle_0$, we have more generally $$\mu_2(w,x)=-\mu_2^*(w,x)$$ for all $w\in W$ and $x\in V\oplus W$, which, together with $\kappa^*_2(x,y)=\mu_2^*(\mu_1(x),y)$, implies that $$\kappa^*_2(v_1,v_2)=-\kappa_2(v_1,v_2)~.$$
The transposed products that cannot be reduced to the products in the Lie 2-algebra are $$\mu_1^*:W\rightarrow V~,~~~\mu_2^*:V\times V\rightarrow W{{\qquad\mbox{and}\qquad}}\mu_3^*:W\times W\times V\rightarrow W~,$$ which have degrees $-1$, $2$ and $1$, respectively. They are defined implicitly via $$\begin{aligned}
\langle \mu_1(v_1),w_1 \rangle_0 =: \langle v_1,\mu_2^*(w_1)\rangle_0~,&~~~
\langle \mu_2(v_1,w),v_2 \rangle_0 =: \langle w,\mu_2^*(v_1,v_2)\rangle_0~,\\
\langle \mu_3(w_1,w_2,w_3),v \rangle_0 &=: \langle w_3,\mu_3^*(w_1,w_2,v)\rangle_0~.
\end{aligned}$$ To simplify notation, we will only denote these three with a star from here on.
Combining our definitions with the homotopy algebra relations, we obtain the following set of equalities: $$\label{eq:starred relations 1}
\begin{aligned}
\mu_2(\mu_1(v_1),v_2)&=\mu_2(v_1,\mu_1(v_2))=\mu_1^*(\mu_2^*(v_2,v_1))~,\\
\mu_2(\mu_1(v),w)&=\mu_1(\mu_2(v,w))=\mu_2^*(\mu_1^*(w),v)~,\\
\mu_1^*(\mu_2(w_1,w_2))&=\mu_2(\mu_1^*(w_1),w_2)=\mu_2(\mu_1^*(w_2),w_1)~,\\
\mu_1^*(\mu_3^*(w_1,w_2,v))&=-\mu_3(\mu_1(v),w_1,w_2)~,\\
\mu_1(\mu_3(w_1,w_2,w_3))&=-\mu_3^*(w_1,w_2,\mu_1^*(w_3))~,\\
\mu_3^*(\mu_1(v_1),w,v_2)&=-\mu_3^*(\mu_1(v_2),w,v_1)~, \\
\mu_3^*(\mu_1(v_1),w,v_2)&=\mu_2^*(v_1,\mu_2(w,v_2))-\mu_2^*(v_2,\mu_2(w,v_1))-\mu_2(w,\mu_2^*(v_1,v_2)~,\\
\end{aligned}$$ as well as $$\label{eq:19}\begin{array}{rl}
\mu_2(w_1,\mu_3^*(w_2,w_3,v))&\!\!\!+\mu_2(w_3,\mu_3^*(w_1,w_2,v))+\mu_2(w_2,\mu_3^*(w_3,w_1,v))=\\
&\hspace{-15ex}\mu_3^*(\mu_2(w_1,w_2),w_3,v)+\mu_3^*(\mu_2(w_3,w_1),w_2,v)+\mu_3^*(\mu_2(w_2,w_3),w_1,v)\\
&\hspace{-17ex}+\mu_3^*(w_1,w_2,\mu_2(w_3,v))+\mu_3^*(w_3,w_1,\mu_2(w_2,v))+\mu_3^*(w_2,w_3,\mu_2(w_1,v))\\
&\hspace{-17ex}-\mu_2^*(\mu_3(w_1,w_2,w_3),v)~.
\end{array}$$
M2-brane model 3-algebras {#ssec:M2-brane_3_algebras}
-------------------------
The currently most successful M2-brane models [@Bagger:2007jr; @Gustavsson:2007vu; @Aharony:2008ug] are given by Chern-Simons matter theories, in which the gauge structure is described by a 3-algebra[^6]. Note that we will use the term [*3-algebra*]{} to collectively describe both the real 3-algebras of [@Cherkis:2008qr] and the hermitian 3-algebras of [@Bagger:2008se] in this paper. These 3-algebras have nothing to do with Lie 3-algebras or other categorifications of the notion of a Lie algebra. Instead, these 3-algebras are readily shown to be equivalent to certain classes of metric differential crossed modules [@Palmer:2012ya]. As we want to identify 3-algebra models in our Lie 2-algebra models later, let us briefly recall this construction.
We start from a strict Lie 2-algebra $L$ endowed with an inner product $\langle{-},{-}\rangle_0$ for which $W={\mathfrak{g}}$ is a real Lie algebra and $V$ is a vector space carrying a faithful orthogonal representation of ${\mathfrak{g}}$. The only non-trivial products are $\mu_2:W\times W\rightarrow W$ and $\mu_2:W\times V\rightarrow W$, which are given by the Lie bracket and the representation of $W$ as endomorphism on $V$, respectively.
As shown in [@deMedeiros:2008zh], isomorphism classes of such data are in one-to-one correspondence to isomorphism classes of real 3-algebras. In particular, we can define implicitly an operator $D:V\times V\rightarrow W$ via $$\langle w, D(v_1,v_2)\rangle_0:=\langle \mu_2(w,v_1),v_2 \rangle_0~.$$ With our above definitions, it follows that $D(v_1,v_2)=-\mu_2^*(v_1,v_2)$. Note that $\mu_2^*(v_1,v_2)$ is antisymmetric. We can then introduce a triple bracket $[{-},{-},{-}]:V^{\wedge 2}\times V\rightarrow V$ by $$\label{eq:triple_bracket}
[v_1,v_2,v_3]:=D(v_1,v_2){\vartriangleright}v_3=-\mu_2(\mu_2^*(v_1,v_2),v_3)~.$$ This bracket satisfies by definition the fundamental identity, cf. , and we therefore arrive at a real 3-algebra. Note that a similar construction exists for hermitian 3-algebras.
As the triple bracket can be defined for any Lie 2-algebra with inner product $\langle{-},{-}\rangle_0$, one can now ask under which condition the fundamental identity is satisfied and the triple bracket yields a real 3-algebra. A short computation reveals that this is only the case for arbitrary strict or skeletal metric Lie 2-algebras.
While there is no connection between the ternary bracket of a 3-Lie algebra and the Jacobiator of a Lie 2-algebra in general, we can construct (at least) one example where they can be essentially identified. Consider the vector space of $n\times n$ matrices ${\mathsf{Mat}}(n)$. Together with the 3-bracket $$[a,b,c]={\,\mathrm{tr}\,}(a)[b,c]+{\,\mathrm{tr}\,}(b)[c,a]+{\,\mathrm{tr}\,}(c)[a,b]~,$$ ${\mathsf{Mat}}(n)$ forms a 3-Lie algebra as shown in [@Awata:1999dz]. There, this 3-Lie algebra was suggested to appear in the quantization of Nambu-Poisson brackets. Interestingly, we can also identify this bracket with the Jacobiator of a reduced Lie 2-algebra $V\rightarrow W$, where $V=W={\mathsf{Mat}}(n)$ and the following higher products are non-vanishing: $$\begin{aligned}
\mu_1(v)&=v~,\\
\mu_2(w_1,w_2)&={\,\mathrm{tr}\,}(w_1)w_2-{\,\mathrm{tr}\,}(w_2)w_1+[w_1,w_2]~,\\
\mu_2(v,w)&=-({\,\mathrm{tr}\,}(v)w-{\,\mathrm{tr}\,}(w)v+[v,w])~,\\
\mu_3(w_1,w_2,w_3)&={\,\mathrm{tr}\,}(w_1)[w_2,w_3]+{\,\mathrm{tr}\,}(w_2)[w_3,w_1]+{\,\mathrm{tr}\,}(w_3)[w_1,w_2]
\end{aligned}$$ for all $v\in V$ and $w\in W$. The higher homotopy relations are readily verified. We will denote this Lie 2-algebra by $2{\mathsf{Mat}}(n)$.
Quantized symplectic and 2-plectic manifolds
============================================
Before coming to physical models, we will briefly review the quantization[^7] of symplectic spaces and discuss generalizations of this to 2-plectic manifolds. The quantized spaces we introduce here will arise as solutions in our Lie 2-algebra models later on.
Quantization of symplectic manifolds
------------------------------------
We start from a symplectic manifold $(M,\omega)$, which is regarded as the phase space of a classical mechanical system. The observables of this system are given by the functions on $M$, which form a commutative algebra under pointwise multiplication. In addition, the symplectic form induces a Lie algebra structure on the vector space of smooth functions on $M$, which turns $M$ into a Poisson manifold. Explicitly, we have for each function $f\in {\mathcal{C}}^\infty(M)$ a corresponding Hamiltonian vector field $X_f$ defined according to $\iota_{X_f}\omega={\mathrm{d}}f$. The Poisson bracket on ${\mathcal{C}}^\infty(M)$ induced by $\omega$ is then given by $$\{f,g\}:=\iota_{X_f}\iota_{X_g}\omega~,$$ and we denote the resulting Poisson algebra by $\Pi_{M,\omega}$. As examples, consider ${\mathbbm{R}}^2$ and $S^2$. On these spaces the symplectic form is the volume form ${\rm vol}$ and the induced Poisson bracket in some coordinates $x^a$, $a=1,2$, reads as $$\{f_1,f_2\}=\frac{{{\varepsilon}}^{ab}}{|{\rm vol}|} {\frac{{\partial}f_1}{{\partial}x^a}}{\frac{{\partial}f_2}{{\partial}x^b}}~.$$
The quantization of a symplectic manifold is given by a Hilbert space ${\mathcal{H}}$ together with a linear map $\hat{-}:{\mathcal{C}}^\infty(M)\rightarrow {\mathsf{End}\,}({\mathcal{H}})$ such that the Poisson algebra $\Pi_{M,\omega}$ is mapped to the Lie algebra ${\mathsf{End}\,}({\mathcal{H}})$ at least to lowest order in some deformation parameter $\hbar$: $$\label{eq:correspondence_principle}
[\hat{f},\hat{g}]=\hat{f}\hat{g}-\hat{g}\hat{f}=\widehat{-{\mathrm{i}}\hbar~\{f,g\}}+{\mathcal{O}}(\hbar^2)$$ for all $f,g\in{\mathcal{C}}^\infty(M)$. Equation is known as the [*correspondence principle*]{}.
2-plectic manifolds {#ssec:2Plectic_Manifolds}
-------------------
Consider a smooth manifold $M$ endowed with a 3-form $\varpi$ that is closed and non-degenerate in the sense that $\iota_X\varpi=0$ implies $X=0$. We call such a 3-form a [*2-plectic form*]{} and say that $M$ is a [*2-plectic manifold*]{}. This can be regarded as a categorification of the notion of a symplectic structure. In particular, three-dimensional manifolds with volume forms $\varpi$ are 2-plectic manifolds.
While a symplectic structure on a manifold $M$ always gives rise to a Poisson structure on $M$ by taking its inverse, a 2-plectic form $\varpi$ gives rise to a [*Nambu-Poisson structure*]{}[^8] only under certain conditions [@springerlink:10.1007/BF00400143]. Therefore, a different analogy should be considered here.
Having discussed categorifications of Lie algebras before, it is natural to expect that there is a categorification of the Poisson algebra in terms of a semistrict Lie 2-algebra [@Baez:2008bu]. Define the set of [*Hamiltonian 1-forms*]{} ${\mathfrak{H}}(M)$ as those forms $\alpha$ for which there is a vector field $X_\alpha$ such that $\iota_{X_\alpha}\varpi=-{\mathrm{d}}\alpha$. Note that for a three-dimensional manifold $M$, ${\mathfrak{H}}(M)=\Omega^1(M)$. We then define the semistrict Lie 2-algebra $\Pi_{M,\varpi}$ as the vector space $V\oplus W:={\mathcal{C}}^\infty(M)\oplus {\mathfrak{H}}(M)$ with non-vanishing products $$\pi_1(f)={\mathrm{d}}f~,~~~\pi_2(\alpha,\beta)=-\iota_{X_{\alpha}}\iota_{X_{\beta}}\varpi~,~~~\pi_3(\alpha,\beta,\gamma)=-\iota_{X_{\alpha}}\iota_{X_{\beta}}\iota_{X_{\gamma}}\varpi~,$$ where $f\in{\mathcal{C}}^\infty(M)$ and $\alpha,\beta,\gamma\in\Omega^1(M)$. Note that the bracket $\pi_2$ is Hamiltonian. That is, $$X_{\pi_2(\alpha,\beta)}=[X_\alpha,X_\beta]~,$$ where the bracket on the right-hand side is the commutator of vector fields. Another useful identity for computations with Hamiltonian vector fields is $$\iota_{[X_\alpha,X_\beta]}={\mathcal{L}}_{X_\alpha}\iota_{X_\beta}-\iota_{X_\beta}{\mathcal{L}}_{X_\alpha}~.$$
A long-standing open question in this context is how to define the analogue of the commutative algebra of observables that on symplectic manifolds was given by the pointwise product of functions on phase space. Ordinary Poisson algebras containing both Lie and associative structure are encoded in a Poisson Lie algebroid. The higher analogue of this structure has been shown to be a so-called Courant Lie 2-algebroid, see [@Rogers:2010ac; @Fiorenza:1304.6292] for more details on this point. To our knowledge, however, an explicit product on ${\mathfrak{H}}(M)$ has not been constructed so far. A solution to this problem might be to switch from the semistrict Lie 2-algebra $\Pi_{M,\varpi}$ to the categorically equivalent, skeletal Lie 2-algebra. Here, the 1-forms form an ordinary Lie algebra, and, if we were able to identify this Lie algebra with a matrix algebra, we could use the ordinary matrix product as a product between observables. Another solution might originate from a comparison with the loop space quantization, cf. [@Saemann:2012ab]. For our purposes, this product is not relevant, and we merely assume that it makes sense to identify observables on 2-plectic manifolds with the vector spaces underlying $\Pi_{M,\varpi}$.
From now on, let us restrict our considerations to three-dimensional Riemannian manifolds $M$ for which $\varpi$ is the volume form. We can endow the Lie 2-algebra $\Pi_{M,\varpi}$ with a metric, following the rules and definitions used in section \[ssec:inner\_products\]. For the two vector subspaces ${\mathcal{C}}^\infty(M)$ and ${\mathfrak{H}}(M)$, we use the usual integrals with respect to the volume form $\varpi$: $$\label{eq:8}
\langle f, g\rangle_0 := \int_{M}\varpi~f\cdot g {{\qquad\mbox{and}\qquad}}\langle \alpha, \beta\rangle_0 := \int_{M}\alpha\wedge \star \beta~,$$ which can be easily checked to be invariant under the action of $\pi_2(\alpha,{-})$. Note that in the non-compact case, finiteness of these integrals becomes an issue. In particular, one should either restrict to classes of functions and 1-forms with finite norm or consider closed subsets of $M$ as integration domain. If possible, one might also consider a 2-plectic form $\varpi$ with appropriate fall-off behavior towards infinity. To avoid boundary contributions, we will always imply a restriction of $\Pi_{M,\varpi}$ to elements with finite norm.
Via the metric, we can now introduce the transposed product $\pi_1^*$ and $\pi_3^*$: $$\label{eq:11}
\langle\pi_1(f),\alpha\rangle_0:=\langle f,\pi_1^*(\alpha)\rangle_0{{\qquad\mbox{and}\qquad}}\langle \pi_3(\alpha, \beta, \gamma), f\rangle_0:= \langle \gamma,\pi_3^*(\alpha,\beta, f)\rangle_0~,$$ which are therefore given by $$\label{eq:12}
\pi_1^*(\alpha)=-\star{\mathrm{d}}\star\alpha{{\qquad\mbox{and}\qquad}}\pi_3^*(\alpha,\beta,f)=\star \,{\mathrm{d}}\, \iota_{X_\beta}\iota_{X_\alpha} \star f~.$$ Note that, by the non-degeneracy of $\varpi$, all combinations of products $$\label{eq:34}
\pi_2(\pi_1(f),\alpha)~,~~~\pi_3(\pi_1(f),\alpha,\beta){{\qquad\mbox{and}\qquad}}\pi_3^*(\pi_1(f),\alpha,g)$$ are identically zero, as well as 2 and 3-products containing more than one $\pi_1(f)$, as easily derived from .
Examples {#ssec:2-plectic_examples}
--------
Let us now review the manifolds ${\mathbbm{R}}^3$ and $S^3$ and their Lie 2-algebras $\Pi_{M,\varpi}$, which will appear in the analysis of the solutions of our model later on.
#### Euclidean space $\mathbb{R}^3$.
We endow three-dimensional Euclidean space $\mathbb{R}^3$ with its canonical volume form $\varpi=\tfrac{1}{3!}{{\varepsilon}}_{ijk}{\mathrm{d}}x^i\wedge{\mathrm{d}}x^j\wedge{\mathrm{d}}x^k$ written in standard Cartesian coordinates $x^i$. All 1-forms are Hamiltonian, and we compute their Hamiltonian vector fields to be $$X_\alpha=X^i_\alpha {\partial}_i=-({{\varepsilon}}^{ijk}{\partial}_j\alpha_k){\partial}_i{{\qquad\mbox{for}\qquad}}\alpha=\alpha_i{\mathrm{d}}x^i~,$$ which leads to the following products: $$\begin{aligned}\label{eq:R^3_products}
&\pi_1(f):={\mathrm{d}}f~,~~\pi_1(\alpha)\stackrel{!}{:=} 0~,\\
&\pi_2(\alpha,\beta):={{\varepsilon}}^{ijk}{\partial}_i\alpha_k({\partial}_j\beta_\ell-{\partial}_\ell\beta_j){\mathrm{d}}x^\ell~,\\
&\pi_3(\alpha,\beta,\gamma):={{\varepsilon}}^{ijk}{{\varepsilon}}^{mnp}{\partial}_m\alpha_n{\partial}_j\beta_k({\partial}_i\gamma_p-{\partial}_p\gamma_i)~.
\end{aligned}$$
The subset of Hamiltonian 1-forms that are constant or linear[^9] together with the set of constant and linear functions and the above defined non-trivial products $\pi_1$ and $\pi_3$ form a [*Heisenberg Lie 2-algebra*]{}, the appropriate categorification of the Heisenberg algebra. Note that higher brackets vanish on constant and exact 1-forms. The remaining linear 1-forms are given by $$\xi_i=\tfrac{1}{2}{{\varepsilon}}_{ijk}x^j{\mathrm{d}}x^k~,$$ whose Hamiltonian vector fields are $-{\partial}_i$ and for which we have $$\label{eq:R^3_special_products}
\pi_2(\xi_i,\xi_j)= {{\varepsilon}}_{ijk}{\mathrm{d}}x^k{{\qquad\mbox{and}\qquad}}\pi_3(\xi_i,\xi_j,\xi_k)=-{{\varepsilon}}_{ijk}~.$$
Note that the 1-forms $\xi_i$ have a special meaning once they are transgressed to loop space. Here, the direction given by the ${\mathrm{d}}x^k$ is interpreted as the tangent to the loop, and one arrives at the following functions on loop space: $$\tfrac{1}{2}{{\varepsilon}}_{ijk}\oint {\mathrm{d}}\tau~ x^j(\tau)~ {\frac{{\mathrm{d}}x^k(\tau)}{{\mathrm{d}}\tau}}~,$$ where $\tau\in S^1$ is the loop parameter. For more details about these functions on loop space, see [@Palmer:2011vx; @Saemann:2012ab].
Assuming finiteness of the norm of the involved functions and 1-forms, we have the following formulas for the transpose product $\pi_3^*$: $$\label{eq:R^3_transposed_product}
\begin{aligned}
\pi_3^*(\alpha,\beta,f)&=-\tfrac{1}{4}{{\varepsilon}}^{ij\ell}{{\varepsilon}}^{mnp}{\partial}_m\alpha_n({\partial}_p\beta_\ell-{\partial}_\ell\beta_p){\partial}_jf{\mathrm{d}}x_i~,\\
\pi_3^*(\xi_i,\xi_j,f)&=-\left(\partial_i f{\mathrm{d}}x_j- \partial_j
f{\mathrm{d}}x_i\right)~.
\end{aligned}$$
#### The sphere $S^3$.
The other example we are interested in is the 3-sphere $S^3$. It will turn out convenient to work in [*Hopf coordinates*]{} $0\leq\eta\leq\tfrac{\pi}{2}$ and $0\leq\theta_i\leq 2\pi$, which parametrize the embedding $S^3{{\hookrightarrow}}{\mathbbm{C}}^2$ via $$z_1={\mathrm{e}}^{{\mathrm{i}}\theta_1}\sin\eta{{\qquad\mbox{and}\qquad}}z_2={\mathrm{e}}^{{\mathrm{i}}\theta_2}\cos\eta~.$$ Note that instead of using the standard range given above, we can also use $0\leq\eta\leq\pi$, $0\leq\theta_1\leq 2\pi$ and $0\leq\theta_2\leq \pi$.
For simplicity, we combine them as $(\eta_1,\eta_2,\eta_3)=(\eta,\theta_1,\theta_2)$. The volume form and the metric read as $$\varpi=\sin\eta_1\cos\eta_1{\mathrm{d}}\eta_1\wedge{\mathrm{d}}\eta_2\wedge{\mathrm{d}}\eta_3{{\qquad\mbox{and}\qquad}}{\mathrm{d}}s^2={\mathrm{d}}\eta_1^2+\sin^2\eta_1\,{\mathrm{d}}\eta_2^2+\cos^2\eta_1\,{\mathrm{d}}\eta_3^2~.$$ For 1-forms $\alpha\in\Omega^1(S^3)$, we compute the following Hamiltonian vector fields $$X_\alpha=X^i_\alpha {\partial}_i=-\frac{1}{\sin\eta_1\cos\eta_1}({{\varepsilon}}^{ijk}{\partial}_j\alpha_k){\partial}_i{{\qquad\mbox{for}\qquad}}\alpha=\alpha_i{\mathrm{d}}\eta_i~,$$ where now ${\partial}_i:={\frac{{\partial}}{{\partial}\eta_i}}$. One readily derives the products: $$\begin{aligned}\label{eq:S^3_products}
&\pi_1(f):={\mathrm{d}}f~,~~\pi_1(\alpha)\stackrel{!}{:=} 0~,\\
&\pi_2(\alpha,\beta):=\frac{1}{\sin\eta_1\cos\eta_1}{{\varepsilon}}^{ijk}{\partial}_i\alpha_k({\partial}_j\beta_\ell-{\partial}_\ell\beta_j){\mathrm{d}}\eta^\ell~,\\
&\pi_3(\alpha,\beta,\gamma):=\frac{1}{\sin^2\eta_1\cos^2\eta_1}{{\varepsilon}}^{ijk}{{\varepsilon}}^{mnp}{\partial}_m\alpha_n{\partial}_j\beta_k({\partial}_i\gamma_p-{\partial}_p\gamma_i)~.
\end{aligned}$$
Here, it is not possible to derive 1-forms from the vector fields $X_{{\partial}_i}$, as $\iota_{X_{{\partial}_1}}\varpi$ is not closed, and therefore it cannot equal ${\mathrm{d}}\xi_1$. Instead, we choose the same vector fields as for ${\mathbbm{R}}^3$, corrected by a factor of $\frac{1}{\sin \eta_1 \cos\eta_1}$. This yields the 1-forms $$\xi_i=\tfrac{1}{2}{{\varepsilon}}_{ijk}\eta^j{\mathrm{d}}\eta^k~,$$ together with the following formulas for the products: $$\pi_2(\xi_i,\xi_j)=\frac{{{\varepsilon}}_{ijk}{\mathrm{d}}\eta^k}{\sin \eta_1 \cos\eta_1}{{\qquad\mbox{and}\qquad}}\pi_3(\xi_i,\xi_j,\xi_k)=-\frac{{{\varepsilon}}_{ijk}{\mathrm{d}}\eta^k}{\sin^2 \eta_1 \cos^2\eta_1}~.$$
The formulas for the transposed product $\pi_3^*$ read as $$\label{eq:S^3_transposed_products}
\begin{aligned}
\pi_3^*(\alpha,\beta,f)&=-\frac{{{\varepsilon}}^{ij\ell}{{\varepsilon}}^{mnp}}{4\sin \eta_1 \cos\eta_1}{\partial}_m\alpha_n({\partial}_p\beta_\ell-{\partial}_\ell\beta_p){\partial}_jf{\mathrm{d}}x_i~,\\
\pi_3^*(\xi_i,\xi_j,f)&=-\frac{1}{\sin \eta_1 \cos\eta_1}\left(\partial_i f{\mathrm{d}}\eta_j- \partial_j
f{\mathrm{d}}\eta_i\right)~.
\end{aligned}$$
Reduction of 2-plectic to symplectic manifolds {#ssec:reduction}
----------------------------------------------
The 2-plectic manifolds we will discuss appear very naturally in the context of M-theory. Roughly speaking, the 2-plectic structure on these spaces arises here as the “dual” of a tri-vector field originating from a non-trivial $C$-field in M-theory, cf. e.g. [@Chu:2009iv]. This is the higher analogue of a symplectic structure arising as a dual to the Seiberg-Witten bivector field [@Seiberg:1999vs]. Our 2-plectic manifolds can be seen as M-theory lifts of symplectic manifolds appearing in string theory. In the following, we briefly comment on taking the inverse of this lift.
To reduce from M-theory to type IIA string theory, we have to identify an M-theory direction along which the 2-plectic form is invariant. Instead of restricting to the usual Kaluza-Klein procedure, we should also allow non-trivial fibrations of the 2-plectic manifold over a symplectic manifold. Since we are mostly interested in three-dimensional spaces, we can regard them as contact manifolds, and, upon reducing along the Reeb vector field corresponding to the contact form, we necessarily obtain a symplectic manifold. In this process, we contract the Hamiltonian 1-forms with the Reeb vector to obtain the Poisson algebra of functions on the underlying symplectic manifold. We will discuss this reduction explicitly for ${\mathbbm{R}}^3$ and $S^3$ in the following.
Another possibility of interpreting this reduction is a slight detour via loop spaces, see e.g. [@Saemann:2012ex; @Saemann:2012ab]: while the boundary of a string on a D-brane yields a point, that of an M2-brane on an M5-brane forms a loop. It is therefore naturally to consider loop spaces of the worldvolume of the M5-brane or submanifolds thereof. Switching to loop space allows us to introduce the so-called transgression map, which reduces the form degree by one: each loop comes with a natural tangent vector, which is given by the loop of the tangent vectors to the loop. Contracting an $n$-form on a manifold with this vector yields an $n-1$-form on loop space. Since this transgression map is a chain map[^10], a 2-plectic form $\varpi$ on a manifold $M$ is mapped to a symplectic form on the corresponding loop space.
To reduce the M-theory loop space to an ordinary space of string theory amounts to restricting to loops that are parallel to the Reeb vector field. Integrating over the loop parameter reduces the dependence of functions on loop space to that of the zero mode of the loop. Therefore, functions on loop space are reduced to functions on the symplectic manifold. Further support of this point of view comes from the observation that the Lie 2-algebra $\Pi_{M,\varpi}$ transgresses to a Poisson algebra on the loop space of $M$. The quantization of $\Pi_{M,\varpi}$ should similarly correspond to a natural quantization of the Poisson algebra on loop space, cf. [@Saemann:2012ab].
Let us now think of the above three-dimensional spaces as contact manifolds. We want to reduce them along the Reeb vectors corresponding to a chosen contact 1-form to obtain two-dimensional manifolds. These manifolds will be endowed with a natural symplectic structure, which is given by the total derivative of the contact 1-form, restricted to the kernel of the same 1-form. Explicitly, after identifying a maximally non-integrable 1-form $\gamma$, which amounts to $\gamma\wedge{\mathrm{d}}\gamma$ being nowhere vanishing, we need to find the corresponding Reeb vector field $X_R$ satisfying $$\label{eq:2}
\iota_{X_R}\gamma=1{{\qquad\mbox{and}\qquad}}\iota_{X_R}{\mathrm{d}}\gamma=0~.$$ Since we are working with three-dimensional manifolds, we can normalize the contact form by imposing the additional condition $$\gamma\wedge {\mathrm{d}}\gamma=\varpi~.$$ Now, every 1-form $\gamma$ in $\Pi_{M,\varpi}$ has its corresponding Hamiltonian vector field $X_\gamma$, and we have also ${\mathrm{d}}\gamma=-\iota_{X_\gamma}\varpi$, so that $\iota_{X_\gamma}{\mathrm{d}}\gamma=0$. That is, $X_\gamma$ satisfies the second requirement of a Reeb vector. Moreover, $\iota_{X_\gamma}\gamma=-1$ since $$\label{eq:4}
0\neq {\mathrm{d}}\gamma=-\iota_{X_\gamma}\varpi=-\iota_{X_\gamma}(\gamma\wedge {\mathrm{d}}\gamma)=-(\iota_{X_\gamma}\gamma){\mathrm{d}}\gamma~.$$ We can therefore take $X_R:=-X_\gamma$ as the Reeb vector corresponding to the contact 1-form $\gamma$. In the M-theory context, the Reeb vector field is a vector field along the ‘M-theory direction’.
The reduction of the 2-plectic manifold together with its induced Lie 2-algebra $\Pi_{M,\varpi}$ to a symplectic manifold with its corresponding Poisson algebra is rather straightforward: all forms are contracted by the Reeb vector field. In particular, we obtain a two-dimensional manifold[^11] $M_R:=M/X_R$, where we divide $M$ by the free abelian action of the Reeb vector field. The symplectic form on $M_R$ is given by $\varpi_R:=\iota_{X_R}\varpi={\mathrm{d}}\gamma$. Moreover, the Hamiltonian 1-forms $\alpha$ on $M$ become functions $f_\alpha:=\iota_{X_R}\alpha$ on $M_R$ and the Lie 2-algebra $\Pi_{M,\varpi}$ reduces to a Poisson algebra $\Pi_{M_R,\varpi_R}$. Hamiltonian 1-forms along (the M-theory direction) $X_R$ are of the form $\alpha=f_\alpha \gamma$. For two such relative forms $\alpha$ and $\beta$, we have $$\label{eq:3}
\iota_{X_R}\pi_2(\alpha,\beta)=-\iota_{X_R}\iota_{X_\alpha}\iota_{X_\beta}\varpi=-\iota_{X_\alpha}\iota_{X_\beta}\varpi_R=-\iota_{X_{f_\alpha}}\iota_{X_{f_\beta}}\varpi_R=\{f_{\alpha},f_{\beta}\}~,$$ where the Hamiltonian vector fields of the functions $f_\alpha$ are defined with respect to $\varpi_R$. Writing ${\mathrm{d}}_R$ for the exterior derivative on $M_R$, we have $$\label{eq:5}
{\mathrm{d}}_R (\iota_{X_R}\alpha)={\mathrm{d}}_R f_{\alpha}=\iota_{X_{f_{\alpha}}}\varpi_R~.$$ Altogether, we recover a two-dimensional symplectic manifold, with all its structure given in terms of our initial 2-plectic one.
#### Reduction of $\mathbb{R}^3$.
To reduce the 2-plectic space $\mathbb{R}^3$ to the symplectic manifold ${\mathbbm{R}}^2$, we use the contact form $\gamma={\mathrm{d}}z- y{\mathrm{d}}x$. The corresponding Reeb vector $X_R$, given by ${\mathrm{d}}\gamma=\iota_{X_R}\varpi$, is therefore $X_R=\partial_z$. Restricting to Hamiltonian 1-forms along the M-theory direction $\gamma$, we recover the usual Poisson algebra for ${\mathbbm{R}}^2$. Consider two such forms $\alpha=f_\alpha\gamma$ and $\beta=f_\beta\gamma$. We have $$\label{eq:9}
\begin{aligned}
\iota_{\partial_z}\pi_2(\alpha,\beta)= \left\{
\iota_{X_R}\alpha,\iota_{X_R}\beta\right\}=&\left\{f_\alpha,f_\beta\right\}=-\iota_{X_{f_\alpha}}\iota_{X_{f_\beta}}\varpi_R\\
\
=&{\frac{{\partial}}{{\partial}x}} f_\alpha~{\frac{{\partial}}{{\partial}y}} f_\beta-{\frac{{\partial}}{{\partial}y}} f_\alpha~ {\frac{{\partial}}{{\partial}x}} f_\beta~.
\end{aligned}$$
We can also reduce the 2-plectic manifold ${\mathbbm{R}}^3\backslash\{0\}$ to $S^2$, recovering the symplectic structure there. The contact form here is given in canonical spherical coordinates by $\gamma=r^2{\mathrm{d}}r-\cos\theta {\mathrm{d}}\phi$. This yields the Reeb vector field $X_R=\frac{1}{r^2}{\partial}_r$. The 2-plectic structure $\varpi$ reduces to the usual symplectic structure of the 2-sphere: $\varpi_R=\sin\theta {\mathrm{d}}\theta\wedge {\mathrm{d}}\phi$. The Lie 2-algebra of Hamiltonian 1-forms $\alpha=f_\alpha \gamma$ on ${\mathbbm{R}}^3\backslash\{0\}$ reduces accordingly to the Poisson algebra of functions on the 2-sphere.
#### Reduction of $S^3$.
Here let us choose the contact form $\gamma=\tfrac{1}{2}{\mathrm{d}}\eta_3+\sin^2\eta_1{\mathrm{d}}\eta_2$, so as to obtain on $S^2$ the symplectic structure $\varpi_R={\mathrm{d}}\gamma=2\sin\eta_1\cos\eta_1{\mathrm{d}}\eta_1\wedge{\mathrm{d}}\eta_2=\sin(2\eta_1){\mathrm{d}}\eta_1\wedge{\mathrm{d}}\eta_2$. The Reeb vector here is $X_R=2\partial_{\eta_3}$, and for Hamiltonian 1-forms $\alpha$, $\beta$ along the M-theory direction we have $$\label{eq:10}
\iota_{X_R}\pi_2(\alpha,\beta)=\frac{2}{\cos\eta_1\sin\eta_1}{{\varepsilon}}^{ij3}\partial_{\eta_i}f_\alpha\partial_{\eta_j}f_\beta=\{f_\alpha,f_\beta\}~,$$ which is the usual Poisson structure on $S^2$.
Lie 2-algebras not originating from 2-plectic manifolds {#pp-wave 2-algebra}
-------------------------------------------------------
Just as a Poisson manifold is not necessarily a symplectic manifold, we should not expect that any interesting Lie 2-algebra of 1-forms comes from a 2-plectic structure. To illustrate this point further, let us consider the categorification of Hpp-waves.
Recall that ten-dimensional homogeneous plane waves arise as the Penrose limit of the near horizon geometry $\mathrm{AdS}_5\times S^5$ in type IIB supergravity [@Blau:2002mw]. If we restrict the plane wave to four dimensions, it can be regarded as the group manifold of a twisted Heisenberg group. Its Lie algebra is the extension of the two-dimensional Heisenberg algebra by one additional generator $J$: $$\label{eq:Nappi_Witten_algebra}
[\lambda_a,\lambda_b]={{\varepsilon}}_{ab}{\mathbbm{1}}~,~~~[J,\lambda_a]={{\varepsilon}}_{ab}\lambda_b~,~~~[{\mathbbm{1}},\lambda_a]=[{\mathbbm{1}},J]=0~.$$ This algebra is also known as [*Nappi-Witten algebra*]{} and it can be regarded as linear Poisson structure on a four-dimensional Hpp-wave. Moreover, it can be obtained in various ways as a solution of the IKKT model, where $J$ and ${\mathbbm{1}}$ are regarded as quantized light-cone coordinates, while $\lambda_a$ are the quantized two remaining spatial coordinates. For further details, including an analogous twisted Nambu-Heisenberg algebra, see [@DeBellis:2010pf].
A categorification of this Poisson structure on a four-dimensional Hpp-wave would clearly correspond to a twist of the Lie 2-algebra induced by the 2-plectic structure on ${\mathbbm{R}}^3$. Although the integration theory of Lie 2-algebras is barely developed, one is led to an interpretation of the twisted Lie 2-algebra as a categorified linear Poisson structure on a five-dimensional Hpp-wave. We start from five coordinates $x^\pm$ and $x^i$, $i=1,\ldots,3$ together with the 1-forms $$\xi_i={{\varepsilon}}_{ijk}x^j{\mathrm{d}}x^k~~~~\text{and} ~~~ \xi^i_\pm=x_\pm {\mathrm{d}}x^i~.$$ The twisted version of the Lie 2-algebra $\Pi_{{\mathbbm{R}}^3,\varpi}$ is given by $$\label{cat.d pp-wave algebra 1}
\pi_2(\xi_i,\xi_j)=-{{\varepsilon}}_{ijk}{\mathrm{d}}x^k~,~~~~~\pi_3(\xi_i,\xi_j,\xi_k)=-{{\varepsilon}}_{ijk}~,$$ where we take the products involving the light-cone sector, parametrized by $x^\pm$, to be: $$\begin{aligned}\label{cat.d pp-wave algebra 2}
\pi_2(\xi_i,\xi_{j-})&={{\varepsilon}}_{ijk}\xi^k~, ~~~~~
\pi_2(\xi^i_-,\xi^j_-)=-{{\varepsilon}}^{ij}_{\phantom{ij}k}\xi^k_-~, ~~~~~\pi_2(\xi^i_+,{-})=0~,
\\ \\
\pi_3(\xi_i,\xi^j_-,\xi^k_-)&=0~,~~~~~
\pi_3(\xi^i_-,\xi_j,\xi_k)=\delta^i_k x^j-\delta^i_j x^k~,~~~~~\pi_3(\xi^i_-,\xi^j_-,\xi^k_-)=0~,
\end{aligned}$$ while all the $\pi_3(\xi^i_+,{-},{-})=0$. The two-products in the above reduce to the Nappi-Witten algebra in 4 dimensions after contraction along one of the ${\mathbbm{R}}^3$ vectors, for instance $\tfrac{\partial}{\partial x^3}$: $$\xi_i\rightarrow \xi_a={{\varepsilon}}_{ab}x^b{\mathrm{d}}x^3~,~~~\text{so that}~~~
\lambda_a\equiv \iota_{\partial_3} \xi_a={{\varepsilon}}_{ab}x^b~,$$ if we further identify $J\equiv -x_-$. In analogy to the symplectic case, we will set all $\pi_2(\xi^i_-,{-})$ and $\pi_2(\xi^i,{-})$ acting on exact 1-forms to zero, in line with the interpretation that they should act as derivations along the direction they define. By combining 2-products we obtain expressions for $\pi_1(\pi_3({-},{-},{-}))$ and thus deduce 3-products $\pi_3$ that are compatible with the Lie 2-algebra structure, given in the second line in . Note that these are only fixed up to constant terms by the Lie 2-algebra equations, so here we chose the simplest possible form for them. We can further take all mixed 2-products $\pi_2(x^\pm,\xi_i)=\pi_2(\xi^i_\pm,x^j)=0$, as well as set $\pi_2(\xi^i_\pm,x^\pm)=0$, since this does not affect the 2-algebra equations, nor do we have any natural reason to expect them to be non-vanishing.
Another example of a Lie 2-algebra that does not arise from a 3-form in the manner described in section \[ssec:2Plectic\_Manifolds\] is that of a twisted Poisson algebra [@Severa:2001qm] arising e.g. in the context of double geometry. This example points towards a more comprehensive mathematical description of higher Poisson structures. A Poisson structure on a manifold is encoded in a corresponding Poisson Lie algebroid. Analogously, one would expect that higher structures are encoded in a Courant Lie 2-algebroid. This is in fact the case for the twisted Poisson algebras discussed in [@Severa:2001qm].
A geometric quantization of twisted Poisson manifolds has been proposed in [@Petalidou:0704.2989] and deformation quantization of these manifolds has been considered in [@Mylonas:2012pg].
Quantization {#ssec:2-plectic_quantization}
------------
The quantization of 2-plectic manifolds remains an open problem. Partial answers have been obtained by quantizing the Nambu-Poisson bracket that arises from a 2-plectic structure under certain conditions, cf. [@DeBellis:2010pf] and references therein. Other approaches use a detour via loop spaces, see e.g. [@Saemann:2012ab]. For a more recent discussions of the general mechanism, see e.g. [@Nuiten:2013aa]. Attacking the quantization of 2-plectic manifolds directly faces the aforementioned problem that even the algebraic structure of classical observables is not fully clarified. Fortunately, we can ignore this problem and regard classical quantization only as a Lie algebra homomorphism to first order in $\hbar$ that maps the Poisson algebra to a Lie algebra of quantum observables. The categorified analogue is then a Lie 2-algebra homomorphism to first order in $\hbar$ that maps a Lie 2-algebra of classical observables - arising e.g. from a 2-plectic structure - to a Lie 2-algebra of quantum observables. Roughly this point of view has been adopted e.g. in [@Rogers:2011zc], see also [@Fiorenza:1304.6292], where prequantization of 2-plectic manifolds has been developed to a considerable amount. Usually, the symplectic form on certain quantizable manifolds defines the first Chern class of the prequantum line bundle. Fully analogously, a 2-plectic structure on certain manifolds defines the Dixmier-Douady class of a prequantum abelian gerbe. Many other ingredients of conventional geometric quantization have natural counterparts in this picture. In particular, the Atiyah algebroid, a symplectic Lie algebroid capturing the Souriau approach to geometric quantization, is replaced by a Courant Lie 2-algebroid, a symplectic Lie 2-algebroid.
Further evidence in favor of quantizing the Lie 2-algebra induced by the 2-plectic structure over the quantization of the Nambu-bracket stems from the above mentioned loop space approach. Both the 2-plectic structure as well as the prequantum abelian gerbe can be consistently mapped to a symplectic form of the loop space of the original manifold. Instead of quantizing the 2-plectic manifold, one can therefore quantize the induced symplectic loop space, cf. [@Saemann:2012ab; @Saemann:2012ex] and references therein. This quantization of loop space is now naturally compatible with the quantization of the 2-plectic structure.
Having established that our notion of quantization will be necessarily incomplete, let us now specify it to the extend we can. Our guiding principle here will be a straightforward analogy with the correspondence principle of ordinary quantization: a quantization of a manifold $M$ endowed with a Lie 2-algebra $\Pi_{M}$ is a semistrict Lie 2-algebra $\hat{\Pi}_{M}$ with products $\mu_i$ together with a map $$\hat{-}:\Pi_{M}\rightarrow \hat{\Pi}_{M}~,$$ which is a Lie 2-algebra homomorphism to lowest order in a deformation parameter $\hbar$. For simplicity, we will restrict our attention to Lie 2-algebra homomorphisms $(\Psi_0,\Psi_{-1},\Psi_2)$ that are purely given in terms of chain maps with $\Psi_2=0$. This results in the following “categorified correspondence principle:” $$\label{eq:2-correspondence_principle}
\begin{aligned}
\mu_1(\hat{X})=\widehat{-{\mathrm{i}}\hbar~\pi_1(X)}+{\mathcal{O}}(\hbar)&~,~~~\mu_2(\hat{X},\hat{Y})=\widehat{-{\mathrm{i}}\hbar ~\pi_2(X,Y)}+{\mathcal{O}}(\hbar^2)~,\\
\mu_3(\hat{X},\hat{Y},\hat{Z})&=\widehat{-{\mathrm{i}}\hbar ~\pi_3(X,Y,Z)}+{\mathcal{O}}(\hbar^2)~.
\end{aligned}$$ For our goals in this paper, this categorified correspondence principle will prove to be sufficient.
Representation of the Heisenberg Lie 2-algebra
----------------------------------------------
While we cannot solve the problem of quantization of 2-plectic manifolds here, we can give some partial insight by regarding the analogue of the Heisenberg algebra, which arises in the quantization of ${\mathbbm{R}}^2$. More specifically, the Heisenberg algebra is spanned by quantized constant and linear functions, $\hat{x}^i$ and $\hat{c}=c{\mathbbm{1}}$, $c\in {\mathbbm{R}}$. These operators satisfy the commutation relation $$\label{eq:Heisenberg_algebra}
[\hat{x}^a,\hat{x}^b]=\widehat{-{\mathrm{i}}\hbar \{x^a,x^b\}}=-{\mathrm{i}}\hbar {{\varepsilon}}^{ab}{\mathbbm{1}}~,~~~a,b=1,2~.$$ Note that the corrections to order ${\mathcal{O}}(\hbar^2)$ in the correspondence principle vanish for coordinate functions. A representation for the Heisenberg algebra is given by $U_3$, the upper triangular $3\times 3$-dimensional matrices: $$\label{eq:rep-Lie}
a \hat{x}^1+b \hat{x}^2-{\mathrm{i}}\hbar c{\mathbbm{1}}~~~\mapsto~~~\left(\begin{array}{ccc}0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0\end{array}\right)~,$$ and the matrix commutator of these upper triangular matrices reproduces the algebra relation .
The Heisenberg Lie 2-algebra is spanned by quantized constant and linear functions as well as constant and linear 1-forms $\hat{x}^i$, $\hat{c}=c{\mathbbm{1}}$ and $\xi^i$, ${\mathrm{d}}x^i$, as defined in section \[ssec:2-plectic\_examples\]. The non-trivial Lie 2-algebra products for the quantized coordinate algebra are $$\mu_1(\hat{x}^i)=\widehat{-{\mathrm{i}}\hbar {\mathrm{d}}x^i}~,~~~\mu_2(\hat{\xi}_i,\hat{\xi}_j)=-\widehat{{\mathrm{i}}\hbar{{\varepsilon}}_{ijk} {\mathrm{d}}x^k}~,~~~\mu_3(\hat{\xi}_i,\hat{\xi}_j,\hat{\xi}_k)=\widehat{{\mathrm{i}}\hbar{{\varepsilon}}_{ijk}}{\mathbbm{1}}~,$$ where we again assumed that the corrections in the correspondence principle to order ${\mathcal{O}}(\hbar^2)$ vanish here.
We represent this Lie 2-algebra on the 2-vector space ${\mathbbm{R}}^4\xrightarrow{\mu_1} U_5$, where ${\mathbbm{R}}^4$ is spanned by basis vectors $e^0,e^i$ and $U_5$ is the vector space of upper triangular $5\times 5$-dimensional matrices. The chain maps of the Lie 2-algebra homomorphism are given by $$\label{eq:rep-2Lie}
\begin{aligned}
-{\mathrm{i}}\hbar c{\mathbbm{1}}+ b_i \hat{x}^i~~~&\mapsto~~~c e^0+b_i e^i~,\\
a^i \hat{\xi}_i-{\mathrm{i}}\hbar b_i \widehat{{\mathrm{d}}x^i}~~~&\mapsto~~~\left(\begin{array}{ccccc}
0 & a^1 & b_3 & 0 & 0 \\
0 & 0 & a^2 & b_1 & 0 \\
0 & 0 & 0 & a^3 & b_2\\
0 & 0 & 0 & 0 & a^1\\
0 & 0 & 0 & 0 & 0
\end{array}\right)~.
\end{aligned}$$ The non-trivial Lie 2-algebra products on this 2-vector space are given by obvious maps $\mu_1:{\mathbbm{R}}^4\rightarrow U_5$ and $\mu_3:U_5^{\wedge 3}\rightarrow {\mathbbm{R}}^4$ together with the map $$\mu_2(u_1,u_2)=[P(u_1),P(u_2)]~,~~~ u_1,u_2\in U_5~,$$ where $$P\left(\begin{array}{ccccc}
0 & a^1 & b_3 & 0 & 0 \\
0 & 0 & a^2 & b_1 & 0 \\
0 & 0 & 0 & a^3 & b_2\\
0 & 0 & 0 & 0 & a^1\\
0 & 0 & 0 & 0 & 0
\end{array}\right):=
\left(\begin{array}{ccccc}
0 & a^1 & 0 & 0 & 0 \\
0 & 0 & a^2 & 0 & 0 \\
0 & 0 & 0 & a^3 & 0\\
0 & 0 & 0 & 0 & a^1\\
0 & 0 & 0 & 0 & 0
\end{array}\right)~.$$ Such brackets containing projectors are quite common in the context of derived brackets and strong homotopy Lie algebras, cf. [@Voronov:math0304038]. Note that the reduction of the representation to is very transparent.
Homogeneous Lie 2-algebra models
================================
Let us now come to the homogeneous Lie 2-algebra models, which are built from the various inner products. As stated before, these models are written in terms of a single type of field $X^a$, $a=1,\ldots,d$, which takes values in a Lie 2-algebra. We start by discussing the difference between the three kinds of metrics. We then consider the classical equations of motion and demonstrate that their solutions contain quantized symplectic and 2-plectic manifolds.
Homogeneous Lie 2-algebra models and the various inner products
---------------------------------------------------------------
The first ingredient are the various non-vanishing products on $L$, which we summarize here for the reader’s convenience: $$\begin{aligned}
\mu_1&:V\rightarrow W~,&\mu_1^*&:W\rightarrow V~,\\
\mu_2&:V\wedge W \rightarrow V~,~~~&\mu_2&:W\wedge W\rightarrow W~,~~~&\mu_2^*&:V\wedge V\rightarrow W~,\\
\mu_3&:W\wedge W\wedge W\rightarrow V~,~~~&\mu_3^*&:W\wedge W \otimes V\rightarrow W~.
\end{aligned}$$ Note that we can neglect the product $\kappa_2$, as it is built from the ones above. Moreover, note that $\mu^*_2(w,\ell)=-\mu_2(w,\ell)$ for any $w\in W$ and $\ell\in L$. The large number of remaining products makes it impossible to discuss a general action, and inspired by the M2-brane models, we will restrict ourselves to actions that are at most sextic in the fields.
In the following, we briefly discuss general Lie 2-algebra models that make use of the three inner products that we introduced in section \[ssec:inner\_products\]. Recall that a key feature of all inner products was the fact that $$\label{eq:W_invariance}
\langle \mu_2(w,\ell_1),\ell_2 \rangle+\langle \ell_1,\mu_2(w,\ell_2)\rangle=0$$ for $w\in W$ and $\ell_1,\ell_2\in L$. This property is required to guarantee that the actions of Lie 2-algebra models exhibit a nice symmetry algebra.
The cyclic metric $\langle {-},{-}\rangle_\infty$ defined in section \[ssec:inner\_products\] is very restrictive. Recall that this metric corresponds to an invariant polynomial, which naturally induces actions for field theories of “Chern-Simons type”, cf. [@Sati:0801.3480]. A typical example is the action discussed in [@Zwiebach:1992ie; @Lada:1992wc], whose stationary points are described by homotopy Maurer-Cartan equations. Here, however, we are more interested in actions of “Yang-Mills type”, of which the IKKT model is an example.
Leaving out the product $\mu_1$, the only non-zero terms we can construct, up to fourth order in $X$, are $$\label{eq:action_cyclic}
S_\infty\:=\ \tfrac{1}{2}m_{ab}\langle X^a,X^b\rangle_\infty + \tfrac{1}{3}c_{abc} \langle
X^a,\mu_2(X^b,X^c)\rangle_\infty + \tfrac{1}{4}\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle_\infty~,$$ where $m_{ab}$ is a ‘mass matrix’ and $c_{abc}\in {\mathbbm{R}}$ is some totally antisymmetric tensor encoding a background yielding a cubic coupling. Higher order terms involving nested $\mu_2$ can be constructed, too. Note, however, that terms involving $\mu_3$ necessarily vanish, cf. . Splitting the fields $X^a$ in the action into the components $X^a=v^a+w^a$ with $v^a\in V$ and $w^a\in W$, we arrive at $$\begin{aligned}
S_\infty\:=\ &\tfrac{1}{2}m_{ab}\langle v^a,v^b\rangle_\infty+\tfrac{1}{2}m_{ab}\langle w^a,w^b\rangle_\infty + \tfrac{1}{3}c_{abc} \langle
w^a,\mu_2(w^b,w^c)\rangle_\infty +\\
&+\tfrac{1}{4}\langle \mu_2(w^a,w^b),\mu_2(w^a,w^b)\rangle_\infty~.
\end{aligned}$$
In the case of the minimally invariant metric, we can write down more general terms. For example, we could consider the following action: $$\label{eq:action_W_invariant}
\begin{aligned}
S_0\ =\ &\tfrac{1}{2}m_{ab}\langle X^a,X^b\rangle_0 + \tfrac{1}{3}c_{abc}\langle
X^a,\mu_2(X^b,X^c))\rangle_0 + \tfrac{1}{4}\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle_0\\
&+d_{abcd}\langle X^a,\mu_3(X^b,X^c,X^d)\rangle_0 + \tfrac{1}{6}\lambda\langle \mu_3(X^a,X^b,X^c),\mu_3(X^a,X^b,X^c)\rangle_0\\
\ =\ &\tfrac{1}{2}m_{ab}\langle v^a,v^b\rangle_0+\tfrac{1}{2}m_{ab}\langle w^a,w^b\rangle_0+\tfrac{2}{3}c_{abc}\langle
v^a,\mu_2(w^b,v^c))\rangle+\tfrac{1}{3}c_{abc}\langle
w^a,\mu_2(w^b,w^c))\rangle\\
&+\tfrac{1}{2}\langle \mu_2(w^a,v^b),\mu_2(w^a,v^b)\rangle+ \tfrac{1}{2}\langle \mu_2(w^a,v^b),\mu_2(v^a,w^b)\rangle+\tfrac14\langle \mu_2(w^a,w^b),\mu_2(w^a,w^b)\rangle\\
&+\tfrac{1}{4}d_{abcd}\langle v^a,\mu_3(w^b,w^c,w^d)\rangle + \tfrac{1}{6}\lambda\langle \mu_3(w^a,w^b,w^c),\mu_3(w^a,w^b,w^c)\rangle~,
\end{aligned}$$ where $c_{abc}\in{\mathbbm{R}}$ and $d_{abcd}\in{\mathbbm{R}}$ encode totally antisymmetric[^12] background tensors and $\lambda\in{\mathbbm{R}}$ is a coupling constant.
In the case of the reduced metric, $V$ is considered as a sub vector space of $W$. Thus, we can replace $X^a$ in the action directly by $w^a$, and we get interaction terms like $$d_{abcd}\langle X^a,\mu_3(X^b,X^c,X^d)\rangle_{\rm red}=d_{abcd}\langle w^a,\mu_3(w^b,w^c,w^d)\rangle_{\rm red}~,$$ which, however, can be rewritten as $3d_{abcd}\langle w^a,\mu_2(\mu_2(w^{[b},w^c),w^{d]})\rangle$.
While actions built from minimally invariant and reduced inner products can contain considerably more interactions than those employing the cyclic inner product, it is not clear to us whether these additional terms are useful. In particular, when considering actions that have quantized symplectic and 2-plectic geometries as solutions, we can restrict ourselves to the terms contained in $S_\infty$.
Symmetries of the models {#ssec:actions_and_symmetries}
------------------------
The symmetries of a general Lie 2-algebra model have to be given by Lie 2-algebra automorphisms. Recall that the symmetry algebra relevant in the IKKT matrix model was the algebra of inner automorphisms of the underlying matrix algebra. We will therefore focus our attention here on [*inner Lie 2-algebra automorphisms*]{}, by which we mean automorphisms $\Psi:L\rightarrow L$ which read infinitesimally as $$\Psi_{-1}(v)=v+\mu_2(\alpha, v)~,~~\Psi_0(w)=w+\mu_2(\alpha,w)~~\mbox{and}~~ \Psi_2(w_1,w_2)=\mu_3(\alpha,w_1,w_2)~,$$ where $v\in V$, $w\in W$, and $\alpha \in W$ is the (infinitesimal) gauge parameter. Under these symmetries, Lie 2-algebra actions remain invariant, independently of the inner product used in their definition. This is due to the invariance described in equation . For example, both the cyclic and minimally invariant inner products split into separate inner products of terms in $W$ and inner products of terms in $V$: $$S=\sum_i\langle w_{1,i},w_{2,i}\rangle + \sum_j\langle v_{1,j},v_{2,j} \rangle~.$$ Each of these terms is invariant under inner Lie 2-algebra automorphisms, e.g. $$\delta \langle w_1,w_2\rangle=\langle \delta w_1, w_2\rangle + \langle w_1,\delta w_2 \rangle
=\langle \mu_2(\alpha,w_1),w_2\rangle + \langle w_1,\mu_2(\alpha,w_2)\rangle=0~.$$ One should stress in this context an important difference to conventional field theories: to propagate the action of the symmetry transformations from a higher product onto the fields, one has to take into account that a Lie 2-algebra automorphism also transforms the higher products themselves. For example, we have $$\delta\mu_2(w_1,w_2)=\mu_2(\delta w_1,w_2)+\mu_2(w_1,\delta w_2)+(\delta \mu_2)(w_1,w_2)~,$$ and the explicit form of $(\delta\mu_2)(w_1,w_2)$ is easily read off equation .
Recall that the IKKT model arose as a dimensional reduction of a ten-dimensional supersymmetric gauge theory. Symmetries of this model are therefore given by residual supersymmetry as well as dimensionally reduced gauge symmetry. We might expect that something similar happens in the case of Lie 2-algebra models, assuming that they arise from a dimensional reduction of semistrict higher gauge theory. While semistrict higher gauge theory has only been developed partially, an attempt to capture its local gauge structure has been made in [@Zucchini:2011aa].
In this framework, gauge symmetry is described by a Lie 2-algebra automorphism $(g_0,g_{-1},g_2)$ together with a flat connection doublet $(\sigma,\Sigma)$ and a 1-form $\tau$ taking values in ${\mathsf{Hom}\,}(W,V)$. The connection doublet and the 1-form are solutions of the consistency relations . For further reference, a concise overview over this gauge structure is included in appendix \[app:B\].
After the dimensional reduction to a point, the consistency relations are satisfied for trivial $(\sigma,\Sigma)$ and $\tau$, and the whole gauge structure therefore reduces to a Lie 2-algebra automorphism. We thus arrive at the symmetries of our Lie 2-algebra model, in analogy with the case of the IKKT model. Note, however, that Lie 2-algebra models arising from dimensionally reducing a semistrict higher gauge theory to a point are more likely to be described by inhomogeneous Lie 2-algebra models, and we will return to this issue in section \[ssec:background\_expansion\_1\].
Reduction to the IKKT model and quantized symplectic manifolds
--------------------------------------------------------------
The reduction to the bosonic part of the IKKT model is a rather trivial affair. Given a (real) Lie algebra ${\mathfrak{g}}$, we can extend it trivially to a Lie 2-algebra $L_{\mathfrak{g}}:V\rightarrow W$ by putting $V=*=\{0\}$ and $W={\mathfrak{g}}$. The only non-trivial higher product is then $\mu_2:{\mathfrak{g}}\times {\mathfrak{g}}\rightarrow {\mathfrak{g}}$, which is given by the commutator. The higher Jacobi identities are trivially satisfied. The Gram-Schmidt inner product yields an inner product on this Lie 2-algebra. This inner product satisfies simultaneously the axioms of cyclic, reduced and minimally invariant inner products, as one readily verifies. We can therefore work with any of the above discussed homogeneous Lie 2-algebra models.
All these models contained the following terms in the action: $$S_0\:=\ \tfrac{1}{2}m_{ab}\langle X^a,X^b\rangle + \tfrac{1}{3}c_{abc} \langle
X^a,\mu_2(X^b,X^c)\rangle + \tfrac{1}{4}\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle~.$$ Assuming that the underlying Lie 2-algebra is the Lie 2-algebra $L_{\mathfrak{g}}$, we recover the bosonic part of the IKKT matrix model together with the bosonic part of the deformation terms . Note that using a T-duality, one can then obtain BFSS matrix quantum mechanics [@Banks:1996vh] in the usual way.
We say that a solution to the IKKT model corresponds to a quantized symplectic manifold, if the matrices $X^a$ describing this solution are given by a complete set of quantized coordinate functions of a noncommutative space. Note that for compact spaces like the fuzzy sphere, these coordinate functions are given by embedding coordinates of the compact manifold $M$ in some ${\mathbbm{R}}^n$. These coordinates should be seen as the pullback of the coordinate functions on ${\mathbbm{R}}^n$ along the embedding[^13] $e:M{{\hookrightarrow}}{\mathbbm{R}}^n$.
Let us briefly recall three important solutions of the IKKT model for future reference. For vanishing masses $m_{ab}$ and cubic couplings $c_{abc}$, we obtain the Moyal plane ${\mathbbm{R}}^{2n}_\theta$, as already mentioned in the introduction. This space is described by quantized coordinate functions $\hat{x}^i$, $i=1,\ldots,2n$, satisfying the Heisenberg algebra, cf. .
The fuzzy sphere $S^2$ is described as a quantized submanifold of ${\mathbbm{R}}^3$ by the quantized coordinate algebra $$[\hat{x}^i,\hat{x}^j]=-{\mathrm{i}}\hbar R {{\varepsilon}}^{ijk}\hat{x}^k~,$$ where $i,j,k=1,2,3$, $R$ is the radius of the fuzzy sphere and $\hbar=\frac{2}{k}$, $k\in {\mathbbm{N}}$, cf. [@DeBellis:2010pf]. As solutions to the IKKT model, it can be obtained in two ways. First of all, we can turn on a mass term $$m_{ij}=-2\hbar^2R^2\delta_{ij}~,$$ as observed in [@Kimura:0103192]. Second, we can tune the cubic coupling proportional to the structure constants of ${\mathfrak{su}}(2)$, $$c_{ijk}=-{\mathrm{i}}\hbar R{{\varepsilon}}_{ijk}~,$$ as discussed in [@Iso:0101102]. Both mass terms and cubic couplings can certainly be combined in a more general fashion.
The quantized Hpp-wave encoded in the Nappi-Witten algebra is obtained as the solution $$\hat{x}^1=\lambda_1~,~~~\hat{x}^2=\lambda_2~,~~~\hat{x}^3=J{{\qquad\mbox{and}\qquad}}\hat{x}^4={\mathbbm{1}}$$ of the action $S_0$ with the following non-trivial mass-terms and couplings: $$\label{eq:background_IKKT_Hpp}
m_{11}=m_{22}=-1{{\qquad\mbox{and}\qquad}}c_{ijk}={{\varepsilon}}_{ijk}~,~~~i,j,k=1,2,3~,$$ see also [@DeBellis:2010sy].
Before coming to the case of 2-plectic manifolds, let us briefly note a subtle point. While the above quantized coordinate algebras do solve the equations of motion resulting from the action $S_0$, they may not correspond to quantized square integrable functions or may yield problematic terms in the action. For example, in the case of the Moyal plane, we have $[X^a,X^b]={{\varepsilon}}^{ab}{\mathbbm{1}}$. The term $\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle={\,\mathrm{tr}\,}([X^a,X^b][X^a,X^b])$ is problematic when evaluated at this solution, as all non-trivial representations of the Heisenberg algebra are necessarily infinite-dimensional and the trace of ${\mathbbm{1}}$ is therefore ill-defined: the operator ${\mathbbm{1}}$ is not [*trace class*]{}. We will encounter the same issue in the case of Lie 2-algebra models. Recall, however, that we are not interested in the value of the action functional. We will first derive the equations of motion assuming our fields have finite norm and then continue the resulting equations to arbitrary Lie 2-algebra elements.
Solutions corresponding to quantized 2-plectic manifolds {#ssec:ingredients:solutions}
--------------------------------------------------------
As recalled above, we call a solution to the IKKT model a quantized symplectic manifold, if it is given in terms of quantized coordinate functions on ${\mathbbm{R}}^n$, into which the symplectic manifold is embedded. Similarly, solutions to the 3-Lie algebra model of [@DeBellis:2010sy] were given by quantized coordinate functions that took values in a 3-Lie algebra. Again, for compact spaces, quantized embedding coordinates of the manifold in some Euclidean space were used.
In the case of Lie 2-algebra models, the coordinate functions should be replaced by the quantization of certain elementary 1-forms. Let us characterize these 1-forms in the following. For compact spaces, we should again consider their embedding in some ${\mathbbm{R}}^n$ and use the pull-back of the elementary 1-forms on ${\mathbbm{R}}^n$ along the embedding. It therefore suffices to characterize elementary one-forms on ${\mathbbm{R}}^n$. There is a number of properties we would like these elementary 1-forms to have:
They should be as simple as possible.
They cannot be exact, as exact forms are central in the Lie 2-algebras induced by 2-plectic structures.
Just as with Cartesian coordinate functions on ${\mathbbm{R}}^n$, the Hamiltonian vector field of the 1-forms should equal the derivative with respect to the Cartesian coordinates on ${\mathbbm{R}}^n$.
Under the reduction procedure outlined in section \[ssec:reduction\], they should reduce to coordinate functions on ${\mathbbm{R}}^{n-1}$.
In Cartesian coordinates $x^i$ on ${\mathbbm{R}}^n$, the simplest 1-forms on ${\mathbbm{R}}^n$ that are not exact are given by $$\label{eq:elementary_1-forms}
\xi^{ij}=x^{[i}{\mathrm{d}}x^{j]}~,$$ and we have encountered these already in section \[ssec:2-plectic\_examples\]. One can easily verify that (iv) on its own would also lead to . Moreover, on spaces ${\mathbbm{R}}^{3n}$ with canonical 2-plectic structure, these elementary 1-forms satisfy (iii).
Another requirement one might impose is on the quantization of elementary 1-forms: the correspondence principle should hold exactly and should not receive any corrections to order ${\mathcal{O}}(\hbar^2)$.
Note that the Lie 2-algebras we obtain from a 2-plectic structure are not reduced, and we do not expect that the corresponding quantized Lie 2-algebra will be reduced. In discussing solutions, we therefore have to restrict ourselves to the cyclic and minimally invariant inner products. In both cases, we are interested in the same action[^14], $$S_1\:=\ \tfrac{1}{2}m_{ab}\langle X^a,X^b\rangle + \tfrac{1}{3}c_{abc} \langle
X^a,\mu_2(X^b,X^c)\rangle + \tfrac{1}{4}\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle~,$$ which, however, leads to different equations of motion. In the cyclic case, we have $$\label{eq:eom_cyclic}
m_{ab}w^b+\mu_2(w^b,\mu_2(w^b,w^a))+c_{abc}\mu_2(w^b,w^c)=0{{\qquad\mbox{and}\qquad}}m_{ab}v^b=0~,$$ while in the minimally invariant case, we have $$\label{eq:eom_0}
\begin{aligned}
m_{ab}v^b+\tfrac{4}{3}c_{abc}\mu_2(w^b,v^c)+\tfrac12\mu_2(w^b,\mu_2(w^b,v^a)) +\tfrac12\mu_2(w^b,\mu_2(v^b,w^a))&=0~,\\
m_{ab}w^b-\tfrac{2}{3}c_{abc}\mu_2^*(v^c,v^b)+c_{abc}\mu_2(w^b,w^c)+\tfrac12\mu_2^*(v^b,\mu_2(v^b,w^a))
+\mu_2(w^b,\mu_2(w^b,w^a))&=0~.
\end{aligned}$$ We now restrict to Lie 2-algebras that arise from the quantization of a Lie 2-algebra $\Pi_{M,\varpi}$ and impose the above mentioned requirement that for elementary functions and 1-forms, the correspondence principle holds precisely without corrections to order ${\mathcal{O}}(\hbar^2)$. This implies that in equations , the terms containing the products $$\mu_2:W\times V\rightarrow V{{\qquad\mbox{and}\qquad}}\mu_2^*:V\times V\rightarrow W$$ vanish on elementary 1-forms and equations reduce to . We can therefore restrict our attention to the latter equations of motion.
Examples of quantized categorified Poisson manifolds as solutions
-----------------------------------------------------------------
As a first example, we consider the quantization of $\Pi_{{\mathbbm{R}}^3,\varpi}$, where $\varpi$ is again the canonical volume form on ${\mathbbm{R}}^3$. Just as the Moyal plane was obtained from the undeformed IKKT model, we expect the quantization $\hat{\Pi}_{{\mathbbm{R}}^3,\varpi}$ of $\Pi_{{\mathbbm{R}}^3,\varpi}$ to arise from the action $S_1$ with $m=c=0$. This is indeed the case: the quantization of the 1-forms $\xi_i=\tfrac{1}{2}{{\varepsilon}}_{ijk}x^i{\mathrm{d}}x^k$ satisfy the following algebra $$\mu_2(\hat{\xi}_i,\hat{\xi}_j)=-{\mathrm{i}}\hbar{{\varepsilon}}_{ijk}\widehat{{\mathrm{d}}x^k}~,$$ where $\widehat{{\mathrm{d}}x^k}$ is central in $\hat{\Pi}_{{\mathbbm{R}}^3,\varpi}$. Putting $$w^i=\hat{\xi}_i{{\qquad\mbox{and}\qquad}}v^i=0~,~~~i=1,\ldots,3~,$$ we obtain a solution to , which we interpret as a quantization ${\mathbbm{R}}^3_\hbar$ of ${\mathbbm{R}}^3$ as 2-plectic manifold.
Note that the solution of the IKKT model corresponding to the Moyal plane trivially extends to Cartesian products ${\mathbbm{R}}^{2n}_\theta={\mathbbm{R}}^2_\theta\times\cdots\times {\mathbbm{R}}^2_\theta$. The same holds here, and we obtain quantized 2-plectic manifolds ${\mathbbm{R}}^{3n}_\hbar={\mathbbm{R}}^3_\hbar\times \cdots\times {\mathbbm{R}}^3_\hbar$.
Note also that as a special case to the above solution, we can use the subalgebra of $\hat{\Pi}_{{\mathbbm{R}}^3,\varpi}$, which corresponds to the reduction to the fuzzy sphere as discussed in section \[ssec:reduction\]. This yields a continuous foliation of quantized ${\mathbbm{R}}^3_\hbar$ by fuzzy spheres, which is different to the discrete foliation given by the space ${\mathbbm{R}}^3_\lambda$ as introduced in [@Hammou:2001cc].
As our second example, let us consider the quantization of the 2-plectic sphere $S^3$. First, note that analogously to the case of the fuzzy sphere solution to the IKKT model, we should embed the 3-sphere into ${\mathbbm{R}}^4$ and describe its quantization as a push-forward on elementary 1-forms on ${\mathbbm{R}}^4$. More specifically, we consider the 1-forms $$\label{eq:1-forms_S^3}
\xi_{\mu\nu}:=\tfrac{1}{2}{{\varepsilon}}_{\mu\nu\kappa\lambda}x^\kappa{\mathrm{d}}x^\lambda~,$$ where $x^\mu,~\mu=1,\ldots,4~,$ are the embedding coordinates of $e:S^3\hookrightarrow {\mathbbm{R}}^4$, where $e(S^3)=\{||x||=1\,|\,x\in {\mathbbm{R}}^4\}$. The higher product $\pi_2$ on these elementary 1-forms is given by $$\label{eq:cat_of_so4}
\pi_2(\xi_{\mu\nu},\xi_{\kappa\lambda})=\delta_{\nu\kappa}\xi_{\mu\lambda}-\delta_{\mu\kappa}\xi_{\nu\lambda}-\delta_{\nu\lambda}\xi_{\mu\kappa}+\delta_{\mu\lambda}\xi_{\nu\kappa}+\pi_1(R_{\mu\nu\kappa\lambda})~,$$ where $$R_{\mu\nu\kappa\lambda}=\tfrac{1}{4}\left({{\varepsilon}}_{\nu\kappa\lambda\rho}x^\rho x^\mu-{{\varepsilon}}_{\mu\kappa\lambda\rho}x^\rho x^\nu-{{\varepsilon}}_{\kappa\mu\nu\rho}x^\rho x^\lambda+{{\varepsilon}}_{\lambda\mu\nu\rho}x^\rho x^\kappa\right)~.$$ Equation shows that this Lie 2-algebra of elementary 1-forms is in fact a categorification of the Lie algebra ${\mathfrak{so}}(4)$, where the usual commutation relations hold up to the isomorphism $\pi_1(R_{\mu\nu\kappa\lambda})$.
Comparing again with the case of the fuzzy sphere arising in the IKKT model, we expect that the quantized 3-sphere arises in two different ways. First, a solution to $S_1$ is given in terms of the quantized 1-forms defined in by $$\label{eq:solution_S^3}
(w^I)=(\hat{\xi}_{12},\hat{\xi}_{13},\hat{\xi}_{14},\hat{\xi}_{23},\hat{\xi}_{24},\hat{\xi}_{34}){{\qquad\mbox{and}\qquad}}v^I=0~,$$ if we set the masses to $$m_{IJ}=-4\hbar^2\delta_{IJ}~,~~~I,J=1,\ldots,6~.$$ Note that the index $I$ should here be regarded as a multi-index $I=([mn])$. Second, is also a solution, if we tune the $c_{IJK}$ to the structure constants of ${\mathfrak{so}}(4)$ in the representation categorified in . Explicitly, we have the following non-trivial entries: $$c_{[124]}={\mathrm{i}}\hbar~,~~~c_{[135]}={\mathrm{i}}\hbar~,~~~c_{[236]}={\mathrm{i}}\hbar~,~~~c_{[456]}={\mathrm{i}}\hbar~.$$ It is quite striking that quantized $S^3$ arises in the same manner in our Lie 2-algebra models as the fuzzy sphere arose in the IKKT model.
As our last example, let us consider the Lie 2-algebra corresponding to a categorification of the Nappi-Witten algebra, which we interpreted as the Lie 2-algebra related to a five-dimensional Hpp-wave. We have nine elementary 1-forms, $$\label{eq:sol_Hpp}
(w^m)=(\hat{\xi}_1,\hat{\xi}_2,\hat{\xi}_3,\hat{\xi}_+^1,\hat{\xi}_+^2,\hat{\xi}_+^3,\hat{\xi}_-^1,\hat{\xi}_-^2,\hat{\xi}_-^3)~,$$ and their non-trivial products $\mu_2$ read as $$\mu_2(\hat{\xi}_i,\hat{\xi}_j)={\mathrm{i}}\hbar{{\varepsilon}}_{ijk}{\mathrm{d}}x^k~,~~~\mu_2(\hat{\xi}_i,\hat{\xi}_j^-)=-{\mathrm{i}}\hbar {{\varepsilon}}_{ijk}\hat{\xi}^k{{\qquad\mbox{and}\qquad}}\mu_2(\hat{\xi}^i_-,\hat{\xi}_-^j)={\mathrm{i}}\hbar{{\varepsilon}}^{ij}{}_k\hat{\xi}^k_-~.$$ The most general action $S_1$ to which is a solution has the following mass parameters and cubic coupling terms: $$\begin{aligned}
m_{mn}={{\mathrm{diag}}}(4\hbar^2,4\hbar^2,4\hbar^2,0,0,0,-2{\mathrm{i}}\hbar\,c_{789},-2{\mathrm{i}}\hbar\,c_{789},-2{\mathrm{i}}\hbar\,c_{789})~,&\\
c_{[ijk_-]}=-{\mathrm{i}}\hbar{{\varepsilon}}_{ijk_-}~,~~~c_{[ijk_+]}=c_{[ijk]}=c_{[ij_-k_-]}=c_{[i_+j_-k_-]}=0~,\hspace{0.1cm}
\end{aligned}$$ while the remaining cubic couplings can be chosen arbitrarily. Here, indices $i_+$ run over $4,5,6$ and indices $i_-$ run over $7,8,9$. Note that these background fields are very similar to those in that gave rise to the Hpp-wave solution in the IKKT model.
Inhomogeneous Lie 2-algebra models {#Inhomogeneous models}
==================================
We now come to inhomogeneous Lie 2-algebra models, in which we have two kinds of fields $\{X^a\}$ and $\{Y^i\}$ taking values in $V$ and $W$, respectively. This class of models includes the homogeneous models as those actions that are written in terms of sums $X^a+Y^a$. Therefore the inhomogeneous models can exhibit all the solutions we found in the previous section. We will start with an inhomogeneous Lie 2-algebra model that reduces for skeletal and strict Lie 2-algebras to zero-dimensional M2-brane models. We then consider a specific inhomogeneous Lie 2-algebra model that results from dimensionally reducing a higher gauge theory and analyze fluctuations around a special solution.
Note that inhomogeneous Lie 2-algebra models are also invariant under the inner Lie 2-algebra automorphisms discussed in section \[ssec:actions\_and\_symmetries\].
Dimensionally reduced M2-brane models
-------------------------------------
We showed in section \[ssec:M2-brane\_3\_algebras\] that Lie 2-algebras that are either skeletal or strict come with a real 3-algebra structure, where the ternary bracket is given by $$\label{eq:ternary_bracket2}
[v_1,v_2,v_3]=-\mu_2(\mu_2^*(v_1,v_2),v_3)~.$$ For $\mu_2^*$ to be non-trivial, we will have to work with the minimally invariant metric $\langle -,-\rangle_0$.
We can now write down inhomogeneous Lie 2-algebra models that make use of this ternary bracket and reduce to previously studied zero-dimensional models related to M2-brane models. The action we are interested in reads as $$\label{eq:M2-brane_model}
\begin{aligned}
S_{\rm M2}=&\tfrac{1}{6}{{\varepsilon}}^{ijk}\langle Y^i,\mu_2(Y^j,Y^k)\rangle_0-\tfrac{1}{2}\langle\mu_2(Y^i,X^a),\mu_2(Y^i,X^a)\rangle_0+\tfrac{{\mathrm{i}}}{2}\langle \bar{\Psi},\mu_2(\Gamma^i Y^i,\Psi)\rangle_0\\&
-\tfrac{{\mathrm{i}}}{4}\langle \bar{\Psi},\mu_2(\mu_2^*(X^a,X^b),\Gamma_{ab}\Psi)\rangle_0-\tfrac{1}{12}\langle \mu_2(\mu_2^*(X^a,X^b),X^c),\mu_2(\mu_2^*(X^a,X^b),X^c)\rangle_0~,
\end{aligned}$$ where the scalars $X^a$, $a=1,\ldots,8$, and the spinors $\Psi$ take values in $V$, while the scalars $Y^i$, $i=0,\ldots,2$, take values in $W$. Our spinor and Clifford algebra conventions are those of [@Bagger:2007jr].
For skeletal or strict Lie 2-algebras, the action equals that of a full dimensional reduction of the ${\mathcal{N}}=2$ M2-brane models discussed in [@Cherkis:2008qr]. Our action then inherits ${\mathcal{N}}=2$ supersymmetry from the 3-dimensional model.
If the ternary bracket happens to be antisymmetric[^15], we recover the 3-Lie algebra models that we discussed in the introduction. In particular, the models of [@Lee:2009ue; @Furuuchi:2009ax; @Sato:2009tr] are obtained by putting $Y_i=0$ and letting $a=0,\ldots,10$, otherwise one arrives at the model discussed in [@DeBellis:2010sy]. It is a trivial exercise to add deformation terms to that are written in terms of 3-brackets.
Note that inhomogeneous Lie 2-algebra models reproducing the dimensionally fully reduced ABJM model can also be written down in a straightforward fashion.
To obtain interesting solutions to the model one should either consider solutions with $Y_i\neq 0$ or solutions that do not arise from 2-plectic manifolds. Otherwise, the equations of motion are trivially satisfied. It is not clear how to interpret solutions that contain both nontrivial $Y_i$ and $X^a$. We therefore refrain from going into any more detail at this point.
Background expansion for higher gauge theory {#ssec:background_expansion_1}
--------------------------------------------
Recall from section \[ssec:background\_IKKT\] that expanding the action of the IKKT model around a solution corresponding to a noncommutative space yields essentially the action for Yang-Mills theory on that noncommutative space [@Aoki:1999vr]. In particular, consider the action . A solution to this action is the Moyal space ${\mathbbm{R}}^{2n}_\theta$ with coordinates satisfying $[\hat{x}^\mu,\hat{x}^\nu]=-{\mathrm{i}}\theta^{\mu\nu}$, $\mu,\nu=1,\ldots,2n$. If we now expand around this solution by writing $X^\mu=\hat{x}^\mu+\hat{A}^\mu$ and observing that $\theta_{\mu\nu}[\hat{x}^\nu,\hat{f}]=\widehat{{\partial}_\mu f}$, we obtain noncommutative Yang-Mills theory on ${\mathbbm{R}}^{2n}_\theta$.
An attempt has been made to reproduce this observation in the context of quantized Nambu-Poisson manifolds using 3-Lie algebras in [@DeBellis:2010sy], but the construction seemed far less natural than in the IKKT case.
Let us now try to obtain field theories on quantized 2-plectic spaces by performing a background expansion. For this, we have to choose the kind of action we expect to reproduce. The most natural candidate here are higher BF-theories as discussed in [@Martins:2010ry].
For simplicity, we will consider higher BF theory on ${\mathbbm{R}}^3$. The field content consists of a 1-form $A$ and a 2-form $B$. Usually, these take values in the vector spaces $W$ and $V$, respectively, that form a strict Lie 2-algebra. Here, however, we immediately allow for a semistrict Lie 2-algebra, and neglect all the technical difficulties that come with a complete discussion of semistrict higher gauge theory, see [@Zucchini:2011aa] and appendix \[app:B\]. If the higher BF theory is supposed to describe a connective structure that captures the parallel transport of an extended object, we have to impose the fake curvature condition $$0={\mathcal{F}}:=F-\mu_1(B):={\mathrm{d}}A+\mu_2(A,A)-\mu_1(B)~.$$ As usual in BF-theory, we also expect the 3-form curvature to vanish: $$0=H:={\mathrm{d}}B+\mu_2(A,B)+\tfrac16\mu_3(A,A,A)~,$$ where we extended the usual definition of the 3-form curvature $H$ in higher gauge theory by a term $\mu_3(A,A,A)$, cf. [@Zucchini:2011aa]. Altogether, we arrive at the action $$S_{\rm BF}=\int_{{\mathbbm{R}}^3} \langle \lambda_1,{\mathcal{F}}\rangle_0+\langle \lambda_0,H\rangle_0~,$$ where $\lambda_0$ and $\lambda_1$ are 0- and 1-forms taking values in $V$ and $W$, respectively.
To obtain a Lie 2-algebra model, we dimensionally reduce the action $S_{\rm BF}$ to a point. We are left with fields $X_{ij}$, $i,j=1,\ldots,3$, taking values in $V$ and fields $Y_i$ with values in $W$ together with additional Lagrange multiplier fields $\lambda_i$ and $\lambda$. Note that we should also twist the BF action by terms $2{\mathrm{i}}\hbar\langle \lambda_i,\widehat{{\mathrm{d}}x^i}\rangle_0$ and ${\mathrm{i}}\hbar\langle \lambda, 1\rangle_0$. This can be easily seen by imagining writing down a BF theory on the Moyal plane; even in the Yang-Mills case we introduce such a twist, cf. . The total action reads as $$\label{eq:BF_red}
\begin{aligned}
S_{0d}=&{{\varepsilon}}^{ijk}\langle \lambda_i,\mu_2(Y_j,Y_k)-\tfrac12\mu_1(X_{jk})\rangle_0+2{\mathrm{i}}\hbar\langle \lambda_i,\widehat{{\mathrm{d}}x^i}\rangle_0\\ &+{{\varepsilon}}^{ijk}\langle \lambda,\tfrac12\mu_2(Y_i,X_{jk})+\tfrac16\mu_3(Y_i,Y_j,Y_k)\rangle_0-{\mathrm{i}}\hbar\langle \lambda,{\mathbbm{1}}\rangle_0~.
\end{aligned}$$ A solution to the corresponding equations of motion is given by elements of the semistrict Lie 2-algebra $\hat{\Pi}_{{\mathbbm{R}}^3,\varpi}$ arising from the quantization of the Lie 2-algebra $\Pi_{{\mathbbm{R}}^3,\varpi}$: $$\label{eq:sol_BF1}
Y_i=\hat{\xi}_i~,~~~X_{ij}=0{{\qquad\mbox{and}\qquad}}\lambda_i=\lambda=0~.$$ We now observe that $${\mathrm{d}}x^i\wedge\pi_2(\xi_i,\alpha)={\mathrm{d}}\alpha~,$$ and therefore $\mu_2(\widehat{\xi_i},{-})$ should be identified with a quantum derivation, at least on 1-forms. Consider the background field expansion $$Y_i=\hat{\xi}_i+\hat{A}_i~,~~~X_{ij}=0+\hat{B}_{ij}~,$$ where $\hat{A}_i$ and $\hat{B}_{ij}$ take values in the obvious vector spaces contained in $\hat{\Pi}_{{\mathbbm{R}}^3,\varpi}$. The action becomes $$\label{eq:2LBF_action}
S_{\rm 2LBF}={{\varepsilon}}^{ijk}\langle \lambda_i,\hat{{\mathcal{F}}}_{jk}\rangle_0 + {{\varepsilon}}^{ijk}\langle \lambda,\hat{H}_{ijk}\rangle_0~,$$ where we defined $$\begin{aligned}
\hat{{\mathcal{F}}}_{ij}&=\mu_2(\hat{\xi}_i,\hat{A}_j)-\mu_2(\hat{\xi}_j,\hat{A}_i)+\mu_2(\hat{A}_i,\hat{A}_j)-\mu_1(\hat{B}_{ij})~,\\
\hat{H}_{ijk}&=\tfrac12\mu_2(\hat{\xi}_{[i},\hat{B}_{jk]})+\tfrac12\mu_2(\hat{A}_{[i},\hat{B}_{jk]})+\tfrac16\mu_3(\hat{\xi}_i+\hat{A}_i,\hat{\xi}_j+\hat{A}_j,\hat{\xi}_k+\hat{A}_k)-{\mathrm{i}}\hbar\hat{{\mathbbm{1}}}~.
\end{aligned}$$ It is interesting to note how the Lie 2-algebra $\Pi_{{\mathbbm{R}}^3,\varpi}$ turned into a gauge Lie 2-algebra of higher BF-theory on a quantized 2-plectic space.
Recall that in the Lie 2-algebra arising from 2-plectic ${\mathbbm{R}}^3$, the higher product between functions and 1-forms, $\pi_2:\Omega^1({\mathbbm{R}}^3)\times {\mathcal{C}}^\infty({\mathbbm{R}}^3)\rightarrow {\mathcal{C}}^\infty({\mathbbm{R}}^3)$, is trivial. We therefore have in the quantized case $$\hat{H}_{ijk}=\tfrac16\mu_3(\hat{\xi}_i+\hat{A}_i,\hat{\xi}_j+\hat{A}_j,\hat{\xi}_k+\hat{A}_k)+{\mathcal{O}}(\hbar^2)~.$$ This interpretation is very close to the one used in [@Chu:2011yd]. There, the 3-form curvature $H$ was identified in a 3-Lie algebra valued model with the product $[\hat{A}_i,\hat{A}_j,\hat{A}_k]$. Considering the 3-Lie algebra $2{\mathsf{Mat}}(n)$ constructed in section \[ssec:M2-brane\_3\_algebras\], where the triple bracket of the 3-Lie algebra can be identified with the higher product $\mu_3$, our $\hat{H}$ essentially matches that of [@Chu:2011yd].
Background expansion using an isomorphic Lie 2-algebra structure
----------------------------------------------------------------
As we saw in the previous section, the Lie 2-algebra ${\mathcal{C}}^\infty(M)\rightarrow {\mathfrak{H}}(M)$ that we obtained from a 2-plectic manifold $M$ in section \[ssec:2Plectic\_Manifolds\] is very restrictive. In particular, elements of $V$ have no possibility of interacting via higher products with $W$. To remedy this, note that we can add the non-trivial product $$\label{eq:mod_product_1}
\pi_2(f,\alpha):=-\iota_{X_\alpha}{\mathrm{d}}f~,~~~f\in {\mathcal{C}}^\infty(M)~,~\alpha\in {\mathfrak{H}}(M)~.$$ This additional product, however, violates the higher homotopy relation $\pi_1(\pi_2(\alpha,f))=\pi_2(\alpha,\pi_1(f))$. To fix this, we modify $\pi_2:{\mathfrak{H}}(M)\times {\mathfrak{H}}(M)\rightarrow {\mathfrak{H}}(M)$ as follows: $$\label{eq:mod_product_2}
\pi_2(\alpha,\beta):=-\iota_{X_\alpha}\iota_{X_\beta}\varpi-{\mathrm{d}}(\iota_{X_\alpha}\beta)+{\mathrm{d}}(\iota_{X_\beta}\alpha)~,~~~\alpha,\beta\in{\mathfrak{H}}(M)~,$$ where we note that $\pi_2(\alpha,\beta)$ is indeed in ${\mathfrak{H}}(M)$. The products $\pi_1$ and $\pi_3$ remain unmodified: $$\pi_1(f):={\mathrm{d}}f{{\qquad\mbox{and}\qquad}}\pi_3(\alpha,\beta,\gamma):=-\iota_{X_\alpha}\iota_{X_\beta}\iota_{X_\gamma}\varpi$$ for $f\in {\mathcal{C}}^\infty(M)$ and $\alpha,\beta,\gamma\in\Omega^1(M)$. We will denote the resulting structure by $\tilde{\Pi}_{M,\varpi}$.
Instead of verifying all the higher homotopy relations for $\tilde{\Pi}_{M,\varpi}$, we can prove a stronger statement: $\tilde{\Pi}_{M,\varpi}$ is isomorphic to the Lie 2-algebras $\Pi_{M,\varpi}$. This is easily seen by giving the explicit Lie 2-algebra homomorphism, cf. section \[ssec:Lie\_2\_algebra\_homomorphism\]: $$\Psi_{-1}={\mathrm{id}}~,~~~\Psi_0={\mathrm{id}}~,~~~\Psi_2(\alpha,\beta)=\iota_{X_\alpha} \beta-\iota_{X_\beta} \alpha~.$$ Equations then yield the higher products and . The higher product $\mu_3$ remains unmodified, as one readily verifies by direct computation using the identity $\iota_{[X_\alpha,X_\beta]}={\mathcal{L}}_{X_\alpha}\iota_{X_\beta}-\iota_{X_\beta}{\mathcal{L}}_{X_\alpha}$.
As an example, let us briefly study the Lie 2-algebra $\tilde{\Pi}_{{\mathbbm{R}}^3,\varpi}$ with 2-plectic form $\varpi={\mathrm{d}}x^1\wedge {\mathrm{d}}x^2\wedge {\mathrm{d}}x^3$. The Hamiltonian vector fields as well as $\pi_1$ and $\pi_3$ are listed in section \[ssec:2-plectic\_examples\]. The formulas for the new products read as $$\begin{aligned}
\pi_2(f,\alpha)&=-{{\varepsilon}}^{ijk}{\partial}_i f {\partial}_j\alpha_k~,\\
\pi_2(\alpha,\beta)&={{\varepsilon}}^{ijk}\left({\partial}_i\alpha_k({\partial}_j\beta_\ell-{\partial}_\ell\beta_j)+{\partial}_\ell(\alpha_i{\partial}_j\beta_k-\beta_i{\partial}_j\alpha_k)\right){\mathrm{d}}x^\ell~.
\end{aligned}$$ For the constant and linear 1-forms ${\mathrm{d}}x^i$ and $\xi_i=\tfrac{1}{2}{{\varepsilon}}_{ijk}x^j{\mathrm{d}}x^k$ we have $$\pi_2(f,{\mathrm{d}}x^i)=0~,~~~\pi_2(f,\xi_i)=-{\partial}_i f~,~~~\pi_2({\mathrm{d}}x^i,\alpha)={{\varepsilon}}^{ijk}{\partial}_\ell {\partial}_j x_k {\mathrm{d}}x^\ell~,~~~\pi_2(\xi_i,\xi_j)=0~.$$ In particular, we see that the operator $\pi_2(\xi_i,f)$ can here be interpreted as a derivation on functions, while it lost its nice derivation property property on 1-forms.
To define a BF-theory via a background expansion using the Lie 2-algebra $\tilde{\Pi}_{{\mathbbm{R}}^3,\varpi}$, we should consider the action without the twist term $2{\mathrm{i}}\hbar\langle \lambda_i,\widehat{{\mathrm{d}}x^i}\rangle_0$. A classical configuration of this action is again and we can follow the discussion of the previous section.
Conclusions
===========
In this paper, we initiated a study of zero-dimensional field theories, in which the fields take values in a semistrict Lie 2-algebra, or, equivalently, a 2-term $L_\infty$-algebra. These Lie 2-algebra models are a categorification of the IKKT matrix model, which is conjectured to provide a background independent formulation of string theory. In particular, Lie 2-algebra models contain the (bosonic part of the) IKKT model and all of its bosonic deformations.
We explored the various notions of inner products on Lie 2-algebras as well as the resulting structure of transposed products. Here, we made an observation concerning the connection between 3-algebras appearing in M2-brane models and categorified Lie algebras. Besides the established link between Lie 2-algebras and the 3-algebras of M2-brane models via differential crossed modules, we also showed that any skeletal Lie 2-algebra with inner product comes with a 3-algebra structure. Moreover, there is a class of reduced Lie 2-algebras, in which the higher product $\mu_3$ can be identified with the 3-bracket of a 3-Lie algebra.
We also pointed out the interaction of inner Lie 2-algebra homomorphisms with the various inner products. This allowed us to examine the symmetries of Lie 2-algebra models, which are compatible with those expected from a dimensional reduction of semistrict higher gauge theory. This is to be compared with the IKKT model, where the symmetry algebra arises from a dimensional reduction of the gauge theory.
We divided the Lie 2-algebra models into two classes: homogeneous Lie 2-algebra models are defined in terms of a single class of fields that take values in the Lie 2-algebra. Inhomogeneous Lie 2-algebra models feature two types of fields, each living within one of the graded vector subspaces of the 2-term $L_\infty$-algebra underlying the model.
Just as in the case of the IKKT model, where solutions to the classical equations of motion can be identified with quantized symplectic manifolds, the homogeneous Lie 2-algebra models we studied have solutions that can be interpreted as quantized 2-plectic manifolds. While the quantization of 2-plectic manifolds is still not fully understood, it was straightforward to outline the expected features of such a quantization that are required for our purposes. In particular, it is expected that under quantization the Lie 2-algebra induced by the 2-plectic structure on a manifold is mapped to a Lie 2-algebra of quantum observables with this map being a Lie 2-algebra homomorphism to lowest order in a deformation parameter $\hbar$. As an example, we examined the Heisenberg Lie 2-algebra, which is contained in the Lie 2-algebra arising from the quantization of ${\mathbbm{R}}^3$. We gave a representation in terms of derived brackets on a 2-vector space.
The quantized symplectic manifolds most readily obtained as solutions in the IKKT model are the Moyal plane and the fuzzy sphere, as well as their Cartesian products. In the Lie 2-algebra models, we found solutions that correspond to the quantizations of ${\mathbbm{R}}^3$ and $S^3$, where the 2-plectic form was given by the canonical volume form on these spaces. Remarkably, these solutions appeared in complete analogy with the above mentioned solutions of the IKKT model.
We also studied solutions given by Lie 2-algebras that do not arise from 2-plectic manifolds. In particular, we considered the Nappi-Witten algebra, which can be regarded as linear Poisson structure on a four-dimensional Hpp-wave. This algebra gives a solution of a particular deformation of the IKKT model. We constructed a categorified analogue of the Nappi-Witten algebra corresponding to a five-dimensional Hpp-wave and again we found that it appears as a solution of our Lie 2-algebra models.
We were able to show that certain inhomogeneous Lie 2-algebra models reproduce previously considered zero-dimensional field theories that are related to M2-brane models. Furthermore, we considered Lie 2-algebra models that arise from a dimensional reduction of a semistrict higher BF-theory in three dimensions. These models contained again a quantization of ${\mathbbm{R}}^3$ as a classical configuration, and expanding around this configuration, we obtained an action that can be interpreted as semistrict higher BF-theory on the quantized ${\mathbbm{R}}^3$. This is fully analogous to the case of the IKKT model, where it is known that expanding around a solution corresponding to a quantized symplectic manifold yields the action of Yang-Mills theory on this noncommutative space. Finally, we considered a Lie 2-algebra that is isomorphic to that obtained from the quantization of ${\mathbbm{R}}^3$, to demonstrate what is to be expected for a more general categorified correspondence principle.
Altogether, we conclude that Lie 2-algebra models are generalizations of the IKKT model that contain various other zero-dimensional models that were proposed in the context of M2-brane models. Moreover, many of the nice features of the IKKT model carry over to these Lie 2-algebra models.
One of our original motivations for studying Lie 2-algebra models was to explore the possibility of supersymmetric such models. This is particularly interesting, as there is more and more evidence that M2- and M5-brane models should be based on semistrict Lie 2-algebras, see e.g. [@Palmer:2012ya; @Palmer:2013pka]. The reason for focusing on the zero-dimensional case instead of the three- and six-dimensional cases is that here the gauge structure severely simplifies. For a brief overview over the complications encountered in the higher dimensional case, see appendix \[app:B\]. The construction of supersymmetric Lie 2-algebra models corresponding to a dimensional reduction of six-dimensional superconformal models is clearly an issue that we plan to attack in future work. Moreover, recall that the IKKT model was connected to type IIB superstring theory via a Schwinger-Dyson equation for the Wilson loops [@Fukuma:1997en]. It would be very interesting to study the corresponding equations for Wilson surfaces in our Lie 2-algebra models.
Further open questions arising from our work concern a potential use of Lie 2-algebras in the regularization of Nambu-Poisson sigma-models as well as the development of our naïve notion of quantization of 2-plectic manifolds to a full quantization. The latter problem would imply to extend our Lie 2-algebras to Poisson 2-algebras or, equivalently, Gerstenhaber algebras, in which also a categorified associative product between observables is realized.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Cedric Troessaert for discussions and the anonymous referee for an extraordinarily helpful report. PR was supported by Fondecyt grant \#3120077. CS was supported by an EPSRC Career Acceleration Fellowship. The Centro de Estudios Cientfícos (CECS) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.
[startsection [section]{}[1]{}[-3.5ex plus -1ex minus -.2ex]{}[2.3ex plus .2ex]{}[****]{}]{}\*[Appendix]{}\[appendices\]
addtoreset[equation]{}[subsection]{}
Useful definitions {#app:A}
------------------
#### Strong homotopy Lie algebras.
An [*$L_\infty$-algebra*]{} or [*strong homotopy Lie algebra*]{} is a graded vector space $L=\oplus_i L_i$ endowed with $n$-ary multilinear totally antisymmetric products $\mu_n$, $n\in{\mathbbm{N}}^*$, of degree $2-n$, that satisfy homotopy Jacobi identities, cf. [@Lada:1992wc; @Lada:1994mn; @0821843621]. These identities read as $$\label{eq:homotopyJacobi}
\sum_{i+j=n}\sum_\sigma\chi(\sigma;x_1,\ldots,x_n)(-1)^{i\cdot j}\mu_{j+1}(\mu_i(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(i+j)})=0$$ for all $n\in {\mathbbm{N}}^*$, where the sum over $\sigma$ is taken over all $(i,j)$ unshuffles. Recall that a permutation $\sigma$ of $i+j$ elements is called an [*$(i,j)$-unshuffle*]{}, if the first $i$ and the last $j$ images of $\sigma$ are ordered: $\sigma(1)<\cdots<\sigma(i)$ and $\sigma(i+1)<\cdots<\sigma(i+j)$. Moreover, the [*graded Koszul sign*]{} $\chi(\sigma;x_1,\cdots,x_n)$, $x_i\in L$ is defined via the equation $$x_1\wedge \cdots \wedge x_n=\chi(\sigma;x_1,\cdots,x_n)\,x_{\sigma(1)}\wedge \cdots \wedge x_{\sigma(n)}$$ in the free graded algebra $\wedge (x_1,\cdots,x_n)$, where $\wedge$ is considered graded antisymmetric.
Note that for elements of $L$ which do not have a definite grading, the above relations have to be resolved to elements of $L$ with homogeneous grading, using linearity of the maps. Note also that we shall denote the grading of an object $x$ by ${\tilde{x}}\in {\mathbbm{Z}}$. For example, we have ${\tilde{x}}=i$ for $x\in L_i$.
Strong homotopy Lie algebras that are concentrated in degrees $-n+1,\ldots,0$, i.e. $L_i=\varnothing$ for $i\notin [-n+1,\ldots,0]$, are categorically equivalent to semistrict Lie $n$-algebras.
#### Nambu-Poisson structures.
A [*Nambu-Poisson structure*]{} [@Nambu:1973qe; @Takhtajan:1993vr] on a smooth manifold ${\mathcal{M}}$ is an $n$-ary, totally antisymmetric multilinear map $\{{-},\dots ,{-}\}:{\mathcal{C}}^\infty({\mathcal{M}})^{\wedge n}\rightarrow{\mathcal{C}}^\infty({\mathcal{M}})$, which satisfies the [*generalized Leibniz rule*]{} $$\{f_1 \,f_2,f_3,\dots ,f_{n+1}\}=f_1\,\{f_2,\dots
,f_{n+1}\}+\{f_1,\dots ,f_{n+1}\} \,f_2$$ as well as the [*fundamental identity*]{} $$\{f_1,\dots ,f_{n-1},\{g_1,\dots ,g_n\}\}=\{\{f_1,\dots ,f_{n-1},g_1\},\dots ,g_n\}+\dots +\{g_1,\dots ,\{f_1,\dots ,f_{n-1},g_n\}\}$$ for all $f_i,g_i\in{\mathcal{C}}^\infty({\mathcal{M}})$. A manifold $M$ endowed with such a [*Nambu $n$-bracket*]{} giving rise to a [*Nambu-Poisson algebra*]{} is called a [*Nambu-Poisson manifold*]{}. Under certain conditions, 2-plectic structures give rise to ternary Nambu-Poisson structures [@springerlink:10.1007/BF00400143].
#### $n$-Lie algebras.
An $n$-Lie algebra[^16] [@Filippov:1985aa] is a vector space ${\mathcal{A}}$ endowed with an $n$-ary, totally antisymmetric and multilinear map $[{-},\cdots,{-}]:{\mathcal{A}}^{\wedge n}\rightarrow {\mathcal{A}}$ that satisfies the [*fundamental identity*]{}: $$\label{eq:FI_algebra}
[a_1,\dots ,a_{n-1},[b_1,\dots ,b_n]]=[[a_1,\dots ,a_{n-1},b_1],\dots ,b_n]+\dots +[b_1,\dots ,[a_1,\dots ,a_{n-1},b_n]]$$ for all $a_i,b_i\in{\mathcal{A}}$. Note that Nambu-Poisson algebras are particular $n$-Lie algebras, and it has been proposed that Nambu-Poisson structures should be quantized in terms of $n$-Lie algebras, cf. [@DeBellis:2010pf] and references therein.
Note that $n$-Lie algebras come with a Lie algebra of inner derivations, which are given by linear combinations of the maps $$D(a_1,\ldots,a_{n-1}){\vartriangleright}x:=[a_1,\ldots,a_{n-1},x]~,$$ where $a_i,x\in {\mathcal{A}}$. The commutator of inner derivations closes on inner derivations because of the fundamental identity .
We can endow an $n$-Lie algebra with a metric, which has to be invariant under the action of inner derivations. In the case of a 3-Lie algebra, this metric induces a metric on the vector space of inner derivations, which is in general indefinite and different from the Killing form.
The first of the recently studied M2-brane models, the Bagger-Lambert-Gustavsson (BLG) model [@Bagger:2007jr; @Gustavsson:2007vu], has a gauge structure that is based on a 3-Lie algebra. This 3-Lie algebra comes with a positive definite metric on the vector space forming the 3-Lie algebra, which induces a metric of split signature on the inner derivations.
#### Generalized 3-algebras.
Because there is essentially only one finite-dimensional 3-Lie algebras with positive definite invariant metric, various generalizations have been proposed. First, there are the hermitian 3-algebras that are based on a complex vector space and that underlie the ABJM M2-brane model [@Aharony:2008ug; @Bagger:2008se]. Second, there are the real 3-algebras, which are relaxed versions of 3-Lie algebras in that their 3-bracket is antisymmetric only in the first two slots [@Cherkis:2008qr]. Both types of 3-algebras can be encoded in terms of Lie algebras and certain representations [@deMedeiros:2008zh], and therefore they form differential crossed modules [@Palmer:2012ya]. Also, as shown in the text, skeletal Lie 2-algebras with inner product come naturally with a generalized 3-algebra structure.
Gauge symmetry in semistrict higher gauge theory {#app:B}
------------------------------------------------
While semistrict higher gauge theory has only been developed partially, an attempt to capture its local gauge structure has been made in [@Zucchini:2011aa]. Below, we will give a rough, quick review of this construction. This serves two purposes. First, we can easily show that it reduces to the Lie 2-algebra homomorphisms describing the symmetries of our Lie 2-algebra models. Second, it demonstrates that it is considerably simpler to study Lie 2-algebra models than to study actual semistrict higher gauge theories.
Let us group the fields we are interested in working with into doublets $(\phi,\Phi)\in\Omega^p(M,W)\times\Omega^{p+1}(M, V)$, where $M$ indicates the manifold they live on and small and capital letters will always indicate $V$- or $W$-valued fields respectively. We will refer to the *degree* of the doublet as the order of the $W$-valued form, in this case $p$. Let us indicate a *connection doublet* by $(a,A)\in\Omega^1(M,W)\times\Omega^2(M,V)$. We can define the curvature of these fields as the doublet $(f,F)\in\Omega^2(M,W)\times\Omega^3(M,V)$: $$\begin{aligned}
\label{eq:29}
f=&{\mathrm{d}}a+\tfrac{1}{2}\mu_2(a,a)-\mu_1(A)~,\\
F=&{\mathrm{d}}A+\mu_2(a,A)-\tfrac{1}{6}\mu_3(a,a,a)~.\end{aligned}$$ The $(f,F)$ doublet can be easily seen to satisfy the *Bianchi identities* $$\begin{aligned}
\label{eq:30}
{\mathrm{d}}f+\mu_2(a,f)+\mu_1(F)=&0~,\\
{\mathrm{d}}F+\mu_2(a,F)-\mu_2(f,A)+\tfrac{1}{2}\mu_3(a,a,f)=&0~.\end{aligned}$$ In analogy to ordinary gauge theory, one would like the Bianchi identities to be given by the requirement that $Df=0=DF$, where $D$ is the *covariant derivative* with respect to the same connection $(a,A)$. This requirement allows one to define the action of $D$ on a generic field doublet $(\phi,\Phi)$ of order $p$ as $$\begin{aligned}
D\phi&={\mathrm{d}}\phi+\mu_2(a,\phi)+(-1)^p\mu_1(\Phi)~,\label{eq:31}\\
D\Phi&={\mathrm{d}}\Phi+\mu_2(a,\Phi)-(-1)^p\mu_2(\phi,A)+\frac{(-1)^p}{2}\mu_3(a,a,\phi)\label{eq:32}~,\end{aligned}$$ forming the $(p+1)$-degree doublet $(D\phi,D\Phi)$. The next step is to define gauge transformations in the semistrict Lie 2-algebra setting. These have to live in the set of automorphisms of the Lie 2-algebra and are expected to satisfy a generalization of the Maurer-Cartan equation ${\mathrm{d}}(g^{-1}{\mathrm{d}}g)+(g^{-1}{\mathrm{d}}g)\wedge(g^{-1}{\mathrm{d}}g)=0$. It is argued in [@Zucchini:2011aa] that the easiest way to generalize traditional gauge theory also makes use of a flat connection doublet $(\sigma,\Sigma)$, which roughly speaking keeps track of how gauge group elements vary with respect to the base manifold coordinates. Overall, the semistrict Lie 2-algebra 1-gauge transformations are defined in [@Zucchini:2011aa] as the following set of ingredients:
a map $g\in\text{Map}(M,{\mathsf{Aut}\,}(L))$, i.e. a set $(g_0,g_{-1},g_2)$ satisfying the requirements elucidated in section \[ssec:Lie\_2\_algebra\_homomorphism\] ;
a flat connection doublet $(\sigma,\Sigma)$:
\[eq:semistict\_gauge\_consistency\] $$\begin{aligned}
\label{eq:33}
{\mathrm{d}}\sigma+\tfrac{1}{2}\mu_2(\sigma,\sigma)-\mu_1(\Sigma)=&0~,\\
{\mathrm{d}}\Sigma +\mu_2(\sigma,\Sigma)-\tfrac{1}{6}\mu_3(\sigma,\sigma,\sigma)=&0~;
\end{aligned}$$
an element $\tau\in\Omega^1(M,{\mathsf{Hom}\,}(W,V))$ satisfying $$\label{eq:35}
{\mathrm{d}}\tau(w)+\mu_2(\sigma,\tau(w))-\mu_2(w,\Sigma)+\tfrac{1}{2}\mu_3(\sigma,\sigma,w)+\tau\left(\mu_2(\sigma,w)+\mu_1(\tau(w))\right)=0~.$$
Note that equation is just the vanishing fake curvature condition as we know it from strict higher gauge theory, while equation is referred to as the *2-Maurer-Cartan equation* in [@Zucchini:2011aa]. Indeed, after introducing a $\tau$-dependent term in the definition of the action of the flat connection, for instance $$\begin{aligned}
\label{eq:36}
g_0^{-1}{\mathrm{d}}g_0(w)-\mu_2(\sigma,w)-\mu_1(\tau(w))=0~\end{aligned}$$ for the $W$ part of the automorphism $g$, one can satisfy the $W$ sector of the Maurer-Cartan equation (and analogously for the $V$-part). Because of the non-vanishing Jacobiator in the semistrict set-up, without $\tau$ this is normally not possible unless one imposes a further condition by hand.\
Apart from equation and its $V$-sector analogue, there is a further condition of compatibility for $\tau$, so that the set of homomorphism rules of section \[ssec:Lie\_2\_algebra\_homomorphism\] are still satisfied for $g$ - the interested reader can find all the details for this construction in [@Zucchini:2011aa].
Now, to see how the above defined gauge transformations act on fields, one requires that, for a given connection doublet $(a,A)$, the covariant derivatives $D$ from and “pull through” all the elements that make up the transformation. That is, if we indicate the full 1-gauge transformation by $(g,\sigma,\Sigma,\tau)$, one requires the derivatives $D$ to act on gauge transformed field doublets $(g{\vartriangleright}\phi,g{\vartriangleright}\Phi)$ as defined in and , but treating all the components $g,\,\sigma,\,\Sigma$ and $\tau$ as covariantly constant. In this way one obtains for connection doublets $(a,A)$ the following action of 1-gauge transformations: $$\begin{aligned}
\label{eq:37}
g{\vartriangleright}a=&g_0(a-\sigma)~,\\
g{\vartriangleright}A=&g_{-1}(A-\Sigma+\tau(a-\sigma))-\tfrac{1}{2}g_2(a-\sigma,a-\sigma)~,\end{aligned}$$ so that its covariant derivative, or curvature doublet $(f,F)\equiv
(Da,DA)$ transforms as $$\begin{aligned}
\label{eq:20}
g{\vartriangleright}f =&g_0(f)~,\\
g{\vartriangleright}F=&g_{-1}(F-\tau(f))+g_2(a-\sigma,f)~.\end{aligned}$$ Similarly one can define *canonical* field doublets $(\phi,\Phi)$, of degree $p$, as those that transform as $$\begin{aligned}
\label{eq:38}
g{\vartriangleright}\phi=&g_0(\phi)~,\\
g{\vartriangleright}\Phi=&g_{-1}(\Phi-(-1)^p\tau(\phi))+(-1)^pg_2(a-\sigma,\phi)~.\end{aligned}$$ Its covariant derivatives then transform as $$\begin{aligned}
\label{eq:61}
g{\vartriangleright}D\phi=&g_0(D\phi)~,\\
g{\vartriangleright}D\Phi=&g_{-1}(D\Phi+(-1)^p\tau(D\phi))-(-1)^pg_2(a-\sigma,D\phi)+(-1)^pg_2(Da,\phi)~.\end{aligned}$$ The obvious thing to notice here is that while in the $W$ sector everything transforms in a “nice” way, that is $\phi\rightarrow
g_0(\phi)$ and $D\phi\rightarrow g_0(D\phi)$, the $V$ sector looks a lot more involved. When constructing actions, these will be based on some inner product that will be invariant under automorphisms of the Lie 2-algebra and therefore under the transformation corresponding to $(g_0,g_{-1})$. In this sense it would be very easy to identify and construct gauge invariant actions if covariant derivatives and curvatures transformed as $D\Phi\rightarrow g_{-1}(D\Phi)$ also in the $V$ sector of actions. Interestingly this can be achieved: the Lie 2-algebra can be *gauge rectified*[^17] by a pair of fields $(\lambda, \rho)$, where $\lambda\in\Omega^0(M,{\mathsf{Hom}\,}(W\wedge
W,V))$ and $\rho\in\Omega^1(M,{\mathsf{Hom}\,}(W,V))$, which have special gauge transformation properties. The products $\mu_2$ and $\mu_3$ can then be corrected by $\lambda$ so that the rectified products $\mu_i^{(\lambda)}$ will transform as $g{\vartriangleright}\mu_i^{(\lambda)}(\ldots)=g_\alpha(\mu_i^{(\lambda)}(\ldots))$, where $\alpha=0,-1$ according to where $\mu_i$ maps to. Similarly field doublets can be rectified, as well as the definition of the covariant derivative, resulting in all fields and covariant derivatives thereof transforming simply by $g_0$ or $g_{-1}$ actions on the objects themselves (e.g. $D^{(\lambda,\rho)}\Phi^{(\lambda,\rho)}\rightarrow g_{-1}(\mu_i^{(\lambda)}(\ldots))$). This means that for actions constructed via a $g_0,\,g_{-1}$ invariant inner product, any terms involving rectified canonical field doublets, covariant derivatives thereof and Lie 2-algebra products $\mu^{(\lambda)}_i$ will be automatically gauge invariant. Moreover, the rectified products $\mu_i^{(\lambda)}(\ldots)$ still form a Lie 2-algebra. To see the details of this procedure we again refer to [@Zucchini:2011aa].
Returning to the gauge transformation setup, upon reduction to zero dimensions all the total derivatives disappear and therefore the auxiliary $\sigma,\,\Sigma$ and $\tau$ can all be set to zero. Also, we cannot talk about field doublets anymore, since all objects we will be considering are of order 0, whether they are valued in $W$ or in $V$. This simplifies matters considerably, as we can now say that the gauge transformations of $w\in W$ and $v\in V$ are given by $$\label{eq:69}
g{\vartriangleright}w=g_0(w) \qquad \text{and}\qquad g{\vartriangleright}v=g_{-1}(v)~,$$ while covariant derivation reduces to $$\label{eq:70}
D w=\mu_2(a,w) \qquad\text{and}\qquad D v=\mu_2(a,v)~,$$ for a 0-degree field $a\in W$. We set $(g_0,g_{-1},g_2)$ as in \[ssec:actions\_and\_symmetries\] to: $$\label{eq:71}
g_0(w):=w+\mu_2(\epsilon,w)~,\quad g_{-1}(v):=v+\mu_2(\epsilon,v)~, \quad g_2(w_1,w_2):=\mu_3(\epsilon,w_1,w_2)~,$$ to first order in the gauge parameter $\epsilon\in W$. It then follows from the homomorphism rules that covariant derivatives transform in the desired way: $$\begin{aligned}
\label{eq:72}
g{\vartriangleright}D w=&g_0(Dw)=\mu_2(g{\vartriangleright}a, g{\vartriangleright}w)=\mu_2(g_0(a),g_0(w))~,\\
g{\vartriangleright}D v=&g_1(Dv)=\mu_2(g{\vartriangleright}a, g{\vartriangleright}v)=\mu_2(g_0(a),g_{-1}(v))~,\end{aligned}$$ as expected. Indeed, the homomorphism rules themselves guarantee that all the 2-algebra products on $w,\,v$ also transform simply by an overall $g_\alpha$, that is $\mu_i(\ldots)\rightarrow g_\alpha\mu_i(\ldots)$, for $\alpha=0,-1$ according to the grading of $\mu_i$. In other words, for zero-dimensional reduced actions, if based on a $\mu_2(w,{-})$ invariant inner product, any terms involving the 2-algebra structures are automatically gauge invariant, without the need to introduce any rectifiers.
[^1]: In this paper, we will use the terms “Lie 2-algebra” and “2-vector spaces” rather freely. Unless stated otherwise, we will use them to refer to 2-term strong homotopy Lie algebras and 2-term chain complexes of vector spaces, respectively.
[^2]: Full supersymmetry, however, seems to be possible only for four scalar fields with a metric of split signature [@Furuuchi:2009ax].
[^3]: To simplify the notation for inner products later on, we restrict ourselves to real vector spaces.
[^4]: Here and in the following, $*$ denotes the trivial Lie algebra $\{0\}$.
[^5]: This definition of a scalar product extends to other $\infty$-algebras. Moreover, it corresponds to the notion of a binary invariant polynomial of the $L_\infty$-algebra.
[^6]: See appendix \[app:A\] for the relevant definitions.
[^7]: In this paper, we will use a very rough notion of quantization that is sufficient for our considerations. For a more detailed discussion, see e.g. [@DeBellis:2010pf] and references therein.
[^8]: See appendix \[app:A\] for a definition and more details.
[^9]: i.e. linear with respect to translations on ${\mathbbm{R}}^3$
[^10]: i.e. it maps closed/exact forms to closed/exact forms
[^11]: In the cases that we are interested in, the quotient space turns out to be a smooth manifold.
[^12]: While only totally antisymmetric parts of $c_{abc}$ contribute to $S_0$, this is not the case for $d_{abcd}$.
[^13]: According to the Whitney embedding theorem, any smooth manifold of dimension $d$ can be smoothly embedded in ${\mathbbm{R}}^{2d}$. This restricts the dimension of the quantized symplectic manifolds that can arise as solutions in the IKKT model. In fact, the Whitney embedding theorem can be improved to ${\mathbbm{R}}^{2d-1}$ unless $d$ is a power of $2$.
[^14]: We were not able to use the additional terms in the action in any sensible way to accommodate the desired solutions of quantized geometries; neither did they seem necessary.
[^15]: which is the case e.g. if the underlying real 3-algebra is $A_4$
[^16]: which is not to be confused with a Lie $n$-algebra arising in the categorification of Lie algebras
[^17]: It has not been shown whether a pair $(\lambda,\rho)$ of gauge rectifiers can always be found, for any Lie 2-algebra.
|
---
abstract: 'In this paper we compare the performance characteristics of our selection based learning algorithm for Web crawlers with the characteristics of the reinforcement learning algorithm. The task of the crawlers is to find new information on the Web. The selection algorithm, called weblog update, modifies the starting URL lists of our crawlers based on the found URLs containing new information. The reinforcement learning algorithm modifies the URL orderings of the crawlers based on the received reinforcements for submitted documents. We performed simulations based on data collected from the Web. The collected portion of the Web is typical and exhibits scale-free small world (SFSW) structure. We have found that on this SFSW, the weblog update algorithm performs better than the reinforcement learning algorithm. It finds the new information faster than the reinforcement learning algorithm and has better new information/all submitted documents ratio. We believe that the advantages of the selection algorithm over reinforcement learning algorithm is due to the small world property of the Web.'
author:
- 'Zs. Palotai$^1$, Cs. Farkas$^2$, A. L[ő]{}rincz$^1$[^1]'
title: 'Selection in Scale-Free Small World'
---
Introduction {#s:intro}
============
The largest source of information today is the World Wide Web. The estimated number of documents nears 10 billion. Similarly, the number of documents changing on a daily basis is also enormous. The ever-increasing growth of the Web presents a considerable challenge in finding novel information on the Web.
In addition, properties of the Web, like scale-free small world (SFSW) structure [@barabasi00scalefree; @Kleinberg01structure] may create additional challenges. For example the direct consequence of the scale-free small world property is that there are numerous URLs or sets of interlinked URLs, which have a large number of incoming links. Intelligent web crawlers can be easily trapped at the neighborhood of such junctions as it has been shown previously [@kokai02learning; @Lorincz02intelligent].
We have developed a novel artificial life (A-life) method with intelligent individuals, crawlers, to detect new information on a news Web site. We define A-life as a population of individuals having both static structural properties, and structural properties which may undergo continuous changes, i.e., adaptation. Our algorithms are based on methods developed for different areas of artificial intelligence, such as evolutionary computing, artificial neural networks and reinforcement learning. All efforts were made to keep the applied algorithms as simple as possible subject to the constraints of the internet search.
Evolutionary computing deals with properties that may be modified during the creation of new individuals, called ’multiplication’. Descendants may exhibit variations of population, and differ in performance from the others. Individuals may also terminate. Multiplication and selection is subject to the fitness of individuals, where fitness is typically defined by the modeler. For a recent review on evolutionary computing, see [@eiben03introduction]. For reviews on related evolutionary theories and the dynamics of self-modifying systems see [@Fryxell98individual; @Clarck00dynamic] and [@kampis91selfmodifying; @Csanyi89evolutionary], respectively. Similar concepts have been studied in other evolutionary systems where organisms compete for space and resources and cooperate through direct interaction (see, e.g., [@taylor02mutualism] and references therein.)
Selection, however, is a very slow process and individual adaptation may be necessary in environments subject to quick changes. The typical form of adaptive learning is the connectionist architecture, such as artificial neural networks. Multilayer perceptrons (MLPs), which are universal function approximators have been used widely in diverse applications. Evolutionary selection of adapting MLPs has been in the focus of extensive research [@yao93review; @yao99evolving].
In a typical reinforcement learning (RL) problem the learning process [@Sutton98Reinforcement] is motivated by the expected value of long-term cumulated profit. A well-known example of reinforcement learning is the TD-Gammon program of Tesauro [@tesauro95temporal]. The author applied MLP function approximators for value estimation. Reinforcement learning has also been used in concurrent multi-robot learning, where robots had to learn to forage together via direct interaction [@mataric97reinforcement]. Evolutionary learning has been used within the framework of reinforcement learning to improve decision making, i.e., the state-action mapping called policy [@stafylopatis98autonomous; @moriarty99evolutionary; @tuyls03extended; @kondo04reinforcement].
In this paper we present a selection based algorithm and compare it to the well-known reinforcement learning algorithm in terms of their efficiency and behavior. In our problem, fitness is not determined by us, but fitness is implicit. Fitness is jointly determined by the ever changing external world and by the competing individuals together. Selection and multiplication of individuals are based on their fitness value. Communication and competition among our crawlers are indirect. Only the first submitter of a document may receive positive reinforcement. Our work is different from other studies using combinations of genetic, evolutionary, function approximation, and reinforcement learning algorithms, in that i) it does not require explicit fitness function, ii) we do not have control over the environment, iii) collaborating individuals use value estimation under ‘evolutionary pressure’, and iv) individuals work without direct interaction with each other.
We performed realistic simulations based on data collected during an 18 days long crawl on the Web. We have found that our selection based weblog update algorithm performs better in scale-free small world environment than the RL algorithm, eventhough the reinforcement learning algorithm has been shown to be efficient in finding relevant information [@Lorincz02intelligent; @rennie99using]. We explain our results based on the different behaviors of the algorithms. That is, the weblog update algorithm finds the good relevant document sources and remains at these regions until better places are found by chance. Individuals using this selection algorithm are able to quickly collect the new relevant documents from the already known places because they monitor these places continuously. The reinforcement learning algorithm explores new territories for relevant documents and if it finds a good place then it collects the existing relevant documents from there. The continuous exploration of RL causes that it finds relevant documents slower than the weblog update algorithm. Also, crawlers using weblog update algorithm submit more different documents than crawlers using the RL algorithm. Therefore there are more relevant new information among documents submitted by former than latter crawlers.
The paper is organized as follows. In Section \[s:related\] we review recent works in the field of Web crawling. Then we describe our algorithms and the forager architecture in Section \[s:architecture\]. After that in Section \[s:experiments\] we present our experiment on the Web and the conducted simulations with the results. In Section \[s:discussion\] we discuss our results on the found different behaviors of the selection and reinforcement learning algorithms. Section \[s:conclusion\] concludes our paper.
Related work {#s:related}
============
Our work concerns a realistic Web environment and search algorithms over this environment. We compare selective/evolutionary and reinforcement learning methods. It seems to us that such studies should be conducted in ever changing, buzzling, wabbling environments, which justifies our choice of the environment. We shall review several of the known search tools including those [@kokai02learning; @Lorincz02intelligent] that our work is based upon. Readers familiar with search tools utilized on the Web may wish to skip this section.
There are three main problems that have been studied in the context of crawlers. Rungsawang et al. [@rungsawang04learnable] and references therein and Menczer [@menczer03complementing] studied the topic specific crawlers. Risvik et al. [@risvik02search] and references therein address research issues related to the exponential growth of the Web. Cho and Gracia-Molina [@cho03effective], Menczer [@menczer03complementing] and Edwards et. al [@edwards01adaptive] and references therein studies the problem of different refresh rates of URLs (possibly as high as hourly or as low as yearly).
Rungsawang and Angkawattanawit [@rungsawang04learnable] provide an introduction to and a broad overview of topic specific crawlers (see citations in the paper). They propose to learn starting URLs, topic keywords and URL ordering through consecutive crawling attempts. They show that the learning of starting URLs and the use of consecutive crawling attempts can increase the efficiency of the crawlers. The used heuristic is similar to the weblog algorithm [@Gabor04value], which also finds good starting URLs and periodically restarts the crawling from the newly learned ones. The main limitation of this work is that it is incapable of addressing the freshness (i.e., modification) of already visited Web pages.
Menczer [@menczer03complementing] describes some disadvantages of current Web search engines on the dynamic Web, e.g., the low ratio of fresh or relevant documents. He proposes to complement the search engines with intelligent crawlers, or web mining agents to overcome those disadvantages. Search engines take static snapshots of the Web with relatively large time intervals between two snapshots. Intelligent web mining agents are different: they can find online the required recent information and may evolve intelligent behavior by exploiting the Web linkage and textual information.
He introduces the InfoSpider architecture that uses genetic algorithm and reinforcement learning, also describes the MySpider implementation of it. Menczer discusses the difficulties of evaluating online query driven crawler agents. The main problem is that the whole set of relevant documents for any given query are unknown, only a subset of the relevant documents may be known. To solve this problem he introduces two new metrics that estimate the real recall and precision based on an available subset of the relevant documents. With these metrics search engine and online crawler performances can be compared. Starting the MySpider agent from the 100 top pages of AltaVista the agent’s precision is better than AltaVista’s precision even during the first few steps of the agent.
The fact that the MySpider agent finds relevant pages in the first few steps may make it deployable on users’ computers. Some problems may arise from this kind of agent usage. First of all there are security issues, like which files or information sources are allowed to read and write for the agent. The run time of the agents should be controlled carefully because there can be many users (Google answered more than 100 million searches per day in January-February 2001) using these agents, thus creating huge traffic overhead on the Internet.
Our weblog algorithm uses local selection for finding good starting URLs for searches, thus not depending on any search engines. Dependence on a search engine can be a suffer limitation of most existing search agents, like MySpiders. Note however, that it is an easy matter to combine the present algorithm with URLs offered by search engines. Also our algorithm should not run on individual users’s computers. Rather it should run for different topics near to the source of the documents in the given topic – e.g., may run at the actual site where relevant information is stored.
Risvik and Michelsen [@risvik02search] mention that because of the exponential growth of the Web there is an ever increasing need for more intelligent, (topic-)specific algorithms for crawling, like focused crawling and document classification. With these algorithms crawlers and search engines can operate more efficiently in a topically limited document space. The authors also state that in such vertical regions the dynamics of the Web pages is more homogenous.
They overview different dimensions of web dynamics and show the arising problems in a search engine model. They show that the problem of rapid growth of Web and frequent document updates creates new challenges for developing more and more efficient Web search engines. The authors define a reference search engine model having three main components: (1) crawler, (2) indexer, (3) searcher. The main part of the paper focuses on the problems that crawlers need to overcome on the dynamic Web. As a possible solution the authors propose a heterogenous crawling architecture. They also present an extensible indexer and searcher architecture. The crawling architecture has a central distributor that knows which crawler has to crawl which part of the web. Special crawlers with low storage and high processing capacity are dedicated to web regions where content changes rapidly (like news sites). These crawlers maintain up-to-date information on these rapidly changing Web pages.
The main limitation of their crawling architecture is that they must divide the web to be crawled into distinct portions manually before the crawling starts. A weblog like distributed algorithm – as suggested here – my be used in that architecture to overcome this limitation.
Cho and Garcia-Molina [@cho03effective] define mathematically the freshness and age of documents of search engines. They propose the Poisson process as a model for page refreshment. The authors also propose various refresh policies and study their effectiveness both theoretically and on real data. They present the optimal refresh policies for their freshness and age metrics under the Poisson page refresh model. The authors show that these policies are superior to others on real data, too.
They collected about 720000 documents from 270 sites. Although they show that in their database more than 20 percent of the documents are changed each day, they disclosed these documents from their studies. Their crawler visited the documents once each day for 5 months, thus can not measure the exact change rate of those documents. While in our work we definitely concentrate on these frequently changing documents.
The proposed refresh policies require good estimation of the refresh rate for each document. The estimation influences the revisit frequency while the revisit frequency influences the estimation. Our algorithm does not need explicit frequency estimations. The more valuable URLs (e.g., more frequently changing) will be visited more often and if a crawler does not find valuable information around an URL being in it’s weblog then that URL finally will fall out from the weblog of the crawler. However frequency estimations and refresh policies can be easily integrated into the weblog algorithm selecting the starting URL from the weblog according to the refresh policy and weighting each URL in the weblog according to their change frequency estimations.
Menczer [@menczer03complementing] also introduces a recency metric which is 1 if all of the documents are recent (i.e., not changed after the last download) and goes to 0 as downloaded documents are getting more and more obsolete. Trivially immediately after a few minutes run of an online crawler the value of this metric will be 1, while the value for the search engine will be lower.
Edwards et al. [@edwards01adaptive] present a mathematical crawler model in which the number of obsolete pages can be minimized with a nonlinear equation system. They solved the nonlinear equations with different parameter settings on realistic model data. Their model uses different buckets for documents having different change rates therefore does not need any theoretical model about the change rate of pages. The main limitations of this work are the following:
- by solving the nonlinear equations the content of web pages can not be taken into consideration. The model can not be extended easily to (topic-)specific crawlers, which would be highly advantageous on the exponentially growing web [@rungsawang04learnable], [@risvik02search], [@menczer03complementing].
- the rapidly changing documents (like on news sites) are not considered to be in any bucket, therefore increasingly important parts of the web are disclosed from the searches.
However the main conclusion of the paper is that there may exist some efficient strategy for incremental crawlers for reducing the number of obsolete pages without the need for any theoretical model about the change rate of pages.
Forager architecture {#s:architecture}
====================
There are two different kinds of agents: the foragers and the reinforcing agent (RA). The fleet of foragers crawl the web and send the URLs of the selected documents to the reinforcing agent. The RA determines which forager should work for the RA and how long a forager should work. The RA sends reinforcements to the foragers based on the received URLs.
We employ a fleet of foragers to study the competition among individual foragers. The fleet of foragers allows to distribute the load of the searching task among different computers. A forager has simple, limited capabilities, like limited number of starting URLs and a simple, content based URL ordering. The foragers compete with each other for finding the most relevant documents. In this way they efficiently and quickly collect new relevant documents without direct interaction.
At first the basic algorithms are presented. After that the reinforcing agent and the foragers are detailed.
Algorithms {#ss:algorithms}
----------
### Weblog algorithm and starting URL selection {#sss:weblog}
A forager periodically restarts from a URL randomly selected from the list of starting URLs. The sequence of visited URLs between two restarts forms a path. The starting URL list is formed from the $START\_SIZE=10$ first URLs of the weblog. In the weblog there are $WEBLOG\_SIZE=100$ URLs with their associated weblog values in descending order. The weblog value of a URL estimates the expected sum of rewards during a path after visiting that URL. The weblog update algorithm modifies the weblog before a new path is started (Algorithm \[t:weblog\_pseudo\]). The weblog value of a URL already in the weblog is modified toward the sum of rewards in the remaining part of the path after that URL. A new URL has the value of actual sum of rewards in the remaining part of the path. If a URL has a high weblog value it means that around that URL there are many relevant documents. Therefore it may worth it to start a search from that URL.
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:weblog\_pseudo\]**Weblog Update**. $\beta$ was set to 0.3
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$visitedURLs\leftarrow$ the steps of the given path\
$values\leftarrow$ the sum of rewards for each step in the given path\
`output`\
starting URL list\
`method`\
$cumValues\leftarrow$ cumulated sum of $values$ in reverse order\
$newURLs\leftarrow visitedURLs$ not having value in $weblog$\
$revisitedURLs \leftarrow visitedURLs$ having value in $weblog$\
`for each` $URL\,\in\,newURLs$\
$weblog(URL)\leftarrow cumValues(URL)$\
`endfor`\
`for each` $URL\,\in\,revisitedURLs$\
$weblog(URL)\leftarrow (1-\beta)\,weblog(URL)\,+$\
$\beta\,cumValues(URL)$\
`endfor`\
$weblog\leftarrow$ descending order of values in $weblog$\
$weblog\leftarrow$ truncate $weblog$ after the $WEBLOG\_SIZE^{th}$\
element\
starting URL list $\leftarrow$ first $START\_SIZE$ elements of $weblog$
------------------------------------------------------------------------
------------------------------------------------------------------------
Without the weblog algorithm the weblog and thus the starting URL list remains the same throughout the searches. The weblog algorithm is a very simple version of evolutionary algorithms. Here, evolution may occur at two different levels: the list of URLs of the forager is evolving by the reordering of the weblog. Also, a forager may multiply, and its weblog, or part of it may spread through inheritance. This way, the weblog algorithm incorporates most basic features of evolutionary algorithms. This simple form shall be satisfactory to demonstrate our statements.
### Reinforcement Learning and URL ordering {#sss:rl}
A forager can modify its URL ordering based on the received reinforcements of the sent URLs. The (immediate) profit is the difference of received rewards and penalties at any given step. Immediate profit is a myopic characterization of a step to a URL. Foragers have an adaptive continuous value estimator and follow the *policy* that maximizes the expected long term cumulated profit (LTP) instead of the immediate profit. Such estimators can be easily realized in neural systems [@Sutton98Reinforcement; @szita03kalman; @schultz00multiple]. Policy and profit estimation are interlinked concepts: profit estimation determines the policy, whereas policy influences choices and, in turn, the expected LTP. (For a review, see [@Sutton98Reinforcement].) Here, choices are based on the greedy LTP policy: The forager visits the URL, which belongs to the *frontier* (the list of linked but not yet visited URLs, see later) and has the highest estimated LTP.
In the particular simulation each forager has a $k(=50)$ dimensional probabilistic term-frequency inverse document-frequency (PrTFIDF) text classifier [@joachims97probabilistic], generated on a previously downloaded portion of the Geocities database. Fifty clusters were created by Boley’s clustering algorithm [@Boley98principal] from the downloaded documents. The PrTFIDF classifiers were trained on these clusters plus an additional one, the $(k+1)^{th}$, representing general texts from the internet. The PrTFIDF outputs were non-linearly mapped to the interval \[-1,+1\] by a hyperbolic-tangent function. The classifier was applied to reduce the texts to a small dimensional representation. The output vector of the classifier for the page of URL $A$ is $\mathbf{state(A)}=(state(A)_1,\ldots , state(A)_k)$. (The $(k+1)^{th}$ output was dismissed.) This output vector is stored for each URL (Algorithm \[t:pageinfo\_URLordering\_pseudo\]).
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:pageinfo\_URLordering\_pseudo\]**Page Information Storage**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$pageURLs\leftarrow$ URLs of pages to be stored\
`output`\
$state\leftarrow$ the classifier output vectors for pages of $pageURLs$\
`method`\
`for each` $URL\,\in\,pageURLs$\
$page\leftarrow$ text of page of $URL$\
$state(URL)\leftarrow$ classifier output vector for $page$\
`endfor`
------------------------------------------------------------------------
------------------------------------------------------------------------
A linear function approximator is used for LTP estimation. It encompasses $k$ parameters, the *weight vector* $\mathbf{weight}=(weight_1,\ldots , weight_k)$. The LTP of document of URL $A$ is estimated as the scalar product of $\mathbf{state(A)}$ and $\mathbf{weight}$: $value(A)=\sum_{i=1}^k weight_i\,state(A)_i$. During URL ordering the URL with highest LTP estimation is selected. The URL ordering algorithm is shown in Algorithm \[t:URLordering\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:URLordering\_pseudo\]**URL Ordering**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$frontier\leftarrow$ the set of available URLs\
$state\leftarrow$ the stored vector representation of the URLs\
`output`\
$bestURL\leftarrow$ URL with maximum LTP value\
`method`\
`for each` $URL\,\in\,frontier$\
$value(URL)\leftarrow\sum_{i=1}^k state(URL)_i\, weight_i$\
`endfor`\
$bestURL\leftarrow$ URL with maximal LTP $value$
------------------------------------------------------------------------
------------------------------------------------------------------------
The weight vector of each forager is tuned by Temporal Difference Learning [@sutton88learning; @szita03kalman; @schultz00multiple]. Let us denote the current URL by $URL_n$, the next URL to be visited by $URL_{n+1}$, the output of the classifier for $URL_j$ by $\mathbf{state(URL_j)}$ and the estimated LTP of a URL $URL_j$ by $value(URL_j) = \sum_{i=1}^k wegiht_i\,state(URL_j)_i$. Assume that leaving $URL_n$ to $URL_{n+1}$ the immediate profit is $r_{n+1}$. Our estimation is perfect if $value(URL_n)=value(URL_{n+1})+r_{n+1}$. Future profits are typically discounted in such estimations as $value(URL_n)=\gamma value(URL_{n+1})+r_{n+1}$, where $0 < \gamma < 1$. The error of value estimation is
$$\delta(n,n+1) = r_{n+1} + \gamma value(URL_{n+1}) - value(URL_n).$$
We used throughout the simulations $\gamma =0.9$. For each step $URL_n \rightarrow URL_{n+1}$ the weights of the value function were tuned to decrease the error of value estimation based on the received immediate profit $r_{n+1}$. The $\delta(n,n+1)$ estimation error was used to correct the parameters. The $i^{th}$ component of the weight vector, $weight_i$, was corrected by
$$\Delta weight_i = \alpha \,\delta(n,n+1) \, state(URL_n)_i$$
with $\alpha=0.1$ and $i=1, \ldots , k$. These modified weights in a stationary environment would improve value estimation (see, e.g, [@Sutton98Reinforcement] and references therein). The URL ordering update is given in Algorithm \[t:URLordering\_update\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:URLordering\_update\_pseudo\]**URL Ordering Update**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$URL_{n+1}\leftarrow$ the step for which the reinforcement is received\
$URL_n\leftarrow$ the previous step before $URL_{n+1}$\
$r_{n+1}\leftarrow$ reinforcement for visiting $URL_{n+1}$\
`output`\
$weight\leftarrow$ the updated weight vector\
`method`\
$\delta(n,n+1) \leftarrow r_{n+1} + \gamma value(URL_{n+1}) -
value(URL_{n})$\
$weight \leftarrow weight\,+\,\alpha \,\delta(n,n+1) \, state(URL_n)$
------------------------------------------------------------------------
------------------------------------------------------------------------
Without the update algorithm the weight vector remains the same throughout the search.
### Document relevancy {#sss:relevant}
A document or page is possibly relevant for a forager if it is not older than 24 hours and the forager has not marked it previously. Algorithm \[t:relevant\_pseudo\] shows the procedure of selecting such documents. The selected documents are sent to the RA for further evaluation.
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:relevant\_pseudo\]**Document Relevancy at a forager**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$pages\leftarrow$ the pages to be examined\
`output`\
$relevantPages\leftarrow$ the selected pages\
`method`\
$previousPages\leftarrow$ previously selected relevant pages\
$relevantPages\leftarrow$ all pages from $pages$ which are\
not older than 24 hours and\
not contained in $previousPages$\
$previousPages\leftarrow$ add $relevantPages$ to $previousPages$
------------------------------------------------------------------------
------------------------------------------------------------------------
### Multiplication of a forager {#sss:multiplication}
During multiplication the weblog is randomly divided into two equal sized parts (one for the original and one for the new forager). The parameters of the URL ordering algorithm (the weight vector of the value estimation) are either copied or new random parameters are generated. If the forager has a URL ordering update algorithm then the parameters are copied. If the forager does not have any URL ordering update algorithm then new random parameters are generated, as shown in Algorithm \[t:multiplication\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:multiplication\_pseudo\]**Multiplication**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$weblog$\
weight vector of URL ordering\
`output`\
$newWeblog$\
$newWeight$\
`method`\
$newWeblog\leftarrow WEBLOG\_SIZE/2$ randomly selected\
URLs and values from $weblog$\
$weblog\leftarrow$ delete $newWeblog$ from $weblog$\
`if` forager has URL ordering update algorithm\
$newWeight\leftarrow$ copy the weight vector of URL ordering\
`else`\
$newWeight\leftarrow$ generate a new random weight vector\
`endif`
------------------------------------------------------------------------
------------------------------------------------------------------------
Reinforcing agent {#ss:ra}
-----------------
A reinforcing agent controls the “life” of foragers. It can start, stop, multiply or delete foragers. RA receives the URLs of documents selected by the foragers, and responds with reinforcements for the received URLs. The response is $REWARD=100$ (a.u.) for a relevant document and $PENALTY=-1$ (a.u.) for a not relevant document. A document is relevant if it is not yet seen by the reinforcing agent and it is not older than 24 hours. The reinforcing agent maintains the score of each forager working for it. Initially each forager has $INIT\_SCORE=100$ score. When a forager sends a URL to the RA, the forager’s score is decreased by $SCORE-=0.05$. After each relevant page sent by the forager, the forager’s score is increased by $SCORE+=1$ (Algorithm \[t:manageURL\_pseudo\]).
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:manageURL\_pseudo\]**Manage Received URL**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xxxxx = `input`\
$URL,forager\leftarrow$ received URL from forager\
`output`\
reinforcement to forager\
updated forager score\
`method`\
$relevants\leftarrow$ relevant pages seen by the RA\
$page\leftarrow$ get page of $URL$\
decrease forager’s score with $SCORE-$\
`if` $page\in relevants$ or page date is older than 24 hours\
send $PENALTY$ to forager\
`else`\
$relevants\leftarrow$ add $page$ to $relevants$\
send $REWARD$ to forager\
increase forager’s score with $SCORE+$\
`endif`
------------------------------------------------------------------------
------------------------------------------------------------------------
When the forager’s score reaches $MAX\_SCORE=200$ and the number of foragers is smaller than $MAX\_FORAGER=16$ then the forager is multiplied. That is a new forager is created with the same algorithms as the original one has, but with slightly different parameters. When the forager’s score goes below $MIN\_SCORE=0$ and the number of foragers is larger than $MIN\_FORAGER=2$ then the forager is deleted (Algorithm \[t:manageForager\_pseudo\]). Note that a forager can be multiplied or deleted immediately after it has been stopped by the RA and before the next forager is activated.
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:manageForager\_pseudo\]**: Manage Forager**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xxxxx = `input`\
$forager\leftarrow$ the forager to be multiplied or deleted\
`output`\
possibly modified list of foragers\
`method`\
`if` ($forager$’s score $\geq$ $MAX\_SCORE$ `and`\
number of foragers $<$ $MAX\_FORAGER$)\
$weblog,URLordering\leftarrow$ call $forager$’s\
**Multiplication, Alg. \[t:multiplication\_pseudo\]**\
$forager$ may modify it’s own weblog\
$newForager\leftarrow$ create a new forager with the received\
$weblog$ and $URLordering$\
set the two foragers’ score to $INIT\_SCORE$\
`else if` ($forager$’s score $\leq$ $MIN\_SCORE$ `and`\
number of foragers $>$ $MIN\_FORAGER$)\
delete $forager$\
`endif`
------------------------------------------------------------------------
------------------------------------------------------------------------
Foragers on the same computer are working in time slices one after each other. Each forager works for some amount of time determined by the RA. Then the RA stops that forager and starts the next one selected by the RA. The pseudo-code of the reinforcing agent is given in Algorithm \[t:reinforcing\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:reinforcing\_pseudo\]**: Reinforcing Agent**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xxxxx = `input`\
seed URLs\
`output`\
$relevants\leftarrow$ found relevant documents\
`method`\
$relevants\leftarrow$ empty set /\*set of all observed relevant pages\
initialize $MIN\_FORAGER$ foragers with the seed URLs\
set one of them to be the next\
`repeat`\
start next forager\
receive possibly relevant URL\
call **Manage Received URL, Alg. \[t:manageURL\_pseudo\]** with URL\
stop forager if its time period is over\
call **Manage Forager, Alg. \[t:manageForager\_pseudo\]** with this forager\
choose next forager\
`until` time is over
------------------------------------------------------------------------
------------------------------------------------------------------------
Foragers {#ss:forager}
--------
A forager is initialized with parameters defining the URL ordering, and either with a weblog or with a seed of URLs (Algorithm \[t:initforager\_pseudo\]). After its initialization a forager crawls in search paths, that is after a given number of steps the search restarts and the steps between two restarts form a path. During each path the forager takes $MAX\_STEP=100$ number of steps, i.e., selects the next URL to be visited with a URL ordering algorithm. At the beginning of a path a URL is selected randomly from the starting URL list. This list is formed from the 10 first URLs of the weblog. The weblog contains the possibly good starting URLs with their associated weblog values in descending order. The weblog algorithm modifies the weblog and so thus the starting URL list before a new path is started. When a forager is restarted by the RA, after the RA has stopped it, the forager continues from the internal state in which it was stopped. The pseudo code of step selection is given in Algorithm \[t:stepselection\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:initforager\_pseudo\]**Initialization of the forager**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
weblog or seed URLs\
URL ordering parameters\
`output`\
initialized forager\
`method`\
set path step number to $MAX\_STEP+1$ /\*start new path\
set the weblog\
either with the input weblog\
or put the seed URLs into the weblog with 0 weblog value\
set the URL ordering parameters in URL ordering algorithm
------------------------------------------------------------------------
------------------------------------------------------------------------
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:stepselection\_pseudo\]**URL Selection**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$frontier\leftarrow$ set of URLs available in this step\
$visited\leftarrow$ set of visited URLs in this path\
`output`\
$step\leftarrow$ selected URL to be visited next\
`method`\
`if` path step number $\leq MAX\_STEP$\
$step\leftarrow$ selected URL by **URL Ordering, Alg. \[t:URLordering\_pseudo\]**\
increase path step number\
`else`\
call the **Weblog Update, Alg. \[t:weblog\_pseudo\]** to update the weblog\
$step\leftarrow$ select a random URL from the starting URL list\
set path step number to 1\
$frontier\leftarrow$ empty set\
$visited\leftarrow$ empty set\
`endif`
------------------------------------------------------------------------
------------------------------------------------------------------------
The URL ordering algorithm selects a URL to be the next step from the frontier URL set. The selected URL is removed from the frontier and added to the visited URL set to avoid loops. After downloading the pages, only those URLs (linked from the visited URL) are added to the frontier which are not in the visited set.
In each step the forager downloads the page of the selected URL and all of the pages linked from the page of selected URL. It sends the URLs of the possibly relevant pages to the reinforcing agent. The forager receives reinforcements on any previously sent but not yet reinforced URLs and calls the URL ordering update algorithm with the received reinforcements. The pseudo code of a forager is shown in Algorithm \[t:forager\_pseudo\].
------------------------------------------------------------------------
------------------------------------------------------------------------
\[t:forager\_pseudo\]**Forager**
------------------------------------------------------------------------
xxx = xx = xx = xx = xx = xx = xx = xx = xx = `input`\
$frontier\leftarrow$ set of URLs available in the next step\
$visited\leftarrow$ set of visited URLs in the current path\
`output`\
sent documents to the RA\
modified $frontier$ and $visited$\
modified $weblog$ and URL ordering weight vector\
`method`\
`repeat`\
$step\leftarrow$ call **URL Selection, Alg. \[t:stepselection\_pseudo\]**\
$frontier\leftarrow$ remove $step$ from $frontier$\
$visited\leftarrow$ add $step$ to $visited$\
$page\leftarrow$ download the page of $step$\
$linkedURLs\leftarrow$ links of $page$\
$newURLs\leftarrow$ $linkedURLs$ which are not $visited$\
$frontier\leftarrow$ add $newURLs$ to $frontier$\
download pages of $linkedURLs$\
call **Page Information Storage, Alg. \[t:pageinfo\_URLordering\_pseudo\]** with $newURLs$\
$relevantPages\leftarrow$ call **Document Relevancy, Alg. \[t:relevant\_pseudo\]** for\
all pages\
send $relevantPages$ to reinforcing agent\
receive reinforcements for sent but not yet reinforced pages\
call **URL Ordering Update, Alg. \[t:URLordering\_update\_pseudo\]** with\
the received reinforcements\
`until` time is over
------------------------------------------------------------------------
------------------------------------------------------------------------
Experiments {#s:experiments}
===========
We conducted an 18 day long experiment on the Web to gather realistic data. We used the gathered data in simulations to compare the weblog update (Section \[sss:weblog\]) and reinforcement learning algorithms (Section \[sss:rl\]). In Web experiment we used a fleet of foragers using combination of reinforcement learning and weblog update algorithms to eliminate any biases on the gathered data. First we describe the experiment on the Web then the simulations. We analyze our results at the end of this section.
Web {#ss:real}
---
We ran the experiment on the Web on a single personal computer with Celeron 1000 MHz processor and 512 MB RAM. We implemented the forager architecture (described in Section \[s:architecture\]) in Java programming language.
In this experiment a fixed number of foragers were competing with each other to collect news at the CNN web site. The foragers were running in equal time intervals in a predefined order. Each forager had a 3 minute time interval and after that interval the forager was allowed to finish the step started before the end of the time interval. We deployed 8 foragers using the weblog update and the reinforcement learning based URL ordering update algorithms (8 WLRL foragers). We also deployed 8 other foragers using the weblog update algorithm but without reinforcement learning (8 WL foragers). The predefined order of foragers was the following: 8 WLRL foragers were followed by the 8 WL foragers.
We investigated the link structure of the gathered Web pages. As it is shown in Fig. \[f:sf\] the links have a power-law distribution ($P(k)=k^\gamma$) with $\gamma = -1.3$ for outgoing links and $\gamma = -2.57$ for incoming links. That is the link structure has the scale-free property. The clustering coefficient [@watts98collective] of the link structure is 0.02 and the diameter of the graph is 7.2893. We applied two different random permutations to the origin and to the endpoint of the links, keeping the edge distribution unchanged but randomly rewiring the links. The new graph has 0.003 clustering coefficient and 8.2163 diameter. That is the clustering coefficient is smaller than the original value by an order of magnitude, but the diameter is almost the same. Therefore we can conclude that the links of gathered pages form small world structure.
![**Scale-free property of the Internet domain**. Log-log scale distribution of the number of (incoming and outgoing) links of all URLs found during the time course of investigation. Horizontal axis: number of edges ($\log k$). Vertical axis: relative frequency of number of edges at different URLs ($\log P(k)$). Dots and dark line correspond to outgoing links, crosses and gray line correspond to incoming links. []{data-label="f:sf"}](sfsw_degree_o_out_x_in){width="2.5in"}
The data storage for simulation is a centralized component. The pages are stored with 2 indices (and time stamps). One index is the URL index, the other is the page index. Multiple pages can have the same URL index if they were downloaded from the same URL. The page index uniquely identifies a page content and the URL from where the page was download. At each page download of any foragers we stored the followings (with a time stamp containing the time of page download):
1. if the page is relevant according to the RA then store “relevant”
2. if the page is from a new URL then store the new URL with a new URL index and the page’s state vector with a new page index
3. if the content of the page is changed since the last download then store the page’s state vector with a new page index but keep the URL index
4. in both previous cases store the links of the page as links to page indices of the linked pages
1. if a linked page is from a new URL then store the new URL with a new URL index and the linked page’s state vector with a new page index
2. if the content of the linked page is changed since the last check then store the page’s state vector with a new page index but same URL index
Simulation {#ss:simulation}
----------
For the simulations we implemented the forager architecture in Matlab. The foragers were simulated as if they were running on one computer as described in the previous section.
### Simulation specification
During simulations we used the Web pages that we gathered previously to generate a realistic environment (note that the links of pages point to local pages (not to pages on the Web) since a link was stored as a link to a local page index):
- Simulated documents had the same state vector representation for URL ordering as the real pages had
- Simulated relevant documents were the same as the relevant documents on the Web
- Pages and links appeared at the same (relative) time when they were found in the Web experiment - using the new URL indices and their time stamps
- Pages and links are refreshed or changed at the same relative time as the changes were detected in the Web experiment – using the new page indices for existing URL indices and their time stamps
- Simulated time of a page download was the average download time of a real page during the Web experiment.
We conducted simulations with two different kinds of foragers. The first case is when foragers used only the weblog update algorithm without URL ordering update (WL foragers). The second case is when foragers used only the reinforcement learning based URL ordering update algorithm without the weblog update algorithm (RL foragers). Each WL forager had a different weight vector for URL value estimation – during multiplication the new forager got a new random weight vector. RL foragers had the same weblog with the first 10 URLs of the gathered pages – that is the starting URL of the Web experiment and the first 9 visited URLs during that experiment. In both cases initially there were 2 foragers and they were allowed to multiply until reaching the population of 16 foragers. The simulation for each type of foragers were repeated 3 times with different initial weight vectors for each forager. The variance of the results show that there is only a small difference between simulations using the same kind of foragers, even if the foragers were started with different random weight vectors in each simulation.
### Simulation measurements
Table \[t:params\] shows the investigated parameters during simulations.
--------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
downloaded is the number of downloaded documents
sent is the number of documents sent to the RA
relevant is the number of found relevant documents
found URLs is the number of found URLs
download efficiency is the ratio of relevant to downloaded documents in 3 hour time window throughout the simulation.
sent efficiency is the ratio of relevant to sent documents in 3 hour time window throughout the simulation.
relative found URL ratio of found URLs to downloaded at the end of the simulation
freshness is the ratio of the number of current found relevant documents and the number of all found relevant documents [@cho03effective]. A stored document is current, up-to-date, if its content is exactly the same as the content of the corresponding URL in the environment.
age A stored current document has 0 age, the age of an obsolete page is the time since the last refresh of the page on the Web [@cho03effective].
--------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Parameter ‘download efficiency’ is relevant for the site where the foragers should be deployed to gather the new information while parameter ‘sent efficiency’ is relevant for the RA. Note that during simulations we are able to immediately and precisely calculate freshness and age values. In a real Web experiment it is impossible to calculate these values precisely, because of the time needed to download and compare the contents of all of the real Web pages to the stored ones.
### Simulation analysis
The values in Table \[t:data\] are averaged over the 3 runs of each type of foragers.
type RL std RL WL std WL
--------------------- -------- -------- -------- --------
downloaded 540636 9840 669673 9580
sent 9747 98 6345 385
relevant 2419 45 3107 60
found URLs 31092 1050 33116 3370
download efficiency 0.0045 0.0001 0.0046 0.0001
sent efficiency 0.248 0.003 0.49 0.031
relative found URL 0.058 0.001 0.05 0.006
freshness 0.7 0.006 0.74 0.011
age (in hours) 1.79 0.04 1.56 0.08
: **Simulation results**. The $3^{rd}$ and $5^{th}$ columns contain the standard deviation of the individual experiment results from the average values. []{data-label="t:data"}
From Table \[t:data\] we can conclude the followings:
- RL and WL foragers have similar download efficiency, i.e., the efficiencies from the point of view of the news site are about the same.
- WL foragers have higher sent efficiencies than RL foragers, i.e., the efficiency from the point of view of the RA is higher. This shows that WL foragers divide the search area better among each other than RL foragers. Sent efficiency would be 1 if none of two foragers have sent the same document to the RA.
- RL foragers have higher relative found URL value than WL foragers. RL foragers explore more than WL foragers and RL found more URLs than WL foragers did per downloaded page.
- WL foragers find faster the new relevant documents in the already found clusters. That is freshness is higher and age is lower than in the case of RL foragers.
![**Efficiency**. Horizontal axis: time in days. Vertical axis: download efficiency, that is the number of found relevant documents divided by number of downloaded documents in 3 hour time intervals. Upper figure shows RL foragers’ efficiencies, lower figure shows WL foragers’ efficiencies. For all of the 3 simulation experiments there is a separate line. []{data-label="f:efficiency"}](dw_efficiency_WL_00_WL_RL_sfsw_rl_wl_1_3_20040810_0818){width="2.5in"}
Fig. \[f:efficiency\] shows other aspects of the different behaviors of RL and WL foragers. Download efficiency of RL foragers has more, higher, and sharper peaks than the download efficiency of WL foragers has. That is WL foragers are more balanced in finding new relevant documents than RL foragers. The reason is that while the WL foragers remain in the found good clusters, the RL foragers continuously explore the new promising territories. The sharp peaks in the efficiency show that RL foragers *find and recognize* new good territories and then *quickly collect* the current relevant documents from there. The foragers can recognize these places by receiving more rewards from the RA if they send URLs from these places.
![**Freshness and Age**. Horizontal axis: time in days. Upper vertical axis: freshness of found relevant documents in 3 hour time intervals. Lower vertical axis: age in hours of found relevant documents in 3 hour time intervals. Dotted lines correspond to weblog foragers, continuous lines correspond to RL foragers. []{data-label="f:freshness"}](freshness_age_2_WL_00_WL_RL_sfsw_rl_wl_1_3_20040810_0818){width="2.5in"}
The predefined order did not influence the working of foragers during the Web experiment. From Fig. \[f:efficiency\] it can be seen that foragers during the 3 independent experiments did not have very different efficiencies. On Fig. \[f:freshness\] we show that the foragers in each run had a very similar behavior in terms of age and freshness, that is the values remains close to each other throughout the experiments. Also the results for individual runs were close to the average values in Table \[t:data\] (see the standard deviations). In each individual run the foragers were started with different weight vectors, but they reached similar efficiencies and behavior. This means that the initial conditions of the foragers did not influence the later behavior of them during the simulations. Furthermore foragers could not change their environment drastically (in terms of the found relevant documents) during a single 3 minute run time because of the short run time intervals and the fast change of environment – large number of new pages and often updated pages in the new site. During the Web experiment foragers were running in 8 WLRL, 8 WL, 8 WLRL, 8 WL, …temporal order. Because of the fact that initial conditions does not influence the long term performance of foragers and the fact that the foragers can not change their environment fully we can start to examine them after the first run of WLRL foragers. Then we got the other extreme order of foragers, that is the 8 WL, 8 WLRL, 8 WL, 8 WLRL, …temporal ordering. For the overall efficiency and behavior of foragers it did not really matter if WLRL or WL foragers run first and one could use mixed order in which after a WLRL forager a WL forager runs and after a WL forager a WLRL forager comes. However, for higher bandwidths and for faster computers, random ordering may be needed for such comparisons.
Discussion {#s:discussion}
==========
Our first conjecture is that selection is efficient on scale-free small world structures. L[ő]{}rincz and Kókai [@Lorincz02intelligent] and Rennie et al. [@rennie99using] showed that RL is efficient in the task of finding relevant information on the Web. Here we have shown experimentally that the weblog update algorithm, selection among starting URLs, is at least as efficient as the RL algorithm. The weblog update algorithm finds as many relevant documents as RL does if they download the same amount of pages. WL foragers in their fleet select more different URLs to send to the RA than RL foragers do in their fleet, therefore there are more relevant documents among those selected by WL foragers then among those selected by RL foragers. Also the freshness and age of found relevant documents are better for WL foragers than for RL foragers.
For the weblog update algorithm, the selection among starting URLs has no fine tuning mechanism. Throughout its life a forager searches for the same kind of documents – goes into the same ‘direction’ in the state space of document states – determined by its fixed weight vector. The only adaptation allowed for a WL forager is to select starting URLs from the already seen URLs. The WL forager can not modify its (‘directional’) preferences according goes newly found relevant document supply, where relevant documents are abundant. But a WL forager finds good relevant document sources in its own direction and forces its search to stay at those places. By chance the forager can find better sources in its own direction if the search path from a starting URL is long enough. On Fig. \[f:efficiency\] it is shown that the download efficiency of the foragers does not decrease with the multiplication of the foragers. Therefore the new foragers must found new and good relevant document sources quickly after their appearances.
The reinforcement learning based URL ordering update algorithm is capable to fine tune the search of a forager by adapting the forager’s weight vector. This feature has been shown to be crucial to adapt crawling in novel environments [@kokai02learning; @Lorincz02intelligent]. An RL forager goes into the direction (in the state space of document states) where the estimated long term cumulated profit is the highest. Because the local environment of the foragers may changes rapidly during crawling, it seems desirable that foragers can quickly adapt to the found new relevant documents. Relevant documents may appear lonely, not creating a good relevant document source, or do not appear at the right URL by a mistake. This noise of the Web can derail the RL foragers from good regions. The forager may “turn” into less valuable directions, because of the fast adaptation capabilities of RL foragers.
Our second conjecture is that selection fits SFSW better than RL. We have shown in our experiments that selection and RL have different behaviors. Selection selects good information sources, which are worth to revisit, and stays at those sources as long as better sources are not found by chance. RL explores new territories, and adapts to those. This adaptation can be a disadvantage when compared with the more rigid selection algorithm, which sticks to good places until ‘provably’ better places are discovered. Therefore WL foragers, which can not be derailed and stay in their found ‘niches’ can find new relevant documents faster in such already known terrains than RL foragers can. That is, freshness is higher and age is lower for relevant documents found by WL foragers than for relevant documents found by RL foragers. Also, by finding good sources and staying there, WL foragers divide the search task better than RL foragers do, this is the reason for the higher sent efficiency of WL foragers than of RL foragers.
We have rewired the network as it was described in Section \[ss:real\]. This way a scale-free (SF) but not so small world was created. Intriguingly, in this SF structure, RL foragers performed better than WL ones. Clearly, further work is needed to compare the behavior of the selective and the reinforcement learning algorithms in other then SFSW environments. Such findings should be of relevance in the deployment of machine learning methods in different problem domains.
From the practical point of view, we note that it is an easy matter to combine the present algorithm with URLs offered by search engines. Also, the values reported by the crawlers about certain environments, e.g., the environment of the URL offered by search engines represent the neighborhood of that URL and can serve adaptive filtering. This procedure is, indeed, promising to guide individual searches as it has been shown elsewhere [@palotai04adaptive].
Conclusion {#s:conclusion}
==========
We presented and compared our selection algorithm to the well-known reinforcement learning algorithm. Our comparison was based on finding new relevant documents on the Web, that is in a dynamic scale-free small world environment. We have found that the weblog update selection algorithm performs better in this environment than the reinforcement learning algorithm, eventhough the reinforcement learning algorithm has been shown to be efficient in finding relevant information [@Lorincz02intelligent; @rennie99using]. We explain our results based on the different behaviors of the algorithms. That is the weblog update algorithm finds the good relevant document sources and remains at these regions until better places are found by chance. Individuals using this selection algorithm are able to quickly collect the new relevant documents from the already known places because they monitor these places continuously. The reinforcement learning algorithm explores new territories for relevant documents and if it finds a good place then it collects the existing relevant documents from there. The continuous exploration and the fine tuning property of RL causes that RL finds relevant documents slower than the weblog update algorithm.
In our future work we will study the combination of the weblog update and the RL algorithms. This combination uses the WL foragers ability to stay at good regions with the RL foragers fine tuning capability. In this way foragers will be able to go to new sources with the RL algorithm and monitor the already found good regions with the weblog update algorithm.
We will also study the foragers in a simulated environment which is not a small world. The clusters of small world environment makes it easier for WL foragers to stay at good regions. The small diameter due to the long distance links of small world environment makes it easier for RL foragers to explore different regions. This work will measure the extent at which the different foragers rely on the small world property of their environment.
Acknowledgement
===============
This material is based upon work supported by the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, Air Force Research Laboratory, under Contract No. FA8655-03-1-3036. This work is also supported by the National Science Foundation under grants No. INT-0304904 and No. IIS-0237782. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, Air Force Research Laboratory.
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[^1]: Corresponding author. email: [email protected]
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abstract: 'We continue the study of hidden ${\mathbb{Z}}_2$ symmetries of the four-point $\hat {sl(2)} _k$ Knizhnik-Zamolodchikov equation iniciated in [@Yo2005]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector $\omega =1$ is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in $AdS_3$ has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the non-violating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, non diagonal functional relations between different solutions of the KZ equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both WZNW and Liouville conformal theories.'
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Departamento de Física, Universidad de Buenos Aires, FCEN UBA
[*Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina*]{}
Introduction
============
The $SL(2,\mathbb{R})_{k}$ WZNW model plays an important role within the context of string theory. This conformal model enters in the worldsheet description of the theory formulated on exact non-compact curved backgrounds, being the most celebrated examples: the string theory on $AdS_{3}$ and on the 2D black hole (the last, by means of its gauged $SL(2,\mathbb{R})_{k}/U(1)$ construction). Consequently, among the main motivations, this topic received particular attention due to its relevance for black hole physics and its relation with the $AdS/CFT$ correspondence. Actually, this conformal field theory raises the hope to work out the details of the correspondence beyond the particle limit approximation [GKS,GK,Oetal]{}. Here, we continue the study of the observables in this theory, focussing out attention on the four-point function.
String amplitudes in $AdS_{3}$
------------------------------
Although the structure of the WZNW model on $SL(2,\mathbb{R})$ (corresponding to strings in Lorentzian $AdS_{3}$) is not yet well understood, the observables of this theory are assumed to be well defined in terms of the analytic continuation of the correlation functions in the Euclidean case, i.e. in the $SL(2,\mathbb{C})_{k}/SU(2)$ WZNW gauged model. Hence, the string scattering amplitudes in Lorentzian $AdS_{3}$ are obtained by integrating over the worldsheet insertions of vertex operators in the model on $SL(2,\mathbb{C})/SU(2)$ and then extending the range of the indices of continuous representations in order to include the discrete representations of $SL(2,\mathbb{R})$ as well. The correlation functions defined in such a way typically develop poles in the space of representation indices and these poles are then interpreted as state conditions of bounded states (called short strings“ in [@MO3]) and yield selection rules [@SHope; @Sope] for the string scattering amplitudes. Besides, the continuous representations (the long strings” in [@MO3]) have a clear interpretation as asymptotic states and define the S-matrix in the Lorentzian target space.
In the last decade, we gained important information on the explicit functional form of the WZNW correlation functions, and this enabled us to study the string theory on $AdS_{3}$ beyond the supergravity approximation, both at the level of the spectrum [@MO1] and at the level of its interactions (see [@GN3; @MO3] and references therein). Originally, some particular cases of the two and three-point functions were explicitly computed in [@B; @BB], and was Teschner who presented the general result in [@Tboost] and carefully described its formal aspects in [Tboost,Tmini,Tope]{}, by employing the boostrap approach and the minisuperspace approximation. Subsequently, other approaches, like the path integral techniques [@Ponjaspath] and the free field representation [GN2,Ponjasfree]{}, were employed to rederive such correlation functions. The operator product expansion was studied in Ref. [@SHope; @Sope] in order to investigate the fusion rules, and the crossing symmetry of the correlation functions was eventually proven in Ref. [@Tcross] by making use of its analogy with the Liouville field theory [@FZ; @Andreev]; see also [Ponsot]{}. The study of the correlation functions was also shown to be useful for consistency checks of the conjectured dualities between this and other conformal models [@FH; @GKnotes; @KKK].
After these objectives were achieved, the study of correlation functions in the $SL(2,\mathbb{R})_{k}$ WZNW model acquired a new dimension since Maldacena and Ooguri pointed out the existence of new representations of $SL(2,\mathbb{R})_{k}$ contributing to the spectrum of the string theory in $AdS_{3}$ [@MO3]. Then, the correlation functions involving these new states had to be analyzed as well. These new representations, obtained from the standard ones by acting with the spectral flow transformation, are semiclassically related to the possibility of the $AdS_{3}$ strings to have a nonzero winding number. This interpretation in terms of winding numbers" does not regard a topological winding, but it turns out to be a consequence of the presence of a non trivial NS-NS $B_{\mu \nu }$ background field. Then, from the beginning, this winding number, as a non topological degree of freedom, was assumed to be possibly violated when the interactions were to take place. This violation was actually first suggested in a unpublished work by Fateev and the brothers Zamolodchikov for the case of the 2D black hole [@FZZ]. A free field computation of three-point function of such winding states, including the violating winding case, appeared in Ref. [@GN3], and a similar free field realization was studied in more detail in [@GN2; @GLopez]. An impressive analysis of the correlation functions in the $SL(2,\mathbb{R})_{k}$ WZNW model was then presented in the paper [@MO3], by Maldacena and Ooguri. There, the pole structure of two, three and four-point functions was discussed in the framework of the semiclassical interpretation and the $AdS/CFT$ correspondence. The exact expressions of two and three-point functions, including the violating winding three-point function, were fully analyzed. Besides, the string theory interpretation of such observables, as representing scattering amplitudes, was given with precision. The four-point function was also studied in [@MO3], and, even though there is still no closed expression for the generic case available, our understanding of its analytic structure was substantially increased due to the results of that work. By making use of the factorization *ansatz* given by Teschner in [@Tope], Maldacena and Ooguri proposed an analytic extension of the expression for the $SL(2,\mathbb{C})_{k}/SU(2)$ conformal blocks. Thus, they gave a precise prescription to integrate the monodromy invariant expression over the space of $SL(2,\mathbb{R})_{k}$ representations and to pick up the pole contribution of the discrete states. Perhaps, the two main observations made in [@MO3] regarding the four-point functions are: the existence of additional poles in the middle of the moduli space, and the fact that the factorization of the four-point function only permits the usual interpretation in terms of physical intermediate states for particular incoming and outgoing kinematic configurations. The analytic structure of particular four-point functions, those leading to logarithmic singularities, was studied in [@GSimeone], showing that this example fits the standard structure of four-point functions in the $AdS/CFT$ correspondence.
Despite all this information we get about four-point functions, it is worth noticing that the mentioned cases only took into account non-flowed representations (representing non-winding string states) as those describing the incoming and outgoing states. For instance, even though the winding strings of the sector $\omega =-1$ were shown to arise in the intermediate channels of four-point functions [@MO3], this was shown by analyzing the processes that only involve incoming and outgoing states of the sector $\omega =0$. Then, it would be interesting to extend the study to the case of four-point functions that involve external states of the sectors $\omega
\neq 0$. Here is where our result enters in the game since it actually permits to get information of the four-point winding violating functions from all what is already known about the conservative case. In fact, in this note we will show how to connect the correlation functions involving one flowed state (winding string state) of the sector $\omega =-1$ (with winding number $\omega =-1$) to the analogous quantity that merely involves non-flowed states (just strings with winding number $\omega =0$). We will also argue that this is actually analogous to the relation existing between the violating winding three-point function and the conservative one. The way of showing such connection takes into account a recent result that presents a new map between WZNW and Liouville correlators. This map, different from that employed in [@Tope] to prove the crossing symmetry in $SL(2,\mathbb{C})/SU(2)$, was discovered by Stoyanovsky some years ago [@S2000], and it was further developed by Teschner and Ribault in Ref. [RT2005,R2005,R2005b]{}. In [@GNakayama2005; @chinitos; @chinitos2; @chinitos3] this map between both CFTs (henceforth denoted SRT map) was analyzed in the context of the implications it has in string theory (see also Ref. [Related]{} for recent works on the relation between Liouville and WZNW correlators). In references [@Yo2005b; @Yo2006], a free field realization of the SRT map was given, and it was shown to reproduce the correct three-point function for the case where one spectral flowed state (string state with winding $\omega =1$) is considered. Such observable had been also computed in [@Yo2005] through the other connection to Liouville theory, discovered by Fateev and Zamolodchikov in [@FZ] and extended in Ref. [Andreev,Tope,Ponsot]{} to the non-compact case (henceforth denoted FZ map). We will be more precise in the following paragraph.
Overview
--------
Recently, a new possibility to study the four-point function involving winding states has raised because of a discovery made by Ribault [@R2005] and Fateev [@Funpublished], stating that correlators involving winding states in WZNW can be written in terms of correlators in Liouville field theory. In the case of four-point function with one state in the sector $\omega =-1$, on which we are interested here, this turns out to correspond to the Liouville five-point function. In principle, this does not seem to imply an actual simplification since five-point function in Liouville theory cannot be simply solved either. However, we noticed that the SRT map is not the only way of mapping the sameLiouville five-point function to a (different) four-point function of the WZNW theory. Indeed, we can also do this by employing the non-compact generalization of the FZ map [@FZ]. If this second map is used, the four-point function reached in the WZNW side is one that contains four non-winding states, enabling us to relate winding violating processes in $AdS_{3}$ with their conservative analogs. However, this is not the whole story since, unlike the SRT map [S2000,RT2005,R2005]{}, the FZ map involves a non-diagonal correspondence between the quantum numbers of both Liouville and WZNW sides. Hence, besides the appropriate combination of the SRT and the FZ maps, a sort of a diagonalization procedure" is also required in order to present the result in a clear form. This diagonalization is eventually achieved by using the FZ map itself and the reflection symmetry of Liouville theory. By doing something similar to that in Ref. [@Yo2005], we will first indicate how such diagonalization is realized due to identities holding between different exact solutions of the KZ equation. In fact, this paper can be regarded as a addendum to Ref. [@Yo2005], being the second part of our study of hidden symmetries in the four-point $\widehat{sl(2)}_{k} $ KZ equation. In [@Yo2005], a set of $\mathbb{Z}_{2}$ symmetry transformations of the KZ equation were studied. Such involutions were realized by means of the action on the four indices of $SL(2,\mathbb{R})$ representations and led to prove identities between different exact solutions of the KZ equation. The main tool for working out such identities was the FZ map, mapping four-point functions of the WZNW theory to a particular subset of five-point functions of the Liouville field theory. Following this line, here we explore the implications of new symmetry transformations on the solutions of KZ equation. The plan of the paper goes as it follows: In the next section we will review the connections between four-point functions in WZNW theory and five-point function in Liouville field theory. The fact that there is not a unique map of this kind[^1] is the reasons for a non trivial relation between correlators of winding and non-winding states to exist. In section 3, as a preliminary result, we first prove a new identity between exact solutions of the KZ equation, complementing the catalog presented in Ref. [@GSimeone; @Yo2005]. This identity is again realized by a non diagonal $\mathbb{Z}_{2}$ transformation of the class studied in Ref. [@Yo2005], acting on the four indices of the representations of $SL(2,\mathbb{R})$. Then, this leads us to show how the four-point correlation function involving one spectral flowed state in the winding sector $\omega =-1$ can be written in terms of the correlation function of non flowed (non winding) states.
Conformal field theory
======================
The WZNW model
--------------
We are interested in the WZNW model formulated on the $SL(2,\mathbb{R})$ group manifold. Its action corresponds to the non-linear $\sigma $-model of strings in Lorentzian $AdS_{3}$ space. On the other hand, its Euclidean version is similarly given by the gauged $SL(2,\mathbb{C})/SU(2)$ model. The states of the Euclidean model are characterized by normalizable operators on the Poincaré hyper half-plane $H_{3}^{+}=SL(2,\mathbb{C})/SU(2)$. These operators can be conveniently written as$$\Phi _{j}(x|z)=\frac{1-2j}{\pi }\left( u^{-1}+u|\gamma -x|^{2}\right) ^{-2j},
\label{OPahora}$$where the variables $\gamma ,\overline{\gamma }$ and $u$ are associated to the $AdS_{3}$ Euclidean metric in Poincaré coordinates; namely$$ds^{2}=k(u^{-2}du^{2}+u^{2}d\gamma d\overline{\gamma });$$while the complex coordinates $x$ and $\overline{x}$ represent auxiliary variables that expand the $SL(2,\mathbb{C})$ representations as it follows$$J^{a}(z_{1})\Phi _{j}(x|z_{2})\sim \frac{1}{z_{1}-z_{2}}D_{x}^{a}\Phi
_{j}(x|z_{2})+...$$for $a=\{3,+,-\}.$ The dots ...“ stand for regular terms” in the OPE, while the differential operators $D_{x}^{a}$ correspond to the realization $$D_{x}^{3}=x\partial _{x}+j,~\qquad D_{x}^{+}=\partial _{x},~\qquad
D_{x}^{-}=x^{2}\partial _{x}+2jx.$$Besides, the operators $J^{a}(z)$ are the local Kac-Moody currents, whose Fourier modes are defined by $J^{a}(z)=\sum_{n\in \mathbb{Z}}J_{n}^{a}$ $z^{-1-n}$ and satisfy the $\widehat{sl(2)}_{k}$ affine Kac-Moody algebra of level $k$. The value of $k$ is related to the string length and the $AdS$ radius through $k=l_{AdS}^{2}/l_{s}^{2}$. This keeps track of the conformal invariance of the WZNW theory. The Sugawara construction yields the stress-tensor that can be used to compute the central charge of this theory, being $$c=3+\frac{6}{k}.$$The formula for the conformal dimension of operators (\[OPahora\]) reads$$h_{j}=\frac{j(1-j)}{k-2}, \label{dddd}$$where the indices take the values $j=\frac{1}{2}+i\mathbb{R}_{>0}$ for $SL(2,\mathbb{C})/SU(2)$. In order to propose a similar algebraic realization for the Lorentzian string theory, it is usually assumed that the observables of the Euclidean theory admits an analytic continuation in the variable $j$, now parameterizing both continuous and discrete representations of $SL(2,\mathbb{R})$. It is worth noticing that the transformation $j\rightarrow 1-j$ is a symmetry of the formula (\[dddd\]); this corresponds to the so called Weyl reflection symmetry and will be important for us in section 3. Operators (\[OPahora\]) represent the vertex operators of the string theory in $AdS_{3}$ and define the correlation functions. Then, after integrating over the variables $z_{\mu }$, the functional form of the $N$-point correlators will still depend on the auxiliary variables $x_{\mu }$, with $\mu =\{1,2,...N\}$. Within the context of the $AdS_{3}/CFT_{2}$ interpretation, those auxiliary variables are interpreted as the coordinates of the dual conformal field theory in the boundary of $AdS_{3}$, and then acquire a geometrical meaning. This picture, employing the variable $x$ to label the representations, is usually referred as the $x$-basis. On the other hand, there exists a different picture that is also convenient, and in which the quantum numbers labeling the representations permits to define string states with well defined momenta in the bulk. This is the often called $m$-basis, and employs the standard way of parameterizing representations of the $SL(2,\mathbb{R})$ group, by using a pair of indices $j,m$. In this frame, the string states in $AdS_{3}$ are given by vectors $|j,m>\otimes $ $|j,\overline{m}>$ of the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ representations. Furthermore, as it was mentioned in the introduction, in Ref. [@MO3] it was shown that an additional quantum number should be included in order to fully characterize the space of states in $AdS_{3}$. This quantum number, denoted $\omega $, is associate to the winding number of strings in $AdS_{3}$, at least in what respects to the states with a suitable asymptotic description (the long strings). From an algebraic point of view, $\omega $ labels the spectral flow transformation that generates new representations of the theory. This is the reason because we will refer to the spectral flowed states“ as winding states”, indistinctly. The vertex operators representing winding states (states with $\omega \neq 0$) in the $m$-basis are to be denoted $\Phi _{j,m,\overline{m}}^{\omega }(z)$ and define the correlation functions in this basis, which we will denote as $\left\langle
\Phi _{j_{1},m_{1},\overline{m}_{1}}(z_{1})\Phi _{j_{2},m_{2},\overline{m}_{2}}(z_{2})...\Phi _{j_{N},m_{N},\overline{m}_{N}}(z_{N})\right\rangle .$ The conformal dimension of the states in the $m$-basis depends both on $m$ and $\omega $ through the expression$$h_{j,m,\omega }=\frac{j(1-j)}{k-2}-m\omega -\frac{k}{4}\omega ^{2}\text{.}$$In the case $\omega =0,$ operators $\Phi _{j,m,\overline{m}}^{\omega =0}(z)$ are related to those of the $x$-basis through the Fourier transform $$\Phi _{j,m,\overline{m}}^{\omega =0}(z)=\int d^{2}x\ \Phi _{j}(x|z)x^{m-j}\overline{x}^{\overline{m}-j} \label{Fourier}$$On the other hand, the definition of string states with $\omega \neq 0$ in the $x$-basis was studied in [@MO3; @Minces]; however, these have not a simple expression. Now, let us discuss the correlation functions in more detail.
The four-point KZ equation
--------------------------
As it was commented, the two and three-point functions in the WZNW model are known [@BB]; and the four-point function in the sector $\omega =0$ was studied in detail in Ref. [@Tope], where a consistent $\mathit{ansatz}$ was proposed based in the analogy with other CFTs; we detail such proposal below.
The four-point correlation functions on the zero-genus topology are determined by conformal invariance up to a factor $f$, which is a function of the cross ratio $z$ and the variables $j_{i},x_{i}$ and $\bar{x}_{i},$ that label the representations; namely$$\left\langle \Phi _{j_{1}}(x_{1}|z_{1})\Phi _{j_{2}}(x_{2}|z_{2})\Phi
_{j_{3}}(x_{3}|z_{3})\Phi _{j_{4}}(x_{4}|z_{4})\right\rangle
=\prod_{a<b}^{4}|x_{a}-x_{b}|^{2J_{ab}}\prod_{a<b}^{4}|z_{a}-z_{b}|^{2h_{ab}}|f_{j_{1},j_{2},j_{3},j_{4}}(x,z)|^{2}
\label{recorda}$$being $$x=\frac{(x_{2}-x_{1})(x_{3}-x_{4})}{(x_{4}-x_{1})(x_{3}-x_{2})},\qquad z=\frac{(z_{2}-z_{1})(z_{3}-z_{4})}{(z_{4}-z_{1})(z_{3}-z_{2})},$$and where $h_{34}=h_{1}+h_{2}-h_{3}-h_{4},$ $h_{14}=-2h_{1},$ $h_{24}=h_{1}-h_{2}+h_{3}-h_{4},$ $h_{23}=h_{4}-h_{1}-h_{2}-h_{3};$ and $J_{34}=j_{1}+j_{2}+j_{3}-j_{4},$ $J_{14}=-2j_{1},$ $J_{24}=j_{1}-j_{2}+j_{3}-j_{4},$ $J_{13}=-j_{1}-j_{2}-j_{3}+j_{4}$.
The function $f=f_{j_{1},j_{2},j_{3},j_{4}}(x,z)$ is then given by certain linear combination of solutions to the Knizhnik-Zamolodchikov partial differential equation (KZ); *i.e.* that combination which turns out to be monodromy invariant. The KZ equation in the case of the $SL(2,\mathbb{R})_{k}$ WZNW model takes the form $$(k-2)z(z-1)\frac{\partial }{\partial z}f_{j_{1},j_{2},j_{3},j_{4}}(x,z)=\left( (z-1){\mathcal{D}}_{1}+z{\mathcal{D}}_{0}\right)
f_{j_{1},j_{2},j_{3},j_{4}}(x,z) \label{balinf}$$where the differential operators are $$\begin{aligned}
{\mathcal{D}}_{1} &=&x^{2}(x-1)\frac{\partial ^{2}}{\partial x^{2}}-\left(
(j_{4}-j_{3}-j_{2}-j_{1}-1)x^{2}+2j_{2}x+2j_{1}x(1-x)\right) \frac{\partial
}{\partial x}+ \\
&&+2(j_{1}+j_{2}+j_{3}-j_{4})j_{1}x-2j_{1}j_{2} \\
{\mathcal{D}}_{0} &=&-(1-x)^{2}x\frac{\partial ^{2}}{\partial x^{2}}+\left(
(-j_{1}-j_{2}-j_{3}+j_{4}+1)(1-x)-2j_{3}-2j_{1}x\right) (x-1)\frac{\partial
}{\partial x}+ \\
&&+2(j_{1}+j_{2}+j_{3}-j_{4})j_{1}(1-x)-2j_{1}j_{3}\end{aligned}$$With [@Tope], we can consider the following *ansatz* for the solution $$f_{j_{1},j_{2},j_{3},j_{4}}(x,z)=\int_{{\mathcal{C}}}dj\frac{C(j_{1},j_{2},j)C(j,j_{3},j_{4})}{B(j)}{\mathcal{G}}_{j_{1},j_{2,}j,j_{3},j_{4}}(x|z)\times \bar{{\mathcal{G}}}_{j_{1},j_{2,}j,j_{3},j_{4}}(\bar{x}|\bar{z}), \label{antonia}$$where the functions $C(j_{1},j_{2},j_{3})$ and $B(j_{1})$ are given by the structure constants and the reflection coefficient of the $SL(2,\mathbb{R})_{k}$ WZNW model, respectively; and where the contour of integration is defined as covering the curve ${\mathcal{C}}=\frac{1}{2}+i\mathbb{R}$. The integration along ${\mathcal{C}}$ turns out to be redundant for a monodromy invariant solution since such particular linear combination is invariant under Weyl reflection $j\rightarrow 1-j$, for which the contour transforms as the complex conjugation ${\mathcal{C}}\rightarrow \overline{{\mathcal{C}}}=-{\mathcal{C}}$. Here, we are interested in the relation between different solutions to (\[balinf\]).
The Liouville field theory
--------------------------
Now, let us briefly review the Liouville field theory, which is the other CFT in which we are interested here. Its action reads
$$S=\frac{1}{4\pi }\int d^{2}z\left( -\partial \varphi \bar{\partial}\varphi
+QR\varphi +\mu e^{\sqrt{2}b\varphi }\right) , \label{action}$$
where the background charge is given by $Q=b+b^{-1}$, and the exponential self-interaction thus corresponds to a marginal deformation. We will set the Liouville cosmological constant $\mu $ to have an appropriate value in order to make the correlation functions acquire a simple form. This is achieved by properly rescaling the zero mode of $\varphi (z)$. The Liouville field theory is reviewed with impressive detail in Ref. [@Yu]; see also [Tliouville,Seiberg]{} and the recent [@Leo]. The central charge of the theory is given by$$c=1+6Q^{2}>1$$and the vertex operators have the exponential form [@Tvertex]$$V_{\alpha }(z)=e^{\sqrt{2}\alpha \phi (z)}, \label{cinc}$$whose conformal dimension is given by$$\Delta _{\alpha }=\alpha (Q-\alpha ). \label{cinci}$$Vertex operators (\[cinc\]) define the Liouville $N$-point correlation functions, which are to be denoted by $\left\langle
V_{a_{1}}(z_{1})V_{a_{2}}(z_{2})...V_{a_{N}}(z_{N})\right\rangle $. Notice that formula (\[cinci\]) remains invariant under $\alpha \rightarrow
Q-\alpha $, henceforth called Liouville reflection. This symmetry induces the identification between both fields $V_{\alpha }(z)$ and $V_{Q-\alpha
}(z) $, yielding the operator valued relation$$\left\langle
V_{a_{1}}(z_{1})V_{a_{2}}(z_{2})...V_{a_{N}}(z_{N})\right\rangle
=R_{b}(\alpha _{1})\left\langle
V_{Q-a_{1}}(z_{1})V_{a_{2}}(z_{2})...V_{a_{N}}(z_{N})\right\rangle ,
\label{whatt}$$which is valid for any vertex $i=\{1,2,...N\}$ (though we exemplified it here for $i=1$), and where $R_{b}(\alpha _{1})$ represents the Liouville reflection coefficient$$R_{b}(\alpha )=\left( \pi \mu \frac{\Gamma (1-b^{2})}{\Gamma (1+b^{2})}\right) ^{\frac{2}{b}\alpha -1-b^{-2}}\frac{\Gamma (2b\alpha -b^{2})\Gamma
(2b^{-1}\alpha -b^{-2})}{\Gamma (2-2b\alpha +b^{2})\Gamma (2-2b^{-1}\alpha
+b^{-2})}, \label{Reflection}$$
Another important feature of the Liouville correlation functions is the fact that those involving states with momentum $\alpha =-1/2b$ satisfy a well known partial differential equation, called the Belavin-Polyakov-Zamolodchikov equation (BPZ). This is similar to that of the minimal models and actually holds for a wider family of correlators, namely all of those involving certain state with momentum $\alpha _{m,n}=\frac{1-m}{2}b+\frac{1-n}{2}b^{-1}$ for any pair $m,n\in \mathbb{Z}_{>0}$. In particular, here we are interested in Liouville five-point correlators of the form$$\left\langle
V_{a_{1}}(z_{1})V_{a_{2}}(z_{2})V_{a_{3}}(z_{3})V_{a_{4}}(z_{4})V_{a_{1,2}=-1/2b}(z_{5})\right\rangle .$$
Now, once both WZNW and Liouville theories were introduced, we move to the following ingredient in our discussion: the close relation existing between correlation functions of each of these two conformal theories. Such relation has multiple aspects indeed; we will discuss two of them in the following two subsections.
The Fateev-Zamolodchikov identity
---------------------------------
The often called FZ map is a dictionary that connects four-point correlation functions in WZNW models to five-point correlation in Liouville field theory. This result was developed in Ref. [@Tcross] (see also Ref. [Ponsot]{}) and, among other ingredients involved in its derivation, is based on the relation existing between solution of the KZ and the BPZ differential ecuations. Such relation was originally noticed by Fateev and Zamolodchikov in Ref. [@FZ] for the case of the compact $SU(2)_{k}$ WZNW case and the minimal models, and it basically states that, starting from any given solution of the four-point KZ equation, a systematical way of constructing a solution of the five-point BPZ equation exists. The FZ map, or strictly speaking its adaptation to the non-compact WZNW model, was employed to investigate several properties of the $SL(2,\mathbb{R})_{k}$ conformal theory, becoming one of the most fruitful tools to this end. In particular, Teschner gave to it its closed form and used it to prove the crossing symmetry of the WZNW model by assuming that a similar relation holds for the conformal blocks. Besides, Andreev rederived the fusion rules of admissible representations of the $\hat{sl(2)}_{k}$ affine algebra by similar means [@Andreev], and Ponsot discussed the monodromy of the theory with such techniques [@Ponsot]. In Ref. [@Yo2005], the FZ map was shown to be useful to prove several identities between exact solutions of the KZ equation, and we will extend such result in the next section here. But first, let us briefly review the FZ statement: The observation made in [FZ]{} is that the KZ equation satisfied by the four-point functions in the WZNW model agrees with the BPZ equation satisfied by a particular set of five-point functions in Liouville field theory. More specifically, Fateev and Zamolodchikov showed that it is possible to get a solution of the KZ equation by starting with one of the BPZ system. This yields the relation
$$\langle \Phi _{j_{2}}(0|0)\Phi _{j_{1}}(x|z)\Phi _{j_{3}}(1|1)\Phi
_{j_{4}}(\infty |\infty )\rangle
=F_{k}(j_{1},j_{2},j_{3},j_{4})X_{k}(j_{1},j_{2},j_{3},j_{4}|x,z)\times$$
$$\times \langle V_{\alpha _{2}}(0)V_{\alpha _{1}}(z)V_{\alpha _{3}}(1)V_{-\frac{1}{2b}}(x)V_{\alpha _{4}}(\infty )\rangle \label{fia}$$
where$$X_{k}(j_{1},j_{2},j_{3},j_{4}|x,z)=|z|^{-4b^{2}j_{1}j_{2}+\alpha _{1}\alpha
_{2}}|1-z|^{-4b^{2}j_{1}j_{3}+\alpha _{1}\alpha _{3}}|x-z|^{-2b^{-1}\alpha
_{1}}|x|^{-2b^{-1}\alpha _{2}}|1-x|^{-2b^{-1}\alpha _{3}},$$$$F_{k}(j_{1},j_{2},j_{3},j_{4})=c_{b}\left( \lambda \pi b^{2b^{2}}\frac{\Gamma (1-b^{2})}{\Gamma (1+b^{2})}\right)
^{1-3j_{1}-j_{2}-j_{3}-j_{4}}\left( \lambda \mu \right) ^{2j_{1}}\prod_{\mu
=1}^{4}\frac{\Upsilon _{b}(2j_{\mu }+1)}{\Upsilon _{b}(2\alpha _{\mu })},
\label{fia2}$$and where the function $\Upsilon _{b}(x)$ is the one introduced in Ref. [ZZ]{} when presenting the explicit form of the Liouville three-point function (see the Appendix for the definition and a survey of functional properties of the $\Upsilon _{b}$-function). The coefficient $c_{b}$ is a $b$-dependent factor, independent of the quantum numbers $j_{\mu }$ and $\alpha _{\mu }$, whose explicit value can be found in Ref. [@Tcross] and references therein, though its explicit value is not important for our purpose here. The relation between the indices $j_{\mu }$, that label the $SL(2,\mathbb{R})_{k}$ representations, and the Liouville momenta $\alpha _{\mu }$ involves a non-diagonal invertible transformation defined through$$2\alpha _{1}=b(j_{1}+j_{2}+j_{3}+j_{4}-1),\qquad 2\alpha
_{i}=b(j_{1}-j_{2}-j_{3}-j_{4}+2j_{i})+Q,\ i=\{2,3,4\}. \label{FZ}$$On the other hand, the relation between the WZNW level $k$ and the Liouville parameter $b$ is given by $$b^{-2}=k-2\in \mathbb{R}_{>0}.$$Now, let us make a remark on the KPZ scaling in (\[fia\]): The parameter $\lambda $ is a free parameter of the WZNW theory and it can be regarded as the mass of the two-dimensional black hole when the gauged $SL(2,\mathbb{R})_{k}/U(1)$ is being considered. This is just a parameter and is free of physical significance since it can be set to any positive value by means of an appropriate rescaling of the zero-mode of the dilaton. Thus, we could absorb the factor $\pi b^{2b^{2}}\Gamma (1-b^{2})/\Gamma (1+b^{2})$ in ([fia2]{}) by simply shifting $\lambda $. This freedom can be used to simplify the functional form of the correlators [@GKnotes].
The Stoyanovsky-Ribault-Teschner identity
-----------------------------------------
Analogously as to the case of the FZ map, the SRT map, discovered by Stoyanovsky in [@S2000], translates solutions of the KZ equation into solutions of the BPZ equation. But this is, in some sense, more general. Unlike the FZ map, the SRT map presents two advantages on its partner: the first is that it involves a diagonal transformation between the quantum numbers $j_{\mu }$ and $\alpha _{\mu }$; the second advantage, and more important indeed, is that it holds for the case of the $N$-point KZ equation for an arbitrary $N$. One of the original versions of the SRT map states the correspondence between $N$-point functions in the WZNW theory and the $2N-2$-point function of the Liouville theory. Furthermore, this was generalized by Ribault in order to connect any $N$-point functions in WZNW to $M$-point functions in Liouville CFT with $N\leq M\leq 2N-2$. Let us describe such generalized form below.
The Ribault formula reads $$\langle \prod_{i=1}^{N}\Phi _{j_{i},m_{i},\bar{m}_{i}}^{\omega
_{i}}(z_{i})\rangle =\mathcal{N}_{k}(j_{1},...j_{N};m_{1},...m_{N})\prod_{r=1}^{M}\int d^{2}w_{r}\ \mathcal{F}_{k}(j_{1},...j_{N};m_{1},...m_{N}|z_{1},...z_{N};w_{1},...w_{M})\times$$$$\times \langle \prod_{t=1}^{N}V_{\alpha _{t}}(z_{t})\prod_{r=1}^{M}V_{-\frac{1}{2b}}(w_{r})\rangle \delta \left( \sum_{\mu =1}^{N}m_{\mu }-\frac{k}{2}M\right) \delta \left( \sum_{\mu =1}^{N}\overline{m}_{\mu }-\frac{k}{2}M\right) , \label{rt}$$where the normalization factor is given by $$\mathcal{N}_{k}(j_{1},...j_{N};m_{1},...m_{N})=\frac{2\pi ^{3-2N}b}{M!\
c_{k}^{M+2}}\prod_{i=1}^{N}\frac{c_{k}\ \Gamma (-m_{i}+j_{i})}{\Gamma
(1-j_{i}+\bar{m}_{i})},$$while the $z$-dependent function is $$\mathcal{F}_{k}(j_{1},...j_{N};m_{1},...m_{N}|z_{1},...z_{N};w_{1},...w_{M})=\frac{\prod_{1\leq r<l}^{N}|z_{r}-z_{l}|^{k-2(m_{r}+m_{l}+\omega _{r}\omega
_{l}k/2+\omega _{l}m_{r}+\omega _{r}m_{l})}}{\prod_{1<r<l}^{M}|w_{r}-w_{l}|^{-k}\prod_{t=1}^{N}\prod_{r=1}^{M}|w_{r}-z_{t}|^{k-2m_{t}}}\times$$$$\times \frac{\prod_{1\leq r<l}^{N}(\bar{z}_{r}-\bar{z}_{l})^{m_{r}+m_{l}-\bar{m}_{r}-\bar{m}_{l}+\omega _{l}(m_{r}-\bar{m}_{r})+\omega _{r}(m_{l}-\bar{m}_{l})}}{\prod_{1<r<l}^{M}(\bar{w}_{r}-\bar{z}_{t})^{m_{t}-\bar{m}_{t}}}. \label{F}$$Here, the Liouville cosmological constant $\mu $ has been set to the value $\mu =b^{2}\pi ^{-2}$ for convenience; this is to make the KPZ scaling of both sides of (\[rt\]) match. It is also important to keep in mind the presence of the free parameter $\lambda $ of the WZNW theory, and the fact that this can still be set to an appropriate value in order to absorb the powers of $\Gamma (b^{2})/\Gamma (1+b^{2})$ in the KPZ overall factor of WZNW correlators [@GKnotes]. For our purpose, we do not need to focus the attention on the specific value of the $j$-independent normalization factor, which we wrote above just for completeness, being equal to $2\pi
^{3-2N}b/M!\ c_{k}^{M+2}$ and where $c_{k}$ represents a $k$-dependent factor whose exact value is discussed in [@R2005] but, again, is not relevant for us.
As in the case of the FZ map, the relation between the level $k$ of the WZNW theory and the parameter $b$ of the Liouville theory is given by $$b^{-2}=k-2\in \mathbb{R}_{>0},$$while the quantum numbers labeling the states of both theories are related through $$\alpha _{i}=-bj_{i}+b+b^{-1}/2=b(k/2-j_{i})\ ,\ \ i=\{1,2,...N\}.
\label{simple}$$We also have$$s=-b^{-1}\sum_{i=1}^{N}\alpha _{i}+b^{-2}\frac{M}{2}+1+b^{-2}\ ,$$and, then, the total winding number is given by$$\sum_{i=1}^{N}\omega _{i}=M+2-N. \label{ese}$$This manifestly shows that scattering processes leading to the violation of the total winding number conservation can occur in principle. In particular, Ribault formula states that the four-point function involving one flowed state of the sector $\omega =-1$ obeys$$\langle \Phi _{J_{2},m_{2},\bar{m}_{2}}^{\omega _{2}=0}(0)\Phi _{J_{1},m_{1},\bar{m}_{1}}^{\omega _{1}=-1}(z)\Phi _{J_{3},m_{3},\bar{m}_{3}}^{\omega
_{3}=0}(1)\Phi _{J_{4},m_{4},\bar{m}_{4}}^{\omega _{4}=0}(\infty )\rangle =\widehat{c}_{b}\prod_{\mu =1}^{N}\frac{\ \Gamma (-m_{\mu }+J_{\mu })}{\Gamma
(1-J_{\mu }+\bar{m}_{\mu })}\times$$$$\begin{aligned}
&&\times \delta (m_{1}+m_{2}+m_{3}-\frac{k}{2})\delta (\overline{m}_{1}+\overline{m}_{2}+\overline{m}_{3}-\frac{k}{2})(z)^{k/2-m_{1}}(1-z)^{k/2-m_{1}}(\overline{z})^{k/2-\overline{m}_{1}}(1-\overline{z})^{k/2-\overline{m}_{1}}\times \\
&&\times \int d^{2}x(x)^{m_{2}-k/2}(\overline{x})^{\overline{m}_{2}-k/2}(1-x)^{m_{3}-k/2}(1-\overline{x})^{\overline{m}_{3}-k/2}(x-z)^{m_{1}-k/2}\ (\overline{x}-\overline{z})^{\overline{m}_{1}-k/2}\times\end{aligned}$$$$\times \langle V_{b\left( k/2-J_{2}\right) }(0)V_{b\left( k/2-J_{1}\right)
}(z)V_{b\left( k/2-J_{3}\right) }(1)V_{-\frac{1}{2b}}(x)V_{b\left(
k/2-J_{3}\right) }(\infty )\rangle , \label{iaf}$$where $\widehat{c}_{b}$ is, again, a $J$-independent factor whose specific value, $\widehat{c}_{b}=2\pi ^{-5}bc_{k}^{N-2-M}$, is known in terms of $c_{k}$, though not important for us. Notice that we have changed the notation here, where we replaced $j_{\mu }$ by $J_{\mu }$; this will be convenient later. Regarding the relation between $J_{\mu }$ and $\alpha
_{\mu }$, let us make some remarks here: Notice that, according to ([simple]{}), the Liouville reflection $\alpha _{\mu }\rightarrow Q-\alpha
_{\mu }$ translates in terms of the WZNW correlators in doing the Weyl reflection $j\rightarrow 1-j.$ This is a simple comment, but is also important. In particular, because of (\[whatt\]), this implies that the correlation function involving the field $\Phi _{j}(x|z)$ and the one involving the field $\Phi _{1-j}(x|z)$ are connected one to each other by the overall factor $R_{b}(b(k/2-j))$. It is not difficult to see that this simple affirmation leads directly to the exact expression for the WZNW two-point function, for instance. This comment will become important below.
The four-point function of winding strings
==========================================
The idea
--------
The idea is to use Eq. (\[fia\]) and Eq. (\[iaf\]) in order to relate the four-point function $\langle \Phi _{J_{1},m_{1},\bar{m}_{1}}^{\omega
_{1}=-1}(z_{1})$ $\prod_{i=2}^{4}\Phi _{J_{i},m_{i},\bar{m}_{i}}^{\omega
_{i}=0}(z_{i})\rangle $ with the four-point function $\langle \prod_{\mu
=1}^{4}\Phi _{j_{\mu }}(x_{\mu }|z_{\mu })\rangle $. The nexus will be the Liouville five-point function $\langle \prod_{\mu =1}^{4}V_{\alpha _{\mu
}}(z_{\mu })V_{-1/2b}(x)\rangle $, appearing in the right hand side of both expressions: this is connected to the first through the SRT map while is connected to the second through the FZ map. After doing this, the second step would be to diagonalize" the relation between indices $j_{\mu }$ and indices $J_{\mu }$. We will do this by using the preliminary result of section 3.2. Notice that, according to Eq. (\[fia\]) and Eq. (\[iaf\]), the relation between momenta $J_{\mu }$ and $j_{\mu }$ is given by (\[J\]), below. The third step would be to perform a Fourier transform in order to translate the operators $\Phi _{j_{\mu }}(x_{\mu }|z_{\mu })$ of the $x$-basis into operators $\Phi _{j_{\mu },m_{\mu },\overline{m}_{\mu
}}(z_{\mu })$ of the $m$-basis.
Non diagonal symmetry of the KZ equation
----------------------------------------
As mentioned, the purpose here is to prove a functional relation obeyed by two solutions of the KZ equation that will be useful further. Specifically, we will see that four-point correlation functions with momenta $j_{1},...j_{4}$ have a simple expression in terms of the analogous quantity with momenta $\widetilde{j}_{1},...\widetilde{j}_{4}$, being related through $$2\widetilde{j}_{\mu }=\sum_{\nu =1}^{4}j_{\nu }-2j_{\mu }.$$To see this, let us begin by observing that, according to (\[FZ\]), the fact of performing the change $\widetilde{j}_{\mu }\rightarrow j_{\mu }$ for all the indices $\mu =\{1,2,3,4\}$ translates in terms of the Liouville momenta $\alpha _{\mu }$ in doing the change $\alpha _{i}\rightarrow
Q-\alpha _{i}$ for $i=\{2,3,4\},$ while keeping $\alpha _{1}$ unchanged. Then, by taking into account the Liouville reflection symmetry (\[whatt\]) and the dictionary (\[FZ\]), we get[^2]$$\left\langle \Phi _{j_{1}}(x|z)\Phi _{j_{2}}(0|0)\Phi _{j_{3}}(1|1)\Phi
_{j_{4}}(\infty |\infty )\right\rangle =\frac{F_{k}(j_{1},j_{2},j_{3},j_{4})X_{k}(j_{1},j_{2},j_{3},j_{4}|x,z)}{F_{k}(\widetilde{j}_{1},\widetilde{j}_{2},\widetilde{j}_{3},\widetilde{j}_{4})X_{k}(\widetilde{j}_{1},\widetilde{j}_{2},\widetilde{j}_{3},\widetilde{j}_{4}|x,z)}\prod_{i=2}^{4}R_{b}(\alpha _{i})\times$$$$\times \left\langle \Phi _{\widetilde{j}_{1}}(x|z)\Phi _{\widetilde{j}_{2}}(0|0)\Phi _{\widetilde{j}_{3}}(1|1)\Phi _{\widetilde{j}_{4}}(\infty
|\infty )\right\rangle \label{wupa}$$The factor $F_{k}(j_{1},j_{2},j_{3},j_{4})/F_{k}(\widetilde{j}_{1},\widetilde{j}_{2},\widetilde{j}_{3},\widetilde{j}_{4})$ involves a quotient of products of $\Upsilon _{b}$-functions that, by making use of the formulae in the Appendix (see (\[magicsuerte\]) below), can be substantially simplified once one takes into account the presence of the three factors $R_{b}(\alpha _{i})$ ($i=2,3,4$). Then, the results reads$$\left\langle \Phi _{j_{1}}(x|z)\Phi _{j_{2}}(0|0)\Phi _{j_{3}}(1|1)\Phi
_{j_{4}}(\infty |\infty )\right\rangle =b^{2\mathcal{P}(j)}|x|^{2b^{-1}(Q-2\alpha _{2})}|1-x|^{2b^{-1}(Q-2\alpha
_{3})}|z|^{b^{2}(j_{1}+j_{2})^{2}-b^{2}(j_{3}+j_{4})^{2}}\times$$$$\times |1-z|^{b^{2}(j_{1}+j_{3})^{2}-b^{2}(j_{2}+j_{4})^{2}}\prod_{\mu
=1}^{4}\frac{\Upsilon _{b}(2j_{\mu }b-b)}{\Upsilon _{b}\left( b\sum_{\nu
=1}^{4}j_{\nu }-2j_{\mu }b-b\right) }\left\langle \Phi _{\widetilde{j}}(x|z)\Phi _{\widetilde{j}_{2}}(0|0)\Phi _{\widetilde{j}_{3}}(1|1)\Phi _{\widetilde{j}_{4}}(\infty |\infty )\right\rangle \label{cardinal}$$where $\mathcal{P}(j)$ is a polynomial in the indices $j_{\mu }$, namely $\mathcal{P}(j)=-3j_{1}+j_{2}+j_{3}+j_{4},$ though is not actually relevant here. Since $j_{1}+j_{2}=\widetilde{j}_{3}+\widetilde{j}_{4}$ and $j_{1}+j_{3}=\widetilde{j}_{2}+\widetilde{j}_{4}$, one immediately notices that the $\mathbb{Z}_{2}$-invariant form under the involution $j_{\mu
}\rightarrow \widetilde{j}_{\mu }$ is given by$$\mathcal{I}_{k}^{\pm }(x,z)=Z_{k}^{\pm
}(j_{1},j_{2},j_{3},j_{4}|x,z)\left\langle \Phi _{j_{1}}(x|z)\Phi
_{j_{2}}(0|0)\Phi _{j_{3}}(1|1)\Phi _{j_{4}}(\infty |\infty )\right\rangle$$where$$\begin{aligned}
Z_{k}^{+}(j_{1},j_{2},j_{3},j_{4}|x,z)
&=&b^{2(j_{1}-j_{2}-j_{3}-j_{4})}|x|^{j_{1}+j_{2}-j_{3}-j_{4}+k-1}|1-x|^{j_{1}-j_{2}+j_{3}-j_{4}+k-1}\times
\\
&&\times
|z|^{+b^{2}(j_{3}+j_{4})^{2}}|1-z|^{+b^{2}(j_{2}+j_{4})^{2}}\prod_{\mu
=1}^{4}\Upsilon _{b}^{-1}(2j_{\mu }b-b) \\
Z_{k}^{-}(j_{1},j_{2},j_{3},j_{4}|x,z)
&=&b^{2(j_{1}-j_{2}-j_{3}-j_{4})}|x|^{j_{1}+j_{2}-j_{3}-j_{4}+k-1}|1-x|^{j_{1}-j_{2}+j_{3}-j_{4}+k-1}\times
\\
&&\times
|z|^{-b^{2}(j_{1}+j_{2})^{2}}|1-z|^{-b^{2}(j_{1}+j_{3})^{2}}\prod_{\mu
=1}^{4}\Upsilon _{b}^{-1}(2j_{\mu }b-b).\end{aligned}$$Formula (\[cardinal\]) will be useful in the next section. Similar functional relations were studied in Ref. [@Yo2005; @Andreev; @Nichols]. Now, let us move to the case of winding strings.
The four-point function of winding strings
------------------------------------------
In this subsection, as an application of our formula (\[cardinal\]), we will employ it to show that, by making use of both FZ and SRT maps, it is feasible to write down a formula that expresses the winding violating four-point functions in terms of the zero-winding four-point function. To do this, let us begin by considering the FZ identity$$\langle \Phi _{j_{2}}(0|0)\Phi _{1-j_{1}}(x|z)\Phi _{j_{3}}(1|1)\Phi
_{j_{4}}(\infty |\infty )\rangle
=F_{k}(1-j_{1},j_{2},j_{3},j_{4})X_{k}(1-j_{1},j_{2},j_{3},j_{4}|x,z)\times$$$$\times \langle V_{\widehat{\alpha }_{2}}(0)V_{\widehat{\alpha }_{1}}(z)V_{\widehat{\alpha }_{3}}(1)V_{-\frac{1}{2b}}(x)V_{\widehat{\alpha }_{4}}(\infty )\rangle , \label{fiaty}$$where the quantum numbers $\widehat{\alpha }_{\mu }$ and $j_{\mu }$ are then related as it follows$$\begin{aligned}
\widehat{\alpha }_{1} &=&\frac{b}{2}(-j_{1}+j_{2}+j_{3}+j_{4}),\qquad
\widehat{\alpha }_{2}=\frac{b}{2}(-j_{1}+j_{2}-j_{3}-j_{4}+k), \\
\widehat{\alpha }_{3} &=&\frac{b}{2}(-j_{1}-j_{2}+j_{3}-j_{4}+k),\qquad
\widehat{\alpha }_{4}=\frac{b}{2}(-j_{1}-j_{2}-j_{3}+j_{4}+k).\end{aligned}$$Now, let us also define quantum numbers $J_{\mu }$ as$$b(k/2-J_{\mu })=\widehat{\alpha }_{\mu }$$for $\mu =\{1,2,3,4\}$. Then, by taking into account (\[whatt\]), we have$$\langle V_{\widehat{\alpha }_{2}}(0)V_{\widehat{\alpha }_{1}}(z)V_{\widehat{\alpha }_{3}}(1)V_{-\frac{1}{2b}}(x)V_{\widehat{\alpha }_{4}}(\infty
)\rangle =\widetilde{A}_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)\langle \Phi
_{j_{2}}(0|0)\Phi _{1-j_{1}}(x|z)\Phi _{j_{3}}(1|1)\Phi _{j_{4}}(\infty
|\infty )\rangle . \label{fiatyu}$$with$$\widetilde{A}_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)=F_{k}^{-1}(1-j_{1},j_{2},j_{3},j_{4})X_{k}^{-1}(1-j_{1},j_{2},j_{3},j_{4}|x,z).
\label{volvio}$$Consequently, we can write the following SRT identity$$\langle \Phi _{J_{2},m_{2},\bar{m}_{2}}^{\omega _{2}=0}(0)\Phi _{J_{1},m_{1},\bar{m}_{1}}^{\omega _{1}=-1}(z)\Phi _{J_{3},m_{3},\bar{m}_{3}}^{\omega
_{3}=0}(1)\Phi _{J_{4},m_{4},\bar{m}_{4}}^{\omega _{4}=0}(\infty )\rangle =\widehat{c}_{b}\prod_{\mu =1}^{N}\frac{\ \Gamma (-m_{\mu }+J_{\mu })}{\Gamma
(1-J_{\mu }+\bar{m}_{\mu })}\times$$$$\begin{aligned}
&&\times \delta (m_{1}+m_{2}+m_{3}-k/2)\delta (\overline{m}_{1}+\overline{m}_{2}+\overline{m}_{3}-k/2)(z)^{k/2-m_{1}}(\overline{z})^{k/2-\overline{m}_{1}}(1-z)^{k/2-m_{1}}(1-\overline{z})^{k/2-\overline{m}_{1}} \\
&&\times \int d^{2}x(x)^{m_{2}-k/2}(1-x)^{m_{3}-k/2}(\overline{x})^{\overline{m}_{2}-k/2}(1-\overline{x})^{\overline{m}_{3}-k/2}(x-z)^{m_{1}-k/2}(\overline{x}-\overline{z})^{\overline{m}_{1}-k/2}\times \end{aligned}$$$$\times \ \langle V_{\widehat{\alpha }_{2}}(0)V_{\widehat{\alpha }_{1}}(z)V_{\widehat{\alpha }_{3}}(1)V_{-\frac{1}{2b}}(x)V_{\widehat{\alpha }_{4}}(\infty )\rangle . \label{iafyy}$$Hence, plugging (\[fiatyu\]) into (\[iafyy\]), we find$$\left\langle \Phi _{J_{1},m_{1},,\overline{m}_{1}}^{\omega _{1}=-1}(z)\Phi
_{J_{2},m_{2},,\overline{m}_{2}}^{\omega _{1}=0}(0)\Phi _{J_{3},m_{3},\overline{m}_{3}}^{\omega _{1}=0}(1)\Phi _{J_{4},m_{4},\overline{m}_{4}}^{\omega _{4}=0}(\infty )\right\rangle =\prod_{\mu =1}^{4}\frac{\Gamma
(J_{\mu }-m_{\mu })}{\Gamma (1-J_{\mu }+\overline{m}_{\mu })}\times$$$$\begin{aligned}
&&\times \delta (m_{1}+m_{2}+m_{3}-k/2)\delta (\overline{m}_{1}+\overline{m}_{2}+\overline{m}_{3}-k/2)(z)^{-m_{1}}(\overline{z})^{-\overline{m}_{1}}(1-z)^{-m_{1}}(1-\overline{z})^{-\overline{m}_{1}}\times \notag \\
&&\times \int d^{2}x\ (x-z)^{m_{1}}(\overline{x}-\overline{z})^{\overline{m}_{1}}(x)^{m_{2}}(\overline{x})^{\overline{m}_{2}}(1-x)^{m_{3}}(1-\overline{x})^{\overline{m}_{3}}\ A_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)\times \notag \\
&&\times \left\langle \Phi _{1-j_{1}}(x|z)\Phi _{j_{2}}(0|0)\Phi
_{j_{3}}(1|1)\Phi _{j_{4}}(\infty |\infty )\right\rangle , \label{A}\end{aligned}$$where we simply defined$$A_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)=\left| \frac{z\ (1-z)}{x\ (1-x)(z-x)}\right| ^{k}\widetilde{A}_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)$$and where, according to the definitions above, the indices $J_{\mu }$ and $j_{\mu }$ turn out to be related by$$2J_{1}=k+j_{1}-j_{2}-j_{3}-j_{4},\qquad 2J_{i}=j_{1}+j_{2}+j_{3}+j_{4}-2j_{i}
\label{J}$$for $i=\{2,3,4\}$. Then, it is feasible to show that$$A_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)\sim \frac{\Upsilon _{b}(2b(k/2-J_{1}))}{\Upsilon _{b}(b(k/2+1-J_{1}-J_{2}-J_{3}-J_{4}))}|x|^{-2J_{2}}|1-x|^{-2J_{3}}|x-z|^{-2J_{1}}\times$$ $$\times \prod_{i=2}^{4}\frac{\Upsilon _{b}(b(2J_{i}-1))}{\Upsilon
_{b}(b(2J_{i}-J_{1}-J_{2}-J_{3}-J_{4}+k/2-1))}|z|^{k+4b^{2}(1-j_{1})j_{2}-4\widehat{\alpha }_{1}\widehat{\alpha }_{2}}|1-z|^{k+4b^{2}(1-j_{1})j_{3}-4\widehat{\alpha }_{1}\widehat{\alpha }_{3}}. \label{elA}$$where the symbol $\sim $ stands just because we are not writing the $k$-dependent overall factor and the precise KPZ scaling here (this can be directly read from Eq. (\[volvio\]) if necessary). For short, in the last equation we have employed the notation$$2\widehat{\alpha }_{1}=b(k/2-J_{1})=b(-j_{1}+j_{2}+j_{3}+j_{4}),\qquad 2\widehat{\alpha }_{i}=b(k/2-J_{i})=b(k+2j_{i}-j_{1}-j_{2}-j_{3}-j_{4}),
\label{alfahat}$$and one could also find convenient to define$$2\alpha _{1}=b(j_{1}+j_{2}+j_{3}+j_{4}-1),\qquad 2\alpha
_{i}=b(j_{1}-j_{2}-j_{3}-j_{4}+2j_{i}+k-1). \label{alfa}$$Identity (\[A\]) does already express the winding violating correlators in terms of the conservative ones. However, such expression is still not fully satisfactory since the right hand side of (\[A\]) contains a four-point correlator that, instead of involving states with momenta $J_{\mu }$, involves states with momenta $j_{\mu }$, $\mu =\{1,2,3,4\}$. Here is where our result of subsection 3.2. becomes useful, since the following step would be to relate the correlator in the $j_{\mu }$-basis with the one that is diagonal in the $J_{\mu }$-basis. This can be simply achieved by making use of the equation (\[cardinal\]) and by taking into account (\[J\]), yielding $$\left\langle \Phi _{j_{1}}(x|z)\Phi _{j_{2}}(0|0)\Phi _{j_{3}}(1|1)\Phi
_{j_{4}}(\infty |\infty )\right\rangle
=B_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)\times$$$$\times \left\langle \Phi _{\frac{k}{2}-J_{1}}(x|z)\Phi _{J_{2}}(0|0)\Phi
_{J_{3}}(1|1)\Phi _{J_{4}}(\infty |\infty )\right\rangle , \label{B}$$with the normalization factor being$$B_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)=b^{2\widehat{\mathcal{P}}(j)}|z|^{b^{2}(J_{3}+J_{4})^{2}-b^{2}(k/2-J_{1}+J_{2})^{2}}|1-z|^{b^{2}(J_{2}+J_{4})^{2}-b^{2}(k/2-J_{1}+J_{3})^{2}}\times$$$$\begin{aligned}
&&\times \frac{\Upsilon _{b}(b(J_{1}+J_{2}+J_{3}+J_{4}-k/2-1))}{\Upsilon
_{b}(b(k-2J_{1}-1))}\prod_{i=2}^{4}\frac{\Upsilon
_{b}(b(2J_{i}-J_{1}-J_{2}-J_{3}-J_{4}+k/2-1))}{\Upsilon _{b}(b(2J_{i}-1))}\times \notag \\
&&\times
|x|^{k-2(J_{1}-J_{2}+J_{3}+J_{4})}|1-x|^{k-2(J_{1}+J_{2}-J_{3}+J_{4})};
\label{elB}\end{aligned}$$here, $\widehat{\mathcal{P}}(j)$ is again a polynomial in the indices $j_{\mu }$ that is related to the $\mathcal{P}(j)$ in (\[cardinal\]) through the replacement $j_{i}\rightarrow J_{i}$, $i=\{2,3,4\}$, $j_{1}\rightarrow k/2-J_{1}$. In writing down (\[elB\]) we used the fact that, as in the previous subsection, the combination of the $\Upsilon _{b}$-functions and the $\Gamma $-functions in $R_{b}(\alpha _{i})$ leads to a very simple expression. In this case, it comes from $$\frac{\Upsilon _{b}(b(j_{1}-j_{2}-j_{3}-j_{4}+2j_{i}))}{\Upsilon
_{b}(b(-j_{1}+j_{2}+j_{3}+j_{4}-2j_{i}))R_{b}\left( \frac{Q}{2}-\frac{b}{2}(j_{1}-j_{2}-j_{3}-j_{4}+2j_{i})\right) }=$$$$=\left( \pi b^{2(b^{2}-1)}\frac{\Gamma (1-b^{2})}{\Gamma (1+b^{2})}\right)
^{J_{1}-J_{2}-J_{3}-J_{4}+2J_{i}-k/2}. \label{magicsuerte}$$Hence, up to the Weyl reflection $j_{1}\rightarrow 1-j_{1}$ that we discuss below, equations (\[B\]) and (\[A\]) represent the result we wanted to prove: the four-point function involving one winding state of the sector $\omega =-1$ admits to be expressed in terms of the four-point function of non-winding states. Let us discuss the final form of this result in the following paragraphs (see (\[DaleR\]) below), where we address the comparison with the case of the three-point function. We will also comment on possible applications and conclude with some remarks in the following section.
Analogy with the case of the three-point function
-------------------------------------------------
The three-point function that includes one spectral flowed state of the sector $\omega =-1$ can be written in terms of the structure constant of three non-flowed states [@FZZ; @MO3]. In the subsection above we proved a similar relation at the level of the four-point functions. Then, the natural question we want to address now is whether both relations are connected in some way. As we will see below, these are indeed analogous. Actually, expressions (\[A\]) and (\[B\]), once considered together, correspond to the generalization of the formula that holds for the three-point structure constants. First, one of the aspects one notices in the formulae above is that the first operator in the right hand side of (\[B\]) represents the state of momentum $\frac{k}{2}-J_{1}$, instead that of momentum $J_{1}$. Actually, this should not be a surprise since it precisely resembles what happens at the level of the three-point function, where the violating winding correlator $$\left\langle \Phi _{J_{1},m_{1},,\overline{m}_{1}}^{\omega _{1}=-1}(0)\Phi
_{J_{2},m_{2},,\overline{m}_{2}}^{\omega _{2}=0}(1)\Phi _{J_{3},m_{3},\overline{m}_{3}}^{\omega _{3}=0}(\infty )\right\rangle$$turns out to be proportional to the integral of the conservative structure constant that corresponds to$$\sim B^{-1}(k/2-J_{1})\left\langle \Phi _{k/2-J_{1}}(0|0)\Phi
_{J_{2}}(1|1)\Phi _{J_{3}}(\infty |\infty )\right\rangle .$$The same association between momenta $J_{1}$ and $k/2-J_{1}$ occurs for the four-point functions here. Actually, such relation between states with quantum numbers $\omega =0$, $J$ and those with $\omega =\pm 1$, $k/2-J$ is understood from the algebraic point of view: This concerns the identification between the discrete series of the $SL(2,\mathbb{R})_{k}$ representations by means of the spectral flow automorphism. This is related to the fact hat both states with $\omega =0$ and $|\omega |=1$ satisfy similar KZ equations. Besides, this is related to the fact that the OPE between the vertex operator $\Phi _{J}(z)$ and the spectral flow operator $\Phi _{k/2}(w)$, which is necessary to provide the winding $\omega _{1}=1$ to the first, yields the single string contribution $\Phi _{k/2-J}(z)$. The involution $J\rightarrow k/2-J$, as a symmetry of the KZ equation, was studied in the first part of our work, [@Yo2005]. As it was detailed in Ref. [@MO3], when presenting the original derivation of [@FZZ], the proportionality factor connecting both violating winding and non-violating winding three-point correlators is basically given by the reflection coefficient $B(J_{1})\sim B^{-1}(k/2-J_{1})$ of the $SL(2,\mathbb{R})_{k}$ WZNW model (see the formula above). This quantity has the form$$B(J)=\frac{1}{\pi b^{2}}\left( \lambda \pi \frac{\Gamma \left(
1-b^{2}\right) }{\Gamma \left( 1+b^{2}\right) }\right) ^{1-2J}\frac{\Gamma
\left( 1+b^{2}-2Jb^{2}\right) }{\Gamma \left( 2Jb^{2}-b^{2}\right) },
\label{bB}$$and therefor$$B(J)=\frac{1}{\pi ^{2}b^{4}}\left( \frac{1}{\lambda \pi }\frac{\Gamma \left(
1+b^{2}\right) }{\Gamma \left( 1-b^{2}\right) }\right)
^{b^{-2}}B^{-1}(k/2-J).$$This permits to show that the picture for the three-point functions turns out to be similar to the one we are obtaining here for the four-point functions. Actually, this is not only manifested in the shifting $J_{1}\rightarrow k/2-J_{1}$, but also in the fact that, when combining both (\[A\]) and (\[B\]), a factor $\frac{\Upsilon _{b}(2b(k/2-J_{1}))}{\Upsilon _{b}(2b(k/2-J_{1})-b)}$ also arises in the product between $A_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)$ and $B_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)$. This factor is precisely proportional to $(2J-1)B(J)$; thus, this is actually in equal footing as to how the WZNW reflection factor stands in the relation between violating and non-violating three-point correlators.
On the other hand, some nice cancellations occur when combining expressions (\[A\]) and (\[B\]). First, we observed that, instead of the operator $\Phi _{j_{1}}(x|z)$, the correlator in the right hand side of (\[A\]) involves its Weyl reflected operator $\Phi _{1-j_{1}}(x|z)$; and, as we commented at the end of section 2, this implies that a factor $R_{b}^{-1}(b(k/2-j_{1}))$ has to be included as well in order to plug ([A]{}) into (\[B\]). Then, such reflection coefficient is eventually simplified due to another contribution standing when multiplying $A_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)$ and $B_{k}(J_{1},J_{2},J_{3},J_{4}|x,z)$; namely, $$\frac{\Upsilon _{b}(b(J_{1}+J_{2}+J_{3}+J_{4}-k/2-1))}{\Upsilon
_{b}(-b(J_{1}+J_{2}+J_{3}+J_{4}-k/2-1))}=\frac{\Upsilon _{b}(2j_{1}b-b)}{\Upsilon _{b}(b-2j_{1}b)}\sim R_{b}(b(k/2-j_{1})).$$Moreover, these are not all the cancellations that take place. Let us also observe that three factors of the form $\frac{\Upsilon _{b}(b(2J_{i}-1))}{\Upsilon _{b}(b(2J_{i}-J_{1}-J_{2}-J_{3}-J_{4}+k/2-1))}$ (with $i=\{2,3,4\}$) and their respective inverses are mutually cancelled as well. These come from the second line of Eq. (\[elA\]) and the second line of Eq. (\[elB\]), respectively. Finally, equations (\[A\]) and (\[B\]), considered together, leads to the following expression$$\left\langle \Phi _{J_{1},m_{1},,\overline{m}_{1}}^{\omega _{1}=-1}(z)\Phi
_{J_{2},m_{2},,\overline{m}_{2}}^{\omega _{1}=0}(0)\Phi _{J_{3},m_{3},\overline{m}_{3}}^{\omega _{1}=0}(1)\Phi _{J_{4},m_{4},\overline{m}_{4}}^{\omega _{4}=0}(\infty )\right\rangle \sim B(J_{1})\prod_{\mu =1}^{4}\frac{\Gamma (J_{\mu }-m_{\mu })}{\Gamma (1-J_{\mu }+\overline{m}_{\mu })}\times$$$$\begin{aligned}
&&\int d^{2}x\ (x-z)^{m_{1}+\Delta _{1}}(\overline{x}-\overline{z})^{\overline{m}_{1}+\Delta _{1}}\prod_{i=1}^{2}(z-z_{i})^{-m_{i}+\Delta _{i}}(\overline{z}-\overline{z_{i}})^{-\overline{m}_{i}+\Delta
_{i}}(x-x_{i})^{m_{i}+\widetilde{\Delta }_{i}}(\overline{x}-\overline{x}_{i})^{\overline{m}_{i}+\widetilde{\Delta }_{i}}\times \notag \\
&&\times \left\langle \Phi _{\frac{k}{2}-J_{1}}(x|z)\Phi _{J_{2}}(0|0)\Phi
_{J_{3}}(1|1)\Phi _{J_{4}}(\infty |\infty )\right\rangle \delta
(m_{1}+m_{2}+m_{3}-k/2)\delta (\overline{m}_{1}+\overline{m}_{2}+\overline{m}_{3}-k/2), \notag \\
&& \label{DaleR}\end{aligned}$$where $B(J_{1})$ is the WZNW reflection coefficient (\[bB\]), while the symbol $\sim $ stands for some $k$-dependent overall factor. For short, in this expression we denoted $x_{2}=z_{2}=0$ and $x_{3}=z_{3}=1$, and exponents $\Delta _{1,2,3}$ and $\widetilde{\Delta }_{2,3}$ refer to $J$-dependent ($m$-independent) linear combinations that are directly given by the exponents arising in (\[A\]) and (\[B\]). Expression (\[DaleR\]) represents the main result here, and it turns out to be actually analogous to the formula connecting winding violating three-point functions to those involving merely three non-winding states. Namely, we showed that the $SL(2,R)_{k}$ WZNW four-point function$$\left\langle \Phi _{J_{1},m_{1},,\overline{m}_{1}}^{\omega _{1}=-1}(z)\Phi
_{J_{2},m_{2},,\overline{m}_{2}}^{\omega _{1}=0}(0)\Phi _{J_{3},m_{3},\overline{m}_{3}}^{\omega _{1}=0}(1)\Phi _{J_{4},m_{4},\overline{m}_{4}}^{\omega _{4}=0}(\infty )\right\rangle ,$$while involving one state of the spectral flowed sector $\omega _{1}=-1,$ can be expressed in terms of the integral of the four-point function$$\sim B^{-1}(k/2-J_{1})\left\langle \Phi _{\frac{k}{2}-J_{1}}(x|z)\Phi
_{J_{2}}(0|0)\Phi _{J_{3}}(1|1)\Phi _{J_{4}}(\infty |\infty )\right\rangle ,$$defined in terms of merely non-spectral flowed states $\omega _{\mu }=0$, $\mu =\{1,2,3,4\}$. The integration over the complex variable $x$ clearly stands for the requirement of the Fourier transform when changing to the $m$-basis. Furthermore, it is worth pointing out that the factor$$\prod_{\mu =1}^{4}\frac{\Gamma (J_{\mu }-m_{\mu })}{\Gamma (1-J_{\mu }+\overline{m}_{\mu })}$$is also present here, completing the analogy with the three-point function case. The fact that a similar pattern is found at the level of the four-point correlators opens a window that would permit to gain information about the violating winding four-point functions by making use of all what is known about the WZNW conservative amplitudes.
Further applications
--------------------
So far, we presented a concise application of our results of [@Yo2005], by showing that the symmetries of the KZ equation, that were inferred by means of its relation to the BPZ equation in Liouville theory, lead to a relation between violating and conserving winding amplitudes in $AdS_{3}$ string theory. Besides, there exist some further application of the formula (\[A\]) we obtained here: One of these feasible applications concerns the integral representation of the four-point conformal blocks and the factorization *ansatz*. Actually, let us return to (\[antonia\]) and expand the chiral conformal blocks as it follows$${\mathcal{G}}_{j_{1},j_{2,}j,j_{3},j_{4}}(x|z)=z^{h_{j}-h_{j_{1}}-h_{j_{2}}}x^{j-j_{1}-j_{2}}\sum_{n=0}^{\infty }{\mathcal{G}}_{j_{1},j_{2,}j,j_{3},j_{4}}^{(n)}(x)z^{n}, \label{niato}$$where, now, $j$ acts as an internal index that labels different solutions to the differential equation. In the *stringy* interpretation it is feasible to assign physical meaning to such index: this is the one parameterizing the intermediate states interchanged in a given four-point scattering process. Then, the integration over the internal index $j$ stands in order to include all the contributions of the conformal blocks ([antonia]{}) to the four-point amplitude (\[recorda\]). By replacing ([niato]{}) into the KZ equation one finds that the leading term contribution in the $z$-power expansion obeys the hypergeometric differential equation. For instance, this permits to analyze the monodromy properties at $x=z$ for leading orders in the large $1/x$ limit. Studying this regime turns out to be important to fully understand the analytic structure of the four-point function [@MO3]. For such purpose, it is useful to study the solution in the vicinity of the point $x=z$. By assuming the extension of the expressions above to the $SL(2,\mathbb{R})$ case, and by deforming the contour of integration as ${\mathcal{C}}\rightarrow \frac{k}{2}-{\mathcal{C}}=\frac{k-1}{2}-i\mathbb{R}$, one can see that, once the integral over $j$ is performed, the solution is actually monodromy invariant at $z=x$. Then, the leading contribution of the solution can be written as it follows [@MO3]$$|f_{j_{1},j_{2},j_{3},j_{4}}(x,z)|^{2}=\frac{1}{2}\int_{\frac{k-1}{2}+i\mathbb{R}}dj|x|^{2(h_{j}-h_{j_{1}}-h_{j_{2}}-j+j_{1}+j_{2})}|zx^{-1}|^{2(h_{j}-h_{j_{1}}-h_{j_{2}})}\frac{C(j_{1},j_{2},j)C(j,j_{3},j_{4})}{B(j)}\times$$$$\times |F(j_{1}+j_{2}-j,j_{3}+j_{4}-j,k-2j,zx^{-1})|^{2}\left( 1+{\mathcal{O}}\left( z^{-1}x\right) \right) +2\pi i\sum_{\{x_{i}\}}Res_{(x=x_{i})}$$where $F(a,b,c;d)$ is the hypergeometric function, and where $\{x_{i}\}$ refers to the set of poles located in the region $1<Re(2j)<k-1$. These poles take the form $j-j_{1}-j_{2}\in \mathbb{N}$ if the constraint $\sum_{i=1}^{4}j_{i}<k$ is assumed (see [@MO3] for the details of the construction and the issue of the integration over the complex variables $z$ and $x$). Taking all this into account, one could try to plug the expression for the conformal blocks (\[antonia\]) in the equation we wrote in (\[A\]). This would lead to an integral realization of the winding violating four-point function. Moreover, such a realization could be then evaluated in a particular case $J_{4}=k/2$ and, by means of the prescription of [@FZZ], be used to compute the three-point function $\langle \Phi
_{J_{1},m_{1},\bar{m}_{1}}^{\omega _{1}=-1}\Phi _{J_{2},m_{2},\bar{m}_{2}}^{\omega _{2}=0}\Phi _{J_{3},m_{3},\bar{m}_{3}}^{\omega _{3}=+1}\rangle
,$ with two winding states of the sectors $\omega =+1$ and $\omega =-1$. This idea does deserve to be explored in future work[^3].
Other possible analysis that can be done regards the Coulomb gas-like integral representation recently presented in [@Yo2005b; @Yo2006]. There, it was proven that the SRT map can be thought of as a free field representation of the $SL(2,\mathbb{R})_{k}$ model in terms of the action$$S=\frac{1}{4\pi }\int d^{2}z\left( -\partial \varphi \bar{\partial}\varphi
+QR\varphi +\mu e^{\sqrt{2}b\varphi }\right) +S_{M},$$where the specific model representing the matter sector $S_{M}$ corresponds to a $c<1$ conformal field theory defined by the action$$S_{M}=\frac{1}{4\pi }\int d^{2}z\left( \partial X^{0}\bar{\partial}X^{0}-\partial X^{1}\bar{\partial}X^{1}-i\sqrt{k}RX^{1}\right) .$$This free field representation leads to an integral expression for the four-point function when the winding number conservation is being maximally violated. On the other hand, for the cases where the winding is conserved, the action $S_{M}$ has to be perturbed by introducing a new term of the form$$\mathcal{O}=\int d^{2}z\ e^{-\sqrt{\frac{k-2}{2}}\varphi (z)+i\sqrt{\frac{k}{2}}X^{1}(z)}.$$This is a perturbation, represented by a primary operator of the matter sector and properly dressed with the coupling to the Liouville field in order to turn it into a marginal deformation. It could be interesting to check the formulas (\[A\])-(\[B\]) in terms of this representation.
Concluding remarks
==================
As an application of our [@Yo2005], in this brief note we proved a new relation existing between the four-point function of winding strings in $AdS_{3}$ (spectral flowed states of the $SL(2,\mathbb{R})_{k}$ WZNW model) and the four-point function of non-winding strings (non-flowed $SL(2,\mathbb{R})_{k}$ states). We showed that the former admits a simple expression in terms of the last, which was studied in more detail in the literature. The fact that such a simple expression exits is mainly due to two facts: first, the non-diagonal $\mathbb{Z}_{2}$ symmetry of the KZ equation realized by (\[cardinal\]); secondly, to the fact that correlators of the WZNW model are connected to those of the Liouville theory in more than one way.
To be more precise, here we have shown the relation that connects the four-point WZNW correlation function involving one $\omega =-1$ spectral flowed state to the four-point function of non-spectral flowed states. This consequently relates the scattering amplitude involving one winding string state in $AdS_{3}$ with the analogous observable for non-winding strings. Such relation is realized by Eq. (\[DaleR\]), and is reminiscent of the relation obeyed by the $SL(2,\mathbb{R})_{k}$ structure constants. On the other hand, it is likely that the four-point function involving one state in the sector $\omega =-1$ corresponds to a limiting procedure of a WZNW five-point function involving one spectral flow operator of momentum $J_{5}=k/2$. Consequently, it is plausible that the connection manifested by equations (\[A\]) and (\[B\]) could then be obtained in an alternative way; for instance, by directly studying the operator product expansion of operators $\Phi _{J_{1}}(x_{1}|z_{1})\Phi _{k/2}(x_{2}|z_{2})$ in the coincidence limit in the five-point conformal blocks. In such case, our result would be seen from a different perspective since, as far as our derivation of (\[A\]) and (\[B\]) is invertible, this would lead to the possibility of deriving the FZ map from a particular case of the SRT map. This was, indeed, one of the main motivations we had for studying this, though the operator product expansion involving an additional spectral flow operator (a fifth operator) turns out to be complicated enough and thus we leave it for the future. Conversely, as far as the relation between both maps is not trivial, as it is emphasized in [@RT2005], our way of proving the connection between violating winding correlators and conserving winding correlators turns out to be an ingenious trick. The conciseness of such a deduction becomes particularly evident once one gets familiarized with the complicated definition of the action of spectral flow operator in the $x$-basis. Moreover, even in the $m$-basis the computation turns out to be simplified by our method because no explicit reference to the decoupling equation of the degenerate state $J=k/2$ was needed at all, since it is was already encoded in the Ribault formula (\[iaf\]). Our hope is that the result of this brief paper might be useful in working out the details of the $SL(2,\mathbb{R})_{k}$ WZNW four-point functions, which is our main challenge within this line of research.
$$$$
**Acknowledgement:** First, I would like to thank Cecilia Garraffo for collaboration in the computations of the section 3. I am also grateful to Université Libre de Bruxelles, and specially thank Glenn Barnich, Frank Ferrari, Marc Henneaux and Mauricio Leston for their hospitality during my stay, where the first part of this work was done. I also thank Pablo Minces for his interest on this work and for very interesting suggestions about possible applications. Last, I thank the Physics Department of New York University for the hospitality during my stay, where the revised version of the papers was finished. This work was partially supported by Universidad de Buenos Aires and CONICET, Argentina.
Appendix: The function $\Upsilon _{b}(x)$ {#appendix-the-function-upsilon-_bx .unnumbered}
=========================================
The function $\Upsilon _{b}(x)$ was introduced by Zamolodchikov and Zamoldchikov in Ref. [@ZZ], and is defined as it follows$$\log \Upsilon _{b}(x)=\frac{1}{4}\int_{\mathbb{R}_{>0}}\frac{d\tau }{\tau }\left( (Q-2x)^{2}e^{-\tau }-\frac{\sinh ^{2}(\frac{\tau }{4}(Q-2x))}{\sinh (\frac{b\tau }{2})\sinh (\frac{\tau }{2b})}\right) , \label{upsilondefintion}$$where $Q=b+b^{-1}$, being $b\in \mathbb{R}_{>0}.$ It also admits a definition in terms of a limiting procedure involving the double Barnes $\Gamma _{2}$-function, though we did not find such a relation necessary here. This function has its zeros at the points $$\begin{aligned}
x &=&mb+nb^{-1} \\
x &=&-(m+1)b-(n+1)b^{-1}\end{aligned}$$for any pair of positive integers $m,n\in \mathbb{Z}_{>0}$. From ([upsilondefintion]{}), it turns out to be evident that this function is symmetric under the inversion of the parameter $b\rightarrow 1/b$, namely$$\Upsilon _{b}(x)=\Upsilon _{1/b}(x).$$This is the first of a list of nice functional properties of this function. The second identity we find useful is the reflection property$$\Upsilon _{b}(x)=\Upsilon _{b}(Q-x), \label{Ur}$$which is keeps track of the reflection symmetry of Liouville structure constants when these are written in terms of $\Upsilon _{b}(x)$. Besides, (\[upsilondefintion\]) also presents the following properties under fixed translations$$\Upsilon _{b}(x+b^{\pm 1})=\Upsilon _{b}(x)\frac{\Gamma (b^{\pm 1}x)}{\Gamma
(1-b^{\pm 1}x)}b^{\pm 1\mp 2b^{\pm 1}x}.$$The above identities are then gathered in the equation$$\Upsilon _{b}(Q\mp x)=\pm \Upsilon _{b}(x)\frac{\Gamma (bx)\Gamma (b^{-1}x)}{\Gamma (\pm bx)\Gamma (\pm b^{-1}x)}b^{2x(b^{\pm 1}-b)},$$which, in particular, includes (\[Ur\]). The relations above also permits to prove that the following relation holds$$\Upsilon _{b}(x)=\Upsilon _{b}(-x)b^{2x(b-b^{-1})}\frac{\Gamma (-bx)\Gamma
(-b^{-1}x)}{\Gamma (bx)\Gamma (b^{-1}x)}.$$All these relations were employed through our computation.
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[^1]: or, more precisely, the fact that the connection between the different maps which are known turns out to be non trivial, see Ref. [@RT2005].
[^2]: I thank Cecilia Garraffo for discussions on this formula.
[^3]: I thank Pablo Minces for pointing me out this possible application.
|
---
address: |
II. Institut für Theoretische Physik, Universität Hamburg,\
Hamburg, Germany
author:
- 'C. EWERZ'
title: 'TOWARDS AN EFFECTIVE THEORY OF SMALL-X QCD [^1] '
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
DESY 97-130
Introduction
============
In this talk I would like to describe some recent developments in the field of small–$x$ perturbative QCD. In particular, I will review the first steps that have been done in the direction of finding an effective theory for small–$x$ QCD.
Regge physics has been around for thirty years now. The focus of Regge physics is the behavior of hadronic scattering amplitudes at very high energy $s$ and fixed momentum transfer $t$ (of the order of some hadronic mass scale), s t M\^2\_ . With the advent of the HERA machine, it has recently attracted new interest. Electron–proton collisions in the now available kinematic range of small Bjorken–$x$ and large momentum transfer $Q^2$ allow for a new test of high energy QCD. The interesting subprocess here is the scattering of a highly virtual photon off the proton. From a theorists point of view it is particularly interesting, because the large photon virtuality $Q^2$ enables us to enter this region of high parton densities using perturbative methods.
Let us first consider hadron–hadron scattering. The optical theorem relates the total cross section to the imaginary part of the forward elastic scattering amplitude, \_ = A\_(s,t=0) . \[opttheo\] It is convenient to perform a Mellin transformation changing from energy $s$ to complex angular momentum $\omega$, A(s,t) = i s \_[-i ]{}\^[+i ]{} ( )\^ A(,t) . \[Mellin\] The high energy behavior of the total cross section is then determined by the singularities of $A(\omega,t)$ in the $\omega$–plane, the so–called Regge poles and Regge cuts. The rightmost singularity gives the leading contribution and is identified with the pomeron. As it describes an elastic amplitude (see (\[opttheo\])) it carries vacuum quantum numbers. The location of the Regge poles in hadron–hadron scattering cannot yet be calculated from first principles, but there are successful phenomeological models that describe the experimental data in this framework.
It follows from unitarity that the total cross section cannot grow infinitely fast at high energy. It has to satisfy the Froissart bound \_ \^2(s) . \[Froissart\]
In the remaining part of my talk I will concentrate on considerations applying to the processes of $\gamma^\ast p$ or $\gamma^\ast \gamma^\ast$ scattering. The high virtuality $Q^2$ of the photon allows a perturbative treatment and thus enables us to get a clear understanding of these processes. $\gamma^\ast$–proton scattering is the basic process in electron–proton collisions. The Bjorken–$x$ variable is at high energy given by $x \simeq \frac{Q^2}{s}$, high energy corresponds to small $x$.
At small $x$ the small value of the coupling constant $\alpha_s$ can be compensated by large logarithms of $x$. This leads us to the leading logarithmic approximation (LLA) \_s 1 ; \_s (1/x) \~1 . In this approximation, the infinite number of contributing diagrams can be resummed. The result is the BFKL equation.[@BFKL] It describes the $t$–channel exchange of a bound state of two reggeized gluons. These reggeized gluons are collective excitations of the Yang–Mills field carrying gluon quantum numbers. They are the relevant degrees of freedom at small $x$.
The BFKL equation is an integral equation in two–dimensional transverse momentum space, because the longitudinal degrees of freedom decouple in the high energy limit. In detail, it has the form \_(,-) = \^0(,-) + (,,’) \_(’,-’) . \[BFKLeq\] $\kf$, $\qf-\kf$ are the momenta of the two gluons, $\phi^0$ is an inhomogeneous term. The integral kernel ${\cal K}$ (the so–called Lipatov kernel) is given by (,’,) &=& N\_c g\^2\
& & - (2)\^3 \^2 (-)\^2 \^[(2)]{}(-’) . The coupling constant is normalized to $\alpha_s=\frac{g^2}{4\pi}$ and $\alpha(\lf^2) = 1 + \beta(\lf^2)$ with (\^2) = g\^2 . is known as the gluon trajectory function. The variable $\omega$ acts as an energy variable in the BFKL equation. It can be shown to be conjugate to rapidity. Without making use of the Mellin transformation we would have found the BFKL equation as an evolution equation in $x$.
The general form of the BFKL amplitude can be derived from the integral equation by iteration. Thus the high energy elastic scattering amplitude is in LLA described by a $t$–channel exchange of gluon ladders:
, the rungs being Lipatov kernels. The BFKL equation has been solved analytically and leads to an increase of the amplitude A \~x\^[-(1+\_)]{}; \_ = 4 2 0.5 for $t=0$ in the limit $x \to 0$. It follows that the Froissart bound is violated at very small $x$, \_\^[\^p]{} \~x\^[-\_]{} \^2(1/x) . This means that the BFKL pomeron violates unitarity at very small $x$.
Unitarization of the BFKL-Pomeron
=================================
To restore unitarity we have to include nonleading (in $\log(1/x)$) corrections to the BFKL equation. The minimal set of corrections restoring unitarity can be identified as the corrections with a larger number of reggeized gluons in the $t$–channel.[@moregluons] Our goal is to find a consistent framework for the description of the infinite number of these terms. This framework will be a more general structure in which the BFKL pomeron appears as the first approximation.
Based on our knowledge of the BFKL equation we can already state some properties of this effective theory. Like in the BFKL case, the longitudinal degrees of freedom can be factored off and the dynamics will take place in two–dimensional transverse momentum space. Again, rapidity will act as the time–like coordinate. The appropriate degrees of freedom will be reggeized gluons.
As I will show, very important new elements are number changing vertices describing the creation and annihilation of $t$–channel gluons. Thus our aim to construct an effective theory of QCD at small $x$ turns out to be that of building a $2+1$–dimensional quantum field theory of reggeized gluons.
The program to be carried out in order to achieve this ambitious goal can be cut into the following pieces. Figure \[fig:structure\] tries to visualize the different elements that have to be extracted from QCD.
- [*The leading order elements*]{} are known: the BFKL equation and the quark loop.
- [*The next–to–leading corrections.*]{} The NLLA corrections to the BFKL equation are needed, for example, to understand which symmetries of the BFKL equation are only due to the LLA and how their breaking occurs. These corrections are currently being calculated by Lipatov and Fadin[@NLLA] and by Ciafaloni and Camici.[@Ciafaloni]
- [*The spectrum of the $n$–gluon state*]{} is a quantum mechanical problem formally described by the so–called BKP equations.[@moregluons; @BKP] Recently there has been a great leap forward in this longstanding problem. Lipatov [@LipXXX], Faddeev and Korchemsky [@XXX] were able to prove that it is, in the large $N_c$ limit, equivalent to the integrable XXX Heisenberg model for noncompact spin zero. This connection opens the field for the application of some very powerful mathematical methods (Bethe ansatz etc.).
- [*Number changing vertices,*]{} which turn the problem into that of a quantum field theory. The vertex $2 \rightarrow 4$ gluons has been found recently,[@BarWust] higher transition vertices, especially the $2\rightarrow 6$ transition, are currently under investigation.[@d6]
- [*Finding symmetries*]{} of the above elements is of course obligatory to understand the structure of the field theory. The most important symmetry of the known elements is their conformal invariance.
The number changing vertices, their emergence and the conformal invariance will be the topic of the following sections.
The Number Changing Vertices
============================
The formal framework for our considerations is made up by amplitudes $D_n$ describing the creation of $n$ reggeized gluons in the $t$–channel from a quark loop at fixed $\omega$ (see fig. \[fig:structure\]). These amplitudes will depend on the two–dimensional transverse momenta and on the color indices of the gluons. The amplitudes obey a tower of coupled integral equations that generalize the BFKL equation. For the purpose of this talk we state the equations only in a graphical form and only up to $n=4$, a more detailed account can be found elsewhere.[@BarWust; @d6] & &
\
& &
\[d3eq\]\
& &
\[d4eq\] The inhomogeneous terms $D_{(n;0)}$ stand for the perturbative coupling of $n$ gluons directly to the light quark loop. The first equation is the usual BFKL equation. The kernels appearing in the other equations can be calculated perturbatively and generalize the Lipatov kernel. They must not be confused with the number changing vertices we want to calculate. The latter emerge when we solve the integral equations which we can do at least partially, as I will explain. The summation symbols indicate that we have to include all possible pairwise interactions of $t$–channel gluons.
The equation for $D_3$ can be solved exactly and the solution is\
& & (D\_2(\_1+\_2,\_3) - D\_2(\_1+\_3,\_2) + D\_2(\_1,\_2+\_3)) , \[d3d2\] where $a_i$ are color indices, $\kf_i$ are the transverse momenta of the gluons and $C_3$ is a normalization constant. This result tells us that, although $D_3$ is formally defined as a 3–gluon amplitude, it is — according to its analytic properties (\[d3d2\]) — actually a superposition of 2–gluon (i.e. BFKL) amplitudes. There is no intermediate 3–gluon state! The details of the calculation show that this fact strongly depends on the inhomogeneous term in the integral equation (\[d3eq\]), namely the coupling of the gluons to the loop containing light (effectively massless) quarks.
The equation (\[d4eq\]) for the 4–gluon amplitude can be solved at least partially. ’Partially’ because the solution involves the full 4–gluon state which is not yet known. Nevertheless, we can extract the structure of the solution from the equation. For simplicity suppressing all color and normalization factors, it has the following form:
\[solutiond4\] The first term is the sum of 2–gluon (BFKL) amplitudes $D_2$. The summation extends over the (seven) possibilities to combine the four momenta into two sums which are then the two arguments of the $D_2$ amplitudes ($C_{1,2}$ indicate color tensors, but again we will not go into the details of $su(3)$ algebra.): D\_4\^R(\_1,\_2,\_3,\_4) &=& C\_1 \[ D\_2(\_1+\_3+\_4,\_2) + D\_2(\_1+\_2+\_4,\_3)\
& & - D\_2(\_1+\_2,\_3+\_4) - D\_2(\_1+\_3,\_2+\_4) \]\
&+& C\_2 \[ D\_2(\_1+\_2+\_3,\_4) + D\_2(\_1,\_2+\_3+\_4)\
& & - D\_2(\_1+\_4,\_2+\_3) \] This first part $D_4^R$ being isolated, the remaining terms in the equation define the transition vertex $V_{2 \rightarrow 4}$. (The full derivation of the vertex is rather lengthy and will not be given here.) We can write the second term as the convolution $G_4 \otimes V_{2 \rightarrow 4} \otimes D_2$ with $G_4$ being the propagator of the 4–gluon state. Starting from the quark loop, we first have a 2–gluon state, then there is a transition from two to four reggeized gluons and at the bottom we have the full, i.e. interacting, 4–gluon state. This teaches us the important lesson that in leading logarithmic order it is not possible to couple the interacting system of four reggeized gluons directly to the quark loop! This coupling always involves the vertex $V_{2 \rightarrow 4}$ and the 2–gluon state.
Let us now have a closer look at the vertex $V_{2 \rightarrow 4}$. It can be written as V\_[2 4]{}\^[a\_1a\_2a\_3a\_4]{}({ [**q**]{}\_j},\_1,\_2,\_3,\_4) &=& \_[a\_1a\_2]{} \_[a\_3a\_4]{} V({ [**q**]{}\_j},\_1,\_2;\_3,\_4)\
& & + \_[a\_1a\_3]{} \_[a\_2a\_4]{} V({ [**q**]{}\_j},\_1,\_3;\_2,\_4)\
& & + \_[a\_1a\_4]{} \_[a\_2a\_3]{} V({ [**q**]{}\_j},\_1,\_4;\_2,\_3) \[colV\] Again, the indices $a_i$ are color labels, $\kf_i$ are the outgoing momenta and $\{ {\bf q}_j\}$ the incoming momenta. The explicit analytic form of the function $V$ can be found in the literature.[@BarWust] The representation (\[colV\]) nicely demonstrates the complete symmetry of the vertex under permutations of the outgoing gluons and displays its simple color structure.
Essentially in the same way, it is possible to extract information about the higher $n$–gluon amplitudes from the corresponding equations even without explicit knowledge of the interacting $n$–gluon systems.
A new result[@d6] I can present here is the following: Like in the case of the 3–gluon amplitude, also the 5–gluon intermediate state is absent, the mechanism being similar to that shown in (\[d3d2\]): the 5–gluon amplitude $D_5$ can be decomposed into a sum of 4–gluon amplitudes $D_4$. Using the same notation as in (\[solutiond4\]),
The summation in the last term now refers to the possibilities to combine two of the five momenta into a sum.
This mechanism seems to apply to every odd number of intermediate gluons, and only even numbers of gluons appear as intermediate states. What we observe here is a generalization of the so–called reggeiziation of the gluon, a rather deep property of QCD in the Regge limit.
The next step will be the calculation of the 6–gluon amplitude. From that we will learn more about the elements of the field theory. The first question to answer is whether there is a new $2\rightarrow 6$ transition vertex. Further, we will be able to understand the transition from the 4–gluon to the 6–gluon system. This should involve the known vertex $V_{2\rightarrow 4}$, but also its generalization to the case in which the 2 gluons above the vertex are not in a color singlet, as has been the case in the 4–gluon amplitude.
Conformal Invariance
====================
The Lipatov kernel and the transition vertex $V_{2\rightarrow 4}$ exhibit a high degree of symmetry in that they are conformally invariant. To explain the meaning of this conformal invariance we go by means of Fourier transformation from transverse momentum space (momenta $\{\kf\}$) to impact parameter space (vectors $\{\vec{\rho} \}$). We then introduce the complex coordinates $\rho = \rho_x +i \rho_y$ and $\rho^\ast = \rho_x - i \rho_y$. This is done for all arguments of the Lipatov kernel and of the vertex.
Möbius (or conformal) transformations are defined by \^= ; ad -bc = 1 . \[mob\] These transformations are characterized by the group (
[cc]{} [a]{}&[b]{}\
[c]{}&[d]{}
) SL(2,[**[C]{}**]{}) / Z\_2 , i.e. the group of projective conformal transformations. The generators of this group form the subalgebra $sl(2,{\bf C})$ of the well-known Virasoro algebra. It was known for some time[@BFKLconf] that the Lipatov kernel is invariant under the transformations (\[mob\]). Recently, also the vertex $V_{2 \rightarrow 4}$ was shown to be symmetric under conformal transformations.[@Vconf] We expect that this will be true also for higher vertices.
The observation that the Lipatov kernel as well as the number changing vertex are conformally invariant naturally leads us to a further property of the effective theory: it will be a conformal field theory. Conformal field theories in two dimensions have been subject to very intense and fruitful investigation in the past years.[@CFT] The applications range from statistical physics to string theory. It turned out that conformal symmetry is an extremely powerful tool. One example which might possibly apply to our considerations is the fact that the $n$–point functions in a conformal field theory are highly restricted. The 3–point function, for instance, is fixed up to a constant. The connection with conformal field theory might be very useful for a deeper understanding of small–$x$ QCD.
It is important to mention that the conformal symmetry described above is not an exact symmetry of Nature even in the Regge limit. It is known that the running of the gauge coupling $\alpha_s$ (the gauge coupling is fixed in LLA) and possibly other next–to–leading corrections will break conformal invariance. Of course, it will be important to understand in detail how the symmetry breaking takes place.
Outlook
=======
The unitarization of the BFKL pomeron is urgently needed to get a deeper understanding of the small–$x$ behavior of structure functions. It is believed that the unitarization of the pomeron will lead us to an effective theory of QCD in the Regge limit.
We are still at the very beginning of constructing such an effective theory of small–$x$ QCD. We have compelling evidence that this theory will be a $2+1$ dimensional conformal field theory. Reggeized gluons have been identified as the correct degrees of freedom. Some of the elements of the effective theory are already known. Other very important elements are still missing, but there has been considerable progress in the last years. So far, the emerging picture is very encouraging.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank J. Bartels, H. Lotter and M. Wüsthoff for many helpful discussions.
References {#references .unnumbered}
==========
[99]{} E.A. Kuraev, L.N. Lipatov, V.S. Fadin, ; Ya.Ya. Balitskii, L.N. Lipatov,
J. Bartels,
V.S. Fadin, L.N. Lipatov, and references therein
G. Camici, M. Ciafaloni, and in preparation
J. Kwieciński, M. Prasza[ł]{}owicz,
L.N. Lipatov,
L.D. Faddeev, G.P. Korchemsky,
J. Bartels, , J. Bartels, M. Wüsthoff,
J. Bartels, C. Ewerz, in preparation
L.N. Lipatov,
J. Bartels, L.N. Lipatov, M. Wüsthoff,
A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, ; P. Ginsparg in [*Les Houches, Session XLIX, 1988*]{}, eds. E. Brézin and J. Zinn–Justin (Elsevier Science Publishers B.V., 1989)
[^1]: Talk presented at the International School of Subnuclear Physics, 34th Course: [*Effective Theories and Fundamental Interactions*]{}, Erice, Sicily, 3–12 July 1996; to appear in the proceedings.
|
---
abstract: 'We studied theoretically the behavior of an injected electron-hole pair in crystalline polyethylene. Time-dependent adiabatic evolution by ab-initio molecular dynamics simulations show that the pair will become self-trapped in the perfect crystal, with a trapping energy of about 0.38 eV, with formation of a pair of trans-gauche conformational defects, three C$_2$H$_4$ units apart on the same chain. The electron is confined in the inter-chain pocket created by a local, 120$^\circ$ rotation of the chain between the two defects, while the hole resides on the chain and is much less bound. Despite the large energy stored in the trapped excitation, there does not appear to be a direct non-radiative channel for electron-hole recombination. This suggests that intrinsic self-trapping of electron-hole pairs inside the ideal quasi-crystalline fraction of PE might not be directly relevant for electrical damage in high-voltage cables.'
address:
- ' International School for Advanced Studies (SISSA) and DEMOCRITOS National Simulation Center, Trieste, Italy.'
- ' Dipartimento di Fisica, Università di Modena, Modena, Italy.'
- '§ International Center for Theoretical Physics (ICTP), Trieste, Italy.'
- '$\sharp$ Pirelli Labs - Materials Innovation, V.le Sarca 222, I-20126, Milano, Italy.'
author:
- 'D. Ceresoli, M. C. Righi, E. Tosatti§, S. Scandolo§, G. Santoro and S. Serra$\sharp$'
title: 'Exciton self-trapping in bulk polyethylene'
---
Introduction
============
Thanks to its large band-gap ($\sim 8.8$ eV), to its chemical inertness, and to its easy processing, polyethylene (PE) is the material of choice in extruded electric cables for high-voltage applications. The highly insulating properties of PE however do not prevent a tiny flow of charge carriers, both electrons and holes, injected at the very high electric fields realized under normal high voltage operation conditions. Understanding and controlling this flow is of course extremely important for improving the cable performances. For example, dielectric breakdown depends crucially on the mobility of the injected charge carriers. Aging and eventual failure of the cable is also believed to be connected to local damage induced by charge flow and/or recombination[@dissado]. From a physicist’s perspective, it is of primary importance to understand the basic electronic properties of PE which underlie the observed phenomenology. In particular it is crucial to explore the detailed nature of the electronic states corresponding to the highest valence-band state (holes), to the lowest conduction band states (electrons), and to the simultaneous presence of holes and electrons whose bound state we will refer to with the standard name of excitons[@bassani].
Excitons are particularly interesting in this context, as it is known [@laurent] that under high field [*and*]{} polarity reversals, electroluminescence – field-induced emission of photons – is observed with particular intensity, plausibly due to radiative electron-hole recombination. However electroluminescence is also known to open the way to electric damage. When in fact the electron-hole recombination energy, amounting to several eV, is released in a non-radiative channel, it is suddenly turned into ion kinetic energy, and will most likely fuel important local structural and eventual electric damage. The possible existence of channels for this kind of damage is as yet unexplored.
Recently, considerable progress was made in the understanding of the nature of, separately, electron and hole states in PE. Photoemission studies [@valence] agree well with electronic structure calculations [@miao; @jones; @serra:interchain] in showing that the filled valence bands are made up of C$-$C (and of C$-$H) bonding states, propagating very effectively along the polymeric chain, but rather ineffectively between the chains. We shall refer to states of this kind, that are heavily chain-localized, as *intrachain* states. The lowest empty conduction band states were found on the contrary (so far only theoretically) to propagate quite well in all directions in three dimensional space, and also to display a maximum amplitude no longer on the chains, but rather in the empty space between the chains. Such a state was referred to as an *interchain* state [@serra:interchain; @cubero]. The interchain nature of the conduction states is held responsible for some uncommon physical properties such as the *negative electron affinity* or NEA [@righi:nea], characteristic of PE.
An extra electron injected in a soft insulator like bulk PE will not remain indefinitely free, delocalized, and conduction band-like. As in other soft materials, the extra electron can be expected to cause some kind of spontaneous static distortion of the lattice – a small polaron – whose effect will be to trap the electron itself in the volume encompassed by the deformation (*self-trapping*). Detailed calculations [@serra:selftrapping] indicated the existence of such a polaron state in bulk crystalline PE. It consists of a short PE chain segment – limited by a pair of *trans-gauche* conformational defects – which undergoes a large rotational distortion, with the extra electron locally bound, and self-trapped. No self-trapping, conversely, was found for holes, owing presumably to the stiffness of the C$-$C and C$-$H covalent bonds in the chains onto which the hole intra-chain wave function is localized.
The scope of this work is to address the issue of how a bound electron-hole pair – an exciton – will behave in this respect once injected in PE. Electron-hole pairs can be created through electrical injection of hot carriers. Although ordinary carrier mobility in real PE is extremely small, of order $10^{-10}$ cm$^2$/V, some electrons and holes do nonetheless get injected at opposite electrodes when the field overcomes the threshold for space charge formation. Once inside, electrons and holes drift toward one another. Due to their (screened) Coulomb attraction they will as soon as possible merge to form bound exciton pairs. Excitons can of course also be created optically, [@bassani1] by absorption of UV photons of energy higher than the PE energy gap.
These considerations and the related questions call for an investigation of the electron-hole pair (*exciton*) state in crystalline PE, in particular of its accompanying structural deformation, and of its possible self-trapped state in the perfect material. We approached this problem theoretically, making use as in our previous electron and hole studies, of constrained density functional electronic structure calculations, that possess enough quantitative accuracy to be predictive.
We will present material in the following order. First we shall describe in Section II the methods and computational techniques used. In Section III we will study the exciton in perfect crystalline PE, and demonstrate its self-trapping, the associated distortion similar to some extent but not identical to that of the isolated electron. Section IV will be devoted to discussion of the possible consequences, and to our conclusions.
Computational techniques
========================
We carried out first-principle electronic structure calculations of PE, using density-functional methods [@DFT], based on plane-wave expansion and pseudopotentials. Technical details are identical to those given in previous papers [@serra:dynamical; @serra:interchain; @serra:selftrapping]. It should be stressed that the choice of a large plane wave basis set in these calculations is particularly important, in order to capture the best possible real nature of conduction states. Restricted local basis sets, otherwise very effective for describing chemical bonding and well localized valence states, should be considered with suspicion, as they may easily fail to describe properly the extremely extended, plane wave-like [*interlayer*]{} conduction states [@serra:interchain]. Calculations were carried out within the BLYP gradient corrected energy functional [@BLYP]. The ion-electron interactions were described by norm-conserving Martins-Troullier pseudopotentials [@MTpseudo]. Wave functions were expanded in plane waves, with a plane wave energy cutoff of 40 Ry. Structural optimization and molecular dynamics (MD) simulations were carried out with a steepest descent and a Car-Parrinello algorithm [@CarParrinello], respectively. The time step for the Car-Parrinello dynamics was set to 5 atomic units (0.6 fs) and the electronic mass to 200 atomic units. Ab initio, finite temperature MD simulations where carried out with the help of a Nosé thermostat with a fictitious mass corresponding to a frequency of 40 THz.
The van der Waals attraction between the chains – a numerically small but qualitatively important contribution to the stability of PE which is not accounted for by quasi-local approximations to the DFT functional such as BLYP – was included empirically through an extra two body interatomic potential with a parametrized $r^{-6}$ tail of the form: $$\label{eq:vdw}
V_{vdw} = - \frac{1}{2} \sum_{i,j} \frac{C_{i,j}}{r^6_{i,j}}f_c(r_{i,j}),$$ where $f_c$ is a function cutting off at short distance. All the parameters entering (\[eq:vdw\]) were given the same values as in Serra *et al.* [@serra:dynamical], optimized to correctly reproduce the equilibrium structure, stability, elastic constants and thermal properties of neutral PE. This ensures that we begin with a sound overall physical description of PE, prior to the excitation, consisting of the simultaneous introduction of an electron and a hole.
A first basic question of the present excited-state calculations is how in fact to introduce the electron-hole pair. A natural choice would be to introduce a triplet exciton. Theoretically, a triplet exciton is the $S=1$ ground state, and can in principle be introduced in a calculation by forcing a total spin of one in a spin-polarized calculation[@marco]. Experimentally, triplet excitons could be created by impact energy loss exchange processes of low energy electrons, [@bocchetta] whereby an incoming spin down electron will fall in energy to occupy an empty conduction state, while kicking away a spin up electron, and thus creating a spin up valence hole. Although its actual energy location and lifetime are presently unknown, a long-lived [*triplet*]{} $S=1$ exciton must surely exist in PE, as in all other molecular solids, with an excitation energy below the ordinary, singlet excitation gap. Practical and computational limitations force us however to abandon this costly spin-polarized procedure, and to consider a simpler alternative solution. We mimic here the presence of an exciton by simply constraining an occupancy of one, instead of two, for the highest pair of occupied Kohn-Sham eigenvalues in a non-spin-polarized calculation (constrained DFT). Spin-contamination problems (related to the fact that such an exciton is, in a non-polarized calculation, neither a singlet nor a triplet pure state [@spincontamination]) are expected to be minor in this case. In fact the hole and the electron states, one intrachain and the other interchain, overlap poorly in space, making the contribution of electron-hole exchange relatively small, and negligible with respect to the large exciton creation energy of many eV (approximately equal to the band gap of about 8.8 eV). This kind of approximation will of course not hold any more when, as [*e.g.,*]{} in aromatics, both electron and hole states share the same electronic nature; or else when, even in impure PE, electron and hole may both become tightly localized onto some chemical defects. However, in the present case of defect free PE we believe that this represents a highly reasonable approximation. It can also be noted that the probable slight overestimate of excitation energy relative to the true triplet state will cancel at least a fraction of the well-known DFT energy gap deficit, thus representing a small improvement, rather than a deficiency.
We carried out first of all the electronic structure calculations and the structural optimization in this model exciton state. A sophisticated many-body description (GW-BS [@louie]) of the electron-hole interaction would be desirable to describe the excitonic energies, but the associated structural relaxations can be obtained quite accurately within a constrained DFT approach, as reported for the self-trapped exciton in a conjugated organic polymer [@artacho]. Subsequently, starting from this state we also carried out a reasonable number of Car-Parrinello molecular dynamics steps, in order to explore effectively the potential energy surface of the system.
Results
=======
Perfect crystalline PE possesses a base-centered orthorhombic crystal structure with lattice parameters $a=$ 4.93 Å, $b=$ 7.4 Å and $c=$ 2.534 Å [@crystalstructure]. The unit cell contains two polymer chains, running parallel to the $c$ direction. The two chains are rotated by $\pm$ 42$^\circ$ around the $c$ axis, and the CH$_2$ units are in a trans-planar conformation, namely the carbon skeleton forms a zig-zag chain lying entirely on a plane (figure 1 of ref. [@serra:interchain]).
The calculated DFT/BLYP energy band gap for ground state PE is 6.46 eV [@serra:dynamical], in perfect agreement with previous calculations [@valence; @jones] but somewhat smaller than the experimental gap of about 8.8 eV. Due to that, the energy available to create a distortion, in our approximation, will represent a lower bound to that in real PE. We modeled the system with four parallel chains, each seven -CH$_2$-CH$_2$- units long in a periodically repeated simulation cell. We first performed a calculation starting from the perfect crystalline PE configuration. However, the self-trapped state of the excess electron is known to involve a structural rearrangement of the chains [@serra:selftrapping], and this rearrangement might be hindered by energy barriers when starting from ideal perfect crystalline PE. To circumvent that, we also performed a series of calculations starting from distorted atomic configurations morphologically similar to that of the self-trapped electron. In particular, we created two gauche defects on the same chain at a distance of one, two, three, and four C$_2$H$_4$ units, the chain segment between the two defects rotated by 120$^\circ$ with respect to the crystalline PE structure.
We start by describing first the results obtained for the undistorted structure. We relaxed first the structure by steepest descent, both in the ground state and in the excited state. In both cases, deviations from the trans-planar conformation were negligible. The ground state HOMO-LUMO gap of 6.46 eV is only slightly reduced after excitation to 6.36 eV: the hole Kohn-Sham (KS) eigenvalue 0.09 eV above the top of the valence band, and the electron KS eigenvalue 0.01 eV below the bottom of the conduction band. The resulting reduction of the energy gap is 0.1 eV. Such a small value may be due to the well known gap-problem of LDA. We did not attempt to perform GW-BS [@louie] calculations in order to compute the exciton binding energy. Starting from this relaxed static structure, we performed next a molecular dynamics simulation at 300 K. The atomic positions oscillated about the trans-planar conformation and no defects were spontaneously created. The behavior with simulation time of the electron and hole Kohn-Sham eigenvalues is shown in fig. \[fig:crystal\_eigen\]: the hole eigenvalue oscillates with a magnitude of about 0.5 eV, but the gap remains substantially unchanged for the duration of the simulation (1.5 ps), showing no evidence for self trapping, at least in this relatively short time.
Similar results were found in the simulations which started from the distorted structures. Annealing the system by molecular dynamics at room temperature for 0.3 ps led to states where the distortion persisted, suggesting that each distortion is a local minimum of the energy landscape, and that different minima are separated by energy barriers, that will take a long time to cross. In order to determine the energetically favored structure, we optimized by steepest-descent the atomic positions of each distorted structure with $n=1,...,4$ rotated C$_2$H$_4$ units. The total energy calculated at the local minimum of each distorted structure is reported in Table I. We found in this way that the self-trapped exciton with lowest energy consisted of *three* rotated C$_2$H$_4$ units (fig. \[fig:3units\_iso\]). Comparison with the self-trapped electron case (without hole), where the distortion consisted of seven rotated units, supports the notion that the electron-hole attraction contributes substantially to the electron localization, reducing the self-trapped polaron size from seven for the electron to three for the electron-hole pair. While in the electron-only case the singly occupied electron level sank about 0.5 eV below the conduction band bottom, that level (upper singly occupied level) is found here to lie a mere 0.04 eV below the conduction band, most likely on account of the extra kinetic energy required by localization in the smaller 3-unit pocket [@serra:selftrapping]. The hole energy level (lower singly occupied level) is 0.01 eV above the top of the valence band, corresponding to a much weaker localization of the hole wavefunction in the neighborhood of the trapped and strongly localized electron, as seen in fig. \[fig:3units\_charge\]. The remainder of the total energy gain, is thus to be ascribed to the screened electron-hole Coulomb attraction. Fig. \[fig:3units\_charge\] shows the electron and hole charge-density profile parallel to the $c$ direction; the dashed line is the square modulus of the hole wave-function; the maxima correspond to the center of the the C$-$C bonds. The solid line is the square modulus of the electron wave-function; the position of the gauche defects is indicated by the two short arrows. The hole is only slightly affected by the presence of the electron but tends to be more localized in the proximity of the electron, due to Coulomb attraction.
Fig. \[fig:3units\_iso\] shows the iso-density surfaces of the hole and electron states together with the polymeric chain. The electron state is clearly *inter-chain* and is trapped in the volume encompassed by the deformation.
Discussion and conclusions
==========================
Our calculations indicate that *self-trapped electron-hole pairs* should exist in crystalline polyethylene, with a distortion pattern qualitatively similar to – although quantitatively different from – that of the self-trapped electron. The necessary energy to detrap the exciton (consisting of reduction of band-gap, increase of kinetic energy due to confinement, creation of two gauche defects, rotation of a segment of chain and electron-hole Coulomb interaction) is about 0.3–0.4 eV. The exciton state, although very high in absolute energy, is apparently long-lived, displaying no apparent direct channel for non-radiative recombination. This suggests that exciton self-trapping in ideal quasi-crystalline PE might not be directly relevant for electrical damage and implies that release of the energy (several eV) stored in the electron-hole pair should take place in other forms. Likely candidates for that would seem conformational or chemical defects, possibly via a combination of non-radiative and radiative decay processes. Further studies will be needed to explore such more complex scenarios.
We thank Dr. G. Perego (Pirelli Cavi) for very fruitful discussions. Work in SISSA was sponsored by MIUR FIRB RBAU017S8R004, FIRB RBAU01LX5H, and COFIN 2003 and COFIN 2004, as well as by INFM Progetto Calcolo Parallelo.
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------------------- ------------------------------
Number of rotated Energy difference respect
C$_2$H$_4$ units to the undistorted case (eV)
1 1.22
2 1.55
[**3**]{} [**-0.38**]{}
4 -0.05
------------------- ------------------------------
: Self-trapped exciton: energy difference respect to the undistorted case.[]{data-label="tab1"}
![Kohn-Sham (KS) eigenvalues of crystalline PE as a function of time, during a constant temperature molecular dynamics run. The two horizontal lines represent the band edges.[]{data-label="fig:crystal_eigen"}](fig1.eps){width="7.5cm"}
![Iso-density surface of the self-trapped exciton; hole-state in light gray; electron state in dark gray. The bare polymeric chain is shown on the left for sake of clarity. The two small arrows indicate the position of the gauche defects.[]{data-label="fig:3units_iso"}](fig2.eps){height="7.5cm"}
![Charge density profile parallel to the $c$ direction. Small arrows indicate the position of the gauche defects.[]{data-label="fig:3units_charge"}](fig3.eps){width="7.5cm"}
|
---
abstract: 'The spin Hall effect as well as the recently discovered magnetic spin Hall effect are among the key spintronics phenomena as they allow for generating a spin current by a charge current. The mechanisms for this charge-spin conversion normally rely on either the relativistic spin-orbit coupling or non-collinear magnetic order, both of which break spin conservation. This limits the spin diffusion length and the charge-spin conversion efficiency connected to these mechanisms due to the internal competition between spin current generation and spin loss. In this work we show that the magnetic spin Hall effect can exist in collinear antiferromagnets and that in some antiferromagnets it exists even without spin-orbit coupling, thus allowing for a spin-charge conversion in a system which conserves spin. We find a very large spin current ($\approx$ 8600 ($\hbar$/e)S/cm) and magnetic spin Hall angle ($\approx 25\%$) in antiferromagnetic RuO$_2$. In addition, we present a symmetry classification of the conventional and magnetic spin Hall effects and identify general symmetry requirements for their existence.'
author:
- 'Rafael González-Hernández'
- Libor Šmejkal
- Karel Výborný
- Yuta Yahagi
- Jairo Sinova
- 'Jakub Železn[ý]{}'
bibliography:
- 'MSHE.bib'
- 'refs\_Jakub.bib'
title: Magnetic Spin Hall Effect in Collinear Antiferromagnets
---
Introduction
============
One of the major recent breakthroughs in spintronics has been the discovery of powerful mechanisms for generating spin currents by charge currents or vice versa through the so-called spin Hall effect (SHE) and the inverse SHE [@Sinova2015]. These processes are commonly referred to as charge-spin or spin-charge conversion. The SHE occurs when an unpolarized charge current transverses a material with strong spin-orbit coupling (SOC) or non-collinear magnetic order [@Zhang2018], leading to a perpendicular spin current where “up” spins accumulate at one edge of the sample and “down” spins accumulate at the other edge [@Sinova2010] (see Fig.\[Fig1\]a,b). Although the SHE was predicted theoretically in 1971 [@Dyakonov1971b], it was only observed experimentally in GaAs in 2004 [@Wunderlich2005; @Kato2004]. SHE borrows directly from the physics and mechanism of the anomalous Hall effect (AHE) and correspondingly much of their description are parallel [@Jungwirth2012]. The SHE and the inverse SHE are widely used in spintronics. For example, the SHE is commonly used to electrically switch the magnetization in ferromagnetic films [@Manchon2018inpress]. ISHE is a practical tool for the electrical detection of spin currents or, for example, for the generation of THz radiation from fs spin current pulses [@Seifert2016].
Recently a new type of a spin to charge conversion mechanism has been theoretically predicted [@Jakub2017] and consequently experimentally observed [@Kimata2019] in non-collinear antiferromagnetic systems. This so-called magnetic spin Hall effect (mSHE) differs from the conventional SHE in that it can only exist in magnetic materials. Specifically this means that whereas the SHE is even under reversal of the magnetic order, the mSHE is odd or equivalently that the SHE is even under time-reversal, whereas the mSHE is odd. This also means that the mSHE is necessarily a dissipative effect, whereas the SHE can be in principle non-dissipative. Microscopically, on the simplest level, the mSHE can be described by a Boltzmann equation and is proportional to the relaxation time, whereas the SHE is in the clean limit independent of the relaxation time and can be described by the intrinsic formula.
The SHE normally originates from the SOC, which is a relativistic interaction that couples spin to the orbital degree of freedom. In non-collinear systems, however, it can exist even in the absence of the SOC, as the non-collinear magnetic order plays a similar role to the SOC by breaking the spin rotation symmetry and coupling the spin to the lattice [@Zhang2018]. Similarly, the mSHE has been predicted to exist in the non-collinear antiferromagnets even in the absence of the SOC, whereas in simple collinear ferromagnets, the SOC is required [@Jakub2017]. Similar phenomenology is observed in other effects such as the AHE [@Taguchi2001; @Zhang2018] or the orbital magnetic moment [@Hanke2016]. These effects require SOC in collinear ferromagnets, but can also exist in the absence of SOC in non-collinear (or specifically non-coplanar) magnetic systems.
In this work we show that the mSHE can also exist in collinear antiferromagnets and that in some it can exist even in absence of the relativistic SOC. In this way it differs from the conventional SHE, which can only exist without SOC in non-collinear magnetic systems. This difference is crucial since in non-collinear magnetic systems no component of spin is a good quantum number and spin is thus not conserved, whereas in collinear magnetic systems without SOC, the component of spin along the magnetic order is a good quantum number and is thus conserved. Therefore, the mSHE in collinear antiferromagnets allows for charge-spin and spin-charge conversion in a system that conserves spin.
The intrinsic contribution to the conventional SHE can be understood as precession of the spin in spin-orbit field. In non-collinear magnetic systems, the origin is similar, except the precession is not due to spin-orbit field, but due to non-collinear magnetic order. In a collinear system without SOC, the spin is conserved and this mechanism is thus not possible. In contrast, the mSHE originates from a redistribution of electrons, which does not rely on a change of the spin. The non-relativistic origin of mSHE in collinear antiferromagnet RuO$_2$ is illustrated in Fig. \[fig:sc\_origin\]. When the relativistic SOC is neglected, the spin is a good quantum number and the wavefunctions of the system can be split up into spin-up and spin-down states. As illustrated in Fig. \[fig:sc\_origin\]a, the spin-up and spin-down Fermi surfaces are anisotropic and rotated with respect to each other. When electric field is applied along the $x$ or $y$ directions, the resulting spin-up and spin-down currents will flow at an angle with respect to the electric field (see Fig. \[fig:sc\_origin\](b)). As a consequence, the charge current flows in the direction of the electric field, whereas the spin current is perpendicular to it. The reason for the anisotropy of the Fermi surfaces is that the Ru sublattices have locally anisotropic crystalline environments, as illustrated on Figs. \[fig:sc\_origin\](c),(d). Because of this crystalline anisotropy, the electrons at the A and B Ru sublattices are flowing at an angle to the electric field and because of the antiferromagnetic order, this is also reflected in the spin-up and spin-down current directions.
The discussion in this work focuses mainly on spin currents that are transverse to the charge current (i.e. flowing in the transverse direction). This is the context in which the term spin-charge conversion is mainly used. However, both the time-reversal-even ($\cal{T}$-even) spin currents (SHE) and the time-reversal-odd ($\cal{T}$-odd) spin currents (mSHE) can also have a longitudinal component. The different types of spin currents are illustrated in Fig. \[Fig1\], however, we note that the terminology of the various contributions is not very well established. In ferromagnets, the $\cal{T}$-odd longitudinal component is usually referred to as the spin-polarized current and is primarily utilized for spin-transfer torque in magnetic junctions [@Ralph2008; @Brataas2012]. In analogy with ferromagnets, the $\cal{T}$-odd longitudinal current in non-collinear antiferromagnets has been referred to as a spin-polarized current [@Jakub2017]. The transverse component has been later referred to as the magnetic SHE and this is a terminology that we also use here. In a recent work, the term mSHE has been used only in relation to the antisymmetric component of the $\cal{T}$-odd spin currents and it has been shown that this antisymmetric component is connected to spin current vorticity in the reciprocal space [@Mook2019arxiv]. Here we use the term mSHE even for the symmetric component since the distinction between the symmetric and antisymmetric component is not as fundamental as in the case of charge conductivity. No special name has been used for the longitudinal component of the $\cal{T}$-even currents [@Wimmer2015]. We simply refer to it as the longitudinal SHE here, although it could be argued that this is not a good term since a longitudinal spin current is not really a “Hall effect”. It is not clear whether it is meaningful to use a different terminology for the longitudinal and transverse components since they can have the same origin, however, from practical point of view, the two are very different, since they are used for different purposes.
We demonstrate the existence of the mSHE and the spin-polarized current in two antiferromagnetic systems: MnTe and RuO$_2$ by utilizing symmetry arguments and *ab-initio* linear response calculations. We find that whereas in MnTe these $\cal{T}$-odd spin currents exist only when the relativistic SOC is included, in RuO$_2$ they are present even in the non-relativistic limit. To understand the general symmetry requirements for the existence of the $\cal{T}$-odd spin currents, we have performed a general classification of the $\cal{T}$-odd and $\cal{T}$-even spin currents. We find that the $\cal{T}$-odd spin currents are allowed in all magnetic systems with broken ${\cal{T}}\tau$ (time-reversal combined with translation) or $\cal{PT}$ (time-reversal combined with inversion) symmetries, or equivalently in all systems with broken time-reversal symmetry in the Laue group. Our result means that the $\cal{T}$-odd spin currents can also exist in truly antiferromagnetic systems, i.e. in systems where no net magnetic moment is allowed.
![Sketch showing the longitudinal and transversal spin-current generated by the $\cal{T}$-even and $\cal{T}$-odd components of the spin conductivity tensor. Note that the spin-polarization of spin currents shown here is just an illustration, other directions of spin-polarization can also occur depending on the symmetry of the material. a) Longitudinal SHE - $\cal{T}$-even current that flows in the same direction as the charge current. b) SHE - $\cal{T}$-even current that flows in a transverse direction to the charge current. c) Spin-polarized current - $\cal{T}$-odd current flowing in the same direction as the charge current. d) mSHE - odd current flowing in the transverse direction to the charge current.[]{data-label="Fig1"}](fig1.pdf){width="47.50000%"}
[Spin currents in RuO2 and MnTe]{}
==================================
In order to study the mSHE in collinear AFMs, we carried out *ab-initio* calculations within the density-functional theory (DFT) framework as implemented in the Vienna *ab-initio* simulation package (VASP) [@PhysRevB.54.11169]. We have studied collinear antiferromagnets RuO$_2$ and MnTe using the GGA+U method, as used in recent reports [@Berlijn2016a; @Kriegner2017]. We have performed both non-relativistic calculations and calculations with the relativistic spin-orbit coupling self-consistently included. Electron wave functions were expanded in plane waves up to a cut-off energy of 500 eV and a grid of 12x12x16 k-point has been used to sample the irreducible Brillouin zone. The calculated band-structures of RuO$_2$ and MnTe are shown in Fig. \[fig:Fermi\_dep\] a),b).
We evaluate the charge and spin conductivities within the linear response theory using the Kubo formula with the constant scaterring-rate $\Gamma$ approximation [@Jakub2017] as implemented in the Wannier-linear-response code [@Jakubcode]. Within the linear response the charge conductivity is described by the conductivity tensor: $j_i = \sigma_{ij}
E_j$, where $\mathbf{j}$ is the current, $\mathbf{E}$ is the electric field and $\sigma_{ij}$ is the conductivity tensor. Similarly, the spin-conductivity is described by a spin-conductivity tensor $\sigma^i_{kj}$, where $i$ corresponds to the spin-polarization of the spin current, $k$ to the direction of spin current flow and $j$ to the direction of electric field. The Kubo formula within the constant $\Gamma$ approximation can be split in two contributions given by
\^[I]{} = - \_[**k**, m, n]{} \[boltzmann\]
\^[II]{}=-2e. \_[**k**,mn]{}\^ \[intrinsic\]
Here $\chi$ is either $\sigma_{ij}$ or $\sigma^i_{kj}$, $u_n(\textbf{k})$ are the Bloch functions of a single band $n$, **k** is the Bloch wave vector, $\varepsilon_{n}$(**k**) is the band energy, $E_F$ is the Fermi energy, $\hat{v}$ is the velocity operator, $\hat{A}$ is the current density operator $\hat{A} = -e\hat{v}_i/V$ in the case of charge conductivity and the spin-current operator $\hat{A} = \frac{1}{2}\{\hat{s}_i,\hat{v}_k\}$ in the case of the spin-conductivity. In order to evaluate the Kubo formula, we have used an effective tight-binding Hamiltonian constructed in the maximally localized Wannier basis [@Mostofi2008] as a post-processing step of the DFT calculations. For the integration, a dense $320^3$ **k**-mesh was used.
For small $\Gamma$, the Eq. \[boltzmann\] scales as $1/\Gamma$ and corresponds to the Boltzmann formula with a constant relaxation time. Eq. is the so-called intrinsic contribution, which is $\Gamma$ independent. As discussed in Ref. [@Jakub2017], for charge conductivity, Eq. \[boltzmann\] is even under time-reversal and Eq. \[intrinsic\] is odd, whereas for the spin conductivity the situation is reversed. For charge conductivity, Eq. \[boltzmann\] corresponds to the ordinary conductivity and Eq. \[intrinsic\] corresponds to the AHE. For spin conductivity, Eq. \[boltzmann\] corresponds to the spin-polarized current and the mSHE and Eq. \[intrinsic\] corresponds to the SHE. In this work we mainly focus on the $\cal{T}$-odd spin conductivity. However, for comparison we also calculate the SHE and the AHE and in order to evaluate the spin-charge conversion efficiency, we also calculate the ordinary charge conductivity.
RuO$_2$ is a conductive transition metal oxide with orthorhombic rutile-type structure [@Berlijn2016a]. The primitive unit cell contains two Ru atoms surrounded by six O atoms that form a distorted octahedron. Recently, it was found that the Ru atoms can exhibit a collinear antiferromagnetic order and it was predicted that the AHE exists in the material [@Smejkal2019]. Because the AHE is in this case caused by a combination of symmetry breaking by the antiferromagnetic order and by the crystal, this effect was referred to as the crystal Hall effect in Ref. [@Smejkal2019]. The origin of antiferromagnetism in RuO$_2$ was recently studied in Ref. [@Ahn2019]. MnTe is a collinear antiferromagnet that crystallizes in the hexagonal NiAs-type structure. It has recently been used to demonstrate anisotropic magnetoresistance in an antiferromagnet [@Kriegner2016a]. It is a semiconductor, with a 1.46 eV indirect bandgap, however, the thin films used for the anisotropic magnetoresistance experiments exhibit an unintentional $p$-doping [@Kriegner2016a]. We model this by shifting the Fermi level, however, since the experimentally observed doping corresponds to only a very small shift, for which it is very difficult to accurately evaluate the response, we instead consider a larger shift of 0.25 eV in most calculations.
In the presence of SOC the symmetry of a material depends on orientation of the magnetic order. In RuO$_2$ the easy axis lies along the \[001\] direction. The AHE is not allowed for this direction of the Néel vector, but it is allowed when the Néel vector is rotated out of the \[001\] direction. *Ab-initio* calculations predict that the easy axis of RuO$_2$ can be switched to the (001) plane by small off-stoichiometry or by alloying with Ir [@Smejkal2019]. The easy axis of MnTe lies in the $c$-plane with much weaker anisotropy in the plane than out-of-plane, which allows for manipulating the Néel vector within the plane by large magnetic fields [@Kriegner2016a; @Kriegner2017]. We find that also in MnTe the AHE is allowed by symmetry, however, unlike in RuO$_2$ it vanishes both for the \[0001\] direction as well as for high-symmetry directions within the plane. This is confirmed by our calculations, shown in Fig. \[fig:angular\_dependence\]c. For comparison we show the same calculation for RuO$_2$ in Fig. \[fig:angular\_dependence\]a. We note that in both materials, the AHE only exists in presence of SOC. This applies generally for any coplanar magnetic material [@Zhang2018].
We now focus on the $\cal{T}$-odd spin currents. For RuO$_2$ calculations we use $\Gamma$ $\approx$ 6.6 $meV$, which is obtained by comparing the calculated conductivity (see Fig. \[fig:cond-ruo2\]) with the average experimental conductivity at 300K ($\sim$ 28400 S/cm)[@Ryden70]. For MnTe we have estimated the $\Gamma$ value to be close to 45 $meV$ by comparing the relaxation time ($\tau=\hbar$/2$\Gamma$) with the experimental value ($\tau=m^*_c\mu$/e) of the electron mobility measures [@ren2016]. We give the general form of the spin-conductivity tensors in RuO$_2$ and MnTe in Table \[table:symmetry\_soc\]. We find that with SOC the $\cal{T}$-odd spin currents are allowed by symmetry in both materials. This is confirmed by calculations shown in Figs. \[fig:angular\_dependence\]b and \[fig:angular\_dependence\]d for RuO$_2$ and MnTe, respectively. Here, we rotate the Néel vector in-plane for both RuO$_2$ and MnTe. We note that unlike the AHE, the $\cal{T}$-odd spin currents do not vanish in RuO$_2$ when the Néel vector is oriented along the $z$ direction, however, in analogy with Ref. [@Smejkal2019] we nevertheless consider the in-plane rotation of the Néel vector. When the SOC is turned off we find that in MnTe no $\cal{T}$-odd spin currents exist. In RuO$_2$ we find that some components of the $\cal{T}$-odd spin currents remain even in the non-relativistic calculation. This is confirmed by symmetry analysis, using the so-called spin groups which describe symmetry of magnetic systems in absence of SOC. The symmetry of the $\cal{T}$-odd spin conductivity tensors in absence of SOC is summarized in Table. \[table:symmetry\_nosoc\] in Appendix \[appendix:A\].
The origin of non-relativistic spin current in RuO$_2$ is illustrated in Fig. \[fig:sc\_origin\]. Since RuO$_2$ is a collinear magnetic material, in absence of SOC, the component of spin along the magnetic order direction is a good quantum number, i.e. it is a conserved quantity. As a consequence the eigenstates of the Hamiltonian can be separated into spin-up and spin-down states and the electrical current can be separated into spin-up and spin-down currents. Since spin-flip scattering is usually much smaller than spin conserving scattering, the two currents can be approximately treated as independent. This is known as the two current model and it is commonly used to describe the spin-polarized current in ferromagnets. In the case of ferromagnets, the two currents typically flow in the same direction, but are different in magnitude, thus they result in a longitudinal spin current (i.e. the spin-polarized current) and no transverse spin current.
In RuO$_2$ when electric field is applied along the \[100\] or \[010\] direction, it creates a spin-up and spin-down currents flowing at an angle with respect to the electric field. This results into an unpolarized longitudinal charge current and pure transverse spin current. A similar result is found for electric field along the \[010\] direction, however, interestingly we find that for the \[110\] and \[-110\] direction, the spin current is flowing in the same direction as the charge current and the charge current is thus spin-polarized. This is fully in agreement with the symmetry analysis shown in Appendix \[appendix:A\].
The magnitude of SHE is often given in terms of the spin Hall angle (SHA), which is defined as $(e/\hbar)\sigma^i_{jk}/\sigma_{kk}$, where $\sigma_{kk}$ is the longitudinal electrical conductivity. The SHA can be used the same way for the magnetic SHE or any other spin current. We find that in RuO$_2$ the magnetic SHA angle for the largest component is very large, $\approx$25%. This component corresponds to a configuration such that the electric field and the spin current are both in the $xy$ plane and the spin-polarization is given by the Néel vector direction. In comparison the SHA in MnTe is much smaller. The likely explanation of this is that in MnTe the origin of the effect is the SOC, which is relatively small effect even in materials containing heavy elements. In contrast, in RuO$_2$, the origin is the exchange interaction only, which is much stronger. In absolute values the spin conductivity in RuO$_2$ is very large $\approx 8600$ ($\hbar$/e)S/cm for the largest component. For comparison, in the widely used Pt, the intrinsic spin Hall conductivity is only 2180 ($\hbar$/e)S/cm and the spin Hall angle is only several percent (though a large variation between experimental results exists). Furthermore, as shown in Fig. \[fig:Fermi\_dep\] the SHC can reach even larger values when the Fermi level is shifted. For a shift of -0.6 eV, the spin conductivity becomes 25000 ($\hbar$/e)S/cm. This is in contrast to Pt, where the Fermi level is exactly at the maximum of the Fermi level conductivity [@Guo2008]. Furthermore, we note that our calculations utilize $\Gamma$ corresponding to room temperature resistivity. In RuO$_2$ the resistance at low temperatures was found to be 100 to 4000 times lower than the room temperature resistivity. Since for small $\Gamma$ the conductivity and $\cal{T}$-odd spin conductivity depend on $\Gamma$ as $1/\Gamma$, this will correspond to an equivalent increase in the spin-conductivity and the $\cal{T}$-odd spin conductivity will thus be much higher at low temperatures.
![Relativistic electronic band structure, anomalous Hall ($\sigma_{ij}$) and magnetic spin Hall conductivity ($\sigma^x_{ij}$) relative to the Fermi energy for AFMs: a) RuO$_2$ and b) MnTe. The vector **n** is $\|$ \[100\] and $\|$ $[1\overline{1}00]$ for RuO$_2$ and MnTe, respectively.[]{data-label="fig:Fermi_dep"}](fig2.pdf){width="47.00000%"}
Symmetry classification
=======================
The $\cal{T}$-odd spin currents have been studied in ferromagnets (primarily in the context of the spin-polarized current), in non-collinear antiferromagnets of the Mn$_3$X type [@Jakub2017; @Kimata2019] and in the collinear antiferromagnets discussed in this work. All of these systems are systems in which the AHE exists. Conversely in simple antiferromagnets no $\cal{T}$-odd spin currents are allowed by symmetry and neither is the AHE. This thus begs the question of whether the symmetry of the $\cal{T}$-odd spin currents is somehow related to the symmetry of the AHE and what are the general requirements for the existence of the $\cal{T}$-odd spin currents. For AHE, the symmetry requirements can be easily formulated. AHE can exist only in systems in which a net magnetic moment is allowed [@Shtrikman1965]. This is because AHE can be expressed as a $\cal{T}$-odd pseudovector and thus it transforms the same as a magnetic moment. We note, however, that in some AHE systems such as the ones discussed here, the net magnetic moment itself is very small and not directly related to AHE.
In general, the shape of linear response tensor for a particular material in an external electric field is determined by the crystal symmetry and magnetic order, which is, in the presence of SOC, described by the magnetic space group. Since the spin conductivity is invariant under translation and inversion, it is sufficient to consider the magnetic Laue groups. These are groups obtained from the magnetic space groups by removing translations and adding inversion to every group (alternatively the inversion could instead be removed). The symmetry of the full spin conductivity tensors was already given in Ref. [@Seemann2015] for all magnetic Laue groups. In that work no separation of the $\cal{T}$-even and $\cal{T}$-odd currents was done and thus no information about the requirements for the existence of $\cal{T}$-odd spin currents can be obtained. Here we study the symmetry of the $\cal{T}$-even and $\cal{T}$-odd spin conductivity tensors separately using a method described in Ref. [@Zelezny2017] and the Symmetr code. We use here the conventional labeling of Laue groups also used in Seemann et at [@Seemann2015], where also the corresponding point groups to each Laue group are given.
The Laue groups can be split into three categories. The groups of category (a) are groups which contain time-reversal as a separate element. These are groups which correspond either to a nonmagnetic crystal or to an antiferromagnet invariant under ${\cal{T}}\tau$ or $\cal{PT}$ symmetries. The groups of category (b) are groups in which no symmetry operation contains time-reversal. The groups of category (c) are groups in which some symmetry operations contain time-reversal, but time-reversal is not present separately. All groups of category (b) and (c) correspond to magnetic systems.
The SHE is allowed by symmetry in any material when SOC is present. We give the general form of the $\cal{T}$-even spin-conductivity tensors for Laue groups of categories (a), (b) and (c) in Tables \[table:even\_a\], \[table:even\_b\], \[table:even\_c\] respectively. Since both the AHE and the $\cal{T}$-odd spin currents are odd under time-reversal, they have to vanish in Laue groups of category (a). We give the general shape of the $\cal{T}$-odd spin conductivity tensors for groups (b) and (c) in Tables \[table:odd\_b\], \[table:odd\_c\] respectively. To compare with AHE, we also give the allowed magnetic moment for each Laue group. From this the AHE can be determined, since it holds for the AHE current that $\mathbf{j} = \mathbf{g} \times \mathbf{E}$, where $\mathbf{g}$ has the same symmetry as $\mathbf{M}$. Interestingly, we find that, unlike for the AHE, the symmetry of the $\cal{T}$-odd spin currents is unrelated to the magnetization and is much less restrictive. Whereas the AHE is present in only 10 of the 21 Laue groups of category (b) and (c), the $\cal{T}$-odd spin-conductivity is present in all 21 groups. This also means that the $\cal{T}$-odd spin currents can exist in truly antiferromagnetic systems in which no net magnetic moment is allowed by symmetry. There results are also evidenced by our calculations. As shown in Fig. \[fig:angular\_dependence\], for example, the AHE vanishes in every high-symmetry direction, whereas the $\cal{T}$-odd spin currents remain. We also find that in all of the 21 Laue groups, apart from the group m-3m, both transverse and longitudinal components are allowed for some directions of electric field. In the group m-3m, the $\cal{T}$-odd spin currents are transverse for any direction of electric field and thus no spin-polarized current can exist in materials with this symmetry.
The symmetry requirements for the existence of $\cal{T}$-odd spin currents can be thus formulated in a simple way. These spin currents will be present in every system in which the $\cal{T}$, ${\cal{T}}\tau$ and $\cal{PT}$ symmetries are broken. They are therefore different than the AHE, which has much more stricter symmetry requirements for its existence. It can be said, however, that in every system in which the $\cal{T}$-odd spin currents exist, the AHE will also exist if the magnetic order is rotated out of high symmetry direction. This is because rotating the magnetic order out of high symmetry direction breaks all rotation symmetries.
Similar classification for existence of spin conserving pure transverse spin currents is significantly more involved and is left to future work. Nevertheless, some rules can be formulated. Since the ordinary SHE can only exist without SOC in a non-collinear magnetic system it cannot exist in a system that conserves spin. Thus only the mSHE can be considered for the transverse spin currents and the above rules, therefore, also apply. This means that a collinear magnetic system with broken ${\cal{T}}\tau$ and $\cal{PT}$ is needed. Although a non-relativistic transverse spin current could also exist in a low symmetry ferromagnet, for spin-charge conversion a pure spin current is utilized, which would be hard to achieve in a ferromagnet. Therefore, we can generally say that to achieve a spin conserving spin-charge conversion, we need a collinear antiferromagnet that breaks the ${\cal{T}}\tau$ and $\cal{PT}$ symmetries and that has asymmetric crystalline environment at each sublattice that allows for existence of a transverse current at each sublattice.
Discussion
==========
The limitations of spin-charge conversion mechanisms based on SOC or non-collinear magnetism are quite fundamental. Since these mechanisms do not conserve spin it is simply not possible to have an efficient spin charge conversion based on these mechanisms in system that approximately conserves spin. This has important consequences. First, it limits the possible functionalities of devices since it means that the spin diffusion length is very short and the spin to charge conversion thus always happens very close to the interface and no long term spin transport through the spin to charge conversion material is possible. Second, theoretical description and experimental study of such spin currents is problematic in absence of spin conservation. This is part of the reason why the origin of the spin-orbit torque in bilayer systems is still not fully understood despite years of intensive research [@Manchon2018inpress]. In contrast the spin-transfer torque, which can be described with the two-current model, is much better understood [@Ralph2008; @Brataas2012]. Third, the short spin diffusion length significantly reduces the efficiency of the spin-charge conversion. Although the spin current itself does not depend on the spin relaxation length, for practical purposes only the spin accumulation or spin torque due to the spin current are utilized and these strongly depend on the spin relaxation length.[@Haney2013].
With the presently proposed mechanism, no such limitations exist. Although no system conserves spin perfectly, the advantage of the mechanism proposed in this work is that the spin to charge conversion efficiency can be tuned independently of the spin conservation and the mechanism could thus in principle exist in systems in which the SOC is small and the spin approximately conserved, which might result in larger efficiency of a current induced spin accumulation or spin torque.
A spin current with a similar origin to the nonrelativistic mSHE in RuO$_2$ has been recently theoretically studied in an organic antiferromagnet [@Naka2019]. Organic materials have very small SOC, which could result in much longer spin relaxation lengths than in RuO$_2$, but on the other hand the studied organic antiferromagnet has very low Néel temperature. We also note than in contrast to what is claimed in Ref. [@Naka2019], the conventional SHE does not require broken inversion symmetry, is not necessarily represented by antisymmetric tensor and does not always originate from SOC. Instead, the core difference between the non-relativistic mSHE in collinear AFMs and the conventional SHE is the transformation under time-reversal.
From a general point of view, utilizing magnetic materials for spin-charge conversion could have some advantages over the common approach based on the SHE in nonmagnetic materials. The symmetry of the conventional SHE in nonmagnetic systems is quite restricted. In most commonly used materials, the SHE has a symmetry such that the direction of electric field, direction of the spin current flow and the spin-polarization of the spin current are all perpendicular. This limits its possible applications. For example, in bilayer systems the spin current flows in the out-of-plane direction, which means that the spin-polarization for the conventional SHE is in-plane. The magnetization of the ferromagnetic layer is typically out-of-plane, however, and then for the most efficient switching based on the antidamping torque, out-of-plane polarization of the spin current is required. In contrast, the symmetry of the ordinary and mSHE in magnetic systems is less restricted. Even more importantly, the spin currents generated in magnetic systems can be controlled by manipulating the magnetic order. This is especially true for the mSHE as illustrated in Fig. \[fig:angular\_dependence\], but also applies for the ordinary SHE, as shown in Appendix \[appendix:C\]. This could allow for completely new functionalities. We note that in the case of the non-relativistic mSHE in RuO$_2$ the only dependence on the Néel vector direction is in that the spin-polarization of the spin current is given by the Néel vector direction. Although this is quite simple dependence, it could also be useful since it allows, for example, for reversing the sign of the spin current by reversing the magnetic order.
The $\cal{T}$-odd spin currents and the AHE share some common symmetry requirements and because of that they often occur in similar materials. The two effects are not otherwise related, however. As discussed in Ref. [@Jakub2017], they have a different microscopic origin and as shown in this work, the general symmetry requirements for their existence are different. The only similarity is that both effects require breaking of the $\cal{T}$, ${\cal{T}}\tau$ and $\cal{PT}$ symmetries. Whereas the $\cal{T}$ symmetry is broken by any magnetic order, the later two are often present in antiferromagnetic systems. In the case of the non-collinear AFMs the ${\cal{T}}\tau$ and $\cal{PT}$ are broken by the non-collinear magnetic order, whereas in the case of RuO$_2$ and MnTe they are broken by the collinear magnetic order together with the nonmagnetic atoms, as discussed in detail in [@Smejkal2019].
The direct mSHE discussed in this work allows for converting a charge current into a spin current. As with the SHE, the existence of the direct mSHE effect directly implies the existence of the inverse effect, which allows for converting a spin current into a current charge current. We focus only on the direct effect here since its theoretical description is simpler, however, the inverse effect can be directly obtained from the direct effect through the Onsager relations [@Seemann2015]. Crucially, the existence of a large direct effect also implies existence of a large inverse effect.
We have mostly discussed the transverse spin currents here (i.e. the mSHE), however, our calculations show that the longitudinal spin currents are also present in the studied systems and are in fact intimately connected to the transverse currents. As mentioned previously, the longitudinal current can be understood as a spin-polarized current. The spin-polarized currents are also very relevant for spintronics since they are utilized for the most widely used spintronics devices: the feromagnetic spin-valves and tunneling junctions. As discussed in Ref. [@Jakub2017] the antiferromagnetic spin-polarized currents could likely be used for the same purpose.
Conclusions
===========
Based on symmetry analysis and $ab$-$initio$ calculations we find that $\cal{T}$-odd spin currents that were recently found in non-collinear antiferromagnets can also exist in collinear antiferromagnetic systems. We identify two antiferromagnetic materials in which they are exist: MnTe and RuO$_2$. Both the longitudinal component, corresponding to the spin-polarized current, and transverse component, corresponding to the mSHE are present, similarly to the non-collinear antiferromagnets. Our results show that in RuO$_2$ the $\cal{T}$-odd spin currents survive even when no relativistic SOC is included. In such a case, since RuO$_2$ is a collinear magnetic system, the net spin is conserved. This thus allows for a spin-charge conversion in a spin-conserving system, which could improve efficiency of spin-charge conversion devices or even allow for new spintronics functionalities. We predict a large $\approx 25\%$ magnetic spin Hall angle in RuO$_2$, which together with the non-relativistic origin of the $\cal{T}$-odd spin currents, makes this a very interesting spin-charge conversion material. We identify a general symmetry requirements for the existence of the $\cal{T}$-odd spin currents. We find that they are allowed in all magnetic materials with broken ${\cal{T}}\tau$ or $\cal{PT}$ symmetries, or equivalently in all materials that have broken time-reversal symmetry in the Laue group.
Parts of this research were conducted using the supercomputer Mogon and/or advisory services offered by Johannes Gutenberg University Mainz (hpc.uni-mainz.de), which is a member of the AHRP (Alliance for High Performance Computing in Rhineland Palatinate, www.ahrp.info) and the Gauss Alliance e.V. The authors gratefully acknowledge the computing time granted on the supercomputer Mogon at Johannes Gutenberg University Mainz (hpc.uni-mainz.de) and the support of Alexander Von Humboldt Foundation. We acknowledge the Grant Agency of the Czech Republic Grant No. 19-18623Y and support from the Institute of Physics of the Czech Academy of Sciences and the Max Planck Society through the Max Planck Partner Group programme. This work was supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project „IT4Innovations National Supercomputing Center – LM2015070“. Yuta YAHAGI acknowledges support from GP-Spin at Tohoku University.
Symmetry of spin-conductivity tensors in RuO$_2$ and MnTe {#appendix:A}
=========================================================
Here we list the general form of spin-conductivity tensors in RuO$_2$ and MnTe, obtained using the Symmetr code [@symcode]. The method is described in Ref. [@Zelezny2017] for symmetry with SOC and Ref. [@Zhang2018] for the symmetry without SOC. All the results are given in cartesian coordinate systems. In RuO$_2$ this is simply a coordinate system oriented along the principal axes of the crystal. In MnTe, this is a coordinate system such that $\mathbf{x}$ is oriented along the \[1000\] direction and $\mathbf{z}$ along the \[0001\] direction.
[p[25mm]{}ccc]{}
------------------------------------------------------------------------
& $\sigma^x$ & $\sigma^y$ & $\sigma^z$\
RuO$_2$ $\mathbf{n} || [001]$& $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & \sigma_{yxz}\\0 & \sigma_{yzx} & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & \sigma_{yxz}\\0 & 0 & 0\\\sigma_{yzx} & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & \sigma_{zyx} & 0\\\sigma_{zyx} & 0 & 0\\0 & 0 & 0\end{matrix}\right)$\
RuO$_2$ $\mathbf{n} || [100]$ & $\left(\begin{matrix}0 & \sigma_{xxy} & 0\\\sigma_{xyx} & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}\sigma_{yxx} & 0 & 0\\0 & \sigma_{yyy} & 0\\0 & 0 & \sigma_{yzz}\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & \sigma_{zyz}\\0 & \sigma_{zzy} & 0\end{matrix}\right)$\
RuO$_2$ $\mathbf{n} \perp [001]$ & $\left(\begin{matrix}\sigma_{xxx} & \sigma_{yyx} & 0\\\sigma_{xyx} & \sigma_{yxx} & 0\\0 & 0 & \sigma_{xzz}\end{matrix}\right)$ & $\left(\begin{matrix}\sigma_{yxx} & \sigma_{xyx} & 0\\\sigma_{yyx} & \sigma_{xxx} & 0\\0 & 0 & \sigma_{xzz}\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & \sigma_{zxz}\\0 & 0 & \sigma_{zxz}\\\sigma_{zzx} & \sigma_{zzx} & 0\end{matrix}\right)$\
MnTe $\mathbf{n} || [0001]$ & $\left(\begin{matrix}\sigma_{xxx} & 0 & 0\\0 & - \sigma_{xxx} & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & - \sigma_{xxx} & 0\\- \sigma_{xxx} & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$\
MnTe $\mathbf{n} || [1000]$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & \sigma_{xyz}\\0 & \sigma_{xzy} & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & \sigma_{yxz}\\0 & 0 & 0\\\sigma_{yzx} & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & \sigma_{zxy} & 0\\\sigma_{zxy} & 0 & 0\\0 & 0 & 0\end{matrix}\right)$\
MnTe $\mathbf{n} \perp [001]$ & $\left(\begin{matrix}0 & 0 & \sigma_{xxz}\\0 & 0 & \sigma_{xyz}\\ \sigma_{xzx} & \sigma_{xzy} & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & \sigma_{yxz}\\0 & 0 & \sigma_{yyz}\\\sigma_{yzx} & \sigma_{yzy} & 0\end{matrix}\right)$ & $\left(\begin{matrix}\sigma_{zxx} & \sigma_{zxy} & 0\\\sigma_{zyx} & \sigma_{zyy} & 0\\0 & 0 & \sigma_{zzz}\end{matrix}\right)$\
[p[30mm]{}ccc]{}
------------------------------------------------------------------------
& $\sigma^x$ & $\sigma^y$ & $\sigma^z$\
RuO$_2$ $\mathbf{n} || [001]$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & \sigma_{zyx} & 0\\\sigma_{zyx} & 0 & 0\\0 & 0 & 0\end{matrix}\right)$\
MnTe & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$ & $\left(\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$\
Symmetry tables {#appendix:B}
===============
All tensors are given in cartesian coordinate systems. These are defined the same way as in Ref. [@Zelezny2017]. The cartesian systems are defined in terms of the conventional basis vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ (see the International Tables for Crystallography [@crystallographictables]). The choice of the cartesian system is straightforward for the orthorhombic, tetragonal and cubic groups. The tensors for the triclinic group $1$ have a completely general form and the choice of the coordinate system is thus irrelevant for this group. For hexagonal and trigonal groups, we choose the right-handed coordinate system that satisfies $\mathbf{x} = \mathbf{a}/|\mathbf{a}|$, $\mathbf{z} = \mathbf{c}/| \mathbf{c}|$. For the monoclinic groups we use the unique axis $b$ setting [@crystallographictables] and choose the right-handed coordinate system that satisfies $\mathbf{x} = \mathbf{a}/|\mathbf{a}|$, $\mathbf{y} = \mathbf{b}/|\mathbf{b}|$.
Angular dependence of SHE {#appendix:C}
=========================
![Angular dependence of SHE components ($\sigma^x$, $\sigma^y$ and $\sigma^z$) for a) RuO$_2$ and b) MnTe AFMs, when the Neel vector **n** is rotated in the $xy$-plane. $\Phi$ denotes the in-plane angle from the $x$-axis. Only the largest components are shown and the energy was set to 0.25 eV below the valence band maximum for the case of MnTe.[]{data-label="fig:she"}](fig3-she.pdf){width="47.50000%"}
|
---
abstract: 'An exact solution of Einstein’s equations which represents a pair of accelerating and rotating black holes (a generalised form of the spinning $C$-metric) is presented. The starting point is a form of the Plebański–Demiański metric which, in addition to the usual parameters, explicitly includes parameters which describe the acceleration and angular velocity of the sources. This is transformed to a form which explicitly contains the known special cases for either rotating or accelerating black holes. Electromagnetic charges and a NUT parameter are included, the relation between the NUT parameter $l$ and the Plebański–Demiański parameter $n$ is given, and the physical meaning of all parameters is clarified. The possibility of finding an accelerating NUT solution is also discussed.'
author:
- |
J. B. Griffiths$^1$[^1] and J. Podolský$^2$[^2]\
\
$^1$Department of Mathematical Sciences, Loughborough University,\
Loughborough, Leics. LE11 3TU, U.K.\
$^2$Institute of Theoretical Physics, Charles University in Prague,\
V Holešovičkách 2, 18000 Prague 8, Czech Republic.
title: '**Accelerating and rotating black holes**'
---
Introduction
============
The Schwarzschild and Kerr metrics, which describe the fields surrounding a static and rotating black hole respectively, are surely the best known solutions of Einstein’s equations. Their charged versions are the Reissner–Nordström and Kerr–Newman solutions respectively. The $C$-metric, together with its charged version, is similarly well known. This represents a pair of causally separated black holes which accelerate away from each other under the action of forces represented by topological singularities along the axis. However, the inclusive solution, which represents a pair of (possibly charged) rotating and accelerating black holes has only recently begun to receive detailed attention.
A large family of electrovacuum solutions of algebraic type D was presented in 1976 by Plebański and Demiański [@PleDem76]. This contains a number of parameters (including a cosmological constant). Special cases of this, after different transformations, include both the Kerr–Newman solution and the $C$-metric. In this sense, it must therefore include a solution which represents an accelerating and rotating black hole. When some of the Plebański–Demiański parameters are set to zero, a solution is obtained that has come to be known as the “spinning $C$-metric”. This has been studied by many authors ([@FarZim80b]–[@Pravdas02]). However, it has very recently been shown by Hong and Teo [@HongTeo05] that a different choice of parameters, which removes the properties associated with a non-zero NUT parameter, is more appropriate to represent an accelerating and rotating pair of black holes.
In this paper, we present the family of solutions which describes the general case of a pair of accelerating and rotating charged black holes in a very convenient form. A generally non-zero NUT parameter is included, but the cosmological constant is taken to be zero. In appropriate limits, this family of solutions explicitly includes both the Kerr–Newman–NUT solution and the $C$-metric, without the need for further transformations. Also, the relation between the Plebański–Demiański parameter $n$ and the NUT parameter $l$ is given explicitly, confirming that these should not generally be identified. This result enables us to discuss the possibility of accelerating NUT solutions, in addition to the accelerating and rotating solution with no NUT parameter that has been identified by Hong and Teo [@HongTeo05].
Our analysis starts with a new form of the Plebański–Demiański metric which includes explicit parameters representing the acceleration of the sources and the twist of the repeated principal null congruences. These parameters are much more helpful than the traditional ones in determining the physical properties of this family of solutions.
An initial form of the metric
=============================
For the case in which the cosmological constant is zero, the Plebański–Demiański metric [@PleDem76] (see also §21.1.2 of [@SKMHH03]) is given by $$\d s^2={1\over(1-\hat p\hat r)^2} \Bigg[
{{\cal Q}(\d\hat\tau-\hat p^2\d\hat\sigma)^2\over\hat r^2+\hat p^2} -{{\cal P}(\d\hat\tau+\hat r^2\d\hat\sigma)^2\over\hat r^2+\hat p^2}
-{\hat r^2+\hat p^2\over{\cal P}}\,\d\hat p^2
-{\hat r^2+\hat p^2\over{\cal Q}}\,\d\hat r^2 \Bigg]
\label{oldPDMetric}$$ where ${\cal P} =\hat k +2\hat n\hat p -\hat\epsilon\hat p^2
+2\hat m\hat p^3-(\hat k+\hat e^2+\hat g^2)\hat p^4$, ${\cal Q} =\hat k+\hat e^2+\hat g^2 -2\hat m\hat r +\hat\epsilon\hat r^2
-2\hat n\hat r^3-\hat k\hat r^4$ and $\hat m$, $\hat n$, $\hat e$, $\hat g$, $\hat\epsilon$ and $\hat k$ are arbitrary real parameters. It is usually assumed that $\hat m$ and $\hat n$ are the mass and NUT parameters although, as will be shown below, this is not generally the case. The parameters $\hat e$ and $\hat g$ represent electric and magnetic charges.
In the case when the parameter $\hat n$ and the charge parameters vanish (i.e. when $\hat n=\hat e=\hat g=0$) with $\hat k>0$ and $\hat\epsilon>0$, the metric (\[oldPDMetric\]), or a transformation of it, is traditionally referred to as the “spinning $C$-metric”. It is considered to represent two uniformly accelerating, rotating black holes, either connected by a conical singularity, or with conical singularities extending from each to infinity. Farhoosh and Zimmerman [@FarZim80b] have investigated the properties of the horizons for these solutions. Bičák and Pravda [@BicPra99] have also transformed the metric into Lewis–Papapetrou form and then expressed it in the canonical form of radiative space-times with boost-rotation symmetry. However, very recently, Hong and Teo [@HongTeo05] have shown that a different non-zero choice of $\hat n$ is more appropriate to describe a pair of accelerating and rotating black holes. (This will be analysed in greater detail below.)
With the metric in the form (\[oldPDMetric\]), however, the acceleration and rotation of the sources are not clearly represented. For ease of interpretation, it is in fact convenient to introduce the rescaling $$\hat p=\sqrt{\alpha\omega}\,p, \qquad \hat r=\sqrt{\alpha\over\omega}\,r, \qquad \hat\sigma=\sqrt{\omega\over\alpha^3}\,\sigma, \qquad
\hat\tau=\sqrt{\omega\over\alpha}\,\tau,
\label{scaling}$$ with the relabelling of parameters $$\hat m+i\hat n=\Big({\alpha\over\omega}\Big)^{3/2}(m+in), \qquad
\hat e+i\hat g={\alpha\over\omega}(e+ig), \qquad
\hat\epsilon={\alpha\over\omega}\,\epsilon, \qquad
\hat k=\alpha^2k.
\label{scaleps}$$ This introduces two additional parameters $\alpha$ and $\omega$ and the associated freedom to choose any two of the parameters in a more convenient way. With these changes, the metric becomes $$\begin{array}{r}
{\displaystyle \d s^2={1\over(1-\alpha pr)^2} \Bigg[
{Q\over r^2+\omega^2p^2}(\d\tau-\omega p^2\d\sigma)^2
-{P\over r^2+\omega^2p^2}(\omega\d\tau+r^2\d\sigma)^2 } \hskip3pc \\[12pt]
{\displaystyle -{r^2+\omega^2p^2\over P}\,\d p^2
-{r^2+\omega^2p^2\over Q}\,\d r^2 \Bigg],}
\end{array}
\label{PleDemMetric}$$ where $$\begin{array}{l}
P=P(p) =k +2\omega^{-1}np -\epsilon p^2 +2\alpha mp^3
-\alpha^2(\omega^2 k+e^2+g^2)p^4, \\[8pt]
Q=Q(r) =(\omega^2k+e^2+g^2) -2mr +\epsilon r^2 -2\alpha\omega^{-1}nr^3
-\alpha^2kr^4,
\end{array}
\label{PQeqns}$$ and $m$, $n$, $e$, $g$, $\epsilon$, $k$, $\alpha$ and $\omega$ are arbitrary real parameters. (It will be shown that the coefficient $\omega^{-1}n$ is in fact well behaved in the limit as $\omega\to0$.)
In terms of a convenient tetrad, the only non-zero component of the Weyl tensor is given by $$\Psi_2= \left( -(m+in) +(e^2+g^2) {1+\alpha pr\over r-i\omega p} \right)
\left({1-\alpha pr\over r+i\omega p}\right)^3.
\label{Weyl1}$$ This confirms that these space-times are of algebraic type D. The only non-zero component of the Ricci tensor is $$\Phi_{11}= {1\over2}\,(e^2+g^2)\,{(1-\alpha pr)^4\over(r^2+\omega^2p^2)^2}.
\label{Ricci1}$$ These components indicate the presence of a curvature singularity at $r=0$, $p=0$. This singularity may be considered as the source of the gravitational field. They also show that the line element (\[PleDemMetric\]) is flat if $m=n=0$ and $e=g=0$. (Notice that the remaining kinematical parameters $\epsilon$, $k$, $\alpha$ and $\omega$, may be non-zero in this flat limit.)
To retain a Lorentzian signature in (\[PleDemMetric\]), it is necessary that $P>0$. Thus, the coordinate $p$ must be restricted to a particular range between appropriate roots of $P$. Specifically, if it is required that a singularity should appear in the boundary of the space-time, then this range must include $p=0$. This would require that $k>0$. However, important non-singular solutions also exist for which the chosen range of $p$ does not include $p=0$.
The points at which $P=0$ correspond to poles of the coordinates. By contrast, surfaces on which $Q=0$ are (Killing) horizons through which coordinates can be extended. Moreover, since $Q(r)=-\alpha^2r^4P(1/\alpha r)$, it is clear that $P$ and $Q$ have the same number of roots. In this paper, it will be assumed that these quartics have four distinct roots. However, other cases, which have less physical significance, also exist. Together with the requirement that $P>0$, the existence of four roots places certain restrictions on the ranges and signs of some of the parameters introduced as coefficients in (\[PQeqns\]). In the case considered here, the parameters acquire the following physical interpretations:
\[$e,g$\]: By examining the electromagnetic field and the curvature tensor components, it is clear that $e$ and $g$ denote the electric and magnetic charges of the sources.
\[$\alpha,\omega$\]: Previously, it has been considered appropriate to use the rescaling (\[scaling\]) with (\[scaleps\]) to set $\alpha=1$ and $\omega=1$. However, the physical interpretation of the solution can be determined more transparently by retaining $\alpha$ and $\omega$ as continuous parameters. In fact, $\alpha$ generally represents the acceleration of the sources. Also, $\omega$ is proportional to the twist of the repeated principal null directions and this relates to both the angular velocity of the sources and the NUT-like properties of the space-time. (It may be noted that these parameters can always be taken to be non-negative. If $\alpha<0$, its sign can always be changed by changing the signs of either $p$ and $n$ or $r$ and $m$. Similarly, if $\omega<0$, its sign can be changed by changing that of either $\sigma$ or $\tau$.) This approach also enables us to obtain the non-accelerating and non-rotating cases by trivially setting either of these parameters to zero.
\[$\epsilon,k$\]: Retaining $\alpha$ and $\omega$ as continuous parameters, we are free to use the rescaling (\[scaling\]) with (\[scaleps\]) in some other way. Actually, it is most convenient to use this freedom to rescale the parameters $\epsilon$ and $k$ to some specific values, although it is not possible to alter their signs. In fact $\epsilon$ and $k$ are related to the discrete parameters that specify the curvature of appropriate 2-surfaces and a canonical choice of coordinates on them. As appropriate for solutions representing black holes, we will only consider the case in which the relevant 2-surfaces have positive curvature. The signs of $\epsilon$ and $k$ have to be consistent with this interpretation.
\[$m,n$\]: For certain choices of the other parameters, it will be shown that $m$ is related to the mass of the source and $n$ to the NUT parameter. However, it should be emphasised that these only acquire their usual specific well-identified meanings in certain special sub-cases.
A more general line element
===========================
It can be shown that, with a transformation of the coordinate $\tau$, the line element (\[PleDemMetric\]) already contains the Kerr–Newman solution when $\alpha=0$ and the $C$-metric when . However, to introduce an explicit NUT parameter, and hence to include the non-singular NUT solution [@NewTamUnt63], it is necessary to include a shift in the coordinate $p$. To include this possibility, we start with the metric (\[PleDemMetric\]) with (\[PQeqns\]), and perform the coordinate transformation $$p=\omega^{-1}(l+a\tilde p), \qquad \tau=t-(l+ap_0)^2a^{-1}\phi,
\qquad \sigma=-\omega a^{-1}\phi,
\label{trans1A}$$ where $a$, $l$ and $p_0$ are arbitrary parameters. By this procedure, we obtain the metric $$\begin{array}{l}
{\displaystyle \d s^2={1\over\Omega^2}\left\{
{Q\over\rho^2}\left[\d t- \left(a(p_0^2-\tilde p^2)
+2l(p_0-\tilde p) \right)\d\phi \right]^2
-{\rho^2\over Q}\,\d r^2 \right.
} \\[8pt]
\hskip8pc {\displaystyle
\left. -{\tilde P\over\rho^2} \Big[ a\d t
-\Big(r^2+(l+ap_0)^2\Big)\d\phi \Big]^2
-{\rho^2\over\tilde P}\,\d\tilde p^2 \right\}, }
\end{array}
\label{altPlebMetric}$$ where $$\begin{array}{l}
\Omega=1-\alpha\omega^{-1} (l+a\tilde p)r \\[6pt]
\rho^2 =r^2+(l+a\tilde p)^2 \\[6pt]
\tilde P= a_0 +a_1\tilde p +a_2\tilde p^2
+ a_3\tilde p^3 +a_4\tilde p^4 \\[8pt]
Q= (\omega^2k+e^2+g^2) -2mr +\epsilon r^2
-2\alpha\omega^{-1}n r^3 -\alpha^2kr^4,
\end{array}$$ and we have put $$\begin{array}{l}
a_0= a^{-2} \Big( \omega^2k +2nl -\epsilon l^2
+2\alpha\omega^{-1} ml^3 -\alpha^2\omega^{-2}(\omega^2k+e^2+g^2)l^4 \Big) \\[6pt]
a_1=2a^{-1}\Big( n -\epsilon l
+3\alpha\omega^{-1} ml^2 -2\alpha^2\omega^{-2}(\omega^2k+e^2+g^2)l^3 \Big) \\[6pt]
a_2= -\epsilon +6\alpha\omega^{-1} ml -6\alpha^2\omega^{-2}(\omega^2k+e^2+g^2)l^2 \\[6pt]
a_3= 2a\alpha\omega^{-1}m -4a\alpha^2\omega^{-2}(\omega^2k+e^2+g^2)l \\[6pt]
a_4= -a^2\alpha^2\omega^{-2}(\omega^2k+e^2+g^2).
\end{array}$$
We can now choose the new parameter $p_0$ to coincide with one of the roots of $\tilde P$. The metric (\[altPlebMetric\]) is then regular at the pole $\tilde p=p_0$ which corresponds to an axis, so that it is appropriate to take $\phi$ as a periodic coordinate. We now consider the case in which $\tilde P(0)>0$ (i.e. $a_0>0$) for which a second root of $\tilde P$ will exist with the opposite sign to that of $p_0$. It is then possible to use the freedom to constrain the parameters $\epsilon$, $k$, $a$ and $l$ to fix this other root at $\tilde p=-p_0$. It can then be seen that the metric component $a(p_0^2-\tilde p^2)$ is regular at this second pole while the component $2l(p_0-\tilde p)$ is not. Thus, the metric is regular on the half-axis $\tilde p=p_0$, but a singularity of some kind occurs on the other half-axis $\tilde p=-p_0$. In fact, with this choice, $a$ corresponds to a Kerr-like rotation parameter for which the corresponding metric components are regular on the entire axis, while $l$ corresponds to a NUT parameter for which the corresponding components are only regular on the half-axis $\tilde p=p_0$.
In addition, the freedom in the choice of $\epsilon$, $k$, $a$ and $l$ can be further used to set $p_0=1$, so that $\tilde P$ has factors $(1-\tilde p)$ and $(1+\tilde p)$. With this, we can write $$\tilde P= (1-\tilde p^2)(a_0-a_3\tilde p-a_4\tilde p^2),$$ so that the conditions $$a_1+a_3=0, \qquad a_0+a_2+a_4=0
\label{Pconditions}$$ must also be satisfied. The available freedom is sufficient to satisfy these two conditions provided the signs of $\epsilon$ and $k$ are consistent. In fact they provide two linear equations which specify the two parameters $\epsilon$ and $n$ as $$\begin{aligned}
&&\epsilon= {\omega^2k\over a^2-l^2}+4{\alpha l\over\omega}\,m
-{\alpha^2(a^2+3l^2)\over\omega^2}(\omega^2k+e^2+g^2), \label{epsilon}\\[8pt]
&&n= {\omega^2k\,l\over a^2-l^2} -{\alpha(a^2-l^2)\over\omega}\,m
+{\alpha^2(a^2-l^2)l\over\omega^2}\,(\omega^2k+e^2+g^2). \label{n}
\end{aligned}$$ Equation (\[n\]) explicitly relates the Plebański–Demiański parameter $n$ to the NUT parameter $l$, while (\[epsilon\]) specifies the required value of $\epsilon$. With these, we obtain that $$a_0={\omega^2k\over a^2-l^2} -2{\alpha l\over\omega}\,m
+3{\alpha^2l^2\over\omega^2}(\omega^2k+e^2+g^2).$$ In fact, it is also possible to choose the parameters such $a_0=1$, provided this is consistent with the signs of $\epsilon$ and $k$. (Otherwise this would imply that $\tilde P$ is not positive between roots at $\pm1$, the geometry would be different and the space-times would not represent black hole-like objects.) The required value of $k$ to achieve $a_0=1$ is given by $$\left( {\omega^2\over a^2-l^2}+3\alpha^2l^2 \right)\,k =1 +2{\alpha l\over\omega}\,m
-3{\alpha^2l^2\over\omega^2}(e^2+g^2).
\label{k}$$
The original metric (\[oldPDMetric\]) contained two kinematical parameters $\hat\epsilon$ and $\hat k$. In the above argument, we have increased these parameters to $\epsilon$, $k$, $\alpha$, $\omega$, $a$ and $l$, but we have introduced three constraints that are effectively represented by (\[epsilon\]), (\[n\]) and (\[k\]). One remaining freedom is therefore still available which could, for example, be used to set $\omega=|a+l|$.
A new form of the metric
========================
With the above conditions satisfied, it is natural to put $\tilde p=\cos\theta$, where $\theta\in[0,\pi]$, so that $\theta$ spans the permitted range of $\tilde P$ between the roots $\tilde p=\pm1$. Substituting also for $\epsilon$ and $n$, the metric (\[altPlebMetric\]) becomes $$\begin{array}{l}
{\displaystyle \d s^2={1\over\Omega^2}\left\{
{Q\over\rho^2}\left[\d t- \left(a\sin^2\theta
+4l\sin^2{\textstyle{\theta\over2}} \right)\d\phi \right]^2
-{\rho^2\over Q}\,\d r^2 \right.
} \\[8pt]
\hskip8pc {\displaystyle
\left. -{\tilde P\over\rho^2} \Big[ a\d t
-\Big(r^2+(a+l)^2\Big)\d\phi \Big]^2
-{\rho^2\over\tilde P}\sin^2\theta\,\d\theta^2 \right\}, }
\end{array}
\label{newMetric}$$ where $$\begin{array}{l}
{\displaystyle \Omega=1-{\alpha\over\omega}(l+a\cos\theta)\,r } \\[6pt]
\rho^2 =r^2+(l+a\cos\theta)^2 \\[6pt]
\tilde P= \sin^2\theta\,(1-a_3\cos\theta-a_4\cos^2\theta) \\[6pt]
Q= {\displaystyle \left[(\omega^2k+e^2+g^2)\bigg(1+2{\alpha l\over\omega}\,r\bigg)
-2mr +{\omega^2k\over a^2-l^2}\,r^2\right] } \\[6pt]
\hskip5pc \times {\displaystyle
\left[1+{\alpha(a-l)\over\omega}\,r\right] \left[1-{\alpha(a+l)\over\omega}\,r\right] }
\end{array}
\label{newMetricFns}$$ and $$\begin{array}{l}
{\displaystyle a_3= 2{\alpha a\over\omega}m -4{\alpha^2 al\over\omega^2}
(\omega^2k+e^2+g^2) } \\[6pt]
{\displaystyle a_4= -{\alpha^2a^2\over\omega^2}(\omega^2k+e^2+g^2) }
\end{array}
\label{a34}$$ with $k$ given by (\[k\]). This contains seven arbitrary parameters $m$, $l$, $e$, $g$, $a$, $\alpha$ and $\omega$. Of these, the first six can be varied independently, and the remaining freedom can be used to set $\omega$ to any convenient value if at least one of the parameters $a$ or $l$ are non-zero.
The non-zero components of the curvature tensor are given by (\[Weyl1\]) and (\[Ricci1\]) in which $\omega p$ is replaced by $l+a\cos\theta$. It can be seen that, if $|l|\le|a|$, the metric (\[newMetric\]) has a curvature singularity when $\rho^2=0$; i.e. at $r=0$, $\cos\theta=-l/a$. However, if $|l|>|a|$, it is non-singular.
We present this metric as the family of solutions which represents a pair of accelerating and rotating charged black holes with a generally non-zero NUT parameter. We will now show that it reduces explicitly to familiar forms of either the Kerr–Newman–NUT solution or the $C$-metric in appropriate cases, without the need for further transformations. After that, we will consider further aspects of the interpretation of these solutions and some additional special cases.
Special cases
=============
It can first be seen that, when $\alpha=0$, we have $\omega^2k=a^2-l^2$ and hence $$\epsilon=1 \qquad \hbox{and} \qquad n=l.$$ The second of these conditions simply identifies the Plebański–Demiański parameter $n$ with the NUT parameter $l$ in this case. Moreover, $\tilde P=\sin^2\theta$, and the metric (\[newMetric\]) becomes $$\begin{array}{l}
{\displaystyle \d s^2= {Q\over\rho^2}\left[\d t- \left(a\sin^2\theta
+4l\sin^2{\textstyle{\theta\over2}} \right)\d\phi \right]^2
-{\rho^2\over Q}\,\d r^2 } \\[8pt]
\hskip8pc {\displaystyle
-{\sin^2\theta\over\rho^2} \Big[ a\d t
-\Big(r^2+(a+l)^2\Big)\d\phi \Big]^2 -\rho^2\,\d\theta^2, }
\end{array}$$ where $$\begin{array}{l}
\rho^2 =r^2+(l+a\cos\theta)^2 \\[6pt]
Q= (a^2-l^2+e^2+g^2) -2mr + r^2 .
\end{array}$$ This is exactly the Kerr–Newman–NUT solution in the form which is regular on the half-axis $\theta=0$. It represents a black hole with mass $m$, electric and magnetic charges $e$ and $g$, a rotation parameter $a$ and a NUT parameter $l$.
Next, let us consider the case in which $\alpha$ is arbitrary but in which $l=0$ so that $\omega^2k=a^2$. In this case, it is convenient to use the remaining scaling freedom to put $\omega=a$, and hence $$\epsilon=1-\alpha^2(a^2+e^2+g^2), \qquad k=1, \qquad n=-\alpha am, \qquad \Omega=1-\alpha r\cos\theta.$$ This will be discussed in detail in section \[SpinC\]. At this point, we will just consider the further limit in which $a\to0$. In this case, the metric (\[newMetric\]) reduces to the form $$\d s^2={1\over(1-\alpha r\cos\theta)^2} \left( {Q\over r^2}\,\d t^2 -{r^2\over Q}\,\d r^2
-\tilde P\,r^2\,\d\phi^2
-{r^2\sin^2\theta\over\tilde P}\,\d\theta^2 \right),$$ where $$\begin{array}{l}
\tilde P=\sin^2\theta\Big(1-2\alpha m\cos\theta +\alpha^2(e^2+g^2)\cos^2\theta\Big), \\[6pt]
Q=(e^2+g^2-2mr+r^2)(1-\alpha^2r^2).
\end{array}$$ This is exactly equivalent to the form for the charged $C$-metric that was introduced recently by Hong and Teo [@HongTeo03] using the coordinates $x=-\cos\theta$ and $y=-1/(\alpha r)$. It describes a pair of black holes of mass $m$ and electric and magnetic charges $e$ and $g$ which accelerate towards infinity under the action of forces represented by a topological (string-like) singularity, for which $\alpha$ is precisely the acceleration.
Interpreting the new metric
===========================
In view of the above limiting cases, it may be argued that (\[newMetric\]) generally describes a pair of accelerating and rotating charged black holes, together with a NUT parameter. The specific relation between the Plebański–Demiański parameter $n$ and the NUT parameter $l$ is determined by (\[n\]) in which $k$ is given by (\[k\]). In addition, the parameters $\alpha$ and $\omega$ are directly related to the acceleration and rotation of the sources respectively.
This interpretation certainly appears to be correct for the case where $|l|\le|a|$, in which the metric (\[newMetric\]) has a curvature singularity at $r=0$, $\cos\theta=-l/a$. Since the space-time is asymptotically flat at conformal infinity, we take $r\in(0,r_\infty)$, where $$r_\infty= \left\{
\begin{array}{cl}
{\displaystyle {\omega\over\alpha(l+a\cos\theta)}} \qquad &\hbox{if} \quad a\cos\theta>-l \\[6pt]
\infty \qquad &\hbox{otherwise}
\end{array} \right.$$ There is an acceleration horizon given by $\alpha r=\omega(|a|+l)^{-1}$. It may be noted that in this case the introduction of the parameter $l$ does not remove the curvature singularity as it does in the standard NUT solution.
By contrast, the case in which $|l|>|a|$ contains no curvature singularity. However, as the space-time is asymptotically flat at conformal infinity, we take $r\in(r_{-\infty},r_\infty)$, where $$\begin{array}{lcc}
\hbox{if} \ l>0, &r_{-\infty}=-\infty, \qquad
&{\displaystyle r_\infty={\omega\over\alpha(l+a\cos\theta)}} \\
\hbox{if} \ l<0, \qquad
&{\displaystyle r_{-\infty}={\omega\over\alpha(l+a\cos\theta)}}, \qquad
&r_\infty=\infty
\end{array}$$ In this case, it can be seen from the form of $Q$ in (\[newMetricFns\]) that outer and inner NUT-like horizons occur at $r=r_\pm$, where $r_\pm$ are the roots of the quadratic $${\omega^2k\over a^2-l^2}\,r^2
-2\bigg(m-{\alpha l\over\omega}(\omega^2k+e^2+g^2)\bigg)r
+(\omega^2k+e^2+g^2) =0.$$ The solution between the horizons ($r_-<r<r_+$) has a completely different structure to that of an accelerating Kerr black hole. It could represent a non-singular Taub-like vacuum cosmological model. There is also an acceleration horizon at $\alpha r=\omega(l\pm a)^{-1}$, taking whichever value is within the permitted range of $r$.
We must now consider the regularity of the metric (\[newMetric\]) on the axis for either of the above cases and for any value of $r$, although we will assume that we are working in the stationary regions.
Let us initially consider a small circle around the half-axis $\theta=0$ in the surface on which $t$ and $r$ are constant. If we take $\phi\in[0,2\pi)$, we would obtain $${\hbox{circumference}\over\hbox{radius}}
=\lim_{\theta\to0} {2\pi\tilde P\over\rho^2} {(r^2+(a+l)^2)\over\theta\sin\theta} =2\pi(1-a_3-a_4),$$ where $a_3$ and $a_4$ are given by (\[a34\]). This would correspond to a conical singularity with a deficit angle of $$\delta=2\pi(a_3+a_4) ={2\pi\alpha a\over\omega} \left( 2m-{\alpha(4l+a)\over\omega}(\omega^2k+e^2+g^2) \right).$$ When we come to consider the half-axis $\theta=\pi$, however, it must be noticed that, near the axis in the stationary regions in which $Q>0$, the lines on which $t$, $r$ and $\theta$ are constant are closed timelike lines if $l\ne0$. We must therefore consider small circles on the surfaces on which $t'$ and $r$ are constant, where $t'=t-4l\phi$. In this case $${\hbox{circumference}\over\hbox{radius}}
=\lim_{\theta\to\pi} {2\pi\tilde P\over\rho^2}
{(r^2+(a-l)^2)\over(\pi-\theta)\sin\theta} =2\pi(1+a_3-a_4).$$ i.e. there is an excess angle in this case if $\phi\in[0,2\pi)$ and $a_3-a_4>0$.
Of course, as in any NUT solution, there are two distinct possible interpretations of the space-time. One, due to Misner [@Misner63], is to consider $t$ as a periodic coordinate with period $8\pi l$, so that the space-time has topology $R\times S^3$. This may be appropriate in the time-dependent region, in which case the solution would represent a generalised Taub universe. The alternative approach, due to Bonnor [@Bonnor69], is to treat the singularity at $\theta=\pi$ as a semi-infinite line singularity which acts as a spike injecting angular momentum into the source. In this case, the topological singularity in the stationary regions is surrounded by a region which contains closed timelike lines. (Bonnor [@Bonnor01] has described these as “torsion singularities”.) Since the emphasis here is on the exterior stationary regions, we will generally adopt the second interpretation below.
By considering the $C$-metric limit, it can be seen that the half-axis $\theta=0$ is that which connects the two distinct black holes to infinity, while the half-axis $\theta=\pi$ corresponds to that between the two black holes. By adjusting the range of $\phi$, it is always possible to remove the conical singularity on either of these half-axes. For example, the singularity on $\theta=\pi$ can be removed by taking $\phi\in\big[0,2\pi(1+a_3-a_4)^{-1}\big)$. The acceleration of the two “sources” would then be achieved by two “strings” of deficit angle $$\delta_0 = {4\pi\,a_3 \over1+a_3-a_4}
\label{def0}$$ connecting them to infinity. Alternatively, the singularity on $\theta=0$ could be removed by taking $\phi\in\big[0,2\pi(1-a_3-a_4)^{-1}\big)$, and the acceleration would then be achieved by a “strut” between them in which the excess angle is given by $$-\delta_\pi = {4\pi\,a_3 \over1-a_3-a_4}.
\label{defpi}$$
It should also be recalled that, when $l\ne0$, the metric (\[newMetric\]) has an additional singularity on $\theta=\pi$ which, in the general case, corresponds to the “axis” between the two “sources”. However, this can be transformed to the other axis by the transformation $t'=t-4l\phi$. It can thus be seen that the topological singularity on the axis which causes the acceleration, and the singularity on the axis associated with the NUT parameter and the existence of closed timelike lines, are mathematically independent. They may each be set on whatever parts of the axis may be considered to be most physically significant.
A pair of accelerating and rotating black holes {#SpinC}
===============================================
Let us now return to the case in which $l=0$ so that $\omega^2k=a^2$ and we may put $\omega=a$. As shown above, the conditions (\[epsilon\]) and (\[n\]) then imply that $\epsilon=1-\alpha^2(a^2+e^2+g^2)$ and $n=-\alpha am$. It can be seen explicitly that the Plebański–Demiański parameter $n$ is non-zero in this case, while the NUT parameter $l$ vanishes. The resulting solution corresponds precisely to that of Hong and Teo [@HongTeo05] which represents an accelerating and rotating pair of black holes without any NUT-like behaviour.
In this case, the metric may be taken in the form $$\d s^2={1\over\Omega^2}\left\{
{Q\over\rho^2}\left[\d t- a\sin^2\theta\,\d\phi \right]^2
-{\rho^2\over Q}\,\d r^2 -{\tilde P\over\rho^2} \Big[ a\d t
-(r^2+a^2)\d\phi \Big]^2
-{\rho^2\over\tilde P}\sin^2\theta\,\d\theta^2 \right\},
\label{HongTeoMetric}$$ where $$\begin{array}{l}
\Omega=1-\alpha r\cos\theta \\[6pt]
\rho^2 =r^2+a^2\cos^2\theta \\[6pt]
\tilde P= \sin^2\theta \Big(1-2\alpha m\cos\theta +\alpha^2(a^2+e^2+g^2)\cos^2\theta\Big) \\[6pt]
Q= \Big((a^2+e^2+g^2) -2mr +r^2\Big) (1-\alpha^2r^2).
\end{array}$$ The only non-zero components of the curvature tensor are given by $$\begin{array}{l}
{\displaystyle \Psi_2= \left(-m(1-i\alpha a)
+(e^2+g^2) {1+\alpha r\cos\theta\over r-ia\cos\theta} \right)
\left({1-\alpha r\cos\theta\over r+ia\cos\theta}\right)^3 } \\[12pt]
{\displaystyle \Phi_{11}= {1\over2}\,(e^2+g^2)\,{(1-\alpha r\cos\theta)^4\over(r^2+a^2\cos^2\theta)^2}.}
\end{array}$$ These indicate the presence of a Kerr-like ring singularity at $r=0$, $\theta={\pi\over2}$. Thus, we may generally take $r\in(0,r_\infty)$.
If $m^2\ge a^2+e^2+g^2$, the expression for $Q$ factorises as $$Q = (r_--r)(r_+-r)(1-\alpha^2r^2),$$ where $$r_\pm = m\pm\sqrt{m^2-a^2-e^2-g^2}.
\label{KerrNewman roots}$$ The expressions for $r_\pm$ are identical to those for the locations of the outer and inner horizons of the non-accelerating Kerr–Newman black hole. However, in this case, there is another horizon at $r=\alpha^{-1}$ which is already familiar in the context of the $C$-metric as an acceleration horizon.
However, the most significant advantage of the form (\[HongTeoMetric\]) is that the range of $\theta$ is bounded by roots which correspond to $\tilde p=1$ and $\tilde p=-1$, and this ensures that closed timelike lines do not appear near the conical singularities on the axis – a property that is usually associated with a non-zero value of the NUT parameter. This confirms that the correct “spinning $C$-metric” without any NUT-like behaviour occurs when $l=0$ (i.e. $n=-\alpha am$) as argued by Hong and Teo [@HongTeo05].
In addition, we also obtain in this case that $$a_3=2\alpha m, \qquad a_4=-\alpha^2(a^2+e^2+g^2),$$ so that the deficit angle of the string pulling the black holes toward infinity, or the excess angle of the strut between them, is obtained immediately using (\[def0\]) or (\[defpi\]) respectively.
Apparently accelerating NUT solutions
=====================================
Let us now consider the complementary case with no Kerr-like rotation in which $a\to0$, but $l$ remains non-zero. In this case $a_3$ and $a_4$ vanish, and we can use the scalings (\[scaling\]) to set the values of $\epsilon$ and $k$ such that $a_0=1$ and $a_2=-1$. The remaining freedom can then be used to set $\omega=l$, and (\[epsilon\]) and (\[n\]) become $$\begin{array}{l}
\epsilon=-k+4\alpha m -3\alpha^2(e^2+g^2+kl^2) \\[6pt]
n= l\,\Big(-k+\alpha m -\alpha^2(e^2+g^2+kl^2)\Big),
\end{array}$$ where $$(1-3\alpha^2l^2)k=-1-2\alpha m+3\alpha^2(e^2+g^2).$$ For the general case in which $\alpha\ne0$, this yields a fairly complicated relation between the parameter $n$ and the NUT parameter $l$.
We now have $\tilde P= \sin^2\theta$, and the metric (\[newMetric\]) becomes $$\d s^2= \tilde Q \left(\d t -4l\sin^2{\textstyle{\theta\over2}}\,\d\phi \right)^2
-{\d r^2 \over\tilde Q\,(1-\alpha r)^4}
-{r^2+l^2\over(1-\alpha r)^2} \left(\d\theta^2 +\sin^2\theta\,\d\phi^2 \right),
\label{a=0Metric}$$ where $$\tilde Q= {(e^2+g^2+kl^2)(1+2\alpha r)-2mr-kr^2 \over r^2+l^2}.$$ The only non-zero components of the curvature tensor are given by $$\begin{array}{l}
{\displaystyle \Psi_2= \left(-(m+in)
+(e^2+g^2) {1+\alpha r\over r-il} \right)
\left({1-\alpha r\over r+il}\right)^3 } \\[12pt]
{\displaystyle \Phi_{11}= {1\over2}\,(e^2+g^2)\,{(1-\alpha r)^4\over(r^2+l^2)^2}, }
\end{array}$$ from which it can be seen that there are no curvature singularities provided $l\ne0$, and we may generally take $r\in(r_{-\infty},r_\infty)$ as defined above.
The metric (\[a=0Metric\]) appears to contain the arbitrary parameter $\alpha$. For the case in which $\alpha=0$, we obtain $k=-1$ and $\tilde Q=\big(r^2-2mr-l^2+e^2+g^2\big)/(r^2+l^2)$. This is precisely the (charged) NUT metric [@NewTamUnt63]. It is therefore natural to initially interpret the metric (\[a=0Metric\]) as a general accelerating NUT solution. However, this is not the correct interpretation for this particular case.
It may first be observed that the deficit angles vanish on the entire axis $\theta=0,\pi$ in this case in which $a=0$; i.e. there is nothing to “cause” an acceleration. More specifically, the transformation $$r={1\over\sqrt{1-\alpha^2L^2}}\left( {\sqrt{1-\alpha^2L^2}\,R-\alpha L^2\over\alpha R+\sqrt{1-\alpha^2L^2}} \right), \qquad
t={T\over\sqrt{1-\alpha^2L^2}}, \qquad l={L\over\sqrt{1-\alpha^2L^2}},
\label{a=0Trans}$$ alters the metric (\[a=0Metric\]) to the form $$\d s^2= F \left(\d T -4L\sin^2{\textstyle{\theta\over2}}\,\d\phi \right)^2
-{\d R^2 \over F}
-(R^2+L^2) \left(\d\theta^2 +\sin^2\theta\,\d\phi^2 \right),$$ where $$F={R^2-2MR-L^2+e^2+g^2\over R^2+L^2},$$ and $$M= { m -2\alpha(1-\alpha^2L^2)(e^2+g^2) +(3-4\alpha^2L^2)\alpha L^2
\over\sqrt{1-\alpha^2L^2}\>(1-4\alpha^2L^2)}.
\label{Mm}$$ This is exactly the charged NUT solution (when $L=0$, it is the usual form of the Reissner–Nordström metric). But the familiar mass parameter is now $M$, and this is related by equation (\[Mm\]) to the Plebański–Demiański parameters $m$, $e$ and $g$ and the parameters $L$ and $\alpha$. In addition, the NUT parameter is now given by $L$, and this is related to $\alpha$ and $l$ as in (\[a=0Trans\]) and hence to the Plebański–Demiański parameter $n$ which is given by (\[n\]). It may also be observed that, in the above transformation, $$R={1\over\sqrt{1+\alpha^2l^2}}\left({r+\alpha l^2\over1-\alpha r}\right),$$ so that $r=\alpha^{-1}$ corresponds to spacelike infinity in the line element (\[a=0Metric\]). But, particularly, it must be concluded that $\alpha$ is a redundant parameter in the metric (\[a=0Metric\]) that can be removed by the above transformation for this case in which $a=0$. Thus, the metric (\[a=0Metric\]) does not represent a new “accelerating” NUT solution.
On the other hand, the general metric (\[newMetric\]) does represent a pair of accelerating sources with a non-zero NUT parameter since it does explicitly contain the $C$-metric. And, just as the accelerating and rotating solution with no NUT features is obtained by putting and not $n=0$, so it may be appropriate to look for an accelerating NUT solution with no rotational features by considering some non-zero value of $a$ rather than putting $a=0$. However, such a solution has to date neither been identified nor proved not to exist.
Conclusions
===========
We have presented, in a most convenient form, the metric (\[newMetric\]) which explicitly represents the complete family of accelerating and rotating black holes with a generally non-zero mass, charge (electric and magnetic) and NUT parameter. The general structure of this family of solutions is indicated in figure 1.
The metric has been obtained after first re-expressing the Plebański–Demiański metric in a more convenient form which explicitly includes the parameters $\alpha$ and $\omega$, which respectively represent the acceleration of the sources and the twist of the repeated principal null congruences (and hence the rotation of the sources and/or the effects of the NUT parameter). The form (\[newMetric\]) excludes the cases of the Plebański–Demiański solution for which the spacelike 2-surfaces spanned by $p$ and $\sigma$ have zero or negative curvature, as these do not represent black hole-like solutions in any limit.
In presenting this form of the metric, we have clarified the physical meaning of the arbitrary parameters involved. As can be seen from the Maxwell field components, $e$ and $g$ are unambiguously the electric and magnetic charges of the sources. The twist of the repeated principal null congruences is proportional to the parameter $\omega$, and this is related to both the Kerr-like rotation parameter $a$ and the NUT parameter $l$. Provided $|a|>|l|$, $\alpha$ is the acceleration of the sources. In particular, we have found an explicit relationship between the parameter $l$ and the Plebański–Demiański parameter $n$ given in (\[n\]). However, the NUT parameter is only identified as $l$ for the non-accelerating case in which $\alpha=0$. When $\alpha\ne0$, it may be more appropriately taken as $L$ which is related to $l$ by (\[a=0Trans\]). But even this is only valid when $a=0$. Thus the all-encompassing physical representation of the NUT parameter when $\alpha$ and $a$ are both non-zero, is still to be determined.
It is also difficult to physically determine the “correct” mass parameter. When $n=0$, the gravitational strength of the singularity is proportional to $m$, but otherwise it is proportional to $\sqrt{m^2+n^2}$. However, when $|l|>|a|$, the space-time is nonsingular and the “mass” of the source is only determined in certain limits. When $\alpha=0$, $m$ is still the familiar mass parameter of the Kerr–Newman–NUT metric. However, when $a=0$, the usual mass parameter of the charged $C$-metric is $M$, which given by (\[Mm\]). Thus, as for the NUT parameter, the physically significant mass parameter is still to be determined when $\alpha$, $L$ and $a$ are all non-zero.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by a grant from the EPSRC.
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[^1]: E–mail: [[email protected]]{}
[^2]: E–mail: [[email protected]]{}
|
---
author:
- |
Titus K Mathew, Aswathy M B and Manoj M\
Department of Physics,\
Cochin University of Science and Technology,\
Kochi-22, India\
E-mail: [email protected]
title: 'Cosmology and thermodynamics of FRW universe with bulk viscous stiff fluid.'
---
Introduction {#intro}
============
The recent discovery on the current acceleration of the universe form type Ia supernovae data [@Perlmutter1; @Riess1; @Hicken1] have shown that about 72$\%$ of the energy density of the universe is in the form of an exotic component, which is capable of producing negative pressure, called dark energy. Cosmological data from other wide range of sources, the cosmic microwave background radiation [@Komatsu1; @Larson2], baryon accoustic oscillations [@Percival1], cluster gas fractions [@Samushia1] and gamma ray burst [@Wang21; @Samushia2] have all confirming this conclusion. In the remaining part of the energy density, it was concluded even earlier to the discovery of dark energy that, 23$\%$ of it consists of weakly interacting matter called dark matter. The evidence for this is coming from variety of observational tests including weak [@Refregier1] and strong [@Tyson1] lensing, large scale structure [@Allen1], as well from supernovae and cosmic microwave background [@Zwicky1; @Rubin1; @Rubin2]. In spite of the fact that all these observational data establishes the existence of the components like dark matter and dark energy, the existence of other exotic fluid components has not been ruled out. For example several models predicts the existence of an exotic component called dark radiation in the universe [@Dutta1]. Another exotic fluid which is predicted by several models is stiff fluid, a fluid with an equation of state $p_s=\rho_s,$ where $p_s$ and $\rho_s$ are the normal pressure and density of the stiff fluid respectively. The equation of state parameter of this fluid is seems to the largest value (equal to 1) consistent with causality, because the speed of sound in this fluid is equal to the speed of light.
The model with stiff fluid was first studied by Zeldovich [@Zeldovich1]. In recent years a large number of models have been proposed in studying the various cosmological properties of stiff fluid. In certain models with self-interacting dark matter components, the self interaction between the dark matter particles is characterized by the exchange of vector mesons via minimal coupling. In such models the self interaction energy is shown to behave like a stiff fluid [@Stiele1]. Stiff fluid is considered in certain cosmological models based on Horava-lifeshitz gravity. In Horava-lifeshitz gravity theories a “detailed balancing” condition was imposed as a convenient simplification and the usefulness of this detailed balancing condition was discussed in references [@Horava1; @Calcagni1; @Kiritsis1]. The stiff is fund to be arised in such models where this detailed balancing condition is relaxed [@Sotiriou1; @Bogdanos1; @Carloni1; @Leon1]. Cosmological models with stiff fluid, based on Horava-Lifeshits gravity, have been studied in references [@Ali1; @Dutta2]. The existence of stiff fluid have been found as exact non-singular solutions in certain inhomogeneous cosmological models [@Fernandez1; @Dadhich1; @Mars1; @Fernanswz2]. The relevance of the stiff fluid equation of state to the matter content of the universe in the early stage of the universe was investigated in reference [@Barrow1]. The decrease in the density of stiff fluid in the universe is found to be faster than that of radiation and matter, hence it’s effect on expansion would be larger in the initial stage of the universe. Primordial Nucleosysnthesis is an event took place in the early phase of the universe, a limit on the density of the stiff fluid can be obtained from big bang nucleosysnthesis constraint as in reference [@Dutta3].
In a homogeneous and isotropic universe bulk viscosity is the unique viscous effect capable to modify the background dynamics. From a theoretical point of view, bulk viscosity arised in a system due to it’s deviations from local thermodynamic equilibrium [@Zimdahl1]. In cosmology, bulk viscosity arised as an effective pressure, restoring the system to its thermal equilibrium, which was broken when universe expands too fast so that the system may not get enough time to restore the local thermal equilibrium [@Wilson1; @Mathews1; @Martens1; @Okamura1]. Several years ago, before the discovery of the present acceleration of the universe, it has been proposed, in the context of the inflationary period of early universe that bulk viscous fluid can produce acceleration in the expansion of the universe [@Heller1; @Heller2; @Waga1; @Beesham1; @Zimdahl3; @Paddy1]. Recently investigations were made on possibility causing the recent acceleration of the universe with bulk viscous matter [@Avelino1; @Avelino2].
In the present work we study a stiff fluid dominated cosmological model with bulk viscosity. We assume a stiff fluid of equation of state $p_s=\rho_s$ and the bulk viscosity is characterized by a constant viscosity coefficient, which is the simplest parametrization for the bulk viscosity. We are deriving the Hubble parameter, density, equation of state, deceleration parameter and analyzing their behaviors for further possibilities including the recent acceleration of the universe. the paper is organized as follows. In section 2 we are giving the basic equations of FRW universe and deriving the general equation for the Hubble parameter in a bulk viscous stiff fluid dominated universe. We are classifying the different cases arised depending the value of the bulk viscous parameter and analyzing the evolution of various cosmological parameters. Section 3 containing the statefinder diagnosis of the model. In section 4, we presents the status of the generalized second law of thermodynamics followed by conclusions in section 5.
Stiff fluid with bulk viscosity
===============================
Stiff fluid cosmological models create interest because in these fluids the speed of light is equal to the speed of sound and its governing equations have the same characteristics as that of the gravitational field [@Wesson1]. The equation of state of the stiff fluid is given as [@Zeldovich1], $$\label{eqn:EOS1}
p_s = \rho_s.$$ This equation of state resembles the equation of state of a special case of models investigated by Masso and others [@Masso1].
In cosmological models the effect of bulk viscosity can be shown to be an added correction to the net pressure $p_s^{'}$ as, $$\label{eqn:CP}
p_s^{'} = p_s - 3 \zeta H,$$ where $\zeta$ is the constant coefficient of viscosity and $H$ is the Hubble parameter. The form of the above equation was originally proposed by Eckart [@Eckart1] in the context of relativistic dissipative process occurring in thermodynamic systems went out of local thermal equilibrium. Later Landau and Lifeshitz proposed an equivalent formulation [@Landau1]. However Eckart theory has got the short comings that, it describes all the equilibria as unstable [@Hiscock1] and signals can propagate through the fluid with superluminal velocities [@Israel1]. Later Israel and Stewart proposed a more general theory which avoids these problems and from which Eckart theory is appearing as the first order limit [@Israel2; @Israel3]. However because of the simple form of Eckart theory, it has been widely used by several authors to characterize the bulk viscous fluid. For example the Eckart approach has been used in models explaining the recent acceleration of the universe with bulk viscous fluid [@Kremer1; @Cataldo1; @Fabris1; @Hu1; @Ren1]. More over Hiscock et. al. [@Hiscock2] have shown that Eckart theory can be favored over the Israel-Stewart model, in explaining the inflationary acceleration of FRW universe with bulk viscous fluid. These motivate the use Eckart results, especially when one try to look at the phenomenon recent acceleration of the universe. At this point one should note the more general formulation than the Israel-Stewart by Pavon et. al. dealing with thermodynamic equilibrium [@Pavon1]
We consider the flat FRW universe favored by the recent WMAP observation [@WMAP1] with the scale factor $$ds^2 = -dt^2 + a^2(t) \left(dr^2 + r^2 d\theta^2 + r^2\sin\theta d\phi^2 \right),$$ where $a(t)$ is the scale factor, $t$ is the cosmic time and $(r,\theta,\phi)$ are the comoving coordinates. The corresponding dynamics equations are, $$\label{eqn:frw1}
H^2 = {\rho \over 3}$$ $$\label{eqn:frw2}
2{\ddot{a} \over a} + \left({\dot{a} \over a}\right)^2 = p^{'}$$ and the conservation equation, $$\label{eqn:con}
\dot{\rho} + 3H \left(\rho + p^{'} \right) = 0,$$ where we have adopted the standard units convention, $8\pi G=1$ and over-dot represent a derivative with respect to cosmic time. From the dynamical equations (\[eqn:frw1\]) and (\[eqn:frw2\]), we can formulate a first order differential equation for the Hubble parameter by using equations (\[eqn:EOS1\]),(\[eqn:CP\]) and (\[eqn:con\]) as, $$\label{eqn:dotH}
\dot{H}={3H\over 2} \left(\zeta -2H \right).$$ The above equation can expressed in terms of the variable $x=\log a,$ suitably integrated and the final result can be written in terms of the scale factor as, $$\label{eqn:hp1}
H = {H_0 \over 6} \left[\bar\zeta+\left(6-\bar\zeta\right) a^{-3} \right],$$ where $\bar\zeta=3\zeta/H_0$ is the dimensionless bulk viscous coefficient, $H_0$ is the present value of the Hubble parameter and we have made the assumption that the present value of the density parameter of the stiff fluid $\Omega_{s0}=1$ for a stiff fluid dominated universe.
Classification and evolution of the bulk viscous stiff fluid dominated model
----------------------------------------------------------------------------
Equation for the Hubble parameter shows that for different value of the viscosity coefficient $\bar\zeta$ we get different models. In this section we are classifying different models of the universe arises due to the different values of the dimensionless viscosity coefficient. We analyse the behavior of the scale factor, density and other parameter of these different cases.
### Case-1: $\bar\zeta=0$
This corresponds to the universe dominated with stiff fluid without bulk viscosity. From equation (\[eqn:hp1\]) the Hubble parameter becomes $H=H_0 a^{-3}.$ From the dynamical equation (\[eqn:frw1\]) the corresponding density of the stiff fluid follows a relation, $$\label{eqn:rhobehav1}
\rho_s \propto a^{-6}.$$ This shows that the density of the non-viscous stiff fluid decays more rapidly than the non-relativistic matter or radiation in a FRW universe, which implies that the effect of the stiff fluid on the expansion of the universe would be the larger at early times. So the limit on the density of the stiff fluid can obtained by considering its effect the Big Bang nucleosysnthesis. Dutta et. al [@Dutta3] made an investigation in this regard and found that the change in the primordial abundance of helium-4 is proportional to the ratio $\rho_s/\rho_R,$ where $\rho_R$ is the radiation density. Consequently they found a limit on the non-viscous stiff fluid density as $\rho_s/\rho_R < 30$ when the temperature of the universe was around 10 MeV.
The evolution of the scale factor can be obtained by integrating the resulting Hubble parameter as, $$a(t) = \left(3 H_0 \left(t - t_0 \right) +1 \right)^{1/3}$$ A second order derivative of the scale factor with time is, $${d^2a \over dt^2} = -{2 H_0^2 \over \left(3H_o(t - t_0) +1 \right)^{5/3} }.$$ This shows that the universe will undergo an eternal deceleration in this case.
The behavior of the density from equation (\[eqn:rhobehav1\]) reveals that as the scale factor $a(t)\to 0,$ the density $\rho_s \to \infty.$ This implies the existence of singularity at the beginning of the universe. This can be further chequed by calculating the curvature scalar for a flat FRW universe using the equation [@Kolb1] $$\label{eqn:curvscalar}
R = \left( {\ddot{a} \over a} + H^2 \right).$$ Using the equation for the Hubble parameter and its time derivative it can easily shown that $R \sim H^2,$ which according to the equation $H=H_0 a^{-3}$ implies that the curvature scale $R \to \infty$ as $a \to 0$ at the origin, confirming the presence of the initial singularity. So it can be concluded that in this case the universe had a Big Bang. The time elapsed since the Big Bang, $t_B,$ is found to be $$t_B = t_0 - {1 \over 3 H_0}.$$ Also it is evident form the behavior of the density that, as $a(t) \to \infty$ the density $\rho_s \to 0.$ In this respect apart form the difference in the dependence on the scale factor, the general behavior of non-viscous stiff fluid is same as that of the non-relativistic matter or relativistic radiation.
### case-2: $0<\bar\zeta<6$ {#zeta}
The Hubble parameter is given by the equation (\[eqn:hp1\]). Following the dynamical equations the density of the bulk viscous stiff fluid in this case is given as, $$\rho_s = 3 \left( \frac{H_0}{6} (\bar\zeta + (6 - \bar\zeta) a^{-3}) \right)^2$$ The evolution of the scale factor is given in figure \[fig:density\]
![Evolution of the density $\rho_s$ with scale factor $a(t).$ The bottommost line is for $\bar\zeta=2,$ the second from the bottom is for $\bar\zeta=4,$ the third line from the bottom is for $\bar\zeta=5,$ the fourth line is for $\bar\zeta=5.7$ and the topmost line is for $\bar\zeta=6.5.$[]{data-label="fig:density"}](density1.pdf)
This equation shows that as the scale factor $a(t) \to 0$ the density $\rho_s \to \infty,$ indicating that there is a singularity at the origin. The presence of the singularity is further confirmed by calculating the curvature scale using equation (\[eqn:curvscalar\]) and is, $$\label{eqn:curvscale2}
R = {3H\bar\zeta \over 2} - H^2,$$ which shows that $R \to \infty$ as $a(t) \to 0,$ confirming the presence of the initial singularity. So the model of the universe in this case does have a Big Bang.
For finding the scale factor equation (\[eqn:hp1\]) can be put in a form, $${da^3 \over dt} - {H_0 \bar\zeta \over 2} a^3 = {H_0\left(6-\bar\zeta\right) \over 2},$$ which can be suitably integrated for the scale factor as, $$\label{eqn:scalefact2}
a(t)=\left({\bar\zeta - 6 + 6 \exp(\bar\zeta H_o [t-t_0]/2) \over \bar\zeta} \right)^{1/3}.$$ This equation for scale factor reveals that, the time elapsed since the Big Bang is, $$t_B = t_0 + {2 \over H_0 \bar\zeta} \ln\left({6 - \bar\zeta \over 6} \right),$$ hence the age of the universe since Big Bang is, $$t_0 - t_B = - {2 \over H_0 \bar\zeta} \ln\left({6 - \bar\zeta \over 6} \right).$$ Taking $H_0 = 100 \, h \, km/sec/Mpc$, with $h=0.74$ the age of the universe is evaluated as per the above equation is around 13.8 Gyr for $\bar\eta=5.7,$ a value which is very closer to that predicted by the CMB anisotropy data [@Tegmark1].
A plot of the evolution of the scale factor is given in figure \[fig:a1\]. The scale factor equation (\[eqn:scalefact2\]) shows that as $t \to \infty$ the scale factor approaches to a form like that of the de Sitter universe, $$a(t) \to \exp(\bar\zeta H_0[t-t_0]/2).$$
![ Evolution of the scale factor with $H_0(t-t_0).$ The lines represents, for $\bar\zeta=2,$ the bottommost line, for $\bar\zeta=4$ second line from the bottom, $\bar\zeta=5$ third form the bottom, $\bar\zeta=5.7$ fourth from bottom and $\bar\zeta=6.5$ the topmost line.[]{data-label="fig:a1"}](scalefactor1.pdf)
While in the early stages of the evolution of the universe, when $\bar\zeta H_0 [t-t_0]/2 <1,$ the scale factor can be approximated as, $$a(t) \sim \left(1+3H_0 [t-t_0] \right)^{1/3}.$$ These equations of the scale factor at the respective limits shows that the universe have an earlier deceleration phase followed by an acceleration phase in the later stage of the evolution. The value of the scale factor or redshift at which the transition form the decelerated to the accelerated expansion occur is depends on the viscosity coefficient $\bar\zeta$ as shown below. From the Hubble parameter we can calculate the derivative of $\dot{a}$ with respect $a(t)$ as, $${d\dot{a} \over da} = \bar\zeta - 2 \left(6-\bar\zeta \right) a^{-3}.$$ Equating this equation to zero, we obtained the transition scale factor as [@Avelino1], $$\label{eqn:trans-a}
a_T = \left({2 (6 - \bar\zeta) \over \bar\zeta } \right)^{1/3},$$ and the corresponding transition redshift is, $$\label{eqn:trans-z}
z_T = \left({\bar\zeta \over 2(6 - \bar\zeta) } \right)^{1/3} - 1.$$ From equations (\[eqn:trans-a\]) and (\[eqn:trans-z\]) it is clear that for $\bar\zeta=4$ the transition from the decelerated phase to the accelerated phase is occur at $z_T=0, \, a_T=1$ corresponds to the present stage of the universe. In the range $0<\bar\zeta<4$ the transition between the decelerated and the accelerated phase takes place in future corresponds to $z_T<0, \, a_T>1.$ The transition takes place in the past of the universe ($z_T>0, \, a_T<1$) when $4<\bar\zeta<6.$ When $\bar\zeta = 0$ the value of $z_T$ becomes -1 and value of scale factor $a_T$ becomes infinity in the future, implies that no transition to accelerated expansion within a finite time and the universe is always decelerating. While for $\bar\zeta =6$ the transition takes place at a time corresponds to $a_T \to 0$ closer to the Big Bang.
As a further clarification of the conclusions in the above paragraph we evaluate the deceleration parameter and the equation of state parameter of the bulk viscous stiff fluid in this case. A positive value of the deceleration parameter characterizes a decelerating universe, while a negative value characterizes an accelerating universe. The deceleration parameter $q$ can be evaluated using the equation, $$q = -1 - {\dot{H} \over H^2}$$ Using the Hubble parameter from equation (\[eqn:hp1\]), the deceleration parameter in terms of the redshift $z$, $$\label{eqn:qparameter}
q = -1 - {3 (\bar\zeta -6) (1+z)^3 \over \bar\zeta + (6-\bar\zeta) (1+z)^3 }$$ where we took $a = (1+z)^{-1}.$ The evolution of the deceleration parameter is shown in figure \[fig:qpara\]. It is clear from the figure that the deceleration parameter $q \to -1$ in the far future of the evolution of the universe as $z \to -1$ for any positive value of the dimensionless bulk viscous parameter $\bar\zeta.$
![ Evolution of the deceleration parameter with redshift The lines represents, for $\bar\zeta=6.5,$ the bottommost line, for $\bar\zeta=5.7$ second line from the bottom, $\bar\zeta=5$ third form the bottom, $\bar\zeta=4$ fourth from bottom and $\bar\zeta=2$ the topmost line.[]{data-label="fig:qpara"}](q1.pdf)
The transition redshift $z_t$ can be obtained by equating $q$ to zero, and it lead to equation (\[eqn:trans-z\]). For $\bar\zeta=0$ the deceleration parameter will be 2, corresponds to a universe dominated with non-viscous stiff fluid. For $\bar\zeta=6$ the parameter $q=-1$ corresponds to the de Sitter phase. So for $0<\bar\zeta<6$ the deceleration parameter is always decreasing form $q(a=0)=2$ to $q(a=\infty)=-1$, with a transition from positive to negative values corresponds to the transition from deceleration to acceleration in the expansion of the universe. The deceleration parameter for today, i.e. for $z=0$ is found to be, $$q(a=1) = 2 - \frac{\bar\zeta}{2}.$$ This is agreeing with our earlier results in equations (\[eqn:trans-a\]) and (\[eqn:trans-z\]) that for $\bar\zeta=4$ the universe would enter the accelerating phase from the decelerated expansion at the present time. For $\bar\zeta<4,$ then $q>0$, we have decelerating universe today and for $\bar\zeta>4$ then $q<0,$ we have accelerating universe today. From the current observational results [@WMAP1; @Tegmark1], the present value of the deceleration parameter is around $-0.64\pm 0.03$, from which the bulk viscous coefficient is to be $\bar\zeta > 4$ for a universe dominated with bulk viscous stiff fluid. These analysis shows that a universe dominated with bulk viscous stiff fluid, it can take the role of the conventional dark energy, to cause the recent acceleration of the universe for a bulk viscous coefficient in the range $4<\bar\zeta<6.$
The evolution of the equation of state $\omega_s$ of the stiff fluid with bulk viscosity can be studied by calculating it using the relation [@tkm2], $$\omega_s = -1 - \frac{1}{3} {d \ln h^2 \over dx},$$ where $h=H/H_0$ the weighted Hubble parameter. Evaluating $\omega_s$ in terms of the redshift $z$ gives, $$\omega_s = -1 - \left( { 2 (\bar\zeta - 6) (1+z)^3 \over \bar\zeta + (6 - \bar\zeta) (1+z)^3} \right).$$ The evolution of the equation of state is as shown in figure \[fig:eos1\].
![ Evolution of the deceleration parameter with redshift The lines represents, for $\bar\zeta=6.5,$ the bottommost line, for $\bar\zeta=5.7$ second line from the bottom, $\bar\zeta=5$ third form the bottom, $\bar\zeta=4$ fourth from bottom and $\bar\zeta=2$ the topmost line.[]{data-label="fig:eos1"}](eos1.pdf)
As $z \to -1, \, (a \to \infty)$ the equation of state parameter $\omega_s \to -1$ in the future corresponds to the de Sitter universe, irrespective of the value of the viscosity coefficient. For $\bar\zeta=0$ the equation of state parameter become $\omega_s=1,$ implying the equation of state for the non-viscous stiff fluid, $p_s=\rho_s.$ For $\bar\zeta=6$ the $\omega_s$ become equal to -1. In the range $0<\bar\zeta <6$ the equation of state varies from $+1$ to $-1,$ and making a transition from positive to negative values. Event though negative value of $\omega_s$ leads to negative pressure, but for universe to be in the accelerating phase, $\omega_s < -1/3.$ The present value of $\omega_s$ is found to be $$\label{eqn:omegapres}
\omega_s (a=1)= 1 - \frac{\bar\zeta}{3}.$$ This equation reveals that $\omega_s$ make a transition from positive values to negative values at the present time if $\bar\zeta = 3.$ While considering the evolution of the $q(a=1)$ parameter, we have shown that, $q$ make a transit to the negative values, giving a universe with accelerated expansion for $\bar\zeta=4.$ The negativity of $q$ parameter implies that the universe is accelerating and at the same time the equation of state parameter must be less than $-1/3$ for the universe to be an accelerated one [@Bamba1]. From equation (\[eqn:omegapres\]) it is clear that $\omega_s(a=1) < -1/3$ only for $\bar\zeta \geq 4.$ The current observational value of equation of state parameter of the fluid responsible for the recent acceleration is around $-0.94\pm 0.1,$ [@Tegmark1] and form equation (\[eqn:omegapres\]) we can infer that in a universe dominated with bulk viscous stiff fluid, the corresponding value of the bulk viscous coefficient is $\bar\zeta>4$ to cause a recent acceleration. So the analysis on the evolution of $\omega_s$ also shows that the bulk viscous stiff fluid can replace the conventional dark energy in causing the recent acceleration, for $4<\bar\zeta<6$.
### Case-3: $\bar\zeta>6$
The equations (\[eqn:hp1\]) and (\[eqn:scalefact2\]) can be used in this case too for assessing the behaviors of the Hubble parameter and scale factor. For $\bar\zeta>6$ these equations shows that the resulting universe will always be accelerating. That is there is no decelerating epoch at all. When $t \to \infty$ the universe tends to the de Sitter phase.But when $t-t_0 \to -\infty$ the scale factor tend to finite minimum value (see figure \[fig:a1\]) instead of zero and is given as, $$\label{eqn:amin1}
\lim_{ t-t_0 \to -\infty} a(t) \equiv a_{min}=\left(1 - \frac{6}{\bar\zeta} \right)^{1/3}$$ The corresponding derivatives $\dot{a}$ and $\ddot{a}$ are zero, hence in this limit the universe become a Einstein static universe. As the universe evolves the scale factor increases monotonically. So there is no Big Bang in this case and the age of the universe is not properly defined.
The curvature scalar can be obtained using equation (\[eqn:curvscale2\]). At $a=a_{min}$, both $\ddot{a}$ and $H$ are zero, hence curvature scalar is also zero and it increases as the universe expands, attains the maximum value $R=\frac{5}{9}\left(H_0\bar\zeta \right)^2$ when $a \to \infty.$ The density of the bulk viscous stiff fluid follows same behavior as the curvature scale (see figure \[fig:density\]), the density is zero when $a=a_{min}$ and attains the maximum value $\left(H_0 \bar\zeta \right)^2 / 12$ as $a \to \infty.$
Statefinder analysis for $4<\bar\zeta<6$
========================================
In the analysis in section \[zeta\] we have concluded that there is a transition form decelerated expansion to accelerated one in the recent past when $4<\bar\zeta<6.$ in the past phase. This gives us hope in considering the discovery of the recent acceleration of the universe in the context of a universe dominated with bulk viscous stiff fluid. The behavior of the scale factor, $q$ parameter and equation of state all shows that
![The $r-s$ plane evolution of the model. The present position in the plane corresponds to values ($r_0,s_0)=(1.25,-0.08.)$ The evolution is the direction as shown by the arrow in the line.[]{data-label="fig:rsplot1"}](rs5.pdf)
the bulk viscous stiff fluid is playing the role of dark energy. So we analyze the model using statefinder parameters to compare it with the standard dark energy models. Statefinder parameters [@Sahni1] are sensitive tool to discriminate various dark energy models, and are defined as $$r= {\ddot{H} \over H^3} - 3q -2$$ and $$s={r - 1 \over 3(q - 1/2)}$$ Using the equations for the Hubble parameter (\[eqn:hp1\]) and deceleration parameter (\[eqn:qparameter\]), the $r-s$ parameter equations can be expressed as, $$\label{eqn:r1}
r={9 (6-\bar\zeta)^2 a^{-6} \over \left(\bar\zeta + (6-\bar\zeta) a^{-3} \right)^2 } + 1$$ and $$\label{eqn:s1}
s= {2 (6 - \bar\zeta)^2 a^{-6} \over (6-\bar\zeta)^2 a^{-6} - \bar\zeta^2}$$ The equations shows that in the limit $a \to \infty$ the statefinder parameters $(r,s) \to (1,0),$ a value similar to the $\Lambda$CDM model of the universe. A plot of the present model in the $r-s$ plane is shown in figure \[fig:rsplot1\], for bulk viscous coefficient $\bar\eta=5,$ and we found that the plot of other values of $\bar\zeta$ are also showing the same behavior, in fact the evolutions are coinciding each other. The plot reveals that the $(r,s)$ trajectory is lying in the region corresponds to $r>1 \, s<0,$ a character similar to that of generalized Chaplygin gas model of dark energy [@Wu1]. On the other hand in comparison with the holographic dark energy model with event horizon as the IR-cut-off [@Huang1; @Wang4] whose $r-s$ evolution starts in the region $s\sim 2/3, \, r\sim 1,$ and end on the $\Lambda$CDM point, the present model starts in the region $r>1 \, s<0$ and end on the $\Lambda$CDM point in the $r-s$ plane. Equations (\[eqn:r1\]) and (\[eqn:s1\]) shows that for $\bar\zeta=0,$ $(r,s)=(10,2)$ and for higher values of $\bar\zeta$ the $(r,s)$ parameter values decreases. The values of the statefinder parameters for the present stage of the universe dominated with bulk viscous stiff fluid, corresponds to $a=1 \, (z=0)$ is, $$r=2 \left(1 - \frac{\bar\zeta}{6} \right)^2 \, \, \, s={(1-\bar\zeta/6) \over 3 (1-\bar\zeta/3)}$$ This shows that as $\bar\zeta$ increases the present values of $(r,s)$ decreases. In figure \[fig:rsplot1\] the present position of the universe is denoted and is corresponds to $(r,s)=(1.25,-0.08),$ which is different form the $\Lambda$CDM model, so the model is well discriminated form the $\Lambda$CDM model of the universe.
Entropy and generalized second law of thermodynamics
====================================================
Bulk viscosity may be the only dissipative effect occurring in a homogeneous and isotropic universe. Any covariant description of dissipative fluids is subjected to the conservation equation, $$\label{eqn:tmunu1}
T^{\mu\nu}_{;\mu}=0,$$ provided there does not occur any matter creation, where the semicolon denote the covariant derivative and $T^{\mu\nu}$ is the energy momentum tensor of the fluid in the universe. The energy momentum tensor in covariant form is given as [@Weinberg1; @Weinberg2], $$T_{\mu\nu}=\rho u_{\mu} u_{\nu}+(g_{\mu\nu}+u_{\mu} u_{\nu}) p^{'},$$ where $u_{\mu}$ is the velocity of the observer who measures the pressure $p^{'},$ whose form is as given in equation (\[eqn:CP\]). The conservation equation with the above form of the energy momentum tensor will lead to the equation (\[eqn:con\]). The bulk viscosity causes the generation of local entropy in the FRW universe [@Weinberg1; @Weinberg2]. The viscous entropy generation in the early universe was studied in reference [@Brevik2]. During the evolution of the universe the sum of the entropies of the fluid within the universe and that of the horizon must always greater than or equal zero, this is well known as the generalized second law (GSL) of thermodynamics. The satatus of the GSL for flat FRW universe with matter and cosmological vacuum was discussed in reference [@tkm1]. The status of the GSL in a flat universe with viscous dark energy was discussed in reference [@Karami2] and the authors have shown that the GSL is valid in FRW universe with apparent horizon as the boundary.
In this section we analyze about the validity of GSL in the present model of the universe dominated with bulk viscous stiff fluid by taking the apparent horizon as the boundary of the universe. The GSL can be formally stated as $$\label{eqn:gsl1}
\dot{S}_s + \dot{S}_h \geq 0,$$ where $S_s$ is the entropy of the stiff fluid and $S_h$ is that of the apparent horizon of the universe. The entropy of the stiff fluid within the horizon of the universe is related to its energy density and pressure through the Gibb’s relation [@Izquierdo1], $$T dS_s = d(\rho_s V) + p^{'} dV,$$ where $V=4\pi/3 H^3$ is the volume of the universe within the apparent horizon with radius $r=H^{-1}$ and $T$ is the temperature of the fluid within the horizon. We take the temperature $T=H/2 \pi$ equal to Hawking temperature of the horizon with the assumption that the fluid within the horizon is in equilibrium with the horizon, so there is no effective flow of the fluid towards the horizon. Using the dynamical equation and the net pressure in equation (\[eqn:CP\]), the time evolution of the entropy of the bulk viscous stiff fluid within horizon become, $$\label{eqn:Ss}
\dot{S}_s = {16\pi^2 \dot{H} \over H^3} - {24\pi^2 \dot{H} \over H^4} \left(2 H - \bar\zeta \right)$$
The entropy of the apparent horizon is given by the Bakenstein-Hawking formula [@Beken1; @Hawking1; @Davies3], $$S_h = 2\pi A$$ where $A=4\pi H^2$ is the area of the apparent horizon. Hence the time rate of the horizon entropy is, $$\label{eqn:Sh}
\dot{S}_h = -{16 \pi^2 \dot{H} \over H^3}.$$ From the equations (\[eqn:Ss\]) and (\[eqn:Sh\]) the GSL condition equation (\[eqn:gsl1\]) is satisfied if $$\dot{H} \left(\bar\zeta - 2 H \right) \geq 0.$$ Using the equation (\[eqn:dotH\]) the above condition become, $$H \left(\bar\zeta - 2 H \right)^2 \geq 0.$$
![ Evolution of the Hubble parameter with scale factor. Bottom line is for $\bar\zeta=2,$ second from bottom for $\bar\zeta=4,$ third from bottom is for $\bar\zeta=5$ and top thick line corresponds to $\bar\zeta=6.2.$[]{data-label="fig:hubble1"}](Hubble1.pdf)
As far as $H$ is positive in an expanding the universe, it is evident that the GSL is satisfied in a bulk viscous stiff fluid dominated universe with apparent horizon as boundary. From equation (\[eqn:hp1\]), the required condition for the validity of GSL is, $$\left(\bar\zeta + (6 - \bar\zeta) a^3 \right) \geq 0.$$ For $\bar\zeta \leq 6$ the above condition is fullfiled consequently the GSL is well satisfied. But when $\bar\zeta>6$, the above condition is satisfied only when $a \geq a_{\min}$ given by equation (\[eqn:amin1\]) and it is clear form the plot, fig. \[fig:hubble1\] of the Hubble parameter with the scale factor.
Conclusions
===========
In this paper we present a study of the bulk viscous stiff fluid dominated universe model with a constant bulk viscous coefficient $\bar\zeta.$ Stiff fluid is an exotic fluid with equation of state parameter $\omega_s=1,$ first studied by Zeldovich [@Zeldovich1]. We analyzed the different possible phases of the model according the value of the dimensionless bulk viscous parameter $\bar\zeta$ and we take $\bar\zeta \geq 0.$ For $\bar\zeta \geq 0$ the model predicts expanding universe in general. For $\bar\zeta=0$ the model reduces to non-viscous stiff fluid dominated universe began with a Big Bang, and is always decelerating with the density varying as $\rho \sim a^{-6}$ .
For $0<\bar\zeta<6$ the model is corresponds to a universe started with a Big Bang and undergoing a decelerated expansion first followed by a transition to the accelerated phase of expansion at later time. For $\bar\zeta=4$ the transition form the decelerated to accelerated expansion epoch takes place today. For $0<\bar\zeta<4$ the transition to the accelerated expansion phase is takes place in the future, but for $4 <\bar\zeta <6$ this transition is found to occur in the past. This shows that the bulk viscous stiff fluid can cause the recent acceleration of the universe. From the behavior of the scale factor we have obtained the age of the universe as $t_0-t_B=-(2/H_0 \bar\zeta) \ln (1 - \bar\zeta/6).$
We have also studied the evolution of the deceleration parameter $q$ and the equation of state parameter $\omega_s$ for $0<\bar\zeta <6.$ For $\bar\zeta=4$ the deceleration parameter enter the negative region today, corresponding to accelerated universe at present. For $\bar\zeta < 4$ the $q$ enter the negative region in the future, while for $4<\bar\zeta <6$ it would enter the negative region in past, implying that the universe make a transition form the decelerated to its accelerated phase in the past. In general the $q \to -1$ as $a \to \infty,$ corresponds to de Sitter model of the universe. However for $\bar\zeta>6,$ $q$ ia always negative, implying eternal acceleration without any transition from the decelerated to accelerated epoch.
Behavior of $\omega_s$ shows that, it’s value is changing from positive to negative when $0<\bar\zeta<6$ implies a transition from decelerated to accelerated epoch and always negative when $\bar\zeta>6$ implies an eternal accelerated universe. But irrespective of the value of the viscous coefficient $\omega_s \to -1$ as $z \to -1 \, (a \to \infty).$ The equation of the today’s value of $\omega_s$ indicating that it would be negative at present if $\bar\zeta>3,$ however that does not corresponds to an accelerating universe. For an accelerating universe $\omega_s <-\frac{1}{3}$ for which $\bar\zeta>4$ according to the equation of today’s value of $\omega_s.$
Statefinder analysis of the model for $4<\bar\zeta<6$ were done, in which range the model predicts recent acceleration of the universe. The today’s position of the model in the $r-s$ plane is found to be $(r_0,s_0)=(1.25,-0.08)$ and different from the $\Lambda$CDM model. However as $a \to \infty$ the statefinder parameters $(r,s) \to (1,0)$ corresponds to the $\Lambda$CDM point.
When $\bar\zeta>6$ we have found that as $(t_0 -t) \to -\infty$ the scale factor tends to a minimum, i.e. $a \to a_{min},$ given by equation (\[eqn:amin1\]), and in this case the model doesn’t have a Big Bang. The density and the curvature scalar are increasing as the universe expands and attains maximum as $a \to \infty.$
We have analyses the status the GSL in the present model, and found that the GSL of thermodynamics is generally valid with apparent horizon as the boundary when $0<\bar\zeta<6.$ But when $\bar\zeta>6$ The GSL is satisfied only if the scale factor, $a > a_{min}$, where $a_{min}$ is given by equation (\[eqn:amin1\]).
Summerising the results, for $\bar\zeta=0$ the model reduces the stiff fluid dominated universe without viscosity. For $0<\bar\zeta<6$ the model predicts a universe with a Big Bang and make transition form the decelerated to the accelerated phase during it’s evolution. For $\bar\zeta>6$ the model doesn’t have a Big Bang, hence age is not properly define and the density and the curvature scalar increases as the universe expands and attains a maximum as $a \to \infty.$
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abstract: |
We propose algorithms that, given the input string of length $n$ over integer alphabet of size $\sigma$, construct the Burrows–Wheeler transform (BWT), the permuted longest-common-prefix (PLCP) array, and the LZ77 parsing in ${\mathcal{O}}(n/\log_{\sigma}n+r\,{\rm polylog}\,n)$ time and working space, where $r$ is the number of runs in the BWT of the input. These are the essential components of many compressed indexes such as compressed suffix tree, FM-index, and grammar and LZ77-based indexes, but also find numerous applications in sequence analysis and data compression. The value of $r$ is a common measure of repetitiveness that is significantly smaller than $n$ if the string is highly repetitive. Since just accessing every symbol of the string requires $\Omega(n/\log_{\sigma}n)$ time, the presented algorithms are time and space optimal for inputs satisfying the assumption $n/r\in\Omega({\rm polylog}\,n)$ on the repetitiveness. For such inputs our result improves upon the currently fastest general algorithms of Belazzougui (STOC 2014) and Munro et al. (SODA 2017) which run in ${\mathcal{O}}(n)$ time and use ${\mathcal{O}}(n/\log_{\sigma}
n)$ working space. We also show how to use our techniques to obtain optimal solutions on highly repetitive data for other fundamental string processing problems such as: Lyndon factorization, construction of run-length compressed suffix arrays, and some classical “textbook” problems such as computing the longest substring occurring at least some fixed number of times.
author:
- Dominik Kempa
bibliography:
- 'paper.bib'
title: |
Optimal Construction of Compressed Indexes\
for Highly Repetitive Texts[^1]
---
Introduction
============
The problem of text indexing is to preprocess the input text $T$ so that given any query pattern $P$, we can quickly (typically ${\mathcal{O}}(|P|+{\rm occ})$, where $|P|$ is the length of $P$ and ${\rm
occ}$ is the number of occurrences of $P$ in $T$) find all occurrences of $P$ in $T$. The two classical data structures for this problem are the suffix tree [@weiner] and the suffix array [@mm1993]. The suffix tree is a trie containing all suffixes of $T$ with each unary path compressed into a single edge labeled by the text substring. The suffix array is a list of suffixes of $T$ in lexicographic order where each suffix is encoded using its starting position. Both data structures take $\Theta(n)$ words of space. In addition to indexing, these data structures underpin dozens of applications in bioinformatics, data compression, and information retrieval. Suffix arrays, in particular, have become central to modern genomics, where they are used for genome assembly and short read alignment, data-intensive tasks at the forefront of modern medical and evolutionary biology [@MBCT2015]. This can be attributed mostly to their space-efficiency and simplicity.
In modern applications, however, which require indexing datasets of size close to the size of available RAM, even the suffix arrays can be prohibitively large, particularly in applications where the text consists of symbols from some alphabet $\Sigma$ of small size $\sigma=|\Sigma|$ (e.g., in bioinformatics $\Sigma=\{{\tt A}, {\tt C},
{\tt G}, {\tt T}\}$ and so $\sigma=4$). For such collections, the classical indexes are $\Theta(\log_{\sigma}n)$ times larger than the text which requires only $\Theta(n \log \sigma)$ bits, i.e., $\Theta(n/\log_{\sigma}n)$ words.
The invention of FM-index [@FM; @fm2005] and the compressed suffix array (CSA) [@CSA; @GrossiV05] at the turn of the millennium addressed this issue and revolutionized the field of string algorithms for nearly two decades. These data structures require only ${\mathcal{O}}(n/\log_{\sigma}n)$ words of space and provide random access to the suffix array in ${\mathcal{O}}(\log^{\epsilon}n)$ time. Dozens of papers followed the two seminal papers, proposing various improvements, generalizations, and practical implementations (see [@nm2007; @Navarro14; @BelazzouguiN14] for excellent surveys). These indexes are now widespread, both in theory where they provide off-the-shelf small space indexing structures and in practice, particularly bioinformatics, where they are the central component of many read-aligners [@bowtie; @bwa].
The other approach to indexing, recently gaining popularity due to the quick increase in the amount of highly repetitive data, such as software repositories or genomic databases is designing indexes specialized for repetitive strings. The first such index [@LZ-index] was based on the Lempel–Ziv (LZ77) parsing [@LZ77], the popular dictionary compression algorithms (used, e.g., in gzip and 7-zip compressors). Many improvements to the basic scheme were proposed since then [@GagieGKNP14; @BilleEGV18; @DCC2015; @GagieGKNP12; @ArroyueloNS12; @ArroyueloN11; @ArroyueloN10], and now the performance of LZ-based indexes is often on par with the FM-index or CSA [@FerradaKP18]. Independently to the development of LZ-based indexes, it was observed that the Burrows–Wheeler transform (BWT) [@bw1994], which underlies the FM-index, produces long runs of characters when applied to highly repetitive data [@MakinenNSV10; @SirenVMN08]. Gagie et al. [@GagieNP17] recently proposed a run length compressed suffix array (RLCSA) that provides fast access to suffix array and pattern matching queries in ${\mathcal{O}}(r\,{\rm polylog}\,n)$ or even ${\mathcal{O}}(r)$ space, where $r$ is the number of runs in the BWT of the text. The value of $r$ is, next to $z$ (the size of LZ77 parsing), a common measure of repetitiveness [@attractors].
Given the small space usage of the compressed indexes, their space-efficient construction emerged as one of the major open problems. A gradual improvement [@LamSSY02; @HonLSS03; @HonSS03] in the construction of compressed suffix array culminated with the work of Belazzougui [@STOC2014] who described the (randomized) ${\mathcal{O}}(n)$ time construction working in optimal space of ${\mathcal{O}}(n/\log_{\sigma}n)$. An alternative (and deterministic) construction was proposed by Munro et al. [@SODA2017]. These algorithms achieve the optimal construction space but their running time is up to $\Theta(\log n)$ times larger than the lower bound of $\Omega(n/\log_{\sigma}n)$ time (required to read the input/write the output).
#### Our Contribution {#our-contribution .unnumbered}
We propose algorithms that, given the input string of length $n$ over integer alphabet of size $\sigma$, construct the Burrows–Wheeler transform (BWT), the permuted longest-common-prefix (PLCP) array, and the LZ77 parsing in ${\mathcal{O}}(n/\log_{\sigma}n+r\,{\rm polylog}\,n)$ time and working space, where $r$ is the number of runs in the BWT of the input.
These are the essential components of nearly every compressed text index developed in the last two decades: all variants of FM-index rely on BWT [@FM; @GagieNP17], compressed suffix arrays/trees rely on $\Psi$ [@CSA; @Sada02] (which is dual to the ${\text{\rm BWT}}$ [@SODA2017; @HonSS03]) and the ${\text{\rm PLCP}}$ array, and LZ77-based and grammar-based indexes rely on the LZ77 parsing [@LZ-index; @Rytter03]. Apart from text indexing, these data structures have also numerous applications in sequence analysis and data compression [@MBCT2015; @navarrobook; @ohl2013].
Since just accessing every symbol of the string requires $\Omega(n/\log_{\sigma}n)$ time, the presented algorithms are time and space optimal for inputs satisfying the assumption $n/r\in\Omega({\rm
polylog}\,n)$ on the repetitiveness. Our results have particularly important implications for bioinformatics, where most of the data is highly-repetitive [@MakinenNSV10; @MBCT2015; @MakinenNSV09] and over small (DNA) alphabet. For such inputs, our result improves upon the currently fastest general algorithms of Belazzougui [@STOC2014] and Munro et al. [@SODA2017] which run in ${\mathcal{O}}(n)$ time and use ${\mathcal{O}}(n/\log_{\sigma} n)$ working space.
We also show how to use our techniques to obtain an ${\mathcal{O}}(n/\log_{\sigma}n+r\,{\rm polylog}\,n)$ time and space algorithms for other fundamental string processing problems such as: Lyndon factorization [@CFL58], construction of run-length compressed suffix arrays [@GagieNP17], and some classical “textbook” problems such as computing the longest substring occurring at least some fixed number of times.
On the way to the above results, we show how to generalize the RLCSA of Gagie et al. [@GagieNP17] to achieve a trade-off between index size and query time. In particular, we obtain a ${\mathcal{O}}(r\,{\rm
polylog}\,n)$-space data structure that can answer suffix array queries in ${\mathcal{O}}(\log n / \log \log n)$ time which improves on the ${\mathcal{O}}(\log n)$ query time of [@GagieNP17].
Preliminaries
=============
We assume a word-RAM model with a word of $w=\Theta(\log n)$ bits and with all usual arithmetic and logic operations taking constant time. Unless explicitly specified otherwise, all space complexities are given in words. All our algorithms are deterministic.
Throughout we consider a string ${T}[1..n]$ of symbols from an alphabet $\Sigma=[1..\sigma]$ of size $\sigma\,{\leq}\, n$. We assume $T[n]\,{=}\,\$$ with a numerical value of $\$$ equal to 0. For $j \in
[1..n]$, we write ${T}[j..n]$ to denote the suffix $j$ of ${T}$. We define the *rotation* of ${T}$ as a string ${T}[j..n]{T}[1..j-1]$ for any position $j\in [1..n]$.
The *suffix array* [@mm1993; @gbys1992] of ${T}$ is an array ${\text{\rm SA}}[1..n]$ which contains a permutation of the integers $[1..n]$ such that ${T}[{\text{\rm SA}}[1]..n] \prec {T}[{\text{\rm SA}}[2]..n] \prec \cdots \prec
{T}[{\text{\rm SA}}[n]..n]$, where $\prec$ denotes the lexicographical order. The inverse suffix array ${\text{\rm ISA}}$ is the inverse permutation of ${\text{\rm SA}}$, i.e., ${\text{\rm ISA}}[j] = i$ iff ${\text{\rm SA}}[i] = j$. The array $\Phi[1..n]$ (see [@KarkkainenMP2009]) is defined by $\Phi[{\text{\rm SA}}[i]]={\text{\rm SA}}[i-1]$ for $i\in[2..n]$, and $\Phi[{\text{\rm SA}}[1]]={\text{\rm SA}}[n]$, that is, the suffix $\Phi[j]$ is the immediate lexicographical predecessor of suffix $j$.
Let ${\text{\rm lcp}}(j_1,j_2)$ denote the length of the longest-common-prefix (LCP) of suffix $j_1$ and suffix $j_2$. The *longest-common-prefix array* [@mm1993; @klaap2001], ${\text{\rm LCP}}[1..n]$, is defined as ${\text{\rm LCP}}[i] = {\text{\rm lcp}}({\text{\rm SA}}[i],{\text{\rm SA}}[i-1])$ for $i
\in [2..n]$ and ${\text{\rm LCP}}[1]=0$. The *permuted LCP array* [@KarkkainenMP2009] ${\text{\rm PLCP}}[1..n]$ is the LCP array permuted from the lexicographical order into the text order, i.e., ${\text{\rm PLCP}}[{\text{\rm SA}}[i]]={\text{\rm LCP}}[i]$ for $i \in [1..n]$. Then ${\text{\rm PLCP}}[j] =
{\text{\rm lcp}}(j,\Phi[j])$ for all $j\in[1..n]$.
The *succinct PLCP array* [@Sada02; @KarkkainenMP2009] ${\mathrm{PLCP_{succ}}}[1..2n]$ represents the PLCP array using $2n$ bits. Specifically, ${\mathrm{PLCP_{succ}}}[j']=1$ if $j'=2j+{\text{\rm PLCP}}[j]$ for some $j\in[1..n]$, and ${\mathrm{PLCP_{succ}}}[j']=0$ otherwise. Any lcp value can be recovered by the equation ${\text{\rm PLCP}}[j]={\text{\rm select}}_{{\mathrm{PLCP_{succ}}}}(1,j)-2j$, where ${\text{\rm select}}_S(c,j)$ returns the location of the $j^{\mbox{\scriptsize
th}}$ $c$ in $S$.
The *Burrows–Wheeler transform* [@bw1994] ${\text{\rm BWT}}[1..n]$ of ${T}$ is defined by ${\text{\rm BWT}}[i] = {T}[{\text{\rm SA}}[i]-1]$ if ${\text{\rm SA}}[i] > 1$ and ${\text{\rm BWT}}[i] = {T}[n]$ otherwise. Let ${\mathcal{M}}$ denote the $n \times n$ matrix, whose rows are lexicographically sorted rotations of ${T}$. We denote the rows by ${\mathcal{M}}[i]$, $i\in [1..n]$. Note that ${\text{\rm BWT}}$ is the last column of ${\mathcal{M}}$.
The *LF-mapping* [@FM] is defined by the equation ${\text{\rm LF}}[{\text{\rm ISA}}[j]] = {\text{\rm ISA}}[j-1]$, $j\in[2..n]$, and ${\text{\rm LF}}[{\text{\rm ISA}}[1]] =
{\text{\rm ISA}}[n]$. By ${\text{\rm $\Psi$}}$ we denote the inverse of ${\text{\rm LF}}$. The significance of ${\text{\rm LF}}$ (and the principle underlying FM-index [@FM]) lies in the fact that, for $i\in[1..,n]$, ${\text{\rm LF}}[i] = {C}[{\text{\rm BWT}}[i]]\,{+}\,
{\text{\rm rank}}_{{\text{\rm BWT}}}({\text{\rm BWT}}[i],i)$, where ${C}[c]$ is the number of symbols in ${T}$ that are smaller than $c$, and ${\text{\rm rank}}_S(c,i)$ is the number of occurrences of $c$ in $S[1..i]$. From the formula for ${\text{\rm LF}}$ we obtain the following fact.
\[lm:lf-in-run\] Let ${\text{\rm BWT}}[b..e]$ be a run of the same symbol and let $i,i'\in[b,e]$. Then, ${\text{\rm LF}}[i]={\text{\rm LF}}[i']+(i-i')$.
If $i$ is the rank (i.e., the number of smaller suffixes) of $P$ among suffixes of ${T}$, then ${C}[c]+{\text{\rm rank}}_{{\text{\rm BWT}}}(c,i)$ is the rank of $cP$. This is called *backward search* [@FM].
We say that an lcp value ${\text{\rm LCP}}[i]={\text{\rm PLCP}}[{\text{\rm SA}}[i]]$ is *reducible* if ${\text{\rm BWT}}[i] = {\text{\rm BWT}}[i-1]$ and *irreducible* otherwise. The significance of reducibility is summarized in the following two lemmas.
\[lm:reducible\] If ${\text{\rm PLCP}}[j]$ is reducible, then ${\text{\rm PLCP}}[j]={\text{\rm PLCP}}[j-1]-1$ and $\Phi[j]=\Phi[j-1]+1$.
\[lm:irreducible\] The sum of all irreducible lcp values is $\le n \log n$.
It can be shown [@MakinenNSV10] that repetitions in $T$ generate equal-letter runs in BWT. By $r$ we denote the number of runs in ${\text{\rm BWT}}$. We can efficiently represent this transform as the list of pairs ${\rm RLBWT} = \langle \lambda_i, c_i \rangle_{i=1,\dots, r}$, where $\lambda_i>0$ is the starting position of the $i$-th run and $c_i\in\Sigma$. Note that $r$ is also the number of irreducible lcp values.
Augmenting RLBWT
================
In this section we present extensions of run-length compressed BWT needed by our algorithms. Each extension expands its functionality while maintaining small space usage and low construction time/space.
Rank and Select Support
-----------------------
One of the basic operations we will need are rank and select queries on BWT. We will now show that a run-length compressed BWT can be quickly augmented with a data structure capable of answering these queries in BWT-runs space.
\[thm:ranksel-support\] Given RLBWT of size $r$ for text ${T}[1..n]$ we can add ${\mathcal{O}}(r)$ space so that, given $i\in[0..n]$ and $c\in[1..\sigma]$, values ${\text{\rm rank}}_{{\text{\rm BWT}}}(c,i)$ and ${\text{\rm select}}_{{\text{\rm BWT}}}(c,i)$ can be computed in ${\mathcal{O}}(\log r)$ time. The data structure can be constructed in ${\mathcal{O}}(r \log r)$ time using ${\mathcal{O}}(r)$ space.
We augment each BWT-run with its length and sort the runs using the symbol as the primary key, and the start of the run as the secondary key. This allows us to compute, for every run $[b..e]$, the value ${\text{\rm rank}}_{{\text{\rm BWT}}}(c,b)$ where $c={\text{\rm BWT}}[b]$. Using this list, both queries can be answered in ${\mathcal{O}}(\log r)$ time using binary search.
LF/PSI and Backward Search Support
----------------------------------
We now show that with the help of the above rank/select data structures we can support more complicated navigational queries, namely, given any $i\in [1..n]$ such that ${\text{\rm SA}}[i]=j$ we can compute ${\text{\rm ISA}}[j-1]$ (i.e., ${\text{\rm LF}}[i]$) and ${\text{\rm ISA}}[j+1]$ (i.e., ${\text{\rm $\Psi$}}[i]$). Note that none of the queries will require the knowledge of $j$. As a simple corollary, we obtain efficient support for backward search on RLBWT.
\[thm:lfpsi-support\] Given RLBWT of size $r$ for text ${T}[1..n]$ we can add ${\mathcal{O}}(r)$ space so that, given $i\in[1..n]$, values ${\text{\rm LF}}[i]$ and ${\text{\rm $\Psi$}}[i]$ can be computed in ${\mathcal{O}}(\log r)$ time. The data structure can be constructed in ${\mathcal{O}}(r \log r)$ time using ${\mathcal{O}}(r)$ working space.
Similarly as in Theorem \[thm:ranksel-support\] we prepare a (sorted) list containing, for each symbol $c$ occurring in ${T}$, the total frequency of symbols smaller than $c$.
To answer ${\text{\rm LF}}[i]$ we first compute ${\text{\rm BWT}}[i]$ (by searching the list of runs), then $C[{\text{\rm BWT}}[i]]$ (by searching the above frequency table), and finally apply Theorem \[thm:ranksel-support\]. To compute $\Psi[i]$ we first determine (using the frequency table) the symbol $c$ following ${\text{\rm BWT}}[i]$ in text and the number $k$ such that this $c$ is the $k$-th occurrence of $c$ in the first column of ${\mathcal{M}}$. It then remains to find the $k$-th occurrence of $c$ in the BWT using Theorem \[thm:ranksel-support\].
\[thm:bs-support\] Given RLBWT of size $r$ for text ${T}[1..n]$ we can add ${\mathcal{O}}(r)$ space so that, given a rank $i\in[0..n]$ of a string $P$ among the suffixes of $T$, for any $c\in [1..\sigma]$ we can compute in ${\mathcal{O}}(\log r)$ time the rank of $cP$. The data structure can be constructed in ${\mathcal{O}}(r \log r)$ time using ${\mathcal{O}}(r)$ working space.
Suffix-Rank Support {#sec:suffix-rank-support}
-------------------
In this section we describe an extension of RLBWT that will allow us to efficiently merge two RLBWTs during the BWT construction algorithm. We start by defining a generalization of BWT-runs and stating their basic properties.
Let ${\text{\rm lcs}}(x,y)$ denote the length of the longest common suffix of strings $x$ and $y$. We define the ${\text{\rm LCS}}[1..n]$ array [@KarkkainenKP12] as ${\text{\rm LCS}}[i]={\text{\rm lcs}}({\mathcal{M}}[i],
{\mathcal{M}}[i-1])$ for $i\in [2..n]$ and ${\text{\rm LCS}}[1]=0$ (recall that $\mathcal{M}$ is a matrix containing sorted rotations of $T$). Let $\tau \geq 1$ be an integer. We say that a range $[b..e]$ of BWT is a *$\tau$-run* if ${\text{\rm LCS}}[b]<\tau$, ${\text{\rm LCS}}[e+1]<\tau$, and for any $i\in[b+1..e]$, ${\text{\rm LCS}}[i]\geq \tau$. By this definition, a BWT run is a 1-run. For $j\geq 0$ let $Q_{j} = \{i\in[1..n] \mid {\text{\rm LCS}}[i]=j\}$ and $R_{\tau}=\bigcup_{j=0}^{\tau-1}Q_j$. Then, $R_{\tau}$ is exactly the set of starting positions of $\tau$-runs.
For any $i\in [2..n]$, $${\text{\rm LCS}}[i] = \left\{
\begin{array}{l l}
0 & \enspace \text{{\rm if}
${\text{\rm BWT}}[i]\neq{\text{\rm BWT}}[i-1]$},\\ {\text{\rm LCS}}[{\text{\rm LF}}[i]]+1 & \enspace
\text{{\rm otherwise.}}\\
\end{array}
\right.$$
Since ${\text{\rm $\Psi$}}$ is the inverse of ${\text{\rm LF}}$ we obtain that for any $j\geq 1$, $Q_{j}=\{{\text{\rm $\Psi$}}[i]\ |\ i\in Q_{j-1}\ {\rm and}\ {\text{\rm $\Psi$}}[i]\notin Q_0\}$. Thus, the set $R_{\tau}$ can be efficiently computed by iterating each of the starting positions of BWT-runs $\tau-1$ times using ${\text{\rm $\Psi$}}$ and taking a union of all visited positions. From the above we see that $|Q_{j+1}| \leq |Q_j|$, which implies that the number of $\tau$-runs satisfies $|R_{\tau}| \leq |Q_0|\tau=r\tau$.
\[thm:rank-support\] Let $S[1..m]$, $S'[1..m']$ be strings with $r$ and $r'$ (respectively) runs in the ${\text{\rm BWT}}$. Given RLBWTs of $S$ and $S'$ it is possible, for any integer $\tau \geq 1$, to build a data structure of size ${\mathcal{O}}(\frac{m}{\tau} + r+r')$ that can, given a rank $i\in [0..m]$ of some suffix $S[j..m]$ among suffixes of $S$, compute the rank of $S[j..m]$ among suffixes of $S'$ in ${\mathcal{O}}(\tau(\log\frac{m}{\tau}+\log r + \log r'))$ time. The data structure can be constructed in ${\mathcal{O}}(\tau^2(r+r') \log (r\tau +
r'\tau) + \frac{m}{\tau}(\log (r\tau) + \log (r'\tau) + \log
\frac{m}{\tau}))$ time and ${\mathcal{O}}(\tau^2(r+r') + \frac{m}{\tau})$ space.
We start by augmenting both RLBWTs with ${\text{\rm $\Psi$}}$ and ${\text{\rm LF}}$ support (Theorem \[thm:lfpsi-support\]) and RLBWT of $S'$ with the backward search support (Corollary \[thm:bs-support\]). This requires ${\mathcal{O}}(r \log r + r' \log r')$ time and ${\mathcal{O}}(r+r')$ space.
We then compute a (sorted) set of starting positions of $\tau$-runs for both RLBWTs. For $S$ this requires answering $r\tau$ ${\text{\rm $\Psi$}}$-queries which takes ${\mathcal{O}}(r\tau \log r)$ time in total, and then sorting the resulting set of positions in ${\mathcal{O}}((r\tau) \log
(r\tau))$ time. Analogous processing for $S'$ takes ${\mathcal{O}}((r'\tau)
\log (r'\tau))$ time. The starting positions of all $\tau$-runs require ${\mathcal{O}}((r+r')\tau)$ space in total.
Next, for any $\tau$-run $[b..e]$ we compute and store the associated $\tau$ symbols. We also store the value ${\text{\rm LF}}^{\tau}[b]$, but only for $\tau$-runs of $S$. Due to simple generalization of Lemma \[lm:lf-in-run\], this will allow us to compute the value ${\text{\rm LF}}^{\tau}[i]$ for *any* $i$. In total this requires answering $\tau^2(r+r')$ ${\text{\rm LF}}$-queries and hence takes ${\mathcal{O}}(\tau^2(r+r')
\log r)$ time. The space needed to store all symbols is ${\mathcal{O}}(\tau^2(r+r'))$.
We then lexicographically sort all length-$\tau$ strings associated with $\tau$-runs (henceforth called *$\tau$-substrings*) and assign to each run the rank of the associated substring in the sorted order. Importantly, $\tau$-substrings of $S$ and $S'$ are sorted together. These ranks will be used as order-preserving names for $\tau$-substrings. We use an LSD string sort with a stable comparison-based sort for each position hence the sorting takes ${\mathcal{O}}\left(\tau^2(r+r') \log (r\tau+r'\tau)\right)$ time. The working space does not exceed ${\mathcal{O}}(\tau(r+r'))$. After the names are computed, we discard the substrings.
We now observe that order-preserving names for $\tau$-substrings allow us to perform backward search $\tau$ symbols at a time. We build a rank-support data structure analogous to the one from Theorem \[thm:ranksel-support\] for names of $\tau$-substrings of $S'$. We also add support for computing the total number of occurrences of names smaller than a given name. This takes ${\mathcal{O}}(r'\tau \log (r'\tau))$ time and ${\mathcal{O}}(r'\tau)$ space. Then, given a rank $i$ of suffix $S[j..m]$ among suffixes of $S'$, we can compute the rank of suffix $S[j-\tau..m]$ among suffixes of $S'$ in ${\mathcal{O}}(\log (r'\tau))$ time by backward search on $S'$ using $i$ as a position, and the name of $\tau$-substring preceding $S[j..m]$ as a symbol.
We now use the above multi-symbol backward search to compute the rank of every suffix of the form $S[m-k\tau..m]$ among suffixes of $S'$. We start from the shortest suffix and increase the length by $\tau$ in every step. During the computation we also maintain the rank of the current suffix of $S$ among suffixes of $S$. This allows us to efficiently compute the name of the preceding $\tau$-substring. The rank can be updated using values ${\text{\rm LF}}^{\tau}$ stored with each $\tau$-run of $S$. Thus, for each of the $m/\tau$ suffixes of $S$ we obtain a pair of integers ($i_S$, $i_{S'}$), denoting its rank among the suffixes of $S$ and $S'$. We store these pairs as a list sorted by $i_S$. Computing the list takes ${\mathcal{O}}\left(\frac{m}{\tau}(\log (r\tau)+\log(r'\tau)) +
\frac{m}{\tau}\log \frac{m}{\tau}\right)$ time. After the list is computed we discard all data structures associated with $\tau$-runs.
Using the above list of ranks we can answer the query from the claim as follows. Starting with $i$, we compute a sequence of $\tau$ positions in the BWT of $S$ by iterating ${\text{\rm $\Psi$}}$ on $i$. For each position we can check in ${\mathcal{O}}(\log \frac{m}{\tau})$ time whether that position is in the list of ranks. Since we evenly sampled text positions, one of these positions has to correspond to the suffix of $S$ for which we computed the rank in the previous step. Suppose we found such position after $\Delta\leq \tau$ steps, i.e., we now have a pair ($i_S$, $i_{S'}$) such that $i_{S'}$ is the rank of $S[j+\Delta..m]$ among suffixes of $S'$. We then perform $\Delta$ steps of the standard backward search starting from rank $i_{S'}$ in the BWT of $S'$ using symbols $S[j{+}\Delta{-}1]$, …, $S[j]$. This takes ${\mathcal{O}}\left(\Delta (\log r + \log
r')\right)={\mathcal{O}}\left(\tau(\log r + \log r')\right)$ time.
Construction of BWT
===================
In this section we show that given the packed encoding of text ${T}[1..n]$ over alphabet $\Sigma=[1..\sigma]$ of size $\sigma\leq n$ (i.e., using ${\mathcal{O}}(n / \log_{\sigma}n)$ words of space), we can compute the packed encoding of ${\text{\rm BWT}}$ of ${T}$ in ${\mathcal{O}}(n/\log_{\sigma}n+r \log^7 n)$ time and ${\mathcal{O}}(n/\log_{\sigma}n+r \log^5 n)$ space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$.
Algorithm Overview
------------------
The basic scheme of our algorithm follows the algorithm of Hon et al. [@HonSS03]. Assume for simplicity that $w/\log {\sigma}=2^k$ for some integer $k$. The algorithm works in $k+1$ rounds, where $k=\log \log_{\sigma}n$. In the $i$-th round, $i\in[0..k]$, we interpret ${T}$ as a string over superalphabet $\Sigma_{i}=[1..\sigma_i]$ of size $\sigma_i=\sigma^{2^i}$, i.e., we group symbols of $T$ into supersymbols consisting of $2^i$ original symbols. We denote this string as ${T}_i$. The rounds are executed in decreasing order of $i$. The input to the $i$-th round, $i\in[0..k{-}1]$, is the run-length compressed BWT of ${T}_{i+1}$, and the output is the run-length compressed BWT of ${T}_i$. We denote the size of RLBWT of $T_i$ by $r_i$. The final output is the run-length compressed BWT of ${T}_0={T}$, which we then convert into packed encoding taking ${\mathcal{O}}(n / \log_{\sigma}n)$ words.
For the $k$-th round, we observe that $|\Sigma_k|=\Theta(n)$ and $|T_k|=\Theta(n/\log_{\sigma}n)$ hence to compute BWT of $T_k$ it suffices to first run any of the linear-time algorithms for constructing the suffix array [@ks2003; @NongZC11; @ka05; @KimSPP03] for $T_k$ and then naively compute the RLBWT from the suffix array. This takes ${\mathcal{O}}(n/\log_{\sigma}n)$ time and space.
Let $S=T_i$ for some $i\in[0..k{-}1]$ and suppose we are given the RLBWT of $T_{i+1}$. Let $S_o$ be the string of length $|S|/2$ created by grouping together symbols $S[2j-1]S[2j]$ for all $j$, and let $S_e$ be the analogously constructed string for pairs $S[2j]S[2j+1]$. Clearly we have $S_o=T_{i+1}$ (recall that we start indexing from 1). Furthermore, it is easy to see that the BWT of $S$ can be obtained by interleaving BWTs of $S_o$ and $S_e$, and discarding (more significant) half of the bits in the encoding of each symbol.
The construction of RLBWT for $S$ consists of two steps: (1) first we compute the RLBWT of $S_e$ from RLBWT of $S_o$, and then (2) merge RLBWTs of $S_o$ and $S_e$ into RLBWT of $S$.
Computing BWT of Se
-------------------
In this section we assume that $S=T_i$ for some $i\in[0..k{-}1]$ and that we are given the RLBWT of $S_o=T_{i+1}$ of size $r_o=r_{i+1}$. Denote the size of RLBWT of $S_e$ by $r_e$. We will show that RLBWT of $S_e$ can be computed in ${\mathcal{O}}(r_e + r_o \log r_o)$ time using ${\mathcal{O}}(r_o + r_e)$ working space.
Recall that both $S_o$ and $S_e$ are over alphabet $\Sigma_{i+1}$. Each of the symbols in that alphabet can be interpreted as a concatenation of two symbols in the alphabet $\Sigma_i$. Let $c$ be the symbol of either $S_o$ or $S_e$ and assume that $c=S[j]S[j+1]$ for some $j\in[1..|S|{-}1]$. By *major subsymbol* of $c$ we denote a symbol (equal to $S[j]$) from $\Sigma_i$ encoded by the more significant half of bits encoding $c$, and by *minor subsymbol* we denote symbol encoded by remaining bits (equal to $S[j+1]$).
We first observe that by enumerating all runs of the RLBWT of $S_o$ in increasing order of their minor subsymbols (and in case of ties, in the increasing order of run beginnings), we obtain (on the remaining bits) the minor subsymbols of the ${\text{\rm BWT}}$ of $S_e$ in the correct order. Such enumeration could easily be done in ${\mathcal{O}}(r_o \log r_o)$ time and ${\mathcal{O}}(r_o)$ working space. To obtain the missing (major) part of the encoding of symbols in the BWT of $S_e$, it suffices to perform the ${\text{\rm LF}}$-step for each of the runs in the BWT of $S_o$ in the sorted order above (i.e., by minor subsymbol), and look up the minor subsymbols in the resulting range of ${\text{\rm BWT}}$ of $S_o$.
The problem with the above approach is the running time. While it indeed produces correct RLBWT of $S_e$, having to scan all runs in the range of BWT of $S_o$ obtained by performing the ${\text{\rm LF}}$-step on each of the runs of $S_o$ could be prohibitively high. To address this we first construct a run-length compressed sequence of minor subsymbols extracted from ${\text{\rm BWT}}$ of $S_o$ and use it to extract minor subsymbols of ${\text{\rm BWT}}$ of $S_o$ in total time proportional to the number of runs in the ${\text{\rm BWT}}$ of $S_e$.
\[lm:inducing\] Given RLBWT of size $r_o$ for $S_o=T_{i+1}$ we can compute the RLBWT of $S_e$ in ${\mathcal{O}}(r_e + r_o \log r_o)$ time and ${\mathcal{O}}(r_o + r_e)$ working space, where $r_e$ is the size of RLBWT of $S_e$.
The whole process requires scanning the ${\text{\rm BWT}}$ of $S_o$ to create a run-length compressed encoding of minor subsymbols, adding the ${\text{\rm LF}}$ support to (the original) RLBWT of $S_o$, sorting the runs in RLBWT of $S_o$ by the minor subsymbol, and executing $r_o$ ${\text{\rm LF}}$-queries on the ${\text{\rm BWT}}$ of $S_o$, which altogether takes ${\mathcal{O}}(r_o \log
r_o)$. All other operations take time proportional to ${\mathcal{O}}(r_o +
r_e)$. The space never exceeds ${\mathcal{O}}(r_o + r_e)$.
Merging BWTs of Se and So
-------------------------
As in the previous section, we assume $S=T_i$ for some $i\in[0..k{-}1]$ and that we are given the RLBWT of $S_o=T_{i+1}$ of size $r_o=r_{i+1}$ and RLBWT of $S_e$ of size $r_e$. We will show how to use these to efficiently compute the RLBWT of $S$ in ${\mathcal{O}}(|S|/\log |S| + (r_o + r_e)\,{\rm polylog}\,|S|)$ time and space.
We start by observing that to obtain ${\text{\rm BWT}}$ of $S$ it suffices to merge the ${\text{\rm BWT}}$ of $S_e$ and ${\text{\rm BWT}}$ of $S_o$ and discard all major subsymbols in the resulting sequence. The algorithm of Hon et al. [@HonSS03] achieves this by performing the backward search. This requires $\Omega(|S|)$ time and hence is too expensive in our case.
Instead, we employ the following observation. Suppose we have already computed the first $t$ runs of the ${\text{\rm BWT}}$ of $S$ and let the next unmerged character in the ${\text{\rm BWT}}$ of $S_o$ be a part of a run of symbol $c_o$. Let $c_e$ be the analogous symbol from the ${\text{\rm BWT}}$ of $S_e$. Further, let $c_e'$ (resp. $c_o'$) be the minor subsymbol of $c_e$ (resp. $c_o$). If $c_o' = c_e'$ then either all symbols in the current run in the ${\text{\rm BWT}}$ of $S_o$ (restricted to minor subsymbols) or all symbols in the current run in the (also restricted) ${\text{\rm BWT}}$ of $S_e$ will belong to the next run in the ${\text{\rm BWT}}$ of $S$. Assuming we can determine the order between any two arbitrary suffixes of $S_o$ and $S_e$ given their ranks in the respective ${\text{\rm BWT}}$s, we could consider both cases and in each perform a binary search to find the exact length of $(t+1)$-th run in the ${\text{\rm BWT}}$ of $S$. We first locate the end of the run of $c_o'$ (resp. $c_e'$) in the BWT of $S_o$ (resp. $S_e$) restricted to minor subsymbols; this can be done after preprocessing input BWTs without increasing the time/space of the merging. We then find the largest suffix of $S_e$ (resp. $S_o$) not greater than the suffix at the end of the run in the BWT of $S_o$. Importantly, the time to compute the next run in the BWT of $S$ does not depend on the number of times the suffixes in that run alternate between $S_o$ and $S_e$. The case $c_e' \neq c_o'$ is handled similarly, except we do not need to locate the end of each run. The key property of this algorithm is that the number of pattern searches is ${\mathcal{O}}(r_i \log |S|)$.
Thus, the merging problem can be reduced to the problem of efficient comparison of suffixes of $S_e$ and $S_o$. To achieve that we augment both RLBWTs of $S_e$ and $S_e$ with the suffix-rank support data structure from Section \[sec:suffix-rank-support\]. This will allow us to determine, given a rank of any suffix of $S_o$, the number of smaller suffixes of $S_e$ and vice-versa, thus eliminating even the need for binary search. Our aim is to achieve ${\mathcal{O}}(|S|/\log |S|)$ space and construction time assuming small $r$ values, thus we apply Theorem \[thm:rank-support\] with $\tau=\log^2|S|$.
\[lm:merging\] Given RLBWT of size $r_e$ for $S_e$ and RLBWT of size $r_o$ for $S_o
= T_{i+1}$ we can compute the RLBWT of $S = T_i$ in ${\mathcal{O}}((r_o+r_e)\log^5|S|+|S|/\log |S|+r_i \log^3|S|)$ time and ${\mathcal{O}}(|S|/\log^2|S| + (r_o + r_e)\log^4|S| + r_i)$ working space.
Constructing the suffix-rank support for $S_o$ and $S_e$ with $\tau=\log^2|S|$ takes ${\mathcal{O}}((r_o+r_e)\log^5|S| + |S|/\log |S|)$ time and ${\mathcal{O}}((r_o+r_e)\log^4|S|+|S|/\log^2|S|)$ working space. The resulting data structures occupy ${\mathcal{O}}(|S|/\log^2|S| +
r_e + r_o)$ space and answer suffix-rank queries in ${\mathcal{O}}(\log^3|S|)$ time. To compute the RLBWT of $S$ we perform $2r_i$ suffix-rank queries for a total of ${\mathcal{O}}(r_i \log^3|S|)$ time.
Putting Things Together
-----------------------
To bound the size of RLBWTs in intermediate rounds, consider the $i$-th round where for $d=2^i$ we group each $d$ symbols of ${T}$ to obtain the string $S={T}_i$ of length $|T|/d$ and let $r_i$ be the number of runs in the BWT of $S$. Recall now the construction of generalized BWT-runs from Section \[sec:suffix-rank-support\] and observe that the symbols of ${T}$ comprising each supersymbol $S[j]$ are the same as the substring corresponding to $d$-run containing suffix ${T}[jd+1..n]$ in the ${\text{\rm BWT}}$ of ${T}$. It is easy to see that the corresponding suffixes of ${T}$ are in the same lexicographic order as the suffixes of $S$. Thus, $r_i$ is bounded by the number of $d$-runs in the ${\text{\rm BWT}}$ of ${T}$, which by Section \[sec:suffix-rank-support\] is bounded by $rd$. Hence, the size of the output RLBWT of the $i$-th round does not exceed $r2^i={\mathcal{O}}(r \log n)$. The analogous analysis shows that the size of RLBWT of $S_e$ has the same upper bound $r2^{i+1}$ as $S_o=T_{i+1}$.
\[thm:bwt\] Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\leq n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words, the ${\text{\rm BWT}}$ of ${T}$ can be computed in ${\mathcal{O}}(n /\log_{\sigma}n + r\log^7
n)$ time and ${\mathcal{O}}(n/\log_{\sigma}n + r\log^5 n)$ working space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$.
The $k$-th round of the algorithm takes ${\mathcal{O}}(n/\log_{\sigma}n)$ time working space and produces a ${\text{\rm BWT}}$ taking ${\mathcal{O}}(n/\log_{\sigma}n)$ words of space. Consider the $i$-th round of the algorithm for $i<k$ and let $S=T_{i}$, and $r_e$ and $r_o$ denote the sizes of RLBWT of $S_e$ and $S_o$ respectively. By the above discussion, we have $r_o,r_e={\mathcal{O}}(r \log n)$. Thus, by Lemma \[lm:inducing\] and Lemma \[lm:merging\] the $i$-th round takes ${\mathcal{O}}(n_i/\log n_i + r\log^6 n_i)={\mathcal{O}}(n/(2^i\log n)+r
\log^6 n)$ time and the working space does not exceed ${\mathcal{O}}(n/\log^2 n + r \log^5 n)$ words, where $n_i\,{=}\,|T_i|\,{=}\,n/2^i$, and we used the fact that for $i<k$, $\log n_i=\Theta(\log n)$. Hence over all rounds we spend ${\mathcal{O}}(n/\log_{\sigma}n + r\log^7 n)$ time and never use more than ${\mathcal{O}}(n/\log_{\sigma}n + r\log^5 n)$ space. Finally, it is easy to convert RLBWT into the packed encoding in ${\mathcal{O}}(n/\log_{\sigma}n+r\log n)$ time.
Thus, we obtained a time- and space-optimal construction algorithm for BWT under the assumption $n/r=\Omega({\rm polylog}\,n)$ on the repetitiveness of the input.
Construction of PLCP {#sec:plcp}
====================
In this section we show that given the run-length compressed representation of ${\text{\rm BWT}}$ of ${T}$, it is possible to compute the ${\mathrm{PLCP_{succ}}}$ bitvector in ${\mathcal{O}}(n/\log n + r \log^{11} n)$ time and ${\mathcal{O}}(n/\log n + r\log^{10} n)$ working space..
The key observation used to construct the PLCP values is that it suffices to only compute the irreducible LCP values. Then, by Lemma \[lm:reducible\], all other values can be quickly deduced. This significantly simplifies the problem because it is known (Lemma \[lm:irreducible\]) that the sum of irreducible LCP values is bounded by ${\mathcal{O}}(n \log n)$.
The main idea of the construction is to compute (as in Theorem \[thm:rank-support\]) names of $\tau$-runs for $\tau=\log^5
n$. This will allow us to compare $\tau$ symbols at a time and thus quickly compute a lower bound for large irreducible LCP values. Before we can use this, we need to augment the BWT with the support for ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries.
Computing SA/ISA Support {#sec:sa-support}
------------------------
Suppose that we are given a run-length compressed ${\text{\rm BWT}}$ of ${T}[1..n]$ taking ${\mathcal{O}}(r)$ space. Let $\tau\geq 1$ be an integer. Assume for simplicity that $n$ is a multiple of $\tau$. We start by computing the sorted list of starting positions of all $\tau$-runs similarly, as in Theorem \[thm:rank-support\]. This requires augmenting the RLBWT with the ${\text{\rm LF}}$/${\text{\rm $\Psi$}}$ support first and in total takes ${\mathcal{O}}(\tau r
\log (\tau r))$ time and ${\mathcal{O}}(\tau r)$ working space. We then compute and store, for the first position of each $\tau$-run $[b..e]$, the value of ${\text{\rm LF}}^{\tau}[b]$. This will allow us to efficiently compute ${\text{\rm LF}}^{\tau}[i]$ for any $i\in[1..n]$.
We then locate the occurrence $i_0$ of the symbol $\$$ in ${\text{\rm BWT}}$ and perform $n/\tau$ iterations of ${\text{\rm LF}}^{\tau}$ on $i_0$. By definition of ${\text{\rm LF}}$, the position $i$ visited after $j$ iterations of ${\text{\rm LF}}^{\tau}$ is equal to ${\text{\rm ISA}}[n-j\tau]$, i.e., ${\text{\rm SA}}[i]=n-j\tau$. For any such $i$ we save the pair $(i,n-j\tau)$ into a list. When we finish the traversal we sort the list by the first component (assume this list is called $L_{{\text{\rm SA}}}$). We then create the copy of the list (call it $L_{{\text{\rm ISA}}}$) and sort it by the second component. Creating the lists takes ${\mathcal{O}}\left((n/\tau)(\log (r\tau) + \log
(n/\tau))\right)$ time and they occupy ${\mathcal{O}}(n/\tau)$ space. After the lists are computed we discard ${\text{\rm LF}}^{\tau}$ samples associated with all runs. Having these lists allows us to efficiently query ${\text{\rm SA}}$/${\text{\rm ISA}}$ as follows.
To compute ${\text{\rm ISA}}[j]$ we find in ${\mathcal{O}}(1)$ time (since we can store $L_{{\text{\rm ISA}}}$ in an array) the pair $(p,j')$ in $L_{{\text{\rm ISA}}}$ such that $j'=\lceil j/\tau \rceil \tau$. We then perform $j'-j<\tau$ steps of ${\text{\rm LF}}$ on position $p$. The total query time is thus ${\mathcal{O}}(\tau \log
r)$.
To compute ${\text{\rm SA}}[i]$ we perform $\tau$ steps of ${\text{\rm LF}}$ (each taking ${\mathcal{O}}(\log r)$ time) on position $i$. Due to the way we sampled ${\text{\rm SA}}$/${\text{\rm ISA}}$ values, one of the visited positions has to be the first component in the $L_{{\text{\rm SA}}}$ list. For each position, we can check this in ${\mathcal{O}}(\log(n/\tau))$ time. Suppose we found a pair after $\Delta<\tau$ steps, i.e., a pair $({\text{\rm LF}}^{\Delta}[i],p)$ is in $L_{{\text{\rm SA}}}$. This implies ${\text{\rm SA}}[{\text{\rm LF}}^{\Delta}[i]]=p$, i.e., ${\text{\rm SA}}[i]=p+\Delta$. The query time is ${\mathcal{O}}\left(\tau (\log r + \log
(n/\tau))\right)$.
\[thm:sa-support\] Given RLBWT of size $r$ for text $T[1..n]$, we can, for any integer $\tau\geq 1$, build a data structure taking ${\mathcal{O}}(r+n/\tau)$ space that, for any $i\in[1..n]$, can answer ${\text{\rm SA}}[i]$ query in ${\mathcal{O}}(\tau
(\log r + \log(n/\tau)))$ time and ${\text{\rm ISA}}[i]$ query in ${\mathcal{O}}\left(\tau \log r \right)$ time. The construction takes ${\mathcal{O}}\left((n/\tau) (\log (r\tau) + \log
(n/\tau))+\tau^2r\log(r\tau)\right)$ time and ${\mathcal{O}}(n/\tau +
r\tau)$ working space.
Computing Irreducible LCP Values {#sec:irreducible}
--------------------------------
We start by augmenting the RLBWT with the ${\text{\rm SA}}$/${\text{\rm ISA}}$ support as explained in the previous section using $\tau_1=\log^2 n$. The resulting data structure answers ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries in ${\mathcal{O}}(\log^3
n)$ time. We then compute $\tau_2$-runs and their names using the technique introduced in Theorem \[thm:rank-support\] for $\tau_2=\log^5 n$.
Given any $j_1,j_2\in[1..n]$ we can check whether it holds ${T}[j_1..j_1+\tau_2-1]={T}[j_2..j_2+\tau_2-1]$ using the above names as follows. Compute $i_1={\text{\rm ISA}}[j_1+\tau_2]$ and $i_2={\text{\rm ISA}}[j_2+\tau_2]$ using the ${\text{\rm ISA}}$ support. Then compare the names of $\tau_2$-substrings preceding these two suffixes. Thus, comparing two arbitrary substrings of ${T}$ of length $\tau_2$, given their text positions, takes ${\mathcal{O}}(\log^3 n)$ time.
The above toolbox allows computing all irreducible LCP values as follows. For any $i\in[1..n]$ such that ${\text{\rm LCP}}[i]$ is irreducible (such $i$ can be recognized by checking if ${\text{\rm BWT}}[i-1]$ belongs to a BWT-run different than ${\text{\rm BWT}}[i]$) we compute $j_1={\text{\rm SA}}[i-1]$ and $j_2={\text{\rm SA}}[i]$. We then have ${\text{\rm LCP}}[i]={\text{\rm lcp}}({T}[j_1..n],{T}[j_2..n])$. We start by computing the lower-bound for ${\text{\rm LCP}}[i]$ using the names of $\tau_2$-substrings. Since the sum of irreducible LCP values is bounded by ${\mathcal{O}}(n \log n)$, over all irreducible LCP values this will take ${\mathcal{O}}(r\log^3 n + \log^3 n \cdot (n \log n) /
\tau_2)={\mathcal{O}}(r\log^3 n + n/\log n)$ time. Finishing the computation of each ${\text{\rm LCP}}$ value requires at most $\tau_2$ symbol comparisons. This can be done by following ${\text{\rm $\Psi$}}$ for both pointers as long as the preceding symbols (found in the ${\text{\rm BWT}}$) are equal. Over all irreducible LCP values, finishing the computation takes ${\mathcal{O}}(r
\tau_2 \log n)={\mathcal{O}}(r \log^6 n)$ time.
\[thm:plcp\] Given RLBWT of size $r$ for ${T}[1..n]$, the ${\mathrm{PLCP_{succ}}}$ bitvector (or the list storing irreducible LCP values in text order) can be computed in ${\mathcal{O}}(n/\log n + r \log^{11} n)$ time and ${\mathcal{O}}(n/\log
n + r\log^{10}n)$ working space.
Adding the ${\text{\rm SA}}$/${\text{\rm ISA}}$ support using $\tau_1=\log^2 n$ takes ${\mathcal{O}}\left(n/\log n + r\log^5 n\right)$ time and ${\mathcal{O}}(n/\log^2 n
+ r\log^2 n)$ working space (Theorem \[thm:sa-support\]). The resulting structure needs ${\mathcal{O}}(r + n/\log^2 n)$ space and answers ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries in ${\mathcal{O}}(\log^3 n)$ time.
Computing the names takes ${\mathcal{O}}(\tau_2^2r \log (\tau_2 r))={\mathcal{O}}(r
\log^{11} n)$ time and ${\mathcal{O}}(\tau_2^2 r)={\mathcal{O}}(r \log^{10} n)$ working space (see the proof of Theorem \[thm:rank-support\]). The names need ${\mathcal{O}}(\tau_2 r)={\mathcal{O}}(r \log^5 n)$ space. Then, by the above discussion, computing all irreducible LCP values takes ${\mathcal{O}}(n/\log n + r \log^6 n)$ time.
By combining with Theorem \[thm:bwt\] we obtain the following result.
\[cor:plcp\] Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\,{\leq}\, n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words, the ${\mathrm{PLCP_{succ}}}$ bitvector (or the list storing irreducible LCP values in text order) can be computed in ${\mathcal{O}}(n/\log_{\sigma} n + r
\log^{11} n)$ time and ${\mathcal{O}}(n/\log_{\sigma} n + r\log^{10}n)$ working space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$.
Construction of RLCSA {#sec:rlcsa}
=====================
In this section, we show how to use the techniques presented in this paper to quickly build the run-length compressed suffix array (RLCSA) recently proposed by Gagie et al. [@GagieNP17]. They observed that if ${\text{\rm BWT}}$ of $T$ has $r$ runs then the arrays ${\text{\rm SA}}/{\text{\rm ISA}}$ and ${\text{\rm LCP}}$ have a bidirectional parse of size ${\mathcal{O}}(r)$ after being differentially encoded. They use a locally-consistent parsing [@BatuES06; @Jez15] to grammar-compress these arrays and describe the necessary augmentations to achieve fast decoding of the original values. This allowed them to obtain a ${\mathcal{O}}(r\,{\rm polylog}\,n)$-space structure that can answer ${\text{\rm SA}}$/${\text{\rm ISA}}$ and ${\text{\rm LCP}}$ queries in ${\mathcal{O}}(\log n)$ time.
The structure described below is slightly different than the original index proposed by Gagie et al. [@GagieNP17]. Rather than compressing the differentially-encoded suffix array, we directly exploit the structure of the array. It can be thought of as a multi-ary block tree [@DCC2015] modified to work with arrays indexed in “lex-order” instead of the original “text-order”. Our data structure matches the space and query time of [@GagieNP17], but we additionally show how to achieve a trade-off between space and query time. In particular, we achieve ${\mathcal{O}}(\log n/\log \log n)$ query time in ${\mathcal{O}}(r\,{\rm polylog}\, n)$ space.
Data Structure {#sec:small-space-sa-support}
--------------
Suppose we are given RLBWT of size $r$ for text $T[1..n]$. The data structure is parametrized by an integer parameter $\tau>1$. For simplicity, we assume that $r$ divides $n$ and that $n/r$ is a power of $\tau$. The data structure is organized into $\log_{\tau}(n/r)$ *levels*. The main idea is, for every level, to store $2\tau$ pointers for each BWT-run boundary. The purpose of pointers is to reduce the ${\text{\rm SA}}$ query near the associated run boundary into ${\text{\rm SA}}$ query at a position that is closer (by at least a factor of $\tau$) to some (usually different) run boundary. Level controls the allowed proximity of the query. At the last level, the ${\text{\rm SA}}$ value at each run boundary is stored explicitly.
More precisely, for $1 \leq k \leq \log_{\tau}(n/r)$, let $b_k=n/(r\tau^k)$ and let ${\text{\rm BWT}}[b..e]$ be one of the runs in the ${\text{\rm BWT}}$. Consider $2\tau$ non-overlapping consecutive blocks of size $b_k$ evenly spread around position $b$, i.e., ${\text{\rm BWT}}[b+ib_k..b+(i+1)b_k-1]$, $i=-\tau,\ldots,\tau-1$. For each block ${\text{\rm BWT}}[s..t]$ we store the smallest $d$ (called *LF-distance*) such that there exists at least one $i\in[s..t]$ such that ${\text{\rm LF}}^d[i]$ is the beginning of the run in the ${\text{\rm BWT}}$ of ${T}$ (note that it is possible that $d=0$). With each block we also store the value ${\text{\rm LF}}^d[s]$ (called *LF-shortcut*), both as an absolute value in $[1..n]$ and as a pointer to the BWT-run that contains it. Due to the simple generalization of Lemma \[lm:lf-in-run\], this allows us to compute ${\text{\rm LF}}^d[i]$ for *any* $i\in[s..t]$. At each level, we store $2\tau$ integers for each of $r$ BWT runs thus in total we store ${\mathcal{O}}(r\tau\log_{\tau}(n/r))$ words.
To access ${\text{\rm SA}}[i]$ we proceed as follows. Assume first that $i$ is not more than $n/r$ positions from the closest run boundary. We first find the ${\text{\rm BWT}}$ run that contains $i$. We then follow the ${\text{\rm LF}}$-shortcuts starting at level 1 down to the last level. After every step, the distance to the closest run boundary is reduced by a factor $\tau$. Thus, after $\log_{\tau}(n/r)$ steps the current position is equal to boundary $b$ of some run ${\text{\rm BWT}}[b..e]$. Let $d_{\rm sum}$ denote the total lengths of ${\text{\rm LF}}$-distances of the used shortcuts. Since ${\text{\rm SA}}[b]$ is stored we can now answer the query as ${\text{\rm SA}}[i]={\text{\rm SA}}[b]+d_{\rm sum}$. To handle positions further than $n/r$ from the nearest run boundary, we add a lookup table $LT[1..r]$ such that $LT[i]$ stores the ${\text{\rm LF}}$-shortcut and ${\text{\rm LF}}$-distance for block ${\text{\rm BWT}}[(i-1)(n/r)+1..i(n/r)]$. The query time is ${\mathcal{O}}(\log_{\tau}(n/r))$, since blocks in the same level have the same length and hence at each level we spend ${\mathcal{O}}(1)$ time to find the pointer to the next level. Note that the lookup table eliminates the initial search of run containing $i$.
The above data structure can be generalized to extract segments of ${\text{\rm SA}}[p..p+\ell-1]$, for any $p$ and $\ell$, faster than $\ell$ single ${\text{\rm SA}}$-accesses, that would cost ${\mathcal{O}}(\ell \log_{\tau}(n/r))$. The main modification is that at level $k$ we instead consider $4\tau-1$ blocks of size $b_k$, evenly spread around position $b$, each overlapping the next by exactly $b_k/2$ symbols, i.e., ${\text{\rm BWT}}[b+ib_k/2..b+(i+2)b_k/2-1]$, $i=-2\tau,\ldots,2(\tau-1)$. This guarantees that any segment-access to ${\text{\rm SA}}$ of length at most $b_k/2$ at level $k$ can be transformed into the segment-access at level $k+1$. We also truncate the data structure at level $k$ where $k$ is the smallest integer with $b_{k} < \log_{\tau}(n/r)$. At that level we store a segment of $2\log_{\tau}(n/r)$ ${\text{\rm SA}}$ values around each BWT run. These values take ${\mathcal{O}}(r \log_{\tau}(n/r))$ space, and hence the two modifications do not increase the space needed by the data structure. This way we can extract ${\text{\rm SA}}[p..p+\alpha-1]$, where $\alpha=\log_{\tau}(n/r)$ in ${\mathcal{O}}(\alpha)$ time, and consequently a segment ${\text{\rm SA}}[p..p+\ell-1]$ in ${\mathcal{O}}((\ell/\alpha+1)\alpha)={\mathcal{O}}(\ell+\log_{\tau}(n/r))$ time.
\[thm:sa\] Assume that ${\text{\rm BWT}}$ of ${T}[1..n]$ consist of $r$ runs. For any integer $\tau{>}1$, there exists a data structure of size ${\mathcal{O}}(r
\tau \log_{\tau}(n/r))$ that, for any $p\in[1..n]$ and $\ell\geq 1$, can compute ${\text{\rm SA}}[p..p+\ell-1]$ in ${\mathcal{O}}(\ell+\log_{\tau}(n/r))$ time.
For $\tau=2$ the above data structure matches the space and query time of [@GagieNP17]. For $\tau=\log^{\epsilon}n$, where $\epsilon>0$ is an arbitrary constant it achieves ${\mathcal{O}}(r \log^{\epsilon} n
\log(n/r))$ space and ${\mathcal{O}}(\log n / \log \log n)$ query time. Finally, for $\tau=(n/r)^{\epsilon}$ it achieves ${\mathcal{O}}(r^{1-\epsilon}n^{\epsilon})$ space and ${\mathcal{O}}(1)$ time query. In particular, if $r=o(n)$ the data structure takes $o(n)$ space and is able to access (any segment of) ${\text{\rm SA}}$ in optimal time.
Construction Algorithm
----------------------
Assume we are given the run-length compressed ${\text{\rm BWT}}$ of ${T}[1..n]$ of size $r$. Consider any block ${\text{\rm BWT}}[s..t]$. Let $d$ be the corresponding ${\text{\rm LF}}$-distance and let ${\text{\rm LF}}^d[i]=b$ for some $i\in[s..t]$ be the beginning of a BWT-run $[b..e]$. We observe that this implies ${\text{\rm LCP}}[b]$ is irreducible and ${\text{\rm LCP}}[b] \geq d$.
We start by augmenting the RLBWT with the ${\text{\rm SA}}$/${\text{\rm ISA}}$ support from Section \[sec:sa-support\] using $\tau_1=\log^2 n$. This, by Theorem \[thm:sa-support\], takes ${\mathcal{O}}\left(n/\log n + r\log^5
n\right)$ time and ${\mathcal{O}}(n/\log^2 n + r\log^2 n)$ working space. The resulting structure needs ${\mathcal{O}}(r + n/\log^2 n)$ space and allows answering ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries in ${\mathcal{O}}(\log^3 n)$ time.
Consider now the sorted sequence $Q$ containing every position $j$ in $T$ such that ${\text{\rm PLCP}}[j]$ is irreducible. Such list can be obtained by computing value ${\text{\rm SA}}[b]$ for every BWT run $[b..e]$ and sorting the resulting values. Computing the list $Q$ takes ${\mathcal{O}}(r \log^3 n)$ time and ${\mathcal{O}}(r)$ working space. The list itself is stored in plain form using ${\mathcal{O}}(r$) space. Next, for any irreducible value ${\text{\rm PLCP}}[j]$ we compute, for any $t=1,\ldots,\lfloor \ell'/\tau_2
\rfloor$ a pair containing ${\text{\rm ISA}}[j+t\tau_2]$ (as key) and $t\tau_2$ (as value), where $\tau_2=\log^4 n$, and $\ell'$ is the distance between $j$ and its successor in $Q$. Since the sum of $\ell'$ values is ${\mathcal{O}}(n)$, computing all pairs takes ${\mathcal{O}}(\log^3 n \cdot (r + n
/ \tau_2))={\mathcal{O}}(n/\log n+r \log^3 n)$ time and ${\mathcal{O}}(n/\tau_2)={\mathcal{O}}(n/\log^4 n)$ working space. The resulting pairs need ${\mathcal{O}}(n / \log^4 n)$ space.
We then sort all the computed pairs by the keys and build a static RMQ data structure over the associated values. This can be done in ${\mathcal{O}}\left(n/\tau_2\right)={\mathcal{O}}(n/\log^4 n)$ time and space so that an RMQ query takes ${\mathcal{O}}(\log n)$ time (using static balanced BST).
Having the above samples augmented with the RMQ allows us to compute ${\text{\rm LF}}$-shortcuts as follows. Let ${\text{\rm BWT}}[s..t]$ be one of the blocks. We perform $\tau_2$ ${\text{\rm LF}}$-steps on position $s$. In step $\Delta$ we first check in ${\mathcal{O}}(\log r)$ time whether the block $[{\text{\rm LF}}^{\Delta}[s]..{\text{\rm LF}}^{\Delta}[s]+(t-s)]$ contains a boundary of a BWT-run. If yes, then we found the ${\text{\rm LF}}$-distance and terminate the procedure. Otherwise, in ${\mathcal{O}}(\log n)$ we compute the minimal value $d_{\min}$ and its position for the block $[{\text{\rm LF}}^{\Delta}[s]..{\text{\rm LF}}^{\Delta}[s]+(t-s)]$ using the RMQ structure (if the block is empty we skip this step). We call $d_{\min}+\Delta$ the *candidate value*. From the way we computed the pairs, the minimum candidate value is equal to the ${\text{\rm LF}}$-distance of ${\text{\rm BWT}}[s..t]$. It is easy to extend this procedure to also return the ${\text{\rm LF}}$-shortcut.
Thus, the ${\text{\rm LF}}$-shortcut for any block can be computed in ${\mathcal{O}}(\tau_2 \log n)={\mathcal{O}}(\log^5 n)$ time. Over all blocks (and including the shortcuts for the lookup table $LT[1..r]$) this takes ${\mathcal{O}}(r \tau \log_{\tau}(n/r) \log^5 n)={\mathcal{O}}(r \tau \log^6 n)$ time. Finally, computing segments of ${\text{\rm SA}}$ values at the last level (after truncating the tree) takes ${\mathcal{O}}(r \log_{\tau}(n/r) \log^3 n)$ time.
\[thm:sa-construct-from-RLBWT\] Given RLBWT of size $r$ for text ${T}[1..n]$ we can build the data structure from Theorem \[thm:sa\] in ${\mathcal{O}}(n/\log n + r\tau\log^6
n)$ time and ${\mathcal{O}}(n/\log^2 n + r(\tau\log_{\tau}(n/r)+ \log^2 n))$ working space.
By combining with Theorem \[thm:bwt\] we obtain the following theorem.
\[thm:sa-construct-from-text\] Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\,{\leq}\,n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words we can build the data structure from Theorem \[thm:sa\] in ${\mathcal{O}}(n/\log_{\sigma} n + r(\tau\log^6 n+\log^{7} n))$ time and ${\mathcal{O}}(n/\log_{\sigma} n + r(\tau\log_{\tau}(n/r)+\log^{5}n))$ working space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$.
Construction of LZ77 Parsing {#sec:lz77}
============================
In this section, we show how to use the techniques introduced in previous sections to obtain a fast and space-efficient LZ77 factorization algorithm for highly repetitive strings.
Definitions
-----------
The LZ77 factorization [@LZ77] uses the notion of the [*longest previous factor*]{} (LPF). The LPF at position $i$ (denoted ${\text{\rm LPF}}[i]$) in ${T}$ is a pair $(p_i,\ell_i)$ such that, $p_i < i$, ${T}[p_i..p_i+\ell_i-1] = {T}[i..i+\ell_i-1]$ and $\ell_i>0$ is maximized. In other words, ${T}[i..i+\ell_i-1]$ is the longest prefix of ${T}[i..n]$ which also occurs at some position $p_i < i$ in ${T}$. If ${T}[i]$ is the leftmost occurrence of a symbol in ${T}$ then such a pair does not exist. In this case we define $p_i = {T}[i]$ and $\ell_i
= 0$. Note that there may be more than one potential $p_i$, and we do not care which one is used.
The LZ77 factorization (or LZ77 parsing) of a string ${T}$ is then just a greedy, left-to-right parsing of ${T}$ into longest previous factors. More precisely, if the $j^{\mbox{{\scriptsize th}}}$ LZ factor (or [*phrase*]{}) in the parsing is to start at position $i$, then we output $(p_i,\ell_i)$ (to represent the $j^{\mbox{{\scriptsize
th}}}$ phrase), and then the $(j+1)^{\mbox{{\scriptsize th}}}$ phrase starts at position $i+\ell_i$, unless $\ell_i = 0$, in which case the next phrase starts at position $i+1$. For the example string ${T}= zzzzzipzip$, the LZ77 factorization produces: $$(z,0),(1,4),(i,0),(p,0),(5,3).$$ We denote the number of phrases in the LZ77 parsing of ${T}$ by $z$. The following theorem shows that LZ77 parsing can be encoded in ${\mathcal{O}}(n \log \sigma)$ bits.
\[thm:lz77-size\] The number of phrases $z$ in the LZ77 parsing of a text of $n$ symbols over an alphabet of size $\sigma$ is ${\mathcal{O}}(n/\log_{\sigma}n)$.
The LPF pairs can be computed using *next and previous smaller values* (NSV/PSV) defined as $$\begin{aligned}
{\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}[i] &= \min \{ j\in [i+1..n] \mid {\text{\rm SA}}[j] < {\text{\rm SA}}[i]\},\\
{\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}[i] &= \max \{ j\in [1..i-1] \mid {\text{\rm SA}}[j] < {\text{\rm SA}}[i]\} .\end{aligned}$$ If the set on the right hand side is empty, we set the value to $0$. We further define $$\begin{aligned}
{\mbox{$\text{\rm NSV}_{\text{\rm text}}$}}[i] &= {\text{\rm SA}}[{\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}[{\text{\rm ISA}}[i]]],\\
{\mbox{$\text{\rm PSV}_{\text{\rm text}}$}}[i] &= {\text{\rm SA}}[{\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}[{\text{\rm ISA}}[i]]].\end{aligned}$$ If ${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}[{\text{\rm ISA}}[i]]=0$ (${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}[{\text{\rm ISA}}[i]]=0$) we set ${\mbox{$\text{\rm NSV}_{\text{\rm text}}$}}[i]=0$ (${\mbox{$\text{\rm PSV}_{\text{\rm text}}$}}[i]=0$).
The usefulness of the NSV/PSV values is summarized by the following lemma.
\[lm:psv-nsv\] For $i\in[1..n]$, let $i_{nsv}={\mbox{$\text{\rm NSV}_{\text{\rm text}}$}}[i]$, $i_{psv}={\mbox{$\text{\rm PSV}_{\text{\rm text}}$}}[i]$, $\ell_{nsv} = {\text{\rm lcp}}(i,i_{nsv})$ and $\ell_{psv} =
{\text{\rm lcp}}(i,i_{psv})$. Then $${\text{\rm LPF}}[i] = \left\{
\begin{array}{ll}
(i_{nsv},\ell_{nsv}) & \text{ if\enspace} \ell_{nsv} > \ell_{psv}, \\
(i_{psv},\ell_{psv}) & \text{ if\enspace} \ell_{psv} =
\max(\ell_{nsv},\ell_{psv}) > 0, \\
({T}[i],0) & \text{ if\enspace} \ell_{nsv} = \ell_{psv} = 0.
\end{array}
\right.$$
Algorithm Overview
------------------
The general approach of our algorithm follows the lazy LZ77 factorization algorithms of [@kkp-jea]. Namely, we opt out from computing all ${\text{\rm LPF}}$ values and instead compute ${\text{\rm LPF}}[j]$ only when there is an LZ factor starting at position $j$.
Suppose we have already computed the parsing of ${T}[1..j-1]$. To compute the factor starting at position $j$ we first query $i={\text{\rm ISA}}[j]$. We then compute (using a small-space data structure introduced next) values $i_{\rm nsv}={\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}[i]$ and $i_{\rm
psv}={\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}[i]$. By Lemma \[lm:psv-nsv\] it then suffices to compute the lcp of ${T}[j..n]$ and each of the two suffixes starting at positions ${\text{\rm SA}}[i_{\rm psv}]$ and ${\text{\rm SA}}[i_{\rm nsv}]$.
It is easy to see that the total length of computed lcps will be ${\mathcal{O}}(n)$, since after each step we increase $j$ by the longest of the two lcps. To perform the lcp computation efficiently we will employ the technique from Section \[sec:plcp\] which allows comparing multiple symbols at a time. This will allow us to spend ${\mathcal{O}}(z\,{\rm polylog}\,n + n/\log n)$ time in the lcp computation. The problem is thus reduced to being able to quickly answer ${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}/{\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}$ queries.
Computing NSV/PSV Support for SA {#sec:nsv-psv}
--------------------------------
Assume that we are given RLBWT of size ${\mathcal{O}}(r)$ for text ${T}[1..n]$. We will show how to quickly build a small-space data structure that, given any $i\in[1..n]$ can compute ${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}[i]$ or ${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}[i]$ in ${\mathcal{O}}({\rm polylog}\,n)$ time.
We split ${\text{\rm BWT}}[1..n]$ into blocks of size $\tau=\Theta({\rm
polylog}\,n)$ and for each $j\in[1..n/\tau]$ we compute the minimum value in ${\text{\rm SA}}[(j{-}1)\tau{+}1..j\tau]$ together with its position. We then build a balanced binary tree over the array of minimas and augment each internal node with the minimum value in its subtree. This allows, for any $j\in[1..n/\tau]$, and any value $x$, to find the maximal (resp. minimal) $j'<j$ (resp. $j'>j$) such that ${\text{\rm SA}}[(j'{-}1)\tau{+}1..j'\tau]$ contains a value smaller than $x$. At query time we first scan the ${\text{\rm SA}}$ positions preceding or following the query position $i\in[1..n]$ inside the block containing $i$. If there is no value smaller than ${\text{\rm SA}}[i]$, we use the RMQ to find the closest block with a value smaller than ${\text{\rm SA}}[i]$. To finish the query it then suffices to scan the ${\text{\rm SA}}$ values inside that block. It takes ${\mathcal{O}}(\log^3 n)$ time to compute ${\text{\rm SA}}$ value (Theorem \[thm:sa-support\]), hence answering a single ${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}$/${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}$ query will take ${\mathcal{O}}(\tau \log^3 n)$.
To compute the minimum for each of the size-$\tau$ blocks of ${\text{\rm SA}}$ we observe that, up to a shift by a constant, there is only $r\tau$ different blocks. More specifically, consider a block ${\text{\rm SA}}[(j{-}1)\tau{+}1..j\tau]$. Let $k$ be the smallest integer such that for some $t\in[(j{-}1)\tau+1..j\tau]$, ${\text{\rm LF}}^k[t]$ is the beginning of a run in ${\text{\rm BWT}}$. It is easy to see that, due to Lemma \[lm:lf-in-run\], ${\text{\rm SA}}[(j{-}1)\tau{+}1..j\tau]=k+{\text{\rm SA}}[{\text{\rm LF}}^k[j\tau]{-}\tau{+}1..{\text{\rm LF}}^k[j\tau]]$, in particular, the equality holds for the minimum element. Thus, it suffices to precompute the minimum value and its position for each of the $r\tau$ size-$\tau$ blocks intersecting a boundary of a ${\text{\rm BWT}}$-run. This takes ${\mathcal{O}}(r\tau \log^3 n)$ time and ${\mathcal{O}}(r\tau)$ working space. The resulting values need ${\mathcal{O}}(r\tau)$ space.
It thus remains to compute the “${\text{\rm LF}}$-distance” for each of the $n/\tau$ blocks of ${\text{\rm SA}}$, i.e., the smallest $k$ such that for at least one position $t$ inside the block, ${\text{\rm LF}}^k[t]$ is the beginning of a ${\text{\rm BWT}}$-run. To achieve this we utilize the technique used in Section \[sec:rlcsa\]. There we presented a data structure of size ${\mathcal{O}}(r + n/\log^2 n + n/\tau_2)$ that can be built in ${\mathcal{O}}(n/\log
n + r\log^5 n + (n \log^3 n) / \tau_2)$ time and ${\mathcal{O}}(n/\log^2 n +
r\log^2 n + n/\tau_2)$ working space, and is able to compute the ${\text{\rm LF}}$-shortcut for any block $[s..t]$ in ${\text{\rm SA}}$ in ${\mathcal{O}}(\tau_2 \log
n)$ time.
\[thm:nsv-psv\] Given RLBWT of size $r$ for text ${T}[1..n]$, we can build a data structure of size ${\mathcal{O}}(r+n/\log^2 n)$ that can answer ${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}$/${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}$ queries in ${\mathcal{O}}(\log^{9} n)$ time. The data structure can be built in ${\mathcal{O}}(n/\log n+r\log^{9}n)$ time and ${\mathcal{O}}(n/\log^2 n + r\log^6 n)$ working space.
We start by augmenting the RLBWT with ${\text{\rm SA}}$/${\text{\rm ISA}}$ support. This takes (Theorem \[thm:sa-support\]) ${\mathcal{O}}(n/\log n+r\log^5 n)$ time and ${\mathcal{O}}(n/\log^2 n + r\log^2n)$ working space. The resulting data structure takes ${\mathcal{O}}(r+n/\log^2 n)$ space and answers ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries in ${\mathcal{O}}(\log^3 n)$ time.
To achieve the ${\mathcal{O}}(n/\log n)$ term in the construction time for the structure from Section \[sec:rlcsa\] we set $\tau_2=\log^4
n$. Then, computing the ${\text{\rm LF}}$-shortcut for any block in ${\text{\rm SA}}$ takes ${\mathcal{O}}(\log^5 n)$ time. Since we have $n/\tau$ blocks to query, we set $\tau=\log^6 n$ to obtain ${\mathcal{O}}(n/\log n)$ total query time. Answering a single ${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}$/${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}$ query then takes ${\mathcal{O}}(\tau \log^3 n)={\mathcal{O}}(\log^{9} n)$.
The RMQ data structure built on top of the minimas of the blocks of ${\text{\rm SA}}$ takes ${\mathcal{O}}(n/\tau)={\mathcal{O}}(n/\log^6 n)$ space, hence the space of the final data structure is dominated by ${\text{\rm SA}}$/${\text{\rm ISA}}$ support taking ${\mathcal{O}}(r+n/\log^2 n)$ words.
The construction time is split between precomputing the minimas in each of the $r\tau$ blocks crossing boundaries of ${\text{\rm BWT}}$-runs in ${\mathcal{O}}(r\tau \log^3 n)={\mathcal{O}}(r \log^{9}n)$ time, and other steps introducing term ${\mathcal{O}}(n/\log n)$.
The working space is maximized when building the ${\text{\rm SA}}$/${\text{\rm ISA}}$ support and during the precomputation of minimas in each of the $r\tau$ blocks, for a total of ${\mathcal{O}}(n/\log^2 n + r\log^6 n)$.
Algorithm Summary
-----------------
\[thm:lz77\] Given RLBWT of size $r$ of ${T}[1..n]$, the LZ77 factorization of ${T}$ can be computed in ${\mathcal{O}}(n/\log n + r \log^{9}n + z\log^{9}n)$ time and ${\mathcal{O}}(n/\log^2 n + z + r\log^8 n)={\mathcal{O}}(n/\log_{\sigma}n +
r\log^8 n)$ working space, where $z$ is the size of the LZ77 parsing of ${T}$.
We start by augmenting the RLBWT with the ${\text{\rm SA}}$/${\text{\rm ISA}}$ support from Section \[sec:sa-support\] using $\tau_1=\log^2 n$. This, by Theorem \[thm:sa-support\], takes ${\mathcal{O}}\left(n/\log n + r\log^5
n\right)$ time and ${\mathcal{O}}(n/\log^2 n + r\log^2 n)$ working space. The resulting structure needs ${\mathcal{O}}(r + n/\log^2 n)$ space and answers ${\text{\rm SA}}$/${\text{\rm ISA}}$ queries in ${\mathcal{O}}(\log^3 n)$ time.
Next, we initialize the data structure supporting the ${\mbox{$\text{\rm PSV}_{\text{\rm lex}}$}}$/${\mbox{$\text{\rm NSV}_{\text{\rm lex}}$}}$ queries from Section \[sec:nsv-psv\]. By Theorem \[thm:nsv-psv\] the resulting data structure needs ${\mathcal{O}}(r+n/\log^2 n)$ space and answers queries in ${\mathcal{O}}(\log^{9}
n)$ time. The data structure can be built in ${\mathcal{O}}(n/\log
n+r\log^{9}n)$ time and ${\mathcal{O}}(n/\log^2 n + r\log^6 n)$ working space. Over the course of the whole algorithm, we ask ${\mathcal{O}}(z)$ queries hence in total we spend ${\mathcal{O}}(z\log^{9}n)$ time.
Lastly, we compute $\tau_3$-runs and their names using the technique introduced in Section \[sec:irreducible\] for $\tau_3=\log^4 n$. This takes ${\mathcal{O}}(\tau_3^2r \log (\tau_3 r))={\mathcal{O}}(r \log^{9} n)$ time and ${\mathcal{O}}(\tau_3^2 r)={\mathcal{O}}(r \log^{8} n)$ working space (see the proof of Theorem \[thm:plcp\]). The names need ${\mathcal{O}}(\tau_3
r)={\mathcal{O}}(r \log^4 n)$ space. The names allow, given any $j_1,j_2\in
[1..n]$, to compute $\ell={\text{\rm lcp}}(j_1,j_2)$ in ${\mathcal{O}}\left(\log^3
n(1+\ell/\tau_3)+\tau_3 \log n\right)={\mathcal{O}}\left(\log^5 n + \ell /
\log n\right)$ time. Thus, over the course of the whole algorithm we will spend ${\mathcal{O}}(z \log^5 n + n/\log n)$ time computing lcp values.
By combining with Theorem \[thm:plcp\] we obtain the following result.
\[thm:lz77-2\] Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\leq n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words, we can compute the LZ77 factorization of ${T}$ in ${\mathcal{O}}(n/\log_{\sigma}n+r\log^{9}n+z\log^{9}n)$ time and ${\mathcal{O}}(n/\log_{\sigma}n+r\log^8 n)$ working space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$ and $z$ is the size of the LZ77 parsing of ${T}$.
Since $z={\mathcal{O}}(r \log n)$ [@GNPlatin18], the above algorithm achieves ${\mathcal{O}}(n/\log_{\sigma}n)$ runtime and working space when $n/r\in\Omega({\rm polylog}\,n)$.
Construction of Lyndon Factorization
====================================
In this section, we show another application of our techniques. Namely, we show that we can obtain a fast and space-efficient construction of Lyndon factorization for highly repetitive strings.
Definitions
-----------
A string $S$ is called a *Lyndon word* if $S$ is lexicographically smaller than all its non-empty proper suffixes. The *Lyndon factorization* (also called *Standard factorization*) of a string $T$ is its unique (see [@CFL58]) factorization $T=f_1^{e_1} \cdots f_m^{e_m}$ such that each $f_i$ is a Lyndon word, $e_{i} \geq 1$, and $f_{i} \succ f_{i+1}$ for all $1 \leq
i < m$. We call each $f_i$ a *Lyndon factor* of $T$, and each $F_i= f_i^{e_i}$ a *Lyndon run* of $T$. The size of the Lyndon factorization is $m$, the number of distinct Lyndon factors, or equivalently, the number of Lyndon runs.
Each Lyndon run can be encoded as a triple of integers storing the boundaries of some occurrence of $f_i$ in $T$ and the exponent $e_i$. Since, for any string, it holds $m<2z$ [@KarkkainenKNPS17] and $z={\mathcal{O}}(n/\log_{\sigma}n)$ [@phdjuha], where $z$ is the number of phrases in the LZ77 parsing, it follows that Lyndon factorization can be stored in ${\mathcal{O}}(n \log \sigma)$ bits.
Algorithm Overview
------------------
Our algorithm utilizes many of the algorithms from the long line of research on algorithms operating on compressed representations such as grammars or LZ77 parsing:
- Furuya et al. [@CPM2018] have shown that given an SLP (i.e., a grammar in Chomsky normal form generating a single string) of size $g$ generating string $T$ of length $n$, the Lyndon factorization of $T$ can be computed in ${\mathcal{O}}(P(g, n) +
Q(g, n)\,g \log \log n)$ time and ${\mathcal{O}}(g \log n + S(g, n))$ space, where $P(g, n)$, $S(g, n)$, $Q(g, n)$ are respectively the pre-processing time, space, and query time of a data structure for longest common extensions (LCE) queries on SLPs. The LCE query, given two positions $i$ and $j$ in the string $T$, returns ${\text{\rm lcp}}(i,j)$, i.e., the length of the longest common prefix of suffixes $T[i..n]$ and $T[j..n]$.
- On the other hand, Nishimoto et al. [@NishimotoIIBT16 Thm 3] have shown how, given an SLP of size $g$ generating string $T$ of length $n$, to construct an LCE data structure in ${\mathcal{O}}(g \log \log g \log n \log^{*}n)={\mathcal{O}}(g \log^3
n)$ time and ${\mathcal{O}}(g \log^{*}n + z \log n \log^{*}n)={\mathcal{O}}(g
\log^2 n)$ space, where $z$ is the size of LZ77 parsing of $T$. The resulting data structure answers a query ${\rm LCE}(i,j)$ in ${\mathcal{O}}(\log n + \log \ell \log^{*}n)={\mathcal{O}}(\log^2 n)$ time, where $\ell={\text{\rm lcp}}(i,j)$. Thus, they achieve $P(g, n)={\mathcal{O}}(g \log^3
n)$, $S(g, n)={\mathcal{O}}(g \log^2 n)$, and $Q(g, n)={\mathcal{O}}(\log^2
n)$. More recently, I [@I17 Thm 2] improved (using different techniques) this to $P(g, n)={\mathcal{O}}(g \log (n/g))$, $S(g,
n)={\mathcal{O}}(g+z\log(n/z))$, and $Q(g, n)={\mathcal{O}}(\log n)$.
- Finally, Rytter [@Rytter03 Thm 2] have shown how, given the LZ77 parsing of string $T$ of length $n$, to convert it into an SLP of size $g={\mathcal{O}}(z \log n)$ in ${\mathcal{O}}(z \log n)$ time and ${\mathcal{O}}(z \log n)$ working space.
The above pipeline leads to a fast and space-efficient algorithm for Lyndon factorization, assuming the compressed representation (such as SLP or LZ77) of text is given *a priori*. It still, however, needs $\Omega(n)$ time if we take into account the time to compute LZ77 or a small grammar using the previously fastest known algorithms. Section \[sec:lz77\] completes this line of research by providing fast and space-efficient construction of the initial component (LZ77 parsing).
Given string $T[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\leq n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words of space, we can compute the Lyndon factorization of $T$ in ${\mathcal{O}}(n/\log_{\sigma}n+r \log^9 n + z\log^9 n)$ time and ${\mathcal{O}}(n/\log_{\sigma}n + r\log^8 n + z\log^2 n)$ working space.
We start by computing the LZ77 parsing using Theorem \[thm:lz77-2\]. This takes ${\mathcal{O}}(n/\log_{\sigma}n+r\log^9 n + z\log^9 n)$ time and ${\mathcal{O}}(n/\log_{\sigma}n+r \log^8 n)$ space. The resulting parsing, by Theorem \[thm:lz77-size\], takes ${\mathcal{O}}(n/\log_{\sigma}n)$ space.
We then use the Rytter’s [@Rytter03] conversion from LZ77 to SLP of size $g={\mathcal{O}}(z \log n)$ that takes ${\mathcal{O}}(z \log n)$ time and ${\mathcal{O}}(z \log n)$ working space. The resulting SLP is then turned into an LCE data structure of I [@I17]; this takes ${\mathcal{O}}(g \log
(n/g))={\mathcal{O}}(z \log^2 n)$ time and ${\mathcal{O}}(g + z\log(n/z))={\mathcal{O}}(z
\log n)$ working space. The resulting LCE data structure takes ${\mathcal{O}}(z \log n)$ space. Finally, we plug this data structure into the algorithm of Furuya [@CPM2018] which gives us the Lyndon factorization in ${\mathcal{O}}(z \log^3 n)$ time and ${\mathcal{O}}(z \log^2 n)$ working space. Thus, the whole pipeline is dominated (in time and space) by the construction of LZ77 parsing.
Similarly as in Section \[sec:lz77\], since $z={\mathcal{O}}(r \log
n)$ [@GNPlatin18], the above algorithm achieves ${\mathcal{O}}(n/\log_{\sigma}n)$ runtime and working space when $n/r\in\Omega({\rm polylog}\,n)$.
Solutions to Textbook Problems
==============================
Lastly, we show how to utilize the techniques presented in this paper to efficiently solve some “textbook” string problems on highly repetitive inputs. Their solution usually consists of computing ${\text{\rm SA}}$ or ${\text{\rm LCP}}$ and performing some simple scan/traversal (e.g., computing the longest repeating substring amounts to finding the maximal value in the ${\text{\rm LCP}}$ array and hence by Theorem \[cor:plcp\] it can be solved efficiently for highly repetitive input), but in some cases requires explicitly applying some of the observations from previous sections. Next, we show two examples of such problems.
Number of Distinct Substrings
-----------------------------
The number $d$ of distinct substrings of a string $T$ of length $n$ is given by the formula $$d = \frac{n(n+1)}{2}-\sum_{i=1}^{n}{\text{\rm LCP}}[i] .$$
Suppose we are given a (sorted) list $(i_1, \ell_1), \ldots, (i_r,
\ell_r)$ of irreducible lcp values (i.e., ${\text{\rm PLCP}}[i_k]=\ell_k$) of string $T$. Since all other lcp values can be derived from this list using Lemma \[lm:reducible\], we can rewrite the above formula (letting $i_{r+1}=n+1$) as: $$d = \frac{n(n+1)}{2}-\sum_{k=1}^{r}f(\ell_k, i_{k+1}-i_{k}) ,$$ where $$f(v,d) = \left\{
\begin{array}{ll}
\frac{v(v+1)}{2} & \text{ if }v < d,\\
d(v-d)+\frac{d(d+1)}{2} & \text{ otherwise. }
\end{array}
\right.$$
Thus, by Theorem \[cor:plcp\] we immediately obtain the following result.
Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma
\leq n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words, we can compute the number $d$ of distinct substrings of $T$ in ${\mathcal{O}}(n/\log_{\sigma} n + r \log^{11} n)$ time and ${\mathcal{O}}(n/\log_{\sigma} n + r\log^{10}n)$ space, where $r$ is the number of runs in the ${\text{\rm BWT}}$ of ${T}$.
Longest Substring Occurring k Times
-----------------------------------
Suppose that we want to find the length $\ell$ of the longest substring of $T$ that occurs in $T$ at least $2 \leq k={\mathcal{O}}(1)$ times. This amounts to computing $$\ell=\max_{i=1}^{n-k+2}\min_{j=0}^{k-2}{\text{\rm LCP}}[i+j].$$
For $k=2$ the above formula can be evaluated by only looking at irreducible lcp values, i.e., using the definition from the previous section, $\ell=\max_{i=1}^{r}\ell_i$. For $k>2$, this does not work, since we have to inspect blocks of LCP values of size $k-1$ in “lex-order”. We instead utilize observations from previous sections. More precisely, recall from Section \[sec:lz77\] that for any $\tau$, up to a shift by a constant, there is only $r\tau$ different blocks of size $\tau$ in ${\text{\rm SA}}$, i.e., for any block block ${\text{\rm SA}}[i..i{+}\tau{-}1]$ there exists $k$ such that ${\text{\rm SA}}[i..i{+}\tau{-}1]=k+{\text{\rm SA}}[j..j{+}\tau{-}1]$ and ${\text{\rm BWT}}[j..j{+}\tau{-}1]$ contains a BWT-run boundary.
We now observe that an analogous property holds for the ${\text{\rm LCP}}$ array: for any block ${\text{\rm LCP}}[i..i{+}\tau{-}1]$ there exists $k$ (the same as above) such that ${\text{\rm LCP}}[i..i{+}\tau{-}1]={\text{\rm LCP}}[j..j{+}\tau{-}1]-k$ and ${\text{\rm BWT}}[j..j{+}\tau{-}1]$ contains boundary of some BWT-run. This implies that we only need to precompute and store the minimum value inside blocks of ${\text{\rm LCP}}$ of length $k-1$ that are not further than $\tau$ positions from the closest BWT-run boundary. All other blocks of ${\text{\rm LCP}}$ can be handled using the above observation and the structure from Section \[sec:rlcsa\] for computing the ${\text{\rm LF}}$-shortcut for any block of ${\text{\rm BWT}}$. More precisely, after a suitable overlap (by at least $k$) of blocks of size $\tau=\Omega({\rm polylog}\,n)$, we can get the answer for all such blocks in ${\mathcal{O}}(n/{\rm polylog}\,n + r\,{\rm
polylog}\,n)$ time.
Given string ${T}[1..n]$ over alphabet $[1..\sigma]$ of size $\sigma\,{\leq}\, n$ encoded in ${\mathcal{O}}(n/\log_{\sigma}n)$ words, we can find the length of the longest substring occurring $\geq $ $k\,{=}\,{\mathcal{O}}(1)$ times in $T$ in ${\mathcal{O}}(n/\log_{\sigma}n+r\,{\rm
polylog}\,n)$ time and space.
Concluding Remarks
==================
An important avenue for future work is to reduce the exponent in the ${\mathcal{O}}(r\,{\rm polylog}\,n)$-term of our bounds and to determine whether the presented algorithms can be efficiently implemented in practice. Another interesting problem is to settle whether the ${\mathcal{O}}(\log n/\log \log n)$ bound obtained in Section \[sec:rlcsa\] is optimal within ${\mathcal{O}}(r\,{\rm polylog}\,n)$ space.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Tomasz Kociumaka for helpful comments and Isamu Furuya, Yuto Nakashima, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda for sharing an early version of their paper [@CPM2018].
[^1]: Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP) and by EPSRC award EP/N011163/1.
|
---
abstract: 'A celebrated result by A. Ionescu Tulcea provides a construction of a probability measure on a product space given a sequence of regular conditional probabilities. We study how the perturbations of the latter in the total variation metric affect the resulting product probability measure.'
address:
- 'Department of Computer Science, University of Oxford'
- 'Delft Institute of Applied Mathematics, TU Delft'
- 'Delft Center for Systems and Control, TU Delft'
author:
- Alessandro Abate
- Frank Redig
- Ilya Tkachev
bibliography:
- '../my\_bib.bib'
title: On the effect of perturbation of conditional probabilities in total variation
---
[ ]{}
Introduction
============
The Ionescu Tulcea extension theorem [@a1972 Section 2.7.2] states that given a sequence of stochastic kernels, there exists a unique probability measure on the product space generated by this sequence, that is a measure whose conditional probabilities equal to these kernels. Such a construction is often used in the theory of general Markov Decision Processes [@bs1978], and general Markov Chains [@r1984] in particular. Hence, it is of a certain interest to study how sensitive the resulting product measure is with respect to perturbations of the generating sequence of kernels. A possible direct application of such result concerns numerical methods, where characteristics of the original stochastic process are studied over its simpler approximations, often defined over a finite state space. Such approximations can be further regarded as a perturbation of the original sequence of kernels [@ta2013] which connects to the original problem.
Here we specifically focus on the metric between kernels and measures given by the total variation norm. Given the pairwise distances between corresponding transition kernels in this metric, we are interested in bounds on the distance between the resulting product measures. A similar study was given in [@rr2013] which used the Borel assumption, that is it assumed that spaces involved are (standard) Borel spaces. However, the bounds obtained in [@rr2013] grow linearly with the cardinality of the sequence and hence are not tight: recall that the total variation distance between two probability measures is always bounded from above by $2$.
In this paper we provide two results: the first generalizes linear bounds in [@rr2013] to the case of arbitrary measurable spaces, whereas the second result shows that under the Borel assumption it is possible to derive much sharper bounds, that appear to be precise in some special cases – e.g. in case of independent products of measures.
The rest of the paper is structured as follows. Section \[sec:prob\] gives a problem formulation together with statements of main results. Proofs are given in Section \[sec:proofs\], which is followed by the discussion in Section \[sec:discus\] and an enlightening example in Section \[sec:example\]. The notation is provided in Appendix, Section \[sec:appendix\].
Problem statement {#sec:prob}
=================
Let us recall the construction of the product measure given the regular conditional probabilities. First of all, we need the following notion of a product of a probability measure and a stochastic kernel which extends a more usual product of two measures.
\[prop:product\] Let $(X,\x)$ and $(Y,\y)$ be arbitrary measurable spaces. For any probability measure $\mu\in \p(X,\x)$ and any stochastic kernel $K:X\to \p(Y,\y)$ there exists a unique probability measure $\q\in \p(X\times Y,\x\otimes \y)$, denoted by $\q := \mu\otimes K$, such that $$\q(A\times B) = \int_A K(x,B)\mu(\d x)$$ for any pair of sets $A\in \x$ and $B\in \y$.
For a proof, see [@a1972 Section 2.6.2].
The construction above immediately extends to any finite sequence of spaces by induction, whereas for the countable products the following result holds true.
\[prop:I-T\] Let $\{(X_k,\x_k)\}_{k \in \N_0}$ be a family of arbitary measurable spaces and let $(\Omega_n,\f_n) = \prod_{k=0}^n(X_k,\x_k)$ be product spaces for any $n\in \bar\N_0$. For any probability measure $P^0\in \p(X_0,\x_0)$ and any sequence of stochastic kernels $(P^k)_{k\in \N}$, where $P^k:\Omega_{k-1}\to \p(X_k,\x_k)$, there exists a unique probability measure ${\mathsf{P}}\in \p(\Omega_\infty,\f_\infty)$, denoted by ${\mathsf{P}}:= \bigotimes_{k=0}^\infty P^k$, such that the finite-dimensional marginal ${\mathsf{P}}^n$ of ${\mathsf{P}}$ on the measurable space $(\Omega_n,\f_n)$ is given by ${\mathsf{P}}^n = \bigotimes_{k=0}^n P^k$ for any $n\in \N_0$.
For a proof, see [@a1972 Section 2.7.2].
In the setting of Proposition \[prop:I-T\], suppose that we are given another sequence of kernels $(\tilde P^k)_{k=0}^\infty$ and let $\tilde{\mathsf{P}}:= \bigotimes_{k=0}^\infty \tilde P^k$ be the corresponding product measure. Given the assumption that $\|P^k - \tilde P^k\| \leq c_k$ for any $k\in \N_0$ and some sequence of reals $(c_k)_{k\in \N_0}$, we study how the distance $\|{\mathsf{P}}^n - \tilde {\mathsf{P}}^n\|$ can be bounded. For the general case of arbitrary measurable spaces, the following result holds true.
\[thm:lin\] Let $\{(X_k,\x_k)\}_{k \in \N_0}$ be any family of measurable spaces and let $\tilde \x_k\subseteq \x_k$ for any $k\in \N_0$. Denote by $(\Omega_n,\f_n) = \prod_{k=0}^n (X_k,\x_k)$ and $(\Omega_n,\tilde \f_n) = \prod_{k=0}^n (X_k,\tilde \x_k)$ the corresponding product spaces for any $n\in \bar\N_0$. Let $P^0 \in \p(X_0,\x_0)$, $\tilde P^0\in \p(X_0,\tilde \x_0)$ and let kernels $P^k :\Omega_{k-1} \to \p(X_k,\x_k)$ and $\tilde P^k:\Omega_{k-1}\to \p(X_k,\tilde \x_k)$ for $k\in \N$ be $\f_{k-1}$- and $\tilde \f_{k-1}$-measurable respectively. If a sequence of reals $(c_k)_{k\in \N_0}$ is such that $\|P^k - \tilde P^k\|\leq c_k$ for all $k\in \N_0$, then for any $n\in \N_0$ it holds that $$\label{eq:thm.lin}
\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\|\leq \sum_{k=0}^n c_k.$$
\[rem:conv\] Through this paper, and in particular in the statement of Theorem \[thm:lin\], we use the following convention. If the domain of one measure is a subset of the domain of another, the total variation distance between them is taken over the smaller domain. For example, in the setting of Theorem \[thm:lin\] we have $\|P^0 - \tilde P^0\| = 2\cdot \sup_{A\in \tilde \x_0}|P^0(A) - \tilde P^0(A)|$.
As it has been mentioned in Introduction, the bounds are not tight. For example, if $c_k = c>0$ for all $k\in \N_0$ then the right-hand side of is $c\cdot n$ and diverges to infinity as $n\to\infty$, whereas the left-hand side stays bounded above by $2$. It appears, that under a rather mild assumption that all involved measurable spaces are (standard) Borel, a stronger result can be obtained.
\[thm:main\] Let $\{X_k\}_{k \in \N_0}$ be a family of Borel spaces and let $\Omega_n = \prod_{k=0}^n X_k$ be product spaces for any $n\in \bar\N_0$. Let further $P^0,\tilde P^0\in \p(X_0)$ and $P^n,\tilde P^k:\Omega_{k-1}\to \p(X_k)$ for $k\in \N$. If a sequence of reals $(c_k)_{n\in \N_0}$ is such that $\|P^k - \tilde P^k\|\leq c_k$ for all $k\in \N_0$, then for any $n\in \N_0$ it holds that $$\label{eq:thm.main}
\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\|\leq 2 - 2\prod_{k=0}^n\left(1-\frac12c_k\right).$$
The proofs of both theorems are given in the next section.
Proofs of the main results {#sec:proofs}
==========================
Proof of Theorem \[thm:lin\] {#ssec:proof.lin}
----------------------------
We prove both theorems by induction, by first studying how the perturbation of a measure and a kernel in Proposition \[prop:product\] propagates to the product measure. The following lemma provides such study in the setting of Theorem \[thm:lin\].
\[lem:lin\] Let $(X,\x)$ and $(Y,\y)$ be two measurable spaces, and let $\tilde \x \subseteq \x$ and $\tilde \y\subseteq \y$. Consider $\mu\in \p(X,\x)$ and $\tilde\mu\in \p(X,\tilde \x)$, and suppose that kernels $K:X\to \p(Y,\y)$ and $\tilde K:X\to \p(Y,\tilde \y)$ are $\x$- and $\tilde \x$-measurable, respectively. Denote by $\q := \mu\otimes K$ and $\tilde \q := \tilde\mu\otimes \tilde K$ the corresponding product measures. It holds that $$\label{eq:lem.lin}
\|\q - \tilde \q\|\leq \|\mu - \tilde\mu\| + \|K - \tilde K\|$$ where in we follow the convention in Remark \[rem:conv\].
Let the set $A\in \tilde \x\otimes \tilde\y$ be arbitrary, and denote by $A_x = \{y\in Y:(x,y)\in A\}$ the $x$-section of $A$ for any $x\in X$. It follows from [@a1972 Section 2.6.2] that $$\q(A) = \int_X K_x(A_x)\mu(\d x),\qquad \tilde\q(A) = \int_X \tilde K_x(A_x)\tilde \mu(\d x)$$ and as a result: $$\begin{aligned}
|\q(A) - \tilde \q(A)| &= \left|\int_X K_x(A_x)\mu(\d x) - \int_X \tilde K_x(A_x)\tilde \mu(\d x)\right|
\\
&\leq \left|\int_X \left(K_x(A_x) - \tilde K_x(A_x)\right)\mu(\d x)\right| + \left|\int_X \tilde K_x(A_x)(\mu - \tilde \mu)(\d x)\right|
\\
&\leq \sup_{x\in X}\sup_{B\in \tilde \y}\left|K_x(B) - \tilde K_x(B)\right| + \sup_{B\in \tilde \x}|\mu(B) - \tilde \mu(B)|
\\
&= \frac12\left(\|\mu - \tilde\mu\| + \|K - \tilde K\|\right)
\end{aligned}$$ which together with implies .
To prove Theorem \[thm:lin\] we are only left to apply the result of Lemma \[lem:lin\] by induction to $\mu = {\mathsf{P}}^n$, $\tilde \mu = \tilde {\mathsf{P}}^n$, $K = P^{n+1}$ and $\tilde K = \tilde P^{n+1}$ for $n\in \N_0$.
Proof of Theorem \[thm:main\]
-----------------------------
Again, we are going to apply induction, however in the current case the analogue of Lemma \[lem:lin\] requires a more intricate proof via the coupling techniques. Let us briefly recall some facts about the coupling. Given an arbitrary measurable space $(\Omega,\f)$, a coupling of two probability measures ${\mathsf{P}},\tilde {\mathsf{P}}\in \p(\Omega,\f)$ is a probability measure $\P\in \p(\Omega^2,\f^2)$ such that the marginals of $\P$ are given by $$\label{eq:coup}
\pi_*\P = {\mathsf{P}},\qquad \tilde\pi_*\P = \tilde {\mathsf{P}},$$ where $\pi(\omega,\tilde \omega) = \omega$ and $\tilde \pi(\omega,\tilde\omega) = \tilde \omega$ for all $(\omega,\tilde\omega)\in \Omega^2$ [@l1992 Section 1.1]. In particular, if $\Omega$ is a Borel space and $\f = \b(\Omega)$, we have the following result: $$\label{eq:coup.in}
\P(\pi = \tilde \pi) \leq \|{\mathsf{P}}\wedge\tilde{\mathsf{P}}\|,$$ The inequality is called the coupling inequality [@l1992 Section 1.2]; it holds true for any coupling measure $\P$ as per . At the same time, thanks to the fact that $\Omega$ is a Borel space, there always exists a maximal coupling: the one for which the equality holds in . As for some pairs of probability measures there may be several choices of their maximal coupling, here we focus on the $\gamma$-coupling [@l1992 Section 1.5].
Let $Z$ be a Borel space and let $\nu, \tilde \nu \in \p(Z)$ be two probability measures on it. The $\gamma$-coupling of $(\nu,\tilde \nu)$ is a measure $\gamma \in \p(Z^2)$ given by $$\gamma(\nu,\tilde \nu):= (\psi_Z)_*(\nu\wedge\tilde \nu) + 1_{[0,1)}(\|\nu\wedge\tilde\nu\|)\cdot\frac{(\nu - \tilde \nu)^+\otimes (\nu-\tilde\nu)^-}{1 - \|\nu\wedge\tilde\nu\|}$$ where $\psi_Z:Z\to Z^2$ is the diagonal map on $Z$ given by $\psi_Z:z\mapsto (z,z)$.
The following lemma is the key in the proof of Theorem \[thm:main\].
\[lem:main.1\] Let $X$ and $Y$ be two Borel spaces, and let $\mu,\tilde \mu\in \p(X)$ and $K,\tilde K:X\to \p(Y)$. Denote by $\q := \mu\otimes K$ and $\tilde \q := \tilde\mu\otimes \tilde K$ the corresponding product measures. It holds that $$\label{eq:lem.main.1}
\|\q \wedge \tilde \q\|\geq \|\mu \wedge \tilde \mu\|\cdot\inf_{x\in X}\|K_x \wedge \tilde K_x\|.$$
The proof is done via coupling of measures $\q$ and $\tilde \q$ in a sequential way.
Firstly, let $m = \gamma(\mu,\tilde\mu) \in \p(X^2)$ be the $\gamma$-coupling of $(\mu,\tilde\mu)$. Secondly, we define a kernel $\kappa:X^2\to \p(Y^2)$ by the following formula: $$\kappa_{x\tilde x} = 1_{\Delta_X}(x,\tilde x)\cdot\gamma\left(K_x,\tilde K_{\tilde x}\right) + 1_{\Delta_X^c}(x,\tilde x)\cdot (K_x\otimes \tilde K_{\tilde x})$$ where $\Delta_X$ is the diagonal of $X^2$ as per Section \[sec:appendix\]. Clearly, the measure $\kappa_{x\tilde x}$ is a coupling of measures $K_x$ and $\tilde K_{\tilde x}$ for any $(x,\tilde x)\in X^2$, which is a maximal coupling on the diagonal, and the independent (or product) coupling off the diagonal. Note that $\kappa:X^2\to \p(Y^2)$ is indeed a kernel. The only non-trivial part of the latter statement concerns the measurability of the positive part and the negative part in the Hahn-Jordan decomposition of the kernel, which follows directly from [@r1984 Lemma 1.5, Chapter 6]. Furthermore, the measurability of $K_x\otimes \tilde K_{\tilde x}$ obviously holds for any measurable rectangle $A\times \tilde A\subseteq Y^2$ and extends to the whole product $\sigma$-algebra by the monotone class theorem (see e.g. [@a1972 Theorem 1.3.9]).
Define $\Q: = m\otimes \kappa \in \p(X^2\times Y^2)$ to be the product measure, and further denote by $\pi_X$, $\tilde \pi_X$, $\pi_Y$ and $\tilde \pi_Y$ the obvious projection maps from that space, e.g. $$\tilde\pi_X(x,\tilde x,y,\tilde y) = \tilde x\in X.$$ We claim that the random element $(\pi_X,\pi_Y)$ is distributed according to $\q$ and $(\tilde \pi_X,\tilde \pi_Y)$ is distributed according to $\tilde \q$. Indeed, for any $A\in \b(X)$ and $B\in \b(Y)$ it holds that $$\begin{aligned}
\Q(\pi_X\in A,\pi_Y\in B) & = \Q((A\times X)\times (B\times Y)) &\text{ by definition of $\pi_X$ and $\pi_Y$}
\\
&= \int_{A\times X}\kappa_{x\tilde x}(B\times Y)m(\d x\times \d \tilde x) &\text{ by Proposition \ref{prop:product}}
\\
& = \int_{A\times X}K_x(B)m(\d x \times \d\tilde x) &\text{ since $\kappa_{x\tilde x}$ is a coupling}
\\
&= \int_A K_x(B)\mu(\d x) &\text{ since $\mu$ is a marginal of $m$}
\\
&= \q(A\times B). &\text{ by Proposition \ref{prop:product}}
\end{aligned}$$ Since both $(\pi_X,\pi_Y)_*\Q$ and $\q$ are probability measures on a Borel space $X\times Y$, and they have been shown to agree on the class measurable rectangles which is closed under finite intersections, they are equal [@r1984 Proposition 3.6, Chapter 0]. Similarly, it holds that $(\tilde \pi_X, \tilde \pi_Y)_*\Q = \tilde \q$. Thus, by the coupling inequality $$\|\q \wedge \tilde \q\|\geq \Q(\pi_X = \tilde \pi_X,\pi_Y = \tilde \pi_Y)$$ On the other hand, for the latter term the following holds: $$\begin{aligned}
\Q(\pi_X = \tilde \pi_X,\pi_Y = \tilde \pi_Y) & = \Q((\Delta_X\times Y^2)\cap (X^2\times\Delta_Y)) = \Q(\Delta_X\times \Delta_Y)
\\
& = \int_{\Delta_X}\kappa_{x\tilde x}(\Delta_Y)m(\d x\times \d\tilde x)
\\
& \geq m(\Delta_X)\cdot \inf_{(x,\tilde x)\in \Delta_X}\kappa_{x\tilde x}(\Delta_Y) = m(\Delta_X)\cdot \inf_{x\in X}\kappa_{xx}(\Delta_Y)
\\
& = \|\mu\wedge\tilde\mu\|\cdot\inf_{x\in X}\|K_x \wedge \tilde K_x\|.
\end{aligned}$$ and the lemma is proved.
As we have mentioned above, Lemma \[lem:main.1\] is an analogue of Lemma \[lem:lin\]. However, unlike in Section \[ssec:proof.lin\], here we cannot go directly from Lemma \[lem:main.1\] to Theorem \[thm:main\] and we need the following auxiliary result first:
\[lem:main.2\] Let $\{X_k\}_{k \in \N_0}$ be a family of Borel spaces and let $\Omega_n = \prod_{k=0}^n X_k$ be product spaces for any $n\in \bar\N_0$. Let further $P^0,\tilde P^0\in \p(X_0)$ and $P^n,\tilde P^n:\Omega_{n-1}\to \p(X_n)$ for $n\in \N$ be initial probability measures and conditional stochastic kernels respectively. Denote: $$a_0 := \|P^0 \wedge \tilde P^0\|,\quad a_k := \inf_{\omega_{k-1}\in \Omega_{k-1}}\|P^k_{\omega_{k-1}} \wedge \tilde P^k_{\omega_{k-1}}\|,\quad k\in \N.$$ and further ${\mathsf{P}}^n := \bigotimes_{k=0}^n P^k$, $\tilde{\mathsf{P}}^n := \bigotimes_{k=0}^n \tilde P^k$. For any $n\in \N_0$ it holds that $$\label{eq:lem.main.2}
\|{\mathsf{P}}^n \wedge \tilde {\mathsf{P}}^n\|\geq \prod_{k=0}^n a_k.$$
The inequality can be proved by induction. Clearly, it holds true for $n = 0$. Suppose it holds true for some $n\in \N_0$, then in the setting of Lemma \[lem:main.1\] put $X = \Omega_n$, $Y = X_{n+1}$, $\mu = {\mathsf{P}}^n$, $\tilde \mu = \tilde {\mathsf{P}}^n$, $K = P^{n+1}$ and $\tilde K = \tilde P^{n+1}$. For the product measures we obtain that $\q = {\mathsf{P}}^{n+1}$ and $\tilde \q = \tilde {\mathsf{P}}^{n+1}$, so now follows immediately from .
To prove Theorem \[thm:main\] we are only left to notice that is equivalent to thanks to the duality argument in .
Discussion {#sec:discus}
==========
Let us discuss the results obtained above and their connection to the literature. First of all, Theorem \[thm:lin\] extends the bounds on the total variation distance between the finite-dimensional marginals ${\mathsf{P}}^n$ and $\tilde{\mathsf{P}}^n$, obtained in [@rr2013 Theorem 1] under the Borel assumption, to the case of arbitrary measurable spaces. Its proof is inspired by the one of [@ta2013 Lemma 1] that focused on the Markovian case $P^1 = P^2 = \dots$ exclusively. The work in [@ta2013] also emphasized the benefit of dealing with sub-$\sigma$-algebras as in Theorem \[thm:lin\]: the perturbation constants $c_k$ in such case are likely to be smaller than those in Theorem \[thm:main\]. Moreover, with focus on numerical methods, it allows dealing with kernels that may not have an integral expression. Although the bounds obtained in [@rr2013] is a special case of Theorem \[thm:lin\], the main focus of [@rr2013] was rather on the construction of the corresponding maximal coupling of the infinite-dimensional product measures ${\mathsf{P}}$ and $\tilde {\mathsf{P}}$. In the setting of Theorem \[thm:lin\] the existence of such coupling is unlikely, as even the measurability of the diagonal, required in the coupling inequality, may be violated in the case of arbitrary measurable spaces.
At the same time, in the setting of [@rr2013] (that is, under the Borel assumption) a stronger result in Theorem \[thm:main\] holds true. Although the latter theorem is only focused on the bounds, Lemma \[lem:main.1\] which is the core in the proof of Theorem \[thm:main\], yields the coupling of finite-dimensional marginals ${\mathsf{P}}^n$ and $\tilde{\mathsf{P}}^n$ that trivially extends to the infinite-dimensional case by Proposition \[prop:I-T\]. With focus on bounds, to show that Theorem \[thm:main\] indeed provides a less conservative estimate than [@rr2013] (or Theorem \[thm:lin\]), let us mention that $$\label{eq:bound.comparison}
2 - 2\prod_{k=0}^n\left(1-\frac12c_k\right) \leq \sum_{k=0}^n c_k$$ for any non-negative sequence $(c_k)_{k\in \N_0}$ and any $n\in \N_0$, which can be shown by induction over $n$.[^1] In particular, there are several cases when bounds in Theorem \[thm:main\] are exact: e.g. when kernels $P^k$ and $\tilde P^k$ are just measures on $X_k$ which corresponds to dealing with a sequence of iid random variables, see also the example in Section \[sec:example\]. In contrast, the bounds in Theorem \[thm:lin\] can grow unboundedly e.g. in case $c_k = c>0$ for all $k\in \N_0$.
Finally, let us mention that it is of interest whether Theorem \[thm:main\] holds true in the general case of arbitrary measurable spaces and arbitrary stochastic kernels – recall however that even in such case the corresponding maximal coupling may not exist. Although we do not provide the answer to this general question, we show that under the assumption that kernels have densities, one can obtain a version of Lemma \[lem:main.1\] for the case of arbitrary measurable spaces: this of course implies the validity of Theorem \[thm:main\] in such case as well, provided that all the kernels involved have densities.
\[lem:int\] Let $(X,\x)$ and $(Y,\y)$ be two arbitrary measurable spaces, and let measures $\mu,\tilde \mu\in \p(X,\x)$ and $\lambda \in\p(Y,\y)$. If kernels $K, \tilde K:X\to \p(Y,\y)$ are given by $$K_x(\d y) := k(x,y)\lambda(\d y),\qquad \tilde K_x(\d y) := \tilde k(x,y) \lambda(\d y)$$ where $k,\tilde k:X\times Y \to [0,\infty)$ are $\x\otimes \y$-measurable functions, then for the product measures $\q := \mu\otimes K$ and $\tilde \q := \tilde\mu\otimes \tilde K$ the inequality holds true.
Note that the following expressions for $\q$ and $\tilde \q$ hold true: $$\begin{aligned}
\q(\d x\times \d y) &= k(x,y) [\mu\otimes \lambda](\d x\times \d y),
\\
\tilde\q(\d x\times \d y) &= \tilde k(x,y) [\tilde \mu\otimes \lambda](\d x\times \d y).
\end{aligned}$$ To make them comparable, let us define $\nu := \mu+\tilde\mu$, so that $$\begin{aligned}
\|\q \wedge \tilde \q\| &= (\q\wedge \tilde \q)(X\times Y)
\\
&= \int_{X\times Y}\min\left(k(x,y)\frac{\d \mu}{\d \nu}(x),\tilde k(x,y)\frac{\d \tilde \mu}{\d \nu}(x)\right)[\nu\otimes\lambda](\d x\times \d y)
\\
&\geq \int_{X\times Y}\min\left(k(x,y),\tilde k(x,y)\right)\min\left(\frac{\d \mu}{\d \nu}(x),\frac{\d \tilde \mu}{\d \nu}(x)\right)[\nu\otimes\lambda](\d x\times \d y)
\\
&= \int_X (K_x\wedge \tilde K_x)(Y) (\mu\wedge\tilde\mu)(\d x) \geq \|\mu\wedge\tilde \mu\|\cdot \inf_{x\in X}\|K_x \wedge \tilde K_x\|
\end{aligned}$$ as desired.
Example {#sec:example}
=======
To enlighten the theoretical results obtained above, we study how conservative are the bounds in Theorem \[thm:main\] on a simple example for which the analytical expression for the total variation distance is available. For that purpose, we consider a Markov Chain with just two states, that is $X = \{0,1\}$ and $P^k = P$ for all $k\in \N$. The kernel $P$ is given by a stochastic matrix, which for simplicity we assume to be diagonal: $P = \left(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \right)$. The initial distribution, denoted by $P^0 = \mu$, is further given by $\mu(\{1\}) = p$, $\mu(\{0\}) = 1-p$. We focus on the case when the perturbed stochastic process is a Markov Chain as well, that is $\tilde P^k = \tilde P$ for all $k\in \N$, and assume that $\tilde P = \left(\begin{smallmatrix} 1-\ve&\ve\\ \ve&1-\ve \end{smallmatrix} \right)$ for some $\ve\in (0,1)$. The perturbed initial distribution we denote by $\tilde P^0 = \tilde \mu$, it is given by $\mu(\{1\}) = p-\delta$ and $\mu(\{0\}) = 1-(p-\delta)$; here $\delta\in (p-1,p)$.
Since the product space $\Omega_n = \{0,1\}^{n+1}$ is finite, the precise value of the total variation distance between the product measures ${\mathsf{P}}^n$ and $\tilde {\mathsf{P}}^n$ can be computed as $$\left\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\right\| = \sum_{\omega_n\in\Omega_n} \left|{\mathsf{P}}^n(\{\omega_n\}) - \tilde{\mathsf{P}}^n(\{\omega_n\})\right|$$ Thanks to the special choice of $P$, the measure ${\mathsf{P}}^n$ is supported only on two points in $\Omega_n$ which makes it is easy to obtain an analytic expression for the sum above. Let us abbreviate: $f_n(\ve):= 1 - (1-\ve)^n$. There are three possible cases depending on the interplay between parameters $p$, $\ve$, $\delta$ and $n$:
- If $\delta < - \frac{pf_n(\ve)}{1-f_n(\ve)}$, then $\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\| = 2(1-p)f_n(\ve) - 2\delta(1-f_n(\ve))$.
- If $- \frac{pf_n(\ve)}{1-f_n(\ve)}\leq \delta \leq \frac{(1-p)f_n(\ve)}{1-f_n(\ve)}$, then $\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\| = 2f_n(\ve)$.
- If $\delta > \frac{(1-p)f_n(\ve)}{1-f_n(\ve)}$, then $\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\| = 2pf_n(\ve) + 2\delta(1-f_n(\ve))$.
According to Theorem \[thm:main\], the bounds are $\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\|\leq 2(f_n(\ve) + \delta(1-f_n(\ve)))$. Clearly, in all cases \[a.\], \[b.\] and \[c.\] the bounds hold true. Moreover, if only the stochastic matrix is perturbed, that is $\delta = 0$ (case \[b.\]), then the total variation is $2f_n(\ve)$ which precisely coincides with the bounds provided by Theorem \[thm:main\]. Similarly, if $p=1$ and $\delta\in (0,1)$ (case \[c.\]), then the total variation $2(f_n(\ve) + \delta(1-f_n(\ve)))$ provides another example when bounds in Theorem \[thm:main\] are exact. In all other cases it is easy to see that there is a strictly positive gap in the inequality .
Unrelated to the precision of bounds in Theorem \[thm:main\], it is interesting to further comment on the phenomenon in case \[b.\]. If $p$, $\ve$ and $\delta$ are fixed, then there exists $N\in \N$ such that for $n\geq N$ always \[b.\] holds: indeed, it follows directly from the fact that the denominator $1-f_n(\ve) = (1-\ve)^n\to 0$ monotonically as $n\to\infty$. As a result, regardless of the value of $\delta\in (0,1)$, for all $n$ big enough $\|{\mathsf{P}}^n - \tilde{\mathsf{P}}^n\|$ does not depend on $\delta$: in particular, it is the same as in the case when $\delta = 0$.
Acknowledgements {#acknowledgements .unnumbered}
================
The last author would like to thank George Lowther for the idea of non-linear bounds, Michael Greinecker for his hints on dealing with kernels, and other users of [MathStackexchange](http://math.stackexchange.com/) and [MathOverflow](http://mathoverflow.net/) for their valuable comments regarding measure theory and coupling.
Appendix {#sec:appendix}
========
For any set $X$ the corresponding diagonal is denoted by $\Delta_X := \{(x,x):x\in X\} \subseteq X^2$. The set of all real numbers is denoted by $\R$ and the set of all natural numbers is denoted by $\N$. We further write $\N_0 := \N\cup \{0\}$ and $\bar\N_0 := \N_0\cup\{\infty\}$. For any $A\subseteq X$ we denote its indicator function by $1_A$.
We say that $X$ is a (standard) Borel space if $X$ is a topological space homeomorphic to a Borel subset of a complete separable metric space. Any Borel space is assumed to be endowed with its Borel $\sigma$-algebra $\b(X)$. An example of a Borel space is the space of real numbers $\R$ endowed with the Euclidean topology.
Given a measurable space $(X,\x)$ we denote the space of all $\sigma$-additive finite signed measures on it by $\bm(X,\x)$. For any set $A\in\x$ we introduce an evaluation map $\theta_A:\bm(X,\x) \to \R$ given by $\theta_A(\mu) := \mu(A)$. We always assume that $\bm(X,\x)$ is endowed with the smallest $\sigma$-algebra that makes all evaluation maps measurable. The subspace of all probability measures in $\bm(X,\x)$ is denoted by $\p(X,\x)$. In case $X$ is a Borel space, we simply write $\bm(X)$ and $\p(X)$ in place of $\bm(X,\b(X))$ and $\p(X,\b(X))$ respectively. If $(Y,\y)$ is another measurable space, by a bounded kernel we mean a measurable map $K:X\to \bm(Y,\y)$ such that $$\sup_{x\in X}\sup_{A\in \y}|K_x(A)|<\infty$$ where we write $K_x$ instead of a more cumbersome $K(x) \in \bm(Y,\y)$ for any $x\in X$. In case $K_x \in \p(Y,\y)$ for any $x\in X$, we say that the kernel $K$ is stochastic. The condition that $K:X\to \bm(Y,\y)$ is measurable is equivalent to $K_{(\cdot)}(A):X\to \R$ being a measurable map for any $A\in \y$ [@k1997a Lemma 1.37]. Clearly, any measure can be considered as a kernel which does not depend on its first argument. Furthermore, any measure $\nu\in \bm(X,\x)$ admits the unique Hahn-Jordan decomposition given by $\nu = \nu^+ - \nu^-$, where $\nu^+$ and $\nu^-$ are two mutually singular non-negative measures on $(X,\y)$ [@a1972 Section 2.1.2]. If $K:X\to \bm(Y,\y)$ is a kernel, then $$\|K\|:= \sup_{x\in X}\left(K^+_x(Y)+K^-_x(Y)\right)$$ defines the total variation of $K$. For any two measures $\nu,\tilde\nu\in \bm(X,\x)$ we denote their minimum by $\nu\wedge\tilde\nu := \nu - (\nu - \tilde \nu)^- \in \bm(X,\x)$. In particular, it holds that $$\label{eq:tv.min}
\|\nu \wedge \tilde \nu\| = 1 - \frac12 \|\nu - \tilde \nu\| = 1 - \sup_{A\in \x}|\nu(A) - \tilde \nu(A)|,$$ for any two probability measures $\nu,\tilde \nu\in \p(X,\x)$.
If $f:X\to Y$ is a measurable map between two measurable spaces $(X,\x)$ and $(Y,\y)$, for any $\nu\in \bm(X,\x)$ the image measure $f_*\nu \in \bm(Y,\y)$ is given by $$(f_*\nu)(A) := \nu\left(f^{-1}(A)\right)$$ for all $A\in \y$. Given a family of measurable spaces $\{(X_k,\x_k)\}_{k\in \N_0}$ and an index set $I\subseteq \bar\N_0$, we denote the product measurable space by $$\prod_{k\in I}(X_k,\x_k) := \left(\prod_{k\in I} X_k,\bigotimes_{k\in I}\x_k\right),$$ where $\bigotimes_{k\in I}\x_k$ is the product $\sigma$-algebra. Let $\tilde I\subseteq I$ and denote $(\Omega,\f) := \prod_{k\in I}(X_k,\x_k)$ and $(\tilde\Omega,\tilde\f) := \prod_{k\in \tilde I}(X_k,\x_k)$. Let further $\pi: \Omega \to \tilde \Omega$ be an obvious projection map. For any $\nu \in \bm(\Omega,\f)$ we say that $\pi_*\mu$ is the marginal of $\mu$ on $(\tilde \Omega,\tilde \f)$.
[^1]: Notice also that the right-hand side of can be considered as the first-order term of the expansion of the product in the left-hand side.
|
---
abstract: 'We calculate the one-loop effective action of a scalar field with general mass and coupling to the curvature in the detuned Randall-Sundrum brane world scenario, where the four-dimensional branes are anti-de Sitter. We make use of conformal transformations to map the problem to one on the direct product of the hyperbolic space $\mathrm{H}^4$ and the interval. We also include the cocycle function for this transformation. This Casimir potential is shown to give a sizable correction to the classical radion potential for small values of brane separation.'
author:
- 'James P. Norman'
bibliography:
- 'ads.bib'
date: 30 June 2004
title: 'Casimir effect between anti-de Sitter braneworlds'
---
Introduction
============
The one- and two- brane Randall-Sundrum models [@Randall:1999ee; @Randall:1999vf] have become one of the most popular scenarios for the discussion of brane world cosmology. The two-brane model was proposed as a solution to the the hierarchy problem without introducing supersymmetry. In the original two-brane Randall-Sundrum model, the brane tensions are tuned so that the branes are flat. However, solutions also exist when the brane tensions are detuned, with either $dS_4$ or $AdS_4$ branes [@Kaloper:1999sm; @Karch:2000ct].
The de Sitter case has been looked at as a model for inflation [@Himemoto:2000nd]. The anti-de Sitter brane case is relevant to a supersymmetric extension of the Randall-Sundrum model which is based on gauged five-dimensional supergravity [@Brax:2002vs; @Bagger:2002rw; @Bagger:2003vc]. In this model, a locally supersymmetric coupling of five dimensional gauged supergravity to branes requires that the branes are $AdS_4$ or $M_4$.
Upon compactification to four dimensions, the separation of the branes is promoted to a field, called the radion. In case where the branes are flat, the classical potential for the radion is zero. This would lead to a massless scalar particle, which is not seen in nature. This is the radion stabilisation problem. One possible solution is that the quantum vacuum energy of the bulk fields could generate a potential, analogous to the Casimir effect, to stabilise of the radion. This mechanism has been looked at in [@Goldberger:2000dv; @Garriga:2000jb; @Flachi:2001pq; @Hofmann:2000cj; @Saharian:2002bw; @Knapman:2003ey] for bulk scalar fields, and [@Flachi:2001bj; @Flachi:2001ke] for fermions, where it was found that it is not possible to simultaneously solve the hierarchy problem and have an acceptable mass for the radion. A related calculation in five dimensional heterotic M theory has been performed in [@Garriga:2001ar; @Moss:2004un].
When the branes are $AdS_4$ or $dS_4$, the brane separation is fixed by the junction conditions and there is a classical potential for the radion [@Bagger:2003dy]. The potential is a stabilizing one in the $AdS_4$ case, and an unstable one in the $dS_4$ case. The Casimir effect would then be a quantum correction to the classical potential. In the $dS_4$ brane case, the quantum effective potential has been calculated for conformally coupled scalar fields in [@Naylor:2002xk] and for massless fermions in [@Moss:2003zk]. Small deviations from the conformal values were considered in [@Elizalde:2002dd].
The Casimir effect between two anti-de Sitter branes has not yet received much attention. This may be because the model is not as relevant to cosmology as its de Sitter cousin. However, there seems to be no *a priori* reason to favour the $dS_4$ case over the $AdS_4$ case, since the brane tensions are put in “by hand”. The Casimir effect in the supersymmetric $AdS_4$ case may be important as it may give a dynamical mechanism for breaking supersymmetry (see [@vonGersdorff:2002tj] for a related calculation on a flat orbifold). However, the calculation of the effective action of the higher spin fields in the supergravity multiplet in this background presents a difficult technical challenge. To simplify the calculation and illustrate our method, here we will consider only a scalar field in the background with $AdS_4$ branes, with the hope that many of the features of the calculation apply to higher spin fields. Some related work on the one-loop effective potential of $AdS_n \times S^n$ was performed in [@Uzawa:2003ji]. Spontaneous generation of $AdS_4$ branes in an $AdS_5$ bulk has been investigated in [@Nojiri:2000eb; @Nojiri:2000bz].
We calculate the one-loop effective action for scalar fields with general mass and coupling to the scalar curvature. Both Dirichlet and Robin type boundary conditions will be considered. It will be shown that the quantum effective potential can give a sizable correction to the classical potential for small values of the radion. Our calculation cannot be carried over easily to the $dS_4$ case, because of the different global properties of $dS_4$ and $AdS_4$.
We choose to work in the “downstairs” picture of a manifold with boundary and impose boundary conditions on the fields, rather than the “upstairs” picture of an orbifold with singular branes. The one-loop effective action of the scalar field is evaluated using $\zeta$ function regularization. We use the properties of the effective action under conformal rescalings of the metric to relate the effective action to that of a scalar field on the direct product of $AdS_4\times I$. $\zeta$ function regularization requires that we rotate to the appropriate Euclidean space, which is $\mathrm{H}^4 \times I$, where $\mathrm{H}^4$ is the four dimensional hyperbolic space. We then calculate the correction factor for this transformation, known as the cocycle function. Our curvature conventions are given in [@Moss:2004un].
Operators and Background Metric
===============================
The background metric for the Randall-Sundrum model with $AdS_4$ branes can be written as $$\label{eq:metric}
ds^2=e^{-2\omega(z)}\left(g^{(4)}_{\mu \nu} dx^\mu dx^\nu + dz^2 \right)$$ where $g^{(4)}_{\mu \nu}$ is the metric of $AdS_4$ with $AdS$ radius $a$, $$\omega(z)=\ln \left(\sqrt{\frac{|\Lambda|}{6}} a \sin \left( z/a\right) \right),$$ and $\Lambda$ is the bulk cosmological constant. In the absence of branes, the conformal coordinate $z$ would run from $0<z<a\pi$. However, in the “downstairs” picture of a manifold with boundaries, the branes cut off the space and $z$ is restricted to the space between the branes. The positions of the branes are fixed at the classical level by junction conditions on the metric. We call the positions of the branes $z_1$ and $z_2$. Explicitly, the classical positions of the branes are $$z^{\text{classical}}_{1,2} = a\arccos{\frac{\sigma_{1,2}}{\sigma}},$$ where $\sigma=\sqrt{|\Lambda|/6}$, $\sigma_1= T_1/6$, $\sigma_2= -T_2/6$, and $T_{1,2}$ are the tensions of the branes at $z_1$ and $z_2$, respectively. However, these values may receive corrections from quantum effects, so we keep $z_1$ and $z_2$ general. One condition we can always impose is that the metric on the brane at $z_1$ has a scale factor of unity. This fixes $\sin(z_1/a)=(\sigma a)^{-1}$. Without loss of generality, we also restrict $z_2>z_1$.
In the limit $a\rightarrow \infty$, the function $w(z)\approx \ln (\sigma z)$, and we recover the flat brane Randall-Sundrum metric. In this limit, the positions of the branes are no longer determined by the junction conditions, with all brane positions being allowed.
We consider the effective action of a scalar field $\phi$ with general mass $m$ and coupling to the scalar curvature $R$. The fluctuation operator is $$\label{eq:operator}
\Delta=-\nabla^2+\xi R + m^2.$$
The one-loop effective action will be regularized using $\zeta$-function regularization. We work in Euclidean space, with $AdS_4$ becoming the hyperbolic space $\mathrm{H}^4$. The $\zeta$ function is defined as the trace of the operator $\Delta$ to some power $-s$, in some region of the complex plane where the trace converges, i.e., $$\zeta(s)=\text{tr} \Delta^{-s}=\sum_n\int_0^\infty d \lambda\ \mu(\lambda) d_n \rho_n(\lambda)^{-s}.$$ Here, the eigenvalues of the operator $\rho_n(\lambda)$ are assumed to split into a continuous part, labeled by the real parameter $\lambda$, and a discrete part, labeled by integers $n$. The spectral function $\mu(\lambda)$ gives a “density of states” in the continuous spectrum, and is analogous to the discrete degeneracy factor $d_n$. The one loop effective action is then defined to be $$W=-\frac12\zeta'(0)-\frac12\zeta(0)\log \mu_R^{2},$$ where we have analytically continued the $\zeta$ function to $s=0$. The renormalisation scale $\mu_R$ has been introduced to make the eigenvalues dimensionless. We can also define the effective potential on the brane at $z_1$ by dividing throughout by the volume of the $\mathrm{H}^4$ space. That is, we define the one-loop effective potential $V$ through $$W=\int|g^{(4)}| d^4 x \ V.$$
Rather than work directly with the eigenvalues of Eq.(\[eq:operator\]), which are difficult to obtain in curved space, we can use the behaviour of the effective action under conformal rescalings of the metric to simplify the problem. Considering operators of Laplace type, with $\Delta = -\nabla^2+X$, we introduce a one-parameter family of metrics $g^\epsilon$ related to the physical metric by a conformal rescaling, so that $$g_{\alpha \beta}^\epsilon={\Omega(\epsilon)}^2 g_{\alpha \beta},
\quad \Omega(\epsilon)=e^{(1-\epsilon)\omega(z)}.$$ The conformally rescaled operator is $\Delta_\epsilon=-\nabla_\epsilon^2+X_\epsilon$, where $X_\epsilon=X\Omega(\epsilon)^{-2} - \frac{3}{16}
\left(R \Omega(\epsilon)^{-2} -R_\epsilon \right)$. One can then show that [@Dowker:1990ue] $$W[\epsilon=1,\Delta]=W[\epsilon=0, \Delta_0]
+C\left[\Omega \right],$$ where the cocycle function $C\left[\Omega \right]$ is given (in five dimensions) in terms of the generalized heat kernel coefficient $B_{5/2}(f,\Delta)$ as $$\label{eq:cocycle}
C[\Omega]=\int_{0}^{1} d\epsilon \
B_{5/2}\left(\omega,\Delta_\epsilon\right).$$ Hence, we can relate the one-loop effective action of the scalar field in the warped metric (\[eq:metric\]) to one on the direct product manifold $\mathrm{H}^4 \times I$.
We will first consider the effective action of the conformally transformed operator. The conformally transformed operator $\Delta_0$ separates into $$\Delta_0 = \Delta_{I}+\Delta^{(4)},$$ where $\Delta_{I}$ contains all dependence on the $z$-direction $$\label{eq:DeltaI}
\Delta_{I} = -\partial_z^2 - \xi \frac{12}{a^2} +
\left(\xi-\frac{3}{16}\right) \left(8 \omega''-12 \omega'^2 \right) +
m^2 e^{-2 \omega},$$ and ${\Delta^{(4)}}$ is the 4-dimensional Laplacian for a massless scalar field on $\mathrm{H}^4$. The eigenvalues and $\zeta$ function for $\Delta^{(4)}$ have been calculated by Camporesi [@Camporesi:1991nw]. The eigenvalues are continuous and labeled by the real parameter $\lambda$. The eigenvalues of $\Delta_I$, which we denote $m_n^2$, are discrete. After performing a separation of variables, we can regard the complete $\zeta$-function as a sum over $\zeta$-functions of scalar fields with mass $m_n$ on $\mathrm{H}^4$ [^1]. Thus, from [@Camporesi:1991nw], we find $$\begin{aligned}
\label{eq:zeta}
\zeta(s)= \frac{a^{2s-4}}{8\pi^2} \int {|g^{(4)}|}^{1/2} \sum_n &&\left\{\frac{b_n^{2-2s}}{8(s-1)}+
\frac{b_n^{4-2s}}{2(s-1)(s-2)} \right. \nonumber \\
&& \left. - 2\int_0^\infty d\lambda
\frac{\lambda\left(\lambda^2+\frac14\right)}
{\left(1+e^{2\pi \lambda}\right){(\lambda^2+b_n^2)}^{s}} \right\},\end{aligned}$$ where we must restrict $s>5/2$, and we have introduced $b_n^2=\frac94 + m_n^2 a^2$.
The eigenvalues $m_n^2$ of the operator $\Delta_{I}$ now need to be found. We consider both Dirichlet and Robin boundary conditions on the field at the boundaries $z_1$ and $z_2$. However, we shall illustrate our method using Dirichlet boundary conditions, and simply quote results for Robin boundary conditions in Appendix \[sec:robin\], as the calculation is similar. The eigenfunctions of Eq. (\[eq:DeltaI\]) can be written in terms of associated Legendre functions of order $\mu=\sqrt{4-20\xi+ m^2/\sigma^2}$ and degree $-1/2+b_n$. For later convenience, we define the functions
$$\begin{aligned}
\label{eq:RandS}
R^{-\mu}_{-1/2+b_n} (\theta) &=& \sqrt{\sin\theta}
\left(\mathrm{P}^{-\mu}_{-1/2+b_n} (\cos\theta)\right),
\\
S^{\mu}_{-1/2+b_n} (\theta) &=& \sqrt{\sin\theta}\left(
\mathrm{P}^{\mu}_{-1/2+b_n} (\cos\theta)
-\frac{2i}{\pi}\mathrm{Q}^{\mu}_{-1/2+b_n} (\cos\theta)\right).\end{aligned}$$
These functions are linearly independent for all $\mu$ and $b_n$. The general solution to $\Delta_I f_n = m_n^2 f_n$ is then a linear combination of $R^{-\mu}_{-1/2+b_n}(z/a)$ and $S^{\mu}_{-1/2+b_n}(z/a)$. Applying Dirichlet boundary conditions $\phi = 0$ on the boundaries $\partial {\cal M}$, leads to an implicit equation for $b_n$ through $$F(b_n)=R^{-\mu}_{-1/2+b_n} (\theta_1)S^{\mu}_{-1/2+b_n}
(\theta_2)-R^{-\mu}_{-1/2+b_n} (\theta_2)S^{\mu}_{-1/2+b_n}(\theta_1)=0,$$ where $\theta_{1,2}=z_{1,2}/a$.
Zeta function on $\mathrm{H}^4 \times I$
========================================
The sums over the discrete eigenvalues $b_n$ in Eq. (\[eq:zeta\]) are complicated by the fact that we only know the $b_n$ through the implicit equation $F(b_n)=0$. However, we can use techniques developed for studying the Casimir effect on balls and spheres [@Elizalde:1993; @Bordag:1996gm] to convert the sums into contour integrals of the function $F$. We will first consider the terms $\sum_n b_n^{2-2s}$ and $\sum_n b_n^{4-2s}$ in the $\zeta$ function. Our procedure follows that of the flat brane case, in that we write the sum as $$\hat{\zeta}(s)=\sum_n b_n^{-s} = \frac{1}{2\pi i} \int_{\cal C} dz z^{-s} \frac{F'(z)}{F(z)},$$ where the contour $\cal C$ encloses all the real positive roots of $F(z)=0$. As in the flat brane case, we wish to write this contour integral as an integral over the imaginary axis. To do this, we need to know the asymptotic behaviour of $F(ix)$ for large $x$. We define the functions $\Sigma^R(x,\theta)$ and $\Sigma^S(x,\theta)$ by $$\label{eq:SigmaR}
R^{-\mu}_{-1/2+ix}
(\theta) = e^{i\pi \mu/2+i\pi/4} \sqrt{\frac{1}{2\pi}} e^{x \theta}
\frac{\Gamma(ix-\mu+1/2)}{\Gamma(ix+1)} \Sigma^R(x,\theta)$$ and $$\label{eq:SigmaS}
S^{\mu}_{-1/2+ix}
(\theta) = e^{i\pi \mu/2-i\pi/4}\sqrt{\frac{2}{\pi}} e^{-x \theta}
\frac{\Gamma(ix+\mu+1/2)}{\Gamma(ix+1)} \Sigma^S(x,\theta).$$ From the representation of the Legendre functions in terms of hypergeometric series [@Bateman page 146], it can be seen that $\Sigma^R(x,\theta)$ and $\Sigma^S(x,\theta)$ have asymptotic series of the Poincaré type — that is, asymptotic expansions in inverse powers of $x$. We can also use properties of the $\Gamma$ function [@Abramowitz page 256] to show $$\ln \left|\frac{\Gamma(ix+\mu+1/2)\Gamma(ix-\mu+1/2)}{\Gamma(ix+1)\Gamma(ix+1)}\right|
=\ln \left|\frac{\sinh{(\pi x)}}{x\cosh{\left[\pi(x-i\mu)\right]}}\right| \sim -\ln(x)+O(e^{-x}).$$ Hence, for large $x$, $$\ln |F(ix)| \sim (\theta_2-\theta_1)x +\chi \ln(x) -\ln \pi + O(x^{-2}),$$ where $$\chi=-1.$$ Therefore, for $s>1$, the contour can be deformed to an integral over the imaginary axis. After a few manipulations, we find $$\begin{aligned}
\label{eq:zetahatreg}
\hat{\zeta}(s)&=&\frac{\sin{\frac{\pi s}{2}}}{\pi} \int_\varepsilon^\infty dx x^{-s} \frac{d}{dx} \ln
\left|\frac{F(ix)}{F_a(ix)}\right|
+\frac{\sin{\frac{\pi s}{2}}}{\pi} \int_\varepsilon^\infty dx x^{-s} \frac{d}{dx} \ln \left|F_a(ix)\right|
\nonumber \\
&&+\frac{1}{2\pi i} \int_{{\cal C}_\varepsilon} dz z^{-s} \frac{F'(z)}{F(z)}\end{aligned}$$ where ${\cal C}_\varepsilon$ is a small semicircle of radius $\varepsilon$ around the origin, and $$F_a(ix)= R^{-\mu}_{-1/2+ix} (\theta_2)S^{\mu}_{-1/2+ix}(\theta_1).$$
The first term on the RHS of Eq. (\[eq:zetahatreg\]) can now be continued to $s=-2$ and $s=-4$. The second term still cannot be analytically continued to these points as the integrals diverge at large $x$ if we take $s<1$. We therefore add and subtract terms which cause the integral to diverge in this region to enable us to analytically continue this term to $s=-2N$, where $N$ is a positive integer. If we define $r_k$ and $s_k$ as the coefficients in the asymptotic expansion of $\ln |\Sigma^R(x,\theta)|$ and $\ln |\Sigma^S(x,\theta)|$, i.e., $$\ln |\Sigma^R(x,\theta)|\sim\sum_{k=1}^\infty r_k (\theta) x^{-k}+O(e^{-x}),
\quad \ln |\Sigma^S(x,\theta)|\sim\sum_{k=1}^\infty s_k (\theta) x^{-k},$$ then we can define $$\begin{aligned}
\label{eq:Fareg}
\ln {|F_a(ix)|}_{\text{reg}}&=&
\ln {|F_a(ix)|}+\ln \pi-(\theta_2-\theta_1)x-\chi\ln(x) \nonumber \\
&&-\sum_{k=1}^{2N-1} \left(r_k(\theta_1)+s_k(\theta_2)\right)
x^{-k} -\left(r_{2N}(\theta_2)+s_{2N}(\theta_1)\right)
x^{-2N}e^{-1/x}.\end{aligned}$$ so that $\ln {|F_a(ix)|}_{\text{reg}} \sim O(x^{-2N-1})$ for large $x$. We can now substitute the for $\ln {|F_a(ix)|}$ in terms of $\ln {|F_a(ix)|}_{\text{reg}}$ in Eq. (\[eq:zetahatreg\]), and analytically perform the integrals of the “extra” terms in the right-hand side of Eq. (\[eq:Fareg\]). We can now analytically continue to $s=-2N$. Provided that $F(x)$ is finite as $x\rightarrow 0$, the contribution from the integral around the small semicircle ${\cal C}_\varepsilon$ vanishes when we take $\varepsilon \rightarrow 0$. Thus, we find $$\hat{\zeta}(-2N)=-(-1)^N N \left(r_{2N}(\theta_2) + s_{2N}(\theta_1) \right),$$ and $$\begin{aligned}
\hat{\zeta}'(-2N)&= -N(-1)^N &\left\{\int_0^\infty dx x^{2N-1} \ln
\left|\frac{F(ix)}{F_a(ix)} \right|+ \int_0^\infty dx x^{2N-1} \ln
{|F_a(ix)|}_{\text{reg}}\right.\nonumber \\
&&\left.-\left(\gamma+\frac{1}{2N}\right)\left(r_{2N}(\theta_1)+s_{2N}(\theta_2)\right)\right\}.\end{aligned}$$ where $\gamma$ is Euler’s constant. We will quote the value of the first four coefficients $r_k$ and $s_k$ in Appendix \[sec:coeff\].
For the third term in Eq. (\[eq:zeta\]), we can interchange the order of integration and summation for large $s$. The resulting sum is not of the form considered above, but is a generalized Epstein type $\zeta$ function. Following Bordag *et al.* [@Bordag:1996gm], we can again write the sum as a contour integral of $F$ around a contour which encloses all the positive zeros of $F$. $$\label{eq:zetaE}
\tilde{\zeta}(s)=\sum_n \left(\lambda^2+b_n^2\right)^{-s}
= \frac{1}{2\pi i} \int_{\cal C} dz \left(\lambda^2+z^2\right)^{-s} \frac{F'(z)}{F(z)}.$$
For $1/2<s<1$, we can proceed in a similar manner to the above and deform the contour to an integral over the imaginary axis. This gives $$\label{eq:zetatildereg}
\tilde{\zeta}(s)=\frac{\sin \pi s}{\pi} \int_{\lambda}^\infty dx \left(x^2-\lambda^2\right)^{-s} \frac{d}{dx} \ln |F(ix)|.$$ We again add and subtract the terms that cause the integral to diverge at $s=0$. We find that the analytic continuation of Eq. (\[eq:zetaE\]) at $s=0$ is $$\begin{aligned}
\tilde{\zeta}(0) &=& \frac{\chi}{2}, \label{eq:epszeta0}\\
\tilde{\zeta}'(0)&=&-\ln |\pi F(i\lambda)| \label{eq:epszetaprime0} . \end{aligned}$$ Hence, we can now write down the complete $\zeta$ function and its derivative at $s=0$. We find $$\begin{aligned}
\zeta(0)&=&\frac{1}{16\pi^2 a^4}\int|g^{(4)}| d^4 x
\left\{-\chi \frac{17}{960} - \sum_{i=1,2}\left(r_4(\theta_i)+\frac14 r_2
(\theta_i) \right)\right\}, \label{eq:zeta0}\\
\zeta'(0)&=&-\frac{a^{-4}}{8\pi^2}\int|g^{(4)}| d^4 x
\left(G(\theta_1,\theta_2) + C_1(\theta_1)
+ C_2(\theta_2)+q\right) \nonumber \\
&&+\ln a^2 \zeta(0) - 2 \gamma \zeta(0),
\label{eq:zetaprime0}\end{aligned}$$ where the “non-local” part in Eq. (\[eq:zetaprime0\]) is $$\begin{aligned}
\label{eq:nonlocal}
{G}(\theta_1,\theta_2)&=&\int_0^\infty dx \ x\left(x^2+\frac14\right)
\ln \left| \frac{F(ix)}{F_a(ix)}\right| \nonumber\\
&&-2\int_0^\infty dx \ x\left(x^2+\frac14\right)
{\left(1+e^{2\pi x}\right)}^{-1}\ln \left|\pi F(ix) \right|,\end{aligned}$$ the “local” functions $C_1(\theta)$ and $C_2(\theta)$ are $$\begin{aligned}
\label{eq:local}
C_1(\theta)&=&\int_0^\infty dx \ x^3
\left\{ \ln |\Sigma^S(x,\theta)|
-\sum_{k=1}^{3}
s_k(\theta)x^{-k}-s_{4}(\theta)x^{-4}e^{-1/x}\right\}\nonumber \\
&&+\frac14\int_0^\infty dx \ x
\left\{ \ln |\Sigma^S(x,\theta)|
-s_1(\theta)x^{-1}-s_{2}(\theta)x^{-2}e^{-1/x}\right\}+\frac12
s_4(\theta),\\
C_2(\theta)&=&\int_0^\infty dx \ x^3
\left\{ \ln |\Sigma^R(x,\theta)|
-\sum_{k=1}^{3}
r_k(\theta)x^{-n}-r_{4}(\theta)x^{-4}e^{-1/x}\right\}\nonumber \\
&&+\frac14\int_0^\infty dx \ x
\left\{ \ln |\Sigma^R(x,\theta)|
-r_1(\theta)x^{-1}-r_{2}(\theta)x^{-2}e^{-1/x}\right\}+\frac12
r_4(\theta),\end{aligned}$$ and the constant part is $$\label{eq:const}
q=\chi \gamma \frac{17}{960} + \int_0^\infty dx \ x\left(x^2+\frac14\right) \ln
\left|\frac{\sinh\pi x}{\cosh \pi(x-i\mu)}\right|.$$
The one-loop effective action in the conformally rescaled metric is $$W_\Omega=-\frac12 \zeta'(0)-\frac12\zeta(0)\ln \mu_R^2,$$ so it can be seen that the terms proportional to $\zeta(0)$ in Eq. (\[eq:zetaprime0\]) can be absorbed into a redefinition of the renormalization scale $\mu_R$.
As a check on our results, the renormalization scale dependent term $\zeta(0)$ can also be calculated directly using heat kernel methods. This is done in Section \[sec:cocycle\].
Cocycle function {#sec:cocycle}
================
The cocycle function for the conformal rescaling was given in Eq. (\[eq:cocycle\]) in terms of the $B_{5/2}(f,\Delta)$ heat kernel coefficient. The $B_{5/2}(f,\Delta)$ heat kernel coefficient has been calculated for general operators of Laplace type with mixed Dirichlet and Robin boundary conditions in [@Branson:1999jz]. It is comprised of curvature terms of order $R^2$ evaluated only on the boundary of the spacetime. The general expression is quite lengthy. However, for a scalar field obeying Dirichlet boundary conditions, many terms are zero. Additionally, the heat kernel coefficients simplify further for the case of a maximally symmetric boundary.
The calculation of the heat kernel coefficient is straightforward but tedious. For the metric (\[eq:metric\]) and operator (\[eq:operator\]), we find that the cocycle function can be written as $$C[\Omega]=\sum_{i=1,2}\frac{1}{a^4(4\pi)^2}\int {|g^{(4)}|}^{1/2} d^4 x
\left\{\omega(z_i){\cal A}_i + {\cal B}_i\right\},$$ where $$\label{eq:calA}
{\cal A}_i= \frac{17}{1920} +
\frac{3}{16}\frac{1+\cos^2\theta_i}{\sin^4\theta_i}\left(\mu^2-1/4\right)
- \frac{1}{8} \frac{1}{\sin^4\theta_i}\left(\mu^2-1/4\right)^2,$$ and $$\label{eq:calB}
{\cal B}_i=-\frac{35}{192}-\frac{889}{1024}\cot^4\theta_i
-\frac{265}{256}\cot^2\theta_i-\frac{1}{64}\frac{\mu^2-4}{\sin^4\theta_i}
\left(17\cos^2 \theta_i+4\right),$$ where we have reintroduced $\mu=\sqrt{4-20\xi+m^2/\sigma^2}$.
One can also use the heat kernel to directly evaluate $\zeta(0)$ through the relation $$\zeta(0)=B_{5/2}(1,\Delta)=\sum_{i=1,2}\frac{1}{a^4(4\pi)^2}\int {|g^{(4)}|}^{1/2} d^4 x
\ {\cal A}_i.$$ Inserting the expressions for $r_2(\theta)$ and $r_4(\theta)$ from Appendix \[sec:coeff\] into the expression for $\zeta(0)$ for the conformally transformed operator (\[eq:zeta0\]), it is easily seen that there is exact agreement between the two calculations. These expressions also reduce to previously calculated values for flat branes [@Garriga:2001ar; @Moss:2004un] in the limit $a\rightarrow \infty$.
It is interesting to notice that the heat kernel coefficient $B_{5/2}$ is comprised entirely of local geometrical objects, while the full $\zeta$ function is a nonlocal quantity as information from both boundaries is required for the eigenvalue problem. It is therefore quite remarkable that $\zeta(0)$ can be obtained from the heat kernel coefficient, and this is a powerful check on our method.
Massless Conformally coupled case
=================================
The integrals in equations (\[eq:nonlocal\]) and (\[eq:local\]) cannot be done analytically in the general case we must resort to numerical methods. However, in the case $m=0$, and $\xi=3/16$, we find $\mu=1/2$ and the Legendre functions simplify to hyperbolic functions and exponentials, so that $$R^{-1/2}_{-1/2+ix}(\theta)= \frac{1}{x}\sqrt{\frac{2}{\pi}}
\sinh(x \theta),$$ and $$S^{1/2}_{-1/2+ix}(\theta)= \sqrt{\frac{2}{\pi}} e^{-x\theta}.$$
All the coefficients $r_k(\theta)$ and $s_k(\theta)$ vanish. The local functions $C_1(\theta)$ and $C_2(\theta)$ also vanish. The constant $q$ can be absorbed into a redefinition of the renormalization scale $\mu_R$. The expression for $\zeta(0)$ also becomes simple. We find $$\zeta(0)=\frac{1}{16\pi^2 a^4} \int |g^{(4)}|^{1/2} d^4 x \ \frac{17}{960}.$$
The integral in the first term for the nonlocal part $G(\theta_1,\theta_2)$ can be done explicitly in terms of the Riemann zeta function $\zeta_R(s)$. The remaining integral in $G(\theta_1,\theta_2)$ must be none numerically. We find $$\begin{aligned}
\label{eq:conscalar}
V_\Omega &= \displaystyle \frac{1}{16\pi^2 a^4}
&\left\{ -\frac{3\zeta_R(5)}{8 L^4} - \frac{\zeta_R(3)}{16 L^2}
- 2\int_0^\infty dx \frac{x \left(x^2 + \frac14\right)}{e^{2\pi x}+1} \ln \frac{2 \sinh{L x}}{x}
\right. \nonumber \\
&&\left.-\frac{17}{1920}\ln \left(a^2 {\mu_R}^2\right)\right\},\end{aligned}$$ where $L=\theta_2-\theta_1=z_2/a-z_1/a$. From the previous equation, it is clear that the effective potential in the conformally rescaled metric is a function solely of the nondimensional conformal distance $L$.
We plot the effective potential $V_\Omega$ as a function of $L$ in Figure \[fig:conformal\]. We have adjusted the renormalisation scale so that the maximum of the potential is at $V_\Omega=0$.
![Numerical results for the effective potential due to a conformally coupled scalar field as a function of the non-dimensional conformal length $L=\theta_2-\theta_1=z_2/a-z_1/a$, where $z_1$ and $z_2$ are the positions of the branes in conformal coordinate and $a$ is the $AdS_4$ radius of the brane at $z_1$[]{data-label="fig:conformal"}](conformal.eps){width="5in"}
To this result, we must also add the cocycle function, discussed in Section \[sec:cocycle\]. The total effective potential is then $$V=V_\Omega+\sum_{i=1,2}\frac{1}{16\pi^2 a^4} \left\{\frac{17}{1920} \ln \sigma a \sin \theta_i
+\frac{5}{96}+\frac{25}{128}
\cot^2\theta_i+\frac{131}{1024}\cot^4\theta_i\right\}.$$
The total effective potential $V$ is now no longer a function of $L=\theta_1-\theta_2$, but now depends explicitly on $\theta_1$ and $\theta_2$. The terms from the cocycle function can dominate the effective potential if $\theta_{1}$ or $\theta_2$ is small or close to $\pi$. Also, the terms due to the cocycle function cannot be absorbed into a redefinition of the brane tensions as is often done in the case of flat branes.
It is worthwhile mentioning here that one cannot obtain the effective action for a conformally coupled scalar field in the background with de Sitter branes, obtained in [@Elizalde:2002dd; @Moss:2003zk], by a simple analytic continuation of $a\rightarrow ia$. This is because of the different global properties of $S^4$ and $\mathrm{H}^4$, as has been noted in previous calculations of $\zeta$ functions on $S^4$ and $\mathrm{H}^4$ [@Camporesi:1991nw].
More general cases {#sec:example}
==================
Numerical Results {#sec:numerical}
-----------------
In the case of general mass and coupling constant $\xi$, the integrals in equations (\[eq:nonlocal\]), (\[eq:local\]) and (\[eq:const\]) must be computed numerically. As an example, we have chosen $m=0$ and $\xi=3/20$, so that $\mu=1$. The integrals that must be evaluated are fairly computationally intensive, but can be performed simply (if slowly) using computer packages such as MAPLE or MATHEMATICA.
The numerical results for the nonlocal part of the effective potential $G(\theta_1,\theta_2$ are shown as a function of $\theta_1=z_1/a$ and $\theta_2=z_2/a$ in Figure \[fig:nonlocal\]. It can be seen from this figure that this part of the effective action is not solely dependent on the difference $\theta_2-\theta_1$, unlike the conformally coupled case considered above. Also, this term in the effective action diverges strongly to negative values as $\theta_2-\theta_1\rightarrow 0$.
![Numerical results for the non-local part of the effective action. $\theta_1$ and $\theta_2$ are $z_1/a$ and $z_2/a$ respectively, where $z_1$ and $z_2$ are the positions of the branes in the conformal coordinate $z$, and $a$ is the $AdS_4$ radius of the brane at $z_1$. The contours are values of $G(\theta_1,\theta_2)$. The shaded region is disallowed, since we have restricted $z_2>z_1$[]{data-label="fig:nonlocal"}](nonlocal.eps){width="5in"}
The local functions $C_1(\theta)$ and $C_2(\theta)$ are shown in Figure \[fig:local\]. These terms can become significant if $\theta_{1}$ or $\theta_2$ are small or close to $\pi$. Finally, We calculate the constant $q$ to be $$q\approx -0.03128.$$
![Numerical results for the local functions $C_1(\theta)$ and $C_2(\theta)$ in the effective potential. Again, $\theta_1=z_1/a$ and $\theta_2=z_2/a$ where $z_1$ and $z_2$ are the positions of the branes in conformal coordinates, and $a$ is the $AdS_4$ radius of the brane at $z_1$.[]{data-label="fig:local"}](local.eps){width="5in"}
These numerical results enable one to calculate the finite part of the effective action of the conformally rescaled operator. However, to this, we must also add the renormalization scale dependent term $\zeta(0)\ln \mu_R^2$. From both Eq. (\[eq:zeta0\]) and the $B_{5/2}(1,\Delta)$ heat kernel coefficient, one can calculate the value of $\zeta(0)$. Using either method, we find $$\zeta(0)=\frac{1}{16\pi^2 a^4}\sum_{i=1,2} \int |g^{(4)}|^{1/2} d^4 x \left\{
\frac{19}{240}+\frac{9}{32}\cot^2\theta_i+\frac{27}{128}\cot^4\theta_i\right\}.$$
We have now calculated all the terms in the effective potential in the conformally rescaled metric $V_\Omega$. The total effective potential including the cocycle function is then $$\begin{aligned}
V=V_\Omega+\sum_{i=1,2}\frac{1}{16\pi^2 a^4} &&
\left\{\left(\frac{19}{240}+\frac{9}{32}\cot^2\theta_i+\frac{27}{128}\cot^4\theta_i\right)
\ln \sigma a \sin \theta_i \right. \nonumber \\
&&\left.+\frac{1}{192}-\frac{13}{256}
\cot^2\theta_i-\frac{73}{1024}\cot^4\theta_i\right\}.\end{aligned}$$
Analytical approximation
------------------------
The numerical results in Section \[sec:numerical\] show that the effective potential becomes very large if $L=\theta_2 -\theta_2\ll 1$. In this case, the dominating contribution to the effective potential is from the integral in the first term in Eq. (\[eq:nonlocal\]). We can approximate this integral using the asymptotic expansion of the Legendre functions. This gives a series in powers of $L$. The first two terms in this expansion of the effective potential are $$\label{eq:approx}
V \approx -\frac{3\zeta_R(5)}{128\pi^2(z_2-z_1)^4}
+\frac{\zeta_R(3)}{64\pi^2(z_2-z_1)^2}\left[\sigma^2 \left(\mu^2-1/4\right) -\frac{1}{4a^2}\right] + O(\ln(z_2-z_1).$$ The first term on the right hand side of Eq. (\[eq:approx\]) is the Casimir potential for two flat branes in flat space, separated by a distance $z_2-z_1$. Note that this leading term is independent of the mass or coupling to the curvature of the scalar field. A similar result was found for the small distance approximation with flat branes in [@Garriga:2001ar].
The second term in Eq. (\[eq:approx\]) is the first order correction to this result due to bulk and brane curvature. This approximation results in a much more manageable expression than the exact expression for the effective potential.
Conclusions
===========
We have calculated the one-loop effective potential for a scalar field with general mass and coupling to the Ricci scalar in the two brane Randall-Sundrum model with detuned brane tensions such that the boundary branes are $AdS_4$. Conformal rescalings of the metric are used to relate the metric to two hyperbolic branes. In general, the resulting expressions contain integrals of Legendre functions which must be performed numerically. We obtain some approximations in the small conformal distance limit, which reduce to the Casimir potential for two flat branes in flat space, and also calculate the first order corrections due to the brane curvature. Our results are checked by comparing the renormalization scale dependence obtained from the conformally rescaled operator with a direct computation using heat kernel coefficients. They are found to agree exactly.
It should be remembered that, in the dimensionally reduced theory, there is also a classical potential for the radion field, unless the brane tensions are tuned. It is interesting to compare the form of the classical radion potential and the quantum effective potential for the radion due to a bulk scalar field. The classical radion effective potential has been analyzed in [@Bagger:2003dy]. The classical potential is most naturally expressed in terms of the proper distance $r$, related to $z_1$ and $z_2$ by $$\pi r=\int_{z1}^{z2} e^{-w(z)} dz = \frac{1}{\sigma} \ln \tan \frac{z_2}{2a}-\frac{1}{\sigma} \ln \tan \frac{z_1}{2a}$$ For small conformal separation, $\pi r \approx z_2-z_1$ and the classical effective potential goes like $$V_\text{classical} \approx \text{const} \ (\sigma a \pi r)^{-2} \approx \text{const} \frac{1}{(\sigma a(z_2-z_1))^2}$$ where “$\text{const}$” is a positive constant. One can see from Eq. (\[eq:approx\]) that the effective potential generated by a quantized bulk scalar field would destabilize the classical potential at small separation, as the quantum effective potential diverges faster than the classical potential in this limit. Of course, there will also be an effective potential generated by graviton fluctuations which have not been included in this discussion.
It would be interesting to extend this calculation to higher spin fields so that one could investigate the effect of the supersymmetry breaking on the Casimir potential. An intriguing possibility in the supersymmetric Randall-Sundrum model is that one-loop effects could generate a potential for the supersymmetry breaking parameter (either the twist angle of the gravitino boundary condition or the v.e.v. of the fifth component of the gauge field) which is a modulus of compactification at the classical level [@Bagger:2003fy]. This situation is reminiscent of supersymmetry breaking in heterotic M theory by gaugino condensation [@Horava:1996vs].
I thank Ian Moss for helpful discussions and comments on an earlier draft of this paper.
Robin boundary conditions {#sec:robin}
=========================
$\zeta$-function on $\mathrm{H}^4 \times I$
-------------------------------------------
Robin boundary conditions have some combination of the field and its normal derivative vanishing on the boundary. We take $$\left(\partial_N + \eta K \right) \phi=0 \quad \text{on} \quad \partial {\cal M}.$$ where $K$ is the trace of the extrinsic curvature of the boundary and $N$ denotes the *outward* pointing unit normal. Under a conformal transformation, this changes to $$\left(\partial_N + \hat{\eta} K \Omega^{-1} \right) \tilde{\phi} =0\quad \text{on}
\quad \partial {\cal M},$$ where $\hat{\eta}=\eta-3/8$ since the field $\phi$ also transforms under the conformal rescaling. Applying this boundary condition to the general solution, the implicit equation for the $b_n$’s can be given in terms of new functions $T^{-\mu}_{\nu}(\theta)$ and $U^{\mu}_{\nu}(\theta)$, defined by
$$\begin{aligned}
T^{-\mu}_{\nu}(\theta)&=&R^{-\mu+1}_\nu(\theta)+\cot \theta
\left(1/2-\mu-4\hat{\eta} \right) R^{-\mu}_\nu(\theta), \\
U^{\mu}_{\nu}(\theta)&=&S^{\mu+1}_\nu(\theta)+\cot \theta \left(1/2+\mu-4\hat{\eta} \right) S^{\mu}_\nu(\theta),\end{aligned}$$
as $$F(b_n)=T^{-\mu}_{-1/2+b_n} (\theta_1)U^{\mu}_{-1/2+b_n}
(\theta_2)-T^{-\mu}_{-1/2+b_n} (\theta_2)U^{\mu}_{-1/2+b_n}(\theta_1)=0.$$ The procedure then follows the Dirichlet case, in that we write the $\zeta$ function as a contour integral, shift the contour to the real axis, and then analytically continue to find the effective action. If we define the functions $\Sigma^T(x,\theta)$ and $\Sigma^U(x,\theta)$ by $$T^{-\mu}_{-1/2+ix}
(\theta) = e^{i\pi \mu/2+i\pi/4} \frac{x}{\sqrt{2\pi }} e^{x \theta}
\frac{\Gamma(ix-\mu+1/2)}{\Gamma(ix+1)} \Sigma^T(x,\theta),$$ and $$U^{\mu}_{-1/2+ix}
(\theta) = -e^{i\pi \mu/2 -i\pi/4}x\sqrt{\frac{2}{\pi}} e^{-x \theta}
\frac{\Gamma(ix+\mu+1/2)}{\Gamma(ix+1)} \Sigma^U(x,\theta),$$ then $\Sigma^T(x,\theta)$ and $\Sigma^U(x,\theta)$ have asymptotic expansions of the Poincaré type. Similar to the Dirichlet case, we define coefficients $t_k(\theta)$ and $u_k(\theta)$ by $$\ln |\Sigma^T(x,\theta)|\sim\sum_{k=1}^\infty t_k (\theta) x^{-k}+O(e^{-x}),
\quad \ln |\Sigma^U(x,\theta)|\sim\sum_{k=1}^\infty u_k (\theta) x^{-k}.$$ We again refer the reader to Appendix \[sec:coeff\] for explicit expressions for $t_k(\theta)$ and $u_k(\theta)$.
Thus, for large $x$, $$\ln |F(ix)| \sim (\theta_2-\theta_1)x +\chi \ln(x) -\ln \pi + O(x^{-2}),$$ where now $$\chi=1.$$ The effective action in the conformally rescaled metric for Robin boundary conditions can now be found by substituting $R$, $S$, $\Sigma^R$, $\Sigma^S$, $r_k$ and $s_k$ in the Dirichlet case by $T$, $U$, $\Sigma^T$, $\Sigma^U$, $t_k$ and $u_k$. Additionally, one must replace $\chi=-1$ for Dirichlet boundary conditions by $\chi=+1$ for Robin boundary conditions.
One subtlety in the Robin case is that there may exist zero or imaginary solutions of $F(b_n)=0$. For example, a conformally coupled scalar field with Robin boundary conditions has a zero mode. This means that there will be a contribution to the $\zeta$-function from integral over the small semi-circle ${\cal C}_\varepsilon$ as $\varepsilon\rightarrow 0$. Imaginary values of $b_n$ signal an instability, and these situations should be physically unacceptable. In our analysis, we assume that the values of $\eta$, $\xi$ and $m$ are such that there are no zero or imaginary solutions of $F(b_n)=0$.
Cocycle function {#cocycle-function}
----------------
The heat kernel coefficient for scalar fields obeying Robin boundary conditions is a little more lengthy than the Dirichlet case. Again, the calculation is straightforward, but messy. For the cocycle function, we find that Eqs. (\[eq:calA\]) and (\[eq:calB\]) become $$\begin{aligned}
{\cal A}_i&=&-\frac{17}{1920} + 64 \cot^4\theta_i \hat{\eta}^4
+2\cot^2\theta_i\hat{\eta}^2
+\frac{1}{8}\frac{1}{\sin^4\theta_i}(\mu^2-1/4)^2 \nonumber \\
&&-\frac{1}{\sin^4\theta_i}\left[\cos^2\theta_i(8\hat{\eta}^2+2\hat{\eta})
+\frac{3}{16}(\cos^2\theta_i+1)\right](\mu^2-1/4),\end{aligned}$$ and $$\begin{aligned}
{\cal B}_i&=&\left(\frac{61}{15360}+\frac{11}{48}\hat{\eta}
-\frac23\hat{\eta}^2-8\hat{\eta}^3\right)\cot^4\theta_i
+\left(\frac{623}{768}+\frac{71}{24}\hat{\eta}
-\hat{\eta}^2\right)\frac{\cos^2\theta_i}{\sin^4\theta_i}
\nonumber \\
&&+\frac{35}{192}\frac{1}{\sin^4\theta_i}+\left[\left(\hat{\eta}+\frac{15}{64}\right){\cos^2\theta_i}
+\frac{1}{16}\right]\frac{\mu^2-4}{\sin^4\theta_i}.\end{aligned}$$ respectively.
Again, these results reduce to the previously known flat brane values calculated in [@Moss:2004un] in the limit $a\rightarrow\infty$.
An example
----------
In some special cases, the implicit equation for the eigenvalues for Robin boundary conditions can be reduced to the one resembling that for a Dirichlet case, plus one “extra” mode. As an example we consider the case where $\mu=0$ and $\eta=1/2$. Then the implicit equation for the $b_n$ reduces to $$\left(b_n^2-\frac14\right)\left(R^{-1}_{-1/2+b_n} (\theta_1)S^{1}_{-1/2+b_n}
(\theta_2)-R^{-1}_{-1/2+b_n}
(\theta_2)S^{1}_{-1/2+b_n}(\theta_1)\right)
=0.$$ This is simply the example considered in Section \[sec:example\], but with the “extra” mode at $b_n=1/2$. We can add this extra mode to the $\zeta$-function considered in Section \[sec:example\] by hand. This gives an extra contribution to $\zeta(0)$ of $-(240\pi^2 a^4)^{-1}$, giving $$\zeta(0)=\frac{1}{16\pi^2 a^4}\sum_{i=1,2} \int |g^{(4)}|^{1/2} d^4 x \left\{
\frac{11}{240}+\frac{9}{32}\cot^2\theta_i+\frac{27}{128}\cot^4\theta_i\right\},$$ again in agreement with the value calculated directly from the heat kernel coefficient $B_{5/2}(1,\Delta)$. The extra mode will also give an extra contribution to $\zeta'(0)$ of a constant (independent of $\theta_1$ or $\theta_2$).
However, the cocycle function cannot be deduced from the Dirichlet case, and must be computed explicitly.
Coefficients in asymptotic expansions of the Legendre functions {#sec:coeff}
===============================================================
Dirichlet boundary conditions {#sec:dirchletcoeff}
-----------------------------
From the asymptotic expansion of the Legendre functions for large degree [@Bateman page 146], the asymptotic expansion of $\ln |\Sigma^R(x,\theta)|$ and $\ln |\Sigma^S(x,\theta)|$ can be shown to be of the form $$\ln |\Sigma^R(x,\theta)|\sim\sum_{k=1}^\infty r_k (\theta) x^{-k},
\quad \ln |\Sigma^S(x,\theta)|\sim\sum_{k=1}^\infty s_k (\theta) x^{-k}.$$ The coefficients are easily evaluated with the help of a computer algebra package such as MAPLE or MATHEMATICA. We obtain the first four coefficients as
$$\begin{aligned}
r_1(\theta) & = & -\frac12 \cot \theta \left(\mu^2-1/4\right), \\
r_2(\theta) & = & -\frac{1}{4} \frac{1}{\sin^2\theta}\left(\mu^2-1/4\right), \\
r_3(\theta) & = & -\frac{1}{96} \frac{\cos \theta}{\sin^3 \theta} \left(\mu^2-1/4\right)
\left(8\mu^2\cos^2 \theta -12\mu^2 - 2\cos^2 \theta + 27 \right),\\
r_4(\theta) & = & \frac{1}{32}\frac{1}{\sin^4\theta}\left(\mu^2-1/4\right)
\left(4\mu^2-8\cos^2\theta-5\right),\end{aligned}$$
and $s_n(\theta)=(-1)^n r_n(\theta)$.
Robin boundary conditions {#sec:robincoeff}
-------------------------
Similarly to the Dirichlet case, for Robin boundary conditions we have $$\ln |\Sigma^T(x,\theta)|\sim\sum_{k=1}^\infty t_k (\theta) x^{-k},
\quad \ln |\Sigma^U(x,\theta)|\sim\sum_{k=1}^\infty u_k (\theta) x^{-k}.$$ This time, we find
$$\begin{aligned}
t_1(\theta) & = & -\frac{1}{2}\cot\theta \left(\mu^2-1/4+8\hat{\eta}\right),\\
t_2(\theta) & = & \frac{1}{4\sin^2\theta}\left(\mu^2-1/4-32\hat{\eta}^2\cos^2\theta
\right),\\
t_3(\theta) & = & -\frac{1}{384}\frac{\cos\theta}{\sin^3\theta}\left(
21-72\mu^2-48\mu^4+2\cos^2\theta-16\mu^2\cos^2\theta\right. \nonumber \\
&&\quad \quad \quad \quad \left.+32\mu^4\cos^2\theta
+192\hat{\eta}+8192\hat{\eta}^3\cos^2\theta-768\mu^2\hat{\eta} \right),\\
t_4(\theta) & = &-\frac{1}{128\sin^4\theta}
\left(5-24\mu^2+16\mu^4+8\cos^2\theta+8192\hat{\eta^4}\cos^4\theta
-32\mu^2\cos^2\theta
\right. \nonumber \\
&&\quad \quad \quad \quad
\left.+256\hat{\eta}^2\cos^2\theta+64\hat{\eta}\cos^2\theta-256\mu^2\hat{\eta}\cos^2\theta-1024\mu^2\hat{\eta}^2\cos^2\theta\right),\end{aligned}$$
and $u_i(\theta)=(-1)^i t_i(\theta)$.
[^1]: In fact, $\Delta_I$ is the Kaluza-Klein mass operator
|
---
abstract: 'We study spin and valley transports in junctions composed of silicene and topological crystalline insulators. We consider normal/magnetic/normal Dirac metal junctions where a gate electrode is attached to the magnetic region. In normal/antiferromagnetic/normal silicene junction, we show that the current through this junction is valley and spin polarized due to the coupling between valley and spin degrees of freedom, and the valley and spin polarizations can be tuned by local application of a gate voltage. In particular, we find a fully valley and spin polarized current by applying the electric field. In normal/ferromagnetic/normal topological crystalline insulator junction with a strain induced in the ferromagnetic segment, we investigate valley resolved conductances and clarify how the valley polarization stemming from the strain and exchange field appears in this junction. It is found that changing the direction of the magnetization and the potential in the ferromagnetic region, one can control the dominant valley contribution out of four valley degrees of freedom. We also review spin transport in normal/ferromagnetic/normal graphene junctions, and spin and valley transports in normal/ferromagnetic/normal silicene junctions for comparison.'
author:
- Takehito Yokoyama
title: Spin and valley transports in junctions of Dirac fermions
---
Introduction
============
There has been a great interest in graphene due to its rich potential from fundamental and applied physics point of view. [@Ando; @Katsnelson; @Castro] Graphene is composed of carbon atoms on a two-dimensional honeycomb lattice. Consequently, electrons in graphene obey the massless Dirac equation. The recent experimetal progress of fabrication of single graphene sheets has triggered tremendous interests from the scientific community.[@novoselov; @zhang; @novoselov_nature] Up to now, many intriguing aspects of graphene have been revealed, such as half integer and unconventional quantum Hall effect[@zhang; @Novoselov2; @Yang], minimum conductivity[@novoselov_nature], and the Klein tunneling[@katsnelson; @Cheianov; @Cayssol; @Huard; @Williams; @Ozyilmaz; @Stander; @Young; @Young2]. Graphene is also a suitable material for applications: it exhibits gate-voltage-controlled carrier conduction, high field-effect mobilities and a small spin-orbit interaction.[@Kane; @Hernando] Therefore, graphene offers a good testing ground for observing spintronics effects. [@Son; @Kan; @Yazyev; @Haugen; @Tombros; @Ohishi; @Cho; @Yokoyama; @Linder2; @Yokoyama2] It has been shown that zigzag edge graphene nanoribbon becomes half-metallic by an external transverse electric field due to the different chemical potential shift at the edges.[@Son; @Kan; @Yazyev] This indicates the high controllability of ferromagnetism in graphene and hence paves the way for spintronics application of graphene. In graphene covered by ferromagnet, spin transport controlled by a gate electrode has been predicted. [@Haugen; @Yokoyama; @Yokoyama2] Also, there are some attempts to use pseudospin (sublattice) degrees of freedom in graphene in order to obtain new functionalities. [@Jose; @Xia; @Majidi]
The goal of valleytronics is to manipulate valley degrees of freedom by electric means and vice versa. This field has developed in graphene[@Rycerz; @Xiao; @Akhmerov], because graphene has two inequivalent Dirac cones at $K$ and $K'$ points, which can be considered as valley degree of freedom. In graphene nanoribbons with a zigzag edge, valley filter and valley valve effect have been predicted.[@Rycerz; @Akhmerov] These stem from intervalley scatterings by a potential step and are thus controllable by local application of a gate voltage.
Silicene is a monolayer of silicon atoms on a two dimensional honeycomb lattice: the silicon analog of graphene.[@Takeda] Recently, it has been reported that this material has been synthesized.[@Lalmi; @Padova; @Padova2; @Vogt; @Lin; @Fleurence] Although silicene is composd of silicon atoms on honeycomb lattice and hence electrons in silicene obey the Dirac equation around the $K$ and $K'$ points at low energy[@Liu; @Liu2], there are a few important differences from graphene: (i) the honeycomb lattice is buckled. Hence, the mass of the Dirac electrons in silicene can be manipulated by external electric field.[@Ezawa; @Ezawa2] The discovery of this property has triggered many intriguing predictions. It has been predicted that there occurs a topological phase transition between topologically trivial and topological insulators by applying electric field. [@Ezawa; @Ezawa2] (ii) silicene has a large spin-orbit coupling compared to graphene which couples spin and valley degrees of freedom. Therefore, one may expect interesting spin and valley coupled physics in silicene.
Topological crystalline insulators are new states of matter defined by a topological invariant constructed by crystal symmetries.[@Fu; @Hsieh; @Slager] Topological crystalline insulators possess even number of gapless surface states on crystal faces that preserve the underlying symmetry. These gapless surface states are topologically protected: they are robust against perturbations as long as the underlying symmetry is preserved. The (001) surface states composed of four Dirac cones in Pb$_x$Sn$_{1-x}$(Te, Se), the first topological crystalline insulator material, have been predicted[@Hsieh] and observed in angle-resolved photoemission spectroscopy experiments[@Tanaka; @Xu; @Dziawa; @Wojek; @Tanaka2]. Recently, measurement of surface transport in epitaxial SnTe thin films has been also reported. [@Taskin] Since these materials have four Dirac cones in contrast to honeycomb systems, topological crystalline insulators have a potential to be placed ahead of graphene for valleytronics applications.[@Ezawa4]
In this paper, we first review spin transport in normal/ferromagnetic/normal graphene junctions, and spin and valley transports in normal/ferromagnetic/normal silicene junctions. Then, we study spin and valley transports in junctions composed of silicene and topological crystalline insulators. We consider normal/magnetic/normal Dirac metal junctions where a gate electrode is attached to the magnetic region. In normal/antiferromagnetic/normal silicene junction, we show that the current through this junction is valley and spin polarized due to the coupling between valley and spin degrees of freedom, and the valley and spin polarizations can be tuned by local application of a gate voltage. In particular, we find a fully valley and spin polarized current by applying the electric field. In normal/ferromagnetic/normal topological crystalline insulator junction with a strain induced in the ferromagnetic segment, we investigate valley resolved conductances and clarify how the valley polarization stemming from the strain and exchange field appears in this junction. It is found that changing the direction of the magnetization and the potential in the ferromagnetic region, one can control the dominant valley contribution out of four valley degrees of freedom.
Graphene
========
Here, we review spin transport in normal/ferromagnetic/normal graphene junctions, following Ref.[@Yokoyama] .
The electrons in graphene obey a massless Dirac equation given by $$H_\pm = v_F (\sigma_x k_x + \eta \sigma_y k_y)$$ with Pauli matrices $\sigma_x$ and $\sigma_y$ which operate on the sublattice space of the honeycomb lattice. The $\eta=\pm$ sign corresponds to the two valleys of $K$ and $K'$ points in the Brillouin zone. Also, there is a valley degeneracy. Hence, one can consider one of the two valleys ($H_\pm$). [@morpurgo2] The linear dispersion relation is valid for Fermi levels up to 1 eV, [@wallace] where the electrons in graphene behave like Weyl fermions in the low-energy regime.
We consider a two dimensional normal/ferromagnetic/normal graphene junction where a gate electrode is attached to the ferromagnetic region. This junction may be realized by putting a ferromagnetic insulator on top of graphene or doping magnetic atoms into graphene. See Figure \[fig1\] for the schematic of the model. We assume that the interfaces are parallel to the $y$-axis and located at $x=0$ and $x=L$. Due to the valley degeneracy, we consider the Hamiltionian $H_+$ with $H_+ = v_F (\sigma_x k_x + \sigma_y k_y) - V(x)$, $V(x)=E_F$ in the normal graphenes and $V(x)= E_F + U \pm H$ in the ferromagnetic graphene. Here, $E_F=v_F k_F$ is the Fermi energy, $U$ is the potential shift controllable by the gate voltage, and $H$ is the exchange field. Here, $\pm$ signs correspond to majority and minority spins. The wavefunctions in each regions can be written as $$\begin{aligned}
\psi_1 = \left( {\begin{array}{*{20}c}
1 \\
{e^{i\theta } } \\
\end{array}} \right)e^{ip\cos \theta x +ip_y y}
+ a_\pm \left( {\begin{array}{*{20}c}
1 \\
{ - e^{ - i\theta } } \\
\end{array}} \right)e^{ - ip\cos \theta x +ip_y y} ,\end{aligned}$$ $$\begin{aligned}
\psi_2 = b_\pm \left( {\begin{array}{*{20}c}
1 \\
{e^{i\theta '} } \\
\end{array}} \right)e^{ip'_\pm \cos \theta 'x +ip_y y}
+ c_\pm \left( {\begin{array}{*{20}c}
1 \\
{ - e^{ - i\theta '} } \\
\end{array}} \right)e^{ - ip'_\pm \cos \theta 'x +ip_y y} ,\end{aligned}$$ $$\begin{aligned}
\psi_3 = d_\pm \left( {\begin{array}{*{20}c}
1 \\
{e^{i\theta } } \\
\end{array}} \right)e^{ip\cos \theta x +ip_y y} \end{aligned}$$ with angles of incidence $\theta$ and $\theta'$, $p = (E + E_F )/v_F$ and $ p'_\pm = (E + E_F + U \pm H)/v_F$. Here, $\psi_1$ and $\psi_3$ denote wavefunctions in the left and right normal graphenes, respectively, while $\psi_2$ is a wavefunction in the ferromagnetic graphene. Due to the translational invariance in the $y$-direction, the momentum parallel to the $y$-axis is conserved: $p_y= p\sin \theta = p'\sin \theta '$.
By matching the wave functions at the interfaces, we obtain the coefficients in the above wavefunctions in Eqs.(2-4). Note that these conditions lead to the current conservation at the interfaces because they are reduced to $\hat v_x \psi_1=\hat v_x \psi_2$ at $x = 0$ and $\hat v_x \psi_2=\hat v_x \psi_3$ at $x = L$ where $\hat v_x $ is the velocity operator given by $\hat v_x = \partial H_ + /\partial k_x = v_F\sigma _x$.
The transmission coefficient is represented as $$\begin{aligned}
d_\pm = \frac{{\cos \theta \cos \theta '{\mathop{\rm e}\nolimits} ^{ - ipL\cos \theta } }}{{\cos(p'_\pm L\cos \theta ')\cos \theta \cos \theta ' - i\sin (p'_\pm L\cos \theta ')(1 - \sin \theta \sin \theta ')}}.\end{aligned}$$
Thus, the dimensionless spin-resolved conductances $G_{\uparrow,\downarrow}$ are obtained as $$\begin{aligned}
G_{\uparrow,\downarrow} = \frac{1}{2}\int_{ - \pi /2}^{\pi /2} {d\theta \cos \theta T_{\uparrow,\downarrow} (\theta )} \end{aligned}$$ with $T_{\uparrow,\downarrow} (\theta ) = \left| {d_\pm(\theta )} \right|^2$. Finally, the spin conductance $G_s$ is defined as $G_s = G_ \uparrow - G_ \downarrow$. Below, we focus on the conductances at zero voltage, setting $E=0$.
First, we will explain the underlying mechanism of spin manipulation by the gate voltage. In the limit of $\left| U \pm H \right| \gg E_F$, we have $\theta' \to 0$, and therefore, the transmission coefficient becomes $$\begin{aligned}
d_\pm \to \frac{{\cos \theta {\mathop{\rm e}\nolimits} ^{ - ipL\cos \theta } }}{{\cos\chi_\pm \cos \theta - i\sin \chi_\pm }}\end{aligned}$$ with $\chi _\pm = \chi \pm \chi _H, \chi = UL/v_F,$ and $\chi _H =HL/v_F$. The transmission probability is thus given by [@katsnelson] $$\begin{aligned}
T_{ \uparrow , \downarrow } (\theta ) \to \frac{{\cos ^2 \theta }}{{1 - \sin ^2 \theta \cos^2 \chi _\pm }}.\end{aligned}$$ From Eq.(8), we find the $\pi$-periodicity with respect to $\chi_\pm$ or $\chi$. [@Haugen; @katsnelson; @linder; @sengupta] It is also seen that $G_{\uparrow,\downarrow} $ has a maximum (minimum) value of 1 (2/3) at $\chi_\pm=0$ $(\pi/2)$. The phase difference between $G_{\uparrow}$ and $G_{\downarrow} $ is given by $\chi _+ - \chi_- = 2\chi _H = 2HL/v_F$. If the phase difference is equal to the half period $\pi/2$ (i.e., $H/E_F= \pi /4 k_F L$), one can expect a large spin current which oscillates with $\chi$, i.e., the gate voltage, because when one of $G_{\uparrow}$ and $G_{\downarrow} $ has a maximum at a certain $\chi$, the other has a minimum at the same $\chi$. As a result, the value of $G_s$ oscillates between $-1/3$ and $1/3$. Notice that the electrical conductance $G_{\uparrow}+G_{\downarrow} $ in the junctions is always positive and hence spin current reversal in our model is not accompanied with the charge current reversal.
In Fig. \[f2\], we show the results in this limiting case. Figure \[f2\] (a) depicts spin resolved conductances as a function of $\chi$. Here, the phases of $G_ \uparrow$ and $G_ \downarrow$ are shifted by half period, $\chi _+ - \chi _-= \pi/2$. As shown in Fig. \[f2\] (b), we obtain a finite spin current. Interestingly, the spin conductance oscillates with the period $\pi$ with respect to $\chi$. This indicates that one can reverse the spin current by changing the gate voltage. Here, we have focused on the limiting case. For more general cases, see Ref.[@Yokoyama].
Silicene
========
Spin and valley transports in normal/ferromagnetic/normal silicene junction have been studied in Ref.[@Yokoyama3]. Here, we review spin and valley transports in this junction and investigate them in a normal/antiferromagnetic/normal silicene junction as shown in Fig. \[fig1\].
Formulation
-----------
The Hamiltonian of the (anti)ferromagnetic silicene is given by [@Liu; @Liu2; @Ezawa; @Ezawa2] $$\begin{aligned}
H = \hbar v_F (k_x \tau _x - \eta k_y \tau _y ) - \Delta _{\eta \sigma } \tau _z - \sigma h\end{aligned}$$ with $\Delta _{\eta \sigma } = \eta \sigma \Delta _{so} - \Delta _z + \sigma h_s $. $\tau$ is the Pauli matrix in sublattice pseudospin space. $\Delta _{so}$ denotes the spin-orbit coupling. $\Delta _z$ is the onsite potential difference between $A$ and $B$ sublattices, which can be manipulated by an electric field applied perpendicular to the plane. $h (h_s)$ is the ferromagnetic (antiferromagnetic or staggered) exchange field in the ferromagnetic (antiferromagnetic) region. $\eta = \pm 1$ corresponds to the $K$ and $K'$ points. $\sigma = \pm 1$ denotes the spin indices. The large value of $\Delta _{so}= 3.9$ meV in silicene [@Liu2] leads to a coupling between valley and spin degrees of freedom, which is a clear distinction from graphene. In the normal regions, we set $\Delta _z=h=h_s=0$. Thus, the gate electrode is attached to the magnetic segment. The eigenvalues of the Hamiltonian in the normal and magnetic silicene are given by $$\begin{aligned}
E = \pm \sqrt {(\hbar v_F k)^2 + (\Delta _N )^2 }
= \pm \sqrt {(\hbar v_F k')^2 + (\Delta _F )^2 } - \sigma h\end{aligned}$$ with $\Delta _N = \eta \sigma \Delta _{so}$ and $\Delta _F = \eta \sigma \Delta _{so} - \Delta _z + \sigma h_s$. $k$ and $k'$ are momenta in the normal and the magnetic regions, respectively. Let $x$-axis perpendicular to the interface and assume the translational invariance along the $y$-axis. The interfaces between the normal and the magnetic silicene are located at $x=0$ and $x=L$ where $L$ is the length of the magnetic silicene. Then, the wavefunctions for valley $\eta$ and spin $\sigma$ in each region can be written as
$$\begin{aligned}
\psi (x < 0) = \frac{1}{{\sqrt {2EE_N } }}e^{ik_x x} \left( {\begin{array}{*{20}c}
{\hbar v_F k_ + } \\
{E_N } \\
\end{array}} \right) + \frac{{r_{\eta ,\sigma } }}{{\sqrt {2EE_N } }}e^{ - ik_x x} \left( {\begin{array}{*{20}c}
{ - \hbar v_F k_ - } \\
{E_N } \\
\end{array}} \right), \\
\psi (0 < x < L) = a_{\eta ,\sigma } e^{ik'_x x} \left( {\begin{array}{*{20}c}
{\hbar v_F k'_ + } \\
{E_F } \\
\end{array}} \right) + b_{\eta ,\sigma } e^{ - ik'_x x} \left( {\begin{array}{*{20}c}
{ - \hbar v_F k'_ - } \\
{E_F } \\
\end{array}} \right), \\
\psi (L < x) = \frac{{t_{\eta ,\sigma } }}{{\sqrt {2EE_N } }}e^{ik_x x} \left( {\begin{array}{*{20}c}
{\hbar v_F k_ + } \\
{E_N } \\
\end{array}} \right)\end{aligned}$$
with $\hbar v_F k'_x = \sqrt {(E + \sigma h)^2 - (\Delta _F )^2 - (\hbar v_F k_y )^2 } $, $ E_N = E + \Delta _N ,E_F = E + \sigma h + \Delta _F$, and $k_ \pm ^{(\prime)} = k_x^{(\prime)} \pm i\eta k_y$. Here, ${r_{\eta ,\sigma } }$ and ${t_{\eta ,\sigma } }$ are reflection and transmission coefficients, respectively. By matching the wavefunctions at the interfaces, we obtain the transmission coefficient: $$\begin{aligned}
t_{\eta ,\sigma } = 4k_x k'_x E_N E_F e^{ - ik_x L} /A,\quad \\
A = (\alpha ^{ - 1} - \alpha )k^2 E_F^2 + (\alpha ^{ - 1} - \alpha )(k')^2 E_N^2
+ E_N E_F \left[ {k_ + (\alpha ^{ - 1} k'_ + + \alpha k'_ - ) + k_ - (\alpha ^{ - 1} k'_ - + \alpha k'_ + )} \right] \end{aligned}$$ with $\alpha = e^{ik'_x L}$.
By setting $k_x = k\cos \phi$ and $k_y = k\sin \phi$, we define normalized valley and spin resolved conductance: $$\begin{aligned}
G_{\eta \sigma } = \frac{1}{2}\int_{ - \pi /2}^{\pi /2} {\left| {t_{\eta, \sigma } } \right|^2 \cos \phi d\phi } .\end{aligned}$$ The valley and spin resolved conductances, $G_{K^{(')}}$ and $G_{ \uparrow ( \downarrow )}$, and valley and spin polarizations, $G_v$ and $G_s$, are defined as follows: $$\begin{aligned}
G_{K^{(')}} = \frac{{G_{{K^{(')}} \uparrow } + G_{{K^{(')}} \downarrow } }}{2}, \\
G_{ \uparrow ( \downarrow )} = \frac{{G_{K \uparrow ( \downarrow )} + G_{K' \uparrow ( \downarrow )} }}{2}, \\
G_v = \frac{{G_K - G_{K'} }}{{G_K + G_{K'} }}, \\
G_s = \frac{{G_ \uparrow - G_ \downarrow }}{{G_ \uparrow + G_ \downarrow }} .\end{aligned}$$
Results
-------
In the following, we fix $L$ and $\Delta_{so}$ as $k_F L=3$ and $\Delta_{so}/E=0.5$ where $k_F=E/(\hbar v_F)$. We consider a finite chemical potential by doping in silicene.
First, let us review the ferromagnetic junctions with $h_s=0$. [@Yokoyama3] Figure \[fig3\] depicts (a) valley resolved conductance $G_{K(K')}$ and (b) spin resolved conductance $G_{ \uparrow ( \downarrow )}$ as a function of $\Delta_z$. As seen from Figure \[fig3\] (a), with increasing $\Delta_z$, the current stemming from the $K'$ point strongly decreases. Then, $G_K$ gives a dominant contribution to the current. We find that $G_{ \uparrow}$ dominates over $G_{\downarrow}$ for large $\Delta_z$ as seen in Fig. \[fig3\](b). These behaviors are attributed to the band structures in the ferromagnetic region.[@Yokoyama3] Figure \[fig4\] illustrates (a) $G_v$ and (b) $G_s$ as functions of $\Delta_z$ and $h$ for $h_s=0$. The valley polarization $G_v$ is odd with respect to $\Delta_z$ and $h$. For large $\Delta_z$, $G_v$ becomes large as we found in Fig. \[fig3\] (a). However, for smaller $\Delta_z$, the magnitude of $G_v$ can be still $\sim 0.5$. We find that even the sign of the valley polarization can be changed by varying $\Delta_z$. It is also seen that $G_v$ changes significantly by varying the exchange field $h$. This indicates that the valley polarization can be manipulated magnetically. The spin polarization $G_s$ is odd in $h$ but even in $\Delta_z$. For large $h$, $G_s$ becomes large as expected. Even for small $h$, $G_s$ can be large for large $\Delta_z$. From Figure \[fig4\], it is also found that fully valley and spin polarized currents are realized for large $\Delta_z$ but relatively high polarizations ($\ge 0.5$) can be realized in a wide parameter regime.
The condition to realize fully valley polarized transport can be obtained as follows. For simplicity, let us focus on the regime with $\Delta _z>0$ and $h>0$. To locate the Fermi level $E (> \Delta_{so})$ within the band gap at the $K'$ point ($\eta = - 1$), $- \left| {\sigma \Delta _{so} + \Delta _z } \right| - \sigma h < \Delta _{so} < \left| {\sigma \Delta _{so} + \Delta _z } \right| - \sigma h$ should be satisfied. Therefore, we obtain the condition necessary for the fully valley polarized transport as $$\begin{aligned}
\Delta _z > \max (h, \Delta _{so} , 2\Delta _{so} - h) .\end{aligned}$$
Next, consider the antiferromagnetic junctions with $h=0$. Figure \[fig5\] depicts (a) valley resolved conductance $G_{K(K')}$ and (b) spin resolved conductance $G_{ \uparrow ( \downarrow )}$ as a function of $\Delta_z$. As seen from Figure \[fig5\] (a), with increasing $\Delta_z$, the current coming from the $K'$ point strongly decreases. Then, $G_K$ gives a dominant contribution to the current. We also find that $G_{ \uparrow}$ dominates over $G_{\downarrow}$ for large $\Delta_z$ as seen from Fig. \[fig5\](b). These behaviors are again attributed to the band structures in the antiferromagnetic region.
Figure \[fig6\] shows (a) $G_v$ and (b) $G_s$ as functions of $\Delta_z$ and $h_s$ for $h=0$. The valley polarization $G_v$ is an even function of $\Delta_z$ but an odd function of $h_s$. For large $\Delta_z$, $G_v$ becomes large, which is consistent with Fig. \[fig5\] (a). We find that the sign of the valley polarization can be changed by varying $\Delta_z$. In constrast to the ferromagnetic case with finite $h$, the $G_v$ can be large for $\Delta_z=0$ but with finite $h_s$. It is also seen that $G_v$ changes significantly by varying the exchange field $h_s$. This again indicates that the valley polarization can be controlled magnetically. The spin polarization $G_s$ is odd in $h_s$ and $\Delta_z$. Thus, the $G_s$ becomes zero at $\Delta_z=0$. This can be also understood from the fact that the bands are spin degenerate for $\Delta_z=0$. Even for small $h_s$, $G_s$ can be large for large $\Delta_z$. From this figure, it is found that fully valley and spin polarized currents are realized for large $\Delta_z$ and $h_s$ regime. Interestingly, we find some parameter regions where $G_v=0$ but $G_s=\pm1$ or $G_v=\pm1$ but $G_s=0$. Namely, by changing the tunable parameter $\Delta_z$, one can realize transitions from a fully valley polarized state without spin polarization to a fully spin polarized state without valley polariztion.
The conditions to realize fully valley or spin polarized transports are obtained as follows. Let us focus on the regime with $\Delta_z, h_s>0$. The gap for valley $\eta$ and spin $\sigma$ is given by $\left| {\Delta _F } \right| = \left| {\eta \sigma \Delta _{so} - \Delta _z + \sigma h_s } \right|$. To realize the fully valley polarized transport $G_v=1$, the gaps at the $K'$ point should be larger than the Fermi energy: $\left| { - \sigma \Delta _{so} - \Delta _z + \sigma h_s } \right| > E
$. Thus, we obtain $$\begin{aligned}
- \Delta _z + E +\Delta _{so}< h_s <\Delta _z - E +\Delta _{so}. \end{aligned}$$ For $\Delta _{so} /E=0.5$, this reduces to $ - \Delta _z /E + 1.5 < h_s/E <\Delta _z /E -0.5$, which is consistent with Fig. \[fig6\] (a). Similarly, to obtain $G_v=-1$, we require $\left| { \sigma \Delta _{so} - \Delta _z + \sigma h_s } \right| > E$, leading to $$\begin{aligned}
\Delta _z + E - \Delta _{so}< h_s .\end{aligned}$$ For $\Delta _{so} /E=0.5$, this reduces to $ \Delta _z / E +0.5< h_s/E$, which is consistent with Fig. \[fig6\] (a). To obtain the fully spin polarized transport, $G_p=1$, the gaps for spin down states are required to be larger than the Fermi energy: $\left| { \Delta _{so} + \Delta _z + h_s } \right|, \left| { \Delta _{so} -\Delta _z -h_s } \right| > E$. Thus, we obtain $$\begin{aligned}
- \Delta _z + E +\Delta _{so}< h_s.\end{aligned}$$ For $\Delta _{so} /E=0.5$, this reduces to $ \Delta _z / E +1.5< h_s/E$, consistent with Fig. \[fig6\] (b). Also, note that when $\Delta _z = h_s$ and $2 \Delta _z > E+\Delta _{so}$ are satisfied, the gaps for spin up states at the $K$ and $K'$ points coincide and the gaps for spin down states are larger than the Fermi energy. Thus, we have $G_v=0$ and $G_p=1$ in this case, as seen in Fig. \[fig6\].
For a ferromagnetic silicene with $k_F L$ = 1 and $E=$10 meV, since $v_F \sim 5 \times 10^5$m/s, we have $L \sim$ 10 nm. For $\Delta_z \sim E$, the electric field applied perpendicular to the plane is estimated as 34 meV/Å$ $ since the distance between the $A$ and $B$ sublattice planes is 0.46 Å. Here, we have assumed the zero temperature limit. This assumption is justified for tempereture regime lower than $\Delta_{so}$, $\Delta_z$, $h$ and $h_s$. Recently, based on first-principles calculations, stability and electronic structures of silicene on Ag(111) surfaces have been investigated.[@Guo; @Wang] It is found that Dirac electrons are absent near Fermi level in all the stable structures due to buckling of the Si monolayer and mixing between Si and Ag orbitals. It is also proposed that either BN substrate or hydrogen-processed Si surface is a good candidate to preserve Dirac electrons in silicene. [@Guo]
A ferromagnetic exchange field could be induced in silicene by the magnetic proximity effect with a magnetic insulator EuO as proposed for graphene, which could be of the order of 1 meV.[@Haugen] Exchange fields on A and B sublattices can be induced by sandwiching silicene by two (different) ferromagnets or attaching a honeycomb-lattice antiferromagnet such as antiferromagnetic manganese chalcogenophosphates (MnPX$_3$, X = S, Se) in monolayer form.[@Ezawa3; @Liang; @Li]
Topological crystalline insulator
=================================
Formulation
-----------
Consider normal/ferromagnetic/normal topological crystalline insulator junctions with flat interfaces at $x=0$ and $x=L$ (See Fig. \[fig1\]). We here study transports on the (001) surface of the topological crystalline insulator. The topological crystalline insulator has four Dirac cones with the same chirality at $\Lambda_X $, $\Lambda'_{X}$, $\Lambda_Y $, and $\Lambda'_{Y}$ points in the (001) surface. [@Liu3; @Fang; @Liu4; @Wang2; @Fan] The effective Hamiltonian of the topological crystalline insulator around the $\Lambda _X $ point is given by [@Liu3; @Fang; @Liu4; @Wang2; @Fan] $$\begin{aligned}
H_X = v_1 \tilde k_x \sigma _y - v_2 \tilde k_y \sigma _x + \tilde m\sigma _z + U\end{aligned}$$ where typically $v_1=1.3$eVÅ, $v_2=0.84$eVÅ, $\sigma$ is the Pauli matrix in spin space, and $$\begin{aligned}
\tilde k_x = k_x + \frac{1}{{v_1 }}\left( {\lambda _{11} \varepsilon _{11} + \lambda _{22} \varepsilon _{22} + \lambda _{33} \varepsilon _{33} + h_y } \right),\;\tilde k_y = k_y - \frac{1}{{v_2 }}\left( {\lambda _{12} \varepsilon _{12} + h_x } \right),\\ \tilde m = \frac{{n'}}{{\sqrt {n^2 + (n')^2 } }}\left( {\lambda _{23} \varepsilon _{23} + h_z } \right) \cong 0.35\left( {\lambda _{23} \varepsilon _{23} + h_z } \right). \end{aligned}$$ Here, $U$ is the potential, $\varepsilon _{ij}$ and $\lambda _{ij}$ $(i, j=1, 2, 3)$ are the strain tensor and electron-phonon couplings of the topological crystalline insulator, respectively. Strain may be induced by substituting Se for Sn,[@Okada] or by attaching a piezoelectric material such as BaTiO$_3$. [@Fan] $n=70$meV and $n'=26$meV describe the intervalley scattering. [@Liu3; @Fang; @Liu4; @Wang2; @Ezawa4] $h_i$ $(i=x, y, z)$ represents the induced exchange field in the ferromagnetic region given by $$\begin{aligned}
(h_x ,h_y ,h_z ) = h(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta ). \end{aligned}$$ We set $\varepsilon _{ij}=h=U=0$ in the normal regions and consider a scattering problem through a barrier region induced by the ferromagnetism and strain.
The wavefunctions in each regions can be written as $$\begin{aligned}
\psi (x \le 0) = \frac{1}{{\sqrt 2 E}}e^{ik_x x} \left( {\begin{array}{*{20}c}
{ - iv_1 k_x - v_2 k_y } \\
E \\
\end{array}} \right) + \frac{r}{{\sqrt 2 E}}e^{ - ik_x x} \left( {\begin{array}{*{20}c}
{iv_1 k_x - v_2 k_y } \\
E \\
\end{array}} \right), \\
\psi (0 < x < L) = \frac{a}{{\sqrt {2E'(E' - \tilde m)} }}e^{ik'_x x} \left( {\begin{array}{*{20}c}
{ - iv_1 k'_x - v_2 \tilde k_y } \\
{E' - \tilde m} \\
\end{array}} \right) + \frac{b}{{\sqrt {2E'(E' - \tilde m)} }}e^{ - ik'_x x} \left( {\begin{array}{*{20}c}
{iv_1 k'_x - v_2 \tilde k_y } \\
{E' - \tilde m} \\
\end{array}} \right), \\
\psi (x \ge L) = \frac{t}{{\sqrt 2 E}}e^{ik_x x} \left( {\begin{array}{*{20}c}
{ - iv_1 k_x - v_2 k_y } \\
E \\
\end{array}} \right) .\end{aligned}$$ Here, $r$ and $t$ are the reflection and transmission coefficients, respectively. We set $E' = E - U$ and assume that the Fermi energy is positive, $E>0$. The dispersion relations are then given by $E = \sqrt {(v_1 k_x )^2 + (v_2 k_y )^2 } = \pm \sqrt {(v_1 k'_x )^2 + (v_2 \tilde k_y )^2 + \tilde m^2 } + U$. Note that due to the translational symmetry in the $y$-direction, the momentum parallel to the $y$-axis is conserved, while the momentum parallel to the $x$-axis is not conserved.
By matching the wavefunctions at the interfaces, $$\begin{aligned}
\psi (+0)=\psi (-0), \; \psi (L+0)=\psi (L-0),\end{aligned}$$ we obtain the transmission coefficient: $$\begin{aligned}
t = \frac{{4pv_1 k'_x Ee^{ - ik_x L} \cos \phi }}{{e^{ - ik'_x L} \left( {iv_1 k'_x + v_2 \tilde k_y + ipe^{i\phi } } \right)\left( {iv_1 k'_x - v_2 \tilde k_y + ipe^{ - i\phi } } \right) + e^{ik'_x L} \left( { - iv_1 k'_x + v_2 \tilde k_y + ipe^{i\phi } } \right)\left( {iv_1 k'_x + v_2 \tilde k_y - ipe^{ - i\phi } } \right)}}.\end{aligned}$$ Here, $p = 1 - (U + \tilde m)/E$, and we set $v_1 k_x = E\cos \phi$ and $v_2 k_y = E\sin \phi$. The normalized conductance stemming from the $\Lambda _X$ point is calculated as $$\begin{aligned}
G_{X} = \frac{1}{2}\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left| t \right|^2 \cos \phi d\phi } .\end{aligned}$$ The conductance coming from the $\Lambda'_X$ point $G_{X'}$ can be obtaind by the substitution $\lambda _{23} \varepsilon _{23} \to - \lambda _{23} \varepsilon _{23}$ in the above result.[@Liu3; @Fang; @Liu4; @Wang2; @Fan]
The effective Hamiltonian around the $\Lambda _Y$ point is given by $$\begin{aligned}
H_Y = v_2 \tilde k_x \sigma _y - v_1 \tilde k_y \sigma _x + \tilde m\sigma _z+U \end{aligned}$$ where $$\begin{aligned}
\tilde k_x = k_x + \frac{1}{{v_2 }}\left( {\lambda _{11} \varepsilon _{22} + \lambda _{22} \varepsilon _{11} + \lambda _{33} \varepsilon _{33} + h_y } \right),\;\tilde k_y = k_y + \frac{1}{{v_1 }}\left( {\lambda _{12} \varepsilon _{12} - h_x } \right),\;\tilde m = \frac{{n'}}{{\sqrt {n^2 + (n')^2 } }}\left( { - \lambda _{13} \varepsilon _{13} + h_z } \right).\end{aligned}$$ The effective Hamiltonian around the $\Lambda' _Y$ point is given by the replacement $\lambda _{13} \varepsilon _{13} \to - \lambda _{13} \varepsilon _{13}$ in $H_Y$.[@Liu3; @Fang; @Liu4; @Wang2; @Fan] The conductances originating from the $\Lambda_Y$ and $\Lambda' _Y$ points, $G_{Y}$ and $G_{Y'}$, can be obtained in a way similar to that from the $\Lambda_X$ point (by replacement of corresponding parameters). Note that $G_{Y}$ is given by $$\begin{aligned}
G_Y = \frac{{v_2 }}{{2v_1 }}\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left| t \right|^2 \cos \phi d\phi } .\end{aligned}$$ The factor of $\frac{{v_2 }}{{v_1 }}$ is included in this expression because the velocity operator for $H_Y$ is given by $$\begin{aligned}
\hat v_x = \frac{{\partial H_Y }}{{\partial k_x }} = v_2 \sigma _y .\end{aligned}$$
Finally, the total conductance $G$ is defined as $$\begin{aligned}
G=G_X+G_{X'}+G_{Y}+G_{Y'} .\end{aligned}$$
Results
-------
In the following, we fix $h/E=\lambda _{12} \varepsilon _{12} /E = 0.2, \lambda _{23} \varepsilon _{23} /E = \lambda _{13} \varepsilon _{13} /E = 0.1, EL/v_1 = 1$ and $v_2 /v_1 = 0.65$.
Figure \[fig7\] shows the valley resolved conductances: (a) $G_{X}$, (b) $G_{X'}$, (c) $G_{Y}$, and (d) $G_{Y'}$ as functions of $\theta$ and $\varphi$ for $U=0$. As shown in Ref.[@Yokoyama4], the inplane exchange field shifts the Fermi surface in momentum space. The conductance is suppressed due to this shift along $k_y$-direction since $k_y$ is conserved. As for $G_{X}$ and $G_{X'}$, when $h_x$ is positive, the shift of the Fermi surface along the $k_y$-direction is enhanced since $\lambda _{12} \varepsilon _{12} >0$. Hence, the conductance is strongly suppressed at $\varphi=0$ as seen from Figures \[fig7\] (a) and (b). On the other hand, when $h_x$ is negative, the shift of the Fermi surface along the $k_y$-direction is canceled. The conductance then becomes large at $\varphi=\pi$. Since the term with $\lambda _{12} \varepsilon _{12}$ in $H_Y$ and $H_{Y'}$ has a sign opposite to that of $h_x$, $G_{Y}$ and $G_{Y'}$ become large at $\varphi=0$ but small at $\varphi=\pi$ as shown in Figures \[fig7\] (c) and (d). As $\theta$ deviates from $\pi/2$, the exchange fields points to $z$-direction, and the dependence of the conductances on $\varphi$ becomes weak. Since $\lambda _{23} \varepsilon _{23} >0$, the mass gap for $H_{X(Y')}$ at $\theta=0$ is larger than that at $\theta=\pi$. Hence, $G_{X(Y')}$ at $\theta=0$ is smaller than that for $\theta=\pi$. With the same reasoning, we find that $G_{X'(Y)}$ at $\theta=0$ is larger than that for $\theta=\pi$. It is also seen that in the parameter region where $G_{X}$ and $G_{X'}$ are large, $G_{Y}$ and $G_{Y'}$ can be small and vice versa, indicative of a valley filtering effect.
In Fig. \[fig8\], the valley resolved conductances are plotted as functions of $\theta$ and $\varphi$ for $U/E=1$. Regarding $G_{X}$ and $G_{X'}$, due to the cancellation of the shift of the Fermi surface, the conductance reaches its maximum at $\varphi=\pi$ as a function of $\varphi$. When $\cos \theta=- \lambda _{23} \varepsilon _{23}/h=-1/2$, namely $\theta=2\pi/3$, $\tilde m$ in $H_X$ becomes zero. Thus, the conductance becomes minimum at this value of $\theta$ as seen in Fig. \[fig8\] (a). In a similar way, we can understand that $G_{X'}$ and $G_{Y}$ become minimum at $\theta=\pi/3$, while $G_{Y'}$ takes a minimum at $\theta=2\pi/3$. In contrast to Fig. \[fig7\], in a parameter region with large $G_{X(Y)}$, $G_{X'(Y')}$ can be small and vice versa for $U/E=1$. This indicates that the valley filtering effects are controllable by varying the potential in the ferromagnetic region and the direction of the magnetization.
In Fig. \[fig9\], we show valley resolved conductances as a function of $U/E$ for (a) $\theta=\varphi=0$ and (b) $\theta=0.5\pi$ and $\varphi=0$. In Fig. \[fig9\] (a), it is found that the relative magnitudes of the conductances depend on $U/E$ which is tunable by gating. For $U/E<0.8$, $G_{X}$ and $G_{X'}$ give dominant contributions. Around $U/E=0.9$, $G_{X'}$ and $G_{Y}$ are dominant, while around $U/E=1$, $G_{X}$ shows a dominant contribution. As shown in Fig. \[fig9\] (b), $G_{Y}$ and $G_{Y'}$ are dominant contributions around $U/E=1$. These results indicate that by changing the direction of the magnetization and the potential in the ferromagnetic region, one can control the dominant valley contribution out of four valley degrees of freedom.
We show the total conductance $G$ as functions of $\theta$ and $\varphi$ in Figs. \[fig10\] (a) and (b), and as a function of $U/E$ in Fig. \[fig10\] (c). At $U=0$, $G$ takes a maximum around $\theta=\varphi=0.5\pi$ and a minimum around $\theta=0.5\pi$ and $\varphi=0$ as seen from Fig. \[fig10\] (a). On the other hand, at $U/E=1$, $G$ takes a maximum for $\theta=0.5\pi$ and $\varphi=\pi$ as shown in Fig. \[fig10\] (b). We also have a large magnetoconductance effect compared to the case with $U=0$. Comparing Figs. \[fig10\] (a) and (b), it is found that by changing $U$, the direction of the magnetization at maximum conductance and that at minimum conductance are exchanged. In Fig. \[fig10\] (c), $G$ is plottd as a function of $U/E$. It is found that the total conductance also depends strongly on the potential in the ferromagnetic region. Therefore, the total conductance is also tunable by electric and magnetic means.
Here, we have considered transport properties on the (001) surface of the topological crystalline insulator. Our formalism is also applicable to the (110) or (111) surfaces of the topological crystalline insulator. Recently, angle-resolved photoemission spectroscopy on the (111) surface of the topological crystalline insulator has been reported. Dirac cones at the $\bar{\Gamma}$ and $\bar{M}$ points have been observed.[@Tanaka3; @Polley] It has been also revealed that the energy location of the Dirac point and the Dirac velocity are different at the $\bar{\Gamma}$ and $\bar{M}$ points.[@Tanaka3] These characteristics can be taken into account in our formalism by changing parameters $v_1$, $v_2$ and $U$ at each valley.
Conclusions
===========
In summary, we have investigated spin and valley transports in junctions composed of silicene and topological crystalline insulators. We have considered normal/magnetic/normal Dirac metal junctions where a gate electrode is attached to the magnetic region. In normal/antiferromagnetic/normal silicene junction, it is shown that the current through this junction is valley and spin polarized due to the coupling between valley and spin degrees of freedom, and the valley and spin polarizations can be tuned by local application of a gate voltage. In particular, we have found a fully valley and spin polarized current by applying the electric field. In normal/ferromagnetic/normal topological crystalline insulator junction with a strain induced in the ferromagnetic segment, we have investigated valley resolved conductances and clarified how the valley polarization stemming from the strain and exchange field appears in this junction. It is found that changing the direction of the magnetization and the potential in the ferromagnetic region, one can control the dominant valley contribution out of four valley degrees of freedom. We have also reviewed spin transport in normal/ferromagnetic/normal graphene junctions, and spin and valley transports in normal/ferromagnetic/normal silicene junctions.
The role of magnetism is different in graphene, silicene and topological crystalline insulator junctions. In graphene junctions, the ferromagnetism induces different chemical potential shifts for up and down spin states, which leads to the shift of the oscillation of the conductances. As a result, a finite spin current appears. In silicene junctions, the (anti)ferromagnetism opens different spin dependent band gaps at $K$ and $K'$ points. This results in spin and valley polarized transports in these junctions. In topological crystalline insulator junctions, the ferromagnetism also induces valley dependent band gaps and inplane “vector potentials" in combination with strain effects. These properties lead to valley dependent transports.
Note added: Recently, we learned of a related work on the ferromagnetic silicene junctions.[@Soodchomshom]
The author thanks S. Murakami, Y. Okada, M. Ezawa, and X. Hu for helpful comments and discussions. This work was supported by Grant-in-Aid for Young Scientists (B) (No. 23740236) and the “Topological Quantum Phenomena" (No. 25103709) Grant-in Aid for Scientific Research on Innovative Areas from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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|
---
abstract: 'Seismic analyses of the CoRoT target HD 49933 have revealed a magnetic cycle. Further insight reveals that frequency shifts of oscillation modes vary as a function of frequency, following a similar pattern to that found in the Sun. In this preliminary work, we use seismic constraint to compute structure models of HD 49933 with the Asteroseismic Modeling Portal (AMP) and the CESAM code. We use these models to study the effects of sound-speed perturbations in near surface layers on p-mode frequencies.'
author:
- 'T. Ceillier$^{*,1}$, J. Ballot$^{2,3}$, R.A. García$^{1}$, G.R. Davies$^1$, S. Mathur$^4$, T.S. Metcalfe$^4$, D. Salabert$^5$'
title: 'HD 49933: A laboratory for magnetic activity cycles'
---
Introduction and observations
=============================
Dynamo processes in the outer convective envelope of solar-like stars generate magnetic fields, which create active regions and spots at the stellar surface (e.g. Lanza 2010). In the Sun, such a dynamo leads to a well known almost regular 22-year magnetic cycle. This dynamo is far from being completely understood, and is difficult to predict as suggested by the recent unusually long solar minimum of cycle 23 (e.g. Salabert et al. 2009). Study of magnetic cycles in other stars should allow us to better understand the physical mechanisms by studying many stars in different evolutionary stages and conditions (e.g. Chaplin et al. 2007; Metcalfe et al. 2007). Variations of magnetic fields modify the stellar structure and therefore frequencies and amplitudes of the acoustic modes (Jiménez-Reyes et al. 2003). Such variations are detectable with current asteroseismological facilities.
HD 49933 is a F5V star observed by CoRoT for 60 and 137 days in 2007 and 2008 (Appourchaux et al. 2008; Benomar et al. 2009). The light curve presents clear signatures of active regions crossing the visible stellar disk that reveal a rotation period of 3.5 days with differential rotation. The analysis of short time series reveals frequency shifts and amplitude variations in acoustic modes unveiling short-term variations of activity, compatible with a short stellar cycle of around 150 days (García et al. 2010). Moreover, the frequency shifts measured in HD 49933 present a clear dependence with increasing frequency, reaching a maximum shift of about 2 $\mu$Hz around 2100 $\mu$Hz (Salabert et al. 2011). Such a dependence is comparable to the one observed in the Sun, which is understood to arise from changes in the outer layers due to its magnetic activity.
Modelling and Discussion
========================
In this work we study the internal structure of HD 49933 paying special attention to properties of the external convection zone. We use the seismic and spectroscopic observations to find a best model with AMP (Metcalfe et al. 2009). We use this model as a starting point to compute two structure models with the CESAM stellar evolution code (Morel 1997). The first model uses the mixing-length theory (MLT, Böhm-Vitense 1958) for treating convection whereas the second use the CGM prescription (Canuto et al 1996). To quantify the sensitivity of modes to the superficial structure, we compute upper turning points of the p modes with frequencies $\nu \in$ \[1400–2600\] $\mu$Hz. Turning points $r_o$ are computed as $2\pi\nu=\omega_c(r_o)$, where $\omega_c=c_s/(2H)$ is the isothermal cut-off frequency ($c_s$ is the sound speed and $H$ the pressure scale height). In Fig. 1a, we see similar results for both models. Nevertheless, in the CGM model, $r_o$ depends less on $\nu$ in the range 1400–2600 $\mu$Hz, because the temperature profile obtained with this prescription is sharper than the one obtained with MLT.
We crudely estimate the impact on frequencies of a perturbation in pressure due to a change in magnetic field. We assume that $c_s$ is perturbed by a change $\delta p$ in pressure, all other quantities staying unchanged. Thus, the travel time of waves is modified by $\delta\tau =\int^{r_o} \delta p / (p c_s)\,\mathrm{d}r$. We then consider the effect on frequencies is $\delta\nu \approx \nu^2\delta\tau$. We consider two profiles for $\delta p$: (A) $\delta p$ is constant in the surface layers; (B) $\delta p$ linearly grows during the first 10 Mm beneath the surface. Results are plotted in Fig. 1b. We use $\delta p = 400\:\mathrm{dyn\,cm^{-2}}$ at the photosphere to recover reasonable values of $\delta\nu$.
|
---
abstract: 'We analyze the perturbative and parametric stability of the QCD predictions for the azimuthal asymmetries in heavy quark leptoproduction. At leading order, the $\cos \varphi $ asymmetry vanishes whereas the $\cos 2\varphi $ one is of leading twist and predicted to be about $15\%$ at energies sufficiently above the production threshold. We calculate the NLO soft-gluon corrections to $\varphi$-dependent leptoproduction to the next-to-leading logarithmic accuracy. The soft-gluon approximation provides a good description of the available exact NLO results at $Q^{2}\lesssim m^{2}$. Our analysis shows that, contrary to the production cross sections, the $\cos 2\varphi $ asymmetry is practically insensitive to soft radiation for $Q^{2}\lesssim m^{2}$ at energies of the fixed target experiments. We conclude that the $\cos 2\varphi $ asymmetry is well defined in pQCD: it is stable both perturbatively and parametrically, and insensitive (in the case of bottom production) to nonperturbative contributions. Measurements of the azimuthal asymmetries would provide an excellent test of pQCD applicability to heavy flavor production.'
address: 'Yerevan Physics Institute, Alikhanian Br.2, 375036 Yerevan, Armenia'
author:
- 'N.Ya. Ivanov[^1]'
title: |
Azimuthal Asymmetries in Heavy Quark Leptoproduction\
as a Test of pQCD\
---
Introduction
============
In the framework of perturbative QCD, the basic spin-averaged characteristics of heavy flavor hadro-, photo- and electroproduction are known exactly up to the next-to-leading order (NLO). During the last ten years, these NLO results have been widely used for a phenomenological description of available data (for a review see [@1]). At the same time, the key question remains open: How to test the applicability of QCD at fixed order to heavy quark production? The problem is twofold. On the one hand, the NLO corrections are large; they increase the leading order (LO) predictions for both charm and bottom production cross sections by approximately a factor of two. For this reason, one could expect that higher-order corrections, as well as nonperturbative contributions, can be essential, especially for the $c$-quark case. On the other hand, it is very difficult to compare pQCD predictions for spin-averaged cross sections with experimental data directly, without additional assumptions, because of a high sensitivity of the theoretical calculations to standard uncertainties in the input QCD parameters. The total uncertainties associated with the unknown values of the heavy quark mass, $m$, the factorization and renormalization scales, $\mu _{F}$ and $\mu _{R}$, $\Lambda _{QCD}$ and the parton distribution functions are so large that one can only estimate the order of magnitude of the pQCD predictions for total cross sections at fixed target energies [@2; @3].
In recent years, the role of higher-order corrections has been extensively investigated in the framework of the soft gluon resummation formalism. For a review see Ref.[@4]. Soft gluon (or threshold) resummation is based on the factorization properties of the cross section near the partonic threshold and makes it possible to resum to all orders in $\alpha _{s}$ the leading (Sudakov double) logarithms (LL) and the next-to-leading ones (NLL) [@5; @6; @7]. Formally resummed cross sections are ill-defined due to the Landau pole contribution, and a few prescriptions have been proposed to avoid the renormalon ambiguities [@8; @9; @10]. Unfortunately, numerical predictions for the heavy quark production cross sections can depend significantly on the choice of resummation prescription [@11]. Another open question, also closely related to convergence of the perturbative series, is the role of subleading logarithms which are not, in principle, under control of the resummation procedure [@11; @12].
For this reason, it is of special interest to study those observables that are well-defined in pQCD. A nontrivial example of such an observable is proposed in [@13; @14], where the single spin asymmetry (SSA) in charm and bottom production by linearly polarized photons, $\gamma ^{\uparrow
}+N\rightarrow Q+X[\overline{Q}]$, was calculated.[^2] It was shown that, contrary to the production cross section, the single spin asymmetry in heavy flavor photoproduction is quantitatively well defined in pQCD: it is stable, both parametrically and perturbatively, and insensitive to nonperturbative corrections. Therefore, measurements of this asymmetry would provide an ideal test of pQCD. As was shown in Ref. [@15], the SSA in open charm photoproduction can be measured with an accuracy of about ten percent in the approved E160/E161 experiments at SLAC [@16] using the inclusive spectra of secondary (decay) leptons.
In the present paper we continue the studies of perturbatively stable observables and calculate the radiative and nonperturbative corrections to the azimuthal asymmetry (AA) in heavy quark leptoproduction: $$l(\ell )+N(p)\rightarrow l(\ell -q)+Q(p_{Q})+X[\overline{Q}](p_{X}).
\label{1}$$ In the case of unpolarized initial states and neglecting the contribution of $Z-$boson, the cross section of the reaction (\[1\]) may be written as $$\begin{aligned}
\frac{\text{d}\sigma _{lN}}{\text{d}x\text{d}Q^{2}\text{d}\varphi }=\frac{%
\alpha _{em}}{(2\pi )^{2}}\frac{1}{xQ^{2}} &&\Big\{ \left[
1+(1-y)^{2}\right] \sigma _{T}( x,Q^{2}) +2\left(1-y\right) \sigma _{L}(
x,Q^{2}) \nonumber \\
&&+2\left(1-y\right) \sigma _{A}( x,Q^{2}) \cos 2\varphi +(2-y)\sqrt{1-y}%
~\sigma _{I}( x,Q^{2}) \cos \varphi \Big\}. \label{2}\end{aligned}$$ The kinematic variables are defined by $$\begin{aligned}
\bar{S}=\left( \ell +p\right) ^{2},\qquad &Q^{2}=-q^{2},\qquad &x=\frac{Q^{2}}{%
2p\cdot q}, \nonumber \\
y=\frac{p\cdot q}{p\cdot \ell },\qquad &Q^{2}=xy\bar{S},\qquad &\rho =\frac{4m^{2}%
}{\bar{S}}. \label{3}\end{aligned}$$ In Eq. (\[2\]), $\sigma _{T}\,(\sigma _{L})$ is the usual $\gamma ^{*}N$ cross section describing heavy quark production by a transverse (longitudinal) virtual photon. The third cross section, $\sigma _{A}$, comes about from interference between transverse states and is responsible for the SSA which occurs in real photoproduction using linearly polarized photons [@13; @14; @15]. The fourth cross section, $\sigma _{I}$, originates from interference between longitudinal and transverse components [@16D]. In the nucleon rest frame, the azimuth $\varphi $ is the angle between the lepton scattering plane and the heavy quark production plane, defined by the exchanged photon and the detected quark $Q$ (see Fig. \[Fg.1\]). The covariant definition of $\varphi $ is $$\begin{aligned}
\cos \varphi &=&\frac{r\cdot n}{\sqrt{-r^{2}}\sqrt{-n^{2}}},\qquad \qquad
\sin \varphi =\frac{Q^{2}\sqrt{1/x^{2}+4m_{N}^{2}/Q^{2}}}{2\sqrt{-r^{2}}%
\sqrt{-n^{2}}}~n\cdot \ell , \label{4} \\
r^{\mu } &=&\varepsilon ^{\mu \nu \alpha \beta }p_{\nu }q_{\alpha }\ell
_{\beta },\qquad \qquad \quad n^{\mu }=\varepsilon ^{\mu \nu \alpha \beta
}p_{\nu }q_{\alpha }p_{Q\beta }. \label{5}\end{aligned}$$ In Eqs. (\[3\]) and (\[4\]), $m$ and $m_{N}$ are the masses of the heavy quark and the target, respectively.
In leading order pQCD, the $\cos \varphi $ dependence vanishes, $\sigma _{I}^{{\rm Born}}(x,Q^{2})=0$. For this reason, in this paper we restrict ourselves to the azimuthal asymmetry, $A(\rho ,x,Q^{2})$, associated with the $\cos 2\varphi $ distribution: $$A(\rho ,x,Q^{2})=\frac{\text{d}^{3}\sigma _{lN}(\varphi =0)+\text{d}%
^{3}\sigma _{lN}(\varphi =\pi )-2\text{d}^{3}\sigma _{lN}(\varphi =\pi /2)}{%
\text{d}^{3}\sigma _{lN}(\varphi =0)+\text{d}^{3}\sigma _{lN}(\varphi =\pi
)+2\text{d}^{3}\sigma _{lN}(\varphi =\pi /2)}=\frac{\varepsilon \,\sigma
_{A}( x,Q^{2}) }{\sigma _{T}( x,Q^{2}) +\varepsilon
\,\sigma _{L}( x,Q^{2}) }, \label{6}$$ where $\varepsilon ={{{{%
{\displaystyle {2(1-y) \over 1+(1-y)^{2}}}%
}}}}$ and $\text{d}^{3}\sigma _{lN}(\varphi )\equiv {%
{\displaystyle {\text{d}^{3}\sigma _{lN} \over \text{d}x\text{d}Q^{2}\text{d}\varphi }}%
}( \rho ,x,Q^{2},\varphi) $. Note that the asymmetry defined by Eq. (\[6\]) is simply related to the mean value of $\cos 2\varphi $:
$$A(\rho ,x,Q^{2})=2\langle \cos 2\varphi \rangle (\rho ,x,Q^{2}),\qquad
\qquad \langle \cos 2\varphi \rangle (\rho ,x,Q^{2})=
\frac{\int\limits_{0}^{2\pi }\text{d}\varphi \cos 2\varphi%
{\displaystyle {\text{d}^{3}\sigma _{lN} \over \text{d}x\text{d}Q^{2}\text{d}\varphi }}%
( \rho ,x,Q^{2},\varphi ) }{\int\limits_{0}^{2\pi }\text{d}\varphi
{\displaystyle {\text{d}^{3}\sigma _{lN} \over \text{d}x\text{d}Q^{2}\text{d}\varphi }}%
( \rho ,x,Q^{2},\varphi ) }. \label{8}$$
In this paper, we calculate the NLO corrections to the $\cos 2\varphi$ asymmetry to the next-to-leading logarithmic accuracy (so-called soft-gluon approximation). Also we analyze the nonperturbative contributions to the AA due to the gluon transverse motion in the target. Our main results can be formulated as follows:
- The azimuthal asymmetry defined by Eq. (\[6\]) is of leading twist; at energies sufficiently above the production threshold, it is predicted to be about $15\%$ for both charm and bottom quark production.
- The soft-gluon approximation provides a good description of the available exact NLO results on leptoproduction in the region of relatively low virtualities, $Q^{2}\lesssim m^{2}$; when $Q^{2}\gg m^{2}$, the quality of the NLL approximation becomes worse.
- Contrary to the production cross sections, the $\cos 2\varphi$ asymmetry in azimuthal distributions of both charm and bottom quark is practically insensitive to radiative corrections at $Q^{2}\lesssim m^{2}$. This implies that large soft-gluon contributions to the $\varphi$-dependent and $\varphi$-integrated cross sections cancel each other in Eq. (\[6\]) with a good accuracy.
- pQCD predictions for the $\cos 2\varphi$ asymmetry are parametrically stable; to within few percent, they are insensitive to standard uncertainties in the QCD input parameters: $\mu _{R}$, $\mu _{F}$, $\Lambda _{QCD}$ and in the gluon distribution function.
- Nonperturbative corrections to the $b$-quark azimuthal asymmetry due to the gluon transverse motion in the target are negligible. Because of the smallness of the $c$-quark mass, the analogous corrections to $A(\rho
,x,Q^{2})$ in the charm case are larger; they are of the order of 20% at $%
Q^{2}\lesssim m^{2}$.
We conclude that, in contrast with the production cross sections, the $\cos 2\varphi$ asymmetry in heavy quark leptoproduction, $A(\rho ,x,Q^{2})$, is an observable quantitatively well defined in pQCD: it is stable, both parametrically and perturbatively, and insensitive (in the case of bottom production) to nonperturbative corrections. Measurements of the AA in bottom production would provide an ideal test of the conventional parton model based on pQCD.
Concerning the experimental aspects, azimuthal asymmetries in bottom production can, in principle, be measured at HERA using the angular distributions of secondary (decay) leptons [@15]. AAs in charm leptoproduction can also be measured in the COMPASS and HERMES experiments. Due to the relatively low $c$-quark mass, data on the $D$-meson azimuthal distributions would make it possible to clarify the role of subleading twist contributions.
The paper is organized as follows. In Section II we analyze the LO and NLO parton level predictions for $\varphi $-dependent leptoproduction of heavy flavor in the single-particle inclusive kinematics. We check the quality of the soft-gluon approach against available exact results and discuss the region of applicability of the NLL approximation. Hadron level predictions for $A(\rho ,x,Q^{2})$ are given in Section III. We consider in detail the pQCD contributions and nonperturbative corrections to the $\cos 2\varphi$ asymmetry at the HERMES, SLAC, COMPASS and HERA energies.
Partonic Cross Sections
========================
Born level predictions
----------------------
At leading order, ${\cal O}(\alpha _{em}\alpha _{s})$, the only partonic subprocess which is responsible for heavy quark leptoproduction is the two-body photon-gluon fusion: $$\gamma ^{*}(q)+g(k_{g})\rightarrow Q(p_{Q})+\overline{Q}(p_{\stackrel{\_}{Q}%
}). \label{10}$$ The $\gamma ^{*}g$ cross sections, $\hat{\sigma}_{k}^{{\rm Born}}$ ($%
k=T,L,A,I$), corresponding to the Born diagrams are [@17a; @17]: $$\begin{aligned}
\hat{\sigma}_{T}^{{\rm Born}}( \hat{\rho},\hat{x}) &=&\frac{\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}}{2m^{2}}\hat{\rho}\Big\{\left[ (1-\hat{x}%
)^{2}+\hat{x}^{2}+\hat{\rho}(1-\hat{x}-\hat{\rho}/2)\right] \ln \frac{%
1+\beta }{1-\beta } \nonumber \\
&&\qquad \qquad \quad {}-\left[ (1-\hat{x})^{2}-\hat{x}(2-3\hat{x})+(1-\hat{x%
})\hat{\rho}\right] \beta \Big\}, \nonumber \\
\hat{\sigma}_{L}^{{\rm Born}}( \hat{\rho},\hat{x}) &=&\frac{\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}}{m^{2}}\hat{\rho}\hat{x}\Big\{ -\hat{\rho}%
\ln \frac{1+\beta }{1-\beta }+2(1-\hat{x})\beta \Big\}, \label{11} \\
\hat{\sigma}_{A}^{{\rm Born}}( \hat{\rho},\hat{x}) &=&\frac{\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}}{2m^{2}}\hat{\rho}^{2}\Big\{ (1-2\hat{x}-%
\hat{\rho}/2)\ln \frac{1+\beta }{1-\beta }-(1-\hat{x})(1-2\hat{x}/\hat{\rho}%
)\beta \Big\}, \nonumber \\
\hat{\sigma}_{I}^{{\rm Born}}( \hat{\rho},\hat{x}) &=&0.
\nonumber\end{aligned}$$ In Eqs. (\[11\]), $e_{Q}$ is the quark charge in units of electromagnetic coupling constant and we use the following definition of partonic kinematic variables: $$\begin{aligned}
s=\left( q+k_{g}\right) ^{2},\qquad \qquad &&\hat{x}=\frac{Q^{2}}{2q\cdot
k_{g}}, \nonumber \\
\beta =\sqrt{1-\frac{4m^{2}}{s}},\qquad \qquad &&\hat{\rho}=\frac{4m^{2}}{%
s+Q^{2}}. \label{12}\end{aligned}$$
Note that the $\cos \varphi $ dependence vanishes due to the $%
Q\leftrightarrow \overline{Q}$ symmetry which, at leading order, requires invariance under $\varphi \rightarrow \varphi +\pi $ [@18].
The hadron level cross sections, $\sigma _{k}( x,Q^{2}) $ ($%
k=T,L,A,I$), have the form $$\sigma _{k}( x,\xi ) =\int\limits_{x+4x/\xi }^{1}\text{d}%
z\,g(z,\mu _{F})\,\hat{\sigma}_{k}\left( \frac{4x}{z\xi },\frac{x}{z}\right)
,\qquad \qquad k_{g}=zp,\qquad \qquad \xi =\frac{Q^{2}}{m^{2}}, \label{13}$$ where $g(z,\mu _{F})$ describes gluon density in a nucleon $N$ evaluated at a factorization scale $\mu _{F}$. The partonic cross sections, $\hat{\sigma}_{k}$, are functions of $\hat{\rho}$ and $\hat{x}$ defined by Eq. (\[12\]).
Soft-gluon corrections at NLO
-----------------------------
To take into account the NLO contributions, one needs to calculate the virtual ${\cal O}(\alpha _{em}\alpha _{s}^{2})$ corrections to the Born process (\[10\]) and the real gluon emission: $$\gamma ^{*}(q)+g(k_{g})\rightarrow Q(p_{Q})+\overline{Q}(p_{\stackrel{\_}{Q}%
})+g(p_{g}). \label{15}$$ The partonic invariants describing the single-particle inclusive (1PI) kinematics are $$\begin{aligned}
s^{\prime }=2q\cdot k_{g}=s+Q^{2}=zS^{\prime },\qquad \qquad &&t_{1}=\left(
k_{g}-p_{Q}\right) ^{2}-m^{2}=zT_{1}, \nonumber \\
s_{4}=s^{\prime }+t_{1}+u_{1},\qquad \qquad &&u_{1}=\left( q-p_{Q}\right)
^{2}-m^{2}=U_{1}, \label{16}\end{aligned}$$ where $s_{4}$ measures the inelasticity of the reaction (\[15\]). The corresponding 1PI hadron level variables describing the reaction (\[1\]) are $$\begin{aligned}
S^{\prime }=2q\cdot p=S+Q^{2},\qquad \qquad &&T_{1}=\left( p-p_{Q}\right)
^{2}-m^{2}, \nonumber \\
S_{4}=S^{\prime }+T_{1}+U_{1},\qquad \qquad &&U_{1}=\left( q-p_{Q}\right)
^{2}-m^{2}. \label{17}\end{aligned}$$ We neglect the photon-(anti)quark fusion subprocesses. This is justified as their contributions vanish at LO and are small at NLO [@19].
The exact NLO calculations of the unpolarized heavy quark production in $%
\gamma g$ [@20; @21], $\gamma ^{*}g$ [@19], and $gg$ [@22; @23] collisions show that, near the partonic threshold, a strong logarithmic enhancement of the cross sections takes place in the collinear, $\vec{p}%
_{g,T} $ $\rightarrow 0$, and soft, $\vec{p}_{g}\rightarrow 0$, limits. This threshold (or soft-gluon) enhancement has universal nature in the perturbation theory and originates from incomplete cancellation of the soft and collinear singularities between the loop and the bremsstrahlung contributions. Large leading and next-to-leading threshold logarithms can be resummed to all orders of perturbative expansion using the appropriate evolution equations [@5; @6; @7]. The analytic results for the resummed cross sections are ill-defined due to the Landau pole in the coupling strength $\alpha _{s}$. However, if one considers the obtained expressions as generating functionals of the perturbative theory and re-expands them at fixed order in $\alpha _{s}$, no divergences associated with the Landau pole are encountered.
Soft-gluon resummation for the photon-gluon fusion has been performed in Ref.[@24] and checked in Refs.[@25; @14]. To NLL accuracy, the perturbative expansion for the partonic cross sections, d$^{2}\hat{\sigma}%
_{k}/$d$t_{1}$d$u_{1}$ ($k=T,L,A,I$), can be written in a factorized form as $$s^{\prime 2}\frac{\text{d}^{2}\hat{\sigma}_{k}}{\text{d}t_{1}\text{d}u_{1}}%
( s^{\prime },t_{1},u_{1}) =B_{k}^{\text{{\rm Born}}}(
s^{\prime },t_{1},u_{1}) \left\{ \delta ( s^{\prime
}+t_{1}+u_{1}) +\sum_{n=1}^{\infty }\left( \frac{\alpha _{s}C_{A}}{\pi
}\right) ^{n}K^{(n)}( s^{\prime },t_{1},u_{1}) \right\} ,
\label{18}$$ with the Born level distributions $B_{k}^{\text{{\rm Born}}}$ given by $$\begin{aligned}
B_{T}^{\text{{\rm Born}}}( s^{\prime },t_{1},u_{1}) &=&\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}\left[ \frac{t_{1}}{u_{1}}+\frac{u_{1}}{t_{1}%
}+4\left( \frac{s}{s^{\prime }}-\frac{m^{2}s^{\prime }}{t_{1}u_{1}}\right)
\left( \frac{s^{\prime }(m^{2}-Q^{2}/2)}{t_{1}u_{1}}+\frac{Q^{2}}{s^{\prime }%
}\right) \right] , \nonumber \\
B_{L}^{\text{{\rm Born}}}( s^{\prime },t_{1},u_{1}) &=&\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}\left[ \frac{8Q^{2}}{s^{\prime }}\left(
\frac{s}{s^{\prime }}-\frac{m^{2}s^{\prime }}{t_{1}u_{1}}\right) \right] ,
\nonumber \\
B_{A}^{\text{{\rm Born}}}( s^{\prime },t_{1},u_{1}) &=&\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}\left[ 4\left( \frac{s}{s^{\prime }}-\frac{%
m^{2}s^{\prime }}{t_{1}u_{1}}\right) \left( \frac{m^{2}s^{\prime }}{%
t_{1}u_{1}}+\frac{Q^{2}}{s^{\prime }}\right) \right] , \label{19} \\
B_{I}^{\text{{\rm Born}}}( s^{\prime },t_{1},u_{1}) &=&\pi
e_{Q}^{2}\alpha _{em}\alpha _{s}\left[ 4\sqrt{Q^{2}}\left( \frac{t_{1}u_{1}s%
}{s^{\prime 2}}-m^{2}\right) ^{1/2}\frac{t_{1}-u_{1}}{t_{1}u_{1}}\left( 1-%
\frac{2Q^{2}}{s^{\prime }}-\frac{2m^{2}s^{\prime }}{t_{1}u_{1}}\right)
\right] . \nonumber\end{aligned}$$
Note that the functions $K^{(n)}( s^{\prime },t_{1},u_{1}) $ in Eq. (\[18\]) originate from the collinear and soft limits. Since the azimuthal angle $\varphi $ is the same for both $\gamma ^{*}g$ and $Q%
\overline{Q}$ center-of-mass systems in these limits, the functions $%
K^{(n)}( s^{\prime },t_{1},u_{1}) $ are also the same for all $%
k=T,L,A,I$. At NLO, the soft-gluon corrections to NLL accuracy in the $%
\overline{\text{MS}}$ scheme are $$\begin{aligned}
K^{(1)}( s^{\prime },t_{1},u_{1}) &=&2\left[ \frac{\ln \left(
s_{4}/m^{2}\right) }{s_{4}}\right] _{+}-\left[ \frac{1}{s_{4}}\right]
_{+}\left\{ 1+\ln \left( \frac{u_{1}}{t_{1}}\right) -\left( 1-\frac{2C_{F}}{%
C_{A}}\right) \left( 1+\text{Re}L_{\beta }\right) +\ln \left( \frac{\mu ^{2}%
}{m^{2}}\right) \right\} \nonumber \\
&&+\delta ( s_{4}) \ln \left( \frac{-u_{1}}{m^{2}}\right) \ln
\left( \frac{\mu ^{2}}{m^{2}}\right) , \label{20}\end{aligned}$$ where we use $\mu =\mu _{F}=\mu _{R}$. In Eq. (\[20\]), $C_{A}=N_{c}$ and $%
C_{F}=(N_{c}^{2}-1)/(2N_{c})$, where $N_{c}$ is number of colors, while $%
L_{\beta }=(1-2m^{2}/s)\{\ln [(1-\beta )/(1+\beta )]+$i$\pi \}$. The single-particle inclusive ”plus“ distributions are defined by $$\left[ \frac{\ln ^{l}\left( s_{4}/m^{2}\right) }{s_{4}}\right]
_{+}=\lim_{\epsilon \rightarrow 0}\left\{ \frac{\ln ^{l}\left(
s_{4}/m^{2}\right) }{s_{4}}\theta ( s_{4}-\epsilon ) +\frac{1}{l+1%
}\ln ^{l+1}\left( \frac{\epsilon }{m^{2}}\right) \delta ( s_{4})
\right\} . \label{21}$$ For any sufficiently regular test function $h(s_{4})$, Eq. (\[21\]) gives $$\int\limits_{0}^{s_{4}^{\max }}\text{d}s_{4}\,h(s_{4})\left[ \frac{\ln
^{l}\left( s_{4}/m^{2}\right) }{s_{4}}\right]
_{+}=\int\limits_{0}^{s_{4}^{\max }}\text{d}s_{4}\left[ h(s_{4})-h(0)\right]
\frac{\ln ^{l}\left( s_{4}/m^{2}\right) }{s_{4}}+\frac{1}{l+1}h(0)\ln
^{l+1}\left( s_{4}^{\max }/m^{2}\right) . \label{22}$$
In Eq. (\[20\]) , we have preserved the NLL terms for the scale-dependent logarithms too. We have checked that the results (\[19\]) and (\[20\]) agree to NLL accuracy with the exact ${\cal O}(\alpha _{em}\alpha _{s}^{2})$ calculations of the photon-gluon cross sections $\hat{\sigma}_{T}$ and $\hat{%
\sigma}_{L}$ given in Ref.[@19].
To perform a numerical investigation of the results (\[19\]) and (\[20\]), it is convenient to introduce for the fully inclusive (integrated over $%
t_{1}$ and $u_{1}$) cross sections, $\hat{\sigma}_{k}$ ($k=T,L,A,I$),[ ]{}the dimensionless coefficient functions $c_{k}^{(n,l)}$, $$\hat{\sigma}_{k}(\eta ,\xi ,\mu ^{2})=\frac{e_{Q}^{2}\alpha _{em}\alpha
_{s}(\mu ^{2})}{m^{2}}\sum_{n=0}^{\infty }\left( 4\pi \alpha _{s}(\mu
^{2})\right) ^{n}\sum_{l=0}^{n}c_{k}^{(n,l)}(\eta ,\xi )\ln ^{l}\left( \frac{%
\mu ^{2}}{m^{2}}\right) , \label{23}$$ where the variable $\eta $ measures the distance to the partonic threshold: $$\eta =\frac{s}{4m^{2}}-1,\qquad \qquad \xi =\frac{Q^{2}}{m^{2}}. \label{23a}$$
Concerning the NLO scale-independent coefficient functions, only $%
c_{T}^{(1,0)}$ and $c_{L}^{(1,0)}$ are known exactly [@19; @26]. As to the $%
\mu $-dependent coefficients, they can by calculated explicitly using the renormalization group equation: $$\frac{\text{d}\hat{\sigma}_{k}(s^{\prime },Q^{2},\mu ^{2})}{\text{d}\ln \mu
^{2}}=-\int\limits_{z_{\min }}^{1}\text{d}z\,\hat{\sigma}_{k}(zs^{\prime
},Q^{2},\mu ^{2})P_{gg}(z), \label{24}$$ where $z_{\min }=(4m^{2}+Q^{2})/s^{\prime }$, $\hat{\sigma}_{k}(s^{\prime
},Q^{2},\mu )$ are the cross sections resummed to all orders in $\alpha _{s}$ and $P_{gg}(z)$ is the corresponding (resummed) Altarelli-Parisi gluon-gluon splitting function. Expanding Eq. (\[24\]) in $\alpha _{s}$, one can find [@24; @14] $$c_{k}^{(1,1)}(s^{\prime },\xi )=\frac{1}{4\pi ^{2}}\int\limits_{z_{\min
}}^{1}\text{d}z\left[ b_{2}\delta (1-z)-\,P_{gg}^{(0)}(z)\right]
c_{k}^{(0,0)}(zs^{\prime },\xi ), \label{25}$$ where $b_{2}=(11C_{A}-2n_{f})/12$ is the first coefficient of the $\beta( \alpha _{s}) $-function expansion and $n_{f}$ is the number of active quark flavors. The one-loop gluon splitting function is [@27]: $$P_{gg}^{(0)}(z)=\lim_{\epsilon \rightarrow 0}\left\{ \left( \frac{z}{1-z}+%
\frac{1-z}{z}+z(1-z)\right) \theta ( 1-z-\epsilon ) +\delta
(1-z)\ln \epsilon \right\} C_{A}+b_{2}\delta (1-z). \label{26}$$
With Eq. (\[25\]) in hand, we are able to check the quality of the NLL approximation against exact answers. In Figs. \[Fg.3\] and \[Fg.4\] we plot the functions $c_{T}^{(n,l)}(\eta ,\xi )$ and $c_{A}^{(n,l)}(\eta ,\xi )$ $(n,l=0,1)$ at $\xi =10^{-2}$ and $\xi =3.16$, respectively. Predictions of the NLL approximation (\[20\]) are given by dotted curves. The available exact results are given by solid lines. One can see a reasonable agreement up to energies $\eta \approx 2$. As to the $%
Q^{2}$-dependence, we have found that the soft-gluon approach reproduces satisfactorily the exact results at $\xi \lesssim 1.$ At high values of $%
\xi $, $Q^{2}\gg m^{2}$, the quality of the NLL approximation becomes worse.[^3]
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Hadron Level Results
====================
pQCD predictions
----------------
Let us now analyze the impact of the approximate NLO perturbative corrections on the AA at hadron level. We will consider the parameters $%
A(Q^{2})$, $A(x)$ and $A(y)$, $$A(Q^{2})=\frac{2\int\limits_{0}^{2\pi }\text{d}\varphi \cos 2\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}Q^{2}\text{d}\varphi }}%
( \rho ,Q^{2},\varphi ) }{\int\limits_{0}^{2\pi }\text{d}\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}Q^{2}\text{d}\varphi }}%
( \rho ,Q^{2},\varphi ) },\qquad \qquad A(x)=\frac{%
2\int\limits_{0}^{2\pi }\text{d}\varphi \cos 2\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}x\text{d}\varphi }}%
( \rho ,x,\varphi) }{\int\limits_{0}^{2\pi }\text{d}\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}x\text{d}\varphi }}%
( \rho ,x,\varphi ) },$$ $$A(y)=\frac{2\int\limits_{0}^{2\pi }\text{d}\varphi \cos 2\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}y\text{d}\varphi }}%
( \rho ,y,\varphi) }{\int\limits_{0}^{2\pi }\text{d}\varphi
{\displaystyle {\text{d}^{2}\sigma _{lN} \over \text{d}y\text{d}\varphi }}%
( \rho ,y,\varphi) }, \label{27}$$ which describe the dependence of the $\cos 2\varphi $ asymmetry on $Q^{2}$, Bjorken $x$ and $y$, respectively. Unless otherwise stated, the CTEQ5M [@28] parametrization of the gluon distribution function is used. The default values of the charm and bottom mass are $m_{c}=$ 1.5 GeV and $m_{b}=$ 4.75 GeV, $\Lambda _{3}=$ 260 MeV and $\Lambda _{4}=$ 200 MeV. For the factorization scale we use $\mu _{F}=\sqrt{%
m_{b}^{2}+Q^{2}/4}$ in the case of bottom production and $\mu _{F}=2\sqrt{m_{c}^{2}+Q^{2}/4}$ in the charm case [@1].
Our results for the $Q^{2}$- and $x$-distributions of the AA in charm leptoproduction at several values of initial energy are presented in Fig. \[Fg.5\] and Fig. \[Fg.6\], respectively. The LO and NLO predictions are given by solid and dotted lines, correspondingly. The lines with label $``1"$ correspond to $\rho _{1}=0.2$, $``2"\rightarrow \rho _{2}=0.1$, $``3"\rightarrow \rho _{3}=0.05$ and $``4"\rightarrow \rho _{4}=0.025$, where $\rho =4m^{2}/\bar{S}$. So, in the charm case, we have: $E_{1}=24$ GeV, $E_{2}=47$ GeV, $E_{3}=95$ GeV and $E_{4}=190$ GeV, where $E$ is the lepton energy in the lab (nucleon rest) frame: $E=(\bar{S}-m_{N}^{2})/(2m_{N})$. The $y$-distribution of the asymmetry at the COMPASS energies is given in Fig. \[Fg.7\]. In Fig. \[Fg.8\] we plot the $A(Q^{2})$, $A(x)$ and $A(y)$ distributions of the asymmetry in bottom production at the same values of $\rho _{i}$ $=\{0.2,0.1,0.05,0.025\}$ which correspond to the following set of initial energies: $E_{i}=\{240,480,960,1920\}$ in units of GeV.
Our calculations given in Figs. \[Fg.5\]$-$\[Fg.8\] represent the central result of this paper. One can see from Figs. \[Fg.5\] and \[Fg.8\] that soft-gluon corrections to $A(Q^{2})$ are about few percent at not large $Q^{2}\lesssim m^{2}$. At fixed values of $x$, the kinematical restriction $Q^{2}\lesssim m^{2}$ leads to $x \lesssim m^{2}/\bar{S}$. For this reason, radiative correction to $A(x)$ are small in the region of $x \lesssim \rho$ (see Figs. \[Fg.6\] and \[Fg.8\]). For comparison, we plot in Fig. \[Fg.10\] the so-called $K$-factors for $\varphi$-integrated cross sections: $K(Q^{2})=\left. \frac{\text{d}\sigma _{lN}^{{\rm NLO}}}{\text{d}%
Q^{2}}\right/ \frac{\text{d}\sigma _{lN}^{{\rm LO}}}{\text{d}Q^{2}}$ and $%
K(x)=\left. \frac{\text{d}\sigma _{lN}^{{\rm NLO}}}{\text{d}x}\right/ \frac{%
\text{d}\sigma _{lN}^{{\rm LO}}}{\text{d}x}$. One can see that large soft-gluon corrections to the production cross sections practically (to within few percent) do not affect the Born predictions for the $\cos 2\varphi $ asymmetry at $Q^{2}\lesssim m^{2}$ and $x \lesssim \rho$.
At fixed values of $y$, the allowed region of $Q^{2}$ is $m^{2}_{l}y^{2}/(1-y)\le Q^{2}\le y\bar{S}-4 m^{2}$, where $m_{l}$ is the initial lepton mass. Since production cross sections rapidly vanish with growth of $Q^{2}$, practically whole contribution to $A(y)$ originates from the low-$Q^{2}$ region. For this reason, radiative corrections to $A(y)$ are negligible practically in the whole region of $y$ (see Figs. \[Fg.7\] and \[Fg.8\]).
Let us now discuss the region of applicability of the adopted soft-gluon approximation. As noted in previous Section, soft radiation reproduces satisfactorily the existing exact NLO results when $\xi $ is not large, $Q^{2}\lesssim m^{2}$. At large $Q^{2}\gg m^{2}$, the hard $(\vec{p}%
_{g,T}\neq 0)$ contributions becomes sizeable and the quality of the NLL approximation becomes worse for both $\varphi $-dependent and $\varphi $-independent cross sections. Moreover, we have observed that soft-gluon approximation overestimates the exact results for $c_{T}^{(1,0)}(\eta ,\xi )$ and $c_{T}^{(1,1)}(\eta ,\xi )$ at $\xi \gg 1$ and, simultaneously, underestimates the corresponding ones for $c_{A}^{(1,1)}(\eta ,\xi )$. For this reason, in the high-$Q^{2}$ region, the full NLO corrections to the $%
\cos 2\varphi $ asymmetry may be smaller than the soft-gluon ones.
As to the energy dependence, one can see from Figs. \[Fg.3\] and \[Fg.4\] that soft radiation describes very well the exact NLO results on $\varphi $-independent photon-gluon fusion at partonic energies up to $\eta \approx 2$. Since the gluon distribution function supports just the threshold region, the soft-gluon contribution dominates the photon-hadron cross sections approximately up to $S/4m^{2}\sim 10$ (and, correspondingly, up to $\rho
\sim 0.1$ for $\sigma _{lN}$). Using the exact results for the $\gamma
^{*}g $ cross sections [@26], we have verified that the contribution originating from the region $\eta >2$ makes only few percent from the NLO hadron-level predictions for $\sigma _{T}(S,Q^{2})$ and $\sigma
_{L}(S,Q^{2}) $ at $S/4m^{2}\lesssim 10$ (and $Q^{2}\lesssim m^{2}$). Results of Ref.[@24] on soft-gluon corrections to $F_{2}^{{\rm charm}%
}(x,\xi )$ confirm our conclusion.
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Presently, exact NLO calculations of the $\varphi $-dependent cross section of heavy flavor production are not completed. However we can be sure that, at energies not so far from the production threshold, the soft radiation is the dominant perturbative mechanism in the case of $\sigma
_{A}(S,Q^{2})$ too. First, LO predictions for the $\cos 2\varphi $-dependent cross section are large and the Sudakov logarithms have universal, $\varphi $-independent structure. For this reason, $\sigma
_{A}(S,Q^{2})$ has also a strong threshold enhancement. Second, our analysis of the exact scale-dependent cross section $c_{A}^{(1,1)}(\eta ,\xi )$ given in Figs. \[Fg.3\] and \[Fg.4\] confirms with a good accuracy the dominance of the soft-gluon contribution. These facts argue that hard and virtual corrections to the $\cos 2\varphi $-dependent cross section cannot affect significantly the soft-gluon predictions for the azimuthal asymmetry at low $Q^{2}$ in the energy region up to $S/4m^{2}\sim 10$.
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Let us briefly discuss the origin of perturbative stability of the $\cos
2\varphi $ asymmetry. Note that the mere $\varphi $-independent structure of the Sudakov logarithms cannot explain our results since perturbative stability does not take place at the parton level. In fact, the ratios $%
\frac{c_{A}^{(1,0)}}{c_{T}^{(1,0)}}(\eta ,\xi )$ and $\frac{c_{A}^{(0,0)}}{%
c_{T}^{(0,0)}}(\eta ,\xi )$ differ essentially from each other even at $\eta
,\xi \lesssim 1$. This is due to the fact that the physical soft-gluon corrections (\[18\]) are determined by a convolution of the Born cross sections with the Sudakov logarithms which, apart from factorized $\delta
(s_{4})$-terms, contain also non-factorizable ones (see Eq. (\[22\])). Kinematically, sizeable values of $\eta \sim 1$ allow $s_{4}/m^{2}\sim 1$ that leads to significant non-factorizable corrections. In other words, collinear bremsstrahlung carries away a large part of initial energy. Since the $\varphi $-dependent and $\varphi $-independent Born level partonic cross sections have different energy behavior, the soft radiation has different impact on these quantities.
Our analysis shows that two more factors are responsible for perturbative stability of the hadron level asymmetry. First, one can see from Figs. \[Fg.3\] and \[Fg.4\] that both $\varphi $-dependent and $\varphi $-independent Born level cross sections take their maximum values practically at the same values of $\eta$. Second, at fixed target energies, the gluon distribution function supports the contribution of the threshold region. In other words, sufficiently soft gluon distribution function makes the collinear gluon radiation effectively soft at the hadron level. In detail, the role of the gluon distribution function in perturbative stability of the azimuthal asymmetry in heavy quark photoproduction is discussed in Ref.[@14].
Another remarkable property of the azimuthal asymmetry closely related to fast perturbative convergence is its parametric stability.[^4] Our analysis shows that the pQCD predictions for the $\cos 2\varphi $ asymmetry are less sensitive to standard uncertainties in the QCD input parameters than the corresponding ones for the production cross sections. For instance, changes of $\mu _{F}$ in the range $m_{c}<\mu _{F}<2%
\sqrt{m_{c}^{2}+Q^{2}/4}$ affect the quantity $A(Q^{2})$ in charm production by less than $7\%$ at $\rho =0.025$ and $\xi \leq 4$. For the $\varphi $-integrated cross section, such changes of $\mu _{F}$ lead to $30\%$ variations in the same kinematics. We have also verified that all the NLO CTEQ5 versions of gluon density as well as the LO parametrization [@28a] lead to asymmetry predictions which coincide with each other with an accuracy of better than $1.5\%$.
Parametric stability of the azimuthal asymmetry leads to the scaling: with a good accuracy the quantity $A(\rho ,x,\xi )$ in Eq. (\[6\]) is a function of three variables, so that $$A^{{\rm {Charm}}}(\rho ,x,\xi )\approx A^{{\rm {Bottom}}}(\rho ,x,\xi )
\label{A1}$$ at the same values of $\rho $, $x$ and $\xi $. To illustrate this property, in Fig. \[Fg.12\] we plot the asymmetry parameter $A(Q^2)$ defined by Eqs. (\[27\]) as a function of $\xi$ at $\rho=10^{-3}$ and $10^{-4}$. Since the soft-gluon approximation is inapplicable to heavy flavor production at high energies, we give only the LO predictions for $A(\xi)$. The LO results for the charm and bottom cases are plotted by solid and dash-dotted lines, respectively. One can see that both curves coincide with each other with an accuracy of better than $1\%$.
It is also seen from Fig. \[Fg.12\] that, at the HERA energy ($\rho \sim 10^{-3}$ and $10^{-4}$ for bottom and charm quark, respectively), the pQCD predictions for the AA in heavy quark leptoproduction are large and can be tested experimentally.
Nonperturbative corrections
---------------------------
Let us discuss how the pQCD predictions for azimuthal asymmetry are affected by nonperturbative contributions due to the intrinsic transverse motion of the gluon in the target. Because of the relatively low $c$-quark mass, these contributions are especially important in the description of the cross sections for charmed particle production [@1].
To introduce $k_{T}$ degrees of freedom, $\vec{k}_{g}\simeq z\vec{p}+\vec{k}%
_{T}$, one extends the integral over the parton distribution function in Eq. (\[13\]) to $k_{T}$-space, $$\text{d}zg(z,\mu _{F})\rightarrow \text{d}z\text{d}^{2}k_{T}f\big( \vec{k}%
_{T}\big) g(z,\mu _{F}). \label{28}$$ The transverse momentum distribution, $f\big( \vec{k}_{T}\big) $, is usually taken to be a Gaussian: $$f\big( \vec{k}_{T}\big) =\frac{{\rm {e}}^{-k_{T}^{2}/\langle
k_{T}^{2}\rangle }}{\pi \langle k_{T}^{2}\rangle }. \label{29}$$ In practice, an analytic treatment of $k_{T}$ effects is usually used. According to [@29], the $k_{T}$-smeared differential cross section of the process (\[1\]) is a 2-dimensional convolution: $$\frac{\text{d}^{4}\sigma _{lN}^{{\rm {kick}}}}{\text{d}x\text{d}Q^{2}\text{d}%
p_{QT}\text{d}\varphi }\left( \vec{p}_{QT}\right) =\int \text{d}^{2}k_{T}%
\frac{{\rm {e}}^{-k_{T}^{2}/\langle k_{T}^{2}\rangle }}{\pi \langle
k_{T}^{2}\rangle }\frac{\text{d}^{4}\sigma _{lN}}{\text{d}x\text{d}Q^{2}%
\text{d}p_{QT}\text{d}\varphi }\Big( \vec{p}_{QT}-\frac{1}{2}\vec{k}%
_{T}\Big) . \label{30}$$ The factor $\frac{1}{2}$ in front of $\vec{k}_{T}$ in the r.h.s. of Eq. (\[30\]) reflects the fact that the heavy quark carries away about one half of the initial energy in the reaction (\[1\]).
Values of the $k_{T}$-kick corrections to LO predictions for the $\cos 2\varphi $ asymmetry in the charm production are shown in Figs. \[Fg.5\]$-$\[Fg.7\] and \[Fg.12\] by dashed curves. Calculating the $k_{T}$-kick effects we use $\langle k_{T}^{2}\rangle =0.5$ GeV$^{\text{2}}$. At fixed target energies, $k_{T}$-smearing for $A(Q^{2})$ and $A(x)$ is about $20$-$25\%$ in the region of low $Q^{2}$ and $x$, respectively, and decreases at large $Q^{2}$ and $x$. In the HERA range, expected values of the $k_{T}$-corrections are systematically smaller (see Fig. \[Fg.12\]).
Analogous calculations for the case of bottom production are presented in Figs. \[Fg.8\] and \[Fg.12\]. It is seen that $k_{T}$-kick corrections to the $b$-quark AA are practically negligible in the whole region of $Q^{2}$ and $x$.
Conclusion
==========
In this paper we have investigated the impact of soft-gluon radiation on the $\cos 2\varphi $ asymmetry in heavy flavor leptoproduction. The NLL approximation provides a good description of the available exact NLO results for $Q^{2}\lesssim m^{2}$ at energies of the fixed target experiments. Our calculations show that the azimuthal asymmetry is practically insensitive to soft-gluon corrections in this kinematics. We conclude that, unlike the $\varphi $-integrated cross sections, the $\cos 2\varphi $ asymmetry in heavy quark leptoproduction is an observable quantitatively well defined in pQCD: it is stable both parametrically and perturbatively, and insensitive (in the case of bottom production) to nonperturbative contributions. This asymmetry is of leading twist and predicted to be about $15\%$ at energies sufficiently above the production threshold for both charm and bottom quark. Measurements of the $\cos 2\varphi $ asymmetry in bottom production at HERA would provide an ideal test of pQCD.
Data on the charm azimuthal distributions from the COMPASS and HERMES experiments would make it possible to clarify the role of subleading twist contributions. Our analysis shows that, in the low-$x$ region, the AA is sensitive to the gluon transverse motion in the target. At high $x$, the intrinsic charm contribution [@30] to the structure functions may be significant [@31; @32]. In detail, the possibility of measuring the intrinsic charm content of the proton using the $\cos 2\varphi $ asymmetry will be considered in a forthcoming publication.
The author would like to thank S.J. Brodsky, A.B. Kaidalov, A.Kotzinian and A.G. Oganesian for useful discussions. I am grateful to High Energy Section of ICTP for hospitality while this work has been completed.
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[^1]: e-mail: [email protected]
[^2]: The well-known examples are the shapes of differential cross sections of heavy flavor production which are sufficiently stable under radiative corrections.
[^3]: Our analysis shows that the same situation takes also place for the energy and $Q^{2}$ behavior of the functions $c_{L}^{(1,l)}(\eta ,\xi )$, $l=0,1$. We do not give corresponding plots for $c_{L}^{(1,l)}(\eta ,\xi )$ because the contribution of the longitudinal cross section to the $\cos 2\varphi $ asymmetry is small numerically.
[^4]: Of course, parametric stability of the fixed order results does not imply a fast convergence of the corresponding series. However, a fast convergent series must be parametrically stable. In particular, it must be $\mu _{R}$- and $\mu _{F}$-independent.
|
---
abstract: 'Tunable metasurfaces have demonstrated the potential for dramatically enhanced functionality for applications including sensing, ranging and imaging. Liquid crystals (LCs) have fast switching speeds, low cost, and mature technological development, offering a versatile platform for electrical tunability. However, to date, electrically tunable metasurfaces are typically designed at a single operational state using physical intuition, without controlling alternate states and thus leading to limited switching efficiencies ($<30\%$) and small angular steering ($<\ang{25}$). Here, we use large-scale computational “inverse design” to discover high-performance designs through adjoint-based local-optimization design iterations within a global-optimization search. We study and explain the physics of these devices, which heavily rely on sophisticated resonator design to fully utilize the very small permittivity change incurred by switching the liquid-crystal voltage. The optimal devices show tunable steering angles ranging from to and switching efficiencies above 80%, exhibiting 6X angular improvements and 6X efficiency improvements compared to the current state-of-the-art.'
author:
- Haejun Chung
- 'Owen D. Miller'
title: |
Tunable metasurface inverse design for\
80% switching efficiencies and 144$^\circ$ angular steering
---
Introduction
============
{width="1.0\linewidth"}
Liquid-crystal (LC) devices promise the possibility for rapid electrical steering of optical beams, yet the complexity of designing for multiple refractive-index states in a single geometry has severely restricted the resulting diffraction efficiencies ($<50\%$), switching efficiencies (defined below, $<25\%$), and steering angles ($\leq \ang{24}$), even in state-of-the-art designs [@li2019phase]. In this work, we use large-scale computational optimization, “inverse design,” to discover fabrication-ready designs with switching efficiencies and diffraction efficiencies above $80\%$ and steering angles up to . We combine adjoint-based gradients [@Bendsoe2013; @jensen2011topology; @yang2009design; @miller2012photonic; @lalau2013adjoint; @piggott2015inverse; @molesky2018inverse; @lin2018topology] for rapid local optimization within a larger global search to discover the high-efficiency and high-steering-angle designs. We compute the complex resonance patterns of the optimal devices, which reveal several competing design requirements that explain the need for computational optimization of many degrees of freedom. Unlike metasurfaces designed for lens-like focusing [@aieta2015multiwavelength; @khorasaninejad2015achromatic; @avayu2017composite; @shrestha2018broadband; @chen2017gan; @paniagua2018metalens; @chen2018broadband; @wang2018broadband] and related applications [@zheng2015metasurface; @ni2013metasurface; @zhao2016full; @arbabi2017planar], we find that the optimal devices should have their field intensities concentrated not in the high-index grating material but instead in the low-index liquid-crystal embedding medium, to enable high switching efficiencies even for the relatively small refractive-index changes of LCs. Our largest-steering-angle devices exhibit $90\%$ diffraction efficiency at $-\ang{72}$ in the off state, and $70\%$ diffraction efficiency at $+\ang{72}$ in the on state, simultaneously exhibiting 6X angular and almost 6X switching-efficiency enhancements over the current state-of-the-art, paving a pathway to efficient liquid-crystal beam-control devices for applications ranging from LIDAR [@poulton2019long; @poulton2017coherent] to spatial light modulator [@shrestha2015high].
Thin optical films with complex lithographic patterns can control phases, amplitudes, diffraction-order excitations, and more general wave dynamics with high efficiency over large-area devices, comprising the basis for the emerging field of metasurfaces [@yu2014flat; @aieta2015multiwavelength]. Metasurfaces have shown significant promise for static (non-tunable) applications such as holography [@zheng2015metasurface; @ni2013metasurface; @zhao2016full], lensing [@aieta2015multiwavelength; @khorasaninejad2015achromatic; @avayu2017composite; @shrestha2018broadband; @chen2017gan; @paniagua2018metalens; @chen2018broadband; @wang2018broadband], and beam converters [@wu2017broadband; @chong2015polarization], in large part due to the use of a relatively simple design principle: for a given frequency of interest, one can specify the desired outgoing phases and amplitudes (and possibly dispersion characteristics [@yu2014flat]) across the device surface, and select from a library of waveguide-like meta-elements to locally approximate those phases and amplitudes. This design principle is not exact—the local-periodicity assumption is a source of error, especially in high-NA lens applications [@chung2019high; @lin2019topology]—and there is significant effort to leverage computation to improve it [@sell2017periodic; @pestourie2018inverse; @lin2019topology], but it has been sufficient for proof-of-principle high-performance devices.
For *dynamic* applications, however, in which the properties of the metasurface are designed to offer varying functionality in multiple operational states, from electrical [@huang2016gate; @yao2014electrically; @sautter2015active; @holsteen2019temporal], mechanical [@ee2016tunable], or thermal [@komar2018dynamic] switching mechanisms, the simple design principle appears to be quite inefficient. One might imagine that multi-state operation would require only small extra considerations in the “library” of designs, accounting for the additional states. However, as we show below, the requirement for high-efficiency resonant behavior in multiple states rapidly leads to highly complex resonant patterns, with individual elements far more complex than those of typical metasurface applications, due to the requirement for the multi-state behavior to be supported by a single geometrical structure. In lieu of a multi-state design principle, previous approaches [@savo2014liquid; @yao2014electrically; @lee2014ultrafast; @komar2017electrically] have simplified the design process by focusing only on high efficiency for a single state, but this naturally leads to lower efficiencies in the switching process over the dynamic range of the devices. (An alternative approach is to use a frequency comb in tandem with a designed metasurface, which can create time-dependent beam profiles albeit without full temporal “steering” control [@shaltout2019spatiotemporal].)
In this work, we show that large-scale computational design, an approach that efficiently optimizes over arbitrarily many degrees of freedom, offers a pathway to high-efficiency dynamic (tunable) metasurfaces. We focus on beam-switching with liquid-crystal devices, which already have significant commercial development and which show promise for applications such as LIDAR. We discuss the complexity of the design space, and describe a combined application of adjoint-based local-optimization techniques within a larger global-optimization platform, and use this approach to discover two-state switching devices with high switching efficiencies and high steering angles.
Computational multi-state design
================================
[Figure \[fig:schematic\]]{}(a,b) is a schematic depiction of a liquid-crystal (LC) beam-switching device. As is typical in LC devices [@bohn2018active; @komar2017electrically], the liquid-crystal layer is embedded between two alignment and contact layers. Within the liquid-crystal region, and above and below the contact layers, complex patterns can be lithographically fabricated, and previous work has designed grating layers for moderate-efficiency electrical [@li2019phase; @komar2017electrically; @buchnev2015electrically] and thermal [@komar2018dynamic] switching of LC devices. The key metric to design for is the *switching efficiency*, i.e. how effectively the device can switch between different optical-beam patterns. For periodic grating and meta-grating structures, diffraction efficiency is an important determinant of the switching efficiency, but not the only one: a device that separates an optical field into a 50% mix of two outgoing diffraction orders, for both voltages of a two-state device, effectively has zero switching efficiency due to the inability to distinguish the two states. Moreover, in many cases back-reflected light represents only a minor loss mechanism, without affecting the *relative* power distribution between the forward-going beams nor the ability to distinguish them, and can be normalized out. Thus, for a two-state optical-switching device operating over frequencies $\omega$ with geometrical degrees of freedom $g$, we define switching efficiency by the expression $$S = \frac{1}{T} \left\{ \frac{1}{2} \sum_{s=\textrm{on,off}}\left[P^s_\textrm{tar}(\omega,g) - \sum_{j\neq \textrm{tar}}P^s_j(\omega,g) \right]\right\},
\label{eq:fom}$$ which is the power in the target (desired) diffraction orders, $P^{s}_{\rm tar}(\omega,g)$, averaged over state $s$, minus the total state-averaged power in all other diffraction orders, $P_{j\neq \textrm{tar}}^s(\omega,g)$, normalized by the forward transmission efficiency $T$. This definition of switching efficiency, which can be easily generalized to more states, linear combinations of diffraction orders, etc., enables comparison among different device designs. [Figure \[fig:schematic\]]{}(c) shows the switching efficiencies of recent state-of-the-art LC beam-switching devices, which show moderate diffraction efficiencies (24–54%, labeled), but somewhat lower switching efficiencies (ranging from $13\%$ to $29\%$)) due to the contamination of unwanted diffraction orders that inhibits the ability to distinguish the on/off states. Included in [Fig. \[fig:schematic\]]{}(c) are the switching efficiencies of the optimal devices that we discover, discussed further below, segregated into three architectures: a class of devices with a single silicon grating in the liquid-crystal region (solid purple line), and two classes of devices with two additional gratings on top and bottom, one for -1 to +1 order steering (solid red line), and one for -2 to +2 order steering (solid blue line). There are many geometrical degrees of freedom in each architecture: the individual“pixels” (78 nm wide) of each grating, the thicknesses of the alignment, contact, and liquid-crystal layers, and the period of the structure. The pixel size is chosen as $\lambda/20$ to provide sufficient control while avoiding features that are too fine for fabrication. We take the switching to occur between two states with the same polarization, in which case the gratings can be chosen to have translation invariance perpendicular to a plane containing both angles, and the system can be modeled in this two-dimensional plane. There are many grating degrees of freedom ($\approx 400$), and to optimize these it is critical to be able to rapidly compute gradients of the switching-efficiency objective. To do so, we use the adjoint method (also known as “topology optimization” [@bendsoe2001topology; @Bendsoe2013] and “inverse design” [@yang2009design; @jensen2011topology; @frandsen2014topology; @lalau2013adjoint; @piggott2015inverse; @su2017inverse; @sell2017large; @callewaert2018inverse; @ganapati2014light] in nanophotonics, and “backpropagation” in the deep-learning community [@Werbos1994; @Rumelhart1986; @LeCun1989]), which is efficient and effective at optimizing many small-scale degrees of freedom [@molesky2018inverse]. Adjoint-based methods exploit reciprocity (or generalized reciprocity [@miller2012photonic]) to convert the process of computing thousands or millions of individual gradient calculations into a single extra simulation, in which “adjoint sources” are specified according to the desired objective, back-propagated through the optical system, and then combined with the “direct” fields excited by the original incident wave to compute all gradients at once. For an objective such as switching efficiency, [Eq. (\[eq:fom\])]{}, that depends on the outgoing electric fields ${{\bm{E}}}$, the general prescription [@miller2012photonic] for each “forward” simulation (in this case, the voltage-on and voltage-off simulations) is to run an “adjoint” simulation with current sources proportional to the derivative of the objective with respect to the electric field (SM): $$\begin{aligned}
{\bm{J}}_{\rm adj}({{\bm{x}}}) &= -i\omega \frac{\partial \mathcal{F}}{\partial {{\bm{E}}}} \nonumber \\
&= -\frac{i\omega}{2}[c_\textrm{tar} \cos\theta_{\rm tar} {\bm{E}}_\textrm{tar}^{*}({{\bm{x}}}) - \sum_{n \neq \textrm{tar}}c_\textrm{n} \cos\theta_n {\bm{E}}_\textrm{n}^{*}({{\bm{x}}})].
\label{eq:Jadj}\end{aligned}$$ In our adjoint equation indicated by [Eq. (\[eq:Jadj\])]{}, we exclude the $1/T$ of [Eq. (\[eq:fom\])]{} to drive the optimization to exhibit high transmission in addition to high switching efficiency. The pixels in the gratings are represented during the design process as grayscale pixels, with refractive indices varying between their minimum and maximum values, and as the local optimization proceeds we penalize intermediate refractive-index values until a binary design is reached. This process is very efficient for the many grating degrees of freedom. However, it is less efficient for variables representing larger geometrical parameters: the thicknesses of the various regions, and the periodicity of the structure. Wave-interference effects create a tremendous number of poor-quality, local optima for these parameters, since varying them even by half a wavelength or less can take one from a field minimum to a maximum.
The many-local-optima problems for these “global” (beyond wavelength-scale) parameters could be significantly compensated by separating them into pixelated local degrees of freedom (DOFs) that vary independently. However, they are fixed by fabrication constraints and must not be separated. Thus, to optimize these parameters, we embedded the grating-DOF local-optimization procedure into a global search to discover optimal thickness and periodicity values. Particle swarm optimization [@robinson2004particle] is used for a global optimization algorithm, initially instantiating many “particles” with random structural parameters (i.e., top TiO$_2$ grating, ITO, alignment, LC, silicon grating and bottom TiO$_2$ grating thicknesses). Within Each “particle” we perform inverse design, computed in a single computational core, optimizing the fine-scale features of the device. Then, new parameters of the next iteration are determined by the optimal figure of merit values. The global optimization is run for 150 iterations, which is sufficient to converge to a set of very similar “particles,” with similar large-scale-feature values. Each iteration of global optimization takes approximately 10 minutes on 25 cores in our computational cluster (Intel Xeon E5-2660 v4 3.2 GHz processors) while each inverse design iteration takes less than 5 seconds in a single core computer.
Optimal Designs
===============
{width="0.8\linewidth"}
We apply the multi-state computational design process described above to discover the single- and multi-grating designs depicted in Figs. \[fig:single\_layer\]–\[fig:wide\_angle\]. We start by designing LC metasurfaces with a single embedded silicon grating, intentionally selecting a platform very similar to that of recent works [@li2019phase; @komar2018dynamic] to show the efficiency gains that are possible through computational design. Then we expand to structures with multiple grating layers, where we show the extensive capability for LC metasurfaces to simultaneously achieve high efficiency and high steering angles. In all of the designs demonstrated below, we use as our design wavelength. For the LC material, we use E7 [@yang2010complex], which has a refractive-index variation $\Delta n$ of about 0.192 between the voltage-on and voltage-off states. TiO$_2$ [@palik1998handbook] is used for top and bottom supportive gratings while we use silicon for the grating inside the LC layer. ITO [@laux1998room] and alignment layers [@ma2018liquid; @bohn2018active; @komar2018dynamic] are included, as typically required. Unlike metasurfaces for lensing and related applications, high-index materials do not appear to be required for high diffraction efficiency nor switching efficiency; we use Si and TiO$_2$ simply because of their common usage [@li2019phase; @komar2017electrically; @komar2018dynamic] and scale-up feasibility. The top ITO works as an electrical contact and the alignment layer coordinates the axis of the LC director into the out-of-plane direction. Of course, a different wavelength, set of materials, or parameter regime can seamlessly be incorporated into our design process.
{width="0.9\linewidth"}
{width="1.0\linewidth"}
Single-grating designs
----------------------
In this section, we design tunable metasurfaces with a single grating layer. Single-grating metasurfaces can be designed by physical intuition using effective medium theory [@choy2015effective], whereby the filling ratio of two materials is adjusted to realize specific transmission phases, or by a unit-cell library approach [@aieta2015multiwavelength], whereby a large design space is decomposed into “unit cells” with a small number of parameters whose entire design space can be stored in a library to design for a small number of criteria. Neither approach is well-suited to designing many parameters for multi-state operation.
[Figure \[fig:single\_layer\]]{} shows an optimal single-grating design for switching between $-\ang{12}$ and $+\ang{12}$, angles chosen to match the current state-of-the-art [@li2019phase]. The optimized single-grating metasurface achieves diffraction efficiencies of 71% in the voltage-on state and 52% in the voltage-off state, with clean outgoing field patterns visible in Figs. \[fig:single\_layer\](c,d). A key determinant of the angular purity is the diffraction efficiency normalized by the total transmission, since reflection does not contribute noise in the outgoing-wave patterns, and the transmission-normalized (TN) efficiencies of this structures are 86% and 63%, respectively. During the optimization, we fix the top and bottom sides have to have thin ITO and alignment layers, while we include the thicknesses of the LC layer and the silicon grating as global design parameters. The beam steering efficiency (48%) shown here is already significantly larger than any other theoretical designs. However, for key applications, one can expect the need for larger steering angles and even high switching efficiencies. Thus, in the next section we explore more complex device architectures.
{width="0.8\linewidth"}
Multi-grating designs
---------------------
In this section, we design tunable metasurfaces with three gratings—one in the silicon, and two in the TiO$_2$ surroundings—to discover an ultra-high-efficiency beam-switching device. Generally, multi-layer metasurface structures offer increased functionality through increased path-length enhancements and multiple-reflection interactions, and multilayer metasurfaces have been proposed for light concentration [@lin2019topology] and flat lens [@lin2018topology] applications. Here, the two TiO$_2$ gratings must enable specific functionality: the bottom grating must be transmissive for the plane wave incident from below, while being highly reflective for all off-angle plane waves reflected from above, and the top grating should either redirect all light to the single desired outgoing diffraction order, or at least restrict transmission through any undesired orders. Though the use of multiple gratings requires precise alignment, the gratings play complementary roles and potentially enable near-unity switching efficiencies even at very high switching angles.
We start by reconsidering the problem of high-efficiency switching from $-\ang{12}$ to $+\ang{12}$. The optimal design, depicted in [Fig. \[fig:narrow\_angle\]]{}(a), achieves diffraction efficiencies of 78% (82% TN) in the voltage-on state and 78% (90% TN) in the voltage-off state, for a switching efficiency of 76%, with very little power in any other outgoing diffraction orders. The clean outgoing waves are depicted in [Fig. \[fig:narrow\_angle\]]{}(c,d). The optimized device has ITO/alignment layer thicknesses of 77, 155 nm, top and bottom grating thicknesses of 155 and 233 nm, a silicon grating thickness of 155 nm, and a liquid-crystal layer thickness of 543 nm.
Among the many designs that were discovered across the single-, double-, and triple-grating architectures, for beam-steering angles from $\ang{24}$ to $\ang{144}$, we highlight here the highest steering-angle designs, which employ a triple-grating structure to achieve steering from $-\ang{72}$ to $+\ang{72}$. By avoiding a design with collections of locally varying “unit cells,” we circumvent the limitations [@chung2019high] arising from breaking the local-periodicity assumption. [Figure \[fig:wide\_angle\]]{} shows an optimal structure with thickness and period (which is $1.05\lambda$). The diffraction efficiencies in the target orders are 62% and 76% for voltage-on and -off states, respectively, while the transmission normalized target efficiencies are 70% and 90%, respectively. These diffraction efficiencies are individually nearly as large as those of state-of-the-art high-angle diffraction gratings that are designed only for a single operational state [@sell2017large]. The real part of the electric-field profile shown in [Fig. \[fig:wide\_angle\]]{}(c),(d) demonstrates the clear angle-directed outgoing wave patterns, and in [Fig. \[fig:wide\_angle\]]{}(d),(h) we simulate a Gaussian beam incident upon the structure to more clearly visualize the high-fidelity switching that is achieved.
To understand the physics of the high-efficiency designs that we discover, we analyze the quality factors ($Q$) and resonance patterns for our high-efficiency structures designed for -2 to +2 order switching. As shown in [Fig. \[fig:resonant\_mode\]]{}(a), we find that the optimal designs are a new kind of *dual-resonance* structure that support one moderate-$Q$ resonance in the on state (red circles), and a different moderate-$Q$ resonance in the off-state (blue circles). In [Fig. \[fig:resonant\_mode\]]{}(b), we depict the resonant field pattern of our $\ang{96}$-steering device, which has 87% switching efficiency and transmission-normalized efficiencies of 86% (voltage-on) and 95% (voltage-off). The field pattern is computed by exciting point dipoles at the high-intensity locations of the plane-wave forward simulation, discovering the mode responsible for the high efficiency. The resonance pattern of the voltage-on state shows strong coupling to $c_{-2}$ channel in transmission direction and $c_{0}$ channel in incidence direction, agreeing well with what one would expect. In the voltage-off state, the new resonant pattern couples to the $c_{+2}$ channel in transmission direction and $c_{0}$ channel in incidence direction. An intriguing trend in [Fig. \[fig:resonant\_mode\]]{}(a) is that when we overlay the diffraction efficiencies (red and blue stars) with the quality factors, we observe a correlation between the two. This suggests that quality factors of at least 30 or so may be necessary to achieve the highest possible diffraction efficiencies in each operational state of a beam-switching or beam-steering device.
Extensions
==========
In this work, we have demonstrated high-efficiency, wide-angle, electrically tunable metasurfaces that operate at wavelength, achieving state-of-the-art steering angles and switching efficiencies. Our inverse-design approach can be applied more broadly to any multi-configuration-state optical functionality, for applications including next-generation LiDAR, spatial light modulators, and free-space data communication. In the liquid-crystal beam-steering design space, natural extensions include many-state operation (towards steering rather than switching) and three-dimensional beam control. In addition to the “bottom-up” large-scale optimization approach presented here, an interesting question is the limits of such design: for a given set of liquid-crystal and semiconductor refractive indices, is it possible to exploit sum rules [@gordon_1963; @purcell_1969; @sohl_gustafsson_kristensson_2007; @sanders_manjavacas_2018; @Shim2019], passivity and convexity [@kwon_pozar_2009; @liberal_ederra_gonzalo_ziolkowski_2014; @miller_polimeridis_reid_hsu_delacy_joannopoulos_soljacic_johnson_2016; @Zhang2019], and/or duality [@angeris2019computational] to map out the limits to maximal performance as a function of the steering angle and the number of operational states?
Funding Information {#funding-information .unnumbered}
===================
H. C. and O. D. M. were partially supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0093.
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---
abstract: 'We propose a new detection method for ultrasound-modulated optical tomography, which allows to perform, using a CCD camera,a parallel speckle detection with an optimum shot noise sensitivity. Moreover,we show that making use of a spatial filter system allows us to fully filter off the speckle decorrelation noise. This method being confirmed by a test experiment.'
author:
- 'M. Gross and P. Goy'
- 'M. Al-Koussa'
title: 'Shot noise detection of the ultrasound tagged photon in ultrasound-modulated optical imaging.'
---
Ultrasound-modulated optical tomography is a new non invasive and non ionizing biological tissues imaging technique. In this technology an ultrasonic wave is focused into a diffusing medium that scatters the incident optical beam. Due to the ultrasonic vibration of the medium, some of the diffused photons are shifted in frequency on an ultrasonic sideband. These are the so called tagged photons [@leveque99], which can be selected to perform imaging. The advantage of the method is its combination of optical contrast and ultrasonic resolution.
Many groups have worked on that field. Marks et al. [@Marks93] investigated the modulation of light in homogeneous scattering media with pulsed ultrasound. Wang et al. [@wang95] performed ultrasound modulated optical tomography in scattering media. Lev and al. made scattering media study in the reflection configuration [@lev2000]. Wang and Ku [@wang98] developed a frequency chirp technique to obtain scalable imaging resolution along the ultrasonic axis by a 1D Fourier transform. Leveque and al. [@leveque99] performed parallel detection of multiple speckles on a video camera and demonstrated an improvement of the detection signal to noise ratio on 1D images of biological tissues. This parallel speckle detection (PSD) is considered to be “so far, the most efficient technique for ultrasound modulated optical tomography” [@wang2002], and is now extensively used in the field, combined [@leveque2000; @wang2000b; @forget2003] or not [@wang2000; @wang2002] with the frequency chirp technique.
By analysing the PSD detection process, we show nevertheless that the PSD sensitivity is far from optimum. Moreover, as noticed by Leveque et al. [@leveque99], PSD is sensitive to the “decorrelation of the speckle pattern, which reduces the signal and increases the noise” (see also [@wang2000; @wang2002]). In this letter, we propose to solve these two problems by adapting our heterodyne technique [@LeClerc2000] to PSD, i.e. by performing heterodyne PSD (or HPSD). In HPSD, the LO beam passes outside the sample. The LO field is thus much larger, and the detection sensitivity is much better. It is then possible to reach the optimum shot noise limit. On the other side, the HPSD LO beam is a plane wave. One can then separate the $k$-space components of the detected field. By using a proper optical arrangement, which reduces the $k$-space extend, we can fully filter off the decorrelation noise.
Let us analyse the PSD detection process. The focus point of the ultrasonic wave can be considered as a source of ultrasonic tagged photons, which are detected coherently by heterodyne detection. For a single pixel detector, the coherent detection selects the field within the spatial mode that fits with the pixel considered as an antenna. The collection efficiency can be characterized by its “optical etendue”, defined as the product of the emitting area by the emission solid angle, which is the two-dimensional generalization of the usual Lagrange invariant of geometrical optics. Due to diffraction, for one single mode, the etendue is about $\lambda^2$. For a $N$ pixels detector, since each pixel is able to perform coherent detection within its mode, the etendue is about $N \lambda ^2$. On the emission side, the etendue is about $\pi S$, since each point of the sample external surface of area $S$ diffuses photons in all outgoing directions (solid angle $\sim
\pi$). The collection efficiency $\eta$, which is the ratio of the etendues is very low : $\eta \approx N\lambda ^2/\pi S$. For a $2
cm \times 2 cm \times 2 cm$ diffusing sample, $\eta \approx
10^{-10}$ for 1 pixel, and $\eta \approx 10^{-4}$ for $N=10^6$ pixels. The meaning of $\eta$ is quite simple. Forget et al. [@forget2003] explained that, in order to detect the speckles efficiently, it is necessary to “position ... the camera ... to match the size of a grain (of speckle) with the size of a pixel”. The camera must be thus placed quite far away from the sample, and the photons, which are diffused by the sample in all directions, have a probability $\sim \eta $ to reach the CCD.
On the CCD surface, each photon is converted into a photo electron with a probability equal to the CCD quantum efficiency $Q$. In the optimal case, the heterodyne detection noise is related to the shot noise of the local oscillator. Accounting of the heterodyne gain, this noise corresponds to 1 photo electron per pixel during the measurement time [@bachor_1998] (since both shot noise and heterodyne gain are proportional to the LO amplitude). PSD is far from this optimum for two reasons. Firstly, the PSD LO beam is obtained by amplitude modulation of the main laser. The noise is thus related to the total intensity (carrier + sideband), while the gain depends on the sideband only. Secondly, the LO beam passes through the sample and is diffused in many modes, which do not match with the mode of the detector. Neglecting absorbtion, and back reflection (which are also present), one looses here a factor $\eta$ on the LO useful intensity. The LO intensity is then too small to get enough heterodyne gain for efficient detection.
To improve the sensitivity, we propose to perform HPSD with the Fig.\[fig\_setup\] a) setup. This setup is similar to the Toida et al. [@inaba91] one, but with a CCD camera in place of the mono pixel detector [@LeClerc2000]. The main laser L is a $\lambda = 850 nm $, $20 mW$ Newport 2010M laser followed by an optical isolator. The mirror MM splits the laser into two beams. The low intensity LO beam is shifted in frequency by $\Delta f$ by the 2 acousto optic modulators AOM1 ($80$ $MHz$) and AOM2 ($80$ $MHz$ $+ \Delta f$). It is expanded by BE ($20 \times $) in order to get a plane wave (diameter $1.5$ $cm$) larger than the CCD area. On the other side, the high intensity signal beam irradiates the sample in a $13 cm$ width water vessel.
The PZT ultrasonic transducer (Parametrics: $f_a=2.2MHz$, diameter $35 mm$, focal length: $50 mm$) generates an ultrasonic wave that is focused into the sample. The signal beam that is diffused toward the CCD ($z$ direction) at both the optical carrier frequency (field $E_L$) and at the ultrasonic sidebands (field $E_A$) interferes with the LO beam (beam splitter BS) on the CCD camera. Accounting of the optical isolator, BS and water losses, the measured laser power reaching the sample is $2.5$ $mW$. The CCD camera (PCO Pixelfly: $1280 \times 1024$ pixels of $6.7 \times
6.7 \mu m $, $f_c=12.5$ $Hz$ , $12 bit$ digital, $2.2 \%$ measured quantum efficiency at $850 nm$) records in real time the interference pattern on a PC computer. MM is adjusted to get an average of $2000$ shots per pixels ($1/2$ full scale) for the LO beam, which remains ever much larger than the signal beam that is strongly attenuated and diffused by the sample. To measure the tagged photons field complex amplitude, we have chosen $\Delta f =
f_a + f_c/4$.
According to the camera gain given by PCO ($2.2$ electrons for 1 LSB: low significant bit), we have measured, without signal beam, a noise corresponding to 1 signal photo electron per pixel (within $10 \%$). Our HPSD setup performs thus a optimal, shot noise limited, heterodyne detection.
Consider now the speckle decorrelation noise. We have first to notice that HPSD is less sensitive to this noise than PSD: the noise is the same in both cases, but the signal is larger in HPSD, because the heterodyne gain is higher. We will see now how to filter off this noise. The setup (Fig.\[fig\_setup\]a dashed rectangle) is modified as shown on Fig.\[fig\_setup\]b. The lens $O$ (focal length $f=250$ $mm$) and the diaphragm $D$ ($\simeq 25
\times 5$ $mm$ located in the $O$ focal plane) collimate the field diffused by the sample ($E_L$ and $E_A$), and reduce the $k$-space extend in the $k_x$ direction. By this way, the speckle grains are enlarged in $x$ and extended over several CCD pixels. Moreover, the LO beam is slightly tilted (in the $x,z$ plane) making a angle $\theta$ with the $z$ direction, so that the tagged photons versus LO beam interference exhibit vertical fringes (along $y$). It is then possible to separate the tagged photon signal (fringes) from the decorrelation noise (no fringes) by a simple Fourier calculation.
To illustrate this point, we have studied the field diffused by a $3.5$ $cm$ phantom sample with a 15 Vpp (Volt peak to peak) $2
MHz$ excitation of the PZT. From $4$ successive CCD images signal $I_1$,$I_2$...$I_4$ (where $I_i$ is proportional to $I=|E|^2$) we have calculated the 4-phases complex signal $S=(I_1 - I_3) +
j.(I_2-I_4)$ where $j^2=-1$. We have calculated $\tilde S
(k_x,k_y)=FFT\left( {S(x,y)} \right)$ by making a $1024 \times
1024$ truncation over the $S(x,y)$ data measured on the CCD ($1280
\times 1024$) followed by a Fast Fourier Transform (FFT). The $\tilde I (k_x,k_y)= |{\tilde S}|^2 $ matrix ($1024 \times 1024$) is imaged on Fig.\[fig\_FFT\] (left hand side) with logarithmic arbitrary gray scale. The sum over the column $\mathcal {I} (k_x)=
\sum\nolimits_{k_y} \tilde I((k_x,k_y))$ is also plotted (right hand side). One can separate here the contributions of the product terms of $I=E.E^*$ (where $E=E_{LO}+E_{L}+E_{A}$: $E_{LO}$ being the LO field, $E_{L}$ the field diffused at the carrier frequency and $E_{A}$ the ultrasonic sideband tagged photons field). The tagged photon heterodyne term $E_{LO}.{E_A}^*$ evolves fast is space (fringes). It yields to the $A$ rectangular bright zone for $\tilde I$, and to the $A$ peak for $\mathcal {I} $ (angular width $d/f$, angular offset $\theta$). The speckle decorrelation noise corresponds to the fluctuations on the $E_L.{E_L}^*$ term, which evolves slowly in space (no fringes). It yields to the bright zone in the center of the $k$-space image ($k_x \approx k_y \approx
0$), and to the triangular (convolution of 2 rectangles) peak in the plot ($B$ and $C$). The fluctuations of the $E_{LO}.{E_{LO}}^*$ term yield to the very narrow peak visible on the 1D plot ($k_x=0$). The other terms give very small contributions. For example, the $E_{LO}.{E_L}^*$ and $E_{L}.{E_A}^*$ terms evolve fast in time ($2 MHz$) and are filter off by the CCD. As seen, a proper choice of the $\theta$ tilt allows us to separate in the $k$-space the tagged photon ($A$) and the speckle decorrelation noise ($B$ and $C$) contributions to signal. For control purpose, we have considered the zone $D$, symmetric to $A$, where the shot noise only contributes.
We have performed a test experiment with the diffusing PSD sample already studied in [@forget2003] (see Fig.2 of [@forget2003]). In the sample, a $4 mm$ diameter vertical ($x$ direction) cylindric black inked zone absorbing the light. The sample is slightly compressed in the $z$ direction and its width is $15 mm$. To get a pertinent information we have summed $\tilde
I (k_x,k_y)$ in the $A$ (tagged photons + shot noise) and $D$ (shot noise alone) zones: $\mathcal {I}_A = \sum\nolimits_{k_x \in
A} \mathcal {I}(k_x) $. We have calculated $\mathcal {I}_{AD} =
\mathcal {I}_{A}- \mathcal {I}_{D}$ (tagged photons alone), and plotted $\mathcal {I}_{AD}/\mathcal {I}_{D}$ (tagged photon signal normalized with respect to shot noise). The Fig.\[fig\_data\]a, b, c and d show the plots obtained by moving the sample ($x=0...20
mm$ with $0.5 mm$ steps) with $1.4$, $2.5$, $4.5$, and $8$ $Vpp$ on the PZT respectively. Each point corresponds to $3$ successive acquisitions of $4$ images ($0.96 s$). The black inked zone near $x=5mm$ is clearly seen. As seen, outside the absorbing zone, $\mathcal {I}_{AD}/\mathcal {I}_{D}$ ($\simeq 0.035$, $0.12$, $0.37$ and $1.2$) is proportional to the square of the PZT applied voltage (i.e. to the ultrasonic power). By making $\Delta f =
f_c/4$, with no ultrasonic wave, we have measured the carrier field $E_L$ signal $\mathcal {I}_{AD}/\mathcal {I}_{D} \simeq
700$. The maximum ($8 Vpp$) ultrasonic conversion factor is then $1.7 \times 10^{-3}$. As seen, curves c and d exhibit roughly the same S/N (signal/noise) ratio. The technical noise is then the limiting factor. On the other side, on curves a and b, the tagged photon signal is lower and the S/N goes down. The shot noise becomes thus the limiting factor. With $2$ zones ($A$ and $D$), with $n_{pix} \approx 2.10^5$ pixels per zone, one expects for $\mathcal {I}_{AD}/\mathcal {I}_{D}$ a shot noise of $\pm \sqrt
{2/n_{pix}}= \pm 0.003$, in good agreement with the noise observed on curve a. This shows that the sensitivity obtained by our technique is truly limited by shot noise.
As seen above, our HPSD detection method presents many advantages for ultrasound-modulated tomography. It allows to perform parallel speckle detection of the ultrasound-modulated component with an optimum shot noise sensitivity, and to fully filter off the speckle decorrelation noise. At the end, many controls are possible on the data. One can measure , for example, both the ultrasound-modulated signal (zone $A$), the shot noise (zone $D$) and the speckle decorrelation noise (zone $B$ and $C$).
We thank the ESPCI group for their help (ultrasonic transducer, diffusing sample), and for fruitful scientific discussions.
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B.C. Forget, F.R. Ramaz, M. Atlan, J. Selb, and A.C. Boccara. High contrast fast fourier transform acousto optical tomography of phantom tissues with a frequency chirp modulation of the ultrasound. , 42:1379, 2003.
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---
abstract: 'We consider the Hamiltonian stationary equation for all phases in dimension two. We show that solutions that are $C^{1,1}$ will be smooth and we also derive a $C^{2,\alpha}$ estimate for it.'
author:
- Arunima Bhattacharya AND Micah Warren
bibliography:
- 'Hamiltonian\_Stationary.bib'
title: Interior Schauder Estimates for the Fourth Order Hamiltonian Stationary Equation in two dimensions
---
Introduction
=============
In this paper, we study the regularity of the Lagrangian Hamiltonian stationary equation, which is a fourth order nonlinear PDE. Consider the function $u:B_{1}\rightarrow\mathbb{R}$ where $B_{1}$ is the unit ball in $\mathbb{R}^{2}$. The gradient graph of $u$, given by $\{(x,Du(x))|x\in
B_{1}\}$ is a Lagrangian submanifold of the complex Euclidean space. The function $\theta$ is called the Lagrangian phase for the gradient graph and is defined by $$\theta=F(D^{2}u)=Im\log\det(I+iD^{2}u)$$ or equivalently, $$\theta=\sum_{i}\arctan(\lambda_{i}) \label{P}$$ where $\lambda_{i}$ represents the eigenvalues of the Hessian.
The nonhomogenous special Lagrangian equation is given by the following second order nonlinear equation $$F(D^{2}u)=f(x). \label{SL}$$
The Hamiltonian stationary equation is given by the following fourth order nonlinear PDE $$\Delta_{g}\theta=0\label{HS}$$ where $\Delta_{g}$ is the Laplace-Beltrami operator, given by: $$\Delta_{g}=\sum_{i,j=1}^{2}\frac{\partial_{i}(\sqrt{detg}g^{ij}\partial_{j})}{\sqrt{detg}}$$ and $g$ is the induced Riemannian metric from the Euclidean metric on $\mathbb{R}^{4}$, which can be written as $$g=I+(D^{2}u)^{2}.$$
Recently, Chen and Warren [@CW] proved that in any dimension, a $C^{1,1\text{ }}$solution of the Hamiltonian stationary equation will be smooth with uniform estimates of all orders if the phase $\theta\geq
\delta+(n-2)\pi/2,$ or, if the bound on the Hessian is small. In the two dimensional case, using [@CW]’s result, we get uniform estimates for $u$ when $\left\vert \theta\right\vert \geq\delta>0$ (by symmetry). In this paper, we consider the Hamiltonian stationary equation for all phases in dimension two without imposing a smallness condition on the Hessian or on the range of $\theta$, and we derive uniform estimates for $u$, in terms of the $C^{1,1}$ bound which we denote by $\Lambda$. We write $||u||_{C^{1,1}(B_{1})}= ||Du||_{C^{0,1}(B_{1})}=\Lambda$. Our main results are the following:
Suppose that $u\in C^{1,1}(B_{1})\cap W^{2,2}(B_{1})$ and satisfies (\[HS\]) on $B_{1}\subset\mathbb{R}^{2}$. Then u is a smooth function with interior Hölder estimates of all orders, based on the $C^{1,1}$ bound of $u$.
Suppose that $u\in C^{1,1}(B_{1})\cap W^{2,2}(B_{1})$ and satisfies (\[SL\]) on $B_{1}\subset\mathbb{R}^{2}$. If $\theta\in C^{\alpha}(B_{1})$, then there exists $R=R(2,\Lambda,\alpha)<1$ such that $u\in C^{2,\alpha}(B_{R})$ and satisfies the following estimate $$|D^{2}u|_{C^{\alpha}(B_{R})}\leq C_{1}(||u||_{L^{\infty}(B_{1})}, \Lambda, |\theta|_{C^{\alpha}(B_{1})}).\label{E}$$
Our proof goes as follows: we start by applying the De Giorgi-Nash theorem to the uniformly elliptic Hamiltonian stationary equation (\[HS\]) on $B_{1}$ to prove that $\theta\in C^{\alpha}(B_{1/2})$. Next we consider the non-homogenous special Lagrangian equation (\[SL\]) where $\theta\in
C^{\alpha}(B_{1/2}).$ Using a rotation of Yuan [@Yuan2002] we rotate the gradient graph so that the new phase $\bar{\theta}$ of the rotated gradient graph satisfies $\left\vert \bar{\theta}\right\vert \geq\delta>0$. Now we apply [@CC] to the new potential $\bar{u}$ of the rotated graph to obtain a $C^{2,\alpha}$ interior estimate for it. On rotating back the rotated gradient graph to our original gradient graph, we see that our potential $u$ turns out to be $C^{2,\alpha}$ as well. A computation involving change of co-ordinates gives us the corresponding $C^{2,\alpha}$ estimate, shown in (\[E\]). Once we have a $C^{2,\alpha}$ solution of (\[HS\]), smoothness follows by [@CW Corollary 5.1].
In two dimensions, solutions to the second order special Lagrangian equation $$F(D^{2}u)=C$$ enjoy full regularity estimates in terms of the potential $u$ [@WarrenYuan2]. For higher dimensions, such estimates fail [@WangYuan] for $\theta=C$ with $\left\vert C\right\vert <(n-2)\pi/2$.
Proof of theorems:
===================
We first prove Theorem 1.2, followed by the proof of Theorem 1.1. We prove Theorem 1.2 using the following lemma.
Suppose that $u\in C^{1,1}(B_{1})\cap W^{2,2}(B_{1})$ satisfies (\[SL\]) on $B_{1}\subset\mathbb{R}^{2}$. Suppose $$0\leq\theta(0)<(\pi/2-\arctan\Lambda)/4\label{C}.$$ If $\theta\in C^{\bar{\alpha}}(B_{1})$, then there exists $0<\alpha<\bar{\alpha}$ and $C_{0}$ such that $$|D^{2}u(x)-D^{2}u(0)|\leq C_{0}(||u||_{L^{\infty}(B_{1})}, \Lambda,|\theta|_{C^{\alpha}(B_{1})})\ast|x|^{\alpha}.$$
Consider the gradient graph $\{(x,Du(x))|x\in B_{1}\}$ where $u$ has the following Hessian bound $$-\Lambda I_{n}\leq D^{2}u\leq\Lambda I_{n}$$ a.e. where it exists.
Define $\delta$ as $$\delta=(\pi/2-\arctan\Lambda)/2>0.$$ Since by (\[C\]) we have $0\leq\theta(0)<\delta/2$, there exists $R^{\prime}(\delta,|\theta|_{C^{\bar{\alpha}}})>0$ such that $$|\theta(x)-\theta(0)|<\delta/2$$ for all $x\in B_{R^{\prime}}\subseteq B_{1}$. This implies for every $x$ in $B_{R^{\prime}}$ for which $D^{2}u$ exists, we have $$\delta>\theta>\theta(0)-\delta/2.$$ So now we rotate the gradient graph $\{(x ,D u (x))\vert x \in B_{R^{ \prime}}\}$ downward by an angle of $\delta$.
Let the new rotated co-ordinate system be denoted by $(\bar{x},\bar{y})$ where $$\begin{aligned}
\bar{x} & =\cos(\delta)x+\sin(\delta)Du(x)\label{R_0}\\
\bar{y} & =-\sin(\delta)x+\cos(\delta)Du(x). \label{R_1}$$
On differentiating $\bar{x}$ (\[R\_0\]) with respect to $x$ we see that $$\frac{d\bar{x}}{dx}=\cos(\delta)I_{n}+\sin(\delta)D^{2}u(x)\leq\cos
(\delta)I_{n}+\Lambda\sin(\delta)I_{n}$$ Thus $$\cos(\delta)I_{n}-\Lambda\sin(\delta)I_{n}\leq\frac{d\bar{x}}{dx}\leq
\cos(\delta)I_{n}+\Lambda\sin(\delta)I_{n}.$$ To obtain Lipschitz constants so that $$\frac{1}{L_{2}}I_{n}\leq\frac{d\bar{x}}{dx}\leq L_{1}I_{n} \label{L}$$ let $$\begin{aligned}
L_{1} & =\cos(\delta)+\Lambda\sin(\delta)\\
L_{2} & =\max\{\genfrac{\vert}{\vert}{}{}{1}{\cos(\delta)I_{n}+D^{2}u(x)\sin(\delta)}|x\in B_{R^{\prime}}\}.\end{aligned}$$ To find the value of $L_{2}$, we see that in $B_{R^{\prime}}$ we have the following:\
let $\min\{\theta_{1},\theta_{2}\}\geq-A$ where $A=\arctan{\Lambda}$. $$\begin{aligned}
\cos(\delta)I_{n}+\sin(\delta)D^{2}u(x) & \geq\cos(\delta)-\sin(\delta)\tan(A)\\
& =\cos(\delta)(1-\tan(\delta)\tan(A))\\
& =\cos(\delta)\frac{\tan(\delta)+\tan(A)}{\tan(\delta+A)}\\
& =\cos(\delta)\frac{\tan(\delta)+\tan(A)}{\tan(\frac{\pi/2-A}{2}+A)}\\
& =\cos(\delta)\frac{\tan(\delta)+\tan(A)}{\tan(\pi/2-\delta)}.\end{aligned}$$ This shows that $$\frac{1}{L_{2}}=\cos(\delta)\frac{\tan(\delta)+\tan(A)}{\tan(\pi/2-\delta)}.$$ Clearly $1/L_{2}$ is positive.
Now, by [@CW Prop 4.1] we see that there exists a function $\bar{u}$ such that $$\bar{y}=D_{\bar{x}}\bar{u}(\bar{x})$$ where $$\bar{u}(x)=u(x)+\sin\delta\cos\delta\frac{|Du(x)|^{2}-|x|^{2}}{2}-\sin
^{2}(\delta)Du(x)\cdot x \label{DU}$$ defines $\bar{u}$ implicity in terms of $\bar{x}$ (since $\bar{x}$ is invertible). Here $\bar{x}$ refers to the rotation map (\[R\_0\]).
Note that $$\bar{\theta}(\bar{x})-\bar{\theta}(\bar{y})=\theta(x)-\theta(y)$$ which implies that $\bar{\theta}$ is also a $C^{\bar{\alpha}}$ function $$\frac{|\bar{\theta}(\bar{x}_{1})-\bar{\theta}(\bar{x}_{2})|}{|\bar{x}_{1}-\bar{x}_{2}|^{\alpha}}=\frac{|\theta(x_{1})-\theta(x_{2})|}{|x_{1}-x_{2}|^{\bar{\alpha}}}\ast\frac{|x_{1}-x_{2}|^{\bar{\alpha}}}{|\bar{x}_{1}-\bar{x}_{2}|^{\bar{\alpha}}}$$ thus,$$|\bar{\theta}|_{C^{\bar{\alpha}}(B_{r_{0}})}\leq L_{2}^{\bar{\alpha}}|\theta|_{C^{\bar{\alpha}}(B_{R^{\prime}})}.$$
Let $\Omega=\bar{x}(B_{R^{ \prime}})$. Note that $B_{r_{0}}\subset\Omega$ where $r_{0} =R^{ \prime}/2L_{2}.$ So our new gradient graph is $\{(\bar{x}
,D_{\bar{x}} \bar{u} (\bar{x}))\vert\bar{x} \in\Omega\}$. The function $\bar{u}$ satisfies the equation $$F(D_{\bar{x}}^{2} \bar{u}) =\bar{\theta} (\bar{x})$$ in $B_{r_{0}}$ where $\bar{\theta} \in
C^{\alpha} (B_{r_{0}})$. Observe that on $B_{r_{0}}$ we have $$\bar{\theta}=\theta-2\delta<\delta-2\delta=-\delta<0$$ as $\theta<\delta$ on $B_{R^{\prime}}$.
: If $\bar{\vert \theta }\vert >\delta $, then $F(D^{2}\bar{u}) =\bar{\theta }$ is a solution to a uniformly elliptic concave equation. \[CL\]
The proof follows from [@CPW16 lemma 2.2] and also from [@CW pg 24].
Now using [@CC Corollary 1.3] we get interior Schauder estimates for $\bar{u}$:
$$|D^{2}\bar{u}(\bar{x})-D^{2}\bar{u}(0)|\leq C(||\bar{u}||_{L^{\infty}(B_{r_{0}/2})}+|\bar{\theta}|_{C^{\alpha}(B_{r_{0}/2})})\label{SC1}$$
for all $\bar{x}$ in $B_{r_{0}/2}$ where $C=C(\Lambda,\alpha)$. This is our $C^{2,\alpha}$ estimate for $\bar{u}$.
Next, in order to show the same Schauder type inequality as (\[SC1\]) for $u$ in place of $\bar{u}$, we establish relations between the following pairs:
1. oscillations of the Hessian of $D^{2}u$ and $D^{2}\bar{u}$
2. oscillations of $\theta$ and $\bar{\theta}$
3. the supremum norms of $u$ and $\bar{u}$ .
We rotate back to our original gradient graph by rotating up by an angle of $\delta$ and consider again the domain $B_{R^{\prime}}(0).$ This gives us the following relations: $$\begin{aligned}
x =\cos(\delta) \bar{x} -\sin(\delta) D_{\bar{x}} \bar{u} (\bar{x})\nonumber\\
y = \sin(\delta) \bar{x} +\cos(\delta) D_{\bar{x}} \bar{u} (\bar{x}).
\label{R}$$ This gives us: $$\begin{aligned}
\frac{dx}{d \bar{x}} =\cos(\delta) I_{n} -\sin(\delta) D_{\bar{x}}^{2} \bar{u}
(\bar{x})\\
D_{\bar{x}} y =\sin(\delta) I_{n} +\cos(\delta) D_{\bar{x}}^{2} \bar{u}
(\bar{x}).\end{aligned}$$ So we have $$D_{x}^{2}u (x) =D_{\bar{x}} y \frac{d\bar{x}}{dx}\newline=[\sin(\delta) I_{n}
+\cos(\delta) D_{\bar{x}}^{2} \bar{u}(\bar{x})][\cos(\delta) I_{n}
-\sin(\delta) D_{\bar{x}}^{2} \bar{u} (\bar{x})]^{-1}.$$ The above expression is well defined everywhere because $D_{\bar{x}}^{2}\bar{u}(\bar{x}) <\cot(\delta) I_{n}$ for all $\bar{x} \in B_{r_{0}}$.
Note that we have $\cos(\delta)I_{n}-D_{\bar{x}}^{2}\bar{u}(\bar{x})\sin(\delta)\geq\frac{1}{L_{1}}$, since $$\frac{dx}{d\bar{x}}=\cos(\delta)I_{n}-\sin(\delta)D_{\bar{x}}^{2}\bar{u}(\bar{x})=\left( \frac{d\bar{x}}{dx}\right) ^{-1}\geq\frac{1}{L_{1}}I_{n}$$ by (\[L\]).
Next, $$\begin{aligned}
D_{x}^{2}u(x)-D_{x}^{2}u(0) & =[\sin(\delta)I_{n}+\cos(\delta)D_{\bar{x}}^{2}\bar{u}(\bar{x})][\cos(\delta)I_{n}-\sin(\delta)D_{\bar{x}}^{2}\bar
{u}(\bar{x})]^{-1}\nonumber\\
& -[\sin(\delta)I_{n}+\cos(\delta)D_{\bar{x}}^{2}\bar{u}(0)][\cos
(\delta)I_{n}-\sin(\delta)D_{\bar{x}}^{2}\bar{u}(0)]^{-1}. \label{Hessian}$$
For simplification of notation we write $$\begin{aligned}
D_{\bar{x}}^{2}\bar{u}(\bar{x}) & =A\\
D_{\bar{x}}^{2}\bar{u}(0) & =B\\
\cos(\delta) & =c,\sin(\delta)=s.\end{aligned}$$ Noting that $[sI_{n}+cA]$ and $[cI_{n}-sA]^{-1}$ commute with each other we can write (\[Hessian\]) as the following equation $$\begin{aligned}
D_{x}^{2}u(x)-D_{x}^{2}u(0) & =\\
& \lbrack cI_{n}-sB]^{-1}[cI_{n}-sB][sI_{n}+cA][cI_{n}-sA]^{-1}-\\
& \lbrack cI_{n}-sB]^{-1}[sI_{n}+cB][cI_{n}-sA][cI_{n}-sA]^{-1}.\end{aligned}$$ Again we see that $$\lbrack cI_{n}-sB][sI_{n}+cA]-[sI_{n}+cB][cI_{n}-sA]=A-B.$$ This means $$D_{x}^{2}u(x)-D_{x}^{2}u(0)=[cI_{n}-sB]^{-1}[A-B][cI_{n}-sA]^{-1}.$$ We have already shown that $$|cI_{n}-sA|\geq\frac{1}{L_{1}}$$ which implies $$|cI_{n}-sA|^{-1}\leq L_{1}.$$ Thus we get $$\begin{aligned}
|D_{x}^{2}u(x)-D_{x}^{2}u(0)| & \leq L_{1}^{2}|D_{\bar{x}}^{2}\bar{u}(\bar{x})-D_{\bar{x}}^{2}\bar{u}(0)|.\nonumber\\
& \leq CL_{1}^{2}(||\bar{u}||_{L^{\infty
}(B_{r_{0}/2})}+|\bar{\theta}|_{C^{\alpha}(B_{r_{0}/2})})|\bar{x}|^{\alpha}\nonumber\\
& \leq CL_{1}^{2+\alpha}(||\bar{u}||_{L^{\infty}(B_{r_{0}/2})}+|\bar{\theta}|_{C^{\alpha}(B_{r_{0}/2})}|x|^{\alpha}\label{bigalpha}$$ where $L_{1}$ is the Lipschitz constant of the co-ordinate change map. This implies $$\frac{1}{L_{1}^{\alpha+2}}|D_{x}^{2}u(x)|_{C^{\alpha}(B_{R})}\leq|D_{\bar{x}}^{2}u(\bar{x})|_{C^{\alpha}(B_{r_{0}/2})}. \label{LH}$$
Recall from (\[DU\]) that $$\bar{u}(x)=u(x)+g(x).$$ This shows $$\begin{aligned}
||\bar{u}(\bar{x})||_{L^{\infty}(B_{r_{0}/2})}=||\bar{u}(x)||_{L^{\infty}(\bar{x}^{-1}(B_{r_{0}/2}))}\leq||\bar{u}(x)||_{L^{\infty}(B_{R^{\prime}})} \nonumber \\
\leq||u(x)||_{L^{\infty}(B_{R^{\prime}})}+||g||_{L^{\infty}(B_{R^{\prime}})}.\label{U}$$ Note that $$||g||_{L^{\infty}(B_{R})}\leq R||Du||_{L^{\infty}(B_{R})}+\frac{1}{2}[R^{2}+||Du||_{L^{\infty}(B_{R})}^{2}] \label{ggg}$$
and combining (\[LH\]), (\[U\]), (\[ggg\]) with (\[bigalpha\]) we get $$\begin{aligned}
& |D_{x}^{2}u(x)-D_{x}^{2}u(0)|\\
& \leq CL_{1}^{\alpha+2}\left\{
\begin{array}
[c]{c}||u||_{L^{\infty}(B_{R^{\prime}})}\\
R||Du||_{L^{\infty}(B_{R})}+\frac{1}{2}[R^{2}+||Du||_{L^{\infty}(B_{R})}^{2}]\\
+L_{2}^{\alpha}r_{0}|\theta|_{C^{\alpha}(B_{R^{\prime}})}\end{array}
\right\} \left\vert x\right\vert ^{\alpha}.\end{aligned}$$ This proves the Lemma.
\[Proof of Theorem 1.2\]First note that the lemma gives a Hölder norm on any interior ball, by a rescaling of the form $$u_{\rho}(x)=\frac{u(\rho x)}{\rho^{2}}$$ for values of $\rho>0$ and translation of any point to the origin. Consider the gradient graph $\{(x,Du(x))|x\in B_{1}\}$ where $u$ satisfies $$F(D^{2}u)=\theta$$ on $B_{1}$ and $\theta\in C^{\bar{\alpha}}(B_{1})\ \text{.}\ $ Then there exists a ball of radius $r$ inside $B_{1}$ on which $osc\theta<\delta/4$ where $\delta$ is as defined in Lemma 2.1. Now this means that either we have $\theta(x)<\delta/2$ in which case, by the above lemma we see that $u\in
C^{2,\alpha}(B_{r})$ satisfying the given estimates; or we have $\theta
(x)>\delta/4$ in which case $u\in C^{2,\alpha}(B_{r})$ with uniform estimates, by claim (\[CL\]) and [@CC Corollary 1.3].
\[Proof of Theorem 1.1\]Since $u\in C^{1,1}(B_{1})\cap W^{2,2}(B_{1})$ satisfies the uniformly elliptic equation $$\Delta_{g}\theta=0,$$ by the De Giorgi-Nash Theorem we have that $\theta\in C^{\alpha}(B_{1/2}).$ This means that u satisfies $$F(D^{2}u)=\theta.$$ By Theorem 1.2 we see that $u\in C^{2,\alpha}(B_{r})$ where $r<1/2$. Smoothness follows by [@CW Corollary 5.1].
|
---
abstract: 'Ge$_{2}$Sb$_{2}$Te$_{5}$ (GST) has been widely used as a popular phase change material. In this study, we show that it exhibits high Seebeck coefficients 200 - 300 $\mu$V/K in its cubic crystalline phase ($\it{c}$-GST) at remarkably high $\it{p}$-type doping levels of $\sim$ 1$\times$10$^{19}$ - 6$\times$10$^{19}$ cm$^{-3}$ at room temperature. More importantly, at low temperature (T = 200 K), the Seebeck coefficient was found to exceed 200 $\mu$V/K for a doping range 1$\times$10$^{19}$ - 3.5$\times$10$^{19}$ cm$^{-3}$. Given that the lattice thermal conductivity in this phase has already been measured to be extremely low ($\sim$ 0.7 W/m-K at 300 K),[@r51] our results suggest the possibility of using $\it{c}$-GST as a low-temperature thermoelectric material.'
author:
- 'Jifeng Sun$^{1,2,3}$'
- 'Saikat Mukhopadhyay$^{1}$'
- 'Alaska Subedi$^{4}$'
- 'Theo Siegrist$^{2,3}$'
- 'David J. Singh$^{1}$'
bibliography:
- 'c-GST.bib'
title: 'Transport Properties of Cubic Crystalline Ge$_{2}$Sb$_{2}$Te$_{5}$: A Potential Low-temperature Thermoelectric Material'
---
\[sec:level1\]INTRODUCTION
==========================
Low-temperature thermoelectric (TE) materials are important for refrigeration applications in electronics, infrared detectors, computers and other areas. [@r49] However, in contrast to high temperature TE materials, there is relatively little progress in the discovery of high performance low temperature TE materials. The most widely used materials at ambient temperature and below are (Bi,Sb)$_{2}$Te$_{3}$ derived compounds. [@r7; @r8] These have figures of merit, ZT, of roughly unity at 300 K with lower ZT values at lower temperatures. Here we show that cubic GST phase change material has potential to become an excellent thermoelectric material in the important temperature range from 200 K - 300 K. Specifically, we find using Boltzmann transport calculations that disordered cubic $\it{p}$-type GST will have high thermopowers in the range 200 $\mu$V/K - 300 $\mu$V/K at remarkably high doping levels consistent with good conductivity. GST already is known to have a very low thermal conductivity. [@r51]
Pseudobinary alloys of (GeTe)$_{m}$-(Sb$_{2}$Te$_{3}$)$_{n}$ have been used in data storage for more than a decade because of their fast phase switching between metastable crystalline (cubic) and amorphous phases.[@r9; @r10; @r11] The above room temperature thermoelectric properties of (GeTe)$_{m}$-(Sb$_{2}$Te$_{3}$)$_{n}$ were previously studied experimentally.[@r12; @r13; @r14; @r15; @r16] In particular, for m = 2 and n = 1; ordered Ge$_{2}$Sb$_{2}$Te$_{5}$ was reported to show interesting thermoelectric properties, even at room temperature. [@r13; @r15; @r16]
Disorder in the crystal structure can be beneficial for thermoelectric properties [@r44; @r45] and hexagonal GST has been already reported to have reasonably good thermoelectric properties. [@r12; @r43] The disorder in phase change materials was discussing in detail by Zhang and co-workers. [@r58] We therefore investigate the thermoelectric properties of cubic disordered GST. Previous measurements of thermoelectric properties in $\it{c}$-GST were restricted to thin-film samples where the thermoelectric properties were found to vary significantly depending on the thickness of the sample.[@r15; @r34] The thermoelectric properties of $\it{c}$-GST in the bulk form have remained unknown. We note that Lee and co-workers have discussed resonant bonding in the context of thermoelectric performance [@r55] and that resonant bonding has been identified in phase change materials. [@r56; @r57]
\[sec:level2\]THEORETICAL APPROACH AND COMPUTATIONAL DETAILS
============================================================
The ab-initio calculations presented here were carried out using the linearized augmented plane-wave (LAPW) method [@r20] as implemented in the WIEN2K code.[@r21] The structures were relaxed using the PBE-GGA functional then a semilocal functional of Tran and Blaha (TB-mBJ) [@r22] was employed to treat the exchange-correlation potential in the calculations of electronic, optical and transport properties. This functional form was shown to better describe the band gaps of a variety of semiconductors and insulators.[@r23; @r24]
The cubic phase of GST occurs in a rock-salt structure with Te occupying one site and Ge, Sb and vacancies randomly occupying the other site. We generated special quasirandom structures (SQS) using the prescription of Zunger *et al.*[@r39] for our first principles study. The SQSs were generated such that all the pair correlation functions (PCF) up to the next nearest neighbors are identical to the average PCF of an infinite random structure. We used the `mcsqs` code as implemented in ATAT software package [@r48] to generate the SQSs using a Monte Carlo algorithm. We considered two SQS structures with 27 atoms (6 Ge, 6 Sb, and 15 Te atoms) and 45 atoms (10 Ge, 10 Sb, and 25 Te atoms) in the unit cell and compared their electronic and transport properties. Despite the relatively poor quality of PCFs for the case with 27-atomic unit cell, we find that the overall calculated properties for these structures are very similar (a detailed comparison of electronic, optical and transport properties in these structures can be found in Ref. [@r54]). This may indicate that the inherent properties in this phase is independent of its PCFs. Thus this 45-atomic unit cell is converged with respect to size to calculate properties of $\it{c}$-GST.
The self-consistent calculations were performed with a 4$\times$4$\times$4 k-point grid of 36 k points in the irreducible Brillouin zone (IBZ) whereas electronic and optical properties were calculated using a 8$\times$8$\times$8 k-mesh. For the calculation of the thermoelectric properties, we employed a denser k-point mesh of 14$\times$14$\times$14. All the calculations were duly tested for convergence with a variety of k-point grids. The thermopower was analysed using the Boltztrap code [@r25] employing Boltzmann transport theory within the constant scattering time approximation (CSTA). This CSTA approach has been shown to be successful in calculating the Seebeck coefficient in a variety of thermoelectric materials.[@r26; @r35; @r36; @r37; @r38] The substance of CSTA is to assume that the energy dependence of the scattering rate is small compared with the energy dependence of the electronic structure. A detailed description of this approach can be found elsewhere.[@r27; @r35] We used well converged basis sets with LAPW basis size corresponding to R$_{MT}$K$_{max}$=9.0, where R$_{MT}$ is the smallest Muffin Tin radius and K$_{max}$ is the plane-wave cutoff parameter. The LAPW sphere radius of 2.47 Bohr was used for Ge whereas 2.5 Bohr was used for Sb and Te. Spin-orbit coupling was included for all the properties reported.
\[sec:level3\]RESULTS AND DISCUSSIONS
=====================================
\[sec:1\]Electronic structure and optical properties
----------------------------------------------------
The total density of states (DOS) of $\it{c}$-GST near the Fermi energy (E$_{F}$) is shown in Fig. \[fig1\]. We find a clear semiconducting behavior with a band gap of 0.17 eV. This is in good agreement with a previous theoretical investigation[@r30] where an indirect band gap of 0.1 eV was noted for $\it{c}$-GST although they used different exchange-correlation potential (local density approximation without spin-orbit coupling) and different method (shifting of the hexagonal structure) to generate the cubic structure. Another previous study considered the cubic GST as a narrow-gap degenerate semiconductor, with E$_{F}$ reside inside the valence band or a defect band.[@r29] The experimentally reported estimated optical gap is $\sim$ 0.5 eV [@r29; @r33] which is larger than what we obtain.
![\[fig1\]Calculated total density of states for $\it{c}$-GST. The Fermi level is set to zero.](TDOS){height="6cm"}
![\[fig2\]Calculated reflectivity of $\it{c}$-GST. Previous experimental result from Ref. [@r32] is also shown for a better comparison.](reflectivity){height="6cm"}
![\[fig3\] Calculated optical absorption coefficient along with experimental data from previous measurements (see Refs. [@r29; @r33])](absorption){height="6cm"}
GST has applications in rewritable DVD and Blu-ray technologies where a large contrast between the optical properties of the cubic and amorphous phases is exploited.[@r9; @r11] In Fig. \[fig2\] and \[fig3\], we present the averaged reflectivity and optical absorption coefficients for $\it{c}$-GST, respectively. It is evident from Fig. \[fig2\] that $\it{c}$-GST displays a high reflectivity ranging from 0.65 to 0.68 within the energy up to 3.5 eV which covers the energy range of DVD (1.91 eV) and blu-ray (3.06 eV)[@r9] technology . This is consistent with the measured optical reflectivity by Garcia-Garcia ${et}$ ${al.}$’s [@r32] \[Fig. \[fig2\]\], though they used thin films of $\it{c}$-GST in their experiments. Fig. \[fig3\]. depicts the calculated optical absorption coefficients together with that from previous measurements. Our calculated absorption coefficient compares well in magnitude with that from Lee ${et}$ ${al.}$ [@r29] and Pirovano ${et}$ ${al.}$ [@r33] for the higher energy regime but disagrees at lower energy reflecting the band gap difference. One possible explanation is that the structure is more complex than the random structure that we assumed, or that larger length scale effects such as local near ferroelectric polarizations play a role in the samples. Additionally, the dimension of the crystals may also play a role since thin films of GST were used in both of the experiments instead of the ideal bulk that we are considering.
\[sec:2\]Thermoelectric properties
----------------------------------
In Fig. \[fig4\], we have plotted the calculated thermopower (S(T)) of $\it{c}$-GST at various temperatures ranging from 100 K up to 500 K, for both electron-doped ($\it{n}$-type) and hole-doped ($\it{p}$-type) cases. It has been reported that it might be possible to dope GST $\it{n}$-type[@r52; @r53], however, keeping in mind that most of the experiments deal with $\it{p}$-type GST, we focus on the thermoelectric properties of $\it{p}$-type $\it{c}$-GST. In fact, there is a rather good electron-hole symmetry in the S(T), so $\it{n}$-type if realized would be similar to $\it{p}$-type. We give the direction average of S(T), i.e. S=$\frac{S_{xx}+S_{yy}+S_{zz}}{3}$, to eliminate artificial anisotropy due to the unit supercell.
![\[fig4\] Calculated thermopowers of $\it{c}$-GST as a function of doping concentration both for $\it{p}$-type and $\it{n}$-type at various temperatures. Solid horizontal line at 200 $\mu$V/K represents the limitation of Seebeck coefficients for good TE materials.](S-avg-np-T){height="6cm"}
We find $\it{c}$-GST exhibits a Seebeck coefficients 200 - 323 $\mu$V/K when doped $\it{p}$-type for doping concentration 1.7$\times$10$^{18}$ - 6.1$\times$10$^{19}$ cm$^{-3}$ at room temperature. More importantly, at low temperature (T = 200 K), the Seebeck coefficient exceeds 200 $\mu$V/K for a doping range of 1$\times$10$^{19}$ - 3.5$\times$10$^{19}$ cm$^{-3}$. A strong bipolar effect was noted for T = 300 K and higher temperatures, however, it was not present at lower temperatures. This is connected with the band gap, which as noted may be too low in the present calculations. In order to assess the possibility of $\it{c}$-GST as a low temperature TE material, it is necessary to compare the calculated Seebeck coefficient with that in popular low-temperature TE materials. For example, Poudel and co-workers[@r7] reported a bulk nano-composite Bi$_{x}$Sb$_{2-x}$Te$_{3}$ with S(T) of 185 $\mu$V/K at 300 K. Chung ${et}$ ${al.}$ [@r8] discovered CsBi$_{4}$Te$_{6}$ with a thermopower of 105 $\mu$V/K at 300 K. Importantly, the high thermopowers are at high carrier concentrations where reasonable electric conductivity may be anticipated.
![\[fig5\]Temperature dependence of the thermopower of $\it{c}$-GST with two $\it{p}$-type doping concentrations and experimental results from previous work. [@r15; @r34]](S-T-Kato){height="6cm"}
We shall now focus on comparing our calculated results with previous experiments on GST thin films. Kato and Tanaka [@r34] investigated thermoelectric properties of Ge$_{2}$Sb$_{2}$Te$_{5}$ thin films with sample dimensions ranging from 50 nm - 600 nm for a temperature range 25 $^{\circ}\mathrm{C}$ to 200 $^{\circ}\mathrm{C}$. For their $\it{c}$-GST samples, they found a pretty high values of S(T) from 280 $\mu$V/K to 380 $\mu$V/K. Lee ${et}$ ${al.}$ [@r15] also measured the temperature dependence of S(T) with 25 nm and 125 nm Ge$_{2}$Sb$_{2}$Te$_{5}$ thin films from 20 $^{\circ}\mathrm{C}$ to 300 $^{\circ}\mathrm{C}$ and observed S(T) as high as 205 $\mu$V/K and 175 $\mu$V/K for 125 nm and 25 nm thick films, respectively. Both of the authors found a decreasing S(T) with increasing temperatures. In order to further compare our calculations with these measurements, we matched the calculated Seebeck coefficients at 300 K with that from Kato’s data for the 550 nm thick thin film and calculated the corresponding doping levels in $\it{c}$-GST. For these particular doping levels, we then plotted the variation of S(T) with temperature together with the respective experimental data, as shown in Fig. \[fig5\]. As seen, a similar decreasing trend of S(T) with respect to temperatures can be found in the calculation for both doping concentrations. However, the agreement of the decreasing feature is more obvious in the low doping level case which falls into the bipolar region and the deviation may stem from the underestimation of the band gap. This may also suggest the fact that the experimental samples are in the low carrier concentrations range where bipolar effect is strong and detrimental to the thermoelectric properties. Therefore a better thermoelectric performance might be feasible in higher doping concentration samples.
\[sec:level4\]SUMMARY$\&$CONCLUSIONS
====================================
To summarize, we performed first-principles calculations to investigate electronic, optical and thermoelectric properties of $\it{c}$-GST with fully random cation site occupancy generated using the SQS scheme. Our results based on Boltzmann transport calculations predict substantially high thermopower at high carrier concentration ($\sim$ 300 $\mu$V/K at 7$\times10^{18}$ cm$^{-3}$) at room temperature. We also find high thermopower at high carrier concentrations going down to 200 K and below suggesting that with heavy hole doping $\it{c}$-GST may be an excellent low-temperature thermoelectric. These results suggest future experiments to better understand the low-temperature thermoelectric properties of $\it{c}$-GST. Specifically, it would be of interest to study the temperature range of 200 K - 300 K with high doping concentrations ranging from 1$\times10^{19}$ cm$^{-3}$ up to 6$\times10^{19}$ cm$^{-3}$.
This work was supported by the Department of Energy through the S3TEC energy frontier research center. JS acknowledges a graduate student fellowship, funded by the Department of Energy, Basic Energy Science, Materials Sciences and Engineering Division, through the ORNL GO! program. A portion of this work was performed at high-performance computing center at the National High Magnetic Field Laboratory.
|
---
abstract: 'We examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z\[i\]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by Hurwitz, and binary quadratic forms over the Gaussian integers. According to this correspondence, a geodesic is called fundamental if the associated binary quadratic form is. Using techniques from sieve theory, symbolic dynamics, and the theory of expander graphs, we show the existence of a compact set in the manifold containing infinitely many fundamental geodesics.'
author:
- Katie McKeon
bibliography:
- 'total\_copy.bib'
date: '7/1/2019'
title: 'Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold'
---
Introduction
============
Closed Geodesics and Dirichlet Forms
====================================
A Symbolic Encoding of Geodesics
================================
Counting Geodesics via Thermodynamic Formalism
==============================================
Counting Geodesics with Congruence Conditions
=============================================
The Growth Parameter $\delta_R$
===============================
Construction of the Multilinear Sifting Set
===========================================
Small Divisors
==============
Large Divisors
==============
Sieve Theorem
=============
Final Estimates
===============
|
---
abstract: 'We present the MDS feature learning framework, in which multidimensional scaling (MDS) is applied on high-level pairwise image distances to learn fixed-length vector representations of images. The aspects of the images that are captured by the learned features, which we call MDS features, completely depend on what kind of image distance measurement is employed. With properly selected semantics-sensitive image distances, the MDS features provide rich semantic information about the images that is not captured by other feature extraction techniques. In our work, we introduce the iterated Levenberg-Marquardt algorithm for solving MDS, and study the MDS feature learning with IMage Euclidean Distance (IMED) and Spatial Pyramid Matching (SPM) distance. We present experiments on both synthetic data and real images — the publicly accessible UIUC car image dataset. The MDS features based on SPM distance achieve exceptional performance for the car recognition task.'
author:
-
- 'Sibgrapi paper ID: [113568]{}\'
bibliography:
- 'example.bib'
title: Feature Learning by Multidimensional Scaling and its Applications in Object Recognition
---
Feature learning; image distance measurement; multidimensional scaling; spatial pyramid matching
Introduction
============
To represent an image by a fixed-length feature vector, there are generally two groups of approaches, often referred to as hand-designed features and feature learning, respectively. In this section, we briefly review several commonly used methods from each group, and relate the proposed MDS feature learning to these existing methods.
Hand-Designed Features
----------------------
Most hand-designed features, or sometimes called hand-crafted features, focus on capturing the color, texture and gradient information in the image. Generally, these features have a closed form to be computed, without looking at other images. Some popular yet simple hand-designed image features include color-histogram, wavelet transform coefficients [@wavelet], scale-invariant feature transform (SIFT) [@sift], color-SIFT [@color-sift], speeded up robust features (SURF) [@surf], histogram of oriented gradients (HOG) [@HOG] and local binary patterns (LBP) [@lbp]. To represent an image with one fixed-length feature vector, there are generally three ways: (1) First, these features can be computed for the entire image, but the resulting feature vector will fail to embed the spatial relationship between different objects or different locations in the image. (2) Second, the image can be first uniformly divided into $M \times N$ blocks. Then these features can be computed for each block, and can be concatenated to make a long feature vector. (3) Further, the division of the image does not have to be uniform, but can be arbitrary. We can just put random rectangular or circular masks onto the image, and compute features for each mask (or “patch”), then concatenate. To do this, the division must be consistent for all images.
The divide-and-concatenate methods will result in very large feature vectors. Given a large dataset, PCA can be used to reduce the dimensionality.
Feature Learning
----------------
Feature learning has often been used as a synonym of deep learning, especially in recent years, and often refers to recent techniques such as sparse coding [@nature; @sparse-coding], auto-encoder [@denoising_autoencoder], convolutional neural networks [@ConvolutionalNN], restricted Boltzmann machines [@science], and deep Boltzmann machines [@dbm]. However, we believe this interpretation of feature learning is literally imprecise. Feature learning should be more generally defined as the opposite to hand-designed features — it should refer to any technique that learns a fixed-length vector representation of each image in the dataset by utilizing the pattern distribution of the entire dataset, or optimizing a target function that is defined on the entire dataset. Any technique that can generate a feature representation of each image without looking at the entire dataset should fail to fall into this category.
We further categorize existing feature learning methods into two subgroups: feature learning with raw intensities, and feature learning with hand-designed features. The proposed MDS feature learning falls into a third new subgroup: feature learning with image distance measurement.
### Feature Learning with Raw Intensities
This subgroup of methods treat the feature learning problem as a dimensionality reduction problem, where the original high-dimensional data are the image intensities, either gray-level or RGB values. Efforts on data dimensionality reduction have a long history [@dim_review], dating from the early work on PCA [@PCA] and its nonlinear form, kernel PCA [@kPCA], to the recent work on sparse coding and deep learning [@nature; @science; @sparse-coding; @ConvolutionalNN; @denoising_autoencoder; @dbm]. In all these methods, high dimensional data, such as an image, is represented by a low dimensional vector. Each entry of this vector describes one salient varying pattern of all images within the training set.
Assume we have a dataset $\mathbf{X}=(\mathbf{x}_1, \mathbf{x}_2, \dots , \mathbf{x}_N){^\mathsf{T}}$, where each $\mathbf{x}_i$ ($1 \leq i \leq N$) is one data point. We briefly review several dimensionality reduction methods below.
- **PCA** linearly projects vector $\mathbf{x}$ to $\mathbf{y}=\mathbf{Ax}$, where $\mathbf{A}$ is obtained by performing eigenvector decomposition on the covariance matrix $\mathbf{S_x=XX}{^\mathsf{T}}$.
- **Kernel PCA** first constructs a kernel matrix $\mathbf{K}$, where each entry of this matrix is obtained by evaluating the kernel function $k(\cdot , \cdot)$ on two data points: $$\mathbf{K}_{i,j}=k(\mathbf{x}_i,\mathbf{x}_j) .$$ Then the Gram matrix is constructed as $$\mathbf{\widetilde K} = \mathbf{K} - \mathbf{1}_N \mathbf{K} - \mathbf{K} \mathbf{1}_N + \mathbf{1}_N \mathbf{K} \mathbf{1}_N ,$$ where $\mathbf{1}_N$ is the $N \times N$ matrix with all elements equal to $1/N$. Next the eigenvector decomposition problem $\mathbf{\widetilde{K}a}_l=\lambda_l N \mathbf{a}_l$ is solved ($\mathbf{a}_l$ is eigenvector and $\lambda_l$ is eigenvalue) and the projected vector $\mathbf{y}$ is computed by $$y_l=\sum \limits _{i=1}^N a_{li} k(\mathbf{x,x}_i).$$
- **Auto-encoders** first normalize all $\mathbf{x}_i$’s to $[0,1]$, and map them to $\mathbf{y}_i=s(\mathbf{Wx}_i+\mathbf{b})$, where $s(\cdot)$ is a sigmoid function. A reconstruction is computed by $\mathbf{z}_i=s(\mathbf{W'y}_i+\mathbf{b'})$. The weight matrices $\mathbf{W}$ and $\mathbf{W'}$, and the bias vectors $\mathbf{b}$ and $\mathbf{b'}$ are obtained by minimizing the average reconstruction error, which can be defined as either traditional square error or cross-entropy.
In PCA and kernel PCA, different entries of $\mathbf{y}$ correspond to eigenvectors of different importance, while in auto-encoder, they are equivalently important.
These techniques have been shown effective on problems such as face recognition [@eigenface; @eigenface2] and even concept recognition [@google-cat]. However, most of these methods require all input data to have exactly the same size. If the input is an image, then the image has to be cropped and resized to be consistent with other images in the dataset. However, cropping the image means loss of information, and resizing the image means change of aspect ratio, which will result in distorted object shapes.
### Feature Learning with Hand-Designed Features {#sec:bov}
One popular method that falls into this subgroup is the bag-of-visual-words (BOV) method [@bov1; @bov2; @feifei]. This method first divides the image into local patches or segments the image into distinct regions, and then extracts hand-designed features for each patch/region. Rather than being concatenated, these feature vectors make an unordered set, or also referred to as “bag”. By performing clustering on the union of all those unordered sets for all training images, a visual vocabulary is established. Now the set of feature vectors previously extracted from each image can be transformed into a “word-frequency” histogram by simply counting which cluster (visual word) is assigned to each patch/region. The “word-frequency” histogram can be optionally normalized to generate the final fixed-length vector.
One extension of BOV is the Fisher Vector (FV) method [@fv1; @fv2]. Rather than simply counting the word frequency, which can be viewed as the 0-order statistics, FV encodes higher order statistics (up to the second order) about the distribution of local descriptors assigned to each visual word. Another extension is the Spatial Pyramid Matching [@spm], which gives different weights to features in different image division levels, and defines an image similarity measurement using the pyramid matching kernel [@pyramid_kernel].
Method
======
In this section, we first review the basics of MDS and its existing solutions, and then introduce our own solution — the iterated Levenberg-Marquardt algorithm (ILMA). Next, we discuss and compare some popular image distance measurement techniques in recent literature.
Multidimensional Scaling: Problem Definition
--------------------------------------------
As a statistical technique for the analysis of data similarity or dissimilarity, multidimensional scaling (MDS) has been well applied to areas such as information visualization [@MDS_book] and surface flattening [@flattening1; @flattening2]. Here we briefly review the basic concepts and definitions of MDS. For convenience, we will use the word “image” instead of “data” or “object” in the context, but we keep in mind that MDS is a technique for general purposes.
Suppose we have a set of $N$ images $\Omega=\lbrace I_1, I_2, ..., I_N \rbrace$, and there is a distance measurement $d(I_i,I_j)$ defined between each pair of images $I_i$ and $I_j$. Note that $d: \Omega \times \Omega \rightarrow \mathbb{R}_{\geq 0}$ is only a measurement of image dissimilarity, not necessarily a metric on set $\Omega$ in the strict sense, since the subadditivity triangle inequality does not necessarily hold. Multidimensional scaling is the problem of representing each image $I_i \in \Omega$ by a point (vector) in a low dimensional space $\mathbf{x}_i \in \mathbb{R}^m$, such that the interpoint Euclidean distance in some sense approximates the distance between the corresponding images [@MDS]. In Section \[sec:image\_distance\] we will discuss how to define the image distance/dissimilarity measurement. Here we focus on the mathematical definitions related to MDS.
For a pair of images $I_i$ and $I_j$, let their low dimensional ($m$-d) representations be $\mathbf{x}_i$ and $\mathbf{x}_j$. The representation error is defined as $e_{ij}=d(I_i,I_j)-||\mathbf{x}_i-\mathbf{x}_j||$, where $||\cdot||$ denotes the $L^2$-norm. The *raw stress* is defined as the sum-of-squares of the representation errors: $$\textrm{Stress}^*=\sum\limits_{1 \leq i < j \leq N}e_{ij}^2 ,$$ while the *normalized stress* (also known as *Stress-1*) is defined as $$\textrm{Stress-1}=\sqrt{\dfrac{\sum\limits_{1 \leq i < j \leq N}e_{ij}^2}{\sum\limits_{1 \leq i < j \leq N}||\mathbf{x}_i-\mathbf{x}_j||^2}} .$$ MDS models require the interpoint Euclidean distances to be “as equal as possible” to the image distances. Thus we can either minimize the raw stress or normalized stress. We compactly represent the image distances by an $N \times N$ symmetric matrix $\mathbf{D}=\left[ d(I_i,I_j) \right] _{N \times N}$ with all diagonal values equal to $0$, and represent the low dimensional vectors by an $N \times m$ matrix $\mathbf{X}=(\mathbf{x}_1, \dots, \mathbf{x}_N){^\mathsf{T}}$. Using the raw stress as the loss function, the MDS problem can be stated as: $$\label{eq:min_raw_stress}
\mathbf{X}^*=\arg \min\limits_{\mathbf{X}} \sum\limits_{1 \leq i < j \leq N}
\left( d(I_i,I_j)-||\mathbf{x}_i-\mathbf{x}_j|| \right) ^2 .$$
Solutions for Multidimensional Scaling
--------------------------------------
There are lots of existing methods for solving Eq. (\[eq:min\_raw\_stress\]), such as Kruskal’s iterative steepest descent approach [@MDS] and de Leeuw’s iterative majorization algorithm (SMACOF) [@SMACOF]. In 2002, Williams demonstrated the connection between kernel PCA and metric MDS [@connection], thus metric MDS problems can also be solved by solving kernel PCA.
In our work, we introduce an iterative least squares solution to the MDS optimization problem. We note that in Eq. (\[eq:min\_raw\_stress\]), the raw stress is minimized with respect to $\mathbf{X}$, which has $N \times m$ entries in total. Thus, when $N$ is large, this nonlinear optimization problem becomes computationally intractable if we attempt to solve for all entries in one step. Inspired by the iterated conditional modes (ICM) method [@ICM], which was developed to solve Markov random fields (MRF), we introduce the two-stage *iterated Levenberg-Marquardt algorithm (ILMA)*. The basic idea of this algorithm is to repeatedly minimize the raw stress with respect to one $\mathbf{x}_i$ while holding all other $\mathbf{x}_i$’s fixed. For this purpose, we maintain a constraining set of the indices of the $\mathbf{x}_i$’s to be fixed. In the initialization stage, indices of all images are selected into the constraining set in a random order. In the adjustment stage, we repeatedly adjust all $\mathbf{x}_i$’s in a randomly permuted order. By doing so, each time we only need to minimize the raw stress with respect to $m$ variables, instead of $N \times m$, which greatly reduces the complexity of the problem. The subproblem can be viewed as a least squares problem, and can be solved by the standard Levenberg-Marquardt algorithm [@levenberg; @marquardt]. Since the total raw stress is monotonically non-increasing through time, the convergence of the adjustment is guaranteed. The details of the two-stage algorithm are given in Algorithm \[alg:Levenberg-Marquardt\]. We will call the low dimensional vectors $\{ \mathbf{x}_i \}$ as *MDS features* or *MDS codes* in the context.
One advantage of our method is that we provide a unified framework for both MDS model training and new data encoding. In MDS model training, pairwise image distances are measured within the training set $\Omega_{\mathrm{train}}$, and Algorithm \[alg:Levenberg-Marquardt\] is applied to encode each training image $I_i \in \Omega_{\mathrm{train}}$ to its MDS code $\mathbf{x}_i$. Now given a new image $\widetilde{I}$, we measure the distance from this image to all training images $d(\widetilde{I},I_i)$, and find its MDS code $\widetilde{\mathbf{x}}$ by: $$\begin{aligned}
\label{eq:encoding}
\min\limits_{\widetilde{\mathbf{x}}} \sum\limits_{I_i \in \Omega_{\mathrm{train}}} ( ||\widetilde{\mathbf{x}}-\mathbf{x}_{i}|| -
d(\widetilde{I},I_i) ) ^2 , \end{aligned}$$ which can be directly solved as a least squares problem using the standard Levenberg-Marquardt algorithm. We follow this practice for the training and testing of MDS models in the experiment in Section \[sec:car\].
**begin** initialization stage:\
Randomly choose a non-diagonal entry $D_{i_0,j_0}$ in $\mathbf{D}$;\
Set $\mathbf{x}_{i_0}=(0, 0, \dots, 0){^\mathsf{T}}$;\
Set $\mathbf{x}_{j_0}=(D_{i_0,j_0}, 0, \dots, 0){^\mathsf{T}}$;\
Initialize the constraining set $A=\lbrace i_0, j_0 \rbrace$;\
**begin** adjustment stage:\
**done**\
Image Distance Measurement {#sec:image_distance}
--------------------------
The measurement of the similarity or dissimilarity between two images is of essential significance in content-based image retrieval [@cbir_survey1; @cbir_survey2]. There are some very simple forms of image distances, such as the traditional Euclidean distance on raw image intensities, and the earth mover’s distance (EMD) on image color histograms [@EMD]. Here, we briefly describe two popular image distance measurement methods: the IMage Euclidean Distance (IMED) [@IMED] and the Spatial Pyramid Matching (SPM) distance [@spm]. These distances will be evaluated in our experiment on real images in Section \[sec:car\].
### IMED
The IMED is a generalized form of the traditional Euclidean distance on raw image intensities. Give two gray-level images $I_1$ and $I_2$ of the same size, the traditional Euclidean distance is defined as the square root of the sum-of-squares of intensity difference at each corresponding image location: $$\label{eq:euclidean_distance}
d_{\mathrm{Euclidean}}^2(I_1,I_2)=\sum\limits_{(r,c)} \left( I_1^{(r,c)}-I_2^{(r,c)} \right)^2,$$ where $I_1^{(r,c)}$ denotes the intensity at row $r$ and column $c$ in image $I_1$. In contrast, IMED also counts for the intensity difference at different locations, but assigns a weight to it, which is a function of the Euclidean distance of the two locations: $$\begin{aligned}
\label{eq:IMED}
d_{\mathrm{IMED}}^2(I_1,I_2)=\sum\limits_{(r,c)}\sum\limits_{(r',c')}
\left( I_1^{(r,c)}-I_2^{(r,c)} \right) \cdot \nonumber \\
g(r,c,r',c') \cdot \left( I_1^{(r',c')}-I_2^{(r',c')} \right),\end{aligned}$$ where $$\label{eq:IMED_g}
g(r,c,r',c')=f(\sqrt{(r-r')^2+(c-c')^2}),$$ and $f(\cdot)$ is a continuous monotonically decreasing function, usually the Gaussian function. An interesting observation by Wang *et al.* [@IMED] is that the IMED (\[eq:IMED\]) on two images is equivalent to the traditional Euclidean distance (\[eq:euclidean\_distance\]) on a blurred version of the two images. The blur operation is called standardizing transform (ST) by the authors.
Although IMED has shown promising performance on some recognition experiments in [@IMED], we can see that it is still a low-level image distance measurement, based on the raw intensities, without embedding any semantic information. Another disadvantage of IMED is that it is only defined on images of the same size. We will apply MDS on IMED distances for the experiment in Section \[sec:car\], where we use Gaussian function for $f(\cdot)$ in Eq. (\[eq:IMED\_g\]) and set $\sigma=1$, and we call this method IMED-MDS.
### SPM Distance {#sec:spm}
The spatial pyramid matching (SPM) [@spm] is based on Grauman and Darrell’s work on pyramid matching kernel [@pyramid_kernel], which measures the similarity of two sets of feature vectors by partitioning the feature space on different levels and taking the sum of weighted histogram intersection functions. Lazebnik *et al.*’s spatial pyramid matching is an “orthogonal” approach — it performs pyramid matching in the 2-d image space, and uses $k$-means for clustering in the feature space (edge points and SIFT features). With a visual vocabulary of size $M$ (number of clusters), and $L$ partition levels, spatial pyramid vectors of dimensionality $M\frac{1}{3}(4^{L+1}-1)$ are generated, and spatial pyramid matching similarities $K^L(I_i,I_j)$ between images $I_i$ and $I_j$ are measured. Authors of [@spm] recommend parameter setting of $M=200$ and $L=2$.
The similarity value $K^L(I_i,I_j)$ lies in $[0,1]$, where 1 is for most similar, and 0 for least similar. We have many ways to define image distances using the similarities, such as: $$\begin{aligned}
\label{eq:spm1}
d_{\mathrm{SPM1}}(I_i,I_j)&=&1-K^L(I_i,I_j) , \\
\label{eq:spm2}
d_{\mathrm{SPM2}}(I_i,I_j)&=&- \ln ((1-\epsilon)K^L(I_i,I_j)+\epsilon) ,\end{aligned}$$ where $\epsilon$ is a small value. We set $\epsilon=0.001$ in (\[eq:spm2\]) for our experiment in Section \[sec:car\].
Unlike IMED, SPM distance is based on hand-designed features such as SIFT and edge points, instead of raw intensities. It models the spatial co-occurrence of different feature clusters, and thus is more semantics-sensitive. Besides, SPM distance does not require the size of images to be the same. We will apply MDS on the two SPM distances defined by Eq. (\[eq:spm1\]) and Eq. (\[eq:spm2\]), and we call them SPM1-MDS and SPM2-MDS, respectively.
Experiments {#sec:experiments}
===========
We present two experiments. The first one is on synthetic data, and is to evaluate the running time performance of different MDS algorithms, and to compare different initialization strategies of our iterated Levenberg-Marquardt algorithm. The second one is a real image object recognition task, in which we compare MDS features with PCA features and kernel PCA features. In the second experiment, we use the UIUC car dataset[^1], and follow a five-fold cross validation to report the classification precision and recall under different feature dimensions.
Synthetic Data Experiment
-------------------------
In this experiment, we use MDS for curved surface flattening [@flattening1] on the manually created *Swiss roll* data, which was introduced in [@LLE], and is known to be complicated due to the highly non-linear and non-Euclidean structure [@SMACOF_multigrid]. The Swiss roll surface contains $591$ points in $\mathbb{R}^3$, as shown in Fig. \[fig:swiss\]. We measure the pairwise interpoint geodesic distances to construct a $591 \times 591$ distance matrix, and re-embed the Swiss roll surface into $\mathbb{R}^3$ by applying MDS on the geodesic distance matrix.
![The Swiss roll surface with 591 points. []{data-label="fig:swiss"}](swiss){width="40.00000%"}
### Running Time
First, we would like to evaluate the running time performance of the proposed iterated Levenberg-Marquardt algorithm and compare with Bronstein’s implementation of the SMACOF algorithm and its variants, including SMACOF with reduced rank extrapolation (RRE) and SMACOF with multigrid [@SMACOF_multigrid; @SMACOF_multigrid2; @SMACOF_multigrid3; @SMACOF_multigrid4]. The results are given in Fig. \[fig:ILMA\_runningtime\], where each number in this plot is averaged on 20 independent repeated experiments, and the running time is reported on a Mac Pro with 2 $\times$ 2.4 GHz Quad-Core Intel Xeon CPU. From Fig. \[fig:ILMA\_runningtime\], we can see that our ILMA is an efficient solution, which runs faster and converges to a smaller raw stress value than other methods. The unrolled surfaces by ILMA in different iterations are shown in Fig. \[fig:ILMA\_swiss\].
![The raw stress *vs.* running time plot of the SMACOF algorithm, its variants, and the proposed iterated Levenberg-Marquardt algorithm (ILMA) on the Swiss roll geodesic distance matrix. []{data-label="fig:ILMA_runningtime"}](ILMA_runningtime){width="47.00000%"}
### Initialization Strategies
Further, we study some modifications to Algorithm \[alg:Levenberg-Marquardt\]. The original algorithm uses a **random order strategy** in the initialization stage, but we can modify it to:
- **Largest-distance-first strategy:** For Algorithm \[alg:Levenberg-Marquardt\], in line 2 we choose the largest non-diagonal entry $D_{i_0,j_0}$ in $\mathbf{D}$ instead of a random one; in line 7, we find the $i^* \in A$ and $j^* \notin A$ that maximize $D_{i^*,j^*}$ rather than a random $j^* \notin A$.
- **Smallest-distance-first strategy:** For Algorithm \[alg:Levenberg-Marquardt\], in line 2 we choose the smallest non-diagonal entry $D_{i_0,j_0}$ in $\mathbf{D}$; in line 7, we find the $i^* \in A$ and $j^* \notin A$ that minimize $D_{i^*,j^*}$.
If we assume that the data to be encoded are comprised of clusters, then an intuitive interpretation of the largest-distance-first strategy is that representatives of each cluster are first encoded, and they are expected to be scattered in the multidimensional space; similarly, the smallest-distance-first strategy encodes all data in one cluster first, and then moves to the nearest cluster.
We have been using the three initialization strategies to solve the MDS problem on the Swiss roll geodesic distance matrix, and it turns out that the random order strategy converges faster than the other two, as shown in Fig. \[fig:init\_strategy\]. Again, each number in this plot is averaged on 20 independent repeated experiments.
![The raw stress in each iteration of the iterated Levenberg-Marquardt algorithm with different initialization strategies. []{data-label="fig:init_strategy"}](init_strategy){width="47.00000%"}
Car Recognition Experiment {#sec:car}
--------------------------
Now we would like to compare the performance of MDS features to the most standard and popular dimensionality reduction algorithms — PCA [@PCA] and kernel PCA [@kPCA] on raw pixel intensities. We use the UIUC car image dataset [@UIUC-car], which contains 550 car and 500 non-car gray-level images of size $40\times100$ (Fig. \[fig:car\_data\]). We can observe that all car images are side-view images, but can be either side, and can be partly occluded. We divide the total of 1050 images into five subsets, each containing 110 car images and 100 non-car images, and each time we use four subsets as training set and one as testing set. We use the following methods to generate fixed-length feature vectors for the images:
1. **PCA** We represent each $40\times100$ gray-level image by a 4000-d vector, and perform standard PCA on such vectors of the training set to get eigenvectors and low dimensional representations of the training images. Then we use the eigenvectors to get the low dimensional representations of the testing images.
2. **kPCA Gaussian** Similar to the above method, but we use Gaussian kernel PCA instead of standard PCA. We follow the automatic parameter selection strategy in [@quan_kPCA] to determine the $\sigma$.
3. **kPCA poly** Similar to the above two methods, but we use third-order polynomial kernel PCA instead of standard PCA.
4. **IMED-MDS** We first measure the IMED between each pair of training images, and run Algorithm \[alg:Levenberg-Marquardt\] to learn the low dimensional MDS features of each training image. Then we measure the IMED from each testing image to each training image, and solve Eq. (\[eq:encoding\]) to obtain the MDS features of each testing image.
5. **SPM1-MDS** Similar to the above method, but we use SPM1 distance (\[eq:spm1\]), instead of IMED, where the SPM parameters are $M=200$ and $L=2$.
6. **SPM2-MDS** Similar to the above method, but we use SPM2 distance (\[eq:spm2\]).
7. **pyramid PCA** Instead of computing MDS features from SPM distances, we can also directly perform PCA on the obtained $M\frac{1}{3}(4^{L+1}-1)$-dimensional spatial pyramid vectors without measuring similarities. In our experiment, we set $M=200$ and $L=2$, and the spatial pyramid vectors are 4200-d. Evaluating this method will allow us to observe whether the MDS on SPM distance measurement captures semantics beyond the spatial pyramids.
![Example car (first row) and non-car (second row) images in UIUC car image dataset.[]{data-label="fig:car_data"}](car_data){width="47.00000%"}
After we have obtained the fixed-length features of all images, we use the features of training images to learn a binary RBF kernel SVM [@SVM; @libsvm], and use it to classify the features of testing images. Each dimension of the feature vector is normalized to 0-mean and unit standard deviation. In the radial basis function $\exp(-||\mathbf{u}-\mathbf{v}||^2/\gamma)$, we set $\gamma$ as the feature vector length. The experiment is repeated for different feature vector lengths from 1 to 20. We show the precision, recall and accuracy in Fig. \[fig:UIUC\_car\_performance\]. We also provide the feature scatter plots of different methods for feature length $m=2$ in Fig. \[fig:car\_scatter\].
In Fig. \[fig:UIUC\_car\_performance\], we can observe that IMED-MDS method performs slightly but not significantly better than directly applying PCA or kernel PCA on raw gray-level intensities, and the superiority of IMED-MDS is more obvious when feature dimension is low. Spatial pyramid based methods do perform much better than other methods. Especially, SPM1-MDS and SPM2-MDS methods outperform all other methods, including pyramid PCA, at all feature dimensions. While the precision and recall of PCA, kernel PCA and IMED-MDS methods saturate at $98\%$ and $96\%$ respectively, the precision and recall of SPM1-MDS and SPM2-MDS saturate at $100\%$ and $99\%$ respectively. At low feature dimensions ($m\leq5$), the accuracy of PCA and kernel PCA are very low, but the SPM1-MDS and SPM2-MDS perform almost as equally well as at very high dimensions.
In Fig. \[fig:car\_scatter\], we can also see that SPM1-MDS and SPM2-MDS separate car and non-car images with very clear class boundary curves in 2-d feature space.
\
Conclusions and Future Work
===========================
In this paper, we have presented a feature learning framework by combining multidimensional scaling with image distance measurement, and compared it with a number of popular existing feature extraction techniques. To the best of our knowledge, we are the first to explore MDS on image distances such as IMage Euclidean Distance (IMED) and Spatial Pyramid Matching (SPM) distance.
We have introduced a unified framework for both MDS model training and new data encoding based on the standard Levenberg-Marquardt algorithm. Our two-stage iterated Levenberg-Marquardt algorithm for MDS model training is an efficient solution, and has shown good running time performance compared with other off-the-shelf implementations (Fig. \[fig:ILMA\_runningtime\]).
In the car recognition experiment, we have demonstrated the power of MDS features. MDS features learned from SPM distances achieve the best classification performance on all feature dimensions. The good performance of MDS features attributes to the semantics-sensitive image distance, since it captures very different information from the images than traditional feature extraction techniques. The MDS further embeds such information into a low-dimensional feature space, which also captures the inner structure of the entire dataset. The MDS embedding is a very necessary step, since in Fig. \[fig:UIUC\_car\_performance\] we can see the performance of MDS features learned from SPM distances is significantly better than simply running PCA on spatial pyramid vectors.
Our ongoing work on this method explores these directions:
1. We study more image distance measurements, such as the Integrated Region Matching (IRM) distance, which was originally designed for semantics-sensitive image retrieval systems [@simplicity]. Performance of MDS codes learned from such distances can be evaluated and compared with the SPM-MDS method in this paper.
2. Our MDS feature learning framework can be validated on larger datasets with many categories of color images of different sizes. For example, we can validate the methodology on the popular Caltech-101 dataset [@caltech101] or the COREL dataset [@simplicity; @corel-divided].
3. In Eq. (\[eq:encoding\]), rather than using the entire training set, we can also use only a subset of the training images to encode new data. It would be interesting to see how the performance varies by applying different subset selection strategies and different sizes of the subset.
4. Currently the two-stage iterated Levenberg-Marquardt algorithm is implemented in MATLAB[^2]. We are also recoding it in C/C++ with the lmfit library[^3], which will be more computationally efficient.
[Quan Wang]{} Quan Wang is currently working towards his Ph.D. degree in Computer and Systems Engineering in the Department of Electrical, Computer, and Systems Engineering at Rensselaer Polytechnic Institute. He received a B.Eng. degree in Automation from Tsinghua University, Beijing, China in 2010. He worked as research intern at Siemens Corporate Research, Princeton, NJ and IBM Almaden Research Center, San Jose, CA in 2012 and 2013, respectively. His research interests include feature learning, medical image analysis, object tracking, content-based image retrieval and photographic composition.
[Kim L. Boyer]{} Dr. Kim L. Boyer is currently Head of the Department of Electrical, Computer, and Systems Engineering at Rensselaer Polytechnic Institute. He received the BSEE (with distinction), MSEE, and Ph.D. degrees, all in electrical engineering, from Purdue University in 1976, 1977, and 1986, respectively. From 1977 through 1981 he was with Bell Laboratories, Holmdel, NJ; from 1981 through 1983 he was with Comsat Laboratories, Clarksburg, MD. From 1986–2007 he was with the Department of Electrical and Computer Engineering, The Ohio State University. He is a Fellow of the IEEE, a Fellow of IAPR, a former IEEE Computer Society Distinguished Speaker, and currently the IAPR President. Dr. Boyer is also a National Academies Jefferson Science Fellow at the US Department of State, spending 2006–2007 as Senior Science Advisor to the Bureau of Western Hemisphere Affairs. He retains his Fellowship as a consultant on science and technology policy for the State Department.
[^1]: <http://cogcomp.cs.illinois.edu/Data/Car/>
[^2]: Code is available at <https://sites.google.com/site/mdsfeature/>
[^3]: <http://joachimwuttke.de/lmfit/>
|
---
abstract: 'This paper provides an overview of the features of fifth generation (5G) wireless communication systems now being developed for use in the millimeter wave (mmWave) frequency bands. Early results and key concepts of 5G networks are presented, and the channel modeling efforts of many international groups for both licensed and unlicensed applications are described here. Propagation parameters and channel models for understanding mmWave propagation, such as line-of-sight (LOS) probabilities, large-scale path loss, and building penetration loss, as modeled by various standardization bodies, are compared over the 0.5-100 GHz range.'
author:
- '[^1] [^2] [^3] [^4] [^5]'
bibliography:
- '5Gpaper\_draft\_v\_1\_0.bib'
title: 'Overview of Millimeter Wave Communications for Fifth-Generation (5G) Wireless Networks-with a focus on Propagation Models'
---
mmWave; 5G; propagation; cellular network; path loss; channel modeling; channel model standards;
Introduction
============
\[sec:intro\]
Wireless data traffic has been increasing at a rate of over 50% per year per subscriber, and this trend is expected to accelerate over the next decade with the continual use of video and the rise of the Internet-of-Things (IoT) [@gubbi2013internet; @rapfcc16]. To address this demand, the wireless industry is moving to its fifth generation (5G) of cellular technology that will use millimeter wave (mmWave) frequencies to offer unprecedented spectrum and multi-Gigabit-per-second (Gbps) data rates to a mobile device [@rappaport2013millimeter]. Mobile devices such as cell phones are typically referred to as user equipment (UE). A simple analysis illustrated that 1 GHz wide channels at 28 or 73 GHz could offer several Gbps of data rate to UE with modest phased array antennas at the mobile handset [@rangan2014millimeter], and early work showed 15 Gbps peak rates are possible with $4 \times 4$ phased arrays antenna at the UE and 200 m spacing between base stations (BSs) [@ghosh2014mmwave; @roh2014millimeter].
Promising studies such as these led the US Federal Communications Commission (FCC) to authorize its 2016 “Spectrum Frontiers” allocation of 10.85 GHz of millimeter wave spectrum for 5G advancements [@FCC16-89], and several studies [@singh2015tractable; @sundaresan2016fluidnet; @banelli2014modulation; @michailow2014generalized] have proposed new mobile radio concepts to support 5G mobile networks.
5G mmWave wireless channel bandwidths will be more than ten times greater than today’s 4G Long-Term Evolution (LTE) 20 MHz cellular channels. Since the wavelengths shrink by an order of magnitude at mmWave when compared to today’s 4G microwave frequencies, diffraction and material penetration will incur greater attenuation, thus elevating the importance of line-of-sight (LOS) propagation, reflection, and scattering. Accurate propagation models are vital for the design of new mmWave signaling protocols (e.g., air interfaces). Over the past few years, measurements and models for a vast array of scenarios have been presented by many companies and research groups [@rangan2014millimeter; @rappaport2013millimeter; @5GCM; @haneda2016indoor; @deng201528; @rappaport201573; @haneda20165g; @nie201372; @haneda2016frequency; @JSAC; @rappaport2015wideband; @maccartney2015indoor; @maccartney2013path; @samimi2015probabilistic; @Mac16c; @sun2016millimeter; @maccartney2015exploiting; @thomas2016prediction; @Sun16b; @samimi20163; @hur2014syn; @rappaport2013broadband; @Koymen15a].
This invited overview paper is organized as follows: Section \[sec:II\] summarizes key 5G system concepts of emerging mmWave wireless communication networks and Section \[sec:III\] presents 5G propagation challenges and antenna technologies. Section \[sec:IV\] gives a thorough compilation and comparison of recent mmWave channel models developed by various groups and standard bodies, while Section \[sec:conc\] provides concluding remarks.
5G System Concepts and Air Interfaces
=====================================
\[sec:II\] 5G promises great flexibility to support a myriad of Internet Protocol (IP) devices, small cell architectures, and dense coverage areas. Applications envisioned for 5G include the Tactile Internet [@fettweis2014tactile], vehicle-to-vehicle communication [@mecklenbrauker2011vehicular], vehicle-to-infrastructure communication [@gozalvez2012ieee], as well as peer-to-peer and machine-to-machine communication [@bhushan2014network], all which will require extremely low network latency and on-call demand for large bursts of data over minuscule time epochs [@maeder2011challenge]. Current 4G LTE and WiFi roundtrip latencies are about 20-60 ms [@nikravesh2016depth; @deng2014wifi], but 5G will offer roundtrip latencies on the order of 1 ms [@andrews2014will]. As shown in Fig. \[fig:network\], today’s 4G cellular network is evolving to support 5G, where WiFi off-loading, small cells, and distribution of wideband data will rely on servers at the edges of the network (edge servers) to enable new use cases with lower latency.
{width="80.00000%"}
Backhaul and Fronthaul
----------------------
Fig. \[fig:network\] shows how backhaul connects the fixed cellular infrastructure (e.g., BSs) to the core telephone network and the Internet. Backhaul carries traffic between the local subnetwork (e.g., the connections between UE and BSs) and the core network (e.g., the Internet and the Mobile Switching Telephone Office). 4G and WiFi backhaul, and not the air interface, are often sources of traffic bottlenecks in modern networks since backhaul connections provided by packet-based Ethernet-over-Fiber links typically provide only about 1 Gbps [@backhaul2015], which may be easily consumed by several UEs. In a typical macrocell site, a baseband unit (BBU) is in an enclosure at the base of a remote cell site and is directly connected to the backhaul. The BBU processes and modulates IP packet data from the core network into digital baseband signals where they are transmitted to remote radio heads (RRHs). The digital baseband signal travels from the BBU to a RRH via a common public radio interface (CPRI) through a digital radio-over-fiber (D-RoF) connection, also known as fronthaul. The RRH converts the digital signal to analog for transmission over the air at the carrier frequency by connecting to amplifiers and antennas to transmit the downlink from the cell tower. The RRH also converts the received radio frequency (RF) uplink signal from the UEs into a digital baseband signal which travels from the RRH to the BBU via the same CPRI and D-RoF connection to the base of the cell tower. The BBU then processes and packetizes the digital baseband signal from the RRH and sends it through a backhaul connection to the core network. In summary, fronthaul is the connection between the RRH and BBU in both directions and backhaul is the connection between the BBU and the core network in both directions.
Modern cellular architectures support a more flexible deployment of radio resources that may be distributed using a cloud radio access network technique, where a BS is split into two parts [@carapellese2014energy], one part where the RRHs are at remote cell sites, and in the other part, one centralized BBU is located up to tens of kilometers away (see Fig. \[fig:network\]). CPRI is used for fronthaul, and interconnects the centralized BBU and multiple RRHs through D-RoF. MmWave wireless backhaul and fronthaul will offer fiber-like data rates and bandwidth to infrastructure without the expense of deploying wired backhaul networks or long-range D-RoF [@hur2013millimeter; @sundaresan2016fluidnet; @H2020].
Small Cells
-----------
An effective way to increase area spectral efficiency is to shrink cell size [@chandrasekhar2008femtocell; @andrews2014will; @dohler2011phy] where the reduced number of users per cell, caused by cell shrinking, provides more spectrum to each user. Total network capacity vastly increases by shrinking cells and reusing the spectrum, and future nomadic BSs and direct device-to-device connections between UEs are envisioned to emerge in 5G for even greater capacity per user [@wang2014cellular]. Femtocells that can dynamically change their connection to the operator’s core network will face challenges such as managing RF interference and keeping timing and synchronization, and various interference avoidance and adaptive power control strategies have been suggested [@chandrasekhar2008femtocell]. An analysis of the wireless backhaul traffic at 5.8 GHz, 28 GHz, and 60 GHz in two typical network architectures showed that spectral efficiency and energy efficiency increased as the number of small cells increased [@ge20145g], and backhaul measurements and models at 73 GHz were made in New York City [@george2014backhaul; @rappaport2015wideband]. Work in [@murdock2014consumption] showed a theory for power consumption analysis, which is strikingly similar to noise figure, for comparing energy efficiency and power consumption in wideband networks. An early small-cell paper [@haider2011spectral] gave insights into enhancing user throughput, reducing signaling overhead, and reducing dropped call likelihoods.
Multi-tier Architecture
-----------------------
The roadmap for 5G networks will exploit a multi-tier architecture of larger coverage 4G cells with an underlying network of closer-spaced 5G BSs as shown in Fig. \[fig:network\]. A multi-tier architecture allows users in different tiers to have different priorities for channel access and different kinds of connections (e.g., macrocells, small cells, and device-to-device connections), thus supporting higher data rates, lower latencies, optimized energy consumption, and interference management by using resource-aware criteria for the BS association and traffic loads allocated over time and space [@hossain2014evolution]. Schemes and models for load balanced heterogeneous networks in a multi-tier architecture are given in [@andrews2014an; @tehrani2014device]. 5G applications will also require novel network architectures that support the convergence of different wireless technologies (e.g., WiFi, LTE, mmWave, low-power IoT) that will interact in a flexible and seamless manner using Software Defined Networking and Network Virtualization principles [@yang2015software; @agyapong2014design].
5G Air Interface
----------------
The design of new physical layer air interfaces is an active area of 5G research. Signaling schemes that provide lower latency, rapid beamforming and synchronization, with much smaller time slots and better spectral efficiency than the orthogonal frequency division multiplexing (OFDM) used in 4G, will emerge. A novel modulation that exploits the dead time in the single-carrier frequency domain modulation method used in today’s 4G LTE uplink is given in [@ghosh2014mmwave]. Work in [@banelli2014modulation] reviews linear modulation schemes such as filter bank multicarrier (FBMC) modulation wherein subcarriers are passed through filters that suppress sidelobes. Generalized frequency division multiplexing (GFDM) is proposed in [@michailow2014generalized], where it is shown that, when compared with OFDM used in current 4G LTE (which has one cyclic prefix per symbol and high out-of-band emissions [@van2008out]), GFDM improves the spectral efficiency and has approximately 15 dB weaker out-of-band emissions. Orthogonal time-frequency-space (OTFS) modulation that spreads the signals in the time-frequency plane has also been suggested, due to superior diversity and higher flexibility in pilot design [@monk2016otfs]. Channel state feedback and management to support directional beam search/steering will also be vital [@sun2014mimo; @haneda2015channel].
5G Unlicensed WiFi {#5GWIFI}
------------------
MmWave WiFi for the 57-64 GHz unlicensed bands has been in development for nearly a decade, with the WirelessHD and IEEE 802.11ad standardization process beginning in 2007, and 2009, respectively [@Rap15a]. IEEE 802.11ad devices, which can reach 7 Gbps peak rates [@yamada2015experimental], and WirelessHD products which can reach 4 Gbps with theoretical data rates as high as 25 Gbps [@siligaris2011a], are both already available in the market. Building on the history of WiFi standard IEEE 802.11n [@charfi2013phy; @perahia2013next], two newer standards, IEEE 802.11ac and 802.11ad, are amendments that improve the throughput to reach 1 Gbps in the 5 GHz band and up to 7 Gbps in the 60 GHz band, respectively. An overview of IEEE Gigabit wireless local area network (WLAN) amendments (IEEE 802.11ac and 802.11ad) [@verma2013wifi; @perahia2011gigabit; @perahia2010ieee] shows the suitability of these two standards for multi-gigabit communications. For the 802.11ad standard [@802.11ad], notable features include fast session transfer for seamless data rate fall back (and rate rise) between 60 GHz and 2.4/5 GHz PHYs, and media access control (MAC) enhancements for directional antennas, beamforming, backhaul, relays and spatial reuse techniques. For enhancements of the PHY layer, beamforming using directional antennas or antenna arrays is used to overcome the increased loss at 60 GHz [@Rap15a]. IEEE 802.11ay standard is an ongoing project with the goal to support a maximum throughput of at least 20 Gbps in the 60 GHz unlicensed band [@maltsev2016channel]. Newer WiFi standards are sure to emerge to exploit the new 64-71 GHz unlicensed spectrum in the US [@FCC16-89].
Vehicular Networks
------------------
Vehicle-to-vehicle (V2V) communications are an important tool for increasing road safety and reducing traffic congestion. Currently the most investigated system is the IEEE 802.11p standard which works in 5.9 GHz band for V2V and vehicle-to-infrastructure (V2I) communication, and is known as dedicated short-range communications (DSRC) [@802.11p]. The mmWave bands (e.g., 24 GHz and 77 GHz [@FCC16-89]) are attractive for V2V and V2I, (e.g., cars, high-speed railway and subway systems) since connected vehicles will need Gbps date rates, which cannot be achieved in the 10 MHz channel bandwidths at 5.9 GHz in current 4G [@moustafa2009vehicular; @bendor2011mmwave; @rappaport2010analysis]. Limitations of V2V connectivity include the difficulty in achieving realistic spatial consistency to sustain the data-link connection for high-speed mobility vehicles [@5GCM; @rap2016ap]. Evaluations have shown that narrow beam directional antennas are more suitable for IEEE 802.11p-based systems [@shivaldova2012roadside], and several schemes aimed at utilizing adaptive antennas for fast moving V2V communications are provided in [@phan2015making].
5G Antenna and Propagation Challenges
=====================================
\[sec:III\] The entire radio spectrum up to 5.8 GHz that has been used for global wireless communications throughout the past 100 years easily fits within the bandwidth of the single 60 GHz unlicensed band, yet there is so much more spectrum still available above 60 GHz [@FCC16-89; @rangan2014millimeter; @Rap15a], as shown in Figure C.1 on page 40 of [@Rap15a]. With radio frequency integrated circuits (RFIC) now routinely manufactured for 24 and 77 GHz vehicular radar, and IEEE 802.11ad WiGig devices now becoming mainstream in high-end laptops and cellphones, low-cost electronics will be viable for the evolution of massively broadband 5G millimeter wave communications [@rappaport2011state].
Today, most spectrum above 30 GHz is used for military applications or deep-space astronomy reception, but the recent FCC Spectrum Frontiers ruling has assigned many bands for mobile and backhaul communications. The various resonances of oxygen and other gasses in air, however, cause certain bands to suffer from signal absorption in the atmosphere. Fig. \[fig:myths\] illustrates how the bands of 183 GHz, 325 GHz, and especially 380 GHz suffer much greater attenuation over distance due to the molecular resonances of various components of the atmosphere, beyond the natural Friis’ free space loss, making these particular bands well suited for very close-in communications and “whisper radio” applications where massive bandwidth channels will attenuate very rapidly out to a few meters or fractions of a meter [@rappaport2013millimeter; @Rap15a]. Fig. \[fig:myths\] also shows many mmWave bands only suffer 1-2 dB more loss than caused by free space propagation per km in air [@ITU-Rattenuation; @sun2017a]. Rain and hail cause substantial attenuation at frequencies above 10 GHz [@xu2000measurements], and 73 GHz signals attenuate at 10 dB/km for a 50 mm/hr rain rate [@rappaport2013millimeter; @ITU-Rspecific; @Rap15a]. Interestingly, as shown in [@rappaport2013millimeter; @rappaport2011state] rain attenuation flattens out at 100 GHz to 500 GHz, and for all mmWave frequencies, rain or snow attenuation may be overcome with additional antenna gain or transmit power. Also, the size and orientation of rain drops and clouds may determine the particular amount of attenuation on air-to-ground links such that satellites could undergo more localized and perhaps less rain attenuation than terrestrial links at mmWave frequencies.
![Atmospheric absorption of electromagnetic waves at sea level versus frequency, showing the additional path loss due to atmospheric absorption [@rappaport2011state]. []{data-label="fig:myths"}](myths.png){width="40.00000%"}
While it is commonly believed that path loss increases dramatically by moving up to mmWave frequencies, extensive work in various environments in [@Sun16b; @maccartney2015indoor; @sun2016propagation; @samimi20163; @sun2015path] shows that Friis’ equation [@friis1946note] dictates this is true only when the antenna gain is assumed to be constant over frequency. If the physical size of the antenna (e.g., effective aperture) is kept constant over frequency at both link ends and the weather is clear, then path loss in free space actually *decreases* quadratically as frequency *increases* [@Rap15a]. The larger antenna gains at higher frequencies require adaptive beam steering for general use at both the BS and UE, compared to legacy mobile antennas with lower gain [@Rap15a]. Beam steerable antenna technologies estimate directions of arrival and adaptively switch beam patterns to mitigate interference and to capture the signal of interest. Adaptive arrays are essential for mmWave communications to compensate the path loss caused by blockage from dynamic obstacles [@Rap15a; @uchendu2016survey; @nitsche2015steering; @samimi20163; @sun2017a; @maccartney2016millimeter].
Penetration into buildings may pose a significant challenge for mmWave communication, and this is a distinct difference from today’s UHF/microwave systems. Measurements at 38 GHz described in [@rodriguez2015analysis] found a penetration loss of nearly 25 dB for a tinted glass window and 37 dB for a glass door. Measurements at 28 GHz [@rappaport2013millimeter] showed that outdoor tinted glass and brick pillars had penetration losses of 40.1 dB and 28.3 dB, respectively, but indoor clear glass and drywall only had 3.6 dB and 6.8 dB of loss. Work in [@jacque2016indoor] shows penetration losses for many common materials and provides normalized attenuation (e.g., in dB/cm) at 73 GHz. MmWave will need to exploit and rapidly adapt to the spatial dynamics of the wireless channel since greater gain antennas will be used to overcome path loss. Diffuse scattering from rough surfaces may introduce large signal variations over very short travel distances (just a few centimeters) as shown in Fig. \[fig:Bristol\]. Such rapid variations of the channel must be anticipated for proper design of channel state feedback algorithms, link adaptation schemes and beam-forming/tracking algorithms, as well as ensuring efficient design of MAC and Network layer transmission control protocols (TCP) that induce re-transmissions. Measurement of diffuse scattering at 60 GHz on several rough and smooth wall surfaces [@rumney2016testing2; @mmMAGIC] demonstrated large signal level variations in the first order specular and in the non-specular scattered components (with fade depths of up to 20 dB) as a user moved by a few centimeters. In addition, the existence of multipath from nearly co-incident signals can create severe small-scale variations in the channel frequency response. As reported in [@rumney2016testing2; @mmMAGIC], measurements showed that reflection from rough materials might suffer from high depolarization, a phenomenon that highlights the need for further investigation into the potential benefits of exploiting polarization diversity for the performance enhancement of mmWave communication systems. Work in [@samimi201628] showed shallow Ricean fading of multipath components and exponential decaying trends for spatial autocorrelation at 28 GHz and quick decorrelation at about 2.5 wavelengths for the LOS environment. Work in [@rap2016ap] shows that received power of wideband 73 GHz signals has a stationary mean over slight movements but average power can change by 25 dB as the mobile transitioned a building cornor from non-line-of-sight (NLOS) to LOS in an urban microcell (UMi) environment [@samimi2016local; @maccartney2016millimeter]. Measurements at 10, 20 and 26 GHz demonstrate that diffraction loss can be predicted using well-known models as a mobile moves around a corner using directional antennas [@sijia2016], and human body blockage causes more than 40 dB of fading [@maccartney2016millimeter; @samimi2016local].
It is not obvious that the stationarity region size or small-scale statistics derived from 3GPP TR 36.873 [@3GPP2014] and other sub-6 GHz channel models, or those used by 3GPP or ITU above 6 GHz are valid for mmWave channels [@sun2017a; @ertel1998overview; @sun2017b; @rappaport20175g; @rappaport2017VTC]. Recent measurements [@rap2016ap; @rumney2016testing2; @samimi2016local] indicate very sharp spatial decorrelation over small distance movements of just a few tens of wavelengths at mmWave, depending on antenna orientation, but more work is needed in this area. The necessity and proper form of spatial consistency, if borne out by measurements, have yet to be fully understood by the research community.
![Results of diffuse scattering measurements at 60 GHz, where smooth surfaces (e.g., windows) offer high correlation over distance, but signals from rough surfaces seem less correlated over distance [@rumney2016testing2; @mmMAGIC]. []{data-label="fig:Bristol"}](Bristol.png){width="40.00000%"}
Channel Modeling
================
\[sec:IV\]
[|c|c|c|]{}
--
--
&
------------------------------------------------------
**LOS probability models (distances are in meters)**
------------------------------------------------------
&
----------------
**Parameters**
----------------
\
---------------------------
3GPP TR 38.901[@3GPP2017]
---------------------------
&
--------------------------------------------------------------------------------
**Outdoor users:**
$P_{LOS}(d_{2D}) = \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)$
**Indoor users:**
Use $d_{2D-out}$ in the formula above instead of $d_{2D}$
--------------------------------------------------------------------------------
&
----------------------
$d_1 =18 \;\text{m}$
$d_2= 36 \;\text{m}$
----------------------
\
-------------
5GCM[@5GCM]
-------------
&
-------------------------------------------------------------------------------------
**$\bm{d_1/ d_2}$ model:**
$P_{LOS}(d_{2D}) = \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)$
**NYU (squared) model:**
$P_{LOS}(d_{2D}) = (\min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2))^2 $
-------------------------------------------------------------------------------------
&
----------------------------
**$\bm{d_1/ d_2}$ model:**
$d_1 =20 \;\text{m}$
$d_2= 39 \;\text{m}$
**NYU (squared) model:**
$d_1 =22 \;\text{m}$
$d_2= 100 \;\text{m}$
----------------------------
\
-------------------
METIS[@METIS2015]
-------------------
&
--------------------------------------------------------------------------------
**Outdoor users:**
$P_{LOS}(d_{2D}) = \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)$
**Indoor users:**
Use $d_{2D-out}$ in the formula above instead of $d_{2D}$
--------------------------------------------------------------------------------
&
-----------------------------
$d_1 =18 \;\text{m}$
$d_2= 36 \;\text{m}$
$10 \;\text{m} \leq d_{2D}$
-----------------------------
\
-------------------
mmMAGIC[@mmMAGIC]
-------------------
&
--------------------------------------------------------------------------
**Outdoor users:**
$P(d_{2D}) = \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)$
**Indoor users:**
Use $d_{2D-out}$ in the formula above instead of $d_{2D}$
--------------------------------------------------------------------------
&
----------------------
$d_1 =20 \;\text{m}$
$d_2= 39 \;\text{m}$
----------------------
\
\
[|c|c|c|]{}
--
--
&
------------------------------------------------------
**LOS probability models (distances are in meters)**
------------------------------------------------------
&
----------------
**Parameters**
----------------
\
---------------------------
3GPP TR 38.901[@3GPP2017]
---------------------------
&
-----------------------------------------------------------------------------------------------------------
**Outdoor users:**
$P_{LOS} = \left( \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)\right) (1+C(d_{2D},h_{UE}))$
where,
$ C(d_{2D},h_{UE})=\begin{cases}
0, & h_{UE} < 13\;\text{m}\\
\left( \frac{h_{UE}-13}{10}\right) ^{1.5}g(d_{2D}), & 13\;\text{m}\leq h_{UE} \leq 23 \;\text{m}
\end{cases}$
and,
$ g(d_{2D})=\begin{cases}
0, & d_{2D} \leq 18\;\text{m}\\
(1.25e-6)(d_{2D})^3 \exp(-d_{2D}/150), & 18\;\text{m} < d_{2D}
\end{cases} $
**Indoor users:**
Use $d_{2D-out}$ in the formula above instead of $d_{2D}$
-----------------------------------------------------------------------------------------------------------
&
----------------------
$d_1 =18 \;\text{m}$
$d_2= 63 \;\text{m}$
----------------------
\
-------------
5GCM[@5GCM]
-------------
&
----------------------------------------------------------------------------------------------------------------------------
**$\bm{d_1/ d_2}$ model:**
$P_{LOS} = \left( \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)\right) (1+C(d_{2D},h_{UE}))$
**NYU (squared) model:**
$P_{LOS} = \left( \left( \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)\right) (1+C(d_{2D},h_{UE}))\right) ^2$
----------------------------------------------------------------------------------------------------------------------------
&
--------------------------
**model:**
$d_1 =20 \;\text{m}$
$d_2= 66 \;\text{m}$
**NYU (squared) model:**
$d_1 =20 \;\text{m}$
$d_2= 160 \;\text{m}$
--------------------------
\
-------------------
METIS[@METIS2015]
-------------------
&
-----------------------------------------------------------------------------------------------------------
**Outdoor users:**
$P_{LOS} = \left( \min(d_1/d_{2D},1)(1-\exp(-d_{2D}/d_2))+ \exp(-d_{2D}/d_2)\right) (1+C(d_{2D},h_{UE}))$
**Indoor users:**
Use $d_{2D-out}$ in the formula above instead of $d_{2D}$
-----------------------------------------------------------------------------------------------------------
&
----------------------
$d_1 =18 \;\text{m}$
$d_2= 63 \;\text{m}$
----------------------
\
Channel models are required for simulating propagation in a reproducible and cost-effective way, and are used to accurately design and compare radio air interfaces and system deployment. Common wireless channel model parameters include carrier frequency, bandwidth, 2-D or 3-D distance between transmitter (TX) and receiver (RX), environmental effects, and other requirements needed to build globally standardized equipment and systems. The definitive challenge for a 5G channel model is to provide a fundamental physical basis, while being flexible, and accurate, especially across a wide frequency range such as 0.5 GHz to 100 GHz. Recently, a great deal of research aimed at understanding the propagation mechanisms and channel behavior at the frequencies above 6 GHz has been published [@rangan2014millimeter; @rappaport2013millimeter; @5GCM; @haneda2016indoor; @deng201528; @rappaport201573; @haneda20165g; @nie201372; @haneda2016frequency; @JSAC; @rappaport2015wideband; @maccartney2015indoor; @maccartney2013path; @samimi2015probabilistic; @Mac16c; @maccartney2015exploiting; @thomas2016prediction; @Sun16b; @samimi20163; @hur2014syn; @rappaport2013broadband; @Koymen15a; @andrews2014will; @rappaport2011state; @sun2016millimeter; @sun2016propagation; @sun2015path; @rodriguez2015analysis; @rumney2016testing2; @mmMAGIC; @samimi201628; @rap2016ap; @samimi2016local; @sijia2016; @3GPP2017; @samimi20153; @xu2000measurements; @bendor2011mmwave; @MiWEBA; @hur2016proposal; @ITU-RM.2135; @jacque2016indoor; @METIS2015; @jarvelainen2016evaluation; @piersanti2012millimeter; @semaan2014outdoor; @haneda2015channel; @maccartney2015millimeter; @maccartney2017study]. The specific types of antennas used and numbers of measurements collected vary widely and may generally be found in the referenced work.
For the remainder of this paper, the models for LOS probability, path loss, and building penetration introduced by four major organizations in the past years are reviewed and compared: (i) the 3rd Generation Partnership Project (3GPP TR 38.901 [@3GPP2017]), which attempts to provide channel models from 0.5-100 GHz based on a modification of 3GPP’s extensive effort to develop models from 6 to 100 GHz in TR 38.900 [@3GPP2016]. 3GPP TR documents are a continual work in progress and serve as the international industry standard for 5G cellular, (ii) 5G Channel Model (5GCM) [@5GCM], an ad-hoc group of 15 companies and universities that developed models based on extensive measurement campaigns and helped seed 3GPP understanding for TR 38.900 [@3GPP2016], (iii) Mobile and wireless communications Enablers for the Twenty-twenty Information Society (METIS) [@METIS2015] a large research project sponsored by European Union, and (iv) Millimeter-Wave Based Mobile Radio Access Network for Fifth Generation Integrated Communications (mmMAGIC) [@mmMAGIC], another large research project sponsored by the European Union. While many of the participants overlap in these standards bodies, the final models between those groups are somewhat distinct. It is important to note that recent work has found discrepancies between standardized models and measured results [@samimi20163; @rappaport2017VTC; @rappaport20175g].
LOS Probability Model
---------------------
The mobile industry has found benefit in describing path loss for both LOS and NLOS conditions separately. As a consequence, models for the probability of LOS are required, i.e., statistical models are needed to predict the likelihood that a UE is within a clear LOS of the BS, or in an NLOS region due to obstructions. LOS propagation will offer more reliable performance in mmWave communications as compared to NLOS conditions, given the greater diffraction loss at higher frequencies compared to sub-6 GHz bands where diffraction is a dominant propagation mechanism [@sijia2016; @rap2016ap], and given the larger path loss exponent as well as increased shadowing variance in NLOS as compared to LOS [@Sun16b]. The LOS probability is modeled as a function of the 2D TX-RX (T-R) separation distance and is frequency-independent, as it is solely based on the geometry and layout of an environment or scenario [@samimi2015probabilistic]. In the approach of 5GCM [@5GCM], the LOS state is determined by a map-based approach in which only the TX and the RX positions are considered for determining if the direct path between the TX and RX is blocked.
### UMi LOS Probability
The UMi scenarios include high user density open areas and street canyons with BS heights below rooftops (e.g., 3-20 m), UE heights at ground level (e.g., 1.5 m) and inter-site distances (ISDs) of 200 m or less [@3GPP2014; @ITU-RM.2135]. The UMi LOS probability models developed by the various parties are provided in Table \[tbl:UMiLOS\] and are detailed below.
#### 3GPP TR 38.901
The antenna height is assumed to be 10 m in the UMi LOS probability model [@3GPP2017] and the model is referred to as the 3GPP/ITU $d_1/ d_2$ model (it originates in [@ITU-RM.2135; @3GPP2014]), with $d_1$ and $d_2$ curve-fit parameters shown in Table \[tbl:UMiLOS\]. In [@3GPP2017], model parameters were found to be $d_1=18 \;\text{m}$ and $d_2 = 36 \;\text{m}$ for UMi. For a link between an outdoor BS and an indoor UE, the model uses the outdoor distance $d_{2D-out}$, which is the distance from the BS to the surface of the indoor building, to replace $d_{2D}$.
#### 5GCM
5GCM provides two LOS probability models, the first one is identical in form to the 3GPP TR 38.901 outdoor model [@3GPP2017], but with slightly different curve-fit parameters ($d_1$ and $d_2$). The second LOS probability model is the *NYU squared* model [@samimi2015probabilistic], which improves the accuracy of the $d_1/ d_2$ model by including a square on the last term. The NYU model was developed using a much finer resolution intersection test than used by 3GPP TR 38.901, and used a real-world database in downtown New York City [@samimi2015probabilistic]. For UMi, the 5GCM $d_1/d_2$ model has a slightly smaller mean square error (MSE), but the *NYU squared* model has a more realistic and rapid decay over distance for urban clutter [@5GCM; @samimi2015probabilistic].
#### METIS
The LOS probability model used in METIS [@METIS2015] is based on the work of 3GPP TR 36.873 [@3GPP2014], and has the same form and the same parameter values as the 3GPP TR 38.901 model in Table \[tbl:UMiLOS\] where the minimum T-R separation distance is assumed to be 10 m in the UMi scenario.
#### mmMAGIC
For the UMi scenario, the mmMAGIC LOS probability model and parameter values are identical to the 5GCM $d_1/d_2$ model [@5GCM].
### UMa LOS Probability
Urban macrocell (UMa) scenarios typically have BSs mounted above rooftop levels of surrounding buildings (e.g., 25-30 m) with UE heights at ground level (e.g., 1.5 m) and ISDs no more than 500 m [@3GPP2014; @ITU-RM.2135]. The UMa LOS probability models are given in Table \[tbl:UMaLOS\] and are identical to the UMi LOS probability models but with different $d_1$ and $d_2$ values.
#### 3GPP TR 38.901
The 3GPP TR 38.901 UMa LOS probability models for outdoor and indoor users are presented in Table \[tbl:UMaLOS\], where for indoor users, $d_{2D-out}$ is used instead of $d_{2D}$ and the models are derived assuming the TX antenna height is 25 m. Due to the larger antenna heights in the UMa scenario, mobile height is an added parameter of the LOS probability as shown in Table \[tbl:UMaLOS\] where $h_{UE}$ represents the UE antenna height above ground.
#### 5GCM
The UMa LOS probability models in the 5GCM white paper [@5GCM] are of the same form as those in 3GPP TR 38.901 [@3GPP2017], but with different $d_1$ and $d_2$ values. The 5GCM includes the *NYU squared* option [@samimi2015probabilistic], similar to the UMi scenario. Differences between the 3GPP TR 38.901 and 5GCM UMa LOS probability models are given via MSE in Fig. \[fig:LOSpro\] for a UE height of 1.5 m. Similar performances are found among the three models, with the *NYU squared* model having the lowest MSE, while also providing the most conservative (e.g., lowest probability) for LOS at distance of several hundred meters [@5GCM; @samimi2015probabilistic].
![Comparison among three different LOS probability models in UMa scenario.[]{data-label="fig:LOSpro"}](PROLOS.png){width="40.00000%"}
#### METIS
The LOS probability model used in [@METIS2015] has the same form as the one in 3GPP TR 38.901 in Table \[tbl:UMaLOS\], and the minimum T-R separation distance is assumed to be 35 m in the UMa scenario.
#### mmMAGIC
The UMa scenario is taken into account in the channel model, however, it is not explicitly mentioned in the table since frequency spectrum above 6 GHz is expected to be used for small cell BSs.
### InH LOS Probability
[|c|c|]{}
-------------------------------------------------------------
**3GPP TR 38.901[@3GPP2017] (all distances are in meters)**
-------------------------------------------------------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
----------------------------------------------------------------------------------
**InH-Mixed office:**
$P_{LOS}=\begin{cases}
1, & d_{2D} \leq1.2\;\text{m}\\
\exp{(-(d_{2D}-1.2)/4.7)} , & 1.2\;\text{m} <d_{2D} < 6.5\;\text{m}\\
\exp{(-(d_{2D}-6.5)/32.6)}\cdot 0.32, & 6.5\;\text{m} \leq d_{2D}
\end{cases} $
**InH-Open office:**
$P_{LOS}^{\text{Open-office}}=\begin{cases}
1, & d_{2D} \leq 5\;\text{m}\\
\exp{(-(d_{2D}-5)/70.8)} , & 5\;\text{m} <d_{2D} < 49\;\text{m}\\
\exp{(-(d_{2D}-49)/211.7)}\cdot 0.54, & 49 \;\text{m} \leq d_{2D}
\end{cases} $
----------------------------------------------------------------------------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
-----------------
**5GCM[@5GCM]**
-----------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
----------------------------------------------------------------------------------
$P_{LOS}=\begin{cases}
1, & d_{2D} \leq1.2\;\text{m}\\
\exp{(-(d_{2D}-1.2)/4.7)} , & 1.2\;\text{m} <d_{2D} < 6.5\;\text{m}\\
\exp{(-(d_{2D}-6.5)/32.6)}\cdot 0.32, & 6.5\;\text{m} \leq d_{2D}
\end{cases} $
----------------------------------------------------------------------------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
-----------------------
**mmMAGIC[@mmMAGIC]**
-----------------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
-------------------------------------------------------------------------
$P_{LOS}=\begin{cases}
1, & d_{2D} \leq1.2\;\text{m}\\
\exp{(-(d_{2D}-1.2)/4.7)} , & 1.2 <d_{2D} < 6.5\;\text{m}\\
\exp{(-(d_{2D}-6.5)/32.6)}\cdot 0.32, & 6.5\;\text{m} \leq d_{2D}
\end{cases} $
-------------------------------------------------------------------------
: LOS probability models in the InH scenario[]{data-label="tbl:InHLOS"}
\
#### 3GPP TR 38.901
The indoor office environment consists of two types: indoor hotspot (InH)-Mixed office and InH-Open office, where the density of obstructions is greater in the mixed office. LOS probability models for a TX antenna height of 3 m for the InH-Mixed office and InH-Open office sub-scenarios are provided in Table \[tbl:InHLOS\].
#### 5GCM
In [@5GCM], different types of indoor office environments were investigated, including open-plan offices with cubicle areas, closed-plan offices with corridors and meeting rooms, and hybrid-plan offices with both open and closed areas, and based on ray-tracing simulations [@jarvelainen2016evaluation]. See Table \[tbl:InHLOS\] and [@5GCM].
#### mmMAGIC
mmMAGIC adopted the 5GCM InH scenario LOS probability model [@5GCM].
### RMa LOS Probability
Rural macrocell (RMa) scenarios typically have BS heights that range between 10 m and 150 m with UE heights at ground level (e.g., 1.5 m) and ISDs up to 5000 m [@3GPP2014; @ITU-RM.2135]. The LOS probabilities for RMa were not specified in METIS or 5GCM channel models. The 3GPP TR 38.901 [@3GPP2017] RMa LOS probability model was adopted from the International Telecommunications Union-Radio (ITU-R) M.2135 [@ITU-RM.2135], which was derived from the WINNER [@WINNERplus] RMa LOS probability model and is given by: $$\label{equ:3GPPRmaLOSprob}
\footnotesize
P_{LOS} = \begin{cases}
1, & d_{2D}\leq 10 \text{ m}\\
\exp\left(-\frac{d_{2D}-10}{1000}\right), & d_{2D} > 10\text{ m}
\end{cases}$$ where $P_{LOS}$ is the LOS probability for a specific T-R pair, $d_{2D}$ is the 2D T-R separation distance (in meters). Similarly, the RMa LOS probability 3GPP TR 38.901 Release 14 channel model [@3GPP2017] is adopted entirely from ITU-R M.2135 [@ITU-RM.2135]. As shown in [@JSAC; @Mac16c], caution is advised since these models were derived from urban (not rural) scenarios below 6 GHz.
Large-Scale Path Loss Models
----------------------------
[|c|c|c|c|]{}
--
--
&
----------------------------------------------------------
**PL \[dB\], $f_c$ is in GHz and $d_{3D}$ is in meters**
----------------------------------------------------------
&
-------------------
**Shadow fading**
**std \[dB\]**
-------------------
&
-------------------------
**Applicability range**
**and Parameters**
-------------------------
\
\
-----------------
5GCM UMi-Street
Canyon LOS
-----------------
&
--------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 21 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
--------------------------------------------------------
&
----------------------
$\sigma_{SF} = 3.76$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
-----------------
5GCM UMi-Street
Canyon NLOS
-----------------
&
----------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 31.7 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
**ABG model:**
$PL = 35.3\log_{10}(d_{3D}) + 22.4 + 21.3\log_{10}(f_c)$
----------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 8.09$\
\
$\sigma_{SF} = 7.82$
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
---------------
5GCM UMi-Open
Square LOS
---------------
&
---------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 18.5\log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
---------------------------------------------------------
&
---------------------
$\sigma_{SF} = 4.2$
---------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
---------------
5GCM UMi-Open
Square NLOS
---------------
&
----------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 28.9 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
**ABG model:**
$PL = 41.4\log_{10}(d_{3D}) + 3.66 + 24.3\log_{10}(f_c)$
----------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 7.1$\
\
$\sigma_{SF} = 7.0$
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
\
-----------------
3GPP UMi-Street
Canyon LOS
-----------------
&
-----------------------------------------------------------
$PL_{UMi-LOS}=\begin{cases}
PL_1, & 10\;\text{m} \leq d_{2D} \leq d_{BP}'\\
PL_2,& d_{BP}' \leq d_{2D} \leq 5 \;\text{km}
\end{cases}$
$PL_1 = 32.4 + 21\log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
$PL_2 = 32.4 + 40 \log_{10}(d_{3D}) + 20\log_{10}(f_c)$
$-9.5\log_{10}((d_{BP}')^2 + (h_{BS}-h_{UE})^2)$
where $d_{BP}'$ is specified in Eq. (\[equ:UMi3GPPbp\])
-----------------------------------------------------------
&
---------------------
$\sigma_{SF} = 4.0$
---------------------
&
---------------------------------------------------
$0.5 < f_c < 100 \;\text{GHz}$
$1.5 \;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 10 \;\text{m}$
---------------------------------------------------
\
-----------------
3GPP UMi-Street
Canyon NLOS
-----------------
&
----------------------------------------------------------------------------
$PL = \max{ \left( PL_{UMi-LOS}(d_{3D}), PL_{UMi-NLOS}(d_{3D}) \right) } $
$PL_{UMi-NLOS} = 35.3\log_{10}(d_{3D}) + 22.4 + 21.3\log_{10}(f_c)$
$ -0.3(h_{UE}-1.5)$
**Option: CI model with 1 m reference distance**
$PL = 32.4 + 20\log_{10}(f_c)+31.9\log_{10}(d_{3D})$
----------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 7.82$\
\
$\sigma_{SF} = 8.2$
&
--------------------------------------------------
$0.5 < f_c < 100 \;\text{GHz}$
$ 10 \;\text{m} < d_{2D} < 5000 \;\text{m}$
$1.5\;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 10 \;\text{m}$
--------------------------------------------------
\
\
------------------
METIS UMi-Street
Canyon LOS
------------------
&
----------------------------------------------------------------------------------
$PL_{UMi-LOS}\begin{cases}
PL_1, & 10\;\text{m} < d_{3D} \leq d_{BP}\\
PL_2,& d_{BP} < d_{3D} \leq 500 \;\text{m}
\end{cases}$
$PL_1 = 22 \log_{10}(d_{3D}) + 28.0
+ 20\log_{10}(f_c) + PL_{0}$
$PL_2 = 40 \log_{10}(d_{3D})+7.8 -18 \log_{10}(h_{BS} h_{UE})$
$ + 2\log_{10}(f_c) + PL_1(d_{BP})$
$d_{BP}$ and $PL_0$ are specified in Eq. (\[equ:METiSBP\]) and (\[equ:METiSPL\])
----------------------------------------------------------------------------------
&
----------------------
$\sigma_{SF} = 3.1 $
----------------------
&
---------------------------------------
$0.8 \leq f_{c} \leq 60 \;\text{GHz}$
---------------------------------------
\
------------------
METIS UMi-Street
Canyon NLOS
------------------
&
-----------------------------------------------------------------------------
$PL = \max{ \left( PL_{UMi-LOS}(d_{3D}), PL_{UMi-NLOS}(d_{3D}) \right) }$
$PL_{UMi-NLOS}=36.7\log_{10}(d_{3D}) +23.15 +26\log_{10}(f_c) -0.3(h_{UE})$
-----------------------------------------------------------------------------
&
----------------------
$\sigma_{SF} = 4.0 $
----------------------
&
--------------------------------------------------
$0.45 \leq f_{c} \leq 6 \;\text{GHz}$
$ 10 \;\text{m} < d_{2D} <2000 \;\text{m}$
$h_{BS}=10 \;\text{m}$
$1.5\;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
--------------------------------------------------
\
\
--------------------
mmMAGIC UMi-Street
Canyon LOS
--------------------
&
------------------------------------------------------------
$PL = 19.2 \log_{10}(d_{3D}) + 32.9 + 20.8\log_{10}(f_c)$
------------------------------------------------------------
&
---------------------
$\sigma_{SF} = 2.0$
---------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
--------------------
mmMAGIC UMi-Street
Canyon NLOS
--------------------
&
------------------------------------------------------------
$PL = 45.0 \log_{10}(d_{3D}) + 31.0 + 20.0 \log_{10}(f_c)$
------------------------------------------------------------
&
----------------------
$\sigma_{SF} = 7.82$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
\
\
There are three basic types of large-scale path loss models to predict mmWave signal strength over distance for the vast mmWave frequency range (with antenna gains included in the link budget and not in the slope of path loss as shown in Eq. (3.9) of [@Rap15a], also see p.3040 in [@rappaport2015wideband]). These include the close-in (CI) free space reference distance model (with a 1 m reference distance) [@sun2016propagation; @rappaport2015wideband; @Sun16b; @sun2015path], the CI model with a frequency-weighted or height weighted path loss exponent (CIF and CIH models) [@maccartney2015indoor; @haneda2016frequency; @Mac16c; @JSAC], and the floating intercept (FI) path loss model, also known as the ABG model because of its use of three parameters $\alpha$, $\beta$, and $\gamma$ [@hata1980; @haneda2016frequency; @maccartney2015indoor; @piersanti2012millimeter; @maccartney2013path; @rappaport2015wideband]. Standard bodies historically create omnidirectional path loss models with the assumption of unity gain antennas for generality. However, it is worth noting that omnidirectional path loss models will not be usable in directional antenna system analysis unless the antenna patterns and true spatial and temporal multipath channel statistics are known or properly modeled [@JSAC; @rappaport2015wideband; @samimi20163; @sun2015synthesizing; @sun2017a; @rappaport20175g; @maccartney2014primrc].
The CI path loss model accounts for the frequency dependency of path loss by using a close-in reference distance based on Friis’ law as given by [@5GCM; @Sun16b; @maccartney2015indoor; @JSAC; @Mac16c]: $$\footnotesize
\label{equ:CI}
\begin{split}
PL^{CI}(f_c,d_{3D})\;\text{[dB]} = \text{FSPL} (f_c,1 \;\text{m}) + 10n\log_{10}\left( d_{3D} \right)+ \chi_{\sigma}^{CI}
\end{split}$$ where $\chi_{\sigma}^{CI}$ is the shadow fading (SF) that is modeled as a zero-mean Gaussian random variable with a standard deviation in dB, $n$ is the path loss exponent (PLE) found by minimizing the error of the measured data to , $d_{3D} > 1m$, $\text{FSPL} (f,1 \;\text{m})$ is the free space path loss (FSPL) at frequency $f_c$ in GHz at 1 m and is calculated by [@friis1946note; @JSAC]: $$\label{FSPL}
\footnotesize
\begin{split}
\text{FSPL}(f_c , 1\;\text{m})= 20 \log_{10}\left( \frac{4 \pi f_c \times 10^9}{c}\right)=32.4 + 20\log_{10}(f_c)\ \text{[dB]}
\end{split}$$ where $c$ is the speed of light, $3 \times 10^8$ m/s. Using it is clear that can be represented as given in Table \[tbl:UMiPL\]. The standard deviation $\sigma$ yields insight into the statistical variation about the distant-dependent mean path loss [@Rap15a].
The CI model ties path loss at any frequency to the physical free space path loss at 1 m according to Friis’ free space equation [@friis1946note], and has been shown to be robust and accurate in various scenarios [@Sun16b; @JSAC; @Mac16c; @thomas2016prediction]. Indoor environments, however, were found to have frequency-dependent loss beyond the first meter, due to the surrounding environment, and work in [@maccartney2015indoor] extended the CI model to the CIF model where the PLE has a frequency-dependent term. Recent work [@JSAC; @Mac16c] has made 73 GHz rural measurements to beyond 10 km and adapted the CIF model form to predict path loss as a function of TX antenna height in RMa scenarios, as path loss was found to be accurately predicted with a height dependency in the PLE, leading to the CIH model[^6], which has the same form of the CIF model given in :
$$\footnotesize
\begin{split}
\label{equ:CIF}
PL^{CIF}(f_c,d)\;\text{[dB]} &= 32.4 + 20\log_{10}(f_c) \\
&+ 10n\left(1+b\left(\frac{f_c-f_0}{f_0} \right) \right) \log_{10}\left( d \right) + \chi_{\sigma}^{CIF}
\end{split}$$
where $n$ denotes the distance dependence of path loss, $b$ is an optimization parameter that describes the linear dependence of path loss about the weighted average of frequencies $f_0$ (in GHz), from the data used to optimize the model [@maccartney2015indoor; @Mac16c; @JSAC].
The CIF model uses two parameters to model average path loss over distance, and reverts to the single parameter CI model when $ b = 0$ for multiple frequencies, or when a single frequency $f=f_0$ is modeled [@5GCM; @haneda2016indoor; @maccartney2015indoor; @haneda20165g; @JSAC].
The FI/ABG path loss model is given as: $$\footnotesize
\label{equ:ABG}
\begin{split}
PL^{ABG}(f_c,d)\;\text{[dB]} = 10\alpha \log_{10}(d) + \beta + 10\gamma \log_{10}(f_c) + \chi^{ABG}_\sigma
\end{split}$$ where three model parameters $\alpha$, $\beta$ and $\gamma$ are determined by finding the best fit values to minimize the error between the model and the measured data. In (\[equ:ABG\]), $\alpha$ indicates the slope of path loss with log distance, $\beta$ is the floating offset value in dB, and $\gamma$ models the frequency dependence of path loss, where $f_c$ is in GHz. Generalizations of the CI, CIF, and FI/ABG models consider different slopes of path loss over distance before and after a breakpoint distance, where the location of the breakpoint depends mostly on the environment. The dual-slope CIF model is: $$\label{equ:CIFdual}
\scriptsize{
PL_{Dual}^{CIF}(d)\;\text{[dB]}=\begin{cases}
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right)\log_{10}( d) , & 1<d \leq d_{BP}\\
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right) \log_{10}(d_{BP})\\
+ 10n_2\left( 1+ b_2\left( \frac{f_c-f_0}{f_0} \right) \right) \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}}$$ The dual-slope ABG model is: $$\label{equ:ABGdual}
\footnotesize{
PL_{Dual}^{ABG}(d)\;\text{[dB]} =\begin{cases}
\alpha_1\ast 10\log_{10} (d) + \beta_1 \\
+ \gamma \ast 10\log_{10}(f_c) , & 1<d \leq d_{BP}\\
\alpha_1\ast 10\log_{10} (d_{BP}) + \beta_1\\
+ \gamma \ast 10\log_{10}(f_c)\\
+\alpha_2 \ast 10 \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}}$$ where the $\alpha_1$ and $\alpha_2$ are the “dual slope” and $d_{BP}$ is the breakpoint distance. Both dual-slope models require 5 parameters to predict distant-dependent average path loss (frequencies are in GHz and distances are in meters).
[|c|c|c|c|]{}
--
--
&
--------------------------------------------------
**PL \[dB\], $f_c$ is in GHz, $d$ is in meters**
--------------------------------------------------
&
-------------------
**Shadow fading**
**std \[dB\]**
-------------------
&
-------------------------
**Applicability range**
**and Parameters**
-------------------------
\
\
----------
5GCM UMa
LOS
----------
&
--------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 20 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
--------------------------------------------------------
&
---------------------
$\sigma_{SF} = 4.1$
---------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
----------
5GCM UMa
NLOS
----------
&
--------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 30 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
**ABG model:**
$PL = 34\log_{10}(d_{3D}) + 19.2 + 23\log_{10}(f_c)$
--------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 6.8$\
\
$\sigma_{SF} = 6.5$
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
\
--------------------
3GPP TR 38.901 UMa
LOS
--------------------
&
----------------------------------------------------------
$PL_{UMa-LOS}=\begin{cases}
PL_1, & 10\;\text{m} \leq d_{2D} \leq d_{BP}'\\
PL_2,& d_{BP}' \leq d_{2D} \leq 5 \;\text{km}
\end{cases}$
$PL_1 = 28.0 + 22 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
$PL_2 = 28.0 + 40 \log_{10}(d_{3D}) + 20\log_{10}(f_c)$
$-9\log_{10}((d_{BP}')^2 + (h_{BS}-h_{UE})^2)$
where $d_{BP}' = 4 h_{BS}'h_{UE}'f_c \times 10^9/c$
----------------------------------------------------------
&
---------------------
$\sigma_{SF} = 4.0$
---------------------
&
---------------------------------------------------
$0.5 < f_c < 100 \;\text{GHz}$
$1.5 \;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 25 \;\text{m}$
---------------------------------------------------
\
--------------------
3GPP TR 38.901 UMa
NLOS
--------------------
&
----------------------------------------------------------------------------
$PL = \max{ \left( PL_{UMa-LOS}(d_{3D}), PL_{UMa-NLOS}(d_{3D}) \right) } $
$PL_{UMa-NLOS} = 13.54+ 39.08\log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
$ -0.6(h_{UE}-1.5)$
**Option: CI model with 1 m reference distance**
$PL = 32.4 + 20\log_{10}(f_c)+30 \log_{10}(d_{3D})$
----------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 6.0$\
\
$\sigma_{SF} = 7.8$
&
--------------------------------------------------
$0.5 < f_c < 100 \;\text{GHz}$
$ 10 \;\text{m} < d_{2D} < 5000 \;\text{m}$
$1.5\;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 25 \;\text{m}$
--------------------------------------------------
\
\
-----------
METIS UMa
LOS
-----------
&
------------------------------------------------------------
$PL_{UMa-LOS}=\begin{cases}
PL_1, & 10\;\text{m} \leq d_{2D} \leq d_{BP}'\\
PL_2,& d_{BP}' \leq d_{2D} \leq 5 \;\text{km}
\end{cases}$
$PL_1 = 28 + 22 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
$PL_2 = 28 + 40 \log_{10}(d_{3D}) + 20\log_{10}(f_c)$
$-9\log_{10}((d_{BP}')^2 + (h_{BS}-h_{UE})^2)$
where $d_{BP}' = 4 (h_{BS}-1)(h_{UE}-1)f_c \times 10^9 /c$
------------------------------------------------------------
&
---------------------
$\sigma_{SF} = 4.0$
---------------------
&
---------------------------------------------------
$0.45 < f_c < 6 \;\text{GHz}$
$ 10 \;\text{m} < d_{2D} < 5000 \;\text{m}$
$1.5 \;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 25 \;\text{m}$
---------------------------------------------------
\
-----------
METIS UMa
NLOS
-----------
&
--------------------------------------------------------------------------------
$PL = \max{ \left( PL_{UMa-LOS}(d_{3D}), PL_{UMa-NLOS}(d_{3D}) \right) } $
$PL_{UMa-NLOS} = 161.94 -7.1 \log_{10}(w) + 7.5 \log_{10}(h)$
$-\left( 24.37-3.7\left( \dfrac{h}{h_{BS}}\right)^2 \right)\log_{10}(h_{BS})$
$+(43.42-3.1\log_{10}(h_{BS}))(\log_{10}(d_{3D})-3)$
$ + 20\log_{10}(f_c)-0.6(h_{UE})$
--------------------------------------------------------------------------------
&
---------------------
$\sigma_{SF} = 6.0$
---------------------
&
--------------------------------------------------
$0.45 < f_c < 6 \;\text{GHz}$
$ 10 \;\text{m} < d_{2D} < 5000 \;\text{m}$
$1.5\;\text{m} \leq h_{UE} \leq 22.5 \;\text{m}$
$h_{BS} = 25 \;\text{m}$
$ w = 20 \;\text{m}$
$h = 20 \;\text{m}$
--------------------------------------------------
\
### UMi Large-Scale Path Loss
#### 5GCM
In the 5GCM white paper [@5GCM], the CI model (\[equ:CI\]) is chosen for modeling UMi LOS path loss, since $\alpha$ in the ABG model (\[equ:ABG\]) is almost identical to the PLE of the CI model, and also $\gamma$ is very close to 2 which is predicted by the physically-based Friis’ free space equation and used in the CI model [@Sun16b]. Both the CI and ABG models were adopted for UMi NLOS in 5GCM, and the parameters values for the CI and ABG models are given in Table \[tbl:UMiPL\]. In the CI path loss model, only a single parameter, the PLE, needs to be determined through optimization to minimize the model error of mean loss over distance, however, in the ABG model, three parameters need to be optimized to minimize the error, but with very little reduction of the shadowing variance compared to the CI model [@sun2016propagation; @Sun16b; @maccartney2015indoor].
![$PL$ vs. T-R distance comparison among four different path loss models in UMi scenario.[]{data-label="fig:PLUMi"}](UMI1PL.png){width="40.00000%"}
#### 3GPP TR 38.901 {#umi3gpp}
Path loss models in [@3GPP2017] use 3D T-R separation distances $d_{3D}$ that account for the BS height ($h_{BS}$) and UE height ($h_{UE}$). The distribution of the shadow fading is log-normal, and the standard deviation for LOS is $\sigma_{SF} = 4.0\;\text{dB}$. The UMi path loss model for LOS is a breakpoint model. For $d_{2D}<d_{BP}'$, the model is essentially a CI model with $n = 2.1$ [@andersen1995; @sun2016propagation; @rappaport2015wideband; @Sun16b; @sun2015path]. The LOS breakpoint distance $d_{BP}'$ is a function of the carrier frequency, BS height, and the UE height [@haneda20165g; @3GPP2017]: $$\label{equ:UMi3GPPbp}
\footnotesize
\begin{split}
d_{BP}' = 4h_{BS}'h_{UE}'f_c \times 10^9/c \\
h_{BS}' = h_{BS}-1.0 \;\text{m}, \\
h_{UE}' = h_{UE}- 1.0 \;\text{m}
\end{split}$$ where $h_{BS}'$ and $h_{UE}'$ are the effective antenna heights at the BS and the UE, and $h_{BS}$ and $h_{UE}$ are the actual antenna heights, respectively. The breakpoint distance in an urban environment [@bullington1947radio] is where the PLE transitions from free space ($n = 2$) to the asymptotic two-ray ground bounce model of $n = 4$ [@feuerstein1994path; @JSAC]. At mmWave frequencies, the use of a breakpoint is controversial as it has not been reported in measurement, but some ray tracing simulations predict that it will occur [@hur2016proposal]. Since the UMi cells radius is typically 500 m or less, the use of a breakpoint and the height factors in (\[equ:UMi3GPPbp\]) are not necessary (the breakpoint distance is larger than 500 m even with the smallest possible breakpoint distance when $h_{BS} = 4$ m and $h_{UE} = 1.5$ m as shown in Fig. \[fig:PLUMi\]). The CI model provides a similar prediction of the path loss with a much simpler equation (\[equ:CI\]) [@sun2015path].
In the NLOS scenarios, the UMi-NLOS model uses the ABG model form [@hata1980], with a frequency-dependent term that indicates path loss increases with frequency and also has an additional height correction term for the UE. Furthermore, a mathematical patch to correct model deficiencies is used to set a lower bound for the NLOS model as the LOS path loss. The shadow fading standard deviation for UMi NLOS is $\sigma_{SF} = 7.82 \;\text{dB}$ [@haneda20165g; @piersanti2012millimeter; @maccartney2013path]. The physically-based CI model is also provided as an optional NLOS path loss model for 3GPP TR 38.901 with parameter values given in Table \[tbl:UMiPL\].
#### METIS {#umimetis}
The path loss model for UMi in METIS [@METIS2015] is a modified version of the ITU-R UMi path loss model [@ITU-RM.2135] and is claimed to be valid for frequencies from 0.8 to 60 GHz (see Table \[tbl:UMiPL\]). Some METIS models include breakpoints based on sub-6 GHz work (see Fig. \[fig:PLUMi\]), yet mmWave measurements to date do not show breakpoints to exist [@hur2016proposal; @JSAC; @METIS2015]. For LOS scenarios, a scaling factor is used, so that the breakpoint distance $d_{BP}$ (in meters) becomes:
$$\footnotesize
\label{equ:METiSBP}
d_{BP} = 0.87 \exp \left( -\frac{\log_{10}(f_c)}{0.65} \right) \frac{4 (h_{BS}-1 \text{m})(h_{UE}-1 \text{m})}{\lambda}$$
and the path loss formula for LOS is written as: $$\footnotesize
\begin{split}
\label{equ:METiSPL}
PL_{LOS}(d_1)~\text{[dB]} = 10n_1\log_{10}\left( d_1\right) + 28.0 + 20\log_{10}\left( f_c \right) + PL_{0}
\end{split}$$ for $10 \text{ m} < d \leqslant d_{BP} $, where $PL_{0}$ is a path loss offset calculated by: $$\label{PLoffset}
\footnotesize
PL_{0}~\text{[dB]} = -1.38 \log_{10}\left( f_c \right) + 3.34$$
Path loss after the breakpoint distance is: $$\label{equ:METiSPL2}
\footnotesize
\begin{split}
PL_{LOS}(d_1)~\text{[dB]} = 10n_2\log_{10}\left( \frac{d_1}{d_{BP}} \right) + PL_{LOS}(d_{BP})
\end{split}$$ for $d_{BP} < d_1 < 500 \text{ m}$ where and represent path loss before and after the breakpoint, respectively. The last term $PL(d_{BP})$ in is derived from (\[equ:METiSPL\]) by substituting $d_1$ with $d_{BP}$ to calculate path loss at the breakpoint distance [@METIS2015].
The UMi NLOS path loss model in METIS is adopted from the 3GPP TR 36.873 [@3GPP2014; @METIS2015] sub-6 GHz model for 4G LTE and is calculated as:
$$\label{equ:METiSPLn}
\footnotesize
\begin{split}
&PL = \max{ \left( PL_{LOS}(d_{3D}), PL_{NLOS}(d_{3D}) \right) }\\
&PL_{NLOS}=36.7\log_{10}(d_{3D}) +23.15 +26\log_{10}(f_c) -0.3(h_{UE})
\end{split}$$
where $f_c$ is in GHz, $10 \text{ m} < d_{3D} < 2000 \text{ m}$, and $1.5 \text{ m} \leq h_{UE} \leq 22.5 \text{ m}$.
#### mmMAGIC
The mmMAGIC project [@mmMAGIC] adopted the ABG path loss model for UMi, similar to that from 5GCM [@5GCM] but with different parameter values (see Table \[tbl:UMiPL\]). Comparisons among the different UMi large-scale path loss models described here are provided in Fig. \[fig:PLUMi\].
### UMa Large-Scale Path Loss
#### 3GPP TR 38.901 {#gpp-tr-38.901-3}
The 3GPP TR 38.901 [@3GPP2017] UMa LOS path loss model is adopted from 3GPP TR 36.873 (below 6 GHz Release 12 for LTE) [@3GPP2014] and TR 38.900 [@3GPP2016; @3GPPTDOC]. For the UMa NLOS scenario, an ABG model and an optional CI model are provided (see Table \[tbl:UMaPL\] for parameters). With respect to the UMa LOS model, 3GPP TR 38.901 inexplicably discards the TR 38.900 [@3GPP2016] model and reverts back to TR 36.873 which is defined only for below 6 GHz [@3GPP2014] while also omitting the InH shopping mall scenario used in TR 38.900. TR 38.901 models omnidirectional path loss from 0.5-100 GHz, but lacks measurement validation in some cases.
#### 5GCM
There are three UMa path loss models used in [@5GCM]: CI, CIF, and ABG [@sun2015path; @Sun16b]. The PLEs of the CI/CIF models for UMa are somewhat lower than for the UMi models indicating less loss over distance, which makes sense intuitively since a larger BS height implies that fewer obstructions are encountered than in the UMi scenario [@thomas2016prediction].
#### METIS {#metis-2}
METIS adopted the sub-6 GHz 3GPP TR 36.873 [@3GPP2014] 3D UMa model that was published in 2014 for LTE, see Table \[tbl:UMaPL\].
### InH Large-Scale Path Loss
[|c|c|c|c|]{}
--
--
&
--------------------------------------------------
**PL \[dB\], $f_c$ is in GHz, $d$ is in meters**
--------------------------------------------------
&
-------------------
**Shadow fading**
**std \[dB\]**
-------------------
&
-------------------------
**Applicability range**
**and Parameters**
-------------------------
\
---------------
5GCM InH
Indoor-Office
LOS
---------------
&
----------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 17.3 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
----------------------------------------------------------
&
----------------------
$\sigma_{SF} = 3.02$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
--------------------
5GCM InH
Indoor-Office
NLOS
single slope (FFS)
--------------------
&
-------------------------------------------------------------------------------------------
**CIF model:**
$PL = 32.4 + 31.9 (1+ 0.06(\frac{f_c-24.2 }{24.2}))\log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
**ABG model:**
$PL = 38.3\log_{10}(d_{3D}) + 17.30 + 24.9\log_{10}(f_c)$
-------------------------------------------------------------------------------------------
&
[@c@]{}\
$\sigma^{CIF}_{SF} = 8.29$\
\
$\sigma^{ABG}_{SF} = 8.03$
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
---------------
5GCM InH
Indoor-Office
NLOS
dual slope
---------------
&
--------------------------------------------------------------------------------------------------------
**Dual-Slope CIF model:**
$PL_{Dual}^{CIF}(d) =\begin{cases}
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right)\log_{10}(d) , & 1<d \leq d_{BP}\\
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right)\log_{10}(d_{BP})\\
+ 10n_2\left( 1+ b_2\left( \frac{f_c-f_0}{f_0} \right) \right) \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}$
**Dual-Slope ABG model:**
$PL_{Dual}^{ABG}(d) =\begin{cases}
\alpha_1\cdot 10\log_{10} (d) + \beta_1 \\
+ \gamma \cdot 10\log_{10}(f_c) , & 1<d \leq d_{BP}\\
\alpha_1\cdot 10\log_{10} (d_{BP}) + \beta_1\\
+ \gamma \cdot 10\log_{10}(f_c)\\
+\alpha_2 \cdot 10 \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}$
--------------------------------------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF}^{CIF} = 7.65$\
\
$\sigma_{SF}^{ABG} = 7.78$
&
------------------------------------------
$6 < f_c < 100 \;\text{GHz}$
**Dual-Slope CIF model:**
$n_1 = 2.51, b=0.06$
$f_0 = 24.1 \;\text{GHz}, n_2 =4.25 $
$b_2 =0.04 , d_{BP} = 7.8 \:\text{m}$
**Dual-Slope ABG model:**
$\alpha_1 =1.7, \beta_1 = 33.0$
$\gamma = 2.49, d_{BP} = 6.9 \;\text{m}$
$\alpha_2 = 4.17$
------------------------------------------
\
---------------
5GCM InH
Shopping-Mall
LOS
---------------
&
----------------------------------------------------------
**CI model with 1 m reference distance:**
$PL = 32.4 + 17.3 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
----------------------------------------------------------
&
----------------------
$\sigma_{SF} = 2.01$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
--------------------
5GCM InH
Shopping-Mall
NLOS
single slope (FFS)
--------------------
&
--------------------------------------------------------------------------------------------
**CIF model:**
$PL = 32.4 + 25.9 (1+ 0.01(\frac{f_c-39.5 }{39.5 }))\log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
**ABG model:**
$PL = 32.1\log_{10}(d_{3D}) + 18.09 + 22.4\log_{10}(f_c)$
--------------------------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF}^{CIF} = 7.40$\
\
$\sigma_{SF}^{ABG} = 6.97$
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
---------------
5GCM InH
Shopping-Mall
NLOS
dual slope
---------------
&
--------------------------------------------------------------------------------------------------------
**Dual-Slope CIF model:**
$PL_{Dual}^{CIF}(d) =\begin{cases}
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right)\log_{10}(d ) , & 1<d \leq d_{BP}\\
FSPL(f_c , 1\;\text{m}) \\
+ 10n_1\left( 1+b_1\left( \frac{f_c-f_0}{f_0}\right) \right)\log_{10}(d_{BP})\\
+ 10n_2\left( 1+ b_2\left( \frac{f_c-f_0}{f_0} \right) \right) \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}$
**Dual-Slope ABG model:**
$PL_{Dual}^{ABG}(d) =\begin{cases}
\alpha_1\cdot 10\log_{10} (d) + \beta_1 \\
+ \gamma \cdot 10\log_{10}(f_c) , & 1<d \leq d_{BP}\\
\alpha_1\cdot 10\log_{10} (d_{BP}) + \beta_1\\
+ \gamma \cdot 10\log_{10}(f_c)\\
+\alpha_2 \cdot 10 \log_{10}(\frac{d}{d_{BP}}), & d>d_{BP}
\end{cases}$
--------------------------------------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF}^{CIF} = 6.26$\
\
$\sigma_{SF}^{ABG} = 6.36$
&
--------------------------------------------
$6 < f_c < 100 \;\text{GHz}$
**Dual-Slope CIF model:**
$n_1 = 2.43, b=-0.01$
$f_0 = 39.5 \;\text{GHz}, n_2 = 8.36 $
$b_2 =0.39 , d_{BP} = 110 \:\text{m}$
**Dual-Slope ABG model:**
$\alpha_1 =2.9, \beta_1 = 22.17$
$\gamma = 2.24, d_{BP} = 147.0 \;\text{m}$
$\alpha_2 = 11.47$
--------------------------------------------
\
[|c|c|c|c|]{}
--
--
&
--------------------------------------------------
**PL \[dB\], $f_c$ is in GHz, $d$ is in meters**
--------------------------------------------------
&
-------------------
**Shadow fading**
**std \[dB\]**
-------------------
&
-------------------------
**Applicability range**
**and Parameters**
-------------------------
\
\
-------------------
3GPP TR 38.901
Indoor-Office LOS
-------------------
&
--------------------------------------------------------------------
$PL_{InH-LOS} = 32.4 + 17.3 \log_{10}(d_{3D}) + 20 \log_{10}(f_c)$
--------------------------------------------------------------------
&
---------------------
$\sigma_{SF} = 3.0$
---------------------
&
---------------------------------
$0.5 < f_c < 100 \;\text{GHz}$
$1 < d_{3D} < 100 \;\text{m} $
---------------------------------
\
--------------------
3GPP TR 38.901
Indoor-Office NLOS
--------------------
&
----------------------------------------------------------------------------
$PL = \max{ \left( PL_{InH-LOS}(d_{3D}), PL_{InH-NLOS}(d_{3D}) \right) } $
$PL_{InH-NLOS} = 17.30+ 38.3\log_{10}(d_{3D}) + 24.9 \log_{10}(f_c)$
**Option: CI model with 1 m reference distance**
$PL = 32.4 + 20\log_{10}(f_c)+31.9 \log_{10}(d_{3D})$
----------------------------------------------------------------------------
&
[@c@]{}\
$\sigma_{SF} = 8.03$\
\
$\sigma_{SF} = 8.29$
&
[@c@]{}$0.5 < f_c < 100 \;\text{GHz}$\
$ 1 < d_{3D} < 86 \;\text{m}$\
\
$ 1 < d_{3D} < 86 \;\text{m}$
\
\
-------------------
METIS
Shopping Mall LOS
-------------------
&
--------------------------------------
$PL = 68.8 + 18.4 \log_{10}(d_{2D})$
--------------------------------------
&
---------------------
$\sigma_{SF} = 2.0$
---------------------
&
------------------------------------
$ f_c =63 \;\text{GHz}$
$1.5 < d_{2D} < 13.4 \;\text{m} $
$h_{BS}=h_{UE}=2 \;\text{m}$
------------------------------------
\
--------------------
METIS
Shopping Mall NLOS
--------------------
&
--------------------------------------
$PL = 94.3 + 3.59 \log_{10}(d_{2D})$
--------------------------------------
&
---------------------
$\sigma_{SF} = 2.0$
---------------------
&
----------------------------------
$ f_c =63 \;\text{GHz}$
$4 < d_{2D} < 16.1 \;\text{m} $
$h_{BS}=h_{UE}=2 \;\text{m}$
----------------------------------
\
\
-------------------
802.11ad
Indoor-Office LOS
-------------------
&
-----------------------------------------------------------------
$PL_{LOS}[dB] = 32.5 + 20 \log_{10}(f_c)+ 20 \log_{10}(d_{2D})$
-----------------------------------------------------------------
&
----------------
$\sigma_{SF} $
----------------
&
-------------------------------
$ 57 < f_c < 63 \;\text{GHz}$
-------------------------------
\
--------------------
802.11ad
Indoor-Office NLOS
--------------------
&
------------------------------------------------------------------
$PL_{NLOS}[dB] = 51.5 + 20 \log_{10}(f_c)+ 6 \log_{10}(d_{2D})$
$PL_{NLOS}[dB] = 45.5 + 20 \log_{10}(f_c)+ 14 \log_{10}(d_{3D})$
------------------------------------------------------------------
&
------------------------------
$\sigma_{SF}^{STA-STA}=3.3 $
$\sigma_{SF}^{STA-AP} = 3$
------------------------------
&
-------------------------------
$ 57 < f_c < 63 \;\text{GHz}$
-------------------------------
\
\
-------------
mmMAGIC InH
LOS
-------------
&
-------------------------------------------------------------------
$PL_{LOS} = 13.8 \log_{10}(d_{3D}) + 33.6 + 20.3 \log_{10}(f_c)$
-------------------------------------------------------------------
&
----------------------
$\sigma_{SF} = 1.18$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
-------------
mmMAGIC InH
NLOS
-------------
&
--------------------------------------------------------------------
$PL = \max{ \left( PL_{LOS}(d_{3D}), PL_{NLOS}(d_{3D}) \right) } $
$PL_{NLOS} = 36.9 \log_{10}(d_{3D}) + 15.2 + 26.8 \log_{10}(f_c)$
--------------------------------------------------------------------
&
----------------------
$\sigma_{SF} = 8.03$
----------------------
&
------------------------------
$6 < f_c < 100 \;\text{GHz}$
------------------------------
\
#### 5GCM
In the InH scenario, besides the CI, CIF, and ABG path loss models, dual-slope path loss models are proposed for different distance zones in the propagation environment and are provided in Table \[tbl:InH5GCMPL\]. For NLOS, both the dual-slope ABG and dual-slope CIF models are considered for 5G performance evaluation, where they each require five modeling parameters to be optimized. Also, a single-slope CIF model that uses only two optimization parameters is considered for InH-Office [@5GCM; @maccartney2015indoor]. The dual-slope model may be best suited for InH-shopping mall or large indoor distances (greater than 50 m), although it is not clear from the data in [@5GCM] that the additional complexity is warranted when compared to the simple CIF model.
#### 3GPP TR 38.901 {#gpp-tr-38.901-4}
The path loss model for the InH-office LOS scenario in 3GPP TR 38.901 [@3GPP2017] is claimed to be valid up to 100 m and has the same form as the CI model in the UMi scenario. The only differences from UMi CI model are that the PLE in InH-office is slightly lower than that in the UMi street canyon due to more reflections and scattering in the indoor environment from walls and ceilings and waveguiding effects down hallways that increase received signal power [@maccartney2015indoor].
The 3GPP TR 38.901 InH-office NLOS path loss model uses the ABG model form similar to its UMi NLOS path loss model, except that there is no height correction term, and the model requires a patch to ensure it is lower-bounded by the LOS path loss as follows: $$\footnotesize
PL~\text{[dB]}= \max{ \left( PL_{InH-LOS}(d_{3D}), PL_{InH-NLOS}(d_{3D}) \right) }$$ $$\footnotesize
\label{equ:3GPPInHN}
PL_{InH-NLOS}~\text{[dB]} = 17.30 + 38.3\log_{10}(d_{3D}) + 24.9\log_{10}(f_c)$$
#### METIS {#metis-3}
In the latest METIS white paper [@METIS2015], the WINNER II path loss model (similar in form to the ABG model) was adopted as the geometry-based stochastic model for short-range 60 GHz (61-65 GHz) links in indoor environments: $$\label{equ:METISInHPL}
\footnotesize
PL~\text{[dB]}= A \log_{10}(d) + B$$ where $A$ and $B$ are curve-fit parameters without the use of Friis’ equation [@friis1946note] (see Table \[tbl:InHPL\] for parameters).
#### mmMAGIC
The InH channel model in mmMAGIC [@mmMAGIC] is adopted from an earlier version of 5GCM [@5GCM], and has the same form as the ABG model. For Indoor-NLOS, the values of the path loss model parameters have been averaged from InH and InH-Shopping Mall.
#### IEEE 802.11ad
In the STA-STA (STA signifies a station, the WiFi term for the UE) LOS scenario [@802.11ad], path loss follows theoretical free space path loss in the CI model form via the Friis’ free space transmission equation as given in Table \[tbl:InHPL\]. No shadowing term is provided in the LOS case, as instantaneous realizations are claimed to be close to the average path loss value over such wideband channel bandwidth. Experiments performed for NLOS situations resulted in path loss for STA-STA as a FI/AB model [@rappaport2015wideband] with the shadow fading standard deviation as $\sigma_{SF}=3.3$ dB. The 2D distance $d_{2D}$ is used for the STA-STA scenario, since it is considered that two stations are deemed to be at the same height above ground.
In the STA-AP (where the AP denotes access point, corresponding to a BS) scenario, the 3D separation distance $d_{3D}$ is used, and the LOS STA-AP path loss model is the same CI model as used in the STA-STA situation but no specific shadow fading term is given. The NLOS STA-AP model takes the same ABG form as that of STA-STA, but with $A_{NLOS} = 45.5$ dB and a shadow fading standard deviation $\sigma_{SF} = 3.0$ dB.
### RMa Large-Scale Path Loss
#### 3GPP TR 38.901 {#gpp-tr-38.901-5}
The 3GPP TR 38.901 RMa path loss model [@3GPP2017] is mostly adopted from sub-6 GHz ITU-R M.2135 [@ITU-RM.2135] as described below, and claims validity up to 30 GHz, based on a single 24 GHz measurement campaign over short distances less than 500 m and without any goodness of fit indication [@TDOC164975]. Work in [@Mac16c; @JSAC] advocates a much more fundamental and accurate RMa model using the CIF model formulation in , where the frequency dependency of the PLE is replaced with a TX height dependency of the PLE, based on many propagation studies that showed UMa and RMa environment did not offer additional frequency dependency of the path loss over distance beyond the first meter of propagation [@Sun16b; @Mac16c; @JSAC; @sun2016propagation].
#### ITU-R
The ITU-R communication sector published guidelines for the evaluation of radio interface technologies for IMT-Advanced in ITU-R M.2135 which is valid for sub-6 GHz [@ITU-RM.2135]. The rural scenario is best described as having BS heights of 35 m or higher, generally much higher than surrounding buildings. The LOS path loss model has a controversial breakpoint distance [@JSAC] and a maximum 2D T-R separation distance of 10 km, while the NLOS path loss model has a maximum 2D T-R separation distance of 5 km with no breakpoint distance. Initial antenna height default values are provided in Table \[tbl:ITURMAappRange\], with the following four correction factor parameters: street width $W$, building height $h$, BS height $h_{BS}$, and UE height $h_{UE}$ (all in meters).
The ITU-R RMa LOS path loss model is quite complex: $$\begin{aligned}
\label{eq:RMaLOS}
\footnotesize
\begin{split}
PL _1~\text{[dB]}& = 20\log(40\pi \cdot d_{3D} \cdot f_c /3)\\
&+\min(0.03h^{1.72},10)\log_{10}(d_{3D}) \\
&-\min(0.044h^{1.72},14.77)+0.002\log_{10}(h)d_{3D}\\
PL_2~\text{[dB]} & = PL_1 (d_{BP})+40\log_{10}(d_{3D}/d_{BP})
\end{split}\end{aligned}$$ where the breakpoint distance $d_{BP}$ is: $$\label{eq:dbp}
\footnotesize
d_{BP} = 2\pi \cdot h_{BS} \cdot h_{UE} \cdot f_c/c$$
It is must be noted that the model reverts to a single-slope model at 9.1 GHz or above, since the breakpoint distance exceeds 10 km (the outer limit of model applicability), thus making the LOS model mathematically inconsistent for mmWave frequencies above 9.1 GHz [@Mac16c; @JSAC].
The NLOS RMa path loss model in is adopted from ITU-R M.2135 and has nine empirical coefficients for various building height and street width parameters [@3GPP2017; @ITU-RM.2135]: $$\begin{aligned}
\label{eq:RMaNLOS}
\footnotesize
\begin{split}
PL~\text{[dB]}& = \max(PL_{RMa-LOS},PL_{RMa-NLOS})\\
PL &_{RMa-NLOS}~\text{[dB]}= 161.04-7.1\log_{10}(W)+7.5\log_{10}(h)\\
&-(24.37-3.7(h/h_{BS})^2)\log_{10}(h_{BS})\\
&+(43.42-3.1\log_{10}(h_{BS}))(\log_{10}(d_{3D})-3)\\
&+20\log_{10}(f_c)-(3.2(\log_{10}(11.75h_{UE}))^2-4.97)
\end{split}\end{aligned}$$
The ITU-R RMa NLOS path loss model from which the 3GPP TR38.901 model is adopted is only specified for frequencies up to 6 GHz and has not been validated in the literature for mmWave frequencies. The ITU-R RMa models were not developed using rural scenarios [@Mac16c; @JSAC], but instead were derived from measurements in downtown Tokyo, making them ill-suited for the RMa case.
#### NYU RMa model
NYU proposed empirically-based CIH RMa path loss models for LOS ($PL_{LOS}^{CIH-RMa}$) and NLOS ($PL_{NLOS}^{CIH-RMa}$) from extensive simulations and 73 GHz field data [@JSAC]:
$$\begin{aligned}
\footnotesize
\begin{split}
PL&_{LOS}^{CIH-RMa}(f_c,d,h_{BS})~\text{[dB]}=32.4+20\log_{10}(f_c)\\
&+23.1\left( 1-0.03\left( \dfrac{h_{BS}-35}{35}\right) \right) \log_{10}(d)+\chi_{\sigma_{LOS}}\\
\end{split}\end{aligned}$$
where $d \geq 1$ , $\sigma_{LOS}=1.7$ , and $10 \text{m} \leq h_{BS} \leq 150~\text{m}$. $$\begin{aligned}
\footnotesize
\begin{split}
PL&_{NLOS}^{CIH-RMa}(f_c,d,h_{BS})~\text{[dB]}=32.4+20\log_{10}(f_c)\\
&+30.7\left( 1-0.049\left( \dfrac{h_{BS}-35}{35}\right) \right) \log_{10}(d)+\chi_{\sigma_{NLOS}}
\end{split}\end{aligned}$$ where $d \geq 1 \text{m}$, $\sigma_{LOS}=6.7~\text{dB}$, and $10 \text{m} \leq h_{BS} \leq 150~\text{m}$.
O2I Penetration Loss
--------------------
### 3GPP TR 38.901 {#gpp-tr-38.901-6}
The overall large-scale path loss models may also account for penetration loss into a building and subsequent path loss inside the building. The O2I path loss model taking account of the building penetration loss according to 3GPP TR 38.901 [@3GPP2017] has the following form: $$\label{equ:O2I}
\footnotesize
PL~\text{[dB]}= PL_b + PL_{tw} + PL_{in} + N(0,\sigma_P^2)$$ where $PL_b$ is the basic outdoor path loss, $PL_{tw}$ is the building penetration loss through the external wall, $PL_{in}$ is the indoor loss which depends on the depth into the building, and $\sigma_P$ is the standard deviation for the penetration loss. The building penetration loss $PL_{tw}$ can be modeled as: $$\footnotesize
PL_{tw}~\text{[dB]} = PL_{npi} - 10\log_{10}\sum_{i=1}^{N}\left( p_i \times 10^{\frac{L_{\text{material}_i}}{-10}} \right)$$ where $PL_{npi}$ is an additional loss which is added to the external wall loss to account for non-perpendicular incidence, $L_{\text{material}_i} =a_{\text{material}_i} +b_{\text{material}_i}\cdot f_c $ is the penetration loss of material $i$, $f_c$ is the frequency in GHz, $p_i$ is the proportion of $i$-th materials, where $\sum p_i =1$, and $N$ is the number of materials. Penetration loss of several materials and the O2I penetration loss models are given in Table \[tbl:O2Imat\].
[|c|c|]{}
--------------
**Material**
--------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
&
----------------------------------------------
**Penetration loss \[dB\], $f_c$ is in GHz**
----------------------------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
\
---------------------------
Standard multi-pane glass
---------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
&
--------------------------------------
$L_{\text{glass}}= 2+ 0.2 \cdot f_c$
--------------------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
\
-----------
IRR glass
-----------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
&
-------------------------------------------
$L_{\text{IRRglass}}= 23 + 0.3 \cdot f_c$
-------------------------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
\
----------
Concrete
----------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
&
----------------------------------------
$L_{\text{concrete}}= 5 + 4 \cdot f_c$
----------------------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
\
------
Wood
------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
&
------------------------------------------
$L_{\text{wood}}= 4.85 + 0.12 \cdot f_c$
------------------------------------------
: O2I penetration loss of different materials [@3GPP2017][]{data-label="tbl:O2Imat"}
\
Rough models are also provided to estimate the building penetration loss in Table \[tab:O2Ipeneloss\]. Both the low-loss and high-loss models are applicable to UMa and UMi-street canyon, while only the low-loss model is applicable to RMa. The O2I car penetration loss included in path loss is determined by:
$$\footnotesize
\label{equ:car}
PL~\text{[dB]}= PL_b + N(\mu,\sigma_P^2)$$
where $PL_b$ is the basic outdoor path loss, and for most cases, $\mu=9 \;\text{dB}$ and $\sigma_P = 5 \;\text{dB}$. An optional $\mu= 20 \;\text{dB}$ is provided for metalized car windows for frequencies ranging from 0.6 to 60 GHz [@3GPP2017].
[|c|c|c|c|]{}
--
--
&
---------------------------------------
**Path loss through external wall:**
$PL_{tw}$ **\[dB\], $f_c$ is in GHz**
---------------------------------------
&
--------------------------------------
**Indoor loss:**
$PL_{in}$ **\[dB\], d is in meters**
--------------------------------------
&
-------------------------
**Standard deviation:**
$\sigma_P$ **\[dB\]**
-------------------------
\
-------------------------------------------
3GPP TR 38.901 Low-loss model [@3GPP2017]
-------------------------------------------
&
-----------------------------------------------------------------------------
$5-10\log_{10}(0.3\cdot 10^{-L_{glass}/10}+0.7\cdot 10^{-L_{concrete}/10})$
-----------------------------------------------------------------------------
&
----------------
$0.5d_{2D-in}$
----------------
&
-----
4.4
-----
\
--------------------------------------------
3GPP TR 38.901 High-loss model [@3GPP2017]
--------------------------------------------
&
--------------------------------------------------------------------------------
$5-10\log_{10}(0.7\cdot 10^{-L_{IRRglass}/10}+0.3\cdot 10^{-L_{concrete}/10})$
--------------------------------------------------------------------------------
&
----------------
$0.5d_{2D-in}$
----------------
&
-----
6.5
-----
\
-----------------------------------------------
5GCM Low-loss model [@5GCM; @rappaport20175g]
-----------------------------------------------
&
---------------------------------------
$ 10 \log_{10}(5 + 0.03 \cdot f_c^2)$
---------------------------------------
&
---------------
Not Specified
---------------
&
-----
4.0
-----
\
------------------------------------------------
5GCM High-loss model [@5GCM; @rappaport20175g]
------------------------------------------------
&
-------------------------------------
$ 10 \log_{10}(10 + 5 \cdot f_c^2)$
-------------------------------------
&
---------------
Not Specified
---------------
&
-----
6.0
-----
\
### 5GCM
The 5GCM adopted the building penetration loss model of 3GPP TR 36.873 which is based on legacy measurements below 6 GHz [@3GPP2014]. Several different frequency-dependent models were also proposed in [@haneda20165g; @5GCM]. In [@semaan2014outdoor], a detailed description of external wall penetration loss using a composite approach is provided. The difference of the building penetration loss model between 5GCM and 3GPP TR 38.901 is that the standard deviation is tentatively selected from the measurement data [@haneda20165g; @semaan2014outdoor]. A very simple parabolic model with a good fit for predicting building penetration loss (BPL) of either high loss or low loss buildings was provided in [@haneda20165g; @rappaport20175g] as: $$\label{equ:nyuO2I}
\footnotesize
\begin{split}
BPL~\text{[dB]} = 10 \log_{10}(A + B \cdot f_c^2)
\end{split}$$ where $f_c$ is in GHz, $A=5$, and $B=0.03$ for low loss buildings and $A=10$ and $B= 5$ for high loss buildings.
### mmMAGIC
The O2I penetration loss model in mmMAGIC has the form of: $$\label{equ:mmO2I}
\footnotesize
\begin{split}
O2I~\text{[dB]} = B_{O2I} + C_{O2I} \cdot \log_{10}\left( f_c \right) \approx 8.5 + 11.2\cdot \log_{10} \left( f_c \right)
\end{split}$$ The advantage of this form is that the coefficients $B_{O2I}$ and $C_{O2I}$ can be added to the existing coefficients in the path loss model of mmMAGIC. A frequency-dependent shadow fading between 8 and 10 dB for the UMi-O2I scenario is presented in [@mmMAGIC]: $$\label{equ:mmO2Isf}
\footnotesize
\begin{split}
\Sigma_{SF}~\text{[dB]} = \sigma_{SF} + \delta_{SF}\cdot \log_{10}\left( f_c \right) \approx 5.7 + 2.3\cdot \log_{10} \left( f_c \right)
\end{split}$$
Spatial consistency {#spatial}
-------------------
Many previous channel models were “drop-based”, where a UE is placed at a random location, random channel parameters (conditioned on this location) are assigned, performance is computed (possibly when moving over a short distance, up to 40 wavelengths), and then a different location is chosen at random. This approach is useful for statistical or monte-carlo performance analysis, but does not provide spatial consistency, i.e., two UEs that are dropped at nearly identical T-R separation distances might experience completely different channels from a system simulator. The importance of spatial consistency is dependent upon the site-specific propagation in a particular location as shown in [@rumney2016testing2; @rap2016ap]. Channel models of 5GCM [@5GCM], 3GPP TR 38.901 [@3GPP2017], METIS [@METIS2015] and MiWEBA [@MiWEBA] provide new approaches for modeling of trajectories to retain spatial consistency.
In 5GCM and 3GPP, both the LOS/NLOS state and the shadowing states are generated on a coarse grid, and spatially filtered. This resulting “map” of LOS states and shadowing attenuations are then used for the trajectories of all UEs during the simulation process. For the implementation of the LOS state filtering, different methods are proposed [@5GCM; @3GPP2017], but the effect is essentially the same. We note that 5GCM and 3GPP also introduce additional procedures to ensure spatial consistencies of the delay and angles, but those considerations are beyond the scope of this paper. The map-based models of METIS [@METIS2015] and MiWEBA [@MiWEBA] inherently provide spatial consistency, as the dominant paths for close-by locations are identical, and their effect is computed deterministically. Generally speaking, spatial consistency is easier to implement in geometry-based models (such as semi-deterministic and geometric-based stochastic channel models) than in tapped-delay line models such as 3GPP. Work in [@rumney2016testing2; @mmMAGIC; @5GCM; @samimi20163; @rap2016ap] shows that the degree of spatial consistency can vary widely at mmWave frequencies.
Conclusion
==========
\[sec:conc\] Often times, standard bodies have additional reasons to adopt particular modeling formulations, beyond physical laws or the fitting of data to observed channel characteristics. Motivations often include ensuring simulations work for legacy software at lower frequencies, or the desire to rapidly converge while preserving legacy approaches (see [@sun2017a; @Sun16b; @maccartney2017study; @JSAC] for example). Channel modeling for 5G is an on-going process and early results show significant capacity differences arise from different models [@rappaport20175g; @sun2017a; @rappaport2017VTC]. Futher work is needed to bolster and validate the early channel models. Many new mmWave channel simulators (e.g., NYUSIM, QuaDRiGa) have been developed and are being used by researchers to evaluate the performance of communication systems and to simulate channel characteristics when designing air interfaces or new wireless technologies across the network stack [@sun2017a; @rajendran2011concepts; @jaeckel2014quadriga; @NYUSIM].
This paper has provided a comprehensive overview of emerging 5G mmWave wireless system concepts, and has provided a compilation of important mmWave radio propagation models developed throughout the world to date. The paper demonstrates early standards work and illustrates the various models obtained by several independent groups based on extensive measurements and ray tracing methods at mmWave frequency bands in various scenarios.
The development of proper propagation models is vital, not only for the long-term development of future mmWave wireless systems but also for fundamental understanding by future engineers and students who will learn about and improve the nascent mmWave mobile industry that is just now being developed. Various companies have started 5G field trials, and some of them have achieved 20 Gbps date rates [@HUAWEI; @ERICSSON]. The fundamental information on path loss and shadowing surveyed in this paper is a prerequisite for moving further along the road to 5G at the unprecedented mmWave frequency bands.
[^1]: T. S. Rappaport (email: [email protected]), Y. Xing (email: [email protected]), G. R. MacCartney , Jr. (email: [email protected]), are with NYU WIRELESS Research Center, and are supported in part by the NYU WIRELESS Industrial Affiliates: AT&T, CableLabs, Cablevision, Ericsson, Huawei, Intel Corporation, InterDigital Inc., Keysight Technologies, L3 Communications, Nokia, National Instruments, Qualcomm Technologies, SiBeam, Straight Path Communications, OPPO, Sprint, Verizon and UMC, in part by the GAANN Fellowship Program, and in part by the National Science Foundation under Grant 1320472, Grant 1237821, and Grant 1302336. NYU Tandon School of Engineering, 9th Floor, 2 MetroTech Center, Brooklyn, NY 11201.
[^2]: A. F. Molisch (email: [email protected]), is with the Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089. His work is supported by the National Science Foundation and Samsung.
[^3]: E. Mellios (email: [email protected] ), is with the Communication Systems & Networks Group, University of Bristol, Merchant Venturers Building, Woodland Road, BS8 1UB, Bristol, United Kingdom
[^4]: J. Zhang (email:[email protected]), is with State Key Lab of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Mailbox NO.92, 100876.
[^5]: The authors thank Shu Sun of NYU for her suggestions on this paper.
[^6]: The CIH model has the same form as except the PLE is a function of the BS height in the RMa scenario instead of frequency, as given by: $PL^{CIH} (f_c,d,h_{BS})~\text{[dB]} = 32.4+20\log_{10}(f_c)+10n\left(1+b_{tx}\left(\frac{h_{BS}-h_{B0}}{h_{B0}}\right)\right)\log_{10}(d)+\chi_{\sigma}, \text{where}~d\geq \text{1 m, and }h_{B0}$ is a reference RMa BS height [@JSAC].
|
---
abstract: |
Assume that $\F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\F$ that is not a root of unity. The universal DAHA (double affine Hecke algebra) $\H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\F$-algebra defined by generators and relations. The generators are $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations assert that $$\begin{gathered}
t_it_i^{-1}=t_i^{-1} t_i=1
\quad
\hbox{for all $i=0,1,2,3$};
\\
\hbox{$t_i+t_i^{-1}$ is central}
\quad
\hbox{for all $i=0,1,2,3$};
\\
t_0t_1t_2t_3=q^{-1}.\end{gathered}$$ In this paper we describe the finite-dimensional irreducible $\H_q$-modules from many viewpoints and classify the finite-dimensional irreducible $\H_q$-modules up to isomorphism. The proofs are carried out in the language of linear algebra.
address: |
Hau-Wen Huang\
Department of Mathematics\
National Central University\
Chung-Li 32001 Taiwan
author:
- 'Hau-Wen Huang'
bibliography:
- 'MP.bib'
title: |
Finite-dimensional irreducible modules of the\
universal DAHA of type $(C_1^\vee,C_1)$
---
[ double affine Hecke algebras, irreducible modules, universal property.]{}
[. 33D45, 33D80.]{}
Introduction
============
In the nineties, the double affine Hecke algebra (DAHA) was introduced by Cherednik in connection with quantum affine Knizhnik–Zamolodchikov equations and Macdonald eigenvalue problems [@DAHA_book; @Cherednik:1992; @Cherednik:1995]. Since that time, DAHAs and their representations have been explored in many other areas such as algebraic geometry [@DAHA:2003; @DAHA:2005; @DAHA:2009; @Oblomkov2009], combinatorics [@Tanaka:2018; @Lee:2017], integrable systems and orthogonal polynomials .
More than 30 years ago, a classification of irreducible modules of affine Hecke algebras was already given in [@AHA87]. Despite this, the classification of finite-dimensional irreducible modules of DAHAs is still open up to now. We are here concerned with the representation theory of DAHA of type $(C_1^\vee,C_1)$. Throughout this paper, let $\F$ denote a field and let $q$ be a nonzero scalar in $\F$. The DAHA of type $(C_1^\vee,C_1)$ is defined as follows:
Given nonzero scalars $k_0,k_1,k_2,k_3\in \F$ the [*DAHA $\H_q(k_0,k_1,k_2,k_3)$ of type $(C_1^\vee,C_1)$*]{} is a unital associative $\F$-algebra generated by $t_0,t_1,t_2,t_3$ subject to the following relations: $$\begin{gathered}
(t_i-k_i)(t_i-k_i^{-1})=0
\quad \hbox{for all $i=0,1,2,3$};
\\
t_0 t_1 t_2 t_3 =q^{-1}.\end{gathered}$$
We restrict our attention to the case when $\F$ is algebraically closed and $q$ is not a root of unity. Relying on the Crawley-Boevey’s results on the Deligne–Simpson problem [@CrawleyBoevey2004], the finite-dimensional irreducible $\H_q(k_0,k_1,k_2,k_3)$-modules was first classified in [@Oblomkov2009]. In this paper, we consider a central extension of DAHA of type $(C_1^\vee,C_1)$ defined as follows:
\[Definition 3.1, [@DAHA2013]\] \[defn:DAHA\] The [*universal DAHA $\H_q$ of type $(C_1^\vee,C_1)$*]{} is a unital associative $\F$-algebra defined by generators and relations. The generators are $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations assert that $$\begin{gathered}
t_it_i^{-1}=t_i^{-1} t_i=1
\quad
\hbox{for all $i=0,1,2,3$};
\\
\hbox{$t_i+t_i^{-1}$ is central}
\quad
\hbox{for all $i=0,1,2,3$};
\\
t_0t_1t_2t_3=q^{-1}.
\label{t0t1t2t3}\end{gathered}$$
In this paper, we describe the finite-dimensional irreducible $\H_q$-modules from many viewpoints and classify the finite-dimensional irreducible $\H_q$-modules without the aid of the Crawley-Boevey’s results. Our proofs mainly use elementary linear algebra techniques. Inspired by [@koo07; @koo08], it was shown that the universal Askey–Wilson algebra $\triangle_q$ is an $\F$-subalgebra of $\H_q$ [@DAHA2013]. In [@Huang:2015], it was given a classification of finite-dimensional irreducible $\triangle_q$-modules. As an application, the present author classifies the lattices of $\triangle_q$-submodules of finite-dimensional irreducible $\H_q$-modules .
The outline of this paper is as follows. Given appropriate parameters $k_0,k_1,k_2,k_3$ we construct an even-dimensional $\H_q$-module $E(k_0,k_1,k_2,k_3)$ and describe how $\{t_i\}_{i=0}^3$ act on a basis of $E(k_0,k_1,k_2,k_3)$. In Theorem \[thm:iso\], we claim that if $E(k_0,k_1,k_2,k_3)$ is irreducible, then it is isomorphic to $E(k_0,k_1^{\pm 1},k_2^{\pm 1},k_3^{\pm 1})$. In Theorem \[thm:even\], we state a classification of even-dimensional $\H_q$-modules via the $\H_q$-modules $E(k_0,k_1,k_2,k_3)$. In Theorems \[thm:iso\_O\] and \[thm:odd\] we state similar results on odd-dimensional $\H_q$-modules. See §\[s:result\] for details. In §\[s:Verma\] we display an infinite-dimensional $\H_q$-module $M(k_0,k_1,k_2,k_3)$ and establish its universal property. Motivated by the universal property, we discuss the existence of the simultaneous eigenvectors of $t_0$ and $t_3$ in finite-dimensional irreducible $\H_q$-modules in §\[s:eigenvector\]. Subsequently we show that $E(k_0,k_1,k_2,k_3)$ is a quotient of $M(k_0,k_1,k_2,k_3)$. We give a necessary and sufficient condition for $E(k_0,k_1,k_2,k_3)$ as irreducible in terms of the parameters $k_0,k_1,k_2,k_3,q$. Moreover we give a proof for Theorem \[thm:iso\]. See §\[section:iso\] for details. In §\[section:even\] we show that any even-dimensional irreducible $\H_q$-module is isomorphic to the $\H_q$-module obtained by pulling back some $\H_q$-module $E(k_0,k_1,k_2,k_3)$ via an automorphism of $\H_q$ and this leads to a proof for Theorem \[thm:even\]. In §\[section:iso\_O\] and §\[section:odd\] we prove Theorems \[thm:iso\_O\] and \[thm:odd\], respectively.
Statement of results {#s:result}
====================
In this section we state our main results (Theorems \[thm:iso\], \[thm:even\], \[thm:iso\_O\] and \[thm:odd\]). In Propositions \[prop:E\] and \[prop:O\], given appropriate parameters $k_0,k_1,k_2,k_3$ we construct an even-dimensional $\H_q$-module $E(k_0,k_1,k_2,k_3)$ and an odd-dimensional $\H_q$-module $O(k_0,k_1,k_2,k_3)$. In Theorems \[thm:iso\] and \[thm:iso\_O\] we display several equivalent descriptions of the finite-dimensional $\H_q$-modules $E(k_0,k_1,k_2,k_3)$ and $O(k_0,k_1,k_2,k_3)$ when they are irreducible. In Theorems \[thm:even\] and \[thm:odd\] we give our classification of finite-dimensional irreducible $\H_q$-modules via the $\H_q$-modules $E(k_0,k_1,k_2,k_3)$ and $O(k_0,k_1,k_2,k_3)$, provided that $\F$ is algebraically closed and $q$ is not a root of unity. In §\[section:iso\]–§\[section:odd\] we will give our proofs for the main results.
Define $$\begin{aligned}
\label{ci}
c_i&=t_i+t_i^{-1}
\qquad
\hbox{for all $i=0,1,2,3$}.\end{aligned}$$ Recall from Definition \[defn:DAHA\] that $c_i$ is central in $\H_q$.
\[prop:E\] Let $d\geq 1$ denote an odd integer. Assume that $k_0,k_1,k_2,k_3$ are nonzero scalars in $\F$ with $$k_0^2=q^{-d-1}.$$ Then there exists a $(d+1)$-dimensional $\H_q$-module $E(k_0,k_1,k_2,k_3)$ satisfying the following conditions:
1. There exists an $\F$-basis $\{v_i\}_{i=0}^d$ for $E(k_0,k_1,k_2,k_3)$ such that $$\begin{aligned}
t_0 v_i
&=
\left\{
\begin{array}{ll}
\textstyle
k_0^{-1} q^{-i} (1-q^i) (1-k_0^2 q^i)
v_{i-1}
+
(
k_0+k_0^{-1}-k_0^{-1}q^{-i}
)
v_i
\qquad
&\hbox{for $i=2,4,\ldots,d-1$},
\\
k_0^{-1} q^{-i-1}
(v_i-v_{i+1})
\qquad
&\hbox{for $i=1,3,\ldots,d-2$},
\end{array}
\right.
\\
t_0 v_0&=k_0 v_0,
\qquad
t_0 v_d=k_0v_d,
\\
t_1 v_i
&=
\left\{
\begin{array}{ll}
-k_1(1-q^i)(1-k_0^2 q^i)v_{i-1}
+k_1 v_i
+k_1^{-1} v_{i+1}
\qquad
&\hbox{for $i=2,4,\ldots,d-1$},
\\
k_1^{-1} v_i
\qquad
&\hbox{for $i=1,3,\ldots,d$},
\end{array}
\right.
\\
t_1 v_0 &=k_1 v_0+k_1^{-1} v_1,
\\
t_2 v_i
&=
\left\{
\begin{array}{ll}
k_0^{-1} k_1^{-1} k_3^{-1} q^{-i-1}
(v_i-v_{i+1})
\qquad
&\hbox{for $i=0,2,\ldots,d-1$},
\\
\textstyle
\frac{(k_0 k_1 k_3 q^i-k_2)
(k_0 k_1 k_3 q^i- k_2^{-1})}
{k_0 k_1 k_3 q^i }
v_{i-1}
+
(k_2+k_2^{-1}-
k_0^{-1} k_1^{-1} k_3^{-1} q^{-i}
) v_i
\qquad
&\hbox{for $i=1,3,\ldots,d$},
\end{array}
\right.
\\
t_3 v_i
&=
\left\{
\begin{array}{ll}
k_3 v_i
\qquad
&\hbox{for $i=0,2,\ldots,d-1$},
\\
-k_3^{-1}(k_0k_1k_3q^i-k_2)
(k_0k_1k_3 q^i-k_2^{-1})
v_{i-1}
+k_3^{-1} v_i
+k_3 v_{i+1}
\qquad
&\hbox{for $i=1,3,\ldots,d-2$}.
\end{array}
\right.
\\
t_3 v_d &=
-k_3^{-1}(k_0k_1k_3q^d-k_2)
(k_0k_1k_3 q^d-k_2^{-1})
v_{d-1}
+k_3^{-1} v_d.\end{aligned}$$
2. The elements $c_0,c_1,c_2,c_3$ act on $E(k_0,k_1,k_2,k_3)$ as scalar multiplication by $$k_0+k_0^{-1},
\quad
k_1+k_1^{-1},
\quad
k_2+k_2^{-1},
\quad
k_3+k_3^{-1}$$ respectively.
It is routine to verify the proposition by using Definition \[defn:DAHA\].
The proof of the following result concerning the isomorphism class of the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is given in §\[section:iso\].
\[thm:iso\] Let $d\geq 1$ denote an odd integer. Assume that $k_0,k_1,k_2,k_3$ are nonzero scalars in $\F$ with $
k_0^2=q^{-d-1}$. If the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible then the following hold:
1. $E(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $E(k_0,k_1^{-1},k_2,k_3)$.
2. $E(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $E(k_0,k_1,k_2^{-1},k_3)$.
3. $E(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $E(k_0,k_1,k_2,k_3^{-1})$.
Let $V$ denote an $\H_q$-module. For any $\F$-algebra automorphism $\e$ of $\H_q$ the notation $$V^\e$$ stands for the $\H_q$-module obtained by pulling back the $\H_q$-module $V$ via $\e$. Let $\Z/4\Z$ denote the additive group of integers modulo $4$. Observe that there exists a unique $\Z/4\Z$-action on $\H_q$ such that each element of $\Z/4\Z$ acts on $\H_q$ as an $\F$-algebra automorphism in the following way:
$\e\in \Z/4\Z$ $t_0$ $t_1$ $t_2$ $t_3$
---------------- ------- ------- ------- -------
${0\pmod 4}$ $t_0$ $t_1$ $t_2$ $t_3$
${1\pmod 4}$ $t_1$ $t_2$ $t_3$ $t_0$
${2\pmod 4}$ $t_2$ $t_3$ $t_0$ $t_1$
${3\pmod 4}$ $t_3$ $t_0$ $t_1$ $t_2$
: The $\Z/4\Z$-action on $\H_q$[]{data-label="Z/4Z-action"}
The classification of even-dimensional irreducible $\H_q$-modules is stated as follows and the proof is given in §\[section:even\].
\[thm:even\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. Let $d\geq 1$ denote an odd integer. Let $\mathbf{EM}_d$ denote the set of all isomorphism classes of irreducible $\H_q$-modules that have dimension $d+1$. Let $\mathbf{EP}_d$ denote the set of all quadruples $(k_0,k_1,k_2,k_3)$ of nonzero scalars in $\F$ that satisfy $
k_0^2=q^{-d-1}$ and $$\begin{gathered}
k_0k_1k_2k_3, k_0k_1^{-1}k_2k_3, k_0k_1k_2^{-1}k_3, k_0k_1k_2k_3^{-1}
\not=
q^{-i}
\qquad
\hbox{for all $i=1,3,\ldots,d$}.\end{gathered}$$ Define an action of the abelian group $\{\pm 1\}^3$ on $\mathbf{EP}_d$ by $$\begin{aligned}
(k_0,k_1,k_2,k_3)^{(-1,1,1)} &= (k_0,k_1^{-1},k_2,k_3),
\\
(k_0,k_1,k_2,k_3)^{(1,-1,1)} &= (k_0,k_1,k_2^{-1},k_3),
\\
(k_0,k_1,k_2,k_3)^{(1,1,-1)} &= (k_0,k_1,k_2,k_3^{-1})\end{aligned}$$ for all $(k_0,k_1,k_2,k_3)\in \mathbf{EP}_d$. Let $\mathbf{EP}_d/\{\pm 1\}^3$ denote the set of the $\{\pm 1\}^3$-orbits of $\mathbf{EP}_d$. For $(k_0,k_1,k_2,k_3)\in \mathbf{EP}_d$ let $[k_0,k_1,k_2,k_3]$ denote the $\{\pm 1\}^3$-orbit of $\mathbf{EP}_d$ that contains $(k_0,k_1,k_2,k_3)$. Then there exists a bijection $\mathcal E:\Z/4\Z \times \mathbf{EP}_d/\{\pm 1\}^3 \to \mathbf{EM}_d$ given by $$\begin{aligned}
(\e,[k_0,k_1,k_2,k_3])
&\mapsto &
\hbox{the isomorphism class of $E(k_0,k_1,k_2,k_3)^\e$}\end{aligned}$$ for all $\e\in \Z/4\Z$ and all $[k_0,k_1,k_2,k_3]\in \mathbf{EP}_d/\{\pm 1\}^3$.
An odd-dimensional $\H_q$-module $O(k_0,k_1,k_2,k_3)$ with four parameters $k_0,k_1,k_2,k_3$ is built as follows:
\[prop:O\] Let $d\geq 0$ denote an even integer. Assume that $k_0,k_1,k_2,k_3$ are nonzero scalars in $\F$ with $$k_0 k_1 k_2 k_3=q^{-d-1}.$$ Then there exists a $(d+1)$-dimensional $\H_q$-module $O(k_0,k_1,k_2,k_3)$ satisfying the following conditions:
1. There exists an $\F$-basis $\{v_i\}_{i=0}^d$ for $O(k_0,k_1,k_2,k_3)$ such that $$\begin{aligned}
t_0 v_i
&=
\left\{
\begin{array}{ll}
\textstyle
k_0^{-1} q^{-i} (1-q^i) (1-k_0^2 q^i)
v_{i-1}
+
(
k_0+k_0^{-1}-k_0^{-1}q^{-i}
)
v_i
\qquad
&\hbox{for $i=2,4,\ldots,d$},
\\
\textstyle
k_0^{-1} q^{-i-1}
(v_i-v_{i+1})
\qquad
&\hbox{for $i=1,3,\ldots,d-1$},
\end{array}
\right.
\\
t_0 v_0&= k_0 v_0,
\\
t_1 v_i
&=
\left\{
\begin{array}{ll}
-k_1(1-q^i)(1-k_0^2 q^i)v_{i-1}
+k_1 v_i
+k_1^{-1} v_{i+1}
\qquad
&\hbox{for $i=2,4,\ldots,d-2$},
\\
k_1^{-1} v_i
\qquad
&\hbox{for $i=1,3,\ldots,d-1$},
\end{array}
\right.
\\
t_1 v_0&= k_1 v_0 +k_1^{-1} v_1,
\qquad
t_1 v_d=-k_1(1-q^d)(1-k_0^2 q^d)v_{d-1}
+k_1 v_d,
\\
t_2 v_i
&=
\left\{
\begin{array}{ll}
k_2 q^{d-i}
(v_i-v_{i+1})
\qquad
&\hbox{for $i=0,2,\ldots,d-2$},
\\
\textstyle
-k_2(1-k_2^{-2}q^{i-d-1})
(1-q^{d-i+1})
v_{i-1}
+
(k_2+k_2^{-1}-
k_2 q^{d-i+1}) v_i
\qquad
&\hbox{for $i=1,3,\ldots,d-1$},
\end{array}
\right.
\\
t_2 v_d &= k_2 v_d,
\\
t_3 v_i
&=
\left\{
\begin{array}{ll}
k_3 v_i
\qquad
&\hbox{for $i=0,2,\ldots,d$},
\\
\textstyle
-k_3^{-1}
(1-k_2^{-2} q^{i-d-1})
(1-q^{i-d-1})
v_{i-1}
+k_3^{-1} v_i
+k_3 v_{i+1}
\qquad
&\hbox{for $i=1,3,\ldots,d-1$}.
\end{array}
\right.\end{aligned}$$
2. The elements $c_0,c_1,c_2,c_3$ act on $O(k_0,k_1,k_2,k_3)$ as scalar multiplication by $$k_0+k_0^{-1},
\quad
k_1+k_1^{-1},
\quad
k_2+k_2^{-1},
\quad
k_3+k_3^{-1}$$ respectively.
It is routine to verify the proposition by using Definition \[defn:DAHA\].
The proof of the following result concerning the isomorphism class of the $\H_q$-module $O(k_0,k_1,k_2,k_3)$ is given in §\[section:iso\_O\].
\[thm:iso\_O\] Let $d\geq 0$ denote an even integer. Assume that $k_0,k_1,k_2,k_3$ are nonzero scalars in $\F$ with $
k_0 k_1 k_2 k_3=q^{-d-1}$. If the $\H_q$-module $O(k_0,k_1,k_2,k_3)$ is irreducible then the following hold:
1. $O(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $O(k_1,k_2,k_3,k_0)^\e$ where $\e=3\pmod{4}$.
2. $O(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $O(k_2,k_3,k_0,k_1)^\e$ where $\e=2\pmod{4}$.
3. $O(k_0,k_1,k_2,k_3)$ is isomorphic to the $\H_q$-module $O(k_3,k_0,k_1,k_2)^\e$ where $\e=1\pmod{4}$.
The classification of odd-dimensional irreducible $\H_q$-modules is stated as follows and the proof is given in §\[section:odd\].
\[thm:odd\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. Let $d\geq 0$ denote an even integer. Let $\mathbf{OM}_d$ denote the set of all isomorphism classes of irreducible $\H_q$-modules that have dimension $d+1$. Let $\mathbf{OP}_d$ denote the set of all quadruples $(k_0,k_1,k_2,k_3)$ of nonzero scalars in $\F$ that satisfy $k_0 k_1 k_2 k_3=q^{-d-1}$ and $$k_0^2,k_1^2,k_2^2,k_3^2\not=
q^{-i}
\qquad
\hbox{for all $i=2,4,\ldots,d$}.$$ Then there exists a bijection $\mathcal O:\mathbf{OP}_d\to \mathbf{OM}_d$ given by $$\begin{aligned}
(k_0,k_1,k_2,k_3)
&\mapsto &
\hbox{the isomorphism class of $O(k_0,k_1,k_2,k_3)$}\end{aligned}$$ for all $(k_0,k_1,k_2,k_3)\in \mathbf{OP}_d$.
An infinite-dimensional $\H_q$-module and its universal property {#s:Verma}
================================================================
In this section, we present an infinite-dimensional $\H_q$-module and its universal property.
\[prop:P\] For any nonzero scalars $k_0,k_1,k_2,k_3\in \F$ there exists an $\H_q$-module $M(k_0,k_1,k_2,k_3)$ satisfying the following conditions:
1. There exists an $\F$-basis $\{m_i\}_{i=0}^\infty$ for $M(k_0,k_1,k_2,k_3)$ such that $$\begin{aligned}
t_0 m_i
&=
\left\{
\begin{array}{ll}
\textstyle
k_0^{-1} q^{-i} (1-q^i) (1-k_0^2 q^i)
m_{i-1}
+
(
k_0+k_0^{-1}-k_0^{-1}q^{-i}
)
m_i
\qquad
&\hbox{if $i=2,4,\ldots$},
\\
k_0^{-1} q^{-i-1}
(m_i-m_{i+1})
\qquad
&\hbox{if $i=1,3,\ldots$},
\end{array}
\right.
\\
t_0 m_0 &= k_0 m_0,
\\
t_1 m_i
&=
\left\{
\begin{array}{ll}
-k_1(1-q^i)(1-k_0^2 q^i)m_{i-1}
+k_1 m_i
+k_1^{-1} m_{i+1}
\qquad
&\hbox{if $i=2,4,\ldots$},
\\
k_1^{-1} m_i
\qquad
&\hbox{if $i=1,3,\ldots$},
\end{array}
\right.
\\
t_1 m_0 &=k_1 m_0+k_1^{-1} m_1,
\\
t_2 m_i
&=
\left\{
\begin{array}{ll}
k_0^{-1} k_1^{-1} k_3^{-1} q^{-i-1}
(m_i-m_{i+1})
\qquad
&\hbox{if $i=0,2,\ldots$},
\\
\frac{(k_0 k_1 k_3 q^i-k_2)(k_0 k_1 k_3 q^i-k_2^{-1})}{k_0 k_1 k_3 q^i}
m_{i-1}
+
(k_2+k_2^{-1}-
k_0^{-1} k_1^{-1} k_3^{-1} q^{-i}
) m_i
\qquad
&\hbox{if $i=1,3,\ldots$},
\end{array}
\right.
\\
t_3 m_i
&=
\left\{
\begin{array}{ll}
k_3 m_i
\qquad
&\hbox{if $i=0,2,\ldots$},
\\
-k_3^{-1}(k_0k_1k_3q^i-k_2)
(k_0k_1k_3 q^i-k_2^{-1} )
m_{i-1}
+k_3^{-1} m_i
+k_3 m_{i+1}
\qquad
&\hbox{if $i=1,3,\ldots$}.
\end{array}
\right.\end{aligned}$$
2. The elements $c_0,c_1,c_2,c_3$ act on $M(k_0,k_1,k_2,k_3)$ as scalar multiplication by $$k_0+k_0^{-1},
\qquad
k_1+k_1^{-1},
\qquad
k_2+k_2^{-1},
\qquad
k_3+k_3^{-1}$$ respectively.
It is routine to verify the proposition via Definition \[defn:DAHA\].
Until the end of this paper we adopt the following conventions: Let $k_0,k_1,k_2,k_3$ denote any nonzero scalars in $\F$. Let $\{m_i\}_{i=0}^\infty$ denote the $\F$-basis for $M(k_0,k_1,k_2,k_3)$ from Proposition \[prop:P\]. In addition, we set the following parameters associated with $k_0,k_1,k_2,k_3,q$: $$\begin{aligned}
\phi_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_0^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(k_0 k_1^{-1} k_3 q^i-k_2)(k_0 k_1^{-1} k_3 q^i-k_2^{-1})
\qquad
&\hbox{if $i$ is odd};
\end{array}
\right.
\label{phi}
\\
\varrho_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_0^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(k_0 k_1 k_3 q^i-k_2)(k_0 k_1 k_3 q^i-k_2^{-1})
\qquad
&\hbox{if $i$ is odd};
\end{array}
\right.
\label{varphi}
\\
\chi_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_0^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(k_0 k_1 k_3^{-1} q^i-k_2)(k_0 k_1 k_3^{-1} q^i-k_2^{-1})
\qquad
&\hbox{if $i$ is odd};
\end{array}
\right.
\label{psi}
\\
\psi_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_1^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(k_0 k_1 k_2 q^i-k_3)(k_0 k_1 k_2 q^i-k_3^{-1})
\qquad
&\hbox{if $i$ is odd}
\end{array}
\right.
\label{varrho}\end{aligned}$$ for all $i\in \Z$.
Define $$\begin{aligned}
X&=t_3t_0,
\label{X}
\\
Y&=t_0t_1.
\label{Y}\end{aligned}$$
\[lem:XYinP\]
1. The action of $X$ on $M(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_3 q^{2\lceil \frac{i}{2}\rceil} X^{(-1)^{i-1}}) m_{i}=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=0$},
\\
\varrho_i
m_{i-1}
\qquad
&\hbox{if $i=1,2,3,\ldots$}.
\end{array}
\right.\end{aligned}$$
2. The action of $Y$ on $M(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) m_i =m_{i+1}
\qquad
\hbox{for all $i=0,1,2,\ldots$}.\end{aligned}$$
Recall $X$ and $Y$ from (\[X\]) and (\[Y\]). Evaluate the actions of $X$ and $Y$ on $M(k_0,k_1,k_2,k_3)$ by Proposition \[prop:P\].
To state the universal property for $M(k_0,k_1,k_2,k_3)$ we are now going to give an alternative description of $M(k_0,k_1,k_2,k_3)$. We begin with the following lemma.
\[Proposition 7.8, [@DAHA2013]\] \[lem:basisH\] The elements $$Y^i X^j t_0^k
c_0^\ell c_1^r c_2^s c_3^t$$ for all integers $i,j,\ell,r,s,t$ and $k\in\{0,1\}$ with $\ell,r,s,t\geq 0$ are an $\F$-basis for $\H_q$.
Let $I(k_0,k_1,k_2,k_3)$ denote the left ideal of $\H_q$ generated by $$\begin{gathered}
t_0-k_0,
\qquad
t_3-k_3,
\label{I1}
\\
c_1-k_1-k_1^{-1},
\qquad
c_2-k_2-k_2^{-1}.
\label{I2}\end{gathered}$$
\[lem:H/I\] The $\F$-vector space $\H_q/I(k_0,k_1,k_2,k_3)$ is spanned by $$Y^i+I(k_0,k_1,k_2,k_3)
\qquad
\hbox{for all $i\in \Z$}.$$
By Lemma \[lem:basisH\] the cosets $$Y^i X^j t_0^k
c_0^\ell c_1^r c_2^s c_3^t
+
I(k_0,k_1,k_2,k_3)$$ for all integers $i,j,\ell,r,s,t$ and $k\in\{0,1\}$ with $\ell,r,s,t\geq 0$ span $\H_q/I(k_0,k_1,k_2,k_3)$. Since $I(k_0,k_1,k_2,k_3)$ contains the elements (\[I1\]) this yields that $$\begin{gathered}
c_0-k_0-k_0^{-1},
\qquad
c_3-k_3-k_3^{-1}
\label{I3}\end{gathered}$$ are in $I(k_0,k_1,k_2,k_3)$. By (\[X\]) and since $I(k_0,k_1,k_2,k_3)$ contains (\[I1\])–(\[I3\]) it follows that $$\begin{aligned}
Y^i+I(k_0,k_1,k_2,k_3)
\qquad
\hbox{for all $i\in \Z$}\end{aligned}$$ span $\H_q/I(k_0,k_1,k_2,k_3)$. The lemma follows.
The $\H_q$-module $M(k_0,k_1,k_2,k_3)$ has the following statement:
\[thm:Piso\] There exists a unique $\H_q$-module homomorphism $$\Phi:\H_q/I(k_0,k_1,k_2,k_3)\to M(k_0,k_1,k_2,k_3)$$ that sends $1+I(k_0,k_1,k_2,k_3)$ to $m_0$. Moreover $\Phi$ is an isomorphism.
Consider the $\H_q$-module homomorphism $\Psi:\H_q\to M(k_0,k_1,k_2,k_3)$ that sends $1$ to $m_0$. By Proposition \[prop:P\] the elements (\[I1\]) and (\[I2\]) are in the kernel of $\Psi$. It follows that $I(k_0,k_1,k_2,k_3)$ is contained in the kernel of $\Psi$. Therefore there exists an $\H_q$-module homomorphism $\H_q/I(k_0,k_1,k_2,k_3)\to M(k_0,k_1,k_2,k_3)$. The existence follows. Since the $\H_q$-module $\H_q/I(k_0,k_1,k_2,k_3)$ is generated by $1+I(k_0,k_1,k_2,k_3)$ the uniqueness follows.
By Lemma \[lem:XYinP\](ii) the homomorphism $\Phi$ sends $$\begin{gathered}
\label{Yi+I}
\prod_{h=0}^{i-1}
(1-k_0 k_1 q^{2\lceil \frac{h}{2}\rceil} Y^{(-1)^{h-1}})
+I(k_0,k_1,k_2,k_3)\end{gathered}$$ to $m_i$ for all $i=0,1,2,\ldots$. Since $\{m_i\}_{i=0}^\infty$ are linearly independent, the cosets (\[Yi+I\]) for all $i=0,1,2,\ldots$ are linearly independent. By Lemma \[lem:H/I\] the cosets (\[Yi+I\]) for all $i=0,1,2,\ldots$ span $\H_q/I(k_0,k_1,k_2,k_3)$. Therefore those cosets form an $\F$-basis for $\H_q/I(k_0,k_1,k_2,k_3)$ and it follows that $\Phi$ is an isomorphism. The result follows.
In light of Theorem \[thm:Piso\] the $\H_q$-module $M(k_0,k_1,k_2,k_3)$ has the following universal property:
\[thm:universal\] If $V$ is an $\H_q$-module which contains a vector $v$ satisfying $$\begin{gathered}
t_0 v=k_0 v,
\qquad
t_3 v=k_3 v,
\\
c_1 v=(k_1+k_1^{-1})v,
\qquad
c_2 v=(k_2+k_2^{-1})v,\end{gathered}$$ then there exists a unique $\H_q$-module homomorphism $M(k_0,k_1,k_2,k_3)\to V$ that sends $m_0$ to $v$.
We finish this section with a comment on the $\H_q$-module $M(k_0,k_1,k_2,k_3)$ and the polynomial representation of $\H_q$. Recall that the $\H_q$-module $P(k_0,k_1,k_2,k_3)$ corresponding to the polynomial representation of $\H_q$ is the ring of Laurent polynomials in one variable $z$ on which the actions of $t_0,t_1,t_2,t_3$ are as follows: $$\begin{aligned}
t_0 : f(z) &\mapsto &
k_0 f(q^2 z^{-1})+\frac{k_0+k_0^{-1}-(k_1+k_1^{-1})q z^{-1}}{1-q^2 z^{-2}}(f(z)-f(q^2 z^{-1})),
\\
t_1 : f(z)
&\mapsto &
\frac{k_1+k_1^{-1}-(k_0+k_0^{-1})q z^{-1}}{1-q^2 z^{-2}}f(z)
+
\frac{k_1+k_1^{-1}-k_0 q z^{-1} - k_0^{-1} q^{-1}z}{1-q^{-2} z^2} f(q^2 z^{-1}),
\\
t_2 : f(z) &\mapsto &
\frac{k_2+k_2^{-1}-(k_3+k_3^{-1})z}{1-z^2} f(z)
+
\frac{k_3 z+k_3^{-1} z^{-1}-k_2-k_2^{-1}}{1-z^2} f(z^{-1}),
\\
t_3 : f(z) &\mapsto &
k_3 f(z^{-1})+\frac{k_3+k_3^{-1}-(k_2+k_2^{-1})z}{1-z^2} (f(z)-f(z^{-1})).\end{aligned}$$ Observe that $t_0,t_3,c_1,c_2$ map $1$ to $k_0,k_3,k_1+k_1^{-1},k_2+k_2^{-1}$ respectively. It follows from Theorem \[thm:universal\] that there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
\label{H->P}
M(k_0,k_1,k_2,k_3)\to P(k_0,k_1,k_2,k_3)\end{gathered}$$ that sends $m_0$ to $1$. The element $Y$ acts on $P(k_0,k_1,k_2,k_3)$ as multiplication by $q^{-1}z$. By Lemma \[lem:XYinP\](ii) the homomorphism (\[H->P\]) sends $m_i$ to $$\begin{gathered}
\label{e:Pbasis}
\prod_{h=0}^{i-1}(1-k_0k_1 q^{2\lceil \frac{h}{2}\rceil} q^{(-1)^h} z^{(-1)^{h-1}})
\qquad
\hbox{for all $i=0,1,2,\ldots$}.\end{gathered}$$ Since the Laurent polynomials (\[e:Pbasis\]) form an $\F$-basis for $P(k_0,k_1,k_2,k_3)$ it follows that (\[H->P\]) is an isomorphism.
Simultaneous eigenvectors of $t_0$ and $t_3$ in $\H_q$-modules {#s:eigenvector}
==============================================================
As suggested by Theorem \[thm:universal\], we examine the existence of the simultaneous eigenvectors of $t_0$ and $t_3$ in finite-dimensional irreducible $\H_q$-modules and similar issues in this section.
\[lem:Schur\] Assume that $\F$ is algebraically closed. If $V$ is a finite-dimensional irreducible $\H_q$-module, then each central element of $\H_q$ acts on $V$ as scalar multiplication.
Apply Schur’s lemma to $\H_q$.
\[lem:theta\] Let $\mu$ denote a nonzero scalar in $\F$ and $$\begin{gathered}
\label{theta}
\theta_i
=\left\{
\begin{array}{ll}
\mu q^i
\quad
&\hbox{if $i$ is even},
\\
\mu^{-1} q^{-i-1}
\quad
&\hbox{if $i$ is odd}
\end{array}
\right.\end{gathered}$$ for all $i\in \Z$. Then the following hold:
1. If $i=j\bmod{2}$ then $\theta_i=\theta_j$ if and only if $q^i=q^j$.
2. If $i\not=j\bmod{2}$ then $\theta_i=\theta_j$ if and only if $\mu^2=q^{-i-j-1}$.
3. For any $k\in \Z$ either of $\{\theta_i\}_{i=k}^{\infty}$ and $\{\theta_i\}_{i=k}^{-\infty}$ contains infinitely many values provided that $q$ is not a root of unity.
(i), (ii): By (\[theta\]) we have $$\begin{gathered}
\theta_i-\theta_j
=
\left\{
\begin{array}{ll}
\mu(q^i-q^j)
\qquad
&\hbox{if $i$ and $j$ are even},
\\
\mu^{-1} q^{-1}(q^{-i}-q^{-j})
\qquad
&\hbox{if $i$ and $j$ are odd},
\\
\mu^{-1} q^{-j-1}(\mu^2 q^{i+j+1}-1)
\qquad
&\hbox{if $i$ is even and $j$ is odd},
\\
\mu^{-1} q^{-i-1}(1-\mu^2 q^{i+j+1})
\qquad
&\hbox{if $i$ is odd and $j$ is even}.
\end{array}
\right.\end{gathered}$$ Therefore (i) and (ii) follow.
(iii): If the scalars $\{\theta_i\}_{i\in \Z}$ are mutually distinct, there is nothing to prove. Assume the contrary. It follows from (i) and (ii) that for any distinct $i,j\in Z$, the scalars $\theta_i$ and $\theta_j$ are equal whenever $i+j$ is equal to a constant. Therefore (iii) follows.
Recall the element $X$ of $\H_q$ from (\[X\]).
\[lem:sim\_equation\] The following equations hold in $\H_q$:
1. $Xt_0-t_0X^{-1}=X c_0-c_3$.
2. $q^{-1} X^{-1} t_2-q t_2 X
=q^{-1} X^{-1} c_2- c_1$.
(i): Observe that the left-hand side of (i) is equal to $$Xt_0-t_3^{-1}.$$ Using (\[ci\]) yields that the right-hand side of (i) is equal to the above. Therefore (i) follows.
(ii): Using (\[t0t1t2t3\]) yields that $$\begin{gathered}
\label{t2X}
t_2 X=q^{-1} t_1^{-1}.\end{gathered}$$ By (\[t2X\]) the left-hand side of (ii) is equal to $$\begin{gathered}
\label{LHS:t2X}
q^{-1} X^{-1} t_2-t_1^{-1}.\end{gathered}$$ By (\[ci\]) the right-hand side of (ii) is equal to $$\begin{gathered}
\label{RHS:t2X}
q^{-1} X^{-1} t_2+q^{-1} X^{-1} t_2^{-1}-t_1-t_1^{-1}.\end{gathered}$$ Using (\[t2X\]) the element (\[RHS:t2X\]) is equal to (\[LHS:t2X\]). Therefore (ii) follows.
\[prop:X\_eigenvector\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. If $V$ is a finite-dimensional irreducible $\H_q$-module, then the following [(i)]{} or [(ii)]{} holds:
1. $t_3$ and $t_0$ have a simultaneous eigenvector in $V$.
2. $t_1$ and $t_2$ have a simultaneous eigenvector in $V$.
Since $\F$ is algebraically closed and $V$ is finite-dimensional, there exists an eigenvalue $\mu$ of $X$ in $V$. Since $X$ is invertible in $\H_q$ the scalar $\mu$ is nonzero. Consider the corresponding scalars $\{\theta_i\}_{i\in \Z}$ given in (\[theta\]). By Lemma \[lem:theta\](iii) there are infinitely many values among $\{\theta_i\}_{i=0}^{-\infty}$. Since $V$ is finite-dimensional there exists an integer $k\leq 0$ such that $\theta_{k-1}$ is not an eigenvalue of $X$ but $\theta_k$ is an eigenvalue of $X$ in $V$. Let $W$ denote the $\theta_k$-eigenspace of $X$ in $V$.
Suppose that $k$ is even. Pick any $v\in W$. Note that $\theta_k^{-1}=\theta_{k-1}$. Applying $v$ to either side of Lemma \[lem:sim\_equation\](i) yields that $$\begin{gathered}
\label{Xt0v}
(X-\theta_{k-1}) t_0 v= ( \theta_k c_0 - c_3) v.\end{gathered}$$ By Lemma \[lem:Schur\] the right-hand side of (\[Xt0v\]) is a scalar multiple of $v$. Left multiplying either side of (\[Xt0v\]) by $X-\theta_k$ yields that $$(X-\theta_{k-1})(X-\theta_k)t_0v=0.$$ Since $\theta_{k-1}$ is not an eigenvalue of $X$ in $V$ it follows that $(X-\theta_k)t_0v=0$ and hence $t_0 v\in W$. This shows that $W$ is $t_0$-invariant. Since $\F$ is algebraically closed there exists an eigenvector $w$ of $t_0$ in $W$. Since $X=t_3t_0$ and $t_0$ is invertible in $\H_q$ it follows that $w$ is also an eigenvector of $t_3$. Therefore (i) follows.
Suppose that $k$ is odd. Pick any $v\in W$. Note that $\theta_k=q^{-2} \theta_{k-1}^{-1}$. Applying $v$ to Lemma \[lem:sim\_equation\](ii) yields that $$\begin{gathered}
\label{X-1t2v}
q^{-1} (X^{-1}-\theta_{k-1}^{-1}) t_2 v
=(q^{-1}\theta_k^{-1} c_2-c_1) v.\end{gathered}$$ By Lemma \[lem:Schur\] the right-hand side of (\[X-1t2v\]) is a scalar multiple of $v$. Left multiplying either side of (\[X-1t2v\]) by $X-\theta_k$ yields that $$(X^{-1}-\theta_{k-1}^{-1})(X-\theta_k)t_2v=0.$$ Since $\theta_{k-1}^{-1}$ is not an eigenvalue of $X^{-1}$ in $V$ it follows that $(X-\theta_k)t_2v=0$ and hence $t_2 v\in W$. This shows that $W$ is $t_2$-invariant. Since $\F$ is algebraically closed there exists an eigenvector $w$ of $t_2$ in $W$. Since $X^{-1}=q t_1 t_2$ by (\[t0t1t2t3\]) and $t_2$ is invertible in $\H_q$ it follows that $w$ is also an eigenvector of $t_1$. Therefore (ii) follows.
\[prop:Y\_eigenvector\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. If $V$ is a finite-dimensional irreducible $\H_q$-module, then the following [(i)]{} or [(ii)]{} holds:
1. $t_0$ and $t_1$ have a simultaneous eigenvector in $V$.
2. $t_2$ and $t_3$ have a simultaneous eigenvector in $V$.
Let $\e=1 \pmod{4}$. The proposition follows by applying Proposition \[prop:X\_eigenvector\] to the $\H_q$-module $V^\e$.
Proof of Theorem \[thm:iso\] {#section:iso}
============================
Throughout this section we use the following conventions: Let $d\geq 1$ denote an odd integer. Assume that $k_0^2=q^{-d-1}$. Let $\{v_i\}_{i=0}^d$ denote the $\F$-basis for the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ from Proposition \[prop:E\](i). Define $$N(k_0,k_1,k_2,k_3)$$ to be the $\H_q$-submodule of $M(k_0,k_1,k_2,k_3)$ generated by $m_{d+1}$.
\[lem:XYinE\]
1. The action of $X$ on $E(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_3 q^{2\lceil \frac{i}{2}\rceil}X^{(-1)^{i-1}}) v_i
=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=0$},
\\
\varrho_i
v_{i-1}
\qquad
&\hbox{if $i=1,2,\ldots,d$}.
\end{array}
\right.\end{aligned}$$
2. The action of $Y$ on $E(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) v_i
=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=d$},
\\
v_{i+1}
\qquad
&\hbox{if $i=0,1,\ldots,d-1$}.
\end{array}
\right.\end{aligned}$$
Recall $X$ and $Y$ from (\[X\]) and (\[Y\]). Evaluate the actions of $X$ and $Y$ on $E(k_0,k_1,k_2,k_3)$ by Proposition \[prop:E\].
\[lem:N\_E\] $\{m_i\}_{i=d+1}^\infty$ is an $\F$-basis for $N(k_0,k_1,k_2,k_3)$.
Since $d$ is odd and $k_0^2=q^{-d-1}$ it follows from Proposition \[prop:P\] that $N(k_0,k_1,k_2,k_3)$ has the $\F$-basis $\{m_i\}_{i=d+1}^\infty$.
\[lem:E\] There exists a unique $\H_q$-module isomorphism $$M(k_0,k_1,k_2,k_3)/N(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2,k_3)$$ that sends $m_i+N(k_0,k_1,k_2,k_3)$ to $v_i$ for all $i=0,1,\ldots,d$.
By Proposition \[prop:E\] the vector $v_0$ satisfies $$\begin{gathered}
t_0 v_0=k_0 v_0,
\qquad
t_3 v_0 =k_3 v_0,
\\
c_1 v_0 =(k_1+k_1^{-1})v_0,
\qquad
c_2 v_0 =(k_2+k_2^{-1})v_0.\end{gathered}$$ Hence there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
\label{M->E}
M(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2,k_3)\end{gathered}$$ that sends $m_0$ to $v_0$ by Theorem \[thm:universal\]. Comparing Lemma \[lem:XYinP\](ii) with Lemma \[lem:XYinE\](ii) the image of $m_i$ under (\[M->E\]) is $v_i$ for all $i=0,1,\ldots,d$ and the image of $m_{d+1}$ under (\[M->E\]) is zero. Hence $N(k_0,k_1,k_2,k_3)$ is contained in the kernel of (\[M->E\]). Therefore (\[M->E\]) induces the $\H_q$-module homomorphism $$\begin{gathered}
\label{M/N->E}
M(k_0,k_1,k_2,k_3)/N(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2,k_3)\end{gathered}$$ that sends $m_i+N(k_0,k_1,k_2,k_3)$ to $v_i$ for all $i=0,1,\ldots,d$. By Lemma \[lem:N\_E\] the cosets $\{m_i+N(k_0,k_1,k_2,k_3)\}_{i=0}^d$ form an $\F$-basis for $M(k_0,k_1,k_2,k_3)/N(k_0,k_1,k_2,k_3)$. Since $\{v_i\}_{i=0}^d$ is an $\F$-basis for $E(k_0,k_1,k_2,k_3)$, it follows that (\[M/N->E\]) is an isomorphism.
\[prop:universal\_E\] Let $V$ denote an $\H_q$-module. If the element $$\begin{gathered}
\label{e:universal_E}
\prod_{i=0}^{d}
(1-k_0 k_1q^{2\lceil \frac{i}{2}\rceil}Y^{(-1)^{i-1}})\end{gathered}$$ vanishes at some vector $v$ of $V$ and there is an $\H_q$-module homomorphism $M(k_0,k_1,k_2,k_3)\to V$ that sends $m_0$ to $v$, then there exists an $\H_q$-module homomorphism $E(k_0,k_1,k_2,k_3)\to V$ that sends $v_0$ to $v$.
Let $\Phi$ denote the $\H_q$-module homomorphism $M(k_0,k_1,k_2,k_3)\to V$ sends $m_0$ to $v$. Since (\[e:universal\_E\]) vanishes at $v$ and by Lemma \[lem:XYinP\](i) the vector $m_{d+1}$ is in the kernel of $\Phi$ and hence $N(k_0,k_1,k_2,k_3)$ is contained in the kernel of $\Phi$. It follows that there is an $\H_q$-module homomorphism $M(k_0,k_1,k_2,k_3)/N(k_0,k_1,k_2,k_3)\to V$ that sends $m_0+N(k_0,k_1,k_2,k_3)$ to $v$. Combined with Lemma \[lem:E\] the proposition follows.
\[lem:irr\_E\] If the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible then the following conditions hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d-1$.
2. $k_0^2\not=q^{-i}$ for all $i=2,4,\ldots,d-1$.
3. $k_0k_1k_2k_3, k_0k_1k_2^{-1}k_3\not=q^{-i}$ for all $i=1,3,\ldots,d$.
By (\[varphi\]) the conditions (i)–(iii) hold if and only if $\varrho_i\not=0$ for all $i=1,2,\ldots,d$. Suppose that there is a $k\in\{1,2,\ldots,d\}$ such that $\varrho_k=0$. Let $W$ denote the $\F$-subspace of $E(k_0,k_1,k_2,k_3)$ spanned by $\{v_i\}_{i=k}^d$. Using Proposition \[prop:E\] yields that $W$ is invariant under $\{t_i^{\pm 1}\}_{i=0}^3$. Hence $W$ is a proper $\H_q$-submodule of $E(k_0,k_1,k_2,k_3)$, a contradiction to the irreducibility of $E(k_0,k_1,k_2,k_3)$. The lemma follows.
\[prop:iso2\] The $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is isomorphic to $E(k_0,k_1^{-1},k_2,k_3)$. Moreover the vectors $$w_i=
\displaystyle
\prod_{h=0}^{i-1}
(1-k_0 k_1^{-1} q^{2\lceil \frac{h}{2}\rceil} Y^{(-1)^{h-1}}) v_0
\qquad
\hbox{for all $i=0,1,\ldots,d$}$$ form an $\F$-basis for $E(k_0,k_1,k_2,k_3)$ and $$\begin{aligned}
(1-k_0 k_3 q^{2\lceil \frac{i}{2}\rceil} X^{(-1)^{i-1}}) w_i
&=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=0$},
\\
\phi_i
w_{i-1}
\qquad
&\hbox{if $i=1,2,\ldots,d$};
\end{array}
\right.
\\
(1-k_0 k_1^{-1} q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) w_i
&=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=d$},
\\
w_{i+1}
\qquad
&\hbox{if $i=0,1,\ldots,d-1$}.
\end{array}
\right.\end{aligned}$$
Let $\{u_i\}_{i=0}^d$ denote the $\F$-basis for $E(k_0,k_1^{-1},k_2,k_3)$ obtained from Proposition \[prop:E\](i). To see the proposition, it suffices to show that there exists an $\H_q$-module homomorphism $E(k_0,k_1,k_2,k_3)\to E(k_0,k_1^{-1},k_2,k_3)$ that sends $w_i$ to $u_i$ for all $i=0,1,\ldots,d$.
By Proposition \[prop:E\] the vector $u_0$ satisfies the following equations: $$\begin{gathered}
t_0 u_0=k_0 u_0,
\qquad
t_3 u_0=k_3 u_0,
\\
c_1 u_0=(k_1+k_1^{-1}) u_0,
\qquad
c_2 u_0=(k_2+k_2^{-1}) u_0.\end{gathered}$$ Hence there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
M(k_0,k_1,k_2,k_3)\to E(k_0,k_1^{-1},k_2,k_3)\end{gathered}$$ that sends $m_0$ to $u_0$ by Theorem \[thm:universal\]. It follows from Lemma \[lem:XYinE\](ii) that $$\begin{gathered}
\label{w_0=0}
\prod_{i=0}^d(1-k_0k_1^{-1}q^{2 \lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}})\end{gathered}$$ vanishes at $u_0$. Since $d$ is odd and $k_0^2=q^{-d-1}$ the $i^{\,{\rm th}}$ term of (\[w\_0=0\]) is equal to $-k_0k_1^{-1} q^{2 \lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}$ times the $(d-i)^{\,{\rm th}}$ term of (\[e:universal\_E\]) for all $i=0,1,\ldots,d$. Hence there exists an $\H_q$-module homomorphism $$\begin{gathered}
\label{E(k1inv)->E}
E(k_0,k_1,k_2,k_3)\to E(k_0,k_1^{-1},k_2,k_3)\end{gathered}$$ that sends $v_0$ to $u_0$ by Proposition \[prop:universal\_E\]. By the construction of $\{w_i\}_{i=0}^d$ the homomorphism (\[E(k1inv)->E\]) maps $w_i$ to $u_i$ for all $i=0,1,\ldots,d$. The proposition follows.
\[lem:irr2\] If the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible then the following conditions hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d-1$.
2. $k_0^2\not=q^{-i}$ for all $i=2,4,\ldots,d-1$.
3. $k_0k_1k_2k_3, k_0k_1^{-1}k_2k_3, k_0k_1k_2^{-1}k_3, k_0k_1k_2 k_3^{-1}\not=q^{-i}$ for all $i=1,3,\ldots,d$.
Suppose that the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible. By Proposition \[prop:iso2\] the $\H_q$-module $E(k_0,k_1^{-1},k_2,k_3)$ is irreducible. Applying Lemma \[lem:irr\_E\] to $E(k_0,k_1^{-1},k_2,k_3)$ yields that $k_0k_1^{-1}k_2k_3,k_0k_1^{-1}k_2^{-1}k_3\not=q^{-i}$ for all $i=1,3,\ldots,d$. Since $k_0^2=q^{-d-1}$ the conditions $k_0k_1^{-1}k_2^{-1}k_3\not=q^{-i}$ for all $i=1,3,\ldots,d$ are equivalent to $k_0k_1k_2k_3^{-1}\not=q^{-i}$ for all $i=1,3,\ldots,d$. The lemma follows.
Define $$\begin{aligned}
R &= \prod_{h=1}^{d}
(1-k_0 k_3 q^{2\lceil \frac{h}{2}\rceil} X^{(-1)^{h-1}}),
\\
S_i &=
\prod_{h=1}^{d-i}
(1-k_0 k_1^{-1} q^{2\lceil \frac{h-1}{2}\rceil} Y^{(-1)^h})
\qquad
\hbox{for all $i=0,1,\ldots,d$}.\end{aligned}$$ It follows from Lemma \[lem:XYinE\](i) that $R v$ is a scalar multiple of $v_0$ for all $v\in E(k_0,k_1,k_2,k_3)$. In particular, for any $i,j\in \{0,1,\ldots,d\}$ there exists a unique $L_{ij}\in \F$ such that $$\begin{gathered}
\label{Lij}
R S_i v_j= L_{ij} v_0.\end{gathered}$$ By Lemma \[lem:XYinE\](ii) the scalars $$\begin{aligned}
L_{ij} &=0 \qquad
\hbox{for all $0\leq i<j\leq d$};
\label{L:lower}\\
L_{ij} &=
\left\{
\begin{array}{ll}
k_1^2
q^{i+j-d-1}
(L_{i-1,j-1}-L_{i,j-1})+L_{i,j-1}
&\hbox{if $i=j \bmod{2}$},
\\
(1-q^{j-i-1}) L_{i,j-1}
+L_{i-1,j}-L_{i-1,j-1}
&\hbox{if $i\not=j \bmod{2}$}
\end{array}
\right.
\label{L:recurrence}\end{aligned}$$ for all $i,j\in\{1,2,\ldots,d\}$. By Proposition \[prop:iso2\] the scalars $$\begin{gathered}
\label{L:init}
L_{i0}=
\prod_{h=1}^{\lfloor \frac{i}{2}\rfloor}
(1-
q^{d-2h+1})
\prod_{h=1}^{\lceil \frac{i}{2}\rceil}
(1-k_3^2
q^{2-2h}
)
\prod_{h=1}^{d-i} \phi_h
\qquad
\hbox{for all $i=0,1,\ldots,d$}.\end{gathered}$$ Solving the recurrence relation (\[L:recurrence\]) with the initial conditions (\[L:lower\]) and (\[L:init\]) yields that $$\begin{aligned}
L_{ij}
&=
\displaystyle
q^{\lfloor\frac{j}{2}\rfloor(d-4\lfloor \frac{i}{2}\rfloor+2\lfloor\frac{j}{2}\rfloor-1)}
\prod_{h=1}^{\lfloor\frac{i-j}{2}\rfloor}
(1-q^{d-2h+1})
\prod_{h=1}^{d-i} \phi_h
\prod_{h=1}^{\lceil\frac{j}{2}\rceil}
\varrho_{2h-1}
\prod_{h=1}^{\lfloor\frac{j}{2}\rfloor}
\varrho_{2(\lfloor \frac{i}{2}\rfloor-h+1)}
\label{e:Lij}
\\
&\qquad\times
\;
\left\{
\begin{array}{ll}
\displaystyle
\prod_{h=1}^{\lceil\frac{i-j}{2}\rceil}
(1-k_3^2 q^{2-2h})
\qquad
&\hbox{if $i$ is odd or $j$ is even},
\\
\displaystyle
q^{d-2i+2j-1}
\varrho_{i-j+1}
\prod_{h=1}^{\lfloor\frac{i-j}{2}\rfloor}
(1-k_3^2 q^{2-2h})
\qquad
&\hbox{if $i$ is even and $j$ is odd}
\end{array}
\right.
\notag\end{aligned}$$ for all $0\leq j\leq i\leq d$.
We now show that the converse of Lemma \[lem:irr2\] is true.
\[thm:irr\] The $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible if and only if the following conditions hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d-1$.
2. $k_0^2\not=q^{-i}$ for all $i=2,4,\ldots,d-1$.
3. $k_0k_1k_2k_3, k_0k_1^{-1}k_2k_3, k_0k_1k_2^{-1}k_3, k_0k_1k_2 k_3^{-1}\not=q^{-i}$ for all $i=1,3,\ldots,d$.
By Lemma \[lem:irr2\] it remains to prove the “if” part. Suppose that (i)–(iii) hold. To see the irreducibility of $E(k_0,k_1,k_2,k_3)$, we let $W$ denote a nonzero $\H_q$-submodule of $E(k_0,k_1,k_2,k_3)$ and we show that $W= E(k_0,k_1,k_2,k_3)$.
Pick any nonzero $w\in W$. Let $[w]$ denote the coordinate vector of $w$ relative to $\{v_i\}_{i=0}^d$. Let $L$ denote the $(d+1)\times (d+1)$ matrix indexed by $0,1,\ldots,d$ whose $(i,j)$-entry $L_{ij}$ is defined as (\[Lij\]) for all $i,j\in\{0,1,\ldots,d\}$. By (\[L:lower\]) the square matrix $L$ is lower triangular. By (\[e:Lij\]) the diagonal entries of $L$ are $$L_{ii}=
q^{\lfloor \frac{i}{2}\rfloor(d-2\lfloor \frac{i}{2}\rfloor-1)}
\prod_{h=1}^{d-i}\phi_h
\prod_{h=1}^{\lceil \frac{i}{2}\rceil} \varrho_{2h-1}
\prod_{h=1}^{\lfloor \frac{i}{2}\rfloor} \varrho_{2(\lfloor \frac{i}{2}\rfloor-h+1)}$$ for all $i=0,1,\ldots,d$. Recall the parameters $\{\phi_i\}_{i\in \Z}$ and $\{\varrho_i\}_{i\in \Z}$ from (\[phi\]) and (\[varphi\]). By (i)–(iii) each of $\{\phi_i\}_{i=1}^d$ and $\{\varrho_i\}_{i=1}^d$ is nonzero. Hence $L$ is nonsingular. Observe that $RS_i w\in W$ is scalar multiplication of $v_0$ by the $i$-entry of $L[w]$. Since $[w]$ is nonzero it follows that $L[w]$ is nonzero and this implies that $v_0\in W$. By Lemma \[lem:XYinE\](ii) the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is generated by $v_0$. Therefore $W=E(k_0,k_1,k_2,k_3)$. The result follows.
We are now ready to give our proof for Theorem \[thm:iso\].
[*Proof of Theorem \[thm:iso\].*]{} (i): Immediate from Proposition \[prop:iso2\].
(ii): Let $\{u_i\}_{i=0}^d$ denote the $\F$-basis for $E(k_0,k_1,k_2^{-1},k_3)$ obtained from Proposiiton \[prop:E\](i). By Proposition \[prop:E\] there is an $\H_q$-module isomorphism $E(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2^{-1},k_3)$ that sends $v_i$ to $u_i$ for all $i=0,1,\ldots,d$. Therefore (ii) follows.
(iii): Let $\{u_i\}_{i=0}^d$ denote the $\F$-basis for $E(k_0,k_1,k_2,k_3^{-1})$ obtained from Proposition \[prop:E\](i). Recall the parameters $\{\chi_i\}_{i\in \Z}$ from (\[psi\]). It is straightforward to verify that each of $$\begin{aligned}
w_i&= k_3^2 \chi_i u_{i-1}
+
(1-k_3^2) u_i
- u_{i+1}
\qquad
\hbox{for $i=1,3,\ldots,d-2$},
\\
w_d&=k_3^2 \chi_d u_{d-1}+(1-k_3^2)u_d.\end{aligned}$$ is a $k_3$-eigenvector of $t_3$ in $E(k_0,k_1,k_2,k_3^{-1})$. Since the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is irreducible it follows from Theorem \[thm:irr\] that $\chi_i\not=0$ for all $i=1,2,\ldots,d$ and $q^i\not=1$ for all $i=2,4,\ldots,d-1$. Thus we may set $$v=
\sum_{i=1}^{\frac{d+1}{2}}
\prod_{h=1}^{i-1}
\frac{k_3^{-2}-q^{2h-2i}}{1-q^{2h-2i}}
\chi_{2i-2h+1}^{-1}
w_{2i-1}.$$ By construction $t_3v=k_3 v$. Using Proposition \[prop:E\] a direct calculation yields that $t_0 v=k_0 v$. By Theorem \[thm:universal\] there exists a unique $\H_q$-module homomorphism $$M(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2,k_3^{-1})$$ that sends $m_0$ to $v$. By Lemma \[lem:XYinE\](ii) the product (\[e:universal\_E\]) vanishes on $E(k_0,k_1,k_2,k_3^{-1})$. Hence there exists an $\H_q$-module homomorphism $$\begin{gathered}
\label{E(k3)->E(k3-1)}
E(k_0,k_1,k_2,k_3)\to E(k_0,k_1,k_2,k_3^{-1})\end{gathered}$$ that sends $v_0$ to $v$ by Proposition \[prop:universal\_E\]. Since the $\H_q$-modules $E(k_0,k_1,k_2,k_3)$ and $E(k_0,k_1,k_2,k_3^{-1})$ are irreducible by Theorem \[thm:irr\], it follows that (\[E(k3)->E(k3-1)\]) is an isomorphism. Therefore (iii) follows. $\square$
Proof of Theorem \[thm:even\] {#section:even}
=============================
In this section we are devoted to proving Theorem \[thm:even\].
\[thm:onto\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. Let $d\geq 1$ denote an odd integer. If $V$ is a $(d+1)$-dimensional irreducible $\H_q$-module, then there exist an $\e\in \Z/4\Z$ and nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ such that the $\H_q$-module $E(k_0,k_1,k_2,k_3)^\e$ is isomorphic to $V$.
Suppose that $V$ is a $(d+1)$-dimensional irreducible $\H_q$-module. According to Propositions \[prop:X\_eigenvector\] and \[prop:Y\_eigenvector\] we may divide the argument into the following cases (i)–(iv):
1. $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V$.
2. $t_1,t_2$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V$.
3. $t_1,t_2$ have a simultaneous eigenvector and $t_2,t_3$ have a simultaneous eigenvector in $V$.
4. $t_0,t_3$ have a simultaneous eigenvector and $t_2,t_3$ have a simultaneous eigenvector in $V$.
(i): In this case we shall show that there are nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ such that the $\H_q$-module $E(k_0,k_1,k_2,k_3)$ is isomorphic to $V$.
Let $v$ denote a simultaneous eigenvector of $t_0$ and $t_3$ in $V$. Choose the scalars $k_0$ and $k_3$ as the eigenvalues of $t_0$ and $t_3$ corresponding to $v$, respectively. Let $w$ denote a simultaneous eigenvector of $t_0$ and $t_1$ in $V$. Choose $k_1$ as the eigenvalue of $t_1^{-1}$ corresponding to $w$. By Lemma \[lem:Schur\] the element $c_1$ acts on $V$ as scalar multiplication by $k_1+k_1^{-1}$ and there exists a nonzero scalar $k_2\in \F$ such that $c_2$ acts on $V$ as scalar multiplication by $k_2+k_2^{-1}$. By Theorem \[thm:universal\] there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
\label{Hmodule_even1}
M(k_0,k_1,k_2,k_3)\to V\end{gathered}$$ that sends $m_0$ to $v$.
Let $v_i$ denote the image of $m_i$ under (\[Hmodule\_even1\]) for each $i=0,1,2,\ldots$. Suppose that there exists an $i\in \{1,2,\ldots,d\}$ such that $v_i$ is an $\F$-linear combination of $v_0,v_1,\ldots,v_{i-1}$. Let $W$ denote the $\F$-subspace of $V$ spanned by $v_0,v_1,\ldots,v_{i-1}$. By construction the dimension of $W$ is less than $d+1$. By Proposition \[prop:P\], $W$ is invariant under $\{t_i^{\pm 1}\}_{i=0}^3$. Hence $W$ is a proper $\H_q$-submodule of $V$, a contradiction to the irreducibility of $V$. Therefore $
\{v_i\}_{i=0}^d$ is an $\F$-basis for $V$. Let $\{\varrho_i\}_{i\in \Z}$ denote the parameters (\[varphi\]) corresponding to the present parameters $k_0,k_1,k_2,k_3$. Observe that $$\label{basis_t1}
v_i
\quad
\hbox{for $i=0,2,\ldots,d-1$},
\quad
v_1,
\quad
v_{i+1}-k_1^2\varrho_i v_{i-1}
\quad
\hbox{for $i=2,4,\ldots,d-1$}$$ form an $\F$-basis for $V$. By Proposition \[prop:P\](i) the matrix representing $t_1$ with respect to the $\F$-basis (\[basis\_t1\]) for $V$ is $$\begin{gathered}
\begin{pmatrix}
k_1 I_{\frac{d+1}{2}} & {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&{\bf 0}
\\
\hline
k_1^{-1} I_{\frac{d+1}{2}} &{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&k_1^{-1} I_{\frac{d+1}{2}}
\end{pmatrix}.\end{gathered}$$ By the rank-nullity theorem $
v_1
$ and $
v_{i+1}-k_1^2\varrho_i v_{i-1}$ for $i=2,4,\ldots,d-1$ form an $\F$-basis for the $k_1^{-1}$-eigenspace of $t_1$ in $V$, as well as $$\begin{gathered}
\label{k1inv-basis}
v_i
\qquad
\hbox{for all $i=1,3,\ldots,d$}. \end{gathered}$$ Since $w$ is a $k_1^{-1}$-eigenvector of $t_1$ in $V$, the vector $w$ is an $\F$-linear combination of (\[k1inv-basis\]).
For any $u\in V$ let $[u]$ denote the coordinate vector of $u$ relative to $\{v_i\}_{i=0}^d$. We are now particularly concerned with $[v_{d+1}]$ and $[w]$. Let $a_i$ and $b_i$ denote the $i^{\rm th}$ entries of $[v_{d+1}]$ and $[w]$ for all $i=0,1,\ldots,d$, respectively. As mentioned earlier the coefficients $b_i=0$ for all $i=0,2,\ldots,d-1$. Since $w$ is an eigenvector of $t_0$ in $V$ it follows from Proposition \[prop:P\] that $b_d\not=0$. Without loss of generality we assume that $$b_d=1.$$ Observe that the first entry of $[t_0 w]$ is equal to $-a_0k_0^{-1} q^{-d-1}$. Since $b_0=0$ it follows that $$a_0=0.$$ We now show that $a_i=0$ for all $i=1,2,\ldots,d$. Using Lemma \[lem:XYinP\](i) yields that $(1-k_0 k_3 q^{d+1} X^{-1}) v_i$ is equal to $$\label{XinV}
\begin{split}
\left\{
\begin{array}{ll}
(1-q^{d+1})v_0,
\qquad
&\hbox{if $i=0$},
\\
(1-k_0^2 k_3^2 q^{d+3}) v_1
-q^{d+1} \varrho_1 v_0
\qquad
&\hbox{if $i=1$},
\\
(1-q^{d-i+1})v_i+q^{d-i+1}\varrho_i v_{i-1}
\qquad
&\hbox{if $i=2,4,\ldots,d-1$},
\\
(1-k_0^2 k_3^2 q^{d+i+2}) v_i
-q^{d-i+2} \varrho_i(v_{i-1}-\varrho_{i-1} v_{i-2})
\qquad
&\hbox{if $i=3,5,\ldots,d$}.
\end{array}
\right.
\end{split}$$ By Lemma \[lem:XYinP\](i) with $i=d+1$ we have $$\begin{aligned}
\label{e1:ud+1}
[(1-k_0 k_3 q^{d+1} X^{-1}) v_{d+1}]
&=
\begin{pmatrix}
0 \\
0 \\
\vdots \\
0 \\
\varrho_{d+1}
\end{pmatrix}.\end{aligned}$$ Evaluating the first $d$ entries of the left-hand side of (\[e1:ud+1\]) by using (\[XinV\]), we obtain that $$\begin{gathered}
\label{ai}
\varrho_i a_i
=
\left\{
\begin{array}{ll}
(q^{i-d-2}-1) a_{i-1}
\qquad
&\hbox{for $i=1,3,\ldots,d$},
\\
(k_0^2 k_3^2 q^{d+i+1}-1) a_{i-1}
\qquad
&\hbox{for $i=2,4,\ldots,d-1$}.
\end{array}
\right.\end{gathered}$$ Suppose on the contrary that there exists an $i\in\{1,2,\ldots,d\}$ with $a_i\not=0$ and $a_{i-1}=0$. By (\[ai\]) the scalar $\varrho_i=0$. Let $W$ denote the $\F$-subspace of $V$ spanned by $v_i,v_{i+1},\ldots,v_d$. It follows from Proposition \[prop:P\] that $W$ is invariant under $\{t_i^{\pm 1}\}_{i=0}^3$. Hence $W$ is a proper $\H_q$-submodule of $V$, a contradiction to the irreducibility of $V$. Therefore $a_i=0$ for all $i=0,1,\ldots,d$. In other words $$\begin{gathered}
\label{ud+1=0}
v_{d+1}=0.\end{gathered}$$
By (\[ud+1=0\]) the left-hand side of (\[e1:ud+1\]) is zero. Hence $\varrho_{d+1}=0$. Since $q$ is not a root of unity, this forces that $$k_0^2=q^{-d-1}.$$ By Lemma \[lem:XYinP\](ii) we have $$\begin{gathered}
\label{e:Yv}
\prod_{i=0}^d(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) v_0=v_{d+1}.\end{gathered}$$ By (\[ud+1=0\]) the right-hand side of (\[e:Yv\]) is zero. Thus, it follows from Proposition \[prop:universal\_E\] that there exists a nontrivial $\H_q$-module homomorphism $$\begin{gathered}
\label{E->V}
E(k_0,k_1,k_2,k_3)\to V.\end{gathered}$$ Since the $\H_q$-module $V$ is irreducible it follows that (\[E->V\]) is onto. Since $E(k_0,k_1,k_2,k_3)$ and $V$ are of dimension $d+1$ it follows that (\[E->V\]) is an isomorphism. Therefore the theorem holds for the case (i).
(ii): Let $\e=1\pmod{4}$. Since $t_0^\e=t_1,t_1^\e=t_2,t_3^\e=t_0$ by Table \[Z/4Z-action\], the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there exist nonzero scalars $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ such that $E(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Therefore the $\H_q$-module $E(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. The theorem holds for the case (ii).
(iii): Let $\e=2\pmod{4}$. Since $t_0^\e=t_2,t_1^\e=t_3,t_3^\e=t_1$ by Table \[Z/4Z-action\], the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there exist nonzero scalars $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ such that $E(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Therefore the $\H_q$-module $E(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. The theorem holds for the case (iii).
(iv): Let $\e=3\pmod{4}$. Since $t_0^\e=t_3,t_1^\e=t_0,t_3^\e=t_2$ by Table \[Z/4Z-action\] the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there exist nonzero scalars $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ such that $E(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Therefore the $\H_q$-module $E(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. The theorem holds for the case (iv).
\[lem:injective\] Let $d\geq 1$ denote an odd integer. For any nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0^2=q^{-d-1}$ the determinants of $t_0,t_1,t_2,t_3$ on $E(k_0,k_1,k_2,k_3)$ are $q^{-d-1},1,1,1$ respectively.
It is routine to verify the lemma by using Proposition \[prop:E\](i).
We are now prepared to prove Theorem \[thm:even\] below.
[*Proof of Theorem \[thm:even\].*]{} By Theorems \[thm:iso\] and \[thm:irr\] the map $\mathcal E$ is well-defined. By Theorem \[thm:onto\] the map $\mathcal E$ is onto. Suppose that there are $\e,\e'\in \Z/4\Z$ and nonzero scalars $k_i,k_i'\in \F$ for all $i=0,1,2,3$ with $k_0^2=k_0'^{2}=q^{-d-1}$ such that the $\H_q$-modules $E(k_0,k_1,k_2,k_3)^\e$ and $E(k_0',k_1',k_2',k_3')^{\e'}$ are isomorphic. Since $q$ is not a root of unity it follows from Lemma \[lem:injective\] that $\e=\e'$. By Proposition \[prop:E\](ii) it follows that $k_0'=k_0$ and $k_i'=k_i^{\pm 1}$ for all $i=1,2,3$. This shows the injectivity of $\mathcal E$. The result follows. $\square$
Proof of Theorem \[thm:iso\_O\] {#section:iso_O}
===============================
Throughout this section we use the following conventions: Let $d\geq 0$ denote an even integer. Assume that $k_0 k_1 k_2 k_3=q^{-d-1}$. Let $\{v_i\}_{i=0}^d$ denote the $\F$-basis for the $\H_q$-module $O(k_0,k_1,k_2,k_3)$ from Proposition \[prop:O\](i). Define $$N(k_0,k_1,k_2,k_3)$$ to be the $\H_q$-submodule of $M(k_0,k_1,k_2,k_3)$ generated by $m_{d+1}$. Under the assumptions we have $$\begin{aligned}
\varrho_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_0^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(q^{i-d-1}-1)(k_2^{-2} q^{i-d-1}-1)
\qquad
&\hbox{if $i$ is odd};
\end{array}
\right.
\label{varphi_even}
\\
\psi_i
&=\left\{
\begin{array}{ll}
(1-q^i) (1-k_1^2 q^i)
\qquad
&\hbox{if $i$ is even},
\\
(q^{i-d-1}-1)(k_3^{-2} q^{i-d-1}-1)
\qquad
&\hbox{if $i$ is odd}.
\end{array}
\right.
\label{varrho_even}\end{aligned}$$
\[lem:XYinO\]
1. The action of $X$ on $O(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_3 q^{2\lceil \frac{i}{2}\rceil}X^{(-1)^{i-1}}) v_i
=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=0$},
\\
\varrho_i
v_{i-1}
\qquad
&\hbox{if $i=1,2,\ldots,d$}.
\end{array}
\right.\end{aligned}$$
2. The action of $Y$ on $O(k_0,k_1,k_2,k_3)$ is as follows: $$\begin{aligned}
(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) v_i
=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=d$},
\\
v_{i+1}
\qquad
&\hbox{if $i=0,1,\ldots,d-1$}.
\end{array}
\right.\end{aligned}$$
Similar to Lemma \[lem:XYinE\].
\[lem:N\_O\] $\{m_i\}_{i=d+1}^\infty$ is an $\F$-basis for $N(k_0,k_1,k_2,k_3)$.
Similar to Lemma \[lem:N\_E\].
\[lem:O\] There exists a unique $\H_q$-module isomorphism $$M(k_0,k_1,k_2,k_3)/N(k_0,k_1,k_2,k_3)\to O(k_0,k_1,k_2,k_3)$$ that sends $m_i+N(k_0,k_1,k_2,k_3)$ to $v_i$ for all $i=0,1,\ldots,d$.
Similar to Lemma \[lem:E\].
\[prop:universal\_O\] Let $V$ denote an $\H_q$-module. If the element $$\begin{gathered}
\prod_{i=0}^{d}
(1-k_0 k_1q^{2\lceil \frac{i}{2}\rceil}Y^{(-1)^{i-1}})\end{gathered}$$ vanishes at some vector $v$ of $V$ and there is an $\H_q$-module homomorphism $M(k_0,k_1,k_2,k_3)\to V$ that sends $m_0$ to $v$, then there exists an $\H_q$-module homomorphism $O(k_0,k_1,k_2,k_3)\to V$ that sends $v_0$ to $v$.
Similar to Proposition \[prop:universal\_E\].
\[lem:irr\_O\] If the $\H_q$-module $O(k_0,k_1,k_2,k_3)$ is irreducible then the following conditions hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d$.
2. $k_0^2,k_2^2\not=q^{-i}$ for all $i=2,4,\ldots,d$.
Similar to Lemma \[lem:irr\_E\].
\[prop:iso2\_O\] Let $\{u_i\}_{i=0}^d$ denote the $\F$-basis for $O(k_1,k_2,k_3,k_0)$ from Proposition [\[prop:O\](i)]{}. Then there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
\label{iso2_O}
O(k_0,k_1,k_2,k_3)\to O(k_1,k_2,k_3,k_0)^\e\end{gathered}$$ that sends $v_0$ to $u_d$, where $\e= 3\pmod{4}$. Moreover [(\[iso2\_O\])]{} is an isomorphism if and only if the following hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d$.
2. $k_1^2,k_3^2\not=q^{-i}$ for all $i=2,4,\ldots,d$.
Note that $(t_0^\e,t_1^\e,t_2^\e,t_3^\e, Y^\e)=(t_3,t_0,t_1,t_2,X)$ by Table \[Z/4Z-action\]. By Proposition \[prop:O\](i) the vector $u_d$ satisfies $$\begin{gathered}
t_0 u_d=k_0 u_d,
\qquad
t_3 u_d= k_3 u_d,
\\
c_1 u_d=(k_1+k_1^{-1}) u_d,
\qquad
c_2 u_d=(k_2+k_2^{-1}) u_d\end{gathered}$$ in $O(k_1,k_2,k_3,k_0)^\e$. By Theorem \[thm:universal\] there exists a unique $\H_q$-module homomorphism $$M(k_0,k_1,k_2,k_3)\to O(k_1,k_2,k_3,k_0)^\e$$ that sends $m_0$ to $u_d$. By Lemma \[lem:XYinO\](i) we have $$\begin{aligned}
\label{YinO_iso2}
(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil}Y^{(-1)^{i-1}}) u_i
=
\left\{
\begin{array}{ll}
0
\qquad
&\hbox{if $i=0$},
\\
\psi_i
u_{i-1}
\qquad
&\hbox{if $i=1,2,\ldots,d$}.
\end{array}
\right.\end{aligned}$$ in $O(k_1,k_2,k_3,k_0)^\e$. In particular $$\prod_{i=0}^d
(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil}Y^{(-1)^{i-1}}) u_d=0.$$ Therefore the $\H_q$-module homomorphism (\[iso2\_O\]) exists by Proposition \[prop:universal\_O\].
Define $w_0=v_0$ and $$w_i=(1-k_0 k_1 q^{d-2\lfloor \frac{i-1}{2}\rfloor}Y^{(-1)^i}) w_{i-1}
\qquad
\hbox{for all $i=1,2,\ldots,d$}.$$ By Lemma \[lem:XYinO\](ii) a routine induction shows that $w_i$ is equal to $$q^{\lceil\frac{i}{2} \rceil
(d-2\lfloor \frac{i}{2}\rfloor)} (v_i-v_{i-1})
+
q^{\lfloor\frac{i}{2} \rfloor (d-2\lceil\frac{i}{2} \rceil+2)}
v_{i-1}$$ plus an $\F$-linear combination of $v_0,v_1,\ldots,v_{i-2}$ for all $i=1,2,\ldots,d$. Hence $\{w_i\}_{i=0}^d$ is an $\F$-basis for $O(k_0,k_1,k_2,k_3)$. Comparing with (\[YinO\_iso2\]) yields that (\[iso2\_O\]) sends $$\begin{aligned}
w_i &\mapsto&
\left(
\prod_{h=1}^i \psi_{d-h+1}
\right) u_{d-i}
\qquad
\hbox{for all $i=0,1,\ldots,d$}.\end{aligned}$$ Hence (\[iso2\_O\]) is an isomorphism if and only if $\psi_i\not=0$ for all $i=1,2,\ldots,d$. The latter is equivalent to the conditions (i) and (ii). The proposition follows.
\[thm:irr\_O\] The $\H_q$-module $O(k_0,k_1,k_2,k_3)$ is irreducible if and only if the following conditions hold:
1. $q^i\not=1$ for all $i=2,4,\ldots,d$.
2. $k_0^2,k_1^2,k_2^2,k_3^2\not=q^{-i}$ for all $i=2,4,\ldots,d$.
($\Rightarrow$): By the irreducibility of $O(k_0,k_1,k_2,k_3)$ the $\H_q$-module homomorphism (\[iso2\_O\]) is an isomorphism. By Lemma \[lem:irr\_O\] and Proposition \[prop:iso2\_O\] the conditions (i) and (ii) follow.
($\Leftarrow$): Consider the operators $$\begin{aligned}
R &= \prod_{h=1}^{d}
(1-k_0^{-1} k_3^{-1} q^{-2\lceil \frac{h}{2}\rceil} X^{(-1)^{h}}),
\\
S_i &=
\prod_{h=1}^{d-i}
(1-k_2^{-1} k_3^{-1} q^{1-2\lceil \frac{h}{2}\rceil} Y^{(-1)^{h}})
\qquad
\hbox{for all $i=0,1\ldots, d$}.\end{aligned}$$ Using Lemma \[lem:XYinO\](i) yields that $R v$ is a scalar multiple of $v_0$ for all $v\in O(k_0,k_1,k_2,k_3)$. In particular, for any $i,j\in \{0,1,\ldots,d\}$ there exists a unique $L_{ij}\in \F$ such that $$\begin{gathered}
\label{Lij_O}
R S_i v_j= L_{ij} v_0.\end{gathered}$$
Using Lemma \[lem:XYinO\](ii) yields that $$\begin{aligned}
L_{ij} &=0 \qquad
\hbox{for all $0\leq i<j\leq d$};
\label{L:lower_O}\\
L_{ij} &=
\left\{
\begin{array}{ll}
(1-k_0^2 k_1^2 q^{i+j}) L_{i,j-1}
+ L_{i-1,j}
-L_{i-1,j-1}
&\hbox{if $i=j \bmod{2}$},
\\
q^{j-i-1}
(L_{i-1,j-1}-L_{i,j-1})+L_{i,j-1}
&\hbox{if $i\not=j \bmod{2}$}
\end{array}
\right.
\label{L:recurrence_O}\end{aligned}$$ for all $i,j\in\{1,2,\ldots,d\}$. By (i) and (ii) it follows from Proposition \[prop:iso2\_O\] that (\[iso2\_O\]) is an isomorphism. From this we obtain that $$\begin{gathered}
\label{Li0:O}
L_{i0}=
\prod_{h=0}^{\lfloor \frac{i-1}{2} \rfloor}
(1-q^{2h-d})
\prod_{h=1}^{\lfloor \frac{i}{2}\rfloor}
(1-k_1^2 k_2^2 q^{d+2h})
\prod_{h=1}^{d-i}\psi_{d-h+1}
\qquad
\hbox{for all $i=0,1,\ldots,d$}.\end{gathered}$$ Solving the recurrence relation (\[L:recurrence\_O\]) with the initial conditions (\[L:lower\_O\]) and (\[Li0:O\]) yields that $L_{ij}$ is equal to $$\begin{aligned}
&
(-q^{d+i-\lceil \frac{j}{2}\rceil} k_1^2 k_2^2)^{\lceil \frac{j}{2}\rceil}
\prod_{h=1}^{\lfloor \frac{j}{2} \rfloor}(1-q^{2h})^{-1}
\prod_{h=\lceil \frac{j}{2}\rceil}^{\lfloor \frac{i-1}{2} \rfloor}
(1-q^{2h-d})
\prod_{h=1}^{\lfloor \frac{i}{2} \rfloor-\lceil \frac{j}{2}\rceil}
(1-k_1^2 k_2^2 q^{d+2h})
\prod_{h=1}^j \varrho_h
\prod_{h=1}^{d-i} \psi_{d-h+1}
\\
&
\qquad
\times
\left\{
\begin{array}{ll}
\displaystyle
(k_3^{-2} q^{j-d-1}-k_0^{2}
q^{j+1}-q^{j-i}+1)
\prod_{h=1}^{\frac{j-1}{2}}
(1-q^{2h-i-1})
\qquad
&\hbox{if $i,j$ are odd and $i>j$},
\\
\displaystyle
-k_0^{2}
q^{i+1}
\prod_{h=1}^{\frac{i-1}{2}}
(1-q^{2h-i-1})
\qquad
&\hbox{if $i,j$ are odd and $i=j$},
\\
\displaystyle
\prod_{h=1}^{\frac{j}{2}}
(1-q^{2h-i-1})
\qquad
&\hbox{if $i$ is odd and $j$ is even},
\\
\displaystyle
\prod_{h=1}^{\lceil \frac{j}{2}\rceil}(q-q^{2h-i-1})
\qquad
&\hbox{if $i$ is even}
\end{array}
\right.\end{aligned}$$ for all $0\leq j\leq i\leq d$. In particular $L_{ii}$ is equal to $$(-q^{d+\lfloor \frac{i}{2}\rfloor} k_1^2 k_2^2)^{\lceil \frac{i}{2}\rceil}
\prod_{h=1}^{\lfloor \frac{i}{2} \rfloor}(1-q^{2h})^{-1}
\prod_{h=1}^i \varrho_h
\prod_{h=1}^{d-i} \psi_{d-h+1}
\times
\left\{
\begin{array}{ll}
\displaystyle
-k_0^{2}
q^{i+1}
\prod_{h=1}^{\frac{i-1}{2}}
(1-q^{2h-i-1})
\qquad
&\hbox{if $i$ is odd},
\\
\displaystyle
\prod_{h=1}^{\frac{i}{2}}(q-q^{2h-i-1})
\qquad
&\hbox{if $i$ is even}.
\end{array}
\right.$$ It follows from (i) and (ii) that $L_{ii}$ is nonzero for all $i=0,1,\ldots,d$. To see the irreducibility of $O(k_0,k_1,k_2,k_3)$ it remains to apply a routine argument similar to the proof of Theorem \[thm:irr\].
We now give our proof for Theorem \[thm:iso\_O\].
[*Proof of Theorem \[thm:iso\_O\].*]{} (i): Immediate from Proposition \[prop:iso2\_O\] and Theorem \[thm:irr\_O\].
(ii): By Theorem \[thm:irr\_O\] the $\H_q$-module $O(k_1,k_2,k_3,k_0)$ is irreducible. It follows from (i) that $O(k_1,k_2,k_3,k_0)$ is isomorphic to the $\H_q$-module $O(k_2,k_3,k_0,k_1)^\e$ where $\e=3\pmod{4}$. Combined with (i) we obtain (ii).
(iii): By Theorem \[thm:irr\_O\] the $\H_q$-module $O(k_2,k_3,k_0,k_1)$ is irreducible.It follows from (i) that $O(k_2,k_3,k_0,k_1)$ is isomorphic to the $\H_q$-module $O(k_3,k_0,k_1,k_2)^\e$ where $\e=3\pmod{4}$. Combined with (ii) we obtain (iii). $\square$
Proof of Theorem \[thm:odd\] {#section:odd}
============================
In the final section we are devoted to proving Theorem \[thm:odd\].
\[thm:onto\_O\] Assume that $\F$ is algebraically closed and $q$ is not a root of unity. Let $d\geq 0$ denote an even integer. If $V$ is a $(d+1)$-dimensional irreducible $\H_q$-module, then there exist nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0 k_1 k_2 k_3=q^{-d-1}$ such that the $\H_q$-module $O(k_0,k_1,k_2,k_3)$ is isomorphic to $V$.
Suppose that $V$ is a $(d+1)$-dimensional irreducible $\H_q$-module. According to Propositions \[prop:X\_eigenvector\] and \[prop:Y\_eigenvector\] we may divide the argument into the following cases (i)–(iv):
1. $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V$.
2. $t_1,t_2$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V$.
3. $t_1,t_2$ have a simultaneous eigenvector and $t_2,t_3$ have a simultaneous eigenvector in $V$.
4. $t_0,t_3$ have a simultaneous eigenvector and $t_2,t_3$ have a simultaneous eigenvector in $V$.
(i): Let $v$ denote a simultaneous eigenvector of $t_0$ and $t_3$ in $V$. Choose $k_0$ and $k_3$ as the eigenvalues $t_0$ and $t_3$ corresponding to $v$, respectively. Let $w$ denote a simultaneous eigenvector of $t_0$ and $t_1$ in $V$. Choose $k_1$ as the eigenvalue of $t_1$ corresponding to $w$. By Lemma \[lem:Schur\] the element $c_1$ acts on $V$ as scalar multiplication by $k_1+k_1^{-1}$ and there exists a nonzero scalar $k_2\in \F$ such that $c_2$ acts on $V$ as scalar multiplication by $k_2+k_2^{-1}$. Moreover we require $k_2$ to satisfy that the algebraic multiplicity of $k_2$ is greater than or equal to the algebraic multiplicity of $k_2^{-1}$ as eigenvalues of $t_2$ in $V$. By Theorem \[thm:universal\] there exists a unique $\H_q$-module homomorphism $$\begin{gathered}
\label{M->V_O}
M(k_0,k_1,k_2,k_3) \to V\end{gathered}$$ that sends $m_0$ to $v$.
Let $v_i$ denote the image of $m_i$ under (\[M->V\_O\]) for all $i=0,1,2,\ldots$. Suppose that there is an $i\in \{1,2,\ldots, d\}$ such that $v_i$ is an $\F$-linear combination of $v_0,v_1,\ldots,v_{i-1}$. Let $W$ denote the $\F$-subspace of $V$ spanned by $v_0,v_1,\ldots,v_{i-1}$. By Proposition \[prop:P\], $W$ is invariant under $\{t_i^{\pm 1}\}_{i=0}^3$. Hence $W$ is a proper $\H_q$-submodule of $V$, a contradiction to the irreducibility of $V$. Therefore $
\{v_i\}_{i=0}^d
$ is an $\F$-basis for $V$.
For $u\in V$ let $[u]$ denote the coordinate vector of $u$ relative to $\{v_i\}_{i=0}^d$. Let $a_i$ and $b_i$ denote the $i$-entries of $[v_{d+1}]$ and $[w]$ for all $i=0,1,\ldots,d$, respectively. Since $w$ is a $k_1$-eigenvector of $t_1$ we have $$\begin{gathered}
\label{t1w=k1w}
[t_1 w]=k_1[w].\end{gathered}$$
We now show that $a_0=0$. To do this, we divide the argument into two cases: (a) $w$ is a $k_0$-eigenvector of $t_0$; (b) $w$ is not a $k_0$-eigenvector of $t_0$.
(a): Using Proposition \[prop:P\](i) yields that the matrix representing $t_0$ with respect to the basis $$\begin{gathered}
v_0,
\quad
v_{i-1}+q^{-i} (v_i-v_{i-1})
\quad \hbox{for $i=2,4,\ldots,d$},
\quad
v_i
\quad \hbox{for $i=1,3,\ldots,d-1$}\end{gathered}$$ for $V$ is $$\begin{gathered}
\begin{pmatrix}
k_0 &{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&{\bf 0} &{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&{\bf 0}
\\
\hline
{\bf 0} &{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&k_0 I_{\frac{d}{2}}
&{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}& -k_0^{-1} I_{\frac{d}{2}}
\\
\hline
{\bf 0} & {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&{\bf 0} & {\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}&k_0^{-1} I_{\frac{d}{2}}
\end{pmatrix}.\end{gathered}$$ By the rank-nullity theorem the $k_0$-eigenspace of $t_0$ in $V$ is spanned by $$\begin{gathered}
\label{k0basis}
v_0,
\qquad
v_{i-1}+q^{-i} (v_i-v_{i-1})
\quad
\hbox{for all $i=2,4,\ldots,d$}.\end{gathered}$$ Hence $w$ is an $\F$-linear combination of (\[k0basis\]). Since $w\not=0$ there exists an $i\in\{0,2,\ldots,d\}$ with $b_i\not=0$. Let $k$ denote the largest even integer such that $b_k\not=0$. Suppose that $k\not=d$. Then $b_{k+2}=0$. Since $q$ is not a root of unity and by (\[k0basis\]) the coefficient $b_{k+1}=0$. By Proposition \[prop:P\](i) the $(k+1)$-entry of $[t_1 w]$ is equal to $k_1^{-1} b_k\not=0$, a contradiction to (\[t1w=k1w\]). Hence $b_d\not=0$. Evaluating the first entry of $[t_1 w]$ by using Proposition \[prop:P\](i), we obtain from (\[t1w=k1w\]) that $$k_1^{-1} a_0 b_d+ k_1 b_0=k_1 b_0.$$ Since $b_d\not=0$ it follows that $a_0=0$.
(b): Observe that $w$ is a $k_0^{-1}$-eigenvector of $t_0$ and $k_0^2\not=1$. Using Proposition \[prop:P\](i) yields that the $k_0^{-1}$-eigenspace of $t_0$ is spanned by $$\begin{gathered}
\label{k0inv-basis}
v_i-(1-k_0^2 q^i) v_{i-1}
\quad
\hbox{for all $i=2,4,\ldots,d$}\end{gathered}$$ Hence $w$ is an $\F$-linear combination of (\[k0inv-basis\]). Evaluating the first entry of $[t_1 w]$ by using Proposition \[prop:P\](i), we obtain from (\[t1w=k1w\]) that $$k_1^{-1} a_0 b_d=0.$$ By a similar argument to the case (a), we see that $b_d\not=0$. Hence $a_0=0$.
We now show that $a_i=0$ for all $i=1,2,\ldots,d$. Let $\{\varrho_i\}_{i\in \Z}$ denote the parameters (\[varphi\]) corresponding to the current parameters $k_0,k_1,k_2,k_3$. Using Lemma \[lem:XYinP\](i) yields that $(1-k_0 k_3 q^{d+2} X) v_i$ is equal to $$\begin{split}\label{XV}
\left\{
\begin{array}{ll}
(1-k_0^2 k_3^2 q^{d+2}) v_0
\qquad
&\hbox{if $i=0$},
\\
(1-q^{d-i+1}) v_i
+q^{d-i+1} \varrho_i v_{i-1}
\qquad
&\hbox{if $i=1,3,\ldots,d-1$},
\\
(1-k_0^2 k_3^2 q^{d+i+2}) v_i
-
q^{d-i+2}\varrho_i
(v_{i-1}-\varrho_{i-1} v_{i-2})
\qquad
&\hbox{if $i=2,4,\ldots,d$}.
\end{array}
\right.
\end{split}$$ By Lemma \[lem:XYinP\](i) with $i=d+1$ we have $$\begin{gathered}
\label{Xud+1}
[(1-k_0 k_3 q^{d+2} X) v_{d+1}]=
\begin{pmatrix}
0
\\
0
\\
\vdots
\\
0
\\
\varrho_{d+1}
\end{pmatrix}.\end{gathered}$$ Evaluating the first $d$ entries of the left-hand side of (\[Xud+1\]) by using (\[XV\]), we obtain that $$\begin{gathered}
\label{ak_O}
\varrho_i a_i
=\left\{
\begin{array}{ll}
(k_0^2 k_3^2 q^{d+i+1}-1) a_{i-1}
\qquad
&\hbox{for $i=1,3,\ldots,d-1$},
\\
(q^{i-d-2}-1) a_{i-1}
\qquad
&\hbox{for $i=2,4,\ldots,d$}.
\end{array}
\right.\end{gathered}$$ Suppose on the contrary that there exists an $i\in\{1,2,\ldots,d\}$ with $a_i \not=0$ and $a_{i-1}=0$. By (\[ai\]) the scalar $\varrho_i=0$. Let $W$ denote the $\F$-subspace of $V$ spanned by $v_i,v_{i+1},\ldots,v_d$. Combined with Proposition \[prop:P\] yields that $W$ is invariant under $\{t_i^{\pm 1}\}_{i=0}^3$. Hence $W$ is a proper $\H_q$-submodule of $V$, a contradiction to the irreducibility of $V$. Therefore $a_i=0$ for all $i=0,1,\ldots,d$. In other words $$\begin{gathered}
\label{ud+1=0_O}
v_{d+1}=0.\end{gathered}$$
Combined with Proposition \[prop:P\](i) yields that the roots of the characteristic polynomial of $t_2$ in $V$ are $$\underbrace{k_2, k_2, \ldots, k_2}_{\hbox{{\tiny $d/2$ copies}}},
\underbrace{k_2^{-1},k_2^{-1},\ldots, k_2^{-1}}_{\hbox{{\tiny $d/2$ copies}}},
k_0^{-1} k_1^{-1} k_3^{-1} q^{-d-1}.$$ By the choice of $k_2$ we have $$k_0 k_1 k_2 k_3=q^{-d-1}.$$ By Lemma \[lem:XYinP\](ii) we have $$\begin{gathered}
\label{e:Yv_O}
\prod_{i=0}^d(1-k_0 k_1 q^{2\lceil \frac{i}{2}\rceil} Y^{(-1)^{i-1}}) v_0=v_{d+1}.\end{gathered}$$ By (\[ud+1=0\_O\]) the right-hand side of (\[e:Yv\_O\]) is zero. Thus, it follows from Proposition \[prop:universal\_O\] that there exists a nontrivial $\H_q$-module homomorphism $$\begin{gathered}
\label{O->V}
O(k_0,k_1,k_2,k_3)\to V.\end{gathered}$$ Since the $\H_q$-module $V$ is irreducible it follows that (\[O->V\]) is onto. Since $O(k_0,k_1,k_2,k_3)$ and $V$ are of dimension $d+1$ it follows that (\[O->V\]) is an isomorphism. Therefore the theorem holds for the case (i).
(ii): Let $\e=1\pmod 4$. Since $t_0^\e=t_1,t_1^\e=t_2,t_3^\e=t_0$ by Table \[Z/4Z-action\] the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there are nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0 k_1 k_2 k_3 =q^{-d-1}$ such that $O(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Hence $O(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. Combined with Theorem \[thm:iso\_O\](i) the theorem holds for the case (ii).
(iii): Let $\e=2\pmod 4$. Since $t_0^\e=t_2,t_1^\e=t_3,t_3^\e=t_1$ by Table \[Z/4Z-action\] the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there are nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0 k_1 k_2 k_3 =q^{-d-1}$ such that $O(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Hence $O(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. Combined with Theorem \[thm:iso\_O\](ii) the theorem holds for the case (iii).
(iv): Let $\e=3\pmod 4$. Since $t_0^\e=t_3,t_1^\e=t_0,t_3^\e=t_2$ by Table \[Z/4Z-action\] the elements $t_0,t_3$ have a simultaneous eigenvector and $t_0,t_1$ have a simultaneous eigenvector in $V^\e$. By (i) there are nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0 k_1 k_2 k_3 =q^{-d-1}$ such that $O(k_0,k_1,k_2,k_3)$ is isomorphic to $V^\e$. Hence $O(k_0,k_1,k_2,k_3)^{-\e}$ is isomorphic to $V$. Combined with Theorem \[thm:iso\_O\](iii) the theorem holds for the case (iv).
\[lem:injective\_O\] Let $d\geq 0$ denote an even integer. For any nonzero $k_0,k_1,k_2,k_3\in \F$ with $k_0 k_1 k_2 k_3=q^{-d-1}$ the determinants of $t_0,t_1,t_2,t_3$ on $O(k_0,k_1,k_2,k_3)$ are $k_0,k_1,k_2,k_3$ respectively.
It is routine to verify the lemma by using Proposition \[prop:O\](i).
We finish this paper with the proof for Theorem \[thm:odd\].
[*Proof of Theorem \[thm:odd\].*]{} By Theorem \[thm:irr\_O\] the map $\mathcal O$ is well-defined. By Theorem \[thm:onto\_O\] the map $\mathcal O$ is onto. Since each element of $\H_q$ has the same determinant on all isomorphic finite-dimensional $\H_q$-modules, it follows from Lemma \[lem:injective\_O\] that $\mathcal O$ is one-to-one. $\square$
|
---
abstract: 'Selected topics of the top-quark mass measurements in well-defined schemes are presented. The measurements have been performed using data recorded with the ATLAS and CMS detectors at the LHC at proton-proton centre-of-mass energies of 7 and 8 TeV. Precision theoretical QCD calculations for both inclusive top-antitop quark pair production and top-antitop quark pair production with an additional jet to extract the top quark mass in the pole-mass scheme have been used.'
author:
- Teresa Barillari
bibliography:
- 'TeresaBarillariLHCP2015.bib'
nocite: '[@*]'
title: 'Measurements of the top-quark mass in fixed schemes and with alternative methods using the ATLAS and CMS detectors at the LHC'
---
INTRODUCTION
============
The top quark is by far the heaviest known fermion and the heaviest known fundamental particle. It plays an important role in the Standard Model (SM). Precise measurements of the top-quark mass (${\ensuremath{m_t}}$) provide a key input to consistency tests of the SM. The mass of the Higgs boson and the top quark are also important parameters in the determination of the vacuum stability [@Degrassi:2012ry; @Baak:2014ora].
Nowadays, the most precise determinations of ${\ensuremath{m_t}}$ have been achieved experimentally from kinematical reconstruction of the measured top-quark decay products, e.g. measuring the semi-leptonic decay channel of top-antitop quark pairs (${\ensuremath{t\overline{t}}}$), where one top quark decays into a b quark, a charged lepton and its neutrino and the other top quark decays into a b quark and two u/d/c/s quarks, yielding a value of ${\ensuremath{m_t}}=$ 172.35 $\pm$ 0.51 GeV [@Khachatryan:2015hba]. These ${\ensuremath{m_t}}$ determinations, however, have not been linked so far in an unambiguous manner to a Lagrangian top-quark mass in a specific renormalization scheme as employed in perturbative calculations in quantum chromodynamics (QCD), electroweak fits, or any theoretical prediction in general [@Moch:1702814; @Buckley:2011ms; @Hoang:2014oea]. The values of ${\ensuremath{m_t}}$ extracted using these schemes are usually identified with the top-quark pole mass, ${\ensuremath{m_t^{pole}}}$. Present studies estimate differences between the two top-quark mass definitions, ${\ensuremath{m_t}}$ and a theoretically well defined short-distance mass definition at a low scale (e.g. ${\ensuremath{m_t^{pole}}}$), of about 1 GeV. In addition to direct ${\ensuremath{m_t}}$ measurements as mentioned above, the mass dependence of the QCD prediction for the cross section ($\sigma_{{\ensuremath{t\overline{t}}}}$) can be used to determine ${\ensuremath{m_t}}$ by comparing the measured to the predicted $\sigma_{{\ensuremath{t\overline{t}}}}$ [@Alekhin:2012py; @Ahrens:2011px; @Langenfeld:2009wd; @Abazov:2011pta; @Beneke:2011mq; @Beneke:2012wb]. Although the sensitivity of $\sigma_{{\ensuremath{t\overline{t}}}}$ to ${\ensuremath{m_t}}$ might not be strong enough to make this approach competitive in precision, it yields results affected by different sources of systematic uncertainties compared to the direct ${\ensuremath{m_t}}$ measurements and allows for extractions of ${\ensuremath{m_t}}$ in theoretically well-defined mass schemes. The values extracted using these methods are usually identified with the top-quark pole mass.
This distinction of the theoretical description of the measured parameter, e.g. either the parameter in the underlying Monte Carlo (MC) generator, $m_t^{MC}$ (or simply ${\ensuremath{m_t}}$), the mass term in the top-quark propagator, ${\ensuremath{m_t^{pole}}}$, or the mass in a well defined low-scale short distance scheme [@Moch:1702814; @Hoang:2008xm], is recently gaining in importance.
In the following, selected ${\ensuremath{m_t^{pole}}}$ measurements performed by the ATLAS [@Aad:2008zzm] and CMS [@Chatrchyan:2008aa] experiments at LHC [@Evans:2008zzb] using data at proton-proton (pp) centre-of-mass energies of 7 and 8 TeV, are presented.
TOP-QUARK POLE MASS MEASUREMENTS
================================
In contrast to the standard kinematical reconstruction of the measured top-quark decay product methods mentioned above, cross-section-like observables can be used to compare QCD predictions depending on ${\ensuremath{m_t^{pole}}}$, with unfolded data. The unfolding removes detector effects, and, in addition these measurements benefit from the larger independence from the mass definition in the used MC generators. For the total cross-section measurements, however, a 5% uncertainty translates into a 1% uncertainty in the top-quark mass [@Alekhin:2013nda] and the difference from going from next-to-leading order (NLO) to next-to-next-to-leading order (NNLO) predictions is even larger ($\sim$ 10%). Experimentally the challenges lie in the unfolding of the data and in the absolute normalization. Furthermore measurements of ${\ensuremath{m_t^{pole}}}$ involving new shape-like observables as proposed in [@Alioli:2013mxa] can help reduce both theoretical and experimental uncertainties.
Measurements of Top-Quark Pole Mass in ${\ensuremath{t\overline{t}}}$ Di-Lepton Events
--------------------------------------------------------------------------------------
The measurements of the ${\ensuremath{t\overline{t}}}$ production cross-section, $\sigma_{{\ensuremath{t\overline{t}}}}$, together with the NNLO prediction in QCD including the resummation of next-to-next-to-leading-logarithmic (NNLL) soft gluon terms [@Czakon:2011xx], are used to determine the top-quark pole mass. Most of such measurements are performed in the electron-muon ($e$ - $\mu$) channel, where each W boson from the top quark decays into a lepton and a neutrino. Events are required to contain an oppositely charged $e$ - $\mu$ pair. The restriction to the di-lepton channel allows obtaining a particular clean ${\ensuremath{t\overline{t}}}$ event sample. The value of ${\ensuremath{m_t^{pole}}}$ is determined from the $\sigma_{{\ensuremath{t\overline{t}}}}$ measurements in pp collisions at centre-of-mass energies of $\sqrt{s} = 7$ TeV and $\sqrt{s} = 8$ TeV with the CMS and ATLAS detector at LHC. Both experiments assume a top-quark mass of $m_t^{MC} = 172.5$ GeV in simulations to extract the reconstruction efficiency.
### CMS Top-Quark Pole Mass Measurements
Compared to previous ${\ensuremath{m_t^{pole}}}$ measurements at 7 TeV [@Chatrchyan:2012bra] and at 8 TeV [@Chatrchyan:2013faa] the latest CMS results [@CMS-PAS-TOP-13-004] include the full CMS data samples with integrated luminosities of 5.0 $fb^{-1}$ (7 TeV) and 19.7 $fb^{-1}$ (8 TeV). The value of ${\ensuremath{m_t^{pole}}}$ at NNLO+NNLL is extracted by confronting the measured cross section $\sigma_{{\ensuremath{t\overline{t}}}}$ at 7 and 8 TeV with predictions employing different parton density function (PDF) sets: NNPDF3.0 [@Ball:2014uwa], CT14 [@Dulat:2015mca], and MMHT2014 [@Harland-Lang:2014zoa]. The obtained ${\ensuremath{m_t^{pole}}}$ values are listed in Table \[tab:mpole\]. The contributions from uncertainties on the CT14 PDF set are scaled to 68% confidence level.
A weighted average is calculated, taking into account all systematic uncertainty correlations between the measured cross sections at 7 and 8 TeV and assuming 100% correlated uncertainties for the theory predictions at the two energies. The combined ${\ensuremath{m_t^{pole}}}$ results are listed in Table \[tab:mpole\] and are in good agreement with each other and the world average value [@ATLAS:2014wva].
PDF ${\ensuremath{m_t^{pole}}}$ ($\sqrt{s} =$ 7 TeV) \[GeV\] ${\ensuremath{m_t^{pole}}}$ ($\sqrt{s} =$ 8 TeV) \[GeV\] ${\ensuremath{m_t^{pole}}}$ ($\sqrt{s} =$ 7 + $\sqrt{s} =$ 8 TeV) \[GeV\]
---------------------------------- ---------------------------------------------------------- ---------------------------------------------------------- ---------------------------------------------------------------------------
NNPDF3.0 [@Ball:2014uwa] 173.4 $\pm^{2.0}_{2.0}$ 173.9 $\pm^{1.9}_{2.0}$ 173.6 $\pm^{+1.7}_{1.8}$
MMHT2014 [@Harland-Lang:2014zoa] 173.7 $\pm^{2.0}_{2.1}$ 174.2 $\pm^{1.9}_{2.2}$ 173.9 $\pm^{+1.8}_{1.9}$
CT14 [@Dulat:2015mca] 173.9 $\pm^{2.3}_{2.4}$ 174.3 $\pm^{2.2}_{2.4}$ 174.1$\pm^{+2.1}_{2.2}$
: Top-quark pole mass measured by CMS at NNLO+NNLL extracted by confronting the measured ${\ensuremath{t\overline{t}}}$ production cross section at 7 and 8 TeV [@Chatrchyan:2012bra; @Chatrchyan:2013faa; @CMS-PAS-TOP-13-004]. The obtained combined ${\ensuremath{m_t^{pole}}}$ results are also listed ($\sqrt{s} =$ 7 + $\sqrt{s} =$ 8 TeV).[]{data-label="tab:mpole"}
Figure \[fig:1\] shows the combined likelihood of the measured and predicted dependence of the ${\ensuremath{t\overline{t}}}$ production cross section on ${\ensuremath{m_t^{pole}}}$ for 7 (left plot) and 8 TeV (right plot).
![Combined likelihood of the measured and predicted dependence of the ${\ensuremath{t\overline{t}}}$ production cross section on the top-quark mass for 7 (left plot) and 8 TeV (right plot) in CMS [@CMS-PAS-TOP-13-004]. The total one standard deviation uncertainty is indicated by a black contour. []{data-label="fig:1"}](CMS-PAS-TOP-13-004-Figure-010-a.eps "fig:"){width="220pt"} ![Combined likelihood of the measured and predicted dependence of the ${\ensuremath{t\overline{t}}}$ production cross section on the top-quark mass for 7 (left plot) and 8 TeV (right plot) in CMS [@CMS-PAS-TOP-13-004]. The total one standard deviation uncertainty is indicated by a black contour. []{data-label="fig:1"}](CMS-PAS-TOP-13-004-Figure-010-b.eps "fig:"){width="220pt"}
In another measurement at $\sqrt{s} =$ 8 TeV CMS [@CMS-PAS-TOP-14-014] uses a folding technique to map fixed order QCD calculations depending on ${\ensuremath{m_t^{pole}}}$ as implemented in the Monte Carlo for Femtobarn calculation MCFM [@Campbell:2010ff], to predict the shape in ${\ensuremath{m_{lb}^{min}}}$. The top quark decay chain considered in this analysis is $t\to W b$ followed by $W \to l\nu$. Neglecting both leptons and b-quark masses, at leading order the quantity ${\ensuremath{m_{lb}}}$ is directly related to ${\ensuremath{m_t}}$ and the mass of the W boson, $m_W$, as follows: ${\ensuremath{m_{lb}}}^2 = \frac{{\ensuremath{m_t}}^2 - m_{w}^2}{2}(1 - \cos\theta_{lb})$. Here, $\theta_{lb}$ is the opening angle between the lepton and the b quark in the W-boson rest frame. The distribution of ${\ensuremath{m_{lb}}}$ has an end point at $max({\ensuremath{m_{lb}}}) \sim \sqrt{{\ensuremath{m_t}}^2 - m_{W'}^2}$, i.e. around 153 GeV for a top-quark mass of 173 GeV. In the analysis, ${\ensuremath{m_{lb}}}$ is reconstructed by choosing the permutation that minimizes the value of ${\ensuremath{m_{lb}}}$ in each event and only the b-jet candidate with the highest transverse momentum $p_{\perp}$ is considered together with both leptons ($e$ and $\mu$). Only one top quark in each event is used. In this particular definition, the combination yielding the smallest ${\ensuremath{m_{lb}}}$ in the event is kept, and referred to as ${\ensuremath{m_{lb}^{min}}}$, shown in Figure \[fig:2\]. The response matrices in ${\ensuremath{m_{lb}^{min}}}$ are obtained from fully simulated events obtained using the matrix element generator MADGRAPH 5.1.5.11 [@Alwall:2011uj] with MADSPIN [@Artoisenet:2012st] for the decay of heavy resonances, PYTHIA 6.426 [@Sjostrand:2006za] for parton showering; the MC events have been passed through a full simulation of the CMS detector based on GEANT [@Agostinelli:2002hh] (combination called MADGRAPH + PYTHIA + GEANT). By using the information on the rate of events alone a value of ${\ensuremath{m_t}}= 171.4 \pm 0.4_{stat} \pm 1.0_{syst}$ GeV is measured. Combining the results obtained using rate+${\ensuremath{m_{lb}^{min}}}$ shape fits one is able to extract ${\ensuremath{m_t}}= 173.1^{1.9}_{1.8}$ GeV. These results can be compared to the mass extraction from the same dataset via the total cross-section calculated at NNLO.
![Normalized event yields obtained by CMS [@CMS-PAS-TOP-14-014], for ${\ensuremath{t\overline{t}}}$ production at the LHC at $\sqrt{s} =$ 8 TeV, presented as a function of ${\ensuremath{m_{lb}^{min}}}$. The bullets are the experimental data points and the error bars indicate their statistical uncertainties. The inset shows the $\chi^2$ distribution as a function of ${\ensuremath{m_t}}$ as determined from the fit of the simulation to the shape of the data.[]{data-label="fig:2"}](CMS-PAS-TOP-14-014-Figure-008.eps){width="250pt"}
### ATLAS Top-Quark Pole Mass Measurements
ATLAS also extracts ${\ensuremath{m_t^{pole}}}$ at NNLO+NNLL by confronting the measured production cross section $\sigma_{{\ensuremath{t\overline{t}}}}$ at 7 and 8 TeV with predictions employing different PDF sets [@Aad:2014kva]: CT10 NLO [@Lai:2010vv], MSTW 2008 68% CL NLO [@Martin:2009iq], and NNPDF 2.3 NLO [@Ball:2012cx]. The extraction of ${\ensuremath{m_t^{pole}}}$ is performed by maximizing a Bayesian likehood function separately for each PDF set and centre-of-mass energy to give ${\ensuremath{m_t^{pole}}}$ values shown in Table \[tab:mpoletop1\].
PDF ATLAS ${\ensuremath{m_t^{pole}}}$ $\sqrt{s} =7$ TeV \[GeV\] ATLAS ${\ensuremath{m_t^{pole}}}$ $\sqrt{s} =8$ TeV \[GeV\]
--------------------------------- ------------------------------------------------------------- -------------------------------------------------------------
C10 NNLOCT10 NLO [@Lai:2010vv] 171.4 $\pm$ 2.6 174.1 $\pm$ 2.6
MSTW 68 % NNLO [@Martin:2009iq] 171.2 $\pm$ 2.4 174.0 $\pm$ 2.5
NNPDF2.3 5f FFN [@Ball:2012cx] $171.3^{+2.2}_{-2.3}$ 174.2 $\pm$ 2.4
: Measurements performed by ATLAS of ${\ensuremath{m_t^{pole}}}$ at NNLO+NNLL extracted by confronting the measured production cross section $\sigma_{{\ensuremath{t\overline{t}}}}$ with predictions employing different PDF sets.[]{data-label="tab:mpoletop1"}
Finally ${\ensuremath{m_t^{pole}}}$ is extracted from the combined $\sqrt{s} =$ 7 and $\sqrt{s} =$ 8 TeV dataset. The resulting value using the envelope of all three considered PDF sets is ${\ensuremath{m_t^{pole}}}= 172.9^{+2.5}_{-2.6}$ GeV. The results are shown in Figure \[fig:3\], together with previous determinations using similar techniques from D0 [@Abazov:2009ae] and CMS [@Chatrchyan:2013haa].
![Comparison of ${\ensuremath{m_t^{pole}}}$ values determined from ATLAS and previous measurements [@Aad:2014kva]. []{data-label="fig:3"}](ATLASarXiv1406-5375fig-08a.eps){width="250pt"}
All extracted values are consistent with the average of measurements from kinematic reconstruction of ${\ensuremath{t\overline{t}}}$ events of $173.34 \pm 0.76$ GeV [@ATLAS:2014wva], showing good compatibility of top-quark masses extracted using very different techniques and assumptions.
Measurements of Top-Quark Pole Mass in ${\ensuremath{t\overline{t}}}\,+$ 1-Jet Events
-------------------------------------------------------------------------------------
The normalized differential cross section for ${\ensuremath{t\overline{t}}}$ production in association with at least 1-jet is studied as a function of the inverse of the invariant mass of the ${\ensuremath{t\overline{t}}}\,+$ 1-jet system. This distribution is used by the ATLAS experiment [@Aad:2015waa] for a precise determination of ${\ensuremath{m_t^{pole}}}$. A new observable suggested in [@Alioli:2013mxa] is used in this measurement: ${\cal R}({\ensuremath{m_t^{pole}}},\rho_s) = \frac{1}{{\ensuremath{\sigma_{t\overline{t} + 1 jet}}}}\frac{d{\ensuremath{\sigma_{t\overline{t} + 1 jet}}}}{d \rho_s}({\ensuremath{m_t^{pole}}},\rho_{s})$. The differential is taken in $\rho_s = 2 m_0 / \sqrt{s_{{\ensuremath{t\overline{t}}}j}}$, that is the ratio of an arbitrary mass scale in the vicinity of ${\ensuremath{m_t}}$, here set to $m_0 = 170$ GeV, over the invariant ${\ensuremath{t\overline{t}}}+ 1\,$jet mass. ${\ensuremath{t\overline{t}}}$ events are selected at $\sqrt{s} = 7$ TeV in a similar way as done for the the lepton+jets analysis [@Aad:2015nba], and an additional central jet with $p_{\perp} > 50$ GeV is added. An SVD unfolding [@Hocker:1995kb] with a response matrix from POWHEG+PYTHIA+GEANT4 [@Sjostrand:2006za; @Agostinelli:2002hh; @Alioli:2010xd] maps the measured $\rho_s$ to parton level. The unfolded distribution of ${\cal R}({\ensuremath{m_t^{pole}}},\rho_s)$ is shown in Figure \[fig:4\] (left). The measurement of ${\ensuremath{m_t^{pole}}}=$ 173.7 $\pm 1.5_{stat} \pm 1.4_{syst}$ GeV is then obtained in a $\chi^2$-fit to $0.25 < \rho_s < 1$ with $\rho_s > 0.675$ being the most sensitive bin, as shown in Figure \[fig:4\] (right).
![Unfolded ${\cal R}({\ensuremath{m_t^{pole}}},\rho_s)$ distribution as measured by ATLAS [@Aad:2015waa] (left). The predictions of the ${\ensuremath{t\overline{t}}}$ + 1-jet calculation at NLO+PS using three different masses (${\ensuremath{m_t^{pole}}}=$ 170, 175 and 180 GeV) are shown with the result of the best fit to the data, ${\ensuremath{m_t^{pole}}}=$ 173.7 $\pm$ 1.5 (stat.) GeV. The value of the most sensitive interval of the ${\cal R}$-distribution $\rho_s >$ 0.65 [@Aad:2015waa] (right). The black point corresponds to the data. The shaded area indicates the statistical uncertainty of this bin.[]{data-label="fig:4"}](1507-01769fig-04.eps "fig:"){width="240pt"} ![Unfolded ${\cal R}({\ensuremath{m_t^{pole}}},\rho_s)$ distribution as measured by ATLAS [@Aad:2015waa] (left). The predictions of the ${\ensuremath{t\overline{t}}}$ + 1-jet calculation at NLO+PS using three different masses (${\ensuremath{m_t^{pole}}}=$ 170, 175 and 180 GeV) are shown with the result of the best fit to the data, ${\ensuremath{m_t^{pole}}}=$ 173.7 $\pm$ 1.5 (stat.) GeV. The value of the most sensitive interval of the ${\cal R}$-distribution $\rho_s >$ 0.65 [@Aad:2015waa] (right). The black point corresponds to the data. The shaded area indicates the statistical uncertainty of this bin.[]{data-label="fig:4"}](ATLAS1507-01769figaux_09.eps "fig:"){width="280pt"}
CONCLUSIONS
===========
Measurements of ${\ensuremath{m_t^{pole}}}$ using alternative methods have been performed by both the ATLAS and CMS experiments using data collected at $\sqrt{s} =$ 7 and 8 TeV at LHC. The latest CMS [@CMS-PAS-TOP-13-004] results obtained using di-lepton ${\ensuremath{t\overline{t}}}$ events at 7 and the 8 TeV give a ${\ensuremath{m_t^{pole}}}=$ 173.6 $\pm^{+1.7}_{1.8}$ GeV. The normalized differential cross section for ${\ensuremath{t\overline{t}}}$ production in association with at least 1-jet studied by the ATLAS experiment [@Aad:2015waa] at 7 TeV give a measurement of ${\ensuremath{m_t^{pole}}}=$ 173.7 $\pm 1.5_{stat} \pm 1.4_{syst}$ GeV. All the extracted values of ${\ensuremath{m_t^{pole}}}$ are consistent with ${\ensuremath{m_t}}$ measurements obtained using standard kinematic reconstruction of ${\ensuremath{t\overline{t}}}$ events.
ACKNOWLEDGMENTS
===============
I would like to thank the ATLAS and CMS Collaborations for giving me the opportunity to give this talk at this conference and the top-quark groups conveners of both experiments for providing me the material presented here.
|
---
title: '*Supplementary Material*: Generative Patch Priors for Practical Compressive Image Recovery'
---
Derivation for $a^*$ and $b^*$
==============================
Consider a vectorized square block of an image $\mathbf{x} \in \mathcal{X} \subset \mathbb{R}^n$ which we want to sense, and denote by $\mathbf{y} \in \mathbb{R}^m$ the compressive measurements obtained by the sensor. Given a measurement matrix $\Phi \in \mathbb{R}^{m \times n}$, with $m < n$ and $\Phi_{i,j} \sim \mathcal{N}(0,1)$, the compressive recovery problem is to estimate $\mathbf{x}$ accurately from $\mathbf{y}$. In the ideal setting, i.e., compressive sensing with known calibration the sensing model is given by $\mathbf{y} = \Phi \mathbf{x}$. Instead we consider a simple calibration model— $\mathbf{y} = (a \Phi + b\mathbf{1})\mathbf{x}$, where $a,b \in \mathbb{R}^1$ are unknown calibration parameters and have to be estimated, and $\mathbf{1} \in \mathbb{R}^{m\times n}$ is a matrix of the same size as $\Phi$ with $1$s.
In order to derive $a$ and $b$, we assume we have a current estimate of the solution ${\mathbf{x}}$ from a pre-trained generator $\mathcal{G}(\mathbf{z})$ for a latent vector $\mathbf{z}$. This is be randomly initialized at the beginning. Under this calibration model, let us define mean squared error loss function as follows: $$\begin{gathered}
\mathcal{L} = ({\mathbf{y}}- {(a\Phi+b\mathbf{1})}{\mathbf{x}})^{\intercal}({\mathbf{y}}- {(a\Phi+b\mathbf{1})}{\mathbf{x}}) \\
\implies \mathcal{L} = {\mathbf{y}}^{\intercal}{\mathbf{y}}- {\mathbf{y}}^{\intercal}{(a\Phi+b\mathbf{1})}{\mathbf{x}}- {\mathbf{x}}^{\intercal}{(a\Phi+b\mathbf{1})}^{\intercal}{\mathbf{y}}+ {\mathbf{x}}^{\intercal}{(a\Phi+b\mathbf{1})}^{\intercal}{(a\Phi+b\mathbf{1})}{\mathbf{x}}\end{gathered}$$ As a result, the derivatives with respect to each unknown $a,b$ are:
$$\begin{split}
\label{eq:derivative_a_and_b}
\frac{\partial \mathcal{L}}{\partial a} = -{\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}- {\mathbf{x}}^T{\Phi}^{\intercal}{\mathbf{y}}+ {\mathbf{x}}^{\intercal}\left[2a{\Phi}^{\intercal}\Phi+b{\Phi}^{\intercal}{\mathbf{1}}+b{\mathbf{1}}^{\intercal}{\Phi}\right]{\mathbf{x}}& \\
\mbox{Similarly,~}\frac{\partial \mathcal{L}}{\partial b} = - {\mathbf{y}}^{\intercal}{\mathbf{1}}{\mathbf{x}}- {\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{y}}+ {\mathbf{x}}^{\intercal}\left[a{\Phi}^{\intercal}{\mathbf{1}}+ {\mathbf{1}}^{\intercal}{\Phi}+ 2b{\mathbf{1}}^{\intercal}{\mathbf{1}}\right]{\mathbf{x}}&
\end{split}$$
By setting these derivatives to zero, we get: $$\begin{aligned}
\frac{\partial \mathcal{L}}{\partial a} = 0
\implies -{\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}-{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{y}}+ 2a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\Phi}{\mathbf{x}}+ b{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}+ b{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\Phi}{\mathbf{x}}= 0.\label{b_0}\\
\implies -2{\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}+ 2a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\Phi}{\mathbf{x}}+ 2b{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}= 0.\label{b_1}\\
\implies b = \frac{{\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}- a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\Phi}{\mathbf{x}}}{{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}}\label{b_2}\end{aligned}$$ Note, in all the terms are scalars and therefore ${\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}={\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{y}}$ etc. Next, we take the partial derivative with respect to $b$. $$\begin{aligned}
\frac{\partial \mathcal{L}}{\partial b} = 0
\implies {\mathbf{y}}^{\intercal}{\mathbf{1}}{\mathbf{x}}-{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{y}}+ a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}+a{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\Phi}{\mathbf{x}}+ 2b{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{1}}{\mathbf{x}}= 0. \label{a_0} \\
\implies -2{\mathbf{y}}^{\intercal}{\mathbf{1}}{\mathbf{x}}+ 2a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}+ 2b{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{1}}{\mathbf{x}}= 0. \label{a_1}\\
\implies b = \frac{{\mathbf{y}}^T{\mathbf{1}}{\mathbf{x}}- a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}}{{\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{1}}{\mathbf{x}}} \label{a_2}\end{aligned}$$ Combining equations and , we get the following: $$\begin{aligned}
\label{ab_0}
\left({\mathbf{y}}^{\intercal}{\Phi}{\mathbf{x}}- a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\Phi}{\mathbf{x}}\right){\mathbf{x}}^{\intercal}{\mathbf{1}}^{\intercal}{\mathbf{1}}{\mathbf{x}}= \left({\mathbf{y}}^T{\mathbf{1}}{\mathbf{x}}- a{\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}\right){\mathbf{x}}^{\intercal}{\Phi}^{\intercal}{\mathbf{1}}{\mathbf{x}}\end{aligned}$$ As in the paper, let us define scalar quantities for notational convenience: $c_{\Phi} = \mathbf{y}^T\Phi\mathbf{x}, c_{\mathbf{1}} = \mathbf{y}^T\mathbf{1}\mathbf{x}, \theta_{\Phi} = (\Phi\mathbf{x})^T(\Phi\mathbf{x}), \theta_{\mathbf{1}} = (\mathbf{1}\mathbf{x})^T(\mathbf{1}\mathbf{x}), \lambda = (\Phi\mathbf{x})^T(\mathbf{1}\mathbf{x})$. This implies, is now reformulated as: $$\begin{aligned}
(c_\Phi - a\theta_\Phi)\theta_\mathbf{1} = (c_\mathbf{1}-a\lambda)\lambda\\
\implies a = \frac{c_\mathbf{1}\lambda-c_{\Phi}\theta_\mathbf{1}}{\lambda^2-\theta_\Phi\theta_\mathbf{1}}\label{a_star}\\
\mbox{~and~} b = \frac{c_{\mathbf{1}}-a^*\lambda}{\theta_{\mathbf{1}}}\label{b_star}\end{aligned}$$ In each step of the alternating minimization, we use the estimates from , and and update the latent vector $\mathbf{z}$, which is repeated until convergence in $a,b,\mathbf{z}$.
Additional results
==================
In figure \[fig:pr10\], we show sample reconstructions for the phase retrieval task at a measurement rate of $10\%$.
{width="0.99\linewidth"}
Self calibration under unknown sensor shift
-------------------------------------------
In figure \[fig:sc\_shift\] we illustrate how reconstruction methods can easily fail to recover the solution when there is even a small shift in the operator. We simulate this using $b = -0.25$ and compare the proposed self calibration approach against no calibration and the untrained network prior (DIP) [@ulyanov2018deep]. We observe that the self calibration is able to successfully correct for the unknown shift, compared to the models that do not account for it.
{width="0.99\linewidth"}
GPP for image inpainting
------------------------
GPP is a generic prior to constrain solutions to the natural image manifold. We show an example here of how it can be used in other challenging inverse problems. In figure \[fig:inpainting\], we illustrate the efficiency of GPP for a for inpainting, where only a small number random pixels are shown, and the task is to recover the original image. Unlike most existing methods, we see that GPP’s solution degrades more gracefully than DIP, even recovering some signal when 99.5% of the pixels are missing.
{width="0.99\linewidth"}
|
---
abstract: |
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one.
We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius $2$, then the supplementary code is also completely regular with covering radius at most $2$. Moreover, in this case, either both codes are completely transitive, or both are not.
With this technique, we obtain infinite families of completely regular and completely transitive codes which are quasi-perfect uniformly packed.
author:
- 'J. Borges, J. Rifà and V. A. Zinoviev'
nocite: '[@*]'
title: On completely regular and completely transitive codes derived from Hamming codes
---
<span style="font-variant:small-caps;">Joaquim Borges</span>
<span style="font-variant:small-caps;">Josep Rifà</span>
<span style="font-variant:small-caps;">Victor Zinoviev</span>
Introduction
============
Let ${\mathbb{F}}_q$ be the finite field of order $q$. The [*weight*]{} of a vector ${{\bf v}}\in{\mathbb{F}}_q^n$, denoted by ${\mbox{wt}}({{\bf v}})$, is the number of nonzero coordinates of ${{\bf v}}$. The vector of weight $0$, or [*zero vector*]{}, is denoted by ${{\mathbf{0}}}$. The [*distance*]{} between two vectors ${{\bf v}},{{\bf w}}\in{\mathbb{F}}_q^n$, denoted by $d({{\bf v}},{{\bf w}})$, is the number of coordinates in which they differ. A subset $C\subset{\mathbb{F}}_q^n$ is called a $q$-[*ary code*]{} of length $n$. Denote by $d$ the [*minimum distance*]{} among codewords in $C$. The [*packing radius*]{} of $C$ is $e=\lfloor (d-1)/2 \rfloor$ and $C$ is said to be an $e$-[*error-correcting*]{} code. Given any vector ${{\bf v}}\in {\mathbb{F}}_q^n$, its [*distance to the code $C$*]{} is $d({{\bf v}},C)=\min_{{{\bf x}}\in C}\{
d({{\bf v}}, {{\bf x}})\}$ and the [*covering radius*]{} of the code $C$ is $\rho=\max_{{{\bf v}}\in {\mathbb{F}}_q^n} \{d({{\bf v}}, C)\}$. Note that $e\leq \rho$. If $e=\rho$, then $C$ is a [*perfect*]{} code. If $e=\rho-1$, then $C$ is called a [*quasi-perfect*]{} code. If $C$ is a $k$-dimensional subspace of ${\mathbb{F}}_q^n$, then $C$ is [*linear*]{} and referred to as an $[n,k,d;\rho]_q$-code. If $C$ is linear of length $n$ and dimension $k$, then a [*generator matrix*]{} $G$ for $C$ is any $k\times n$ matrix with $k$ linearly independent codewords as rows. A [*parity-check matrix*]{} for $C$ is an $(n-k)\times n$ matrix $H$ such that $C$ is the null space of $H$, i.e. $H{{\bf x}}^T={{\mathbf{0}}}^T$ if and only if ${{\bf x}}\in C$. The [*dual code*]{} $C^\perp$ is the orthogonal subspace to $C$. Hence, $H$ generates $C^\perp$ and $G$ is a parity-check matrix for $C^\perp$.
A linear single-error-correcting ($e=1$) perfect code is called a [*Hamming*]{} code. Such a code has parameters $$[n=(q^m-1)/(q-1), k=n-m, d=3;\rho=1]_q\;\;\;(m>1)$$ and is denoted by ${\cal H}_m$. A parity-check matrix for ${\cal H}_m$, denoted by $H_m$, contains a maximal set of $n=(q^m-1)/(q-1)$ pairwise linearly independent column vectors of length $m$ [@MacW]. The dual code ${\cal H}_m^\perp$ generated by $H_m$ is called [*simplex*]{} and it is a constant-weight code, that is, all nonzero codewords have the same weight $q^{m-1}$.
We denote by $~D=C+{{\bf x}}~$ a [*coset*]{} of $C$, where $+$ means the componentwise addition in ${\mathbb{F}}_q$.
For a given $q$-ary code $C$ of length $n$ and covering radius $\rho$, define $$C(i)~=~\{{{\bf x}}\in {\mathbb{F}}_q^n:\;d({{\bf x}},C)=i\},\;\;i=0,1,\ldots,\rho.$$ The sets $C(0)=C,C(1),\ldots,C(\rho)$ are called the [*subconstituents*]{} of $C$.
Say that two vectors ${{\bf x}}$ and ${{\bf y}}$ are [*neighbors*]{} if $d({{\bf x}},{{\bf y}})=1$. Given two vectors ${{\bf x}}=(x_1,\ldots,x_n),{{\bf y}}=(y_1,\ldots,y_n)\in{\mathbb{F}}_q^n$, we say that ${{\bf y}}$ [*covers*]{} ${{\bf x}}$ if $y_i=x_i$, for all $i$ such that $x_i\neq 0$.
\[de:1.1\] A $q$-ary code $C$ of length $n$ and covering radius $\rho$ is [*completely regular*]{}, if for all $l\geq 0$ every vector ${{\bf x}}\in C(l)$ has the same number $c_l$ of neighbors in $C(l-1)$ and the same number $b_l$ of neighbors in $C(l+1)$. Define $a_l = (q-1){\cdot}n-b_l-c_l$ and set $c_0=b_\rho=0$. The parameters $a_l$, $b_l$ and $c_l$ ($0\leq l\leq \rho$) are called [*intersection numbers*]{} and the sequence ${\operatorname{IA}}=\{b_0, \ldots, b_{\rho-1}; c_1,\ldots,
c_{\rho}\}$ is called the [*intersection array*]{} of $C$.
Let $M$ be a monomial matrix, i.e. a matrix with exactly one nonzero entry in each row and column. Such a matrix can be written as $M=DP$, where $D$ is a monomial diagonal matrix and $P$ is permutation matrix. If $q$ is prime, then the automorphism group of $C$, ${\operatorname{Aut}}(C)$, consists of all monomial $n\times n$ matrices $M$ over ${\mathbb{F}}_q$ such that ${{\bf x}}M \in C$ for all ${{\bf x}}\in C$. If $q$ is a power of a prime number, then the monomial automorphism group of $C$ is denoted by ${\operatorname{MAut}}(C)$, however, ${\operatorname{Aut}}(C)$ also contains any field automorphism of ${\mathbb{F}}_q$ which preserves $C$.
\[transdual\] If $DP$ is the corresponding matrix to an automorphism $\alpha$ of a code (where $D$ is a monomial diagonal matrix and $P$ is a permutation matrix), then $D^{-1}P$ corresponds to an automorphism $\alpha'$ of the dual code.
See [@HP Thm. 1.7.9, p. 27].
\[transitius\] As a consequence of Lemma \[transdual\], $\alpha$ and $\alpha'$ are both transitive on the set of one-weight vectors, or both are not. Note also that if, for a code $C$, ${\operatorname{MAut}}(C)$ is transitive, then so is ${\operatorname{Aut}}(C)$ since ${\operatorname{MAut}}(C)\subseteq{\operatorname{Aut}}(C)$.
It is well known, e.g. see [@MacW], that the monomial automorphism group of a Hamming code ${\cal H}_m$ is isomorphic to the general linear group ${\operatorname{GL}}(m,q)$, which acts transitively on the set of one-weight vectors. In the binary case, the action of ${\operatorname{GL}}(m,2)$ on the set of coordinate positions is even doubly transitive.
The group ${\operatorname{Aut}}(C)$ acts on the set of cosets of $C$ in the following way: for all $\pi\in {\operatorname{Aut}}(C)$ and for every vector ${{\bf v}}\in {\mathbb{F}}_q^n$ we have $\pi({{\bf v}}+ C) = \pi({{\bf v}}) + C$.
\[de:1.3\] Let $C$ be a linear code over ${\mathbb{F}}_q$ with covering radius $\rho$. Then $C$ is [*completely transitive*]{} if ${\operatorname{Aut}}(C)$ has $\rho +1$ orbits when acts on the cosets of $C$.
Since two cosets in the same orbit have the same weight distribution, it is clear that any completely transitive code is completely regular.
Completely regular and completely transitive codes are classical subjects in algebraic coding theory, which are closely connected with graph theory, combinatorial designs and algebraic combinatorics. Existence, construction and enumeration of all such codes are open hard problems (see [@BRZ2; @BRZ3; @BCN; @Koo; @Neum; @Dam] and references there).
It is well known that new completely regular codes can be obtained by the direct sum of perfect codes or, more general, by the direct sum of completely regular codes with covering radius $1$ [@BZZ; @Sole].
In the current paper, starting from Hamming codes and choosing appropriate columns of their parity-check matrix, we obtain parity-check matrices for completely regular codes. More precisely, given the parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns of $H_m$ into two subsets. We consider these two subsets of columns as parity-check matrices of two codes, $A$ and $B$. We say that $B$ is the [*supplementary code*]{} of $A$ (and $A$ is the supplementary code of $B$). If $A$ or $B$ is a Hamming code, then the supplementary code is also completely regular and completely transitive. We point out that, in this case, the dual code of the supplementary code belongs to the family SU1 in [@Cald]. If $A$ or $B$ is a completely regular code with covering radius $2$, then the supplementary code is completely regular with covering radius at most $2$. Moreover, in this situation both codes are completely transitive or are not, simultaneously.
In this way, we construct infinite families of $q$-ary completely regular and completely transitive codes. It is worth mentioning that for fixed $q$, we obtain a growing number of completely regular codes as the length of the starting Hamming code increases.
In the next section, we recall several known results on completely regular codes, which we shall use later. The main results and constructions are presented in .
Preliminary results {#preliminars}
===================
In this section we see several results we will need in the next sections.
\[graph\] Let $C$ be a completely regular code with covering radius $\rho$ and intersection array $\{b_0,\ldots,b_{\rho-1};c_1,\ldots,c_\rho\}$. If $C(i)$ and $C(i+1)$, $0\leq i <\rho$, are two subconstituents of $C$, then $$b_i|C(i)|=c_{i+1}|C(i+1)|.$$
Let $C\subset{\mathbb{F}}_q^n$ be a code. For any vector ${{\bf x}}\in{\mathbb{F}}_q^n$ and for all $j=0,\ldots,n$, define $B_{{{\bf x}},j}$ as the number of codewords at distance $j$ from ${{\bf x}}$: $$B_{{{\bf x}},j}=|\{{{\bf z}}\in C\mid d({{\bf x}},{{\bf z}})=j\}|.$$
\[def:2.5\] A quasi-perfect $e$-error-correcting $q$-ary code $C$ is called [*uniformly packed*]{} if there exist natural numbers $\lambda$ and $\mu$ such that for any vector ${{\bf x}}$: $$B_{{{\bf x}},e+1} = \left\{
\begin{array}{cl}
\lambda & \mbox{ if } d({{\bf x}},C)=e, \\
\mu & \mbox{ if } d({{\bf x}},C)=e+1.
\end{array}\right.$$
Van Tilborg [@vTi] (see also [@Lind; @SZZ]) showed that no nontrivial codes of this kind exist for $e>3$.
\[UPCR\] A uniformly packed code is completely regular.
For a code $C$, we denote by $s+1$ the number of nonzero terms in the dual distance distribution of $C$, obtained by the MacWilliams transform. The parameter $s$ was called [*external distance*]{} by Delsarte [@Del], and is equal to the number of nonzero weights of $C^\perp$ if $C$ is linear. The following properties show the importance of this parameter.
\[params\] If $C$ is any code with packing radius $e$, covering radius $\rho$, and external distance $s$, then
- [[@Del]]{} $\rho \leq s$.
- [[@Del]]{} $C$ is perfect ($e=\rho$) if and only if $e=s$.
- [[@GvT]]{} $C$ is quasi-perfect uniformly packed if and only if $s=e+1$.
- [[@Sole]]{} If $C$ is completely regular, then $\rho=s$.
The new construction of completely regular codes {#main}
================================================
Let $H_m$ be the parity-check matrix of a $q$-ary Hamming code ${\cal H}_m$ of length $n=(q^m-1)/(q-1)$, where $m>1$. Take a non-empty subset of $n_A<n$ columns of $H_m$ as the parity-check matrix of a code $A$. Call $B$ the supplementary code that has as parity-check matrix the remaining $n_B=n-n_A$ columns of $H_m$. In this section, we see that if $A$ or $B$ is a completely regular code with covering radius $\rho(A)\leq 2$, then so is the supplementary code, under certain conditions.
For the rest of this section, we write $n_j=(q^j-1)/(q-1)$, for any integer value $j>0$.
The case $\rho(A)=1$ {#subseccio1}
--------------------
Since there are no two linearly dependent columns in $H_m$, we have that, for $n_A\geq 3$, the minimum distance of $A$ (and of $B$, for $n_A\leq n-3$) is at least three and thus, the packing radius is at least 1. If $\rho(A)=1$ (hence $n_A\neq 2$), then $e=\rho(A)=1$ for $n_A\geq 3$, by . Therefore, $A$ is a perfect Hamming code for $n_A>1$.
For $u\in\{1,\ldots,m-1\}$, $H_m$ can be written as:
$$\label{form1}
H_m=\left[
\begin{array}{c|c}
H^*_u & H_{u,m}
\end{array}
\right],$$
where the first $u$ rows of $H^*_u$ are as the parity-check matrix of ${\cal H}_u$ and the remaining $m-u$ rows are all-zero vectors. For the case $u=1$, the matrix $H^*_u$ is simply the column vector $(1,0,\ldots,0)^T$. We call $B_{u,m}=B$ the code that has parity-check matrix $H_{u,m}$. Note that for $n_A>1$, we have $A={\cal H}_u$.
\[pesos\] The dual code of $B_{u,m}$, i.e. the code $B_{u,m}^\perp$ generated by $H_{u,m}$ has exactly two nonzero weights, namely, $w_1=q^{m-1}$ and $w_2=q^{m-1}-q^{u-1}$.
Clearly, $H^*_u$ generates the simplex code, i.e. the dual of the Hamming code, of length $n_u=(q^u-1)/(q-1)$. Hence any vector generated by $H^*_u$ has weight $0$ or $q^{u-1}$. Since any nonzero vector generated by $H_m$ has weight $q^{m-1}$, the result follows.
\[parB\] The code $B_{u,m}$ has parameters $$[n_B=(q^m-q^u)/(q-1),k=(q^m-q^u)/(q-1)-m,d;\rho=2]_q,\;\;\mbox{where }$$ $$d=\left\{\begin{array}{cl}
4 & \mbox{if } u=m-1,q=2;\\
3 & \mbox{otherwise.}
\end{array}\right.$$
The length $n_B$ of $B_{u,m}$ is simply the length of ${\cal H}_m$ minus the number of columns of $H^*_u$. The dimension $k$ is the length of $B_{u,m}$ minus the number of rows of $H_{u,m}$ (or $H_m$).
Of course, $H_{u,m}$ has no scalar multiple columns, hence $d>2$. Given two columns ${{\bf h}}_i$ and ${{\bf h}}_j$ of $H_{u,m}$ we know that there is a column ${{\bf h}}_\ell$ in $H_m$ which is linearly dependent with ${{\bf h}}_i$ and ${{\bf h}}_j$. If $u<m-1$ or $q>2$, we can choose ${{\bf h}}_i$ and ${{\bf h}}_j$ such that the last $m-u$ entries are linearly independent, then ${{\bf h}}_\ell$ cannot be one of the first $n_u$ columns of $H_m$. Indeed, those columns have zeros in the last $m-u$ entries. Hence, $B_{u,m}$ contains codewords of weight $3$. For the case $u=m-1$ and $q=2$, the previous argument does not work since the last row of $H_{u,m}$ is the all-ones vector. Thus, $H_m$ can be written as:
$$\label{form2}
H_m=\left[
\begin{array}{c|c|c}
H_{u} & H_{u} & {{\mathbf{0}}}^T\\
{{\mathbf{0}}}& {{\mathbf{1}}}& 1
\end{array}
\right].$$
In fact, in this case, $B_{u,m}$ is the binary extended Hamming code of length $2^u$ and, therefore, it has minimum weight $4$.
Finally, since $B_{u,m}$ is not perfect, $\rho > e=1$ and, by Lemma \[pesos\], $B_{u,m}$ has external distance $s=2$, hence $\rho\leq 2$ by Theorem \[params\].
\[comptes\] The number of vectors at distance 1 and at distance 2 from $B_{u,m}$ are, respectively: $$\begin{aligned}
|B_{u,m}(1)| &=& q^{n_B-m}(q^m-q^u), \mbox{ and }\\
|B_{u,m}(2)| &=& q^{n_B-m}(q^u-1),\end{aligned}$$ where $n_B=(q^m - q^u)/(q-1)$ is the length of $B_{u,m}$.
The number of vectors of weight 1 is $(q-1)n_B$. All these vectors are at distance 1 from exactly one codeword (the zero vector). Thus, $|B_{u,m}(1)|=(q-1)n_B|B_{u,m}|=q^{n_B-m}(q^m-q^u)$.
Since the covering radius of $B_{u,m}$ is $\rho=2$, we have that $$\begin{split}
|B_{u,m}(2)|=&|{\mathbb{F}}_q^{n_B}|-|B_{u,m}(1)|-|B_{u,m}|=\\&q^{n_B}-q^{n_B-m}(q^u-1)-q^{n_B-m}=q^{n_B-m}(q^u-1).
\end{split}$$
The code $B_{u,m}$ is quasi-perfect uniformly packed (hence completely regular) with intersection array: $${\operatorname{IA}}=\{q^m-q^u, q^u-1;1,q^m-q^u\}.$$
Since $s=\rho=e+1$, $B_{u,m}$ is a quasi-perfect uniformly packed code, by Theorem \[params\]. Since $d\geq 3$, it is clear that $b_0=(q-1)n_B=q^m-q^u$ and $c_1=1$. Given a vector ${{\bf x}}$ of weight 1, the vectors ${{\bf y}}$ of weight 2 covering ${{\bf x}}$ not at distance one from $B_{u,m}$ are those which are covered by codewords of ${\cal H}_m$ of weight 3, but not in $B_{u,m}$, hence with the third nonzero coordinate in the first $n_u$ positions. In other words, for ${{\bf x}}$ we can choose anyone of these $n_u$ first positions and, for each of these positions, anyone of the $q-1$ multiples. Therefore ${{\bf x}}$ is covered by $(q-1)n_u=q^u-1$ vectors of weight 2 at distance 2 from $B_{u,m}$. Thus, we obtain $b_1=q^u-1$.
By Lemma \[graph\], we know that $b_1|B_{u,m}(1)|=c_2|B_{u,m}(2)|$. Applying Lemma \[comptes\], we obtain: $$c_2=\frac{(q^u-1)q^{n_B-m}(q^m-q^u)}{q^{n_B-m}(q^u-1)}=q^m-q^u.$$
It is not difficult to prove directly that given a vector ${{\bf x}}\in B_{u,m}(2)$, any neighbor of ${{\bf x}}$ must be in $B_{u,m}(1)$, obtaining the value of $c_2$.
Denote by $({{\bf x}}\mid {{\bf x}}')=(x_1,\ldots,x_{n_u}\mid x'_{n_u+1},\ldots,x'_{n_m})$ a vector in ${\mathbb{F}}_q^{n_m}$ such that ${{\bf x}}\in{\mathbb{F}}_q^{n_u}$ and ${{\bf x}}'\in{\mathbb{F}}_q^{n_m-n_u}$. Let $e_j$ denote any one-weight vector with its nonzero coordinate at position $j$.
\[cosets\] The number of cosets of $B_{u,m}$ of minimum weight $2$ is $q^u-1$. Moreover, for any vector ${{\bf x}}'\in{\mathbb{F}}_q^{n_m-n_u}$ in one such coset, the vector $({{\mathbf{0}}}\mid {{\bf x}}')$ is contained in a coset of weight $1$ of ${\cal H}_m$ with leader $e_j$, which has its nonzero coordinate at position $j\in\{1,\ldots,n_u\}$.
The total number of cosets of $B_{u,m}$ is $q^{n_B}/q^{n_B-m}=q^m$. Since there are one coset of minimum weight 0 (the code $B_{u,m}$) and $(q-1)n_B=q^m-q^u$ cosets of minimum weight 1, we obtain that the number of cosets of minimum weight 2 is $q^m-(q^m-q^u)-1=q^u-1$.
Since $d({{\bf x}}',B_{u,m})=2$, we have that there is some codeword ${{\bf c}}'\in B_{u,m}$ such that ${{\bf y}}'={{\bf x}}'-{{\bf c}}'$ has weight $2$. Hence, $({{\mathbf{0}}}\mid y')$ is covered by some codeword (of weight $3$) $(e_j\mid {{\bf y}}')\in {\cal H}_m$. Thus, $({{\mathbf{0}}}\mid {{\bf y}}')=({{\mathbf{0}}}\mid {{\bf x}}' - {{\bf c}}')\in {\cal H}_m-e_j$. Note that $({{\mathbf{0}}}\mid {{\bf c}}')\in {\cal H}_m$. Then, $({{\mathbf{0}}}\mid {{\bf x}}' - {{\bf c}}') + ({{\mathbf{0}}}\mid c')\in {\cal H}_m - e_j$, implying $({{\mathbf{0}}}\mid {{\bf x}}')\in {\cal H}_m - e_j$.
The matrix $H_m$ (\[form1\]) can be written as: $$\label{form3}
H_m=\left[
\begin{array}{c|c|c|c|c}
H_u & H_u & \cdots & H_u & {{\mathbf{0}}}_{u,n_{m-u}} \\
\hline
{{\mathbf{0}}}_{m-u,n_u} & G_1 & \cdots & G_{q^{m-u}-1} & H_{m-u}
\end{array}
\right],$$ where ${{\mathbf{0}}}_{i,j}$ stands for the all-zero matrix of size $i\times j$ and $G_1,\ldots,G_{q^{m-u}-1}$ are $m-u \times n_u$ matrices, each one with identical nonzero columns and such that no two columns of distinct $G_i's$ are equal. To see that the matrix (\[form3\]) is equivalent to the matrix (\[form1\]), note that no two columns of the matrix (\[form3\]) are linearly dependent. Therefore, the matrix (\[form3\]) is a parity-check matrix for ${\cal H}_m$. Indeed the total number of columns is $q^{m-u}n_u + n_{m_u}=n_m$.
For $i=0,\ldots,q^{m-u}$, we call $i$-[*block*]{} of coordinate positions the set $\{in_u+1,\ldots,(i+1)n_u\}$. Thus, the first block, or $0$-block, corresponds to $\{1,\ldots,n_u\}$. For $i=1,\ldots,q^{m-u}-1$, the $i$-block corresponds to the set of coordinates of the matrix $G_i$. Finally, the last block, or $q^{m-u}$-block, corresponds to the coordinates of the matrix $H_{m-u}$.
\[autos\] If $\alpha\in{\operatorname{Aut}}({\cal H}_u)$ (acting on the coordinates $\{1,\ldots,n_u\}$), then there exists $\beta\in{\operatorname{Aut}}(B_{u,m})$ (acting on the coordinates $\{n_u+1,\ldots,n_m\}$) such that $\gamma=(\alpha\mid \beta)\in {\operatorname{Aut}}({\cal H}_m)$.
Given $\alpha\in{\operatorname{Aut}}({\cal H}_u^\perp)$, consider $\gamma=(\alpha\mid\alpha_1\mid\cdots\mid\alpha_{q^{m-u}-1}\mid id)$, where the action of each $\alpha_i$ is identical to the action of $\alpha$ but on the corresponding $i$-block of coordinate positions, and $id$ is the identity on the last block of coordinates. Clearly, $\gamma\in{\operatorname{Aut}}({\cal H}_m^\perp)$ and $\beta=(\alpha_1\mid\cdots\mid\alpha_{q^{m-u}-1}\mid id)\in{\operatorname{Aut}}(B_{u,m}^\perp)$. By Lemma \[transdual\], the result follows.
\[transitiu\] The automorphism group ${\operatorname{Aut}}(B_{u,m})$ is transitive (on the set of one-weight vectors with coordinates in $\{n_u+1,\ldots,n_m\}$).
Recall that the automorphism group of a Hamming code ${\cal H}_m$ is isomorphic to ${\operatorname{GL}}(m,q)$, which acts transitively on the set of one-weight vectors.
Consider the parity-check matrix of ${\cal H}_m$ given in $(\ref{form2})$. Consider the $m\times m$ matrices $H_{K,M,N}$, where $K,M$ are $u\times u$, $(m-u)\times (m-u)$, nonsingular matrices, respectively, and $N$ is a $u\times (m-u)$ matrix. $$H_{K,M,N} = \begin{pmatrix}
K & N\\
0 & M
\end{pmatrix}.$$
The matrices $H_{K,M,N}$ are in ${\operatorname{GL}}(m,q)$ and act on $H_m$ as monomial automorphisms, stabilising the Hamming code ${\cal H}_u$, so we can consider these matrices as automorphisms of $B_{u,m}$. Now, we want to show that these matrices assure the transitivity of ${\operatorname{Aut}}(B_{u,m})$. Take the $i$th and $j$th columns, say ${{\bf h}}_i$ and ${{\bf h}}_j$, respectively, where $i,j\in\{n_u+1,\ldots,n_m\}$. We want to find appropriate matrices $K,M,N$ such that $H_{K,M,N}({{\bf h}}_i)=\lambda{{\bf h}}_j$, for any $\lambda\in{\mathbb{F}}_q$.
Take the projections of both ${{\bf h}}_i$, ${{\bf h}}_j$ on the first $u$ coordinates, say ${{\bf h}}_i^{(u)}$ and ${{\bf h}}_j^{(u)}$, respectively. And also let ${{\bf h}}_i^{(m-u)}$ and ${{\bf h}}_j^{(m-u)}$ be the respective projections on the last $m-u$ coordinates.
First of all, consider the case when $i$ and $j$ are not in the last block of coordinate positions, so that ${{\bf h}}_i^{(u)}$ and ${{\bf h}}_j^{(u)}$ are nonzero vectors. Now, take $N=0$, take the matrix $K$ such that $K({{\bf h}}_i^{(u)})=\lambda{{\bf h}}_j^{(u)}$ and the matrix $M$ such that $M({{\bf h}}_i^{(m-u)})=\lambda{{\bf h}}_j^{(m-u)}$. Indeed, we can do these last assignations since the matrix $K$ is in ${\operatorname{GL}}(u,q)$, the matrix $M$ is in ${\operatorname{GL}}(m-u,q)$ and the monomial automorphism group of a $q$-ary Hamming code is transitive on the set of one-weight vectors. Hence, we have $H_{K,M,N}({{\bf h}}_i)=\lambda{{\bf h}}_j$.
Secondly, consider the case when $i$ and $j$ belong to the last block of coordinate positions. Then, ${{\bf h}}_i^{(u)}$ and ${{\bf h}}_j^{(u)}$ are the all-zeros vector. Now, take $N=0$, any nonsingular matrix $K$ and the matrix $M$ such that $M({{\bf h}}_i^{(m-u)})=\lambda{{\bf h}}_j^{(m-u)}$. Hence, we have $H_{K,M,N}({{\bf h}}_i)=\lambda{{\bf h}}_j$.
Finally, consider the case when $i$ is in the last block of coordinate positions and $j$ is not. In this case, ${{\bf h}}_i^{(u)}$ is the all-zeros vector and ${{\bf h}}_j^{(u)}$ is a nonzero vector. Now, take as matrix $K$ any nonsingular matrix and the matrix $M$ such that $M({{\bf h}}_i^{(m-u)})=\lambda{{\bf h}}_j^{(m-u)}$. Let $\ell$ be anyone of the nonzero coordinates of ${{\bf h}}_i^{(m-u)}$ and say $\gamma$ its value. Take the matrix $N$ with all columns equal to the all-zeros vector, except the $\ell$th column which is $\lambda\gamma^{-1}{{\bf h}}_j^{(u)}$. Hence, we have $H_{K,M,N}({{\bf h}}_i)=\lambda{{\bf h}}_j$. For the inverse case, when ${{\bf h}}_i^{(u)}$ is a nonzero vector and ${{\bf h}}_j^{(u)}$ is the all-zeros vector, we can use the same argumentation and finally take the inverse matrix of $H_{K,M,N}$.
In fact, Proposition \[transitiu\] shows that the action of ${\operatorname{MAut}}(B_{u,m})$ on the set of one-weight vectors is transitive. As a consequence, see Remark \[transitius\], the full automorphism group ${\operatorname{Aut}}(B_{u,m})$ is also transitive.
The code $B_{u,m}$ is completely transitive.
By , $\rho(B_{u,m})=2$. Hence, we have to see that the cosets of weight $i$ are in the same orbit, for $i=1$ and $i=2$.
Since ${\operatorname{Aut}}(B_{u,m})$ is transitive by Proposition \[transitiu\], we have that all the cosets of $B_{u,m}$ with minimum weight one are in the same orbit.
By Lemma \[autos\], it follows that ${\operatorname{Aut}}({\cal H}_u)={\operatorname{GL}}(u,q)$ acting on the first $n_u$ coordinates is contained in ${\operatorname{Aut}}({\cal H}_m)={\operatorname{GL}}(m,q)$, acting on the full set of $n_m$ coordinate positions. Let $B_{u,m}+{{\bf x}}$ and $B_{u,m}+{{\bf y}}$ be two cosets of minimum weight 2, where we assume that ${{\bf x}}$ and ${{\bf y}}$ have weight two. Let ${\cal H}_m +e_i$ and ${\cal H}_m + e_j$, with $i,j\in\{1,\ldots,n_u\}$, be the corresponding cosets of ${\cal H}_m$, according to Lemma \[cosets\]. Since ${\operatorname{Aut}}({\cal H}_u)={\operatorname{GL}}(u,q)$ is transitive, and by Lemma \[autos\], there is an automorphism $\gamma\in{\operatorname{Aut}}({\cal H}_m)$ fixing setwise the first $n_u$ coordinates (and the last $n_m-n_u$) such that $\gamma({\cal H}_m +e_i)={\cal H}_m + e_j$. By Lemma \[autos\], it is clear that the action of $\gamma$ in the last $n_m-n_u$ coordinates sends $B_{u,m}+{{\bf x}}$ to $B_{u,m}+{{\bf y}}$. Indeed, if $\gamma(B_{u,m}+{{\bf x}})=B_{u,m}+{{\bf z}}$, for some ${{\bf z}}$ of weight two, then $e_j+{{\bf y}}$ and $e_j+{{\bf z}}$ are codewords in ${\cal H}_m$. Thus, ${{\bf y}}$ and ${{\bf z}}$ are in the same coset. Therefore, all the cosets of $B_{u,m}$ of weight two are in the same orbit.
The case $\rho(A)=2$
--------------------
For this case, we have the following result.
\[rho2\] If the code $A$ has dimension $n_A-m$ and is completely regular with $\rho(A)=2$, then the supplementary code $B$, of length $n_B$, is completely regular with $\rho(B)\leq 2$.
If $A$ is completely regular with $\rho(A)=2$ then, by , the external distance of $A$ is $s(A)=2$. Hence, $A^\perp$ has two nonzero weights, say $w_1$ and $w_2$. Consider any nonzero vector ${{\bf z}}=({{\bf x}}\mid{{\bf y}})\in {\cal H}^\perp_m$, where ${{\bf x}}\in A^\perp$ and ${{\bf y}}\in B^\perp$. Since ${{\bf z}}$ is a nonzero codeword of the simplex code of length $n_m$, we know that the weight of ${{\bf z}}$ is ${\mbox{wt}}({{\bf z}})=q^{m-1}$. Also, ${\mbox{wt}}({{\bf z}})={\mbox{wt}}({{\bf x}})+{\mbox{wt}}({{\bf y}})$ and thus we obtain that ${\mbox{wt}}({{\bf y}})=q^{m-1}-w_1$ or ${\mbox{wt}}({{\bf y}})=q^{m-1}-w_2$. Note that ${{\bf x}}$ cannot be the zero vector because the dimension of $A^\perp$ is $m$. We conclude that $B^\perp$ has at most two nonzero weights (if $w_1$ or $w_2$ equals $q^{m-1}$, then $B^\perp$ has only one nonzero weight). Therefore $s(B)\leq 2$, implying $\rho(B)\leq 2$, by .
If $s(B)=1$, then $B$ is the trivial code of length 1, $B=\{(0)\}$, or $B$ is a Hamming code, by . In any case, $B$ is completely regular. In fact, if $s(B)=1$, we are in the situation of , interchanging the roles of $A$ and $B$.
If $s(B)=2$ and $\rho(B)=2$, then $B$ is a quasi-perfect uniformly packed code, by . Therefore, $B$ is completely regular by .
Finally, note that $s(B)=2$ and $\rho(B)=1$ is not possible:
- If $n_B=1$, then $s(B)$ cannot be $2$.
- If $n_B=2$, then $B=\{(0,0)\}$, which has $\rho(B)=2$.
- If $n_B\geq 3$, then $B$ has packing radius $e\geq 1$. Since $e\leq \rho(B)$, if we assume $\rho(B)=1$, then we have $e=\rho(B)< s(B)$ contradicting .
If the length of $A$ verifies $n_A>n_{m-1}$, then the zero vector cannot be a row of the parity-check matrix of $A$, otherwise $H_m$ would have two linearly dependent columns. Hence, the zero vector could not be generated by the rows of the parity-check matrix of $A$ and, as a consequence, the dimension of $A^\perp$ would be $m$. Therefore, the condition $n_A>n_{m-1}$ implies that the dimension of $A$ is $n_A-m$. Note that the converse statement is not true (see the next example).
\[GolayTernari\] Let $A$ be the ternary Golay $[11,6,5;2]_3$ code. Consider the ternary matrix $H_5$, which is the parity-check matrix of a ternary Hamming $[121,116,3;1]_3$ code. Let $B$ be the supplementary code which has length $n_B=110$.
Since $A$ is perfect (so completely regular) with covering radius $\rho(A)=2$, we have that $B$ is a completely regular code. Clearly, $B$ is not perfect, thus $\rho(B)=2$. Therefore, the parameters of $B$ are $[110,105,3;2]_3$. Moreover, we have computationally verified that $B$ is completely transitive and with intersection array $${\operatorname{IA}}=\{220,20;1,200\}.$$
Note that the hypothesis about the dimension of $A$ in cannot be relaxed, as the next example shows.
\[GolayTernari2\] Let $A$ be the punctured ternary Golay $[10,6,4;2]_3$ code. As in , consider $H_5$, the parity-check matrix of a ternary Hamming $[121,116,3;1]_3$ code. Now, let $B$ be the supplementary code with length $n_B=111$. In this case, the dimension of $A$ is $6\neq n_A-m=5$.
The code $A$ is completely regular and completely transitive with intersection array $${\operatorname{IA}}=\{20,18;1,6\}.$$ The code $B$ has parameters $[111,106,3;2]_3$ and it is not completely regular since its external distance is $s(B)=4$.
The construction described in does not work for covering radius $\rho(A)=3$. For example, let $A$ be the extended ternary Golay code and consider the ternary matrix $H_6$, which is the parity-check matrix of a ternary Hamming $[364,258,3;1]_3$ code. Let $B$ be the supplementary code.
The code $A$ is completely transitive with $\rho(A)=3$. The code $B$ has parameters $[352,346,3;2]_3$ and it is not completely regular since its external distance is $s(B)=3$.
We also give the expressions of the intersection numbers of $A$ and $B$ in terms of the lengths $n_A$ and $n_B$ and the parameter $b_1$.
\[IAs\] Assume that the code $A$ is completely regular with dimension $n_A-m$, covering radius $\rho(A)=2$, and the supplementary code $B$ has also covering radius $\rho(B)=2$.
- The code $B$ is completely regular with dimension $n_B-m$.
- The code $A$ has intersection array $${\operatorname{IA}}(A)=\{b_0,b_1;c_1,c_2\}=\{(q-1)n_A,b_1;1,\frac{n_A}{n_B}b_1\}.$$
- The code $B$ has intersection array $${\operatorname{IA}}(B)=\{b_0',b_1';c_1',c_2'\}=\{(q-1)n_B,(q-1)n_A-\frac{n_A}{n_B}b_1;1,(q-1)n_B-b_1\}.$$
For (i), we already know that $B$ is completely regular, by . Assume that the dimension of $B$ is less than $n_B-m$. Hence, the parity-check matrix of $B$ can be written containing at least one zero row. Since $s(B)=\rho(B)=2$, the dual code $B^\perp$ contains two nonzero weights, say $w_1$ and $w_2$. But, in this case, $A^\perp$ would contain three nonzero weights: $q^{m-1}$, $q^{m-1}-w_1$ and $q^{m-1}-w_2$; leading to a contradiction because $A$ has external distance $s(A)=2$.
For (ii) and (iii), with similar computations as in , we have: $$\begin{aligned}
|A(1)| &=& (q-1)n_A|A|,\\
|A(2)| &=& q^{n_A}-(q-1)n_A|A|-|A|.\end{aligned}$$ By , we obtain $$b_1(q-1)n_A|A|=c_2\left(q^{n_A}-\left((q-1)n_A+1\right)|A|\right).$$ Taking into account that $|A|=q^{n_A-m}$ and $n_B=n_m-n_A$, the expression simplifies to $$\label{primera}
b_1n_A=c_2n_B.$$ By (i), we have that $|B|=q^{n_B-m}$, thus we symmetrically obtain $$\label{segona}
b_1'n_B=c_2'n_A.$$
Let $J_A$ (respectively $J_B$) be the set of $n_A$ (resp. $n_B$) coordinate positions corresponding to the code $A$ (resp. $B$). Define $X_A$ (resp, $X_B$) as the set of one-weight vectors with coordinates in $J_A$ (resp. $J_B$). Define also $Y_A$ (resp. $Y_B)$ as the set of two-weight vectors in $A(2)$ (resp. $B(2)$) with coordinates in $J_A$ (resp. $J_B$). Consider the bipartite graph $\Gamma_A$ (resp. $\Gamma_B$) with vertex set $X_A\cup Y_A$ (resp. $X_B\cup Y_B$) and edges joining pairs of vertices ${{\bf x}}$, ${{\bf y}}$, where ${{\bf x}}\in X_A$, ${{\bf y}}\in Y_A$ (resp. ${{\bf x}}\in X_B$, ${{\bf y}}\in Y_B$), if $d({{\bf x}},{{\bf y}})=1$.
The degree of any vertex in $X_A$ (resp. $X_B$) is $b_1$ (resp. $b'_1)$ by definition. Hence, the total number of edges in $\Gamma_A$ (resp. $\Gamma_B$) is $b_1(q-1)n_A$ (resp. $b'_1(q-1)n_B$).
Consider the set of all two-weight vectors with one nonzero coordinate in $J_A$ and one nonzero coordinate in $J_B$. There are $(q-1)^2n_An_B$ such vectors. If $e_i+e_j$ is one of these vectors ($i\in J_A$, $j\in J_B$), then there is exactly one codeword ${{\bf z}}\in {\cal H}_m$ of weight three which covers $e_i+e_j$, say ${{\bf z}}=e_i+e_j+e_k$. If $k\in J_A$, then $e_i+e_k\in A(2)$ and $e_i+e_k$ is a neighbor of $e_i$. Else, if $k\in J_B$, then $e_j+e_k\in B(2)$ and $e_j+e_k$ is a neighbor of $e_j$. Therefore, the vector $e_i+e_j$ induces either one edge of $\Gamma_A$, or one edge of $\Gamma_B$.
We conclude that $$b_1n_A(q-1)+b_1'n_B(q-1)=(q-1)^2n_An_B,$$ which simplifies to $$\label{tercera}
b_1n_A+b_1'n_B=(q-1)n_An_B.$$
The values $b_0=(q-1)n_A$, $b'_0=(q-1)n_B$, $c_1=c_1'=1$ are trivial since $A$ and $B$ have minimum distance at least three. From , we obtain $c_2=n_Ab_1/n_B$. Using , we compute $b_1'=(q-1)n_A-n_Ab_1/n_B$, and by , $c_2'=(q-1)n_B-b_1$.
Finally, we will also show that the codes $A$ and $B$, under the hypothesis of , are both completely transitive or both are not.
\[subgroup\] Assume that the code $A$ has dimension $n_A-m$, covering radius $\rho(A)=2$, and let $B$ be the supplementary code. Then ${\operatorname{Aut}}(A)$ is a subgroup of ${\operatorname{Aut}}(B)$.
Let $\phi\in {\operatorname{Aut}}(A)$. Let $H_A$ (respectively $H_B$) be the parity-check matrix of $A$ (resp. $B$). Note that since the dimension of $A$ is $n_A-m$, the minimum distance of $A$ is not less than three. Let ${{\bf h}}_1,\ldots, {{\bf h}}_s$ be a set of $s$ columns in $H_A$ such that $\sum_{i=1}^{s} \alpha_i {{\bf h}}_i=0$, where $\alpha_i\in {\mathbb{F}}_q$. Hence, we are assuming that the columns ${{\bf h}}_1,\ldots, {{\bf h}}_s$ are the support of a codeword in $A$. Since $\phi$ is an automorphism of $A$ we should have $\sum_{i=1}^{s} \alpha_i \phi({{\bf h}}_i)=0$ and so the action of $\phi$ is linear over the columns in $H_A$. Since the dimension of $A$ is $n_A-m$ we can extend, by linearity, the action of $\phi$ over all columns in $H_m$ obtaining $\phi^{(e)}\in {\operatorname{Aut}}({\cal H}_m)$. The projection of $\phi^{(e)}$ over the columns of $H_B$ gives $\phi^{(e)}_B\in {\operatorname{Aut}}(B)$. It is clear that if $\phi,\psi \in {\operatorname{Aut}}(A)$ with $\phi\not= \psi$, then $\phi^{(e)}_B \not= \psi^{(e)}_B$.
If the dimension of $A$ is $n_A-m$ and the dimension of $B$ is $n_B-m$ then ${\operatorname{Aut}}(A)$ and ${\operatorname{Aut}}(B)$ are isomorphic as abstract groups.
If the code $A$ is completely transitive with dimension $n_A-m$ and covering radius $\rho(A)=2$, then the supplementary code $B$ is also completely transitive.
From , when $A$ is a completely regular code with $\rho(A)=2$, then $B$ is completely regular with $\rho(B)\leq 2$. When $\rho(B)=1$ the code $B$ is a completely transitive code, so we are interested in proving that $B$ is a completely transitive code in the case when $\rho(B)=2$. Since $A$ is completely regular, we have that the dimension of $B$ is $n_B-m$, by . Hence, the minimum distances of $A$ and $B$ are not less than three.
Take two pairs of columns in the parity-check matrix $H_B$ of the code $B$. Say ${{\bf h}}_{i1}, {{\bf h}}_{i2}$ and ${{\bf h}}_{j1},{{\bf h}}_{j2}$. Each pair represents a vector of weight two and we assume that both vectors are at distance two from $B$ and also they are not in the same coset (modulo the code $B$). Therefore, we obtain two different columns ${{\bf h}}_i, {{\bf h}}_j$ in $H_A$ (the parity-check matrix of $A$), in such a way that both triples ${{\bf h}}_{i1}, {{\bf h}}_{i2}, {{\bf h}}_i$ and ${{\bf h}}_{j1},{{\bf h}}_{j2}, {{\bf h}}_j$ are the support of codewords of weight three in ${\cal H}_m$. Since $A$ is completely transitive, there exists $\phi\in {\operatorname{Aut}}(A)$ taking one of these columns to the other and, from , we have an automorphism in ${\operatorname{Aut}}(B)$ taking the pair ${{\bf h}}_{i1}, {{\bf h}}_{i2}$ to ${{\bf h}}_{j1},{{\bf h}}_{j2}$. Now, to finish the proof we need to show that taking ${{\bf h}}_i, {{\bf h}}_j$ two different columns in $H_B$, we have an automorphism in ${\operatorname{Aut}}(B)$ which leads ${{\bf h}}_i$ to ${{\bf h}}_j$. It is easy to see that there are $\frac{n-1}{2}(q-1)^2$ codewords of minimum weight in the Hamming code ${\cal H}_m$ containing in its support the coordinate position corresponding to ${{\bf h}}_i$. Of those codewords, there are $(a'_1 -(q-2))(q-1)/2$ such that they are also codewords of weight three in $B$ ($a'_1$ is the corresponding parameter of the code $B$ in ). Also there are $b'_1(q-1)$ codewords sharing exactly two coordinates with the support of the code $B$. Hence, $$\label{eq}
\begin{split}
\frac{n-1}{2}(q-1)^2 &-b'_1(q-1)-\frac{(a'_1-(q-2))(q-1)}{2}=\\ &\frac{q-1}{2}\big[(n-1)(q-1)-2b'_1 -(n_B(q-1)-b'_1-1-(q-2))\big]=\\
&\frac{q-1}{2}\big[(n_A-1)(q-1)-b'_1 +(q-1)\big]=\\
&\frac{q-1}{2}\big[n_A(q-1)-b'_1\big]= \frac{(q-1)}{2}\frac{n_Ab_1}{n_B}=\frac{(q-1)c_2}{2}
\end{split}$$ is the number of codewords of weight three from ${\cal H}_m$ with the coordinate corresponding to ${{\bf h}}_i$ in its support and the other two coordinates in the support of the code $A$. The code $B$ will be a completely transitive code when gives a number greater than or equal to one, which is obvious. This proves the statement.
It is worth mentioning that, in some cases, the construction used here can be equivalent to the construction described in [@BRZ4]. But this is not always the case as the following examples show.
\[Ex1\] Consider the binary matrix $H_6$, which the parity-check matrix of the binary Hamming code of length $63$. Take $35$ columns of $H_6$ following the procedure described in [@BRZ4], as the parity-check matrix of the code $A$. The code $A$ has parameters $[35,29,3;2]_2$ and the supplementary code $B$ has parameters $[28,22,3;2]_2$. Computationally, we have verified that both codes are completely regular but not completely transitive.
\[Ex2\] Now, consider the binomial code $A'=C^{(7,4)}$, whose parity-check matrix $H^{(7,4)}$ contains as columns all the binary vectors of length $7$ and weight $4$ (see [@RZ]). Note that adding all the rows of $H^{(7,4)}$ gives the zero vector, hence the dimension of $(A')^\perp$ is $6$. The code $A'$ has parameters $[35,29,3,2]_2$ and the supplementary code $B'$ has parameters $[28,22,3;2]_2$. In this case, we have computationally verified that both codes are completely regular and also completely transitive.
The codes $A$ and $A'$ (respectively $B$ and $B'$) in and are completely regular codes with the same parameters. However, $A$ and $A'$ (resp. $B$ and $B'$) are not equivalent since $A'$ (resp. $B'$) is completely transitive, but $A$ (resp. $B$) is not.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been partially supported by the Spanish grants TIN2016-77918-P, (AEI/FEDER, UE). The research of the third author was carried out at the IITP RAS at the expense of the Russian Fundamental Research Foundation (project no. 19-01-00364).
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---
abstract: |
Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in ${\operatorname{Aut}}G$) of $c'$ is contained in the stabiliser of $c$.
We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges.
Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.
author:
- 'Florian Lehner[^1] and Simon M. Smith[^2]'
bibliography:
- 'ref.bib'
title: On symmetries of edge and vertex colourings of graphs
---
Introduction
============
This paper concerns the symmetries of colourings of graphs. A large number of research papers have been written on this topic; mostly, these have focused on finding colourings with few symmetries and a small number of colours. In particular, the *distinguishing number* $D(G)$ of a graph is the least number of colours such that there is a vertex colouring which is not preserved by any non-identity automorphism. Motivated by a recreational mathematics problem, this notion was introduced by Albertson and Collins in 1996 [@albertsoncollins-distinguishing] and has since received considerable attention.
Recently, Kalinowski and Pilśniak [@kalinowski-distindex] suggested the following edge version. The *distinguishing index* $D'(G)$ of a graph $G$ is the smallest number of colours such that there is an edge colouring which is not preserved by any non-identity automorphism. Many results about distinguishing numbers hold for distinguishing indices as well, sometimes with almost identical proofs (see [@Broere-distindex; @imrich-distindex; @kalinowski-distindex], for example). Furthermore, there are problems such as Tucker’s Infinite Motion Conjecture [@tucker-infinitemotion] that are still wide open for vertex colourings, but whose edge colouring version has a relatively simple proof [@lehner-edgemotion]. This suggests that finding edge colourings with few symmetries is generally easier than finding such vertex colourings. This suggestion is further supported by the following result. It was first proved for finite graphs in [@kalinowski-distindex]. See [@imrich-distindex] for an extension to infinite graphs and [@lehner-edgemotion] for an alternative proof for both finite and infinite graphs.
\[thm:distindexdistnumber\] If $G$ is a connected graph of order at least $3$, then $D'(G) \leq D(G) + 1$.
It was shown in [@imrich-distindex] that Theorem \[thm:distindexdistnumber\] remains true even if $D(G)$ is an arbitrary infinite cardinal. Since $\alpha = \alpha + 1$ for any infinite cardinal $\alpha$, this implies that for any graph $G$ with infinite distinguishing number we have $D'(G) \leq D(G)$.\
In this paper we thoroughly investigate the relationship between $D'(G)$ and $D(G)$. For this purpose, we say that an (edge or vertex) colouring $c'$ is *less symmetric* than an (edge or vertex) colouring $c$, if the stabiliser of $c'$ (that is, the setwise stabiliser of those edges or vertices coloured $c'$) is a subgroup of the stabiliser of $c$. With this notion we have the following results.
\[thm:vertextoedgecolouringinfinite\] Let $G$ be a connected graph and let $c$ be a $k$-vertex colouring of $G$ where $k$ is an infinite cardinal. Then there is a $k$-edge colouring $c'$ of $G$ which is less symmetric than $c$.
\[thm:vertextoedgecolouring\] Let $G$ be a connected graph and let $c$ be an arbitrary $k$-vertex colouring of $G$ with $k \in \mathbb{N}$. If $G$ is not a bicentred tree, then there is an $k$-edge colouring $c'$ which is less symmetric than $c$.
In particular, this implies that the inequality in Theorem \[thm:distindexdistnumber\] can only be sharp if $G$ is a bicentred tree with finite distinguishing number. Note that examples of trees where Theorem \[thm:distindexdistnumber\] holds with equality are known, see [@kalinowski-distindex], whence Theorem \[thm:vertextoedgecolouring\] does not extend to all graphs. In fact, in the case of trees we show that the converse of Theorem \[thm:vertextoedgecolouring\] is true.
\[thm:edgetovertexcolouring\] Let $T$ be a tree and let $c'$ be an arbitrary $k$-edge colouring of $T$. Then there is a $k$-vertex colouring $c$ which is less symmetric than $c'$.
Finally, as an application of our results we characterise all graphs for which Theorem \[thm:distindexdistnumber\] is sharp.
Preliminaries
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Basic notions and notations
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Throughout this paper $G = (V,E)$ will denote a graph with vertex set $V$ and edge set $E$. We follow [@diestel-buch] for notions that are not explicitly defined. Note that we do not make assumptions on the cardinality of $V$ or the degree of a vertex. We will, however, require our graphs to be simple (i.e. no multiple edges or loops) and undirected. The automorphism group of $G$ will be denoted by ${\operatorname{Aut}}G$, and for $\gamma \in {\operatorname{Aut}}G$ and $x \in V \cup E$ we will denote by $\gamma x$ the image of $x$ under the automorphism $\gamma$.
A *vertex colouring* of $G$ is a function from $V$ to a set $C$ of colours. If $|C| = k$ we will speak of a $k$-vertex colouring and if $k$ is finite, we will usually assume that $C = \{1,2,\ldots,k\}$. We say that $\gamma \in {\operatorname{Aut}}G$ *preserves* a vertex colouring $c$, if for every $v \in V$ ($e \in E$) we have $c(v) = c(\gamma v)$. The *stabiliser* of a vertex colouring is the set of all automorphisms which preserve it. Analogous definitions can be made for edge colourings. We say that a vertex or edge colouring $c'$ is *less symmetric* than another vertex or edge colouring $c$, if the stabiliser of $c'$ is contained in the stabiliser of $c$. Clearly, the relation of being less symmetric is transitive, and if $c'$ is less symmetric than $c$ and $c$ is less symmetric than $c'$, then the stabilisers of $c$ and $c'$ coincide.
A *centre* of a graph is a vertex which minimises the maximal distance to other vertices. It is a well known fact that every finite tree has either one centre or two adjacent centres. In the first case we call the tree *unicentred*, in the second case we call it *bicentred*. Note that every automorphism of a unicentred tree fixes the centre whereas every automorphism of a bicentred tree fixes the *central edge*, i.e. the edge connecting its two centres.
While the notion of a centre is not always well defined for infinite graphs, we can still define an analogous notion for certain infinite trees. A *ray* is a one sided infinite path, a graph is called *rayless*, if it does not contain a ray. By results of Schmidt [@schmidt-rayless] (see [@halin-rayless] for an exposition in English) every rayless tree has a canonical finite subtree (its kernel) which is preserved by every automorphism. Hence we can define a centre of a rayless tree to be a centre of its kernel and call a rayless tree unicentred or bicentred, depending on whether we get a unique centre or a two adjacent centres.
Canonical colourings
--------------------
The following way to obtain an edge colouring from a vertex colouring was introduced in [@lehner-edgemotion] in order to prove Theorem \[thm:distindexdistnumber\]. Let $G$ be a graph and let $c\colon V \to C$ be a colouring of the vertex set of $G$ with colours in $C$. Without loss of generality assume that $C$ carries the additional structure of an Abelian group—by [@hajnal-algebraicAC], any set can be endowed with an Abelian group structure as long as we assume the Axiom of Choice. Now we can obtain a colouring of the edge set by $e \mapsto c(u) + c(v)$ for $e = uv$. We will call such an edge colouring a *canonical* edge colouring.
\[prp:preservecanonical\] Any vertex colouring is less symmetric than the corresponding canonical edge colouring.
We have to show that any automorphism $\gamma$ preserving the vertex colouring $c$ also preserves the canonical edge colouring $c'$. Let $e = uv$ be an edge, then $$c'(\gamma(e)) = c(\gamma(u)) + c(\gamma(v)) = c(u) + c(v) = c'(e).
{\color{black}}$$
\[lem:nofixedpoint\] Let $G$ be a connected graph, let $c$ be a vertex colouring of $G$ and let $c'$ be the corresponding canonical edge colouring. If an automorphism $\gamma$ preserves $c'$ and $c(v) = c(\gamma v)$ for some vertex $v$, then $\gamma$ preserves $c$.
Let $u$ be a neighbour of $v$ and let $e = uv$. Then $c(\gamma u) = c'(\gamma e)- c(\gamma v) = c'(e)-c(v) = c(u)$. Now use induction over the distance from $v$ to show that the same is true for every vertex of $G$.
\[lem:changecolouring\] Let $G$ be a connected graph, let $c$ be a vertex colouring and let $c'$ be the corresponding canonical edge colouring. Let $c''$ be an edge colouring with the following properties
- there is at least one edge $e$ such that $c''(e) \neq c'(e)$, and
- for every edge $e=uv$ with $c''(e) \neq c'(e)$ and every $\gamma$ which preserves $c''$ we have $c(u) = c(\gamma u)$ and $c(v) = c(\gamma v)$.
Then $c''$ is less symmetric than $c$.
Let $\gamma$ be an automorphism which preserves $c''$. First we show that $\gamma$ preserves $c'$, that is, $c'(e) = c'(\gamma e)$ for every edge $e = uv$. If $c''(e) = c'(e)$ and $c''(\gamma e) = c'(\gamma e)$ then this follows from the fact that $\gamma$ preserves $c''$. If $c''(e) \neq c'(e)$, then $c'(\gamma e) = c(\gamma u) + c(\gamma v) = c(u) + c(v) = c'(e)$. If $c''(\gamma e) \neq c'(\gamma e)$ we can use an analogous argument replacing $\gamma$ by $\gamma^{-1}$.
So every automorphism which preserves $c''$ also preserves $c'$. Note that from the two conditions it is clear that there is a vertex $v$ such that $c(v) = c(\gamma v)$. Hence we can apply Lemma \[lem:nofixedpoint\] to conclude that every automorphism which preserves $c''$ also preserves $c$ whence $c''$ is less symmetric than $c$.
Proof of the main results
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In this section we prove Theorems \[thm:vertextoedgecolouringinfinite\], \[thm:vertextoedgecolouring\], and \[thm:edgetovertexcolouring\]. The proofs of Theorems \[thm:vertextoedgecolouringinfinite\] and \[thm:edgetovertexcolouring\] are simple applications of results from the previous section. Before giving the proofs we recall the statements of the theorems.
Let $G$ be a connected graph and let $c$ be a $k$-vertex colouring of $G$ where $k$ is an infinite cardinal. Then there is a $k$-edge colouring $c'$ of $G$ which is less symmetric than $c$.
Let $C$ be the set of colours used by $c$ and let $a, b \notin C$. Since $C$ is infinite, we have that $|C \cup \{a,b\}| = |C|$. Let $c'$ be the canonical edge colouring corresponding to $c$ and let $v$ be a vertex with at least two neighbours. Let $c''$ be the edge colouring obtained from $c'$ by changing the colours of two edges incident to $v$ to $a$ and $b$ respectively. It is easy to see that if an automorphism preserves $c''$ then it must fix the two recoloured edges, and consequently $c''$ satisfies the conditions of Lemma \[lem:changecolouring\].
Let $T$ be a tree and let $c'$ be an arbitrary $k$-edge colouring of $T$. Then there is a $k$-vertex colouring $c$ which is less symmetric than $c'$.
The theorem follows from Proposition \[prp:preservecanonical\] and the following lemma.
\[lem:edgetovertexcolouring\] Let $T$ be a tree, let $c'$ be an edge colouring of $T$, and let $v$ be an arbitrary vertex. Then there is a unique vertex colouring $c$ such that $c(v) = 0$ and $c'$ is the canonical edge colouring corresponding to $c$.
We inductively construct the colouring $c$ and (simultaneously) show that it is unique. First, define $c(v) := 0$ and suppose $\bar{c}$ that is some other vertex colouring satisfying the conditions of the lemma, with $\bar{c}(v) = 0$ and $c'$ the canonical edge colouring corresponding to $\bar{c}$.
Inductively, suppose we have defined $c$ on all vertices up to distance $r$ from $v$, and that $c$ and $\bar{c}$ agree on these vertices. Now every vertex $u$ at distance $r+1$ from $v$ has a unique neighbour $w$ at distance $r$ from $v$, and we define $c(u) := c'(uw) - c(w)$. However, the colouring $\bar{c}$ cannot have $c'$ as its canonical edge colouring unless it also assigns to $u$ the colour $c'(uw) - c(w)$. Whence $c$ and $\bar{c}$ agree on vertices up to distance $r+1$ from $v$. By induction, we have constructed $c$ and shown that $c$ is unique.
Besides Theorem \[thm:edgetovertexcolouring\], the above lemma has another implication which will be useful later.
\[lem:colouringbijection\] Let $T$ be a tree and let $\varphi$ be the function which maps to every vertex colouring its canonical edge colouring.
1. If $T$ is rooted/unicentred, then $\varphi$ is a bijection between vertex colourings in which the root/centre receives colour $0$ and edge colourings.
2. If $T$ is bicentred, then $\varphi$ is a bijection between vertex colourings in which both centres receive colour $0$ and edge colourings in which the central edge has colour $0$.
In both cases, $\varphi$ preserves stabilisers, i.e. the vertex colourings to which we restrict and their corresponding edge colourings have the same stabiliser.
Lemma \[lem:edgetovertexcolouring\] implies that the maps are bijective. By Proposition \[prp:preservecanonical\], any vertex colouring $c$ is less symmetric than $\varphi(c)$. Conversely, since any automorphism must map centres to centres, Lemma \[lem:nofixedpoint\] implies that for the colourings satisfying the condition $\varphi(c)$ is less symmetric than $c$, hence $\varphi$ preserves stabilisers as claimed.
The remainder of the section is devoted to the proof of Theorem \[thm:vertextoedgecolouring\]. While the proof is divided up into several cases, each of the cases follows the same rough idea: if the canonical edge colouring corresponding to $c$ is not yet less symmetric, apply some minor modifications to it in order to obtain less symmetric colouring. Except in the case where $G$ is a tree, these modifications will only affect a finite number of edges making them easy to track.
Let $G$ be a connected graph and let $c$ be an arbitrary $k$-vertex colouring of $G$ with $k \in \mathbb N$. If $G$ is not a bicentred tree, then there is an $k$-edge colouring $c'$ which is less symmetric than $c$.
A prime number of colours is enough
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We first show that it is enough to prove Theorem \[thm:vertextoedgecolouring\] when the number of colours is a prime. Indeed, assume that the theorem was true for all primes and assume that we are given a colouring with $k = p_1p_2\cdots p_r$ colours. Then we can translate this colouring into a colouring $c$ with colours in $\mathbb Z_{p_1} \times \cdots \times \mathbb Z_{p_r}$. Let $c_i$ be the projection of this colouring onto the $\mathbb Z_{p_i}$ component.
For every $i$ find an edge colouring $c_i'$ such that every automorphism preserving $c_i'$ also preserves $c_i$. Define $c' = c_1' \times \cdots \times c_r'$. Then an automorphism preserving $c'$ has to preserve every $c_i'$, hence it also preserves every $c_i$ and thus also $c$.
From now on, we will generally make the following assumption:
\[ass:general\] The colouring $c$ uses colours in $\mathbb Z_p$ where $p$ is a prime.
The tree case
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Theorem \[thm:vertextoedgecolouring\] is true for trees.
If $T$ is a bicentred tree there is nothing to show. If $T$ has a unique central vertex, then Lemma \[lem:colouringbijection\] (potentially after swapping colours) implies Theorem \[thm:vertextoedgecolouring\].
Hence we may assume that $T$ contains a ray. If $T$ does not contain a double ray, then by results from [@halin-automorphism_oneended] every automorphism must fix some vertex. We can again apply Lemma \[lem:nofixedpoint\] to conclude that any automorphism preserving the canonical edge colouring also preserves $c$.
Finally assume that $T$ contains double rays. Define the *spine* $S$ of $T$ as the subgraph induced by all vertices which lie on a double ray. Clearly, $S$ is setwise fixed by every automorphism of $T$. Observe that $S$ is a leafless tree. Pick an arbitrary root $r$ of $S$ and define an edge colouring $c_s'$ on $S$ as follows.
Assign colour $0$ to all edges incident to $r$ and colour $1$ to all edges at odd distance from $r$ (i.e. edges connecting vertices at distance $2n-1$ from $r$ to vertices at distance $2n$). For an edge at even distance $2n$ from $r$ define the *ancestor* of $e$ as the the unique vertex $v$ at distance $n$ from $r$ which lies on a path connecting $r$ to $e$. Now colour every edge at even distance from $r$ by the colour $c(v)$ of its ancestor.
We claim that any automorphism $\gamma$ which preserves $c_s'$ on $S$ also preserves $c$ on $S$. Indeed, any such $\gamma$ must fix $r$ because it is the only vertex for which all incident edges are coloured $0$. Now if $\gamma$ moves a vertex $v$ to a vertex $w$, then it also moves the edges whose ancestor is $v$ to edges whose ancestor is $w$. By definition of $c_s'$ this is only possible if $c(v) = c(w)$.
Now consider the edge colouring $c''$, where $S$ is coloured according to $c_s'$ and the remaining edges according to the canonical edge colouring $c'$ of $c$. If $\gamma$ preserves $c''$ then it preserves $c$ on $S$ and therefore, by Proposition \[prp:preservecanonical\], it preserves $c'$ on $S$. Hence $\gamma$ preserves $c'$. Since $\gamma$ preserves $c$ on $S$, we can apply Lemma \[lem:nofixedpoint\] and deduce that $\gamma$ preserves $c$.
Graphs containing cycles
------------------------
In this section assume that $G$ is a graph containing a cycle and $c$ is a vertex colouring of $G$ satisfying Assumption \[ass:general\] . Furthermore we will always assume that $c'$ is the canonical edge colouring corresponding to $c$. We start with some observations on $c'$. Let $W = e_1e_2\dots e_l$ be the edge sequence of a closed walk starting at a vertex $v$.
1. \[obs:even\] If $l$ is even, then $\sum_{i=1}^l (-1)^i c'(e_i) = 0$.
2. \[obs:odd\] If $l$ is odd, then $\sum_{i=1}^l (-1)^i c'(e_i) = - 2 c(v)$.
Note that sum in the second observation evaluates to $0$ for $p=2$. If $p$ is odd, however, this sum can be used to find $c(v)$. Consequently, any automorphism which preserves $c'$ must map $v$ to a vertex with the same colour. Hence by Lemma \[lem:nofixedpoint\] every such automorphism preserves $c$. We have thus proved the following.
\[prp:Not\_bipartite\_p\_odd\] If $G$ is not bipartite and $p$ is odd, then any automorphism which preserves the canonical edge colouring must also preserve $c$.
We now define an equivalence relation on a subset of the edge set of $G$. Let $E_C$ be the set of edges of $G$ which are contained in some cycle. Define a relation on $E_C$ by $$e \sim f \Leftrightarrow \text{every cycle containing $e$ also contains $f$}.$$
The relation $\sim$ is an equivalence relation.
The relation clearly is reflexive and transitive. To show that it is symmetric assume for a contradiction that every cycle containing $e$ must also contain $f$, but there is a cycle $C$ which contains $f$ but not $e$. Let $D$ be any cycle containing $e$. Then $(C \cup D) - \{f\}$ contains a cycle through $e$ which does not contain $f$.
Every equivalence class of the relation $\sim$ is a subset of some cycle (in fact it is the intersection of all cycles containing any and hence all of its members). In particular, every class forms either a cycle or is a disjoint union of paths. A [*path equivalence class*]{} (resp. [*cycle equivalence class*]{}) is a path (resp. cycle) $P$ in $G$ such that the edges in $P$ are an entire equivalence class for the above relation. An equivalence class that is a disjoint union of (at least two) paths is a [*disjoint equivalence class*]{}.
\[lem:existence-pathclass\] Every cycle which is not a cycle equivalence class contains at least two distinct path equivalence classes.
We first claim that different equivalence classes cannot *cross* on a cycle, i.e. if $e \sim e' \nsim f \sim f'$, then the cyclic order of the four edges on a cycle cannot be $(e,f,e',f')$. Indeed, if $e$ and $e'$ are not equivalent to $f$, then there is a cycle containing $e$ and $e'$, but not $f$. In particular, there is a path $P$ between endpoints of $e$ and $e'$ not containing $f$. If there is a cycle $C$ such that the cyclic order on $C$ is $(e,f,e',f')$, then $P \cup C$ contains a cycle which contains $f'$ but not $f$, showing that $f \nsim f'$.
Now let $C$ be a cycle that is not a cycle equivalence class. If all equivalence classes contained in $C$ are path equivalence classes, then we are done since there are at least two of them. So assume that there is a disjoint equivalence class $A$ in $C$. Since equivalence classes cannot cross, every other equivalence class is contained in a unique component of $C - A$. Let $X$ be a component of $C - A$ and let $B$ be a disjoint equivalence class contained in $X$. Then at least one component of $X - B$ is a path whose endpoints are not incident with any edge in $A$. Assume that $B$ be an equivalence class which minimises the length of the shortest such path. Any equivalence class that meets this shortest path has to be completely contained in it. By the minimality condition on $B$ each such equivalence class must be connected. Whence every component of $C - A$ contains at least one path equivalence class.
We now distinguish cases by the different types of equivalence classes: First we deal with the case where there is a cycle equivalence class. If there is no cycle equivalence class, then by Lemma \[lem:existence-pathclass\] there is a path equivalence class. We then consider the case where there are path equivalence classes of length at least $2$ and finally we look at the case where all path equivalence classes have length $1$.
Theorem \[thm:vertextoedgecolouring\] is true if there is a cycle equivalence class.
Let $C$ be a cycle equivalence class. First assume that $C$ has odd length. If $p$ is odd, then we already know that the canonical edge colouring has the desired properties, so we can assume that $p=2$.
We define an edge colouring $c''$ on $G$ as follows. For all edges $e \in C$ let $v_e$ be the vertex opposite to $e$ in $C$. If $\sum_{v \in C} c(v) = 1$, then define $c''(e) = c(v_e)$ for all $e \in C$; otherwise set $c''(e) = c(v_e)+1$ for all $e \in C$. For all other edges in $G$ set $c''$ equal to $c'$. With this definition, the sum of the edge colours of $C$ is $1$. From \[obs:odd\] we deduce that $C$ is the only odd cycle with this property. In particular, every $c''$-preserving automorphism $\gamma$ must preserve $C$ setwise. Furthermore, such an automorphism must preserve the vertex colouring $c$ on $C$ since equal edge colours of $e,f \in C$ imply equal vertex colours of $v_e$ and $v_f$. Hence $\gamma$ preserves the canonical edge colouring on $C$ and thus on all of $G$. By Lemma \[lem:nofixedpoint\] any automorphism which preserves $c$ on $C$ and $c'$ on $G$ must preserve $c$ on all of $G$.
So we may assume that the length of $C$ is even. If $C$ contains at least $6$ edges, then we instead take $c''$ to be the edge colouring of $G$ in which the edges of $C$ are coloured by the sequence $(1,1,0,1,0,0,\ldots,0)$ while all other edges in $G$ are coloured according to $c'$. Then $C$ is the only cycle violating \[obs:even\]. Hence every $c''$-preserving automorphism $\gamma$ must fix $C$ setwise, and since the sequence gives a distinguishing colouring, $C$ is fixed pointwise. Thus, by Lemma \[lem:nofixedpoint\] such an automorphism must preserve $c$ on all of $G$.
Finally consider the case where $C$ has length $4$. If $C$ is the only cycle equivalence class of length $4$, then it must be setwise fixed by every automorphism. It is an easy exercise to find an edge colouring $c''$ of $C$ such that every automorphism of $C$ which preserves $c''$ also preserves the vertex colouring on $C$. Hence in this case we can again apply Lemma \[lem:nofixedpoint\] to conclude that we have found a colouring with the desired properties.
We may hence assume that there is more than one cycle equivalence class of length $4$. Let $D$ be another such cycle. Note that removing the edges of $C$ disconnects $G$ into four components, since otherwise $C$ wouldn’t form its own equivalence class. There is a unique vertex $v_D$ of $C$ which lies in the same component as $D$. Analogously we define the vertex $v_C \in D$. Now if we colour $C$ by the sequence $(1,0,0,0)$ and $D$ by the sequence $(1,1,1,0)$, then $C$ and $D$ are the only cycles violating \[obs:even\], so both of them must be fixed setwise by any automorphism which preserves the resulting edge colouring. This implies that $v_C$ and $v_D$ must be fixed by each such automorphism, and consequently every automorphism which preserves the edge colouring must fix $C$ and $D$ pointwise. By the same arguments as before we conclude that every automorphism preserving this edge colouring must also preserve $c$.
Theorem \[thm:vertextoedgecolouring\] is true if there is a path equivalence class of length at least $2$.
Let $P$ be an arbitrary path equivalence class of length at least $2$, and let $\tilde{c}$ be the colouring obtained by colouring all vertices of $P$ with colour $0$ and all others by $c$.
Let $\tilde{c}'$ be the canonical edge colouring corresponding to $\tilde{c}$ and let $\tilde{c}''$ be the colouring obtained from $\tilde{c}'$ by adding $1$ to the colour of the first edge of $P$.
Then a cycle violates \[obs:even\] or \[obs:odd\] with respect to $\tilde{c}''$ if and only if it contains $P$. Since $P$ is the unique equivalence class with this property, any automorphism which preserves $\tilde{c}''$ must fix $P$ setwise. By Lemma \[lem:nofixedpoint\], every automorphism which fixes $\tilde{c}''$ must thus fix $\tilde{c}$.
Furthermore, since all vertices of $P$ hat the same colour in $\tilde{c}$, all edges had the same colour in $\tilde{c}'$. Since we only changed the colour of the first edge, this implies that $P$ must be fixed pointwise by every automorphism which preserves $\tilde{c}''$. Since the only vertices where $c$ and $\tilde{c}$ potentially differ are those of $P$, this implies that every automorphism which fixes $\tilde{c}''$ must also fix $c$.
\[clm:Path\_classes\_One\] Theorem \[thm:vertextoedgecolouring\] is true if all path equivalence classes have length $1$.
Denote by $E_P$ the set of edges contained in path equivalence classes. Clearly every automorphism must fix $E_P$ setwise. Let us call an edge [*$c$-monochromatic*]{} if its vertices both receive the same colour in $c$.
We begin with an observation: if there is $c$-monochromatic edge $e\in E_P$, then (using the same colours as $c$) we can describe an edge-colouring $c''$ that is less symmetric than $c$. Indeed, suppose that $e \in E_P$ is $c$-monochromatic. Let $c'$ be the canonical edge colouring corresponding to $c$ and let $c''$ be obtained from $c'$ by adding $1$ to the colour of $e$. Then a cycle violates \[obs:even\] or \[obs:odd\] with respect to $c''$ if and only if it contains $e$. Since $e$ is the unique equivalence class with this property, any automorphism which preserves $c''$ must fix $e$. Since the endpoints of $e$ have the same colour in $c$ we conclude by Lemma \[lem:nofixedpoint\] that any such automorphism preserves $c$, whence $c''$ is less symmetric than $c$.
From now on we assume that $E_P$ contains no $c$-monochromatic edges. We next consider the case where there exists a 3-cycle in $E_P$. If $p = 2$ then there would be a $c$-monochromatic edge in $E_P$, which cannot happen by assumption. On the other hand, if $p$ is odd then $G$ is not bipartite and we can apply Proposition \[prp:Not\_bipartite\_p\_odd\] to deduce that the canonical edge colouring corresponding to $c$ is less symmetric than $c$. Hence, if $E_P$ contains a $3$-cycle then our claim holds.
From now on we assume that $E_P$ contains no $3$-cycles. Let us now address the case where there are two distinct incident edges $e = uv$ and $f= vw$ in $E_P$ such that $c(u) = c(w)$. By assumption, $uw \not \in E_P$. Let $\tilde{c}$ be a new vertex colouring of $G$ in which $\tilde{c}(v) := c(w)$ and for all vertices $x \not = v$ we have $\tilde{c}(x) := c(x)$. Take $\tilde{c}'$ to be the canonical edge colouring of $G$ corresponding to $\tilde{c}$. Let $\tilde{c}''$ be obtained from $\tilde{c}'$ by adding $1$ to the colour of $e$. Then a cycle violates \[obs:even\] or \[obs:odd\] with respect to $\tilde{c}''$ if and only if it contains $e$. Therefore, if $\gamma$ is a $\tilde{c}''$-preserving automorphism then $\gamma$ must fix $e$ setwise. Since $\tilde{c}''$ and $\tilde{c}'$ differ only in $e$, the automorphism $\gamma$ must also preserve $\tilde{c}'$. Since $\tilde{c}(v) = c(w) = c(u)$ and $\tilde{c}(u) = c(u)$, we have that $\gamma$ preserves $\tilde{c}$ by Lemma \[lem:nofixedpoint\]. Now $uv$ and $vw$ are $\tilde{c}$-monochromatic edges and no vertex other than $u,v$ or $w$ lies in a $\tilde{c}$-monochromatic edge in $E_P$. Since $uw \not \in E_P$, we conclude that $v$ is fixed by $\gamma$. Since $\tilde{c}$ and $c$ differ only at $v$, it follows that $\gamma$ preserves $c$. Hence $\tilde{c}''$ is less symmetric than $c$, and our claim holds.
From now on we assume that if there are two distinct incident edges $e = uv$ and $f= vw$ in $E_P$, then $c(u) \not = c(w)$. Now consider the general case where there are two distinct incident edges $e = uv$ and $f= vw$ in $E_P$. By assumption we have that $c(u) \not = c(w)$ and $uw \not \in E_P$. Let $\tilde{c}$ be a new vertex colouring of $G$ in which $\tilde{c}(u) := c(w)$ and $\tilde{c}(v) := c(w)$ and for all vertices $x$ distinct from $u$ and $v$ we have $\tilde{c}(x) := c(x)$. Take $\tilde{c}'$ to be the canonical edge colouring of $G$ corresponding to $\tilde{c}$. Let $\tilde{c}''$ be obtained from $\tilde{c}'$ by adding $1$ to the colour of $f$. Then a cycle violates \[obs:even\] or \[obs:odd\] with respect to $\tilde{c}''$ if and only if it contains $f$. Therefore, if $\gamma$ is a $\tilde{c}''$-preserving automorphism then $\gamma$ must fix $f$ setwise. It follows then that $\gamma$ must also preserve $\tilde{c}'$. Since $f$ is $\tilde{c}$-monochromatic, we have that $\gamma$ preserves $\tilde{c}$ by Lemma \[lem:nofixedpoint\]. Notice that all edges in $E_P$ that do not contain $u$ or $v$ are not $\tilde{c}$-monochromatic. Furthermore, if there is a $\tilde{c}$-monochromatic edge in $E_P$ not containing $u$ or $w$, say $vu'$, then $\tilde{c}(u') = c(u')$ must equal $\tilde{c}(v) = c(w)$. Hence $wv$ and $vu'$ are two distinct incident edges in $E_P$ with $c(w) = c(u')$, which cannot happen by assumption. Therefore, the only $\tilde{c}$-monochromatic edges in $E_P$ are $uv$ and $vw$ and, possibly, some other edges containing $u$. Since $\gamma$ preserves $\tilde{c}$ and fixes $f$ setwise, it must fix $v$ and therefore it must also fix $w$. Furthermore, $e$ and $f$ are the only $\tilde{c}$-monochromatic edges containing $v$, so $u$ must also be fixed by $\gamma$. Since $\tilde{c}$ and $c$ differ only at $u$ and $v$, it follows that $\gamma$ preserves $c$. Hence $\tilde{c}''$ is less symmetric than $c$ and our claim holds.
From now on we assume no two edges in $E_P$ are incident. We claim that we can find $e,f \in E_P$ with the following property.
1. \[clm:star\] There is no automorphism which fixes $e$ and $f$, swaps the endpoints of $e$ or $f$, and preserves the colouring $c$ on all vertices not incident to $e$ or $f$.
Once we have found such edges, let $\tilde{c}$ be the colouring obtained from $c$ by changing the colour of the endpoints of $e$ to $0$ and the colour of the endpoints of $f$ to $1$. Since $e$ and $f$ are the only $\tilde{c}$-monochromatic edges in $E_P$, they must be fixed by any $\tilde{c}$-preserving automorphism. By \[clm:star\] every $\tilde{c}$-preserving automorphism must fix the endpoints of both $e$ and $f$ and thus $\tilde{c}$ is less symmetric than $c$. Since there are $\tilde{c}$-monochromatic edges we can apply the observation we made at the beginning of our proof of Claim \[clm:Path\_classes\_One\], using $\tilde{c}$ instead of $c$. Hence there is an edge colouring which is less symmetric than $\tilde{c}$ and thus also less symmetric than $c$.
It remains to show that we can find edges satisfying \[clm:star\]. Since there are no cycle equivalence classes, every cycle in $G$ contains at least one edge in $E_P$, hence $G - E_P$ is a forest. No edge of $E_P$ has both its endpoints in the same tree $T$, otherwise we would obtain a cycle in $G$ that only contains one path equivalence class, and this is impossible by Lemma \[lem:existence-pathclass\].
Recall that, by assumption, no two edges in $E_P$ are incident. Let $e_0 = u_0v_0 \in E_P$, let $T$ be the tree in $G-E_P$ containing $u_0$, and let $C_1$ and $C_2$ be two cycles in $G$ that both contain $e_0$ but differ in the other edge incident to $u_0$. Such cycles exist because otherwise there would be an edge equivalent to $e_0$ contradicting the assumption that $e_0$ forms its own equivalence class. Assuming that $C_i$ traverses $e_0$ from $v_0$ to $u_0$, let $e_i = u_iv_i$ be the next edge in $E_P$ visited by $C_i$ for $i \in \{1,2\}$.
Denote by $d$ the distance between vertices in $G-E_P$, allowing the value $\infty$ for vertices in different components.
If there are $i,j \in \{0,1,2\}$ such that $d(u_i,u_j) \neq d(v_i,v_j)$, then an automorphism that swaps $u_i$ and $v_i$ must move $u_j$ to a vertex $x$ with $d(x,v_i) = d(u_j,u_i)$. But $x = u_j$ is impossible since $v_j \notin T$ and thus $d(u_j,v_i) = \infty$, and $x = v_j$ is impossible since $d(u_i,u_j) \neq d(v_i,v_j)$. Hence there is no automorphism which swaps $u_i$ and $v_i$ and fixes $e_j$. Analogously, there is no automorphism which swaps $u_j$ and $v_j$ and fixes $e_i$. Hence $e = e_i$ and $f = e_j$ satisfy \[clm:star\].
Finally assume that $d(u_i,u_j) = d(v_i,v_j)$ for all $i,j$. Since $u_0$ lies on the unique path connecting $u_1$ and $u_2$ in $T$. we know that $d(u_1,u_2) = d(u_1,u_0) + d(u_0,u_2)$. The same equation hence also holds for $v_0$, $v_1$, and $v_2$ showing that $v_0$ lies on the unique path connecting $v_1$ and $v_2$ in some other tree. Now an automorphism swapping $u_1$ and $v_1$ and fixing $e_2$ would have to swap these two paths. Hence such an automorphism would have to swap $u_0$ and $v_0$, but since $c(u_0) \neq c(v_0)$ this means that it would not preserve the colouring. Since an analogous statement is true for an automorphism swapping $u_2$ and $v_2$ and fixing $e_1$, we conclude that in this case $e = e_1$ and $f = e_2$ satisfy \[clm:star\].
Distinguishing colourings
=========================
As mentioned previously, our main results can be used to characterise the graphs for which equality holds in Theorem \[thm:distindexdistnumber\]. In [@alikhani-distindexdistnumber], the following family $\mathcal T$ of trees is introduced. A tree $T$ is contained in $\mathcal T$ if it satisfies the following requirements:
1. $T$ is bicentred with central edge $e$,
2. the rooted trees $T_1$ and $T_2$ attached to the endpoints of $e$ are isomorphic, and
3. $T_1$ has a unique distinguishing $k$-edge colouring up to isomorphism.
It is then proved that the finite members of this class are the only finite trees achieving equality in Theorem \[thm:distindexdistnumber\]. We extend this result and show that a graph $G$ satisfies Theorem \[thm:distindexdistnumber\] with equality if and only if $G \in \mathcal T$.
The following example shows how to construct infinitely many members of $T$.
Let $\{T_i\mid i \in I \}$ be a (finite or infinite) collection of (finite or infinite) rooted trees all of which admit a distinguishing $k$-colouring. Assume further that each $T_i$ only admits finitely many non-isomorphic distinguishing $k$-colourings and let $n_i$ be the number of such colourings.
Take a disjoint union of an edge $e=uv$ and $2 k n_i$ copies of each $T_i$. For every $i$ draw an edge from $u$ to the roots of $k n_i$ copies of $T_i$. For the remaining copies of $T_i$ draw an edge from $v$ to the root.
Figure \[fig:bad\] shows a tree obtained by this construction for $k=2$ obtained in this way, where the collection consisted of a single vertex and a path of length $1$.
Let $T$ be a bicentred tree, let $e=uv$ be its central edge, and let $T_1$ and $T_2$ be the components of $T-e$. Then for any integer $k \geq 2$ the following are equivalent.
1. \[itm:distnumberdistindex\] $D(T) = k$ and $D'(T)= k+1$.
2. \[itm:centraledge\] $D(T) = k$ and there is no distinguishing $k$-vertex colouring $c$ with $c(u) = c(v)$.
3. \[itm:tbad\] $T \in \mathcal T$.
We first show that \[itm:distnumberdistindex\] $\implies$ \[itm:centraledge\]. If there was a distinguishing $k$-vertex colouring with $c(u) = c(v)$, then (by swapping colours), there is such a colouring with $c(u) = c(v) = 0$, and thus by Lemma \[lem:colouringbijection\] also a distinguishing $k$-edge colouring.
For the implication \[itm:centraledge\] $\implies$ \[itm:tbad\] first observe that if $T_1$ and $T_2$ are not isomorphic, then every automorphism would fix $u$ and $v$. Thus in any distinguishing $k$-vertex colouring we can change the colours of $u$ and $v$ to obtain a distinguishing vertex colouring with $c(u) = c(v)$. Further, if there is no distinguishing vertex colouring with $c(u) = c(v)$, then any two distinguishing vertex colourings of $T_1$ with $c(u) = 0$ are isomorphic. Clearly, this implies that their canonical edge colourings must be isomorphic as well and by Lemma \[lem:colouringbijection\] there are no further distinguishing $k$-edge colourings of $T_1$.
Finally, for \[itm:tbad\] $\implies$ \[itm:distnumberdistindex\], note that if $T_1$ has an (up to isomorphism) unique distinguishing $k$-edge colouring, then every $k$-edge colouring of $T$ is either not distinguishing in $T_1$ or $T_2$, or has a colour preserving automorphism swapping $T_1$ and $T_2$. Hence $D'(T) > k$. On the other hand, Lemma \[lem:colouringbijection\] implies that there is a distinguishing vertex colouring of $T_1$ with $c(u) = 0$ and by changing the colour of the root we obtain $k$ non-isomorphic distinguishing $k$-vertex colourings. By colouring $T_1$ and $T_2$ by two non-isomorphic distinguishing vertex colourings we get a distinguishing vertex colouring of $T$, whence $D(T) \leq k$. By Theorem \[thm:distindexdistnumber\] the only possibility for $D'(T) > k$ and $D(T) \leq k$ is $D(T) = k$ and $D'(T)= k+1$.
[^1]: Address: Mathematics Institute, University of Warwick, U.K. Florian Lehner was supported by the Austrian Science Fund (FWF), grant J 3850-N32
[^2]: Address: Charlotte Scott Research Centre for Algebra, School of Mathematics and Physics, University of Lincoln, U.K.
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abstract: 'The zero locus of a function $f$ on a graph $G$ is defined as the graph for which the vertex set consists of all complete subgraphs of $G$, on which $f$ changes sign and where $x,y$ are connected if one is contained in the other. For $d$-graphs, finite simple graphs for which every unit sphere is a $d$-sphere, the zero locus of $(f-c)$ is a $(d-1)$-graph for all $c$ different from the range of $f$. If this Sard lemma is inductively applied to an ordered list functions $f_1,\dots,f_k$ in which the functions are extended on the level surfaces, the set of critical values $(c_1,\dots,c_k)$ for which $F-c=0$ is not a $(d-k)$-graph is a finite set. This discrete Sard result allows to construct explicit graphs triangulating a given algebraic set. We also look at a second setup: for a function $F$ from the vertex set to $R^k$, we give conditions for which the simultaneous discrete algebraic set $\{ F=c \}$ defined as the set of simplices of dimension $\in \{k, k+1,\dots,n\}$ on which all $f_i$ change sign, is a $(d-k)$-graph in the barycentric refinement of $G$. This maximal rank condition is adapted from the continuum and the graph $\{ F=c \}$ is a $(n-k)$-graph. While now, the critical values can have positive measure, we are closer to calculus: for $k=2$ for example, extrema of functions $f$ under a constraint $\{g=c\}$ happen at points, where the gradients of $f$ and $g$ are parallel $\nabla f = \lambda \nabla g$, the Lagrange equations on the discrete network. As for an application, we illustrate eigenfunctions of $d$-graphs and especially the second eigenvector of $3$-spheres, which by Courant-Fiedler has exactly two nodal regions. The separating nodal surface of the second eigenfunction $f_1$ always appears to be a 2-sphere in experiments. By Jordan-Schoenfliess, both nodal regions would then be 3-balls and the double nodal curve $\{f_1=0,f_2=0\}$ would be an un-knotted curve in the 3-sphere. Graph theory allows to approach such unexplored concepts experimentally, as the corresponding question are open even classically for nodal surfaces of the ground state of the Laplacian of a Riemannian $3$-sphere $M$.'
address: |
Department of Mathematics\
Harvard University\
Cambridge, MA, 02138
author:
- Oliver Knill
date: 'August 23, 2015'
title: A Sard theorem for graph theory
---
Introduction
============
We explore vector-valued functions $F$ on the vertex set of a finite simple graph $G =(V,E)$. Most of the notions introduced here are defined for general finite simple graphs. But as we are interested in Lagrange extremization, Morse and Sard type results in graph theory as well as questions in the spectral theory of the Laplacian on graphs related to Laplacians of Riemannian manifolds, we often assume $G$ to be a $d$-graph, which is a finite simple graph, for which all unit spheres are $(d-1)$-spheres in the sense of Evako [@KnillJordan]. In a first setup, more suited for Sard, for all except finitely many choices of $c \in R^d$, the graph $\{ F=c \} = \{ f_1=c_1, \dots , f_k = c_k \}$ is a $(d-k)$-graph, in a second setup, closer to classical calculus, we need to satisfy locally a maximal rank condition to assure that the graph representing the discrete algebraic set $\{F = c \}$ is a $(d-k)$-graph.\
The first part of the story parallels classical calculus and deals with the concept of level surface. It starts with a pleasant surprise when looking at a single function: there is a strong Sard regularity for a $d$-graph $G$: given a hyper-surface $G_f(x) = \{ f = c \}$ defined as the graph with vertex set consisting of all simplices on which $f-c$ changes sign, the graph $\{ f=c \}$ is a $(d-1)$-graph if $c$ is not a value taken by $f$. The topology of $G_f(c)$ changes only for parameter values $c$ contained in the finite set $f(V)$. This observation was obtained when studying coloring problems [@knillgraphcoloring; @knillgraphcoloring2; @KnillNitishinskaya], as a locally injective function on the vertex set is the same than a vertex coloring. Before, in [@indexexpectation; @indexformula], we just looked at the edges, on which $f$ changes sign and then completed the graph artificially. Now, we have this automatic. We hope to apply this to investigate nodal regions of eigenfunctions of the Laplacian of a graph, where we believe the answers to be the same for compact $d$-dimensional Riemannian manifolds or finite $d$-graphs. With the context of level surfaces one has the opportunity to look at nodal surfaces of eigenfunctions, which are also known as Chladni surfaces bounding nodal regions.\
With more than one function, the situation changes as the singular set typically becomes larger in the discrete. This is where the story splits. In the commutative setup, where we look at the zero locus of all functions simultaneously, Sard fails, while in the setup, where an ordered set of functions is considered, Sard will be true: as in the classical Sard theorem [@Morse1939; @Sard1942], the set of critical values has zero measure. The difference already is apparent if we take two random functions on a discrete $3$-sphere for example. Then simultaneous level set $\{ f=0, g=0 \; \}$ is a graph without triangles but it is rarely a $1$-graph, a finite union of circular graphs. The reason is that the tangent space is a finite set and the probability of having two parallel gradients at a vertex does not have zero probability. However, if we look at $g=0$ on the two dimensional surface $f=0$, we get a finite union of cyclic graphs. In general, we salvage regularity and Sard by defining the algebraic set $\{ F=c \}$ in a different way by recursively building hyper surfaces: start with the hyper surface $\{ f_1=c_1 \}$, then extend $f_2$ to the new graph and look at $\{ f_2=c_2 \}$ inside $\{f_1 =c_1 \}$. Now a vector value $c \in R^k$ is always regular if $c$ is not in the image of $F$ applied to recursively defined graph. This is the discrete analog of the multi-dimensional Sard theorem in classical analysis. The order in which the functions are taken, matters in the discrete. But this is not a surprise as we refine in each step the graph and therefore have to extend the functions to the barycentric refinements. Its only for sufficiently smooth functions like eigenfunctions of low energy eigenvalues of the graph that the answer can be expected to be independent of the ordering.\
The graph theoretical approach is useful for making experiments: take a 3-graph $G=(V,E)$ for example and take two real valued functions $f,g$ on the vertex set $V$. If $c$ is not in the image of $f$, we can look at the level surface $f=c$ and extend $g$ to a function there (vertices are now simplices and we just average the function value $g$ to extend the function). If $d$ is not in the image of $g$, then the $1$-graph $Z=\{ g=d \}$ is a subgraph of $\{ f=c \}$. It is a finite set of closed curves in $G_2$. In other words, each connected component is a knot in the 3-sphere. Unlike in the continuum, we do not have to worry about cases, where the knot intersects or self intersects. The Sard theorem assures that this will never happen in the discrete. We only have to assure that the value $c$ is not in $f(V)$ and $d$ is not in $g(V)$.\
The second setup, which defines discrete algebraic sets $Z=\{ f_1=c_1,\dots ,f_{k}=c_{k} \}$ in the Barycentric refinement $G_1$ of $G$ requires a few definitions. We need conditions under which these sets are nonsingular in the sense that they again form a $(d-k)$-graph, graphs for which the unit sphere $S(x)$ is a sphere at every vertex. In general, the situation is the same as in the continuum, where varieties are not necessarily manifolds. The definition is straightforward: define $\{F=c \; \}$ as the set of simplices of dimension in $\{ k,\dots ,n \}$ in $G$ on which all functions $f_j$ change sign simultaneously. The graph $\{F = c\}$ is a subgraph of the Barycentric refinement $G_1 = G \times 1$ of $G$. Conditions for regularity could be formulated locally in terms of spheres in spheres. In some sense, a locally projective tangent bundle with discrete projective spaces associated to unit spheres replaces the tangent bundle. It is still possible to define a vector space structure at each point and define a discrete gradient $\nabla f$ of a function. If $G$ is a $d$-graph, $x$ is a complete $K_{d+1}$ subgraph, then $\nabla f(x,x_0)$ is defined as the vector $\nabla f = \langle {\rm sign}(f(x_1)-f(x_0)), \dots,{\rm sign}(f(x_d)-f(x_0)) \rangle$. The local injectivity condition means that $\nabla f(x,x_0)$ is not zero for all $d$-simplices containing $x_0$. This leads to the Sard Lemma ([\[sardlemma\]]{}). But now, when changing $c$, we also need to look at the values $c_i=f(x_i)$ taken by the function $f$ and investigate whether they are critical. What happens if $c$ passes a value $c_i = f(x_i)$? If $f<c_i$ is homotopic to $f \leq c_i$, then the set $\{ f = c \} \cap S(x)$ is a $(d-1)$-sphere which by Jordan-Brower-Schoenflies [@KnillJordan] divides $S(x)$ into two complementary balls $S^-(x)$, $S^+(x)$. The Poincaré-Hopf index $1-\chi(S^-(x)) = i_f(x)$ [@poincarehopf] is then zero. At a local minimum of $f$ for example, the graph $S^-(x)$ is empty so that the index is $1$. At a local maximum, $S^+(x)$ is empty and the index is either $1$ or $-1$ depending on whether the dimension of $G$ is even or odd. In two dimensions, where we look at discrete multi-variable calculus, maxima and minima have index $1$ and saddle points have index $-1$.\
The set of hypersurfaces $M_j=\{ f =c_j \}$ form a contour map for which the individual leaves in general have different homotopy types. Topological transitions can happen only at values $f(V)$ of $f$ on the vertex set $V$ of the graph. We can relate the symmetric index $j_f(x)=[i_f(x)+i_{-f}(x)]/2$ at a vertex $x$ of $G$ with the $(d-2)$-dimensional graph $\{ f=c_k \}$ in the unit sphere $S(x)$. We have called this the graph $B_f(x)$ and showed that the graph can be completed to become a $d$-graph. This completion can be done now more elegantly. As pointed out in [@eveneuler], for $d=4$ for example, we get for every locally injective function $f$ a $2$-dimensional surface $B_f$, the disjoint union of $B_f(x)$. We see that a function $f$ not only defines $(d-1)$-graphs $G_f(c)=\{f=c\}$ but also $(d-2)$-graphs $B_f(x)$ called “central surfaces" for every vertex $x$.\
If we have more than one function as constraints, the singularity structure of $\{f_1=c_1, \dots, f_k =c_k \;\}$ is more complicated and resembles the classical situation, where singularities can occur. Also the Lagrange setup, where we mazimize or minimize functions under constraints, is very similar to the classical situation. The higher complexity entering with 2 or more functions is no surprise as also for classical algebraic sets defined as the zero locus of finitely many polynomials, the case of several functions is harder to analyze. In calculus, when studying extrema of a function $f$ under constraints $g$, following Lagrange, one is interested in critical points of $f$ and $g$ as well as places, where the gradients are parallel. Lets assume that we have two functions $f,g$ on the vertex sets of a geometric graph. The intersection $\{ f=c, g=d \}$ with $S(x)$ is then a sphere of co-dimension $2$. If we have $2$ constraints and $G$ is $4$-dimensional for example, then $S(x)$ is a $3$-sphere and $S(x) \cap \{ F=0 \}$ is a knot inside $S(x)$. This can already be complicated. By triangulating Seifert surfaces associated by a knot, one can see that any knot can occur as a co-dimension $2$ curve of a graph. Briskorn sphere examples allow to make this explicit in the case of torus knots. With more than one function, Poincaré-Hopf indices form a discrete $1$-form valued grid because changing any of the $c_j$ can change the Euler characteristic. This allows to express the Euler characteristic as a discrete line integral in the discrete target set of $F$. There are now many Poincaré-Hopf theorems, for every deformation path, there is one.\
If we look at the set $F=c$, where all functions change sign, singular values for $F=(f_1, \dots ,f_k)$ are vectors $c=(c_1,\dots,c_k)$ in $R^k$ for which the graph $\{ F=c \}$ is not a $(d-k)$-graph or values $c=F(x)$ taken on by $F$ on vertices $x$. Lets look at the very special example, where all $f_j$ are the same function. Now, near the diagonal $c_i=c$, there is an entire neighborhood of parameter values, where regularity fails. We see that the Sard statement does not hold in this commutative setting. We therefore also look at the non-commutative setup, where we fix $f_1=c_1$ first, then look at the level surface $\{ f_2=c_2 \}$ on the surface $\{f_1=c_1 \}$ and proceed inductively. Sard is now more obvious, but the sets $F=0$ depend on the order, in which the functions $f_j$ have been chosen. This order dependence is no surprise in a quantum setting, if we look at $f_j$ as observables. It simply depends in which order we measure and fix the $f_j$.\
There are analogies to Morse theory in the continuum: in the case of one single function, we can single out a nice class of graphs and functions, which lead to a discrete Morse theory. The geometric graph $G$ as well as functions on vertices are the only ingredients. One can now assign a Morse index $m(x)$ at a critical point and have $i_f(x) = (-1)^{m(x)}$. The requirement is an adaptation of a reformulation of the definition of being Morse means in the continuum that for a small sphere $S(x)$, the set $S(x) \cap \{ f(y)=f(x) \}$ is a product $S_{m-1} \times S_{d-1-m}$. For example for $m=1,d=2$, we have a saddle point, where $S(x)$ intersects the level surface in $4$ points forming $S_0 \times S_0$. In the enhanced picture, where we look at the graph $G_1$, we have this regularity more likely. One can extend the function to the new graph and repeat until one get a Morse function.\
We can use graphs described by finitely many equations in order to construct examples of graphs to illustrate classical calculus like Stokes or Gauss theorem or surfaces to illustrate classical multivariable calculus. The notion of $d$-graphs has evolved from [@elemente11; @indexexpectation; @eveneuler; @knillgraphcoloring] to [@knillgraphcoloring2], where it reached its final form. While finishing up [@KnillJordan] a literature search showed that spheres had been defined in a similar way already by Evako earlier on.\
The enhanced Barycentric refinement graph $G_1 = G \times K_1$ is a regularizes graph which helps to study Jordan-Brouwer questions in graph theory [@KnillJordan] and introduce a product structure on graphs which is compatible with cohomology [@KnillKuenneth]. It also illustrates the Brouwer fixed point theorem [@brouwergraph], as the fixed simplices on $G$ can be seen as fixed points on $G_1$. The simplex picture is also useful as the Dirac operator $D=d+d^*$ on a graph builds on it [@DiracKnill; @knillmckeansinger]. The graph spectra of successive Barycentric refinements converges universally, only depending on the size of the largest complete subgraph [@KnillBarycentric].\
An other application of the present Sard analysis is a simplification of [@indexformula] which assures that the curvature $K$ is identically zero for $d=(2m+1)$-dimensional geometric graphs: write the symmetric index $j_f(x)=i_f(x)+i_{-f}(x)$ in terms of the Euler characteristic of the $(d-2)$-dimensional graph $G_f(x)$ obtained by taking the hypersurface $\{ y \; | \; f(y)=f(x) \}$ in the unit sphere $S(x)$. For odd-dimensional geometric $d$-graphs, this symmetric index is is $-\chi(B_f(x))/2=j_f(x)$, while for even-dimensional graphs, it is $1-\chi(B_f(x))/2=j_f(x)$. One of the corollaries given here is that if $f$ is locally injective, then $B_f(x)$ is always a geometric $(d-2)$-graph if $G$ is a $d$-graph. In [@indexexpectation], we called the uncompleted graphs polytopes and showed that one can complete them. Now, since the expectation of $j_f(x)$ is curvature, [@indexexpectation; @colorcurvature], this gives an immediate proof that odd-dimensional $d$-graphs have constant $0$ curvature. Zero curvature follows for odd-dimensional graphs. The regularity allows to interpret Euler curvature as an average of two-dimensional sectional curvatures. Euler characteristic written as the expectation of these sectional curvature averages is close to Hilbert action. It shows that the quantized functional “Euler characteristic" is not only geometrically relevant but that it has physical potential.
Level surfaces
==============
The study of level surfaces $\{ f=c \}$ in a graph is not only part of discrete differential topology. It belongs already to discretized multivariable calculus, where surfaces $\{ f(x,y,z) = 0 \}$ in space or curves $\{ f(x,y) = 0 \}$ in the plane are central objects. Some would call calculus on graphs “quantum calculus". We want to understand under which conditions a sequence of $k$ real valued functions $F=(f_1,\dots,f_k)$ on the vertex set of a graph leads to co-dimension $k$ graphs $\{ F=0 \}$. We start with the case $k=1$, where we have a level surface $\{ f = c \}$.
Given a finite simple graph $G=(V,E)$, define the [**level hyper surface**]{} $\{ f=c \}$ as the graph $G'=(V',E')$ for which the vertex set $V'$ is the set of simplices $x$ in $G$, on which the function $f-c$ changes sign in the sense that there are vertices in $x$ for which $f-c < 0$ and vertices in $x$ for which $f-c>0$. A pair of simplices $(x,y) \in V' \times V'$ is in the edge set $E'$ of $G'$ if $x$ is a subgraph of $y$ or if $y$ is a subgraph of $x$. In the case $c=0$, the graph $\{ f = 0 \}$ is also called the [**zero locus**]{} of $f$.
[**Remarks**]{}.\
[**1)**]{} Instead of taking a real-valued function, we could take a function taking values in an ordered field. We need an ordering as we need to tell, where a function “changes sign".\
[**2)**]{} Most of the time we will assume $c$ is not a value taken by $f$. An alternative would be to include all simplices which contain a vertex on which $f=0$.
[**Examples.**]{}\
[**1)**]{} Let $G$ be a $2$-sphere like for example an icosahedron. Let $f$ be $1$ on exactly one vertex $x$ (a discrete Dirac delta function) and $0$ everywhere else. Now $\{ f=1/2 \}$ consists of all edges and triangles containing $x$. They form a circular graph and $\{ f = 0 \} = C_{2 {\rm deg}(x)}$.\
[**2)**]{} Let $G= C_n \times C_n \times C_n$ be a discrete $3$-torus and let $f$ be a function which is $1$ on a circular closed graph and $0$ else. Then $ \{ f =1/2 \}$ is a $2$-dimensional torus.
The Sard lemma
==============
The following definitions are recursive and were first put forward by Evako. See [@KnillJordan] for our final version.
A [**$d$-sphere**]{} is a finite simple graph for which unit sphere $S(x)$ is a $(d-1)$-sphere and such that removing a single vertex from the graph renders the graph contractible. Inductively, a graph is [**contractible**]{}, if there exists a vertex such that both $S(x)$ and the graph generated with vertices without $x$ are contractible.
A [**$d$-graph**]{} $G$ is a finite simple graph for which every unit sphere $S(x)$ is a $(d-1)$-sphere.
[**Examples.**]{}\
[**1)**]{} A $1$-graph is a finite union of circular graphs, for which each connectivity component has $4$ or more vertices.\
[**2)**]{} The icosahedron and octahedron graphs are both $2$-graphs. In the first case, the unit spheres are $C_5$, in the second case, the unit spheres are $C_4$.\
[**3)**]{} In [@KnillEulerian] we have classified all [**Platonic d-graphs**]{} using Gauss Bonnet [@cherngaussbonnet]. Inductively, a $d$-graph $G$ is called Platonic, if there exists a $(d-1)$-graph $H$ which is Platonic such that all unit spheres of $G$ are isomorphic to $H$. In dimension $d=3$, there are only two Platonic graphs, the 16 and 600 cell. In dimensions $d \geq 4$, only the cross polytopes are platonic.\
The following Sard lemma shows that we do not have to check for the geometric condition if we look at level surfaces: it is guaranteed, as long as we avoid function values in $f(V)$. Not even local injectivity is needed:
\[sardlemma\] Given a function $f:V \to {\bf R}$ on the vertex set of a $d$-graph $G=(V,E)$. For every $c \notin f(V)$, the level surface $\{ f=c \}$ is either the empty graph or a $(d-1)$-graph.
We have to distinguish various cases, depending on the dimension of $x$. If $x$ is an edge, where $f$ changes sign, then $f$ changes sign on each simplex containing $x$. The set of these simplices is a unit sphere in the Barycentric refinement $G_1$ of $G$ and therefore a $(d-1)$ sphere. If $x$ is a triangle, then there are exactly two edges contained in $x$, on which $f$ changes sign. The sphere $S(x)$ in $\{ f=c \}$ is a suspension of a $(d-2)$-sphere: this is the join of $S_0$ with $S_{d-2}$. In general, if $x$ is a complete subgraph $K_k$ then the unit sphere $S(x)$ is a join of a $(k-2)$-dimensional sphere and a $(d-k)$-dimensional sphere, which is a $(d-2)$-dimensional sphere. As each unit sphere $S(x)$ in $\{ f = c \}$ is a $(d-2)$-sphere, the level surface $\{ f=c \}$ is a $(d-1)$-graph.
[**Examples.**]{}\
[**1)**]{} If $d=2$, and $f$ changes sign on a triangle $K_3$, then it changes sign on exactly two of its edges. If $f$ changes sign on an edge, then it changes sign on exactly two of its adjacent triangles. We see that the level surface $\{ f=c \}$ is a graph for which every vertex has exactly two neighbors. In other words, each unit sphere is the $0$-sphere.\
[**2)**]{} If $d=3$, and $f$ changes sign on a tetrahedron $x=K_4$, then there are two possibilities. Either $f$ changes sign on three edges connected to a vertex in which case we have $3$ edges and $3$ triangles in the unit sphere of $K_4$ with $4$ vertices. A second possibility is that $f$ changes on $4$ edges and $2$ triangles in which case the unit sphere consists of $6$ vertices. Now look at a triangle $x=K_3$. It is contained in exactly two tetrahedra and contains two edges. The unit sphere is $C_4$. Finally, if $x=K_2$ is an edge, then all triangles and tetrahedra attached to $x$ form a cyclic graph of degree $C_{2n}$ where $n$ is the number of tetrahedra hinging on $x$.\
The Sard lemma can be used for minimal colorings for which the number of colorings is exactly known:
If $G$ is a $d$-graph and $c$ is not in the range of $f$, then the surface $H=\{f=c\}$ is a $(d-1)$-graph which is $d$-colorable. The chromatic polynomial $p(x)$ of these graphs satisfies $p(d)=d!$.
To every vertex of $G_1$, we can attach a “dimension" which is the dimension of the simplex in $G$ it came from. This dimension is the coloring. It remains a coloring when looking at subgraphs.
[**Examples.**]{}\
[**1)**]{} For a level surface $f=0$ on a $d$-dimensional graph, we get a $(d-1)$ graph which is $(d+1)$-colorable. For example, for $d=2$, the graph can be colored with $3$ colors. This is minimal as any triangle already needs $3$ colors. It implies the graph is Eulerian: the vertex degree is even everywhere.\
[**2)**]{} If $\Omega$ is the number of colorings with minimal color of $G$, we can for every vertex $x$ look at the [**index**]{} $i_f(x) = 1-\chi(S^-_f(x))$ where $S^-_f(x)$ is the set of vertices $y$ on the sphere $S(x)$, where $f(y)<f(x)$. Given a geometric graph $G$ of dimension $d$, then $G_1$ is $d+1$ colorable. The number of colorings is $(d+1)!$. We can look at the list of indices which are possible on each point and call this the index spectrum. The set of vertices $x$ where $\{ f<c \}$ changes the homotopy type are called [**critical points**]{} of $f$. If the index is nonzero, then we have a critical point because the Euler characteristic is a homotopy invariant. But there are also critical points with zero index, as in the continuum. Finding extrema of $f$ can be done by comparing the function values of all vertices where $i_f(x)$ is not zero or more generally, where $S^-_f(x)$ is not contractible. At a local minimum $S^-f(x)$ is empty.
The central surface
===================
Besides the surface $G_f(c) = \{ f=c \}$, there is for every vertex $x$ a central surface of co-dimension $2$ which is obtained by looking at level surfaces $B_f(x)$ obtained by looking at $\{ f = c \}$ inside the unit sphere $S(x)$. This object was introduced in [@indexformula; @eveneuler] for $4$-graphs $G$, where the central surface is a $2$-dimensional graph, a disjoint union of $2$-dimensional subgraphs $B_f(x)$ of the $3$-dimensional unit spheres $S(x)$.
A real-valued function $f$ on the vertex set of a graph $G=(V,e)$ is called [**locally injective**]{}, if $f(x) \neq f(y)$ for $(x,y) \in E$. An other word for a locally injective function is a [**coloring**]{}.
Each of these surfaces are subgraphs of their unit sphere $S(x_0)$. We have one surface for each vertex $x_0$. On each sphere $S(x,y) = S(x) \cap S(y)$, we can look at the intersection of $B_f(x)$ and $B_f(y)$. It consists of all simplices in $S(x,y)$ where both $f(z)-f(y)$ and $f(z)-f(x)$ change sign. Sometimes they can be joined together along a circle. For example, given a $3$-graph $G$, then the union $B_f$ of all $B_f(x)$ consists of all edges and triangles so that the max and min on each larger tetrahedron are attained in the edge or triangle. The following definition was first done in [@poincarehopf]:
Given a finite simple graph $G$ and a locally injective real-valued function $f$ on the vertex set $V$, the [**Poincaré-Hopf index**]{} is defined as $i_f(x) = 1-\chi(S^-_f(x))$, where $S^-_f(x)$ is generated by $\{ y \in S(x) \; | \; f(y)<f(x) \}$. The [**symmetric index**]{} is defined as $j_f(x) = (i_f(x) + i_{-f}(x))/2$.
The Poincaré-Hopf theorem [@poincarehopf] tells that $\sum_{x \in V} i_f(x)=\chi(G)$. Since this also holds for the function $-f$, we have $$\sum_{x \in V} j_f(x) = \chi(G) \; .$$ The following remark made in [@indexformula] expresses $j_f(x)$ as the Euler characteristic of a central surface, provided the graph is geometric:
Given a $d$-graph $G$ and a locally injective function $f$. If $B_f(x)$ is the central surface, then for odd $d$, we have $$j_f(x) = -\chi(B_f(x))/2 \; ,$$ for even $d$, we have $$j_f(x) = 1-\chi(B_f(x))/2 \; .$$
$B_f(x)$ is a $(d-2)$-graph in $S(x)$, which by assumption is a $(d-1)$-graph. Since $f$ is locally injective, the function $g(y)=f(y)-f(x)$ does not take the value $0$ on $S(x)$. By the Sard lemma, the graph $B_f(x)$ is a $(d-2)$-graph.
For a $d$-graph $G$ with odd dimension $d$, the curvature $$K(x) = 1- \frac{S_0(x)}{2} + \frac{S_1(x)}{3}- \frac{S_2(x)}{4} + \dots$$ (where $S_k(x)$ are the number of $K_{k+1}$ subgraphs of the unit sphere $S(x)$) has the property that it is constant zero for every vertex $x$.
The expectation $j_f(x)$ is curvature [@indexexpectation; @colorcurvature]. Since $j_f(x)$ is identically zero as the Euler characteristic of an odd dimensional $(d-2)$-graph, also curvature is identically zero.
Note that in the continuum, the Euler curvature is not even defined for odd dimensional graphs as the definition involves a Pfaffian [@Cycon]. Having the value $0$ in the discrete is only natural.\
A function $f$ on the vertex set is called a [**Morse function**]{} if it is locally injective and if at every critical point, there is a positive integer $m$, such that $B_f(x) = \{ f(y)=f(x) \}$ within $S(x)$ is a product $S_{m-1} \times S_{d-1-m}$ or the empty graph if $m=0$ or $m=d$. The integer $m$ is called [**Morse index**]{} of the critical point $x$.
Depending on whether $m$ is odd or even, we have $\chi(B_f(x)) = 4$ or $0$ so that the index $1-\chi(B_f(x))/2=-1$ if $m$ is odd and $1$ if $m$ is even. When adding a critical point, this corresponds to add a $m$-dimensional handle. It changes the Euler characteristic by $(-1)^m$ and changes the $m$’th cohomology by $1$.
Lagrange
========
In this section we try to follow some of the standard calculus setup when extremizing functions with or without constraints. But it is done in a discrete setting, where space is a graph. As school calculus mostly deals with functions of two variables, we illustrate things primarily for $2$-dimensional graphs, even so everything can be done in any dimensions.\
There are three topics related to critical points in two dimensions: A) extremizations without constraints, B) equilibrium points of vector fields and C) extremization problems with constraints which are called Lagrange problems. In the case A), can look at extrema of a function $f$ on the vertex set of a $2$-graph, in the case B) we look at equilibrium points of a pair $F=(f,g)$ of functions on the vertex set of a $2$-graph, and finally in the case C), we look at extrema of a function $f$ on the vertex set under the constraint $g=c$ on a $2$-graph.\
Lets first look at the “second derivative test" on graphs. Recall that a vertex $x$ in a graph is a critical point of a function $f$, if $\{ f<f(x) \}$ and $\{f \leq f(x) \}$ are not homotopic. This is equivalent to the statement that $S^-_f(x) = \{ y \in S(x)\; | \; f(y)<f(x) \}$ is a graph which is not contractible. The analogue of the discriminant $D$ is the Poincaré-Hopf index $i_f(x) = 1-\chi(S^-_f(x))$. Here is the analogue of the second derivative test:
Let $G$ be a $d$-graph and assume $f$ is locally injective and $x$ is a critical point. There are three possibilities:\
a) If $S^-_f(x)$ is a $(d-1)$-sphere, then $x$ is a local maximum.\
b) If $i_f(x)$ is positive and $S^-_f(x)$ is empty then $x$ is a local minimum.\
c) If $d=2$ and if $i_f(x)$ is negative, then $x$ is a type of saddle point.
For a $2$-dimensional graph, the index is nonzero if and only if $x$ is a critical point because a subgraph of a circular graph has Euler characteristic $1$ if and only if it is a contractible graph. It has Euler characteristic $0$ if and only if it is either empty or the full circular graph. In all other cases of subgraphs of a circular graph, the Euler characteristic counts the number of connectivity components.\
In higher dimensions, there are cases of graphs $S_f^-(x)$ having Euler characteristic $1$ but not being contractible. In higher dimensions, $S_f^-(x)$ can have negative Euler characteristic so that the index $i_f(x)$ can become larger than $1$.
[**Example.**]{}\
[**1)**]{} For $d=2$, the standard saddle point is $i_f(x)=-1$. The function $f$ changes sign on 4 points. A discrete “Monkey saddle" has index $i_f(x)=-2$. It is obtained for example at a vertex $x$ for which the unit ball is a wheel graph with $C_6$ boundary such that $f(y)$ is alternating smaller or bigger than $f(x)$.\
[**2)**]{} If $d$ is odd and $f$ has a local maximum, $i_f(x)=-1$. This is analogue to the continuum, where $D$ is the determinant of the Hessian.\
For simplicity, we restrict to a simple $2$-dimensional situation, where we have two functions $f,g$ on the vertex set $V$ of a $2$-graph $G=(V,E)$. We can think of $F=(f,g)$ it as a vector field and see the equilibrium points are the intersection of null-clines as in the continuum. Classically, the critical points of $f$ under the constraint $g=c$ are the places, where these null-clines are tangent or are degenerate in that one of the gradients is zero.
In more generality we have defined the set $G_F=\{ f_1=0, \dots ,f_k=0 \}$ as the graph whose vertex set consists of the set of simplices of dimension in $\{ k,\dots ,d\}$ on which all functions $f_j$ change sign and where two simplices are connected, if one is contained in the other. It is possible for example that for a $d$-graph, the set $G_F$ consists of [**all**]{} $d$-dimensional simplices in $G$. This is still a $0$-graph, a graph with no edges because no $d$-simplex is contained in any other $d$-simplex.
[**Example.**]{}\
[**1)**]{} If $G$ is a $2$-graph and $x$ belongs to a triangle $t$ containing two edges which both contain $x$, then $\nabla f(x,t) = \langle 1,1 \rangle$. Two functions have a parallel gradient in $t$, if and only the sign changes in $t$ happen on the same edges of the triangle.\
[**Remarks**]{}:\
[**1)**]{} A function $f$ on the vertex set $V$ is a $0$-form. The exterior derivative $df$ is a function is a 1-form, a function on the edges of the graph. The exterior derivative depends on a choice of orientations on complete subgraphs. For $1$ forms in particular, where edges have been ordered at first, the situation at a vertex $x$ defines then $df( (x,y)) = f(y)-f(x)$. When restricting to a $d$-simplex, we get $d$ real numbers, we as in the continuous forms the [**gradient**]{}.\
[**2)**]{} The ordering of the coefficients of the gradient vector depends on the orientation of the triangle. This is similar to the classical case, where a triangle in a triangulation of a surface defines a normal vector at $x$, once a vertex $x$ of the triangle and an orientation of the triangle is given.\
[**3)**]{} The analogue of rank $d(f,g)=2$ means that simultaneous sign changes happen on one edge of a triangle only, so that the two sign change is in the same direction on that edge. Geometrically this implies that the spheres $f=0,g=0$ in $S(x)$ intersect transversely. Two discrete gradients in $Z_2^d$ are parallel if and only they are the same because the only nonzero scalar is $1$.\
The classical Lagrange analysis shows that the critical points of $F$, the place where the Jacobean $dF$ has rank $0$ or $1$ are candidates for maxima of $f$ under the constraint $g=c$. In the same way as in the continuum, we can write down Lagrange equations for any number of functions on a $d$-graph. The most familiar case for two functions:
Let $G$ be a $d$-graph. If the discrete gradients $\nabla f$, $\nabla g$ are nowhere parallel, then $F = \{ f = 0, g = 0 \}$ is a $(d-2)$-graph.
It should be possible to estimate the measure of the set of global critical values if $F=(f_1,\dots,f_k)$ are functions on the vertex set of a finite simple graph $G$. Instead of doing so, we will look later at a Sard setup for which the critical values have zero measure.\
Lets look at the example of two functions $f,g$ on a $3$-graph. The set $\{ F=0 \}$ is the set of simplices, where both $f,g$ change sign. The condition $dF$ having maximal rank means that the surface $f=0$ on $S(x)$ and the sphere $g=0$ in $S(x)$ intersect in a union of $1$-spheres. Here is the Lagrange setup with two functions in three dimensions:
If $F=(f,g)$ are two functions on the vertex set of a $3$-graph and if $dF$ has maximal rank at every $x$ and $(c,d)$ is not in $F(V)$, then $F=c$ is either empty of a $1$-graph, a finite collection of circles. \[case2-3\]
The set $\{ F=c \}$ is a graph whose vertices are the triangles ($K_3$ subgraphs of $G$) or tetrahedra ($K_4$ subgraphs of $G$), where both $f$ and $g$ change sign. Let $x$ be a tetrahedron in $\{F =c \}$. The maximal rank condition prevents parallel gradients like $\nabla f=\langle 1,1,1 \rangle = \nabla g = \langle 1,1,1 \rangle$ so that it is impossible to have 3 triangles in $\{ F=c \}$ inside $x$. Assume a single triangle is present where both $f,g$ change sign, then there is a common edge, where both $f,g$ change sign and a second triangle must also be in $\{ F = c \}$. We see that there are exactly two triangles $y,z$ present on which both $f,g$ change sign. This means the vertex $x$ has exactly two neighbors $y,z$. Each of the triangles $y,z$ has two neighbors, the tetrahedra attached to them. We see that $\{ F =c \}$ is a graph with the property that every unit sphere is the zero sphere $S_0$. Therefore, it is a finite collection of circular graphs.
The Lagrange problem extremizing $f$ under the constraint $g=c$ is the following:
Given two functions $f,g$ on a $2$-graph. Extremizing $f$ under the constraint $g=c$ happens on triangles, where $df$ and $dg$ are parallel (which includes the case when $df=0$ or $dg=0$). In other words, extrema happen on Lagrange critical points.
The following maximal rank condition is the same as in the continuum. It tells that the graph formed by the simplices having dimension in $\{ k, \dots, d \}$ on which all functions $f_k$ change sign simultaneously is a $(n-k)$-graph if the rank of the set of gradients in $Z_2^k$ is maximal:
Assume $G$ is a $d$-graph and $F$ is a $R^k$-valued function on the vertex set. If $\nabla F$ has maximal rank on every $d$-simplex, then $\{ F=c \}$ is a $(d-k)$-graph.
For a fixed vertex $x$ and $d$-simplex $X$, we have a tangent space $Z_2^d$ for which every function $f_j$ contributes a vector $\nabla f_i$ telling on which edges emanating from $x$ inside $X$, the sign of $f_i$ changes. By assumption, the $k$ vectors $\nabla f_i$ are linearly independent.\
Use induction with respect to $d$: if $k=d$, there is nothing to show because by definition, the set $\{ F=c \}$ is the set of $d$-simplices, where all functions change sign and this is a $0$-graph as there are no edges. We claim that in the unit sphere $S(x)$, the functions $f_1,\dots ,f_k$ induce $k$ linearly independent gradients $\nabla f_j$ on the $(d-1)$-simplex $Y=X \setminus x$. Indeed, on each triangle, the sum of the $df_j$ values is zero (as ${\rm curl}({\rm grad}(f))=0$). A nontrivial relation between the gradients on $Y$ would induced a nontrivial additive relation between the gradients on $X$. Now, by induction, the $k$ functions on $S(x)$ define a $(d-1-k)$-graph.
[**Remark.**]{} One could also try induction with respect to $k$ and consider $\{ f_1 = c_1 \}$ which is a $(d-1)$-graph, by the Sard lemma. The problem is that one has to extend $f_2,\dots,f_d$ in such a way on the simplices so that one has still maximal rank condition. The problem is that $\{ f_1 = c_1 \}$ has now a different vertex set than $G$ and that linking things is difficult.\
[**Examples:**]{}\
[**1)**]{} The case $d=3$, $k=2$ was discussed in Proposition \[case2-3\].\
[**2)**]{} In the case $d=4$, $k=2$, we want the two functions both to be locally injective and the gradients of the two functions $f,g$ not to be parallel. The set $F=c$ consists of all tetrahedra $K_4$ and hypertetrahedra $K_5$ on which both $f$ and $g$ change sign. The gradients restricted to the tangent space on $S(x)$ are not parallel and we can apply the analysis of the previous case to each unit sphere which shows that in $S(x)$ the set $F=c$ is a collection of circular graphs. This shows that the unit spheres of $F=c$ have the property that each unit sphere there is a circular graph. Therefore $F=c$ is a geometric 2-graph, a surface if $\nabla f,\nabla g$ are nowhere parallel.\
[**3)**]{} In the case $d=4$, $k=3$, it is the first time that the maximal rank condition is not just a parallel condition. Given a vertex $x$ and a tetrahedron $t=(x,y_1,y_2,y_3)$. The discrete gradient is $\langle f(y_1)-f(x), f(y_2)-f(x), f(y_3)-f(x) \rangle$. An example of a violation of the maximal rank condition for three functions $f,g,h$ would be $\nabla f = \langle 1,0,1 \rangle$, $\nabla g = \langle 1,1,0 \rangle$, $\nabla h = \langle 0,1,1 \rangle$. They are pairwise not parallel but $\nabla f + \nabla g + \nabla h = 0$.\
[**4)**]{} The non-degeneracy condition is not always needed: for $d$ functions $F=(f_1,\dots,f_d)$ on a $d$-graph, we look at the simplices on which all functions change sign. The condition $dF \neq 0$ implies that the set of $d$-dimensional simplices on which all functions $f_j$ change sign are isolated.
Sard theorem
============
Since the set of critical values can hove positive measure if we look at the simultaneous solution, we change the setup and look at hypersurfaces in hypersurfaces. This will lead to a Sard theorem as in the continuum. We will have to pay a prize: the order with which we chose the hypersurfaces within hypersurfaces now will matter.
A function $f$ on the vertex set is called [**strongly injective**]{} if all function values $f(x_i)$ are rationally independent. It is [**strongly locally injective**]{} if in each complete subgraph, the values are rationally independent. A list of functions $f_1,\dots, f_k$ is called [**strongly injective**]{} if the union of all function values $f_j(x_i)$ are rationally independent.
Strongly injective functions are generic from the measure and Baire point of view: given a finite simple graph $G$ with $n$ vertices. Look at the probability space $\Omega$ of all functions from the vertex set to $[-1,1]$, where the probability measure is the product measure on $[-1,1]^n$.
For any $k$, with probability one, a random sample $f_1,\dots,f_k$ in $\Omega^k$ is strongly locally injective.
There are $v=v_0+v_1+\cdots+v_d$ complete subgraphs in $G$. They define the $\sum_i i \cdot v_i$ numbers $c_{ij} = f_i(x_j)$. There is a countably many rational independence conditions $\sum_{ij} a_{ij} c_{ij} =0$ to be avoided, where $a_{ij}$ are integers. The complement of a countable union of such hyperplane sets of zero measure in $[-1,1]^n$ and consequently has zero measure.
Given an ordered list of functions $f_1,f_2,\dots, f_k$ from the vertex list $V$ of a finite simple graph $G=(V,E)$ and $c_1, \dots, c_k$ be $k$ values. Let $\overline{f}_1=f_1$. Denote by $\overline{f}_2$ the function $f_2$ extended to $\{ \overline{f}_1 = c_1 \}$ and by $\overline{f}_3$ the function $f_3$ extended to $\{ \overline{f}_1 = c_1, \overline{f}_2 = c_2 \}$ etc, always assuming that $\overline{f}_{j+1}$ defined on $\overline{f}_1 = c_1, \dots, \overline{f}_j = c_j \}$ does not take the value $c_{j+1}$. We call the sequence $c_1,\dots, c_k$ [**compatible**]{} with $f_1,\dots,f_k$ if a sequence $\overline{f}_j$ can be defined so that none of them are not constant.
Given $k$ strongly injective functions $f_{1},\dots,f_k$. For all except a finite set of vectors $(c_1,\dots,c_k)$ the sequence $c_1,\dots,c_k$ is compatible and the set $\{ f_1 = c_1, \dots, f_k = c_k \}$ is a geometric $(d-k)$-graph.
This follows inductively from the construction. In each step, only a finite set of $c$ values are excluded.
The assumption is stronger than what we need. It sometimes even works in the extreme case of two identical functions: Let $f$ be the function on the octahedron given the values $f_1(1)=13,f_1(2)=15,f_1(3)=17,f_1(4)=19$ on the equator and the value $f_1(5)=1$ on the north pole and the value $f_1(6)=31$ on the south pole. Lets take $c_1=2$. Now, $\{ f_1 = c_1 \}$ is the cyclic graph with vertices $\{ (51)$, $(512)$, $(52)$, $(523)$, $(53)$, $(534)$, $(54)$, $(541) \}$. The function $\overline{f}_2$ takes there the values $\overline{f}_2(51)=(f_2(5)+f_2(1))/2=(1+13)/2=7$, $\overline{f}_2(52)=(f_2(5)+f_2(2))/2=(1+15)/2=8$, $\overline{f}_2(53)=(f_2(5)+f_2(3))/2=(1+17)/2=9$, $\overline{f}_2(54)=(f_2(5)+f_2(4))/2=(1+19)/2=10$, $\overline{f}_2(512)=(f_2(5)+f_2(1) f_2(2))/2=(1+13+15)/3=29/3$, $\overline{f}_2(523)=(f_2(5)+f_2(2) f_2(3))/2=(1+15+17)/3=35/3$, $\overline{f}_2(534)=(f_2(5)+f_2(3) f_2(4))/2=(1+17+19)/3=37/3$, $\overline{f}_2(523)=(f_2(5)+f_2(2) f_2(3))/2=(1+19+13)/3=11$. Now for example, for $c_2=8.5$ the set $\{ f_2=c_2 \}$ is a $0$-graph.\
As an example, lets look at the [**double nodal surface**]{} $f_3 =0$ in $f_2=0$, where $f_3$ is the third eigenvector. By Sard, we know:
If $G$ is a $d$-graph and the eigenfunctions $f_2,f_3$ are strongly injective not having the value $0$, then the double nodal surface is a $(d-2)$-surface in $G_2$.
[**Example**]{}:\
1) For the octahedron, the smallest $2$-sphere, the spectrum is of the Laplacian is $$\{ 0,4,4,4,6,6 \}$$ The ground state space is the eigenspace to the eigenvalue $4$ and three dimensional and spanned by $(-1,0,0,0,0,1)$, $(0,-1,0,0,1,0)$, $(0,0,-1,1,0,0) \}$. The eigenspace to the eigenvalue $6$ is spanned by $(1,0,-1,-1,0,1), (0,1,-1,-1,1,0)$. In both eigenspaces, there are injective functions $f_2=(-1,-2,-3,3,2,1),f_3=(1,2,-3,-3,2,1)$ in the eigenspace. The graph $\{ f_2=0\}$ is the cyclic graph $C_{12}$ while the graph $\{ f_3 =0 \}$ is $C_8 \cup C_8$. The graph $\{ f_3=0\}$ within $\{f_2=0\}$ is a two point graph. The graph $\{ f_2=0 \}$ within $\{ f_3=0 \}$ is not defined as $f_2$ extended to the simplex set in a linear way produces a lot of function values $0$.
It leads to a generalization of a result we have shown for $2$-graphs:
Any compact $d$-manifold $M$ has a finite triangulation which is $(d+1)$-colorable.
By Nash-Tognoli, any compact manifold can be written as a variety $F=c$ in some $R^d$. Now just rewrite this in the discrete as the zero locus of $F=c$.
The curvature at a vertex $x$ of such a graph triangulation can be written as the expectation of $d!$ Poincaré-Hopf indices.
If $\{f=0\}$ be the zero locus of $f:G \to R$. It would be nice to find a smaller homeomorphic graph which represents this set.
Nodal sets
==========
To illustrate a possible application, lets look at the problem of nodal sets of the Laplacian $L$ of a graph. Understanding the [**Chladni patterns**]{} of the Laplacian on a manifold with or without boundary is a classical problem in analysis. The nodal region theorem of Courant in the discrete also follows from the min-max principle [@Fiedler1975; @VerdiereGraphSpectra; @Spielman2009]. For a general graph $G$, and an eigenfunction $f$, one looks at the number of connectivity components of $Z_f^+ = \{ \pm f \geq 0 \}$. Let $v_k$ be the $k$’th eigenvector. Since both for compact Riemannian manifolds as well as for finite graphs, the zero eigenvalues are not interesting as harmonic functions are locally constant, we look primarily at the second or third eigenvalue. The Fiedler nodal theorem assures then that the graph generated by $\vec{v}_k > 0$ has maximally $k-1$ components. Especially, the second eigenvector, the “ground state", always has exactly two nodal components. If an eigenfunction $f$ of the Laplacian is locally injective and has no roots, we can look at its nodal surfaces $f=0$ separating the nodal regions. I $f$ does not take the value $0$, then $\{f = 0 \}$ is defined even if $f$ is not locally injective. As in the continuum, one can ask how big the set $\{ v \in V \; | \; f(v)=0 \}$ can become.
If $G$ is a $2$-sphere, then the two nodal surface is a simple closed curve in $G$. How common is the situation that the ground state does not take the value $0$ and is locally injective? The situation that $0$ is in the range of the ground state appears to be rare. For random 2-spheres (of the order of 500 vertices generated by random edge refinements from platonic and Archimedean solids) we get a typical ground state energy in the order of $0.08$ and the third eigenvalue in the order $0.2$.\
[@BLS] note the following [**eigenfunction principle**]{}: any eigenfunction to an eigenvalue $0<\lambda<n$ takes the value $0$ on every vertex of degree $n-1$, if $n$ is the number of vertices: Proof: let $f$ be the eigenfunction to an eigenvalue $\lambda$. The function $f$ is perpendicular to the harmonic constant function so that $\sum_{x \in V} f(x)=0$. From $Lf(v) = (n-1) f(v) - \sum_{x \neq v} f(x) = \lambda f(v)$, we get $n f(v) = \lambda f(v)$ which by assumption implies $f(v)= 0$.\
[**Example.**]{}\
For a wheel graph with $n$ vertices, there is one eigenvalue $n$ with eigenvector $(1-n,1,1, \dots, 1)$, an eigenvalue $0$ with eigenvector $(1,1, \dots, 1)$. All eigenfunctions to eigenvalues between take the zero value somewhere. For example, in the case $n=7$, the eigenvalues are $\{ 0,2,2,4,4,5,7 \}$ with eigenvectors $[0,-1,-1,0,1,1,0]$, $[0,1,0,-1,-1,0,1]$, $[0,-1,1,0,-1,1,0]$, $[0, -1, 0, 1,-1,0,1]$, $[0,-1,1,-1,1,-1,1]$ and $[-6,1,1,1,1,1,1]$.\
Let $G$ be a $d$-graph. Let $f$ be the ground state, the eigenvector to the first nonzero eigenvalue $\lambda$, the spectral gap. Let $d$ denote the exterior derivative. There will be no confusion with $d$ also denoting the dimension of $G$. We assume that the eigenvalue $\lambda$ is simple, that $f$ has no roots and that $df$ has no roots. This assures that $Z=0$ is a geometric $(d-1)$-graph. As $L=d d^*$ on $0$ forms, we get from $d^* d f = \lambda f$ that $d d^* d f = \lambda df$ so that $df$ is an eigenfunction to the one form Laplacian $L_1=d d^* + d^* d$. The set $\{ df=0 \}$ of all triangles and simplices where both $f$ and $df$ changes sign is the same than $f=0$.\
What is the topology of the hypersurface $Z_2 = \{ \vec{v}_2=0 \}$? For a $2$-sphere $G$, we know that there are two components so that the principal nodal curve $Z_2$ has to be a circle. Is the nodal curve to the second eigenvalue a $d$-sphere, if $G$ is a $d$-graph? We believe that the answer is yes and robust. If $f$ should have roots, we can add a small random function with $|g(x)| \leq \epsilon$. We expect that for sufficiently small $\epsilon>0$ and almost all $g$, the surface $Z = \{ f+g =0 \; \}$ has the same topology.
Does the topology of the nodal manifold to the second eigenvalue depend on the topology of $G$ only?\
We think the answer is yes as $Z$ has to be a connected surface and that going from a genus $k$ to a genus $k+1$ surface can not happen so easily. More generally, we expect: if $G$ and $H$ are homotopic graphs of the same dimension, then the second nodal manifolds of $G$ and $H$ are homotopic.\
Summary
=======
A $d$-graph is a finite simple graph for which every unit sphere is a $(d-1)$ sphere. Given $k$ locally injective real-valued functions $f_1,\dots,f_k$ on the vertex set of a $d$-graph, we can define the zero locus $Z=\{\vec{F}=\vec{0}\}$ of $F$ in $G$ as the graph $(V,E)$ with vertex set $V$ consisting of complete subgraphs in $G$, on which all the functions $f_j$ change sign and for which two vertices are connected if one is a subgraph of the other. The graph $Z$ is a subgraph of the barycentric refinement $G_1$ but like varieties in the continuum, the graph might be singular and not be a $(d-k)$-graph. We can define a discrete tangent bundle on $G_1$ so that the projection of $\nabla f$ in $T_xG = Z_2^d$ to $T_xH$ is zero, establishing the graph analogue of “gradients are perpendicular to level surfaces". If every unit sphere $S(x)$ in $G_1$ has the property that $Z \cap S(x)$ is a $(d-k-1)$-sphere in the $(d-1)$-sphere $S(x)$, then $F=c$ is a $(d-k)$-graph $H$. The later plays the role of $\{f_1=c_1, \dots =f_k=c_k \}$ for $k$ differentiable functions $f$ on a smooth manifold $M$ for which the maximal rank condition ${\rm rank}(dF)(x)=d-k$ for $F=(f_1,\dots,f_k), x \in Z$ is satisfied. Discrete Lagrange equations help to maximize or minimize $f$ under a constraint $g=c$: as in the continuum, critical points need parallel gradients $\nabla f, \nabla g$.\
If the [**ordered zero locus**]{} is defined by looking at hypersurfaces on successive barycentric refinements, we never run into singularities if the functions are strongly locally injective in the sense that the function values of $f_j$ on the vertex set are rationally independent on each complete subgraph. The [**Sard theorem**]{} states then that the graph $\{f_1=c_1,\dots,f_{k}=c_{k} \}$ is a $(d-k)$-graph inside the $d$-graph $G_k$ for all $\vec{c}$ not in a finite set. An application is the observation that any $d$-dimensional projective algebraic set admits an approximation by manifolds with a triangulation which is a minimally $(d+1)$-colorable graph, using an explicitly constructed $d$-graph determined by the equations defining $V$.\
The possibility to define level surfaces in discrete setups can be illustrated in the case of eigenfunctions of the Laplacian. Pioneered by Chladni, the geometry of nodal surfaces is important for the physics of networks. Of special interest is the [**ground state**]{} $f_2$ of a compact Riemannian manifolds or finite graph. The nodal surface $f_2=0$ is a $(d-1)$-dimensional hypersurface and the double nodal surface $f_2=0,f_3=0$ is generically a codimension $2$ manifold. If the manifold or graph is a 3-sphere, the nodal surface is two dimensional and the double nodal surface is a collection of closed curves in $S^3$. We asked here whether $f_2=0$ have positive genus and whether $f_2=0,f_3=0$ can be knotted. Graph theory allows to investigate this experimentally. The answers for Riemannian manifolds and graphs are expected to be the same.\
Lets look at some [**multi-variable calculus terminology**]{} in a $2$-graph translated to graph theory:\
------------------------ -------------------------------------------------------
Critical point $x$ $S^-_f(x)$ is not contractible
Discriminant $D$ Poincaré-Hopf index $i_f(x)=1-\chi(S^-_f(x))$
$D>0,f_{xx}<0$ $i_f(x)=1$, $S^-_f(x) = S_f(x)$
$D>0,f_{xx}>0$ $i_f(x)=1$, $S^-_f(x) = \emptyset$
$D<0$ $i_f(x)<0$
$D=0$ not locally injective
Level curve $f(x,y)=0$ zero locus $f=0$ in $G_1$.
$T_xM$ maximal simplex $t$ in $G$ containing $x$
Tangent vector $T_xM$ $\nabla f = {\rm sign}(f(y)-f(x)) \; | \; y \in t \}$
Lagrange equations $\nabla f = \lambda \nabla g$ or $\nabla g=(0,0)$
------------------------ -------------------------------------------------------
Questions
=========
A\) In the commutative case, where $F=c$ is a subgraph of the barycentric refinement $G_1$, the set of critical values can have positive measure. It would be nice to find an upper bound on the measure of constants $c$ for which the graph $\{ F=c \} \subset G_1$ can be singular.\
B) In the context of graph colorings, the following question is analog to the [**Nash embedding problem:**]{} Is every $d$-graph a subgraph of a barycentric refinement $G_n$ of some geometric $d$-graph? If the answer were yes, we would have a bound $d+1$ for the chromatic number of $G$. A special case $d=3$ would prove the $4$-color theorem. We approached this problem by writing the sphere as embedded in a $3$ sphere then making homotopy deformations to render the sphere Eulerian and so $4$ colorable coloring in turn the embedded sphere. While a Whitney embedding of a graph is possible if we aim for a homeomorphic image, we don’t have yet a sharp discrete analogue of the classical Whitney embedding theorem, fixing the dimension. Realizing a graph as a subgraph of a product of linear graphs is analogue to an isometric embedding of a Riemannian manifold in some Euclidean space is a discrete Nash problem.\
C) If $f,g$ are two strongly locally injective function on some $d$-graph. Under which conditions is it true that the level set $g=d$ in $f=c$ is topologically equivalent to the level set $g=d$ in $f=c$? Random examples in a 3-sphere show that the answer is no in general. It might therefore be possible that $f_2=0$ in $f_3=0$ is different that $f_3=0$ in $f_2=0$ but we have not yet an example, where this difference takes place.\
D) Assume $G$ is a $3$-sphere and the ground state $f_1$ (eigenvector to the smallest positive eigenvalue) does not take the value $0$. Is the nodal hyper surface $\{f=0\}$ always a $2$-sphere? The case where $0$ is in the image of the ground state $f_1$ only appears in very rare cases. As long as the $0$’s are isolated, we can randomly change the value on such places and not change the topology of the nodal surface $\{ f = 0 \}$.
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---
abstract: 'We prove that $\mathsf{RCA}_0+\mathsf{RT}_2^2\not\rightarrow \mathsf{WKL}_0$ by showing that for any set $C$ not of -degree and any set $A$, there exists an infinite subset $G$ of $A$ or $\overline{A}$, such that $G\oplus C$ is also not of -degree.'
address: 'Department of Mathematics, Central South University, South campus, south dormitory No. 6 Room. 619, Changsha, 410083, China'
author:
- Jiayi Liu
bibliography:
- 'RT2.bib'
title: '$\mathsf{RT}_2^2$ does not imply $\mathsf{WKL}_0$'
---
Introduction {#sec1}
============
Reverse mathematics studies the proof theoretic strength of various second order arithmetic statements. Several statements are so important and fundamental that they serve as level lines. Many mathematical theorems are found to be equivalent to these statements and they are unchanged under small perturbations of themselves. The relationships between these statements and “other” statements draw large attention. $\mathsf{WKL}_0$ is one of these statements. $\mathsf{WKL}_0$ states that every infinite binary tree admits an infinite path. It is well known that as a second order arithmetic statement, $\mathsf{WKL}_0$ is equivalent to the statement that for any set $C$ there exists $B\gg C$, where $B\gg C$ means $B$ is of -degree relative to $C$. A good survey of reverse mathematics is [@simpson1999subsystems] or [@friedman1974some], [@friedman1976systems]. One of the second order arithmetic statements close to $\mathsf{WKL}_0$ is $\mathsf{RT}_2^2$.
Let $[X]^k$ denote $\{F\subseteq X: |F|=k\}$. A k-coloring $f$ is a function, $[X]^n\rightarrow \{1,2,\ldots, k\}$. A set $H\subseteq [X]^k$ is homogeneous for $f$ iff $f$ is constant on $[H]$. A stable coloring $f$ is a 2-coloring of $[\mathbb{N}]^2$ such that $(\forall n\in\mathbb{N})(\exists N)(\forall m>N)$ $f(\{m,n\})=f(\{N,n\})$. For a stable coloring $f$, $f_1=\{n\in\mathbb{N}: (\exists N)(\forall
m>N),f(m,n)=1\}$, $f_2=\mathbb{N}-f_1$.
For any n and k, every k-coloring of $[\mathbb{N}]^n$ admits an infinite homogeneous set.
Let $\mathsf{RT}_k^n$ denote the Ramsey’s theorem for k-coloring of $[\mathbb{N}]^n$. And $\mathsf{SRT}_k^2$ denotes the Ramsey’s theorem restricted to stable coloring of pair.
Jockusch [@jockusch1972ramsey] showed that for $n>2$ $\mathsf{RT}_2^n$ is equivalent to $\mathsf{ACA}_0$, while Seetapun and Slaman [@seetapun1995strength] showed that $\mathsf{RT}_2^2$ does not imply $\mathsf{ACA}_0$. As to $\mathsf{WKL}_0$, Jockusch [@jockusch1972ramsey] proved that $\mathsf{WKL}_0$ does not imply $\mathsf{RT}_2^2$. Whether $\mathsf{RT}_2^2$ implies $\mathsf{WKL}_0$ remained open. A more detailed survey of Ramsey’s theorem in view of reverse mathematics can be found in Cholak Jockusch and Slaman [@Cholak2001]. Say a set $S$ cone avoid a class $\mathcal{M}$ iff $(\forall C\in\mathcal{M})[ C\not\leq_T S]$.
The problem has been a major focus in reverse mathematics in the past twenty years. The first important progress was made by Seetapun and Slaman [@seetapun1995strength], where they showed that
For any countable class of sets $\{C_j\}$ $j\in\omega$, each $C_i$ is non-computable, then any computable 2-coloring of pairs admits an infinite cone avoiding (for $\{C_j\}$) homogeneous set.
Parallel this result, using Mathias Forcing in a different manner Dzhafarov and Jockusch [@Dzhafarov2009] Lemma 3.2 proved that
For any set $A$ and any countable class $\mathcal{M}$, each member of $\mathcal{M}$ is non-computable, there exists an infinite set $G$ contained in either $A$ or its complement such that $G$ is cone avoiding for $\mathcal{M}$.
The main idea is to restrict the computational complexity (computability power) of the homogeneous set as much as possible, with complexity measured by various measurements. Along this line, with simplicity measured by extent of lowness, Cholak Jockusch and Slaman [@Cholak2001] Theorem 3.1 showed, by a fairly ingenious argument,
\[th1\] For any computable coloring of the unordered pairs of natural numbers with finitely many colors, there is an infinite $low_2$ homogeneous set $X$.
Here we adopt the same idea to prove that
\[th2\] For any set $C$ not of -degree and any set $A$. There exists an infinite subset $G$ of $A$ or $\overline{A}$, such that $G\oplus C$ is also not of -degree.
$\mathsf{RT}_2^2\not\rightarrow \mathsf{WKL}_0$
It suffices to construct a countable class $\mathcal{M}$ satisfying the following four conditions (a)$C,B\in\mathcal{M}\rightarrow
C\oplus B\in\mathcal{M}$; (b)$(C\in\mathcal{M}\wedge B\leq_T C)\rightarrow B\in\mathcal{M}$; (c)$(\forall
C\in\mathcal{M})[C\not\gg 0]$; (d)$\mathcal{M}\vdash \mathsf{RT}_2^2$. It is shown in [@Cholak2001] Lemma 7.11 that $\mathsf{RCA}_0+\mathsf{RT}_2^2$ is equivalent to $\mathsf{RCA}_0+\mathsf{SRT}_2^2+\mathsf{COH}$. Moreover, it’s easy to prove that for any $C$-uniform sequence $C_1,C_2,\ldots$, $C$ being non--degree, there exists an infinite set $G$ cohesive for $C_1,C_2,\ldots$ such that $G\oplus C$ is not of -degree. This can be proved using finite extension method as following. Here and below $\sigma\prec\rho$ means $\sigma$ is an initial part of $\rho$; $\sigma\subseteq \rho$ means $\{n\leq |\sigma|: \sigma(n) = 1\}\subseteq \{n\leq|\rho|:\rho(n)=1\}$. At stage $s$, we define $Z_s= \begin{cases} Z_{s-1}\cap C_s \textrm{ if }Z_{s-1}\cap C_s\textrm{ is infinite;}\\ Z_{s-1}\cap \overline{C_s}\textrm{ else; }\end{cases}$ ($Z_0=C_0$ if $C_0$ is infinite $\overline{C_0} $ if else), $\rho_s\succ\rho_{s-1}$ with $\rho_{s-1}\subsetneq\rho_s\subseteq Z_s/\rho_{s-1}$. And whenever possible we also require $(\exists n)[\Phi_s^{C\oplus \rho_s}(n)=\Phi_n(n)\downarrow]$. We argue $G=\cup\rho_s$ is one of the desired sets. Clearly $G$ is infinite since $(\forall s)\rho_{s-1}\subsetneq\rho_{s}$. The cohesiveness of $G$ follows from $(\forall s)[G\subseteq^* Z_s]$ and $Z_s\subseteq^* C_s\vee Z_s\subseteq^* \overline{C_s}$. Furthermore, $(\forall s)[\Phi_s^{C\oplus G}\textrm{ is not a 2-DNR}]$. For else, suppose contradictory $\Phi_s^{C\oplus G}$ is a 2-DNR. Therefore $(\forall \rho\succeq\rho_{s-1},\rho\subseteq Z_s/\rho_{s-1})[\Phi_s^{C\oplus \rho}(n)\downarrow\wedge\Phi_n(n)\downarrow \Rightarrow \Phi_s^{C\oplus \rho}(n)\ne\Phi_n(n)]$. Since $\Phi_s^{C\oplus G}$ is total so $(\forall n)(\exists \rho\succeq \rho_{s-1},\rho\subset Z_s/\rho_{s-1})[\Phi_{s}^{C\oplus \rho}(n)\downarrow]$. Thus we could compute a 2-DNR using $Z_s$, but $Z_s\leq_T C$ contradict the fact that $C\not\gg 0$.
Let $B_0=\emptyset$. Let $f\in\Delta_2^{0,B_0}$ be a stable coloring, by Theorem \[th2\] there exists an infinite $G_0$, $G_0\subseteq f_1\vee G_0\subseteq f_2$ such that $B_0\oplus G_0\not\gg 0$, note that such $G_0$ computes an infinite homogeneous set of $f$. Let $B_1=B_0\oplus G_0$, $\mathcal{M}_1 = \{X\in 2^\omega:X\leq_T B_1\}$. Clearly $\mathcal{M}_1$ satisfies (a)(b)(c). Let $G_1$ be cohesive for a sequence of uniformly $\mathcal{M}_1$-computable sets (where $\mathcal{M}_1$-computable means computable in some $C\in\mathcal{M}_1$), furthermore $G_1\oplus B_1\not\gg 0$. Let $B_2=B_1\oplus G_1$, $\mathcal{M}_2=\{X\in
2^\omega: X\leq_T B_2\}$. Clearly $\mathcal{M}_2$ also satisfies (a)(b)(c). Iterate the above process in some way that ensures (1) for any uniformly $\mathcal{M}_j$-computable sequence $C_{1},C_{2}\ldots$, there exists $G_{i-1}\in\mathcal{M}_i$ cohesive for $C_{1},C_{2},\ldots$ and (2) for any $C\in\Delta_2^{0,\mathcal{M}_j}$, there exists an infinite $G_{i-1}\in\mathcal{M}_i$, $G_{i-1}\subseteq C\vee
G_{i-1}\subseteq \overline{C}$, while preserving the fact that for all resulted $B_i = B_{i-1}\oplus G_{i-1}$, $B_i\not\gg 0$. It follows that $\mathcal{M}=\bigcup\limits_{i=0}^{\infty}\mathcal{M}_i$ $\vdash \mathsf{RCA}_0+\mathsf{SRT}_2^2\leftrightarrow \mathsf{RT}_2^2$ but clearly $\mathcal{M}$ satisfies (a)(b)(c). The conclusion so follows.
The organization of this paper is as following. In Section \[sec2\] we introduce some notations and the requirements we use. In Section \[sec3\] we give some intuition about the proof by demonstrating the construction of the first step. Section \[sec4\] defines the forcing conditions and shows how to use these conditions to obtain a desired set $G$. Section \[sec5\] is devoted to the most important construction, i.e. how to construct a successive condition to force the requirements.
Preliminaries {#sec2}
=============
We say $X$ codes an ordered $k$-partition of $\omega$ iff $X=X_1\oplus X_2\oplus\cdots \oplus X_k$ and $\bigcup\limits_{i=1}^k X_i=\omega$, (*not necessarily* with $X_i\cap X_j = \emptyset$). A *$k$-partition class* is a non-empty collection of sets, where each set codes a $k$-partition of $\omega$. A tree $T\subseteq 2^{<\omega}$ is an ordered $k-$partition tree of $\omega$ iff every $\sigma\in T$ codes an ordered $k$-partition of $\{0,1,\ldots, |\sigma|\}$. Note that the class of all ordered $k-$partitions of $\omega$ is a $\Pi_1^0$ class.
For $n$ many ordered $k-$partitions, $X^{0},\ldots, X^{n-1}$ $$Cross(X^{0},X^{2},\ldots, X^{n-1};2)= \bigoplus\limits_{j< k, p<q\leq n-1}
Y^{(p,q)}_{j}$$ where $Y^{(p,q)}_{j}=X^{p}_j\cap X^{q}_j$, i.e. $Y^{(p,q)}_{j}$ is the intersection of those $X^{p}$ and $X^{q}$’s $j^{th}$ part, with $p\ne q$. For $n$ classes of ordered $k-$partitions $S_0,S_1,\ldots, S_{n-1}$
$$\begin{split}
Cross(S_0,S_1,\ldots, S_{n-1};2)=\{& Y\in 2^\omega:\textrm{ there exists } X^{i}\in S_i \textrm{ for each }i\leq n-1,
\\
&Y=Cross(X^{0},\ldots, X^{n-1};2)\}
\end{split}$$ Note that if each $S_i$ is a $\Pi_1^0$ class, then let $T_i$ be computable tree with $[T_i] = S_i$, operation $Cross$ can be defined on strings of $\{0,1\}$ in a nature way, therefore there exists a computable tree $T\subseteq 2^{<\omega}$ such that $T=Cross(T_{0},T_{1},\ldots, T_{n-1};2)$. So $[T]=Cross(S_0,S_1,\ldots, S_{n-1};2)$ i.e. $Cross(S_0,S_1,\ldots, S_{n-1};2)$ is a $\Pi_1^0$ class.
\[disag\]
1. A *valuation* is a finite partial function $\omega \rightarrow
2$.
2. A valuation $p$ is *correct* if $p(n)\ne\Phi_{n}(n)\!\downarrow$ for all $n \in \operatorname{dom}p$.
3. Valuations $p,q$ are *incompatible* if there is an $n$ such that $p(n) \neq q(n)$.
We try to ensure that $G$ satisfies the following requirements. To ensure that $(G \cap A)$ and $(G \cap \overline{A})$ are infinite, we will satisfy the requirements $$Q_m : |G\cap A|\geq m\wedge |G\cap \overline{A}|\geq m.$$ To ensure that $(G \cap A)
\oplus C$ does not have -degree, we would need to satisfy the requirements $$R^A_e : \Phi_e^{(G \cap A) \oplus C} \textrm{ total} \Rightarrow (\exists n)
[\Phi_e^{(G \cap A) \oplus C}(n) = \Phi_n(n)\!\downarrow].$$ Intuitively, $R^A_e$ is to ensure $(G\cap A)\oplus C$ does not compute any 2- via $\Phi_e$. (Without loss of generality we assume all $\Phi_0,\Phi_1,\ldots$ in this paper are $\{0,1\}$-valued functionals.) Similarly, to ensure $(G\cap\overline{A})\oplus C$ does not compute any 2- via $\Phi_e$, we try to make $G$ satisfy $$R^{\overline{A}}_i : \Phi_i^{(G \cap \overline{A}) \oplus C} \textrm{
total} \Rightarrow (\exists n)[ \Phi_i^{(G \cap \overline{A}) \oplus
C}(n) = \Phi_n(n)\!\downarrow].$$ Thus we will satisfy the requirements $$R_{e,i}: R^A_e \, \vee \, R^{\overline{A}}_i.$$
These requirements suffice to provide a desired $G$. Note that if there is some $e$ that $G$ does not satisfy $R^{A}_e$ then $G$ must satisfy all $R_i^{\overline{A}}$ since $G$ satisfy $R_{e,i}$ for all $i$. This implies $G\cap\overline{A}$ is not of -degree. See also [@Cholak2001], [@Dzhafarov2009].
Before we introduce the forcing condition, to get some intuition, we firstly demonstrate the construction of the first step.
First step {#sec3}
==========
Suppose we wish to satisfy $R_{e,i}$ that is:
either $(\exists n)[(\Phi_e^{(G\cap A)\oplus C}(n)= \Phi_n(n)\!\downarrow)\vee \Phi_e^{(G\cap A)\oplus C}$ is not total\],
or $(\exists n)[(\Phi_i^{(G\cap \overline{A})\oplus C}(n)= \Phi_n(n)\!\downarrow)\vee \Phi_i^{(G\cap \overline{A})\oplus C}$ is not total\].
**Case i.** *Try* to find a correct $p$ such that
$$\begin{split}\label{pro1}
&(\forall X=X_0\oplus X_1, X_0\cup X_1=\omega)( \exists \rho\exists n\in \operatorname{dom}p)
\\
&[\Phi_e^{(\rho\cap X_0)\oplus C} (n)\downarrow =\Phi_n(n)\downarrow \ne p(n)\vee \Phi_i^{(\rho\cap X_1)\oplus C} (n)\downarrow =\Phi_n(n)\downarrow\ne p(n)]
\end{split}$$
\
Note that substitute $X_0=A,X_1=\overline{A}$ in above sentence, there is a $\rho\in2^{<\omega}$ such that $\Phi_e^{(\rho\cap A)\oplus C} (n)\downarrow = \Phi_n(n)\downarrow \vee \Phi_i^{(\rho\cap \overline{A})\oplus C} (n)\downarrow = \Phi_n(n)\downarrow$. Therefore finitely extend initial segment requirement to $\rho$ and set $P_1=\{\omega\}$. *To satisfy $R_{e,i}$*, we ensure $G\succ \rho$. Clearly all $G\succ \rho$ satisfy $R_{e,i}$.
**Case ii.** *Try* to find three pairwise incompatible partial functions $p_i:\omega\rightarrow \{0,1\}$, $i=0,1,2$ that ensure the following $\Pi_1^0$ classes are non-empty: $$\begin{split}
S_{i}=&\{X=X_0\oplus X_1: X_0\cup X_1=\omega \wedge
\\
&[(\forall Z)(\forall n\in \operatorname{dom}p_i)\ \neg(\Phi_e^{(Z\cap X_0)\oplus C}(n)\!\downarrow\ne p_i(n))\wedge \neg(\Phi_i^{(Z\cap X_1)\oplus C}(n)\!\downarrow \ne p_i(n))\ ]
\}
\end{split}$$ Let $$P_1=Cross(S_{0},S_1,S_2;2)$$
i.e.
$(\forall Y\in P_1)\ Y=Y_0\oplus Y_1\oplus Y_2\oplus Y_3\oplus Y_4\oplus Y_5$
$(\exists X^0\in S_0\ \exists X^1\in S_1\ \exists X^2\in S_2)$ $X^i = X^i_0\oplus X^i_1$ for $i=0,1,2$ such that\
$Y_0=X^0_0\cap X^1_0, $ $Y_1=X^1_0\cap X^2_0, $ $Y_2=X^2_0\cap X^0_0, $\
$Y_3=X^0_1\cap X^1_1, $ $Y_4=X^1_1\cap X^2_1, $ $Y_5=X^2_1\cap X^0_1, $\
Note:
1. $S_i$ is a $\Pi_1^0$ class of ordered 2-partitions for all $i\leq 2$;\
2. $\Phi_e^{G\oplus C}$ is not total on any $G\subseteq Y_{i}$, for $i=0,1,2$ and $\Phi_i^{G\oplus C}$ is not total on any $G\subseteq Y_{i}$, for $i=3,4,5$. To see this, suppose for some $G\subseteq Y_0$, $\Phi_e^{G\oplus C}$ outputs on both $\operatorname{dom}p_0,\operatorname{dom}p_1$. Let $p_0(n)\ne p_1(n)$ then either $\Phi_e^{G\oplus C}(n)\ne p_0(n)$ or $\Phi_e^{G\oplus C}(n)\ne p_1(n)$. (Recall that we assume that all $\Phi$ are $\{0,1\}-$valued.) Suppose it is the former case, but $G\subseteq Y_0\subseteq
X^0_0$, $X^0_0\oplus X^0_1\in S_0$, by definition of $S_0$ $\Phi_e^{G\oplus C}(n)\downarrow\Rightarrow \Phi_e^{G\oplus C}(n)= p_0(n)$;\
3. $P_1$ is a $\Pi_1^0$ class. Though seemingly not, but note that each $S_i$ is a $\Pi_1^0$ class therefore there are computable trees $T_i$, $i\leq 2k$, such that $[T_i]=S_i$ for all $i$, furthermore $Cross$ can be applied to binary strings and is computable in this sense, thus there exists some computable tree $T'_1=Cross(T_0,T_1,T_2;2)$ with $P_1=[T'_1]$;\
4. $\bigcup_{i=0}^5 Y_i=\omega$. (See Lemma \[lem1\], this is just the pigeonhole principle. This is why we choose *three* pairwise incompatible valuations at *this* step.)
*To satisfy $R_{e,i}$*, we ensure that for some path $Y\in P_1$, $Y=Y_0\oplus Y_1\oplus \cdots \oplus Y_5$, $G$ will be contained in some $Y_i$. By item 2 in above note, $R_{e,i}$ is satisfied.
We will show in Lemma \[altlem\] that if there is no correct valuation as in case i then there must exist such three incompatible valuations i.e., either case i or case ii occurs.
Now we give the framework of our construction i.e. the forcing conditions.
Tree forcing {#sec4}
============
Let $\sigma \in 2^{<\omega}$ and let $X$ be either an element of $2^\omega$ or an element of $2^{<\omega}$ of length at least the same as that of $\sigma$. Here and below, we write $X/\sigma$ for the set obtained by replacing the first $|\sigma|$ many bits of $X$ by $\sigma$.
We will use conditions that are elaborations on Mathias forcing conditions. Here a *Mathias condition* is a pair $(\sigma,X)$ with $\sigma \in 2^{<\omega}$ and $X \in 2^\omega$. The Mathias condition $(\tau,Y)$ *extends* the Mathias condition $(\sigma,X)$ if $\sigma \preceq \tau$ and $Y/\tau \subseteq X/\sigma$. A set $G$ *satisfies* the Mathias condition $(\sigma,X)$ iff $\sigma \prec
G$ and $G \subseteq X/\sigma$.
We will be interested in $\Pi^{0,C}_1$ $k$-partition classes, that is, $\Pi^{0,C}_1$ classes that are also $k$-partition classes.
A *condition* is a tuple of the form $(k,\sigma_0,\ldots,\sigma_{k-1},P)$, where $k>0$, each $\sigma_i \in
2^{<\omega}$, and in this paper $P$ is a non-empty $\Pi^{0,C}_1$ $k$-partition class. We think of each $X_0 \oplus \cdots \oplus X_{k-1} \in P$ as representing $k$ many Mathias conditions $(\sigma_i,X_i)$ for $i < k$.
\[def-ext\] A condition $$d=(m,\tau_0,\ldots,\tau_{m-1},Q) \textrm{\emph{ extends }}
c=(k,\sigma_0,\ldots,\sigma_{k-1},P),$$ also denoted by $d\leq c$, iff there is a function $f : m\rightarrow k$ with the following property: for each $Y_0 \oplus\cdots \oplus Y_{m-1} \in Q$ there is an $X_0 \oplus \cdots \oplus
X_{k-1} \in P$ such that each Mathias condition $(\tau_i,Y_i)$ extends the Mathias condition $(\sigma_{f(i)},X_{f(i)})$. In this case, we say that $f$ *witnesses* this extension, and that *part $i$ of $d$ refines part $f(i)$ of $c$*. (Whenever we say that a condition extends another, we assume we have fixed a function witnessing this extension.)
\[def-sat\] A set $G$ *satisfies* the condition $(k,\sigma_0,\ldots,\sigma_{k-1},P)$ iff there is an $X_0 \oplus
\cdots \oplus X_{k-1} \in P$ such that $G$ satisfies some Mathias condition $(\sigma_i,X_i)$. In this case, we also say that $G$ satisfies this condition *on part $i$*.
\[def-for\]
1. A condition $(k,\sigma_0,\ldots,\sigma_{k-1},P)$ *forces $Q_m$ on part $i$* iff $|\sigma\cap A|\geq m \wedge |\sigma\cap \overline{A}|\geq m$. Clearly, if $G$ satisfies such a condition on part $i$, then $G$ satisfies requirement $Q_m$. (Note that if $c$ forces $Q_m$ on part $i$, and part $j$ of $d$ refines part $i$ of $c$, then $d$ forces $Q_m$ on part $j$.)
2. A condition *forces $R_{e,i}$ on part $j$* iff every $G$ satisfying this condition on part $j$ also satisfies requirement $R_{e,i}$. A condition *forces $R_{e,i}$* iff it forces $R_{e,i}$ on each of its parts. (Note that if $c$ forces $R_{e,i}$ on part $i$, and part $j$ of $d$ refines part $i$ of $c$, then $d$ forces $R_{e,i}$ on part $j$. Therefore, if $c$ forces $R_{e,i}$ and $d$ extends $c$, then $d$ forces $R_{e,i}$.)
For a condition $c=(k,\sigma_0,\ldots,\sigma_{k-1},P)$, we say that *part $i$ of $c$ is acceptable* if there is an $X_0
\oplus \cdots \oplus X_{k-1} \in P$ such that $X_i \cap A$ and $X_i
\cap \overline{A}$ are both infinite.
For example, in the first step, $P_0 = \{\omega\},k_0 = 1,\sigma_0=\lambda$, and for every $Y\in P_1$ $Y$ is of the form $Y=\bigoplus_{i=0}^5 Y_i$. Clearly $Y_i\subseteq X_{f_1(i)} $, where $X_{f_1(i)} = \omega\in P_0$. $f_1(i) = 0$ for all $i$ witnesses this extension relation.
Note that it is *not* the case that for every $X'\in P'$ there exists a single $X\in P$ such that $(\forall i\leq
k'-1)[ (\sigma_{i}',X_i')$ $\leq $ $(\sigma_{f(i)},X_{f(i)})]$.
The general plan.
-----------------
The proof will consist of establishing the following two lemmas. The proof of the second lemma is the core of the argument.
\[P-lem1\] Every condition has an acceptable part. Therefore for every condition $c$ and every $m$, there is a condition $d$ extending $c$ such that $d$ forces $Q_m$ on each of its acceptable parts.
\[R-lem\] For every condition $c$ and every $e$ and $i$, there is a condition $d$ extending $c$ that forces $R_{e,i}$.
*Proof of Theorem \[th2\].*
Given these lemmas, it is easy to see that we can build a sequence of conditions $c_0,c_1,\ldots$ with the following properties.
1. Each $c_{s+1}$ extends $c_s$.
2. If $s = \langle e,i \rangle$ then $c_s$ forces $R_{e,i}$.
3. Each $c_s$ has an acceptable part.
4. If part $i$ of $c_s$ is acceptable, then $c_s$ forces $Q_s$ on part $i$.
Clearly, if part $j$ of $c_{s+1}$ refines part $i$ of $c_s$ and is acceptable, then part $i$ of $c_s$ is also acceptable. Thus we can think of the acceptable parts of our conditions as forming a tree under the refinement relation. This tree is finitely branching and infinite, so it has an infinite path. In other words, there are $i_0,i_1,\ldots$ such that for each $s$, part $i_{s+1}$ of $c_{s+1}$ refines part $i_s$ of $c_s$, and part $i_s$ of $c_s$ is acceptable, which implies that $c_s$ forces $Q_s$ on part $i_s$. Write $c_s=(k_s,\sigma^s_0,\ldots,\sigma^s_{k_s-1},P_s)$. Let $G=\bigcup_s
\sigma^s_{i_s}$. Let $U_s$ be the class of all $Y$ that satisfy $(\sigma^s_{i_s},X_{i_s})$ for some $X_0 \oplus \cdots \oplus X_{k_s-1}
\in P_s$. Note that
- $U_0 \supseteq U_1 \supseteq \cdots$; Since $G\in U_{s+1}\Leftrightarrow (\exists X\in P_{s+1})[G$ satisfies $(\sigma_{i_{s+1}}^s,X_{i_{s+1}})]$ $\Rightarrow (\exists Z\in P_{s})[(\sigma_{i_{s+1}}^{s+1},X_{i_{s+1}})\leq (\sigma_{i_s}^s,Z_{i_s})\wedge$ $G$ satisfies $(\sigma_{i_s}^s,Z_{i_s})]$ $\Leftrightarrow G\in U_s$.
- Each $U_s$ contains an extension of $\sigma^s_{i_s}$ i.e. $U_s\ne\emptyset$;
- Each $U_s$ is closed;
By compactness of $2^\omega$ $\bigcap\limits_{s=0}^\infty U_s\ne\emptyset$. But clearly $(\forall Z\in \bigcap\limits_{s=0}^\infty U_s) [Z\succ \sigma^s_{i_s}]$ for all $s$. Thus $G$ is the unique element of $\bigcap\limits_{s=0}^\infty U_s$. In other words, $G$ satisfies each $c_s$ on part $i_s$, and hence satisfies all of our requirements.
Proof of Lemma \[P-lem1\]
=========================
It is here that we use the assumption that $A \nleq{_{\textrm{\tiny{\fontfamily{cmr}\selectfont T}}}} C$. Let $c=(k,\sigma_0,\ldots,\sigma_{k-1},P)$ be a condition. Write $P_\tau$ for the set of all $X \in P$ that extend $\tau$.
For each $\tau=\tau_0\oplus \cdots \oplus \tau_{k-1}$, if $P_\tau \neq \emptyset$ then there is an $X_0 \oplus \cdots \oplus X_{k-1} \in P_\tau$ and an $i<k$ such that $X_i$ contains elements $m \in A$ and $n \in \overline{A}$ such that $m,n \geq |\tau_i|$.
Assuming the claim for now, we build a sequence of strings as follows. Let $\rho^0$ be the empty string. Given $\rho^s=\rho^s_0\oplus \cdots \oplus
\rho^s_{k-1}$ such that $P_{\rho^s}$ is non-empty, let $X=X_0 \oplus
\cdots \oplus X_{k-1} \in P_{\rho^s}$ and $i_s<k$ be such that $X_{i_s}$ contains elements $m \in A$ and $n \in \overline{A}$ with $m,n \geq
|\rho^s_{i_s}|$. Then there is a $\rho_{s+1}=\rho^{s+1}_0\oplus \cdots
\oplus \rho^{s+1}_{k-1} \prec X$ such that, thinking of strings as finite sets, $\rho^{s+1}_{i_s} \setminus \rho^s_{i_s}$ contains elements of both $A$ and $\overline{A}$. Now let $Y =\bigcup_s \rho_s$ and let $i$ be such that $i=i_s$ for infinitely many $s$. Then $Y \in P$ and $Y$ witnesses the fact that part $i$ of $c$ is acceptable.
Fix $m$. To obtain the desired $d\leq c$ that forces $Q_m$ on each of its acceptable part. It is enough to show that for the condition $c=(k,\sigma_0,\ldots,\sigma_{k-1},P)$, if part $i$ of $c$ is acceptable, then there is a condition $d_0=(k,\tau_0,\ldots,\tau_{k-1},Q)$ extending $c$ such that $d_0$ forces $Q_m$ on part $i$, where the extension of $c$ by $d_0$ is witnessed by the identity map. (Note that if part $i$ of $d_0$ is acceptable, then so is part $i$ of $c$.) Then we can iterate this process, forcing $Q_m$ on each acceptable part in turn, to obtain the condition $d$ in the statement of the lemma.
So fix an acceptable part $i$ of $c$. Then there is a $\tau \succ
\sigma_i$ with $|\tau\cap A|\geq m$ and $|\tau\cap \overline{A}|\geq m$, and there is an $X_0 \oplus
\cdots \oplus X_{k-1} \in P$ with $\tau \prec X_i/\sigma_i$. Let $Q =
\{X_0 \oplus \cdots \oplus X_{k-1} \in P : \tau \prec X_i/\sigma_i\}$. Let $d_0 =(k,\sigma_0,\ldots,\sigma_{i-1},\tau,\sigma_{i+1},\ldots,\sigma_{k-1},Q)$. Then $d_0$ is an extension of $c$, with the identity function $id:k\rightarrow k$ witness this extension and it clearly forces $Q_m$ on part $i$.
Thus we are left with verifying the claim.
Assume for a contradiction that there is a $\tau=\tau_0 \oplus \cdots \oplus \tau_{k-1}$ such that $P_\tau \neq \emptyset$ and for every $X_0 \oplus \cdots \oplus
X_{k-1} \in P_\tau$ and every $i<k$, either $X_i {\upharpoonright}_{\geq
|\tau_i|}{} \subseteq A$ or $X_i {\upharpoonright}_{\geq |\tau_i|}{} \subseteq
\overline{A}$. It is easy to see that $\tau$ has an extension $\nu=\nu_0 \oplus \cdots \oplus \nu_{k-1}$ such that $P_\nu \neq
\emptyset$ and for each $i<k$, either $\nu_i(m_i)=1$ for some $m_i
\geq |\tau_i|$ or for every $X_0 \oplus \cdots \oplus X_{k-1} \in
P_\nu$, we have $X_i {\upharpoonright}_{\geq |\tau_i|}{} = \emptyset$. In the latter case, let $m_i$ be undefined. Let $S_A$ be the set of all $i<k$ such that $m_i$ is defined and is in $A$, and let $S_{\overline{A}}$ be the set of all $i<k$ such that $m_i$ is defined and is in $\overline{A}$. If $X_0 \oplus \cdots \oplus X_{k-1} \in P_\nu$, then $X_i {\upharpoonright}_{\geq |\tau_i|}{} \subseteq A$ for all $i \in S_A$, and $X_i
{\upharpoonright}_{\geq |\tau_i|}{} \subseteq \overline{A}$ for all $i \in
S_{\overline{A}}$.
We now claim we can compute $A$ from $C$, contrary to hypothesis. To see that this is the case, let $T$ be a $C$-computable tree such that $P_\nu$ is the set of infinite paths of $T$. For $\rho \in T$, write $T_\rho$ for the tree of all strings in $T$ compatible with $\rho$. Suppose we are given $n \geq |\tau|$. Let $j>|\nu|$ be such that for each $\rho = \rho_0\oplus \cdots \oplus \rho_{k-1} $ of length $j$, we have $n < |\rho_i|$ for all $i<k$. Let $L_A$ be the set of all $\rho
\in T$ of length $j$ such that $\rho_i(n)=1$ for some $i \in S_A$ and let $L_{\overline{A}}$ be the set of all $\rho \in T$ of length $j$ such that $\rho_i(n)=1$ for some $i \in S_{\overline{A}}$. If $\rho
\in L_A$ and $T_\rho$ has an infinite path then, by the definition of $S_A$, we have $n \in A$. Similarly, if $\rho \in L_{\overline{A}}$ and $T_\rho$ has an infinite path then $n \in \overline{A}$. Thus, if $\rho \in L_A$ and $\rho' \in L_{\overline{A}}$, then at least one of $T_\rho$ and $T_{\rho'}$ must be finite. So if we $C$-compute $T$ and start removing form $L_A$ and $L_{\overline{A}}$ every $\rho$ such that $T_\rho$ is found to be finite, one of $L_A$ or $L_{\overline{A}}$ will eventually be empty. They cannot both be empty because $P_\nu$ is non-empty. If $L_A$ becomes empty, then $n \in
\overline{A}$. If $L_{\overline{A}}$ becomes empty, then $n \in A$.
We now turn to the proof of Lemma \[R-lem\].
Forcing $R_{e,i}$ {#sec5}
=================
\[def1\]
1. $\Phi_e^{\rho\oplus C}$ *disagrees* with a valuation $p$ on a set $X$ iff there is a $Y\subseteq X$ and an $n\in\operatorname{dom}p$, $\Phi_e^{Y/\rho\oplus C}(n)\ne p(n)$;
2. Let $c=(k,\sigma_0,\ldots,\sigma_{k-1},P)$ be a condition, $p$ be a valuation and $U\subseteq \{0,1,\ldots, k-1\}$. We say that $c$ *disagrees* with $p$ on $U$ if for every $X_0 \oplus \cdots \oplus
X_{k-1} \in P$ and every $Z_0,Z_1,\ldots, Z_{2k-1}$ with $(\forall l)[ X_l=Z_{2l} \cup Z_{2l+1}]$, there is a $Y$, a $j\in U(c)$, and an $n \in \operatorname{dom}p$ such that either $\Phi_e^{((Y \cap
Z_{2j})/\sigma_j^A) \oplus C}(n)\!\downarrow \neq p(n)$ or $\Phi_i^{((Y
\cap Z_{2j+1})/\sigma_j^{\overline{A}}) \oplus C}(n)\!\downarrow \neq
p(n)$.
The following facts illustrate the central idea of the construction.
\[fac1\] For two pairwise incompatible valuations $p_0,p_1$, if $\Phi^\rho$ does not disagree with *both* $p_0,p_1$, on set $X$. Then for any $Y\subseteq X$, $\Phi^{Y/\rho}$ is not total on $\operatorname{dom}p_0\cup \operatorname{dom}p_1$.
\[fac2\] If $\Phi^\rho$ does not disagree with $p$ on a set $X$ then for any $Y\subseteq X$, $\Phi^\rho$ does not disagree with $p$ on set $Y$.
Therefore,
\[fac6\] For two incompatible valuations $p_0$, $p_1$. If $\Phi^\rho$ does not disagree with $p_0$ on a set $X_0$, and does not disagree with $p_1$ on a set $X_1$ then for any $Y\subseteq X_0\cap X_1$, $\Phi^{Y/\rho}$ is not total on $\operatorname{dom}p_0\cup\operatorname{dom}p_1$.
The following lemma tells how to ensure that the tree of each condition is an ordered partition tree.
\[lem1\] For any $n$ many ordered $2k-$partitions of $\omega$, namely $X^{0}$, $X^{1}$,$\ldots$, $X^{n-1}$, if $n > 2k$ then $Cross(X^{0},X^1,\ldots, X^{n-1};2)$ is a $2k \binom{n}{2}$-partition. Therefore if $S_0,S_2,\ldots, S_{n-1}$ are $n$ classes of ordered $2k$-partitions of $\omega$ then\
$Cross(S_0,S_1,\ldots, S_n;2)$ is a class of $2k\binom{n}{2}$-partition of $\omega$.
Straightforward by pigeonhole principle. It suffices to show that for any $x\in \omega$, there is some $i\leq
2k-1$, some $X^{p},X^{q},p\ne q$, such that $X^p = \bigoplus_{i=0}^{2k-1} X_i^p$, $X^q = \bigoplus_{i=0}^{2k-1} X_i^q$, $x\in X_i^p\cap X_i^q$. For $i=0,2,\ldots, 2k-1$ let $F_{i}=\{p\leq n-1:x\in X^p_i\}$. Since each $X^{p}$ is an ordered partition, therefore for each $p$ there exists some $i$ such that $p\in F_{i}$. So $\bigcup\limits_{i=0}^{2k-1} F_{i}=\{0,1,2,\ldots, n-1\}$. But $n>2k$ thus there is some $i\leq 2k-1$ such that $F_{i}$ contains two elements say $p,q$, thus $x\in X^p_i\cap
X^q_i$.
Construction {#subsec-con}
------------
Fix $e,i$ and a condition $c=(k,\sigma_0,\ldots,\sigma_{k-1},P)$. For any condition $d$, let $U(d)$ be the set of all $j$ such that part $j$ of $d$ does not force $R_{e,i}$ on part $j$. If $U(d)=\emptyset$ then there is nothing to prove, so we assume $U(d) \neq \emptyset$. It is clearly enough to obtain a condition $d$ extending $c$ such that $|U(d)|<|U(c)|$. Then one could simply iterate this process. Here and below, we write $\sigma^A$ for the string of the same length as $\sigma$ defined by $\sigma^A(n)=1$ iff $\sigma(n)=1\wedge n \in A$, and similarly for $\sigma^{\overline{A}}$.
We will use two ways to extend conditions.
*Begin construction:*
**Case i.** $c$ disagrees with some correct valuation $p$ on $ U(c)$.
Let $X_0 \oplus \cdots \oplus X_{k-1} \in P$. For $j=0,1,\ldots, k-1$ let $Z_{2j}=X_j \cap
A$ and $Z_{2j+1}=X_j \cap \overline{A}$. By the definition of disagreeing with a correct valuation on $U(c)$, there exists a $j\in U(c)$, an $n\in \operatorname{dom}p$ and a $Y$ such that either $\Phi_e^{((Y \cap
Z_{2j})/\sigma_j^A) \oplus C}(n)\!\downarrow = \Phi_n(n)\!\downarrow$ or $\Phi_i^{((Y \cap Z_{2j+1})/\sigma_j^{\overline{A}}) \oplus C}(n)\!\downarrow
= \Phi_{n}(n)\!\downarrow$. In other words, either $\Phi_e^{(Y/\sigma_j \cap A)
\oplus C}(n)\!\downarrow = \Phi_n(n)\!\downarrow$ or $\Phi_i^{(Y/\sigma_j \cap
\overline{A}) \oplus C}(n)\!\downarrow = \Phi_n(n)\!\downarrow$.
If $\tau$ is a sufficiently long initial segment of $Y$, then for every $Z$ extending $\tau$, we have either $\Phi_e^{(Z \cap A) \oplus
C}(n)\!\downarrow = \Phi_n(n)\!\downarrow$ or $\Phi_i^{(Z \cap
\overline{A}) \oplus C}(n)\!\downarrow = \Phi_n(n)\!\downarrow$. We may assume that $\tau \succeq \sigma_j$. Let $Q$ be the class of all $W_0
\oplus \cdots \oplus W_{k-1} \in P$ such that $\tau$, thought of as a finite set, is a subset of $W_j/\sigma_j$ and let $d=(k,\sigma_0,\ldots,\sigma_{j-1},\tau,\sigma_{j+1},\ldots,\sigma_{k-1},Q)$. Note that $Q$ is a non-empty $\Pi_1^{0,C}$ class since it contains $X_0 \oplus \cdots \oplus X_{k-1}$. Clearly $d$ is an extension of $c$, with the identity function $id:k\rightarrow k$ witnessing this extension relation, and clearly $d$ forces $R_{e,i}$ on part $j$, so that $|U(d)|<|U(c)|$.
**Case ii.** There are pairwise incompatible valuations $p_0,\ldots,p_{2k}$ such that $c$ does not disagree with any $p_l$ on $U(c)$. We will show in Lemma \[altlem\] that these are the only two cases that will occur.
For each $l<2k$ let $S_l$ be the class of all sets of the form $Z_0 \oplus \cdots \oplus
Z_{2k-1}$ such that $(Z_0 \cup Z_1) \oplus (Z_2 \cup Z_3) \oplus \cdots \oplus (Z_{2k-2}
\cup Z_{2k-1}) \in P$ and for all $j\in U(c)$, every $n\in \operatorname{dom}p_l$, every $Y$ we have, neither $\Phi_e^{(Y\cap Z_{2j})/\sigma_j^A \oplus C}(n)\!\downarrow \neq p_l(n)$ nor $\Phi_i^{(Y\cap Z_{2j+1})/\sigma_j^{\overline{A}} \oplus C}(n)\!\downarrow \neq
p_l(n)$.
Since $c$ does not disagree with any of the $p_l$ on $U(c)$, all $S_l$ are non-empty. It is then easy to see that each $S_l$ is in fact a $\Pi^{0,C}_1$ $2k$-partition class.
Let $Q=Cross( S_0,\ldots,S_{2k};2)$ and let $$d=\left(2k\binom{2k+1}{2}, \sigma_0,\ldots,\sigma_0,
\sigma_1,\ldots,\sigma_1,
\ldots,\sigma_{k-1},\ldots,\sigma_{k-1},Q\right),$$ where each $\sigma_i$ appears $2\binom{2k+1}{2}$ many times. We show that $d$ is a condition extending $c$, and $d$ forces $R_{e,i}$.
1. Since each $S_i$ is non-empty therefore $Q$ is non-empty. Furthermore, since each $S_i$ is a $\Pi_1^{0,C}$ class then $Q$ is also a $\Pi_1^{0,C}$ class. Because $Cross$, when applied to strings, is computable therefore by applying $Cross$ to the $2k+1$ computable trees $T_{i}$ with $[T_{i}]=S_i$ one obtains a computable tree $T$ with $[T]=Q$.
2. $Q$ is a class of ordered $2k\binom{2k+1}{2}$-partitions of $\omega$. To see this, note that $S_i$, $i\leq 2k$, are $2k+1$ classes of ordered 2k-partitions of $\omega$, by Lemma \[lem1\] $Q$ is a class of ordered $2k\binom{2k+1}{2}$-partitions of $\omega$. Therefore combine with item 1 and recall the fact that the initial segments in $d$ are not changed, it follows that $d$ *is* a condition.
3. For each new part $i'$ of $d$ and every $ W_0\oplus W_1\oplus\cdots\oplus W_{k'-1}\in Q$, where $k'=2k\binom{2k+1}{2}$, there exists $X_0\oplus X_1\oplus\cdots\oplus X_{k-1}\in P$, and $i\leq k-1$ with $W_{i'}/\sigma_{i'}\subseteq
X_{i}/\sigma_{i}$, and $\sigma_i=\sigma_{i'}$, i.e. each new part is contained in an old part of some path through $P$. It follows that $d$ extends $c$. To see this, note that by definition of $P$ for each $i'\leq k'-1$ there exist $p,q\leq 2k,p\ne q$ and $j\leq 2k-1$ determined by $i'$, such that $(\forall W\in Q)( \exists X^{p}\in
S_p\ \exists X^{q}\in S_q)$ $[W_{i'}=X_{j}^p\cap X_{j}^q]$. Furthermore, by definition of $S_p$, $X^p_{j}\cup X^{p}_{j'}= X_i$ for some $j'\leq 2k-1$, and some $X=X_0\oplus X_1\oplus\cdots\oplus X_{k-1}\in P$. Therefore $$W_{i'}=X^p_{j}\cap X^q_j\subseteq X^p_j\subseteq X^p_j\cup X^p_{j'}=X_i$$ i.e. each part $i'$ of each $W\in Q$ is contained in some part $i$ of some $X\in P$.
4. $d$ forces $R_{e,i}$. To see this, let $G$ satisfy $d$. Then there is some $j<k$, some $a \neq b<2k+1$, some $Z_0 \oplus \cdots \oplus Z_{2k-1} \in S_a$, and some $W_0 \oplus \cdots
\oplus W_{2k-1} \in S_b$ such that $G$ satisfies one of the Mathias conditions $(\sigma_j,Z_{2j} \cap W_{2j})$ or $(\sigma_j,Z_{2j+1} \cap
W_{2j+1})$. Then $G$ satisfies $c$ on part $j$, so if $j \notin U(c)$, then $G$ satisfies $R_{e,i}$. So assume $j \in U(c)$.
Let us suppose $G$ satisfies $(\sigma_j,Z_{2j} \cap W_{2j})$, the other case being similar. Then $(G \cap A)/\sigma_j$ satisfies both of the Mathias conditions $(\sigma_j,Z_{2j})$ and $(\sigma_j,W_{2j})$. Let $n$ be such that $p_a(n) \neq p_b(n)$. By the definitions of $S_a$ and $S_b$, we have $\neg(\Phi_e^{(G \cap A) \oplus C}(n)\!\downarrow \neq p_a(n))$ and $\neg(\Phi_e^{(G \cap A)\oplus C}(n)\!\downarrow \neq p_b(n))$. Hence we must have $\Phi_e^{(G \cap A) \oplus C}(n)\!\uparrow$. Thus $d$ forces $R_{e,i}$.
*End of construction*
It remains to prove that
\[altlem\] For a valuation $p$, let $S_p$ be the $\Pi_1^{0,C}$ class of all $Z_0\oplus\cdots\oplus Z_{2k-1}$ with $Z_{0}\cup Z_{1}\oplus\cdots\oplus Z_{2k-2}\cup Z_{2k-1}\in P$ such that for every $j\in U(c)$, every $\mu\in 2^{\omega}$, and every $n \in \operatorname{dom}p$,
- neither $\Phi_e^{((\mu\cap Z_{2j})/\sigma^A_j) \oplus C}(n)[|\mu|]\!\downarrow \neq
p(n)$,
- nor $\Phi_i^{((\mu\cap Z_{2j+1})/\sigma^{\overline{A}}_j) \oplus
C}(n)[|\mu|]\!\downarrow \neq p(n)$.
One of the following must hold.
1. There is a correct valuation $p$ such that $S_p$ is empty i.e. $c$ disagrees with the correct $p$ on $U(c)$.
2. There are pairwise incompatible valuations $p_0,\ldots,p_{2k}$ such that $S_p$ is not empty i.e. $c$ does not disagree with $p_l$ on $U(c)$ for all $l\leq 2k$.
We note that item 1 and item 2 are equivalent to case i and case ii respectively. Furthermore $S_p$ is a $\Pi_1^{0,C}$ class uniformly in $p$. Consequently for each $j<k$, the set of all valuations $p$ such that $c$ disagrees with $p$ on $U(c)$ is $C$-c.e. Let $E$ denote this $C$-c.e. set of valuations.
Assume that alternative 1 above does not hold. Since $C$ does not have -degree, there is no $C$-computable function $h$ such that if $\Phi_n(n)\!\downarrow$ then $h(n)\neq\Phi_n(n)$.
Let $S$ be the collection of all finite sets $F$ such that for each $n
\notin F$, either $\Phi_n(n)\!\downarrow$ or there is a $p \in E$ such that $F \cup \{n\} \subseteq \operatorname{dom}p$ and for every $m \in \operatorname{dom}p \setminus
F \cup \{n\}$, we have $p(m)\neq \Phi_m(m)\!\downarrow$. If $F \notin S$, then there is at least one $n \notin F$ for which the above does not hold. We say that any such $n$ *witnesses* that $F \notin S$.
First suppose that $\emptyset \in S$. Then for each $n$, either $\Phi_{n}(n)\!\downarrow$ or there is a $p \in E$ such that $n \in
\operatorname{dom}p$ and for every $m \neq n$ in $\operatorname{dom}p$, we have $p(m)\neq\Phi_m(m)\!\downarrow$. Then we can define $h \leq{_{\textrm{\tiny{\fontfamily{cmr}\selectfont T}}}} C$ by waiting until either $\Phi_n(n)\!\downarrow$, in which case we let $h(n)=1-\Phi_n(n)$, or a $p$ as above enters $E$, in which case we let $h(n)=1-p(n)$. Since no element of $E$ is correct, in the latter case, if $\Phi_n(n)\!\downarrow$ then $p(n) = \Phi_n(n)$, so $h(n)=\Phi_n(n)$. Since $C$ does not have -degree, this case cannot occur.
Thus $\emptyset \notin S$. Let $n_0$ witness this fact. Given $n_0,\ldots,n_j$, if $\{n_0,\ldots,n_j\} \notin S$, then let $n_{j+1}$ witness this fact. Note that if $n_j$ is defined then $\Phi_{n_j}(n_j)\!\uparrow$.
Suppose that for some $j$, we have $\{n_0,\ldots,n_j\} \in S$. Then $\{n_0,\ldots,n_{j-1}\} \notin S$, as otherwise $n_j$ would not be defined. We define $h \leq{_{\textrm{\tiny{\fontfamily{cmr}\selectfont T}}}} C$ as follows. First, let $h(n_l)=0$ for $l \leq j$. Given $n \notin \{n_0,\ldots,n_j\}$, we wait until either $\Phi_n(n)\!\downarrow$, in which case we let $h(n)=1-\Phi_n(n)$, or a $p$ enters $E$ such that $\{n_0,\ldots,n_j,n\} \subseteq \operatorname{dom}p$ and for every $m \in \operatorname{dom}p \setminus \{n_0,\ldots,n_j,n\}$, we have $p(m)\ne\Phi_m(m)\!\downarrow$. If $\Phi_n(n)\!\uparrow$ then the latter case must occur, since $\{n_0,\ldots,n_j\} \in S$. In this case, we cannot have $p(n)\neq\Phi_n(n)\!\downarrow$, as then $p$ would be a counterexample to the fact that $n_j$ witnesses that $\{n_0,\ldots,n_{j-1}\} \notin S$. Thus we can let $h(n)=1-p(n)$. Again, since $C$ does not have -degree, this case cannot occur.
Thus $\{n_0,\ldots,n_j\} \notin S$ for all $j$. There are $2^{j+1}$ many valuations with domain $\{n_0,\ldots,n_j\}$, and they are all pairwise incompatible. None of these valuations can be in $E$, as that would contradict the fact that $n_j$ witnesses that $\{n_0,\ldots,n_{j-1}\} \notin S$. Taking $j$ large enough, we have $2k+1$ many pairwise incompatible valuations, none of which are in $E$.
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=1200 =16.0truecm =24.0truecm =0 \_8[\_8]{} CERN-TH/95-214
BI-TP 95/30 2.5truecm
**CHARMONIUM COMPOSITION AND NUCLEAR SUPPRESSION**
1.5 truecm
**D. Kharzeev and H. Satz**
Theory Division, CERN, CH-1211 Geneva, Switzerland
and
Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany
2 truecm
**Abstract:**
We study charmonium production in hadron-nucleus collisions through the intermediate next-to-leading Fock space component $|(\c)_8g\rangle$, formed by a colour octet $\c$ pair and a gluon. We estimate the size of this state and show that its interaction with nucleons accounts for the observed charmonium suppression in nuclear interactions.
------------------------------------------------------------------------
CERN-TH/95-214
BI-TP 95/30
August 1995
Charmonium production in hadronic collisions inherently involves different energy and time scales; for an extensive recent treatment, see \[Bodwin\]. The first step is the creation of a heavy $\c$ pair, e.g., by gluon fusion; this takes place in a very short time, $\t_{\rm pert} \simeq m_c^{-1}$. The pair is in a colour octet state; to neutralise its colour and yield a resonance state of quantum numbers, it has to absorb or emit an additional gluon (Fig. 1). The time $\t_8$ associated to this process is determined by the virtuality of the intermediate $\c$ state. In the rest frame of the $\c$, it is approximately \[KS1\] $$\t_8 \simeq {1\over \sqrt \Delta}, \eqno(1)$$ where $\Delta \equiv [(p+k)^2-m_c^2]= 2pk$. For massless quarks, this gives the familiar $1/k_T$ for the time uncertainty associated with gluon emission/absorption; for charm quarks of (large) mass $m_c$, we get $$\t_8 \simeq {1 \over \sqrt{2m_c k_0}} \eqno(2)$$ where $k_0$ is the energy of the additional gluon. If it is sufficiently soft, $\t_8 > \t_{\rm pert}$. For production at mid-rapidity of a nucleon-nucleon collision, the colour neutralisation time becomes $$t_8 \simeq \t_8 [1+(P_A/2m_c)^2]^{1/2} \eqno(3)$$ in the rest frame of target or projectile nucleon, with $P_A$ denoting the momentum of the $\c$ pair in this frame. As seen from the nucleon, colour neutralisation of fast $\c$ pairs will thus take a long time. Equivalently, a fast $\c$ travels in the time $t_8$ a long distance, $$d_8 \simeq \t_8 (P_a/2m_c), \eqno(4)$$ in the rest frame of the nucleon. On the basis of the process shown in Fig. 1, this seems to imply the existence of a coloured $\c$ state of well-defined quantum numbers over times or distances much greater than the confinement scale of about 1 fm (corresponding to $\la^{-1}$ with $\la\simeq
0.2$ GeV). This problem is particularly evident for production at low transverse momentum, for which the additional gluon has to be quite soft; in this case, however, perturbative arguments become in any case questionable. The problem seems avoidable for production at sufficiently high $p_T$, i.e., for large enough $k_0$. Hence the colour singlet model , which treats the complete colour singlet formation process in Fig. 1 within perturbative QCD, was usually restricted to high $p_T$ production. The low $p_T$ problem was until now usually “solved" by noting that inclusive production occurs in the colour field of the collision, leaving the form of the colour neutralisation unspecified (“colour evaporation" ).
Recent data \[Fermi\] have shown, however, that also at high $p_T$ non-perturbative long-time features seem to be essential for charmonium production . The most important outcome of these studies (see \[Mangano\] for a recent review) is that higher Fock space components of charmonium states play a dominant role in their production. We thus decompose the state $|\psi \rangle$ $$|\psi\rangle = a_0 |(\c)_1\rangle + a_1 |(\c)_8 g\rangle
+ a_2 |(\c)_1 gg \rangle + a'_2|(\c)_8 gg \rangle + ... \eqno(5)$$ into a pure $\c$ colour singlet component ($^3S_1$), into a component consisting of $\c$ colour octet ($^1S_0$ or $^3P_J$) plus a gluon, and so on. The higher Fock space coefficients correspond to an expansion in the relative velocity $v$ of the charm quarks. For the wave function of the , the higher components thus correspond to (generally small) relativistic corrections. For production, however, their role can become decisive. While in short-time production the lowest component is the most important, in long-time processes the next higher term becomes dominant.
Analogous decompositions hold for the other charmonium states . In all cases, the first higher Fock space state consists of a colour octet $\c$ plus a gluon. For the ¶, the next-to-leading terms again contain a colour octet ($^3P_J$ or $^1S_0$ $\c$) plus a gluon; for the $\chi$’s, a $^3S_1$ colour octet $\c$ is combined with a gluon.
This sheds some light onto the unspecified colour evaporation process. When the colour octet $\c$ leaves the field of the nucleon in which it was produced, it will in general neutralise its colour by combining non-perturbatively with an additional collinear gluon, thus producing the $(\c)_8 g$ component of the or the other charmonium states (Fig. 2). A necessary prerequisite for this is the small size of the $\c$, due to the heavy quark mass; it is only because of this that the soft gluon interacts with the $\c$ as a colour octet and not with the individual quarks. After the “relaxation time" $\t_8$, the $(\c)_8 g$ will then absorb the accompanying gluon to revert to the dominant $(\c)_1$ charmonium mode (Fig. 3). Note that we are here considering those $\c$ pairs which will later on form charmonia. The $(\c)_8$ could as well neutralise its colour by combining with a light quark-antiquark pair, but this would result predominantly in open charm production.
The production of charmonia in a kinematic regime involving long time scales thus implies the production of the composite and hence extensive state $(\c)_8g$. Its intrinsic transverse size $r_8$ can be estimated through the uncertainty in the transverse momentum induced in the charm quark when it absorbs the accompanying gluon to go from $|(\c)_8g \rangle$ to the basic Fock component $|(\c)_1\rangle$ (Fig. 3). From the non-relativistic form for heavy quarks, $${p^2 \over 2m_c} \simeq k_0, \eqno(6)$$ where $p$ is the quark transverse momentum in the $(\c)_8 g$ cms, we obtain from the lowest allowed gluon energy $k_0=\la$ the intrinsic size $$r_8 \simeq {1\over \sqrt{2m_c\la}} \simeq 0.25~{\rm fm}. \eqno(7)$$ Since this size is determined only by the $(\c)_8 g$ composition of the next-to-leading Fock space state and the gluon momentum cut-off in confined systems, it is the same for all charmonium states. In general, the time and momentum uncertainty would be given by the binding energy and the size of the state, i.e., for the by the mass difference $\e_0 =2M_D - M_{\psi} \simeq 0.64$ GeV between it and open charm. This would imply different sizes for the $(\c)_8 g$ component of different resonance states; for a related discussion, see \[Mueller\]. In the case of all charmonium and bottonium states, however, the common confinement cut-off $\sqrt{2m_Q
\la}$ is more stringent and hence the relevant one.
We now want to study the effect of these considerations on production in proton-nucleus collisions. For low $P_T$ production, with $k_0 \simeq
\la$ in Eq. (2), $\t_8$ would exceed the size of the even heavy nuclei for ’s of sufficiently high lab momenta. Maintaining an approach based on the colour singlet model would thus require a colour octet $\c$ to pass through the entire nuclear medium \[KS1\]. The composition of the colour-neutralising cloud needed for this was so far undetermined, and hence estimates for the resulting cross sections were generally obtained by assuming such a dressed $(\c)_8$ to be of hadronic size . It was also left open why interactions with the surrounding cloud would not destroy the spatial and quantum number structure of the $(\c)_8$. Quarkonium production through higher Fock space components now provides a specific description of the $(\c)_8$ passage through the nucleus. The system leaves the nucleon in whose field it was formed as a colour singlet $(\c)_8 g$ and continues as such through the nuclear medium. The size of this charmonium state $(\c g) \equiv |(\c)_8g \rangle$ is given by Eq. (7) and is thus essentially that of a . Also its interaction with hadrons is similar to the -hadron interaction \[KS3\], but with two important distinctions. The gluon exchanged between the two colliding systems will now couple predominantly to the gluon or to the $(\c)_8$ component of the $(\c g)$. Since both are colour octets, in contrast to the colour triplet components of the , the coupling is correspondingly enhanced by a factor 9/4. Such an interaction will render the $(\c)_8 g$ system coloured. Due to the repulsive one-gluon exchange interaction, the colour octet $(\c_8)$ is not bound, in contrast to the colour singlet $(\c)_1$. Moreover, the probability of the $(\c)_8$ encountering another collinear gluon to again form a colour singlet system of quantum numbers is minimal. Hence any $(\c g)$ interaction will generally lead to its break-up, so that there is no threshold factor. The cross section for $(\c g)$-hadron interactions is thus just the geometric -hadron cross section, $\sigma_{\psi N}$, increased by the enhanced coupling, $$\sigma_{(\c g) N} = {9\over 4} \sigma_{\psi N}.\eqno(8)$$ Short-distance QCD calculations \[KS3\] lead to $\sigma_{\psi N} \simeq$ 2.5 - 3 mb, in good agreement with photoproduction data \[photo\]. From this we get $\sigma_{(\c g) N} \simeq$ 6 - 7 mb. Because of the general nature of the arguments leading to this value, it holds equally for the interactions of the next-to-leading Fock space components of ¶ and $\chi$ with hadrons.
This cross section, while larger than the high energy -hadron cross section by about a factor two, is very much smaller than the hadronic value of 20 - 50 mb previously assumed for the colour octet $\c$ passing through the nucleus . This has immediate consequences. The mean free path of the $(\c g)$ in nuclear matter, $\lambda_{(\c g)}=1/n_0
\sigma_{(\c g) N} \simeq 10$ fm is larger than the radius of even the heaviest nuclei. Moreover, since the $\c$ combines with an already existing collinear gluon, there is no coherence length associated with any $(\c g)$ formation. Hence shadowing is excluded as dominant quarkonium suppression mechanism in $p-A$ collisions.
In passing through the nucleus, the small physical state $(\c g)$ will interact incoherently with the nucleons along its trajectory. The charmonium production probability on nuclei relative to that on nucleons, valid for , $\chi$ and ¶ production, thus becomes $$S_A = exp\{-n_0\sigma_{(\c g) N} L\} , \eqno(9)$$ where $L$ denotes the length of the path through nuclear matter of standard density $n_0=0.17$ fm$^{-3}$, and $\sigma_{(\c g) N} \simeq$ 6 - 7 mb is the inelastic $(\c g)$-nucleon cross section obtained above.
With Eq. (9) we have derived the Gerschel-Hüfner fit \[Gerschel\], introduced as a phenomenological description of hadron-nucleus data on production. In \[Gerschel\], it was attempted to interpret the cross section entering in Eq. (9) as the physical -hadron cross section, which led to theoretical as well as experimental problems. The fit value was 5 - 7 mb; both short distance QCD \[KS3\] and photoproduction experiments \[photo\] give a -hadron cross section smaller by at least a factor two. Moreover, $p-A$ data lead to exactly the same suppression of ¶ and production for all $A$, with an $A$-independent production ratio ¶/()$\simeq 0.15$. Since the geometric size of the ground state and the next radial excitation differ by more than a factor four, an equal suppression contradicts the interaction of fully formed physical resonances. We find here that all aspects of the observed suppression arise naturally in charmonium production through the next higher Fock space component $(\c g)$. A composite state $(\c g)$, which is of the same size for all charmonia, passes through the nuclear medium and hence leads to equal ¶ and suppression. The value of the cross section $\sigma_{(\c g)N}$ thus obtained is in good agreement with that obtained from a fit to the data \[Gerschel\].
The same argumentation also provides the suppression of bottonium production in $p-A$ collisions. The radius of the $(\b)_8g$ state is (see Eq. (7)) a factor $\sqrt{m_c/m_b} \simeq
\sqrt 3$ smaller than that of the $(\c g)$, so that $\sigma_{(\b g) N} \simeq (1/3)\sigma_{(\c g)N} \simeq 2$ mb.
To illustrate how well both and production in $p-A$ collisions are described by this scenario, we show in Fig. 4 recent high energy data at $\sqrt s \simeq 20$ \[Carlos\], 30 \[Fredj\] and 40 GeV \[E772\]. With an average path length $L_A=(3/4)[(A-1)/A][1.12~A^{1/3}]$ we obtain excellent agreement for $\sigma_{(\c g)N} \simeq 6$ mb and $\sigma_{(\b g) N} \simeq 2$ mb. In \[Gerschel\] it was shown that production data from $\pi-A$ collisons lead to very similar values for the cross section in Eq.(9).
With quarkonium suppression in hadron-nucleus collisions thus accounted for in terms of interaction between $(\Q)_8g$ states and nucleons, we can now also consider nucleus-nucleus interactions. For collision energies of $\sqrt s \simeq 20$ GeV, the centers of the two colliding nuclei have at time $t_8$ separated in the cms by about 5 fm. With the nuclei Lorentz-contracted to a thickness of about 0.5 fm or less, this means that a produced at mid-rapidity in the cms has experienced in its early phase an effect corresponding to that obtained by superimposing the passages through the two nuclei \[Gerschel\]. Hence the charmonium suppression now is given by $$S_{(A-B)} = exp\{-n_0\sigma_{(\c g)N}(L_A+L_B)\} , \eqno(10)$$ again in accord with the phenomenological fit of \[Gerschel\]. The path length $L=L_A+L_B$ in Eq. (10) varies with impact parameter and hence with the associated transverse energy $E_T$ produced in the collision. A relation between $L$ and $E_T$ can thus be obtained through a detailed study of the collision geometry \[Salmeron\]. An alternative is given by determining $L$ through the broadening of the average transverse momentum of or Drell-Yan dileptons, since this broadening also depends on the average path length \[NA38G\]. In Fig. 5 we show the resulting values \[Carlos\], normalised to the $(\c g)$ suppression of Eqs. (8) and (9), for both $p-A$ and $S-U$ data. We conclude that also the suppression observed so far in $S-U$ collisions is completely accounted for by $(\c
g)$ suppression in standard nuclear matter.
The comparison of and ¶ production in nucleus-nucleus collisions provides a test for the presence of a medium in the later stages. As long as the interaction leading to charmonium suppression is determined by the $(\c g)$ state, and ¶ must be equally suppressed. To make and ¶ suppression different, the medium must see the fully formed resonances and distinguish between them. In Fig. 5 we have included also the ¶ suppression divided by the $(\c g)$ suppression (8) . It is evident that in $S-U$ collisions and ¶ are not affected equally, the ¶ being much stronger suppressed. This establishes the presence of a medium at a time late enough for fully formed charmonium resonances to exist. On the other hand, this medium breaks up only the ¶, leaving the unaffected. It was shown \[KS3\] that interactions with hadrons in the range of present collision energies cannot dissociate a , while for the much more loosely bound ¶ this is readily possible. We therefore conclude that the medium probed by charmonium production in present $S-U$ collisions is of hadronic nature, i.e., confined; it could consist either of stopped nucleons or of secondary hadrons produced in the collision \[Wong\]. Note that we include the ¶ data as function of $L$ in Fig. 5 simply to show the additional suppression present in this case. It is not at all clear that $L$ is a meaningful variable for the effect of such a confined environment on charmonia. We also note that the appearent absence of any effect of this medium on the observed , even though this is to about 40 % produced through $\chi$ decay, is in accord with the short distance QCD form of the $\chi$-hadron cross section \[KS-CD\].
To establish colour deconfinement, either in equilibrium or pre-equilibrium systems, nucleus-nucleus collisions have to produce a suppression beyond that given by Eq. (10), i.e., beyond that found in $p-A$ collisions \[Gerschel\], and different from ¶ suppression. If such an additional suppression were found, the results for inelastic -hadron collisions obtained from short distance QCD \[KS1\] would rule out a confined medium, and the difference in and ¶ suppression would exclude $(\c g)$ interactions as a source.
In summary: we study quarkonium production in hadron-nucleus collisions through the intermediate next-to-leading Fock space state consisting of a colour octet $\c$ and a gluon. We estimate the inelastic $(\c g)$-nucleon cross section and use this to
[–]{}[exclude shadowing as main origin for the quarkonium suppression observed in hadron-nucleus interactions;]{}
[–]{}[derive the Gerschel-Hüfner fit describing such suppression, both for $\c$ and $\b$ states;]{}
[–]{}[conclude that the suppression observed in $S-U$ collisions is fully accounted for by early $(\c g)$ interactions with standard nuclear matter;]{}
[–]{}[conclude that the stronger ¶ suppression found in $S-U$ collisions is due to an additional confined medium present at a later stage.]{}
It will be interesting to see if the results of forthcoming $Pb-Pb$ collisions can provide first indications for deconfinement – in particular, if they show a stronger dissociation than accounted for by $(\c g)$ interactions.
**Acknowledgements**
We thank A. Capella, C. Gerschel, A. Kaidalov, C. Lourenco and M. Mangano for stimulating and helpful discussions. The support of the German Research Ministry BMFT under contract 06 BI 721 is gratefully acknowledged.
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[)]{}[C. Lourenco (NA38/51), “Recent Results on Dimuon Production from the NA38 Experiment", CERN-PPE/95-72, May 1995, and Doctorate Thesis, Universidade Técnica de Lisboa, Portugal, January 1995.]{}
[)]{}[L. Fredj (NA38/51), Doctorate Thesis, Université de Clermont-Ferrand, France, September 1991.]{}
[)]{}[R. Salmeron, B 389 (1993) 301.]{}
[)]{}[C. Baglin et al., B 268 (1991) 453.]{}
[)]{}[C. Baglin et al., B 345 (1995) 617.]{}
[)]{}[C.-Y. Wong, “Suppression of ¶ and in High Energy Heavy Ion Collisions", Oak Ridge Preprint ORNL-CTP-95-04, June 1995.]{}
[)]{}[D. Kharzeev and H. Satz, “Colour Deconfinement and Quarkonium Dissociation", Preprint CERN-TH/95-117, May 1995; to appear in R. C. Hwa (Ed.), [*Quark-Gluon Plasma II*]{}, World Scientific Publ. Co., Singapore.]{}
**Figure Captions**
=0.6 truecm
[Fig. 1:]{}[ production through gluon fusion.]{}
[Fig. 2:]{}[ production via $(\c g)$ colour singlet.]{}
[Fig. 3:]{}[Transition from $(\c g)$ colour singlet to $\c$ colour singlet.]{}
[Fig. 4:]{}[ and suppression in $p-A$ collisions; data are compared to $(\c g)$ suppression (Eq. (9)) in nuclear matter and to the corresponding form for $(\b g)$.]{}
[Fig. 5:]{}[, ¶ and suppression in $p-A$ and $S-U$ collisions, normalised to $(\c g)$ and $(\b g)$ suppression in nuclear matter.]{}
|
---
abstract: 'We present results on the static potential, and torelon and glueball masses from simulations of QCD with two flavours of dynamical Wilson fermions on $16^3\times 32$ and $24^3\times 40$ lattices at $\beta=5.6$.'
author:
- |
SESAM and T$\chi$L collaborations: G.S. Bali$^{\rm a}$, N. Eicker$^{\rm b}$, L. Giusti$^{\rm c}$, U. Glässner$^{\rm d}$, S. Guesken$^{\rm d}$, H. Hoeber$^{\rm d}$, P. Lacock$^{\rm b}$, T. Lippert$^{\rm b}$, G. Martinelli$^{\rm c}$, F. Rapuano$^{\rm c}$, G. Ritzenhöfer$^{\rm e}$, K. Schilling$^{\rm b,d}$, G. Siegert$^{\rm b}$, A. Spitz$^{\rm b}$, P. Ueberholz$^{\rm d}$ and J. Viehoff$^{\rm d}$\
[$^a$]{}Department of Physics, The University, Highfield, Southampton SO17 1BJ, UK\
[$^b$]{}HLRZ c/o Forschungszentrum Jülich, D-52425 Jülich and DESY, D-22603 Hamburg, Germany\
[$^c$]{}Dip. di Fisica, Univ. “La Sapienza” and INFN, Sezione di Roma, P’lle A. Moro, I-00185 Rome, Italy\
[$^d$]{}Physics Department, University of Wuppertal, D-42097 Wuppertal, Germany\
[$^e$]{}MIT, Center for Theoretical Physics, Cambridge, Massachusetts 02139, USA
title: 'Glueballs and string breaking from full QCD[^1]'
---
SIMULATION
==========
The present simulations have been performed at various $\kappa$ values and lattice volumes at $\beta=5.6$ (Table \[tab1\]). The effective lattice resolution ranges from $a^{-1}\approx 2.0$ GeV ($\kappa=0.156$) down to $a^{-1}\approx 2.5$ GeV ($\kappa=0.158$) while the ratio $m_{\pi}/m_{\rho}$ varies from 0.83 to 0.55, corresponding to sea quarks that are slightly heavier than the strange quark and of about one quarter of its mass, respectively. In addition, results from quenched reference simulations at $\beta= 6.0$ ($a^{-1}\approx
2.1$ GeV) and $\beta= 6.2$ ($a^{-1}\approx 3.1$ GeV) are presented.
-.6cm ‘?=?
$\kappa$ $V$ $r_0a^{-1}$ $n_{\mbox{\scriptsize glue}}$ $n_{\mbox{\scriptsize pot}}$
------------- ---------- ------------- ------------------------------- ------------------------------
0.1560 $16^332$ 5.11(3) 2129 236
0.1565 $16^332$ 5.28(5) — 323
0.1570 $16^332$ 5.46(5) 2039 240
0.1575 $16^332$ 5.98(7) 2272 270
0.1575 $24^340$ 5.93(4) 1243 122
0.1580 $24^340$ 6.33(7) 743 95
$\beta=6.0$ $16^332$ 5.33(3) — 570
$\beta=6.2$ $32^4$ 7.29(4) — 116
: Simulation parameters.[]{data-label="tab1"}
-.7cm
On each of the small volumes, about 5000 thermalized trajectories have been generated. One half of this amount has been achieved on the $24^3\times 40$ lattice at $\kappa=0.1575$ while the $\kappa=0.158$ run has not been completed yet. Glueball and torelon measurements ($n_{\mbox{\scriptsize glue}}$) have been taken every 2 trajectories while smeared Wilson loops ($n_{\mbox{\scriptsize pot}}$) were measured every 20 trajectories (16 at $\kappa=0.1565$). Prior to statistical analysis, all data have been binned into blocks whose extent was large compared to the relevant autocorrelation time (see Ref. [@lippert]).
THE STATIC POTENTIAL
====================
An increase of the strength of the Coulomb-like attractive force between static sources at small separations, in respect to the quenched limit, has been observed previously [@wachter]; the QCD coupling is running slower with the energy scale, once sea quarks are switched on. We confirm these findings with increased statistical accuracy and full control over systematic uncertainties (see Ref. [@guesken]). Moreover, in extrapolating the Sommer scale $r_0^{-1}$ as a polynomial in the quark mass to the physical limit, we find a $\beta$ shift, $\beta_{n_f=0}
-\beta_{n_f=2}=0.57(2)$. This corresponds to an increase in the coupling of about 10% which is close to the naïve perturbative ratio $33/(33-2n_f)$.
With dynamical fermions, the static meson can decay into two static-light mesons. Ignoring meson-meson interactions, we expect the QCD string to “break” as soon as the potential exceeds twice the static-light mass, i.e. at about 1.5 fm. Neglecting quark mass effects on the dynamics of the binding problem (which is a reasonable assumption once this mass is small compared to a typical binding energy of 500 MeV), the string breaking distance should be shifted by an amount $\Delta r\approx 2\Delta m/\sigma$ when changing the quark mass by $\Delta m$. $\sigma = Ka^{-2}$ denotes the [*effective*]{} string tension. From these considerations, we find $\Delta r < 0.25$ fm for $\kappa \geq 0.156$. However, up to a separation of 2 fm no indications of a flattening of the potential are found. We suspect that this is due to a bad overlap of the Wilson loop with the two-meson state.
-.75cm =7.5cm -1.2cm
Reggè trajectories suggest values of 420 to 440 MeV for the string tension. Fits of potential models to the bottomonium spectrum yield similar values. However, such parametrizations face difficulties in setting a reliable upper bound on $\sigma$; effects of an increased slope of the potential at large distance can be absorbed by decreasing the strength of the short range interaction. This is due to the fact that the spectrum is rather insensitive towards the shape of the potential outside of a region $0.2$ fm $<r<1$ fm. In contrast, the scale $r_0$ which is defined through the interquark force at intermediate distance can be constrained to $r_0^{-1}\approx 395$ MeV rather accurately. In Fig. \[fig1\], we display the value $r_0\sqrt{\sigma}$ against $\kappa$. While a quenched value $\sqrt{\sigma}\approx 460$ MeV appears to be reasonable, with two sea quarks this is reduced to about 450 MeV such that for three active sea quarks a picture, consistent with the slope of Reggè trajectories, is likely to emerge.
TORELON STATES
==============
Torelons are flux tubes encircling the periodic spatial lattice boundaries. We restrict ourselves to torelons with winding number one. As a consequence of the center group symmetry all such torelons are degenerate in the quenched case. Moreover, the mass of such a state is expected to equal $L_Sa\sigma$ (up to tiny finite size corrections), where $L_Sa$ denotes the spatial extent of the lattice.
-1cm =7.5cm -1.2cm
With sea quarks, the $Z_3$ symmetry is only approximate and various torelon states, that can be classified in accord with irreducible representations of the cubic symmetry group, can obtain different masses. Moreover, mixing (or decays) between torelon states and isoscalar mesons might occur. The $T_1^{+-}$ torelon is accompanied by the lightest isoscalar, the continuum $J=1$ $h_1(1190)$ meson.
-.15cm =7.5cm -1.2cm
In Fig. \[fig2\] we display effective masses of three different torelon states at $\kappa=0.156$. Note that all effective masses are strict upper limits on the real mass value. All states are degenerate and in agreement with the expectation $L_S a\sigma$. The same holds true for $\kappa=0.157$. However, at $\kappa=0.1575$ and $\kappa=0.158$ we see indications of certain torelon states becoming lighter than expected. This situation is depicted in Fig. \[fig3\] for $\kappa=0.158$, where the $T_1^{+-}$ state falls below $24a\sigma$. Signals of the other states disappear into noise.
GLUEBALLS
=========
At sufficiently light sea quark masses we expect the $0^{++}$ glueball to mix with the two mesonic $I=0$ states of the $L=1$ nonet. Moreover, decays into two $\pi$’s become possible. Results on the scalar ($A_1^{++}$) and tensor ($E^{++}$) glueballs are displayed in Table \[tab2\] and Fig. \[fig4\]. No indications of the mentioned effects are found.
-.6cm ‘?=?
$\kappa$ $L_S$ $m_{0^{++}}r_0$ $m_{2^{++}}/m_{0^{++}}$
---------------- ------- ----------------- -------------------------
0.1560 16 3.56(12) 1.62(07)
0.1570 16 3.01(18) 1.89(13)
0.1575 16 3.26(25) 1.86(13)
0.1575 24 4.27(23) 1.46(28)
0.1580 24 4.49(41) 1.54(21)
$\beta=6.0$ 20 3.69(23) 1.67(15)
$\beta=\infty$ — 4.22(14) 1.40(15)
: Glueball masses.[]{data-label="tab2"}
-.7cm
-.15cm =7.5cm -1.2cm
In quenched simulations, finite size effects on the scalar glueball are small [@michael2] as long as $m_{0^{++}}<2m_t\approx 2L_Sa\sigma$ where $m_t$ denotes the torelon mass. When including sea quarks, the glueball might break up and decay into a single torelon [@michael3] through an intermediate mesonic state. The expected torelon masses $L_Sa\sigma$ are included into Fig. \[fig4\] for $L_S=16$ (dashed error bars). Contrary to the quenched case, we indeed find significant FSE at $\kappa=0.1575$ (and probably $\kappa=0.157$). In general, the large volume data is consistent with quenched results. The $0^{++}$ mass appears to exceed the $\beta=6.0$ reference value but agrees with the continuum extrapolated result.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We acknowledge support by the DFG (grants Schi 257/1-4 and Schi 257/3-2), the EU (contracts SC1\*-CT91-0642, CHRX-CT92-0051 and CHBG-CT94-0665) and PPARC (grant GR/K55738).
[9]{} T$\chi$L collaboration: T. Lippert et al., these proceedings.
SESAM collaboration: U. Glässner et al., Phys. Lett. B383 (1996) 98.
S. Guesken, these proceedings.
C. Michael and M. Teper, Nucl. Phys. B314 (1989) 347.
J. Kripfganz and C. Michael, Nucl. Phys. B314 (1989) 25.
[^1]: Presented by Gunnar Bali.
|
---
abstract: |
Let $G_m$ resp. $G_f$ be the minimal resp. formal Kac-Moody group, associated to a symmetrizable generalized Cartan matrix, over a field ${\mathbb{F}}$ of characteristic 0. Let ${\mbox{${\mathbb{F}}\,[G_m]$}}$ be the algebra of strongly regular functions on $G_m$.\
We denote by $\widehat{G_m}$ resp. $\widehat{G_f}$ certain monoid completions of $G_m$ resp. $G_f$, build by using the faces of the Tits cone.\
We show that there is an action of $\widehat{G_f}\times\widehat{G_f}$ on the spectrum of ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$. As a $\widehat{G_f}\times\widehat{G_f}$-set it can be identified with a certain quotient of the $\widehat{G_f}\times\widehat{G_f}$-set $\widehat{G_f}\times\widehat{G_f}$, build by using $\widehat{G_m}$.\
We prove a Birkhoff decomposition for the ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$.\
We describe the stratification of the spectrum of ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$ in $G_f\times G_f$-orbits. We show that every orbit can be covered by suitably defined big cells.
author:
- |
Claus Mokler\
\
Universität Wuppertal, Fachbereich Mathematik\
Gaußstraße 20\
D-42097 Wuppertal, Germany\
[email protected]
title: 'The ${\mathbb{F}}$-valued points of the algebra of strongly regular functions of a Kac-Moody group'
---
\[section\] \[Theorem\][Definition]{} \[Theorem\][Proposition]{} \[Theorem\][Proposition+Definition]{} \[Theorem\][Corollary]{} \[Theorem\][Remark]{} \[Theorem\][Remarks]{} \[Theorem\][Lemma]{}
[**Mathematics Subject Classification 2000:**]{} 17B67, 22E65.\
[**Keywords:**]{} Kac-Moody groups, algebra of strongly regular functions.
Introduction {#introduction .unnumbered}
============
The minimal Kac-Moody group $G_m$, which V. Kac and D. Peterson associated in [@KP1] to a Kac-Moody algebra ${\mbox{$\bf g$}}$ over a field ${\mathbb{F}}$ of characteristic 0, is a group analogue of a semisimple simply connected algebraic group.\
For a symmetrizable minimal Kac-Moody group, Kac and Peterson defined and investigated in [@KP2] the algebra of strongly regular functions ${\mbox{${\mathbb{F}}\,[G_m]$}}$ on $G_m$. This algebra has many properties in common with the coordinate ring of a semisimple simply connected algebraic group. It is an integrally closed domain, even a unique factorization domain. It admits a Peter and Weyl theorem, i.e., $$\begin{aligned}
{\mbox{${\mathbb{F}}\,[G_m]$}} &\cong & \bigoplus_{{\Lambda}\in P^+} L^*({\Lambda})\otimes L({\Lambda})\end{aligned}$$ as $G_m\times G_m$-modules. But the following things, which hold in the non-classical case, are different:\
1) Assigning to every element of $G_m$ its point evaluation, $G_m$ embeds in the set of ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$, which we denote by ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}}$. But this map is not surjective.\
2) There exists no comultiplication of ${\mbox{${\mathbb{F}}\,[G_m]$}}$, dual to the multiplication of $G_m$. The situation is not too bad, left and right multiplications with elements of $G_m$ induce comorphisms. A more serious difference, the inverse map of $G_m$ does not induce a comorphism.\
In particular there is no natural group structure, even no natural monoid structure on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}}$.\
Kac and Peterson posed the problem to determine ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}}$, or at least a certain part of it, [@KP2].\
\
The tensor category ${\cal O}_{adm}$ of admissible modules of $\cal O$ generalizes the category of finite dimensional representations of a semisimple Lie algebra, keeping the complete reducibility theorem.\
In a way similar to the reconstruction process of the Tannaka-Krein duality, a suitable category of representations of a Lie algebra, together with a suitable category of duals, determines a monoid with coordinate ring. In some sense, this monoid is the biggest monoid acting reasonably on the representations. The coordinate ring is a coordinate ring of matrix coefficients.\
We determined and investigated in [@M] the monoid $\widehat{G_m}$ corresponding to ${\cal O}_{adm}$ and its category of restricted duals. Equipped with its coordinate ring ${\mbox{${\mathbb{F}}\,[\widehat{G_m}]$}}$ of matrix coefficients, the monoid $\widehat{G_m}$ contains $G_m$ as Zariski open, dense unit group. It has similar properties as a reductive algebraic monoid. But it is a purely non-classical phenomenon, its classical analogue is a semisimple, simply connected algebraic group. For generalizing results of classical invariant theory, this monoid is more fundamental than the Kac-Moody group itself. For its history in connection with Slodowy and Peterson we refer to the introduction of [@M].\
The coordinate ring ${\mbox{${\mathbb{F}}\,[\widehat{G_m}]$}}$ is isomorphic to the algebra of strongly regular functions ${\mbox{${\mathbb{F}}\,[G_m]$}}$ by the restriction map. Therefore the monoid $\widehat{G_m}$ embeds in ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}}$, but also this map is not surjective.\
To investigate the ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$, we define and investigate a monoid $\widehat{G_f}$, which is build in a similar way as $\widehat{G_m}$, but the minimal Kac-Moody group $G_m$ replaced by the formal Kac-Moody group $G_f$. In a subsequent paper we will prove that this monoid corresponds to ${\cal O}_{adm}$ and its category of full duals.\
In this paper we obtain the following description of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}}$: We get, in a natural way, an action $\pi$ of $\widehat{G_f}\times \widehat{G_f}$ on ${\mbox{${\mathbb{F}}\,[G_m]$}}$ by homomorphisms of algebras. Therefore we also obtain a $\widehat{G_f}\times \widehat{G_f}$-action on the spectrum of ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G_m]$}}$ from the right. Composing the evaluation map at the unit of $G_m$ with $\pi$, we get a $\widehat{G_f}\times \widehat{G_f}$-equivariant map $$\begin{aligned}
{{\,\bf \diamond\,}}:\,\widehat{G_f}\times \widehat{G_f} \,\;\to \;\, {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}} & \quad,\quad & (x,y) \mapsto x{{\,\bf \diamond\,}}y \;\;.\end{aligned}$$ We show that this map factors to a $\widehat{G_f}\times \widehat{G_f}$-equivariant bijection with a quotient set of $\widehat{G_f}\times \widehat{G_f}$, which is obtained as follows: The Chevalley involution of $G_m$ extends to an involution $*$ of $\widehat{G_m}$. We factor $\widehat{G_f}\times \widehat{G_f}$ by the $\widehat{G_f}\times \widehat{G_f}$-equivariant equivalence relation generated by $$\begin{aligned}
(x,zy) \;\sim\;(z^*x,y ) &\quad,\quad & x,y\in \widehat{G_f}\;,\; z\in \widehat{G_m}\;.\end{aligned}$$ Due to this description, we can use structural properties of $\widehat{G_m}$ and $\widehat{G_f}$ to prove properties of the $\widehat{G_f}\times \widehat{G_f}$-space ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. In particular we prove the Birkhoff decomposition $$\begin{aligned}
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}} &=& \dot{\bigcup_{\hat{w}\in \widehat{\cal W}}} B_f{{\,\bf \diamond\,}}\hat{w} B_f\;\;.\end{aligned}$$ Here $B_f$ is the formal Borel group, and the Weyl monoid ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$ is a certain monoid containing the Weyl group.\
We determine the stratification of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ in $G_f\times G_f$-orbits: $$\begin{aligned}
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_m]$}} &=& \dot{\bigcup_{{\Theta}\,special}} G_f{{\,\bf \diamond\,}}e(R({\Theta})) G_f\;\;.\end{aligned}$$ Here ${\left\{\left.\;e(R({\Theta}))\;\right|\; {\Theta}{\mbox}{ special }\;\right\}}$ is a finite set of certain idempotents of $\widehat{G_m}$. We show that each orbit is locally closed and irreducible. We determine the closure relation of the orbits. We describe the big cell $B_f{{\,\bf \diamond\,}}e(R({\Theta})) B_f$ of the orbit $G_f {{\,\bf \diamond\,}}e(R({\Theta})) G_f$, and also the covering of $G_f {{\,\bf \diamond\,}}e(R({\Theta})) G_f$ by this big cell. We give stratified transversal slices to the orbits.\
The following work is in relation to these results:\
Let $(M,{\mbox{${\mathbb C}\,[M]$}})$ be a connected reductive algebraic monoid. Denote by $G$ its reductive unit group. Let $T$ be a maximal torus of $G$. Let $B$ and $B^-$ be opposite Borel subgroups containing $T$.\
Assigning to every element of $M$ its point evaluation, $M$ identifies with the ${\mathbb{C}}$-valued points of ${\mbox{${\mathbb C}\,[M]$}}$.\
L. Renner showed in [@Re2], that $M$ admits Bruhat decompositions. Due to the existence of a longest element of the Weyl group, these are equivalent to the Birkhoff decompositions. In particular $M=\dot{\bigcup}_{r\in {\cal R}} B^- r B$, where ${\cal R}$ is the Renner monoid. Due to the work of M. Putcha and L. Renner in [@Pu2], [@Re1] is the decomposition $M=\dot{\bigcup}_{e\in \Lambda} GeG$. The set $\Lambda$ is a certain set of idempotents, a cross-section lattice, which has been introduced by Putcha in [@Pu1]. Renner found and described in [@Re2] the big cell $B^-eB$ in $GeG$.\
The full spectrum of the algebra of strongly regular functions has been used by M. Kashiwara in [@Kas] for his infinite dimensional algebraic geometric approach to the flag variety of a Kac-Moody group. In contrary to Kac and Peterson, he constructs the algebra of strongly regular functions without using the minimal Kac-Moody group. He uses the coalgebra structure of the universal enveloping algebra of ${\mbox{$\bf g$}}$, to construct the algebra of strongly regular functions as a certain subalgebra of the corresponding dual algebra. Kashiwara defines an open subscheme of the full spectrum, which has a countable covering by suitably defined big cells. To obtain his flag variety, Kashiwara factors the subscheme by the action of $B_f\times \{1\}$.\
Taking ${\mathbb{F}}={\mathbb{C}}$, there exists a real unitary form $K$ of $G_m$, which coincides with the compact form in the classical case. D. Pickrell conjectures in [@Pic] certain $K$-biinvariant resp. $K$-invariant measures for Kac-Moody groups. He proves the existence of these measures in the affine case.\
Important for the construction of the biinvariant measures is a $G_f\times G_f$-space $G_f\times_{G_m}G_f$. He equips this space with a proalgebraic complex manifold structure, using a covering of big cells as an atlas. He also equips $G_f\times_{G_m}G_f$ with an algebra of matrix coefficients, isomorphic to the algebra of strongly regular functions. Important for the construction of the invariant measures is the flag $G_f$-space $\{1\}\times B_f\,\backslash\, G_f\times_{G_m}G_f$, also equipped with a proalgebraic manifold structure.\
It is not difficult to see, that the ${\mathbb{C}}$-valued points of the subscheme of Kashiwara, as well as the $G_f\times G_f$-space $G_f\times_{G_m}G_f$ of Pickrell identify with the biggest $G_f\times G_f$-orbit of ${\mbox{Specm\,}}{\mbox{${\mathbb C}\,[G_m]$}}$. In a subsequent paper, we will investigate the completed flag varieties in the setting of Kashiwara and in the setting of Pickrell. Presumably the extended Bruhat order of [@M2] will be important.\
Pickrell also proved a Birkhoff decomposition for $G_f\times_{G_m}G_f$. He noted, that in the affine case, many interesting completions of $G_m$ corresponding to loop groups are embedded in the set $G_f\times_{G_m}G_f$. The Birkhoff decomposition of $G_f\times_{G_m}G_f$ induces the Birkhoff decomposition of these completions.\
Now certain functional analytical closures, for example the closure used by Peterson in [@KP4] for his KAK-decomposition, are also embedded in the full set ${\mbox{Specm\,}}{\mbox{${\mathbb C}\,[G_m]$}}$. Hopefully this will help to make their structure more explicit.\
Preliminaries
=============
In this section we collect some basic facts about Kac-Moody algebras, minimal and formal Kac-Moody groups, the algebra of strongly regular functions, and the corresponding monoid completion, which are used later.\
One aim is to introduce our notation. Another aim is to put these things, which can be found in the literature, on equal footing appropriate for our goals.\
The minimal Kac-Moody group, given in [@KP1], [@KP3], corresponds to the derived Kac-Moody algebra. We work with a slightly enlarged group corresponding to the full Kac-Moody algebra as in [@Ti1], [@MoPi]. The algebra of strongly regular functions on this group is slightly larger, than the algebra of [@KP2]. We introduce this algebra as the restricted coordinate ring of a monoid. The formal Kac-Moody group has been constructed in [@Sl1], starting with a realization, glueing parabolic subgroups of finite type, which are equipped with a proalgebraic structure. We only need the formal Kac-Moody group corresponding to a simply connected minimal free realization, and we introduce this group by a representation theoretic construction.\
All the material stated in this subsection about Kac-Moody algebras can be found in the books [@K] (most results also valid for a field of characteristic zero with the same proofs), [@MoPi], about the minimal Kac-Moody group in [@KP1], [@KP3], [@MoPi], about the formal Kac-Moody group in [@Sl1], about the algebra of strongly regular functions in [@KP2], about the faces of the Tits cone in [@Loo], [@Sl1], [@M], and about the monoid completion of the minimal Kac-Moody group in [@M].\
We denote by ${\mathbb{N}}={\mathbb{Z}}^+$, ${\mathbb{Q}}^+$, resp. ${\mathbb{R}}^+$ the sets of strictly positive numbers of ${\mathbb{Z}}$, ${\mathbb{Q}}$, resp. ${\mathbb{R}}\,$, and the sets ${\mathbb{N}}_0={\mathbb{Z}}^+_0$, ${\mathbb{Q}}^+_0$, ${\mathbb{R}}^+_0$ contain, in addition, the zero.\
In the whole paper, ${\mathbb{F}}$ is a field of characteristic 0 and ${\mathbb{F}}^\times$ its group of units.\
[**Generalized Cartan matrices:**]{} Starting point for the construction of a Kac-Moody algebra, and its associated simply connected minimal and formal Kac-Moody groups is a [*generalized Cartan matrix*]{}, which is a matrix $A=(a_{ij})\in M_{n}({\mathbb{Z}})$ with $a_{ii}=2$, $a_{ij}\leq 0$ for all $i\neq j$, and $a_{ij}=0$ if and only if $a_{ji}=0$. Denote by $l$ the rank of $A$, and set $I:=\{1,2,\ldots, n\}$.\
For the properties of the generalized Cartan matrices, in particular their classification, we refer to the book [@K]. In this paper we assume $A$ to be symmetrizable.\
[**Realizations:**]{} A [*simply connected minimal free realization*]{} of $A$ consists of dual free $\mathbb{Z}$-modules $H$, $P$ of rank $2n-l$, and linear independent sets $\Pi^\vee =\{h_1,\ldots, h_n\}\subseteq H $, $\Pi=\{{\alpha}_1,\ldots,{\alpha}_n\}\subseteq P$ such that ${\alpha}_i(h_j)=a_{ji}\,$, $i,j=1,\dots, n$. Furthermore there exist (non-uniquely determined) fundamental dominant weights ${\Lambda}_1,\ldots, {\Lambda}_n\in P$ such that ${\Lambda}_i(h_j) ={\delta}_{ij}$, $i,j=1,\ldots, n$.\
$P$ is called the [*weight lattice*]{}, and $Q:={\mathbb{Z}}{\mbox}{-span}{\left\{\left.\;{\alpha}_i\,\;\right|\; \,i\in I\;\right\}}$ the [*root lattice*]{}.\
Set $Q^\pm_0:={\mathbb{Z}}^\pm_0{\mbox}{-span}{\left\{\left.\;{\alpha}_i\,\;\right|\; \,i\in I\;\right\}}$, and $Q^\pm:=Q^\pm_0\setminus\{0\}$.\
We fix a system of fundamental dominant weights ${\Lambda}_1,\ldots, {\Lambda}_n$, and extend $h_1,\ldots, h_n\in H$, ${\Lambda}_1,\ldots, {\Lambda}_n\in P$ to a pair of dual bases $h_1,\ldots, h_{2n-l}\in H$, ${\Lambda}_1,\ldots, {\Lambda}_{2n-l}\in P$. We set $H_{rest}:={\mathbb{Z}}{\mbox}{-}span{\left\{\left.\;h_i\;\right|\; i=n+1,\ldots,2n-l\;\right\}}$.\
[**The Weyl group, the Tits cone and its faces:**]{} Identify $H$ and $P$ with the corresponding sublattices of the following vector spaces over ${\mathbb{F}}\,$: $$\begin{aligned}
{\mbox{\textbf{h}}}\;\,:=\;\, {\mbox{\textbf{h}}}_{\mathbb{F}}\;\,:=\;\, H \otimes_{\mathbb Z} {\mathbb{F}}&\quad,\quad &
{\mbox{\textbf{h}}}^* \;\,:=\;\, {\mbox{\textbf{h}}}^*_{\mathbb{F}}\;\,:=\;\, P \otimes_{\mathbb Z} {\mathbb{F}}\;\;.\end{aligned}$$ ${\mbox{\textbf{h}}}^*$ is interpreted as the dual of ${\mbox{\textbf{h}}}$. Order the elements of ${\mbox{\textbf{h}}}^*$ by ${\lambda}\leq{\lambda}'$ if and only if ${\lambda}'-{\lambda}\in Q_0^+$.\
Choose a symmetric matrix $B\in M_n ({\mathbb{Q}})$ and a diagonal matrix $D={\mbox}{diag}({\epsilon}_1,\ldots, {\epsilon}_n)\,$, ${\epsilon}_1,\ldots,{\epsilon}_n\in{\mathbb{Q}^+}$, such that $A=DB$. Define a nondegenerate symmetric bilinear form on ${\mbox{\textbf{h}}}$ by: $$\begin{aligned}
{\left(h_i\midh\right)} \;=\; {\left(h\midh_i\right)} \;:=\; {\alpha}_i(h)\,{\epsilon}_i &\qquad & i\in I\,,
\quad h\in{\mbox{\textbf{h}}}\;,\\
{\left(h'\midh''\right)} \;:=\; 0 \qquad\qquad &\qquad & h',\,h'' \in {\mbox{\textbf{h}}}_{rest}:=H_{rest}\otimes{\mathbb{F}}\;.\end{aligned}$$ Denote the induced nondegenerate symmetric form on ${\mbox{\textbf{h}}}^*$ also by ${\left(\;\mid\;\right)}$.\
The [*Weyl group*]{} ${\mbox{$\cal W$}}={\mbox{$\cal W$}}(A)$ is the Coxeter group with generators $\sigma_i\,$, $i\in I$, and relations $$\begin{aligned}
\sigma_i^2 \;=\; 1 \qquad (i\in I) \;\:&\;,\;&\;\:
{(\sigma_i\sigma_j)}^{m_{ij}} \;=\; 1 \qquad (i,j\in I,\,i\ne j)\;\;.\end{aligned}$$ The $m_{ij}$ are given by: $\quad \begin{tabular}{c|ccccc}
$a\_[ij]{}a\_[ji]{}$ & 0 & 1 & 2 & 3 & $$ 4 \\[0.5ex] \hline
$m\_[ij]{}$ & 2 & 3 & 4 & 6 & no relation between $\_i$ and $\_j$
\end{tabular}$\
The Weyl group ${\mbox{$\cal W$}}$ acts faithfully and contragrediently on ${\mbox{\textbf{h}}}$ and ${\mbox{\textbf{h}}}^*$ by $$\begin{aligned}
\sigma_i h \;:=\; h - {\alpha}_i\,(h) h_i & \qquad i\in I,\quad h\in {\mbox{\textbf{h}}}\;\;,\\
\sigma_i {\lambda}\;:=\; {\lambda}- {\lambda}(h_i)\,{\alpha}_i & \qquad i\in I,\quad {\lambda}\in {\mbox{\textbf{h}}}^*\;\;,\end{aligned}$$ leaving the lattices $H$, $Q$, $P$, and the forms invariant.\
$\Delta_{re}:={\mbox{$\cal W$}}{\left\{\left.\;{\alpha}_i\;\right|\; i\in I\;\right\}}$ is called the set of [*real roots*]{}, and $\Delta_{re}^\vee:={\mbox{$\cal W$}}{\left\{\left.\;h_i\;\right|\; i\in I\;\right\}}$ the set of [*real coroots*]{}. The map ${\alpha}_i\mapsto h_i\,$, $i\in I$, can be extended to a ${\mbox{$\cal W$}}$-equivariant bijection ${\alpha}\mapsto h_{\alpha}$.\
To illustrate the action of ${\mbox{$\cal W$}}$ on ${\mbox{\textbf{h}}}^*_{\mathbb{R}}$ geometrically, for $J\subseteq I$ define $$\begin{aligned}
F_J &:=& {\left\{\left.\;{\lambda}\in{\mbox{\textbf{h}}}^*_{\mathbb{R}}\;\right|\; {\lambda}(h_i)\,=\,0 \;{\mbox}{ for }\; i\in J\,,\;\;\;
{\lambda}(h_i)\,>\,0\; {\mbox}{ for }\;i\in I\setminus J\;\right\}}\;\;,\\
\overline{F_J} &:=& {\left\{\left.\;{\lambda}\in{\mbox{\textbf{h}}}^*_{\mathbb{R}}\;\right|\; {\lambda}(h_i)\,=\,0\; {\mbox}{ for }\;i\in J
\,,\;\;\;{\lambda}(h_i)\,\geq\,0\; {\mbox}{ for }\;i\in I\setminus J\;\right\}}\;\;. \end{aligned}$$ $\overline{F_J}$ is a finitely generated convex cone with relative interior $F_J$. The parabolic subgroup ${\mbox{$\cal W$}}_J$ of ${\mbox{$\cal W$}}$ is the stabilizer of every element ${\lambda}\in F_J$. For $\sigma\in{\mbox{$\cal W$}}$ call $\sigma F_J$ a [*facet*]{} of [*type*]{} $J$.\
The [*fundamental chamber*]{} $\overline{C} \,:=\, {\left\{\left.\;{\lambda}\in{\mbox{\textbf{h}}}^*_{\mathbb{R}}\;\right|\; {\lambda}(h_i)\,\geq\,0\; {\mbox}{ for }\;i\in I\;\right\}}$ is a fundamental region for the action of ${\mbox{$\cal W$}}$ on the convex cone $X:={\mbox{$\cal W$}}\,\overline{C}\,$, which is called the [*Tits cone*]{}. The partition $\overline{C}=\dot{\bigcup}_{J\subseteq I} F_J$ induces a ${\mbox{$\cal W$}}$-invariant partition of $X$ into facets.\
A set ${\Theta}\subseteq I$ is called [*special*]{}, if either ${\Theta}=\emptyset$, or else all connected components of the generalized Cartan submatrix $(a_{ij})_{i,j\in{\Theta}}$ are of non-finite type. Set ${\Theta}^\bot:={\left\{\left.\;i\in I\;\right|\; a_{ij}=0 {\mbox}{ for all } j\in{\Theta}\;\right\}}$. Every face of the Tits cone $X$ is ${\mbox{$\cal W$}}$-conjugate to exactly one of the faces $$\begin{aligned}
R({\Theta}) \;\,:=\;\, X\,\cap\,{\left\{\left.\;{\lambda}\in {\mbox{\textbf{h}}}_{{\mathbb{R}}}^*\;\right|\; {\lambda}(h_i)=0 {\mbox}{ for all } i\in{\Theta}\;\right\}}
\;\,=\;\,{\mbox{$\cal W$}}_{{\Theta}^\bot} \overline{F}_{\Theta}&\;,\; & {\Theta}\;{\mbox}{ special}\;\,.\end{aligned}$$ The parabolic subgroup ${\mbox{$\cal W$}}_{\Theta}$ is the pointwise stabilizer of $R({\Theta})$, and the parabolic subgroup ${\mbox{$\cal W$}}_{{\Theta}\cup{\Theta}^\bot}$ is the stabilizer of the set $R({\Theta})$ as a whole.\
The relative interior of $R({\Theta})$ is given by the union of the facets $\sigma F_{{\Theta}\cup{\Theta}_f}$, where $\sigma\in{\mbox{$\cal W$}}_{{\Theta}^\bot}$, and ${\Theta}_f$ is a subset of ${\Theta}^\bot$, which is either empty, or else for which all connected components of $(a_{ij})_{i,j\in{\Theta}_f}$ are of finite type.\
[**The Kac-Moody algebra:**]{} The [*Kac-Moody algebra*]{} ${\mbox{$\bf g$}}={\mbox{$\bf g$}}(A)$ is the Lie algebra over ${\mathbb{F}}$ generated by the abelian Lie algebra ${\mbox{\textbf{h}}}$ and $2n$ elements $e_i,f_i$, ($i\in I$), with the following relations, which hold for any $i,j \in I$, $h \in {\mbox{\textbf{h}}}\,$: $$\begin{aligned}
\left[ e_i,f_j \right] \,=\, \delta_{ij} h_i \;\;,\;\;
\left[ h,e_i \right] \,=\, {\alpha}_i(h) e_i \;\;,\;\;
\left[ h,f_i \right] \,=\, -{\alpha}_i(h) f_i \;\;, \\
\left(ad\,e_i\right)^{1-a_{ij}}e_j \,=\, \left(ad\,f_i\right)^{1-a_{ij}}f_j \,=0 \qquad (i\neq j)\;\;.
\end{aligned}$$ The [*Chevalley involution*]{} $*:{\mbox{$\bf g$}}\to{\mbox{$\bf g$}}$ is the involutive anti-automorphism determined by $ e_i^*=f_i$, $ f_i^*=e_i$, $h^*=h$, ($i\in I$, $h\in {\mbox{\textbf{h}}}$).\
The nondegenerate symmetric bilinear form [( $|$ )]{} on ${\mbox{\textbf{h}}}$ extends uniquely to a nondegenerate symmetric invariant bilinear form [( $|$ )]{} on ${\mbox{$\bf g$}}$. We have the [*root space decomposition*]{} $$\begin{aligned}
{\mbox{$\bf g$}}=\bigoplus_{{\alpha}\in {\mbox{\textbf{h}}}^*}{\mbox{$\bf g$}}_{{\alpha}} \quad {\mbox}{where} \quad
{\mbox{$\bf g$}}_{\alpha}:= {\left\{\left.\;x\in {\mbox{$\bf g$}}\;\right|\; [h,x]={\alpha}(h)\,x\;{\mbox}{ for all }\;h\in {\mbox{\textbf{h}}}\;\right\}}\;\;. \end{aligned}$$ In particular ${\mbox{$\bf g$}}_0 = {\mbox{\textbf{h}}}$, ${\mbox{$\bf g$}}_{{\alpha}_i}={\mathbb{F}}e_i$, and ${\mbox{$\bf g$}}_{-{\alpha}_i}={\mathbb{F}}f_i$, $i\in I $.\
The set of roots ${\mbox{$\Delta$}}:={\left\{\left.\;{\alpha}\in{\mbox{\textbf{h}}}^*\setminus\{0\}\;\right|\; {\mbox{$\bf g$}}_{\alpha}\ne \{0\}\;\right\}}$ is invariant under the Weyl group, ${\mbox{$\Delta$}}=-{\mbox{$\Delta$}}$, and ${\mbox{$\Delta$}}$ spans the root lattice $Q\,$. We have ${\mbox{$\Delta_{re}$}}\subseteq {\mbox{$\Delta$}}\,$, and ${\mbox{$\Delta_{im}$}}:={\mbox{$\Delta$}}\setminus{\mbox{$\Delta_{re}$}}$ is called the set of [*imaginary roots*]{}.\
${\mbox{$\Delta$}}$, ${\mbox{$\Delta_{re}$}}$, and ${\mbox{$\Delta_{im}$}}$ decompose into the disjoint union of the sets of [*positive*]{} and [*negative*]{} roots ${\mbox{$\Delta$}}^\pm:={\mbox{$\Delta$}}\cap Q^\pm$, ${\mbox{$\Delta_{re}$}}^\pm:={\mbox{$\Delta_{re}$}}\cap Q^\pm$, ${\mbox{$\Delta_{im}$}}^\pm:={\mbox{$\Delta_{im}$}}\cap Q^\pm$.\
There is the [*triangular decomposition*]{} ${\mbox{$\bf g$}}= {\mbox{$\textbf{n}^-$}}\oplus {\mbox{\textbf{h}}}\oplus {\mbox{$\textbf{n}^+$}}$, where ${\mbox{\textbf{n}}}^\pm:=\bigoplus_{{\alpha}\in {\Delta}^\pm} {\mbox{$\bf g$}}_{\alpha}$.\
[**Irreducible highest weight representations:**]{} For every ${\Lambda}\in{\mbox{\textbf{h}}}^*$ there exists, unique up to isomorphism, an irreducible representation $(L({\Lambda}),\pi_{\Lambda})$ of ${\mbox{$\bf g$}}$ with highest weight ${\Lambda}$. It is ${\mbox{\textbf{h}}}$-diagonalizable, and we denote its set of weights by $P({\Lambda})$.\
Any such representation carries a nondegenerate symmetric bilinear form ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}:L({\Lambda})\times L({\Lambda})\to{\mathbb{F}}$ which is contravariant, i.e., ${\left\langle \left\langle v\mid xw
\right\rangle \right\rangle}={\left\langle \left\langle x^*v\mid w
\right\rangle \right\rangle}$ for all $v,w\in L({\Lambda})$, $x\in{\mbox{$\bf g$}}$. This form is unique up to a nonzero multiplicative scalar.\
[**The minimal and the formal Kac-Moody group:**]{} We say that a Lie algebra ${\bf l}$ acts [*locally nilpotent*]{} on an ${\bf l}$-module $V$, if for every $v\in V$, there exists a positive integer $m\in{\mathbb{N}}$, such that for all $x_1,\,x_2,\ldots,x_m\in {\bf l}$ we have $x_1x_2\cdots x_m v=0$.\
Call a ${\mbox{$\bf g$}}$-module $V$ [*m-admissible*]{}, if $V$ is ${\mbox{\textbf{h}}}$-diagonalizable with set of weights $P(V)\subseteq P$, and ${\mbox{$\bf g$}}_{\alpha}$ acts locally nilpotent on $V$ for all ${\alpha}\in{\mbox{$\Delta_{re}$}}$.\
Examples are the adjoint representation (${\mbox{$\bf g$}}\,$, $ad\,$), and the irreducible highest weight representations ($L({\Lambda})$, $\pi_{\Lambda}$), ${\Lambda}\in P^+:=P\cap{\mbox{$\overline{C}$}}$.\
(Note that m-admissible is slightly different from [*integrable*]{}, which means $V$ is ${\mbox{\textbf{h}}}$-diagonalizable, and ${\mbox{$\bf g$}}_{\alpha}$ acts locally nilpotent on $V$ for all ${\alpha}\in{\mbox{$\Delta_{re}$}}$. The weights of an integrable module can be contained in ${\left\{\left.\;{\lambda}\in{\mbox{\textbf{h}}}^*\;\right|\; {\lambda}(h_i)\in{\mathbb{Z}},\;i=1,\,\ldots,\,n\;\right\}}$. If the generalized Cartan matrix is degenerate, then this set is no lattice.)\
The [*minimal Kac-Moody group*]{} $G=G_m=G_m(A)$ can be characterized in the following way:\
$\bullet$ The group $G$ acts on every m-admissible representation. Two elements $g,g'\in G$ are equal if and only if for all m-admissible modules $V$, and for all $v\in V$, we have $gv=g'v$.\
$\bullet$ (1) For every $h\in H$, $s\in{\mathbb{F}}^\times$ there exists an element $t_h(s)\in G$, such that for any m-admissible representation $(V,\pi)$ we have $$\begin{aligned}
t_h(s)v_{\lambda}&=& s^{{\lambda}(h)}v_{\lambda}\quad,\quad v_{\lambda}\in V_{\lambda}\;\,,\;\,{\lambda}\in P(V)\;.\end{aligned}$$ (2) For every $x\in{\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in{\mbox{$\Delta_{re}$}}$, there exists an element $\exp(x)\in G$, such that for any m-admissible representation $(V,\pi)$ we have $$\begin{aligned}
\exp(x)v &=& \exp(\pi(x)) v \quad,\quad v\in V\;.\end{aligned}$$ $G$ is generated by the elements of (1) and (2).\
Call a ${\mbox{$\bf g$}}$-module $V$ [*f-admissible*]{}, if $V$ is m-admissible, and ${\mbox{\textbf{n}}}^+$ acts locally nilpotent on $V$.\
Examples are the representations ($L({\Lambda})$, $\pi_{\Lambda}$), ${\Lambda}\in P^+=P\cap{\mbox{$\overline{C}$}}$.\
Set ${\mbox{\textbf{n}}}_f:=\prod_{{\alpha}\in \Delta^+}{\mbox{$\bf g$}}_{\alpha}$ and ${\mbox{$\bf g$}}_f:={\mbox{\textbf{n}}}^-\oplus{\mbox{\textbf{h}}}\oplus{\mbox{\textbf{n}}}_f$. The Lie bracket of ${\mbox{$\bf g$}}$ extends in the obvious way to a Lie bracket of ${\mbox{$\bf g$}}_f$. Every f-admissible ${\mbox{$\bf g$}}$-module can be extended to a ${\mbox{$\bf g$}}_f$-module. The Lie algebra ${\mbox{$\bf g$}}_f$ should be interpreted as the Lie algebra of the [*formal Kac-Moody group*]{} $G_f=G_f(A)$, which can be characterized in the following way:\
$\bullet$ The group $G_f$ acts on every f-admissible representation. Two elements $g,g'\in G_f$ are equal if and only if for all f-admissible modules $V$, and for all $v\in V$ we have $gv=g'v$.\
$\bullet$ (3) $G_f$ contains $G$.\
(4) For every $x\in{\mbox{\textbf{n}}}_f$ there exists an element $\exp(x)\in G_f$, such that for any f-admissible representation $(V,\pi)$ we have $$\begin{aligned}
\exp(x)v &=& \exp(\pi(x)) v \quad,\quad v\in V\;\;.\end{aligned}$$ $G_f$ is generated by $G$ and the elements of (4).\
The ${\mbox{$\bf g$}}$-module ${\mbox{$\bf g$}}_f$ is not f-admissible. Nevertheless $G_f$ acts on ${\mbox{$\bf g$}}_f$, extending the adjoint action of $G$ on ${\mbox{$\bf g$}}$, compare [@Sl1], Section 5.11.\
Both Kac-Moody groups act faithfully on $\bigoplus_{{\Lambda}\in P^+}L({\Lambda})$. They have the following important structural properties:\
1) The elements of (1) induce an embedding of the torus $H\otimes_{\mathbb{Z}}{\mathbb{F}}^\times$ into $G\subseteq G_f$. Its image is denoted by $T$.\
For ${\alpha}\in{\mbox{$\Delta_{re}$}}$ the elements of (2) induce an embedding of $({\mbox{$\bf g$}}_{\alpha},+)$ into $G\subseteq G_f$. Its image $U_{\alpha}$ is called the root group belonging to ${\alpha}$.\
Let ${\alpha}\in {\mbox{$\Delta_{re}^+$}}$ and $x_{\alpha}\in{\mbox{$\bf g$}}_{{\alpha}}$, $x_{-{\alpha}}\in{\mbox{$\bf g$}}_{-{\alpha}}$ such that $[x_{\alpha},x_{-{\alpha}}]=h_{\alpha}$. There exists an injective homomorphism of groups $\phi_{\alpha}:\,{\mbox}{SL}(2,{\mathbb{F}}) \to G$ with $$\begin{aligned}
\phi_{\alpha}\left(\begin{array}{cc}
1 & s\\
0 & 1
\end{array}\right) \;:=\; \exp(s x_{\alpha}) \;\,,\,\;
\phi_{\alpha}\left(\begin{array}{cc}
1 & 0\\
s & 1
\end{array}\right) \;:=\; \exp(sx_{-{\alpha}})\;\,,\,\;(s\in {\mathbb{F}}^\times)\;.\end{aligned}$$ 2) Denote by $N$ the subgroup generated by $T$ and $
n_{\alpha}:= \phi_{\alpha}\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right)$, ${\alpha}\in \Delta_{re}$. The Weyl group ${\mbox{$\cal W$}}$ can be identified with the group $N/T$ by the isomorphism $\kappa:\,{\mbox{$\cal W$}}\to N/T$, which is given by $\kappa(\sigma_{\alpha}):=n_{\alpha}(1) T$, ${\alpha}\in {\mbox{$\Delta_{re}$}}$.\
We denote an arbitrary element $n\in N$ with $\kappa^{-1}(nT)=\sigma\in{\mbox{$\cal W$}}$ by $n_\sigma$. The set of weights $P(V)$ of an m-admissible ${\mbox{$\bf g$}}$-module $(V,\pi)$ is ${\mbox{$\cal W$}}$-invariant, and $ n_\sigma V_{\lambda}=V_{\sigma {\lambda}}$, ${\lambda}\in P(V)$.\
3) Let $U^\pm$ be the subgroups generated by $U_{\alpha}$, ${\alpha}\in\Delta_{re}^\pm $. Let $U_f:=\exp({\mbox{\textbf{n}}}_f)$. Then $U^\pm$ and $U_f$ are normalized by $T$. Set $$\begin{aligned}
B^\pm\;\,:=\,\; T\ltimes U^\pm &\quad,\quad & B_f\;\,:=\,\; T\ltimes U_f\;\;.\end{aligned}$$ The pairs ($B^\pm$, $N$) are twinned BN-pairs of $G$ with the property $B^+\cap B^-= B^\pm\cap N = T$. The pair ($B_f$, $N$) is a BN-pair of $G_f$ with $B_f\cap N=T$. We have the following decompositions, called [*Bruhat*]{} and [*Birkhoff decompositions*]{}: $$\begin{aligned}
G \;\,=\;\, \dot{ \bigcup_{\sigma\in{\cal W}}} B^\epsilon\sigma B^\delta \quad,\quad
G_f \;\,=\;\, \dot{ \bigcup_{\sigma\in{\cal W}}} B^\epsilon\sigma B_f \qquad,
\qquad \epsilon,\delta\;\in\;\{\,+\,,\,-\,\}\;\;.\end{aligned}$$ 4) There are also Levi decompositions of the standard parabolic subgroups. In this paper we only use the corresponding decompositions for the groups $U^\pm$ and $U_f$: Set ${\mbox{$\Delta$}}^\pm_J:={\mbox{$\Delta$}}^\pm\cap \sum_{j\in J}{\mathbb{Z}}\,{\alpha}_j$, and ${({\mbox{$\Delta$}}^J)}^\pm:={\mbox{$\Delta$}}^\pm\setminus\sum_{j\in J}{\mathbb{Z}}\,{\alpha}_j$. Similarly define $({\mbox{$\Delta$}}_J)^\pm_{re}$ and $({\mbox{$\Delta$}}^J)^\pm_{re}$ by replacing ${\mbox{$\Delta$}}^\pm$ by ${\mbox{$\Delta$}}_{re}^\pm$. Set $({\mbox{\textbf{n}}}_J)^\pm:=\bigoplus_{{\alpha}\in \Delta_J^\pm}{\mbox{$\bf g$}}_{\alpha}$, $({\mbox{\textbf{n}}}_f)_J:=\prod_{{\alpha}\in\Delta_J^+}{\mbox{$\bf g$}}_{\alpha}$, and $({\mbox{\textbf{n}}}_f)^J:=\prod_{{\alpha}\in(\Delta^J)^+}{\mbox{$\bf g$}}_{\alpha}$. We have: $$\begin{aligned}
U^\pm\;\,=\;\, U^\pm_J \ltimes (U^J)^\pm &\quad,\quad & U_f\;\,=\;\, (U_f)_J \ltimes (U_f)^J\;\;.\end{aligned}$$ Here $U_J^\pm$ is the group generated by the root groups $U_{\alpha}$, ${\alpha}\in ({\mbox{$\Delta$}}_J)^\pm_{re}$. $(U^J)^\pm$ is the smallest normal subgroup of $U^\pm$, containing the root groups $U_{\alpha}$, ${\alpha}\in ({\mbox{$\Delta$}}^J)^\pm_{re}$. This group equals $\,\bigcap_{\sigma\in{\cal W}_J}\sigma U^\pm\sigma^{-1}$. Furthermore $(U_f)_J:=\exp(({\mbox{\textbf{n}}}_f)_J)$ and $(U_f)^J:=\exp(({\mbox{\textbf{n}}}_f)^J)$.\
The derived minimal Kac-Moody group $G'$ is identical with the Kac-Moody group as defined in [@KP1]. It is generated by the root groups $U_{\alpha}$, ${\alpha}\in {\mbox{$\Delta_{re}$}}$. We have $G = G'\rtimes T_{rest}$, where $T_{rest}:=H_{rest}\otimes_{\mathbb Z} {\mathbb{F}}$ is a subtorus of $T$.\
The group $G_f$ is identical with the Kac-Moody group of [@Sl1] for a simply connected minimal free realization.\
[**The monoid ${\mbox{$\widehat{G}$}}$:**]{} The category $\cal O$ is defined as follows: Its objects are the ${\mbox{$\bf g$}}$-modules $V$, which have the properties:\
(1) $V$ is ${\mbox{\textbf{h}}}$-diagonalizable with finite dimensional weight spaces.\
(2) There exist finitely many elements ${\lambda}_1,\,\ldots,\,{\lambda}_m\in{\mbox{\textbf{h}}}^*$, such that the set of weights $P(V)$ of $V$ is contained in the union $\bigcup_{1=1}^m D({\lambda}_i)$, where $D({\lambda}_i):={\left\{\left.\;{\lambda}\in{\mbox{\textbf{h}}}^*\;\right|\; {\lambda}\leq {\lambda}_i\;\right\}}$.\
The morphisms of $\cal O$ are the morphisms of ${\mbox{$\bf g$}}$-modules.\
For $V$ a module of ${\cal O}$ and $v\in V$, we denote by $supp(v)$ the set of weights of the nonzero weight space componsnts of $v$.\
A module of ${\cal O}$ is m-admissible if and only if it is f-admissible, and we call such a module [*admissible*]{}. We denote by ${\cal O}_{adm}$ the full subcategory of the category $\cal O$, whose objects are admissible modules.\
There is a complete reducibility theorem. Every object of ${\cal O}_{adm}$ is isomorphic to a direct sum of the admissible irreducible highest weight modules $L({\Lambda})$, ${\Lambda}\in P^+$.\
The set of weights of a module of ${\cal O}_{adm}$ is contained in $X\cap P$, because we have $$\begin{aligned}
\bigcup_{{\Lambda}\in P^+} P({\Lambda}) &=& X\cap P\;\;.\end{aligned}$$ Let ${\Lambda}\in P^+$, and ${\Theta}$ be special. Because the set of weights $P({\Lambda})$ is contained in the convex hull of ${\mbox{$\cal W$}}{\Lambda}\subseteq {\mbox{\textbf{h}}}_{\mathbb{R}}^*$, we find easily $$\begin{aligned}
P({\Lambda})\cap R({\Theta}) \;\,=\;\,\emptyset &{\mbox}{ if and only if }& {\Lambda}\notin R({\Theta})\;\;.\end{aligned}$$ The monoid ${\mbox{$\widehat{G}$}}$ can be characterized in the following way:\
$\bullet$ The monoid ${\mbox{$\widehat{G}$}}$ acts on every module of ${\cal O}_{adm}$. Two elements $\hat{g},\hat{g}'\in {\mbox{$\widehat{G}$}}$ are equal if and only if for all modules $V$ of ${\cal O}_{adm}$, and for all $v\in V$, we have $\hat{g}v=\hat{g}'v$.\
$\bullet$ (1) ${\mbox{$\widehat{G}$}}$ is an extension of the minimal Kac-Moody group $G$.\
(2) For every face $R$ of the Tits cone there exists an element $e(R)\in {\mbox{$\widehat{G}$}}$, such that for every module $V$ of ${\cal O}_{adm}$ we have $$\begin{aligned}
e(R)v_{\lambda}&=& \left\{ \begin{array}{ccc}
v_{\lambda}&\;&{\lambda}\in R\\
0 &\;& {\mbox}{else}
\end{array}\right. \quad,\quad \;v_{\lambda}\in V_{\lambda}\,,\; \;{\lambda}\in P(V)\;\;.\end{aligned}$$ ${\mbox{$\widehat{G}$}}$ is generated by $G$ and the elements of (2).\
Note that the monoid ${\mbox{$\widehat{G}$}}$ acts faithfully on the sum $\bigoplus_{{\Lambda}\in P^+}L({\Lambda})$.\
The [*Chevalley involution*]{} $*:\,{\mbox{$\widehat{G}$}}\to {\mbox{$\widehat{G}$}}$ is the involutive anti-isomorphism determined by $\exp(x_{\alpha})^*:=\exp(x_{\alpha}^*)$, $t^*:= t$, $e(R)^* := e(R)$, where $x_{\alpha}\in{\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in{\mbox{$\Delta_{re}$}}$, $t\in T$, and $R\in{\mbox{${\cal R}(X)$}}$.\
It is compatible with any nondegenerate symmetric contravariant form ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$ on any module $V$ of ${\cal O}_{adm}$, i.e., ${\left\langle \left\langle xv\mid w
\right\rangle \right\rangle}={\left\langle \left\langle v\mid x^*w
\right\rangle \right\rangle}$, $v,w\in V$, $x\in{\mbox{$\widehat{G}$}}$.\
The following formulas are useful for computations in ${\mbox{$\widehat{G}$}}$:\
$\bullet$ Let $R$, $S$ be faces of the Tits cone, and $n_\sigma\in N$. Then $$\begin{aligned}
e(R)e(S)=e(R\cap S) \qquad ,\qquad n_\sigma e(R) n_\sigma^{-1}\;\,=\;\, e(\sigma R)\;\;.\end{aligned}$$ $\bullet$ An element $g$ of $T$, $N$, $U$, $U^-$, resp. $G$ satisfies $$\begin{aligned}
e(R({\Theta})) g &=& e(R({\Theta}))\end{aligned}$$ if and only if it satisfies $$\begin{aligned}
g^* e(R({\Theta})) &=& e(R({\Theta}))\end{aligned}$$ if and only if it is contained in $T_{\Theta}$, $N_{\Theta}$, $U_{\Theta}$, $U_{\Theta}^-\ltimes (U^{{\Theta}\cup{\Theta}^\bot})^{-}$, resp. $G_{\Theta}\ltimes U^{{\Theta}\cup{\Theta}^\bot}$. Here $T_{\Theta}$ is the subtorus of $T$ generated by $t_{h_j}(s)$, $j\in {\Theta}$, $s\in{\mathbb{F}}^\times$, $N_{\Theta}$ is the subgroup of $N$ generated by $T_{\Theta}$ and $n_{{\alpha}_j}$, $j\in {\Theta}$, and $G_{\Theta}$ is the subgroup of $G$ generated by $U_{{\alpha}_j}^\pm$, $j\in {\Theta}$.\
$\bullet$ An element $g$ of $T$, $N$, $U$, $U^-$, resp. $G$ satisfies $$\begin{aligned}
g e(R({\Theta})) g^{-1} &=& e(R({\Theta}))\end{aligned}$$ if and only if it is contained in the groups $T$, $ N_{{\Theta}\cup{\Theta}^\bot} T$, $U_{{\Theta}\cup{\Theta}^\bot}$, $U_{{\Theta}\cup{\Theta}^\bot}^-$, resp. $G_{{\Theta}\cup{\Theta}^\bot} T$.\
$\bullet$ In particular we have $$\begin{aligned}
U e(R({\Theta})) &=& U_{{\Theta}^\bot} e(R({\Theta}))\;\,=\;\,
e(R({\Theta}))U_{{\Theta}^\bot}\;\;,\\
e(R({\Theta}))U^- &=& e(R({\Theta}))U^-_{{\Theta}^\bot}\;\,=\;\,
U^-_{{\Theta}^\bot}e(R({\Theta}))\;\;.\end{aligned}$$ The minimal Kac-Moody group $G$ is the unit group of ${\mbox{$\widehat{G}$}}$. Every idempotent is $G$-conjugate to some idempotent $e(R({\Theta}))$, ${\Theta}$ special. We have $$\begin{aligned}
{\mbox{$\widehat{G}$}}&=& \dot{\bigcup_{{\Theta}\,special}} G e(R({\Theta})) G\;\;.\end{aligned}$$ The Weyl group acts on the monoid (${\mbox{${\cal R}(X)$}}\,,\,\cap$). The semidirect product ${\mbox{${\cal R}(X)$}}\rtimes{\mbox{$\cal W$}}$ consists of the set ${\mbox{${\cal R}(X)$}}\times{\mbox{$\cal W$}}$ with the structure of a monoid given by $$\begin{aligned}
(R,\sigma)\cdot(S,\tau) &:=& (R\cap\sigma S,\sigma \tau)\;\;.\end{aligned}$$ For $R\in{\mbox{${\cal R}(X)$}}$ let $Z_{\cal W}(R):={\left\{\left.\;\sigma\in {\mbox{$\cal W$}}\;\right|\; \sigma{\lambda}={\lambda}{\mbox}{ for all }{\lambda}\in R\;\right\}}$ be the pointwise stabilizer of $R$. The Weyl monoid ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$ is defined as the monoid ${\mbox{${\cal R}(X)$}}\rtimes{\mbox{$\cal W$}}$ factored by the congruence relation $$\begin{aligned}
(R,\sigma) \sim (R',\sigma') &:\iff & R\:=\:R'\quad{\mbox}{and}\quad\sigma'\sigma^{-1}\in Z_{\cal W}(R)\;\;.\end{aligned}$$ We denote the congruence class of $(R,\sigma)$ by ${\mbox{$\varepsilon\left(R\right)$}}\sigma$.\
Assigning to $\sigma\in{\mbox{$\cal W$}}$ the element $\sigma:=\sigma e(X)\in{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$, the Weyl group ${\mbox{$\cal W$}}$ identifies with the unit group of ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$. The partition of ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$ into ${\mbox{$\cal W$}}\times{\mbox{$\cal W$}}$-orbits is given by $$\begin{aligned}
{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}&=& \dot{\bigcup_{{\Theta}\;special}} \,{\mbox{$\cal W$}}\,{\mbox{$\varepsilon\left(R({\Theta})\right)$}}\,{\mbox{$\cal W$}}\;\;.\end{aligned}$$ Assigning to $e(R)\in {\mbox{${\cal R}(X)$}}$ the element ${\mbox{$\varepsilon\left(R\right)$}}:={\mbox{$\varepsilon\left(R\right)$}}1\in{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$, the monoid $(\,{\mbox{${\cal R}(X)$}}\,,\,\cap \,)$ embedds into ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$. Its image are the idempotents of ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$.\
For $J\subseteq I$ denote by ${\mbox{$\cal W$}}^J$ the minimal coset representatives of ${\mbox{$\cal W$}}/{\mbox{$\cal W$}}_J$, and denote by ${\mbox}{}^J{\mbox{$\cal W$}}$ the minimal coset representatives of ${\mbox{$\cal W$}}_J\backslash {\mbox{$\cal W$}}$. It is easy to see that there are the following uniquely determined normal forms of an element $\hat{\sigma}\in {\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$: $$\begin{aligned}
\hat{\sigma}\;\,=\;\, \sigma_1 {\mbox{$\varepsilon\left(R({\Theta})\right)$}} \sigma_2 &{\mbox}{ with }&
{\Theta}{\mbox}{ special }\,,\;\sigma_1\in {\mbox{$\cal W$}}^{{\Theta}\cup{\Theta}^\bot}\,,\;
\sigma_2\in{\mbox}{}^{{\Theta}}{\mbox{$\cal W$}}\;. \\
\hat{\sigma}\;\,=\;\, \tau_1 {\mbox{$\varepsilon\left(R({\Theta})\right)$}} \tau_2 &{\mbox}{ with }&
{\Theta}{\mbox}{ special }\,,\;\tau_1\in {\mbox{$\cal W$}}^{{\Theta}}\,,\;
\tau_2\in{\mbox}{}^{{\Theta}\cup{\Theta}^\bot}{\mbox{$\cal W$}}\;.\end{aligned}$$ Let $J\subseteq I$. We call the submonoid ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_J$, which is generated by ${\mbox{$\cal W$}}_J$ and the elements ${\mbox{$\varepsilon\left(R({\Theta})\right)$}}$, ${\Theta}\subseteq J$ special, a parabolic submonoid. Denote by $J^\infty$ the union of all connected components of nonfinite type of $J$. We have $$\begin{aligned}
{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_J \;\,=\;\, \dot{\bigcup_{R\,a\,face\,of\,X\atop R\supseteq R(J^\infty)}} {\mbox{$\cal W$}}_J \,{\mbox{$\varepsilon\left(R\right)$}}\;\,=\;\,
\dot{\bigcup_{\Xi \,special\atop \,\Xi\subseteq J^\infty}} {\mbox{$\cal W$}}_J \,{\mbox{$\varepsilon\left(R(\Xi)\right)$}}\,{\mbox{$\cal W$}}_J\;\;.\end{aligned}$$ We get an abelian submonoid of ${\mbox{$\widehat{G}$}}$ by ${\mbox{$\widehat{T}$}}:=\dot{\bigcup}_{R\in{\cal R}(X)} T e(R) $. We get a submonoid of ${\mbox{$\widehat{G}$}}$ by ${\mbox{$\widehat{N}$}}:=\dot{\bigcup}_{R\in{\cal R}(X)} N e(R)$. Define a congruence relation on ${\mbox{$\widehat{N}$}}$ as follows: $$\begin{aligned}
\quad\hat{n}\:\sim\:\hat{n}'\quad:\iff\quad\hat{n}T\:=\:\hat{n}'T\quad\iff \quad\hat{n}'\:\in\:\hat{n}T\quad\iff\quad\hat{n}\in\hat{n}'T\end{aligned}$$ The Weyl monoid ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}$ is isomorphic to the monoid ${\mbox{$\widehat{N}$}}/T$, an isomorphism $\kappa:{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}\to {\mbox{$\widehat{N}$}}/T$ given by $\kappa(\sigma{\mbox{$\varepsilon\left(R\right)$}})=n_\sigma e(R) T$.\
${\mbox{$\widehat{G}$}}$ has [*Bruhat*]{} and [*Birkhoff decompositions*]{}: $$\begin{aligned}
{\mbox{$\widehat{G}$}}&=& \dot{ \bigcup_{\hat{n}\in \widehat{N}}}\,\; U^\epsilon\,\hat{n}\, U^\delta \;\,=\;\,
\dot{ \bigcup_{\hat{\sigma}\in \widehat{\cal W}}}\,\; B^\epsilon\,\hat{\sigma}\, B^\delta \quad,\quad \epsilon,\,\delta \;\in \;\{\,+\,,\,-\,\}\;\;.\end{aligned}$$ [**The coordinate ring of ${\mbox{$\widehat{G}$}}$, and the algebra of strongly regular functions:**]{} For a module $V$ of ${\cal O}_{adm}$, $v,w\in V$, and ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$ a nondegenerate symmetric contravariant bilinear form on $V$, call the function ${\tilde}{f}_{vw}:\,{\mbox{$\widehat{G}$}}\to{\mathbb{F}}$ defined by ${\tilde}{f}_{vw}(x):={\left\langle \left\langle v\mid xw
\right\rangle \right\rangle}$, $x\in{\mbox{$\widehat{G}$}}$, a [*matrix coefficient*]{} of ${\mbox{$\widehat{G}$}}$. The set of all such matrix coefficients ${\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$ is an algebra. It is an integral domain, and admits a [*Peter-Weyl theorem*]{}: Equip ${\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$ with an action $\pi$ of ${\mbox{$\widehat{G}$}}\times {\mbox{$\widehat{G}$}}$, and an involutive automorphism $*$ by: $$\begin{aligned}
\begin{array}{ccc}
(\pi(g,h)\,f)(x) &:=& f(g^* x h) \\
f^* (x) &:=& f(x^*)
\end{array} &, & g,x,h\in {\mbox{$\widehat{G}$}}\,,\quad f\in{\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}\;\;.\end{aligned}$$ For every ${\Lambda}\in P^+$ fix a nondegenerate symmetric contravariant bilinear form on $L({\Lambda})$. The map $\bigoplus_{{\Lambda}\in P^+} L({\Lambda})\otimes L({\Lambda}) \to {\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$ induced by $v\otimes w\mapsto {\tilde}{f}_{vw}\,$ is an isomorphism of ${\mbox{$\widehat{G}$}}\times {\mbox{$\widehat{G}$}}$-modules. It identifies the direct sum of the switch maps of the factors with the involution.\
The monoids ${\mbox{$\widehat{T}$}}$, ${\mbox{$\widehat{N}$}}$, ${\mbox{$\widehat{G}$}}$ are the Zariski closures of $T$, $N$, $G$, and $G$ is the Zariski open dense unit group of ${\mbox{$\widehat{G}$}}$.\
The algebra of [*strongly regular functions*]{} ${\mbox{${\mathbb{F}}\,[G]$}}$ is obtained by restricting the functions of ${\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$ onto $G$. The restriction map is an isomorphism from ${\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$ to ${\mbox{${\mathbb{F}}\,[G]$}}$.\
Restricting the functions of ${\mbox{${\mathbb{F}}\,[G]$}}$ onto $G'$, resp. $T_{rest}$ gives the algebras ${\mbox{${\mathbb{F}}\,[G']$}}$, resp. ${\mbox{${\mathbb{F}}\,[T_{rest}]$}}$, the first identical with the algebra of strongly regular functions as defined in [@KP2], the second the classical coordinate ring of the torus $T_{rest}\,$. ${\mbox{${\mathbb{F}}\,[G]$}}$ is isomorphic to ${\mbox{${\mathbb{F}}\,[G']$}}\otimes {\mbox{${\mathbb{F}}\,[T_{rest}]$}}$, by the comorphism dual to the multiplication map $G'\times T_{rest}\to G\,$.\
[**Substructures:**]{} For $\emptyset\neq J\subseteq I$ the submatrix $A_J:=(a_{ij})_{i,j\in J}$ of $A$ is a generalized Cartan matrix. There exist saturated sublattices $H(A_J)\subseteq H$, $P(A_J)\subseteq P$ with $(h_j)_{j\in J}\subseteq H(A_J)$, $({\alpha}_j)_{j\in J}\subseteq P(A_J)$, giving a simply connected minimal free realization of $A_J$. We have $P=P(A_J)\oplus H(A_J)^\bot$, and the projections of ${\Lambda}_j$, $j\in J$, to $P(A_J)$ are a system of fundamental dominant weights.\
The corresponding Kac-Moody algebra ${\mbox{$\bf g$}}(A_J)$ embeds in ${\mbox{$\bf g$}}$. If we identify ${\mbox{$\bf g$}}(A_J)$ with its image, then the set of roots ${\mbox{$\Delta$}}(A_J)$ identifies with ${\mbox{$\Delta$}}_J:={\mbox{$\Delta$}}\cap \sum_{j\in J}{\mathbb{Z}}\,{\alpha}_j$, and the Weyl group ${\mbox{$\cal W$}}(A_J)$ identifies with the parabolic subgroup ${\mbox{$\cal W$}}_J$.\
The face lattice of the Tits cone of $A_J$ embeds onto a sublattice of the face lattice of the Tits cone of $A$, and the Weyl monoid $\widehat{{\mbox{$\cal W$}}(A_J)}$ identifies with the parabolic submonoid ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_J$.\
The minimal and formal Kac-Moody groups $G(A_J)$, $G_f(A_J)$, the monoid $\widehat{G(A_J)}$ embed in $G$, $G_f$, ${\mbox{$\widehat{G}$}}$ in the obvious way.\
The images of these embedding depend on the choice of the sublattice $H(A_J)$, only $H_J:={\mathbb{Z}}{\mbox}{-}span{\left\{\left.\;h_j\;\right|\; j\in J\;\right\}}$ is uniquely determined by $A_J$. Denote by $\widehat{G'}$ the submonoid of ${\mbox{$\widehat{G}$}}$, which is generated by $G'$ and the elements $e(R)$, $R$ a face of $X$. The images of ${\mbox{$\bf g$}}(A_J)'$, $G(A_J)'$, $G_f(A_J)'$, and $\widehat{G(A_J)'}$ are independent of this choice, and denoted by ${\mbox{$\bf g$}}_J$, $G_J$, $(G_f)_J$, and $\widehat{G}_J$.\
An admissible irreducible highest weight module $L({\Lambda})$ of ${\mbox{$\bf g$}}$, equipped with a nondegenerate contravariant symmetric bilinear form, decomposes as a ${\mbox{$\bf g$}}(A_J)$-module into an orthogonal direct sum of admissible irreducible highest weight modules of ${\mbox{$\bf g$}}(A_J)$, which are ${\mbox{\textbf{h}}}$-invariant. In particular $L_J({\Lambda}):=U({\mbox{\textbf{n}}}_J^-)L({\Lambda})_{\Lambda}$ is an admissible irreducible highest weight module of ${\mbox{$\bf g$}}(A_J)$, its highest weight given by the projection of ${\Lambda}$ to $P(A_J)$, (${\Lambda}\in P^+$).\
The coordinate rings ${\mbox{${\mathbb{F}}\,[\widehat{G(A_J)'}]$}}$, ${\mbox{${\mathbb{F}}\,[G(A_J)']$}}$ identify with the restrictions of ${\mbox{${\mathbb{F}}\,[{\mbox{$\widehat{G}$}}]$}}$, ${\mbox{${\mathbb{F}}\,[G]$}}$ to ${\mbox{$\widehat{G}$}}_J$, $G_J$. (A similar statement for ${\mbox{${\mathbb{F}}\,[\widehat{G(A_J)}]$}}$, ${\mbox{${\mathbb{F}}\,[G(A_J)]$}}$ is not valid.)\
To simplify the notation of many formulas, it is useful to set ${\mbox{$\bf g$}}_\emptyset:=\{0\}$, and $G_\emptyset:={\mbox{$\widehat{G}$}}_{\emptyset}:=(G_f)_\emptyset:=\{1\}$.\
For $M\subseteq {\mbox{$\widehat{G}$}}$ and $J\subseteq I$ set $M_J:=M\cap {\mbox{$\widehat{G}$}}_J$, and similarly for $M\subseteq G_f$ set $M_J:=M\cap (G_f)_J$.
An easy algebraic geometric setting \[Setting\]
===============================================
In this section we develop an easy algebraic geometric setting, which is useful to determine the ${\mathbb{F}}$-valued points of the algebra of strongly regular functions.\
This section is a complement to [@M], Section 3, but it can be read independently. The definition of a morphism in [@M], Section 3, is more restrictive, because it has to preserve an extra structure, which we do not need here.\
We will consider certain nonempty sets $A$ equipped with point separating algebras of functions ${\mbox{${\mathbb{F}}\,[A]$}}$, which we call coordinate rings.\
The closed sets of the Zariski topology on such a set $A$ are given by the zero sets of the functions of ${\mbox{${\mathbb{F}}\,[A]$}}$. Note that $A$ is irreducible if and only if ${\mbox{${\mathbb{F}}\,[A]$}}$ is an integral domain.\
A morphism of sets with coordinate rings ($A,{\mbox{${\mathbb{F}}\,[A]$}}$) and ($B,{\mbox{${\mathbb{F}}\,[B]$}}$) consists of a map $\phi:A\to B$, whose comorphism $\phi^*:{\mbox{${\mathbb{F}}\,[B]$}}\to{\mbox{${\mathbb{F}}\,[A]$}}$ exists. In particular a morphism is Zariski continuous.\
If ($B,{\mbox{${\mathbb{F}}\,[B]$}}$) is a set with coordinate ring, and $A$ is a nonempty subset of $B$, then we get a coordinate ring on $A$ by restricting the functions of ${\mbox{${\mathbb{F}}\,[B]$}}$ to $A$.\
If ($A,{\mbox{${\mathbb{F}}\,[A]$}}$) is a set with coordinate ring and $f\in{\mbox{${\mathbb{F}}\,[A]$}}\setminus\{0\}$, the principal open set $D_A(f):={\left\{\left.\;a\in A\;\right|\; f(a)\neq 0\;\right\}}$ is equipped with a coordinate ring by identifying the localization ${\mbox{${\mathbb{F}}\,[A]$}}_f$ in the obvious way with an algebra of functions on $D_A(f)$. If $A$ is irreducible, then also $D_A(f)$ is irreducible.\
If ($A,{\mbox{${\mathbb{F}}\,[A]$}}$) and ($B,{\mbox{${\mathbb{F}}\,[B]$}}$) are sets with coordinate rings, then $A\times B$ is equipped with a coordinate ring by identifying the tensor product ${\mbox{${\mathbb{F}}\,[A]$}}\otimes{\mbox{${\mathbb{F}}\,[B]$}}$ in the obvious way with an algebra of functions on $A \times B$. If $A$ and $B$ are irreducible, then also $A\times B$ is irreducible.\
To construct the sets with coordinate rings, which we will use later, fix nondegenerate symmetric contravariant forms ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$ on all modules $L({\Lambda})$, ${\Lambda}\in P^+$, and extend to a form on $\bigoplus_{{\Lambda}\in P^+} L({\Lambda})$, also denoted by ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$, by requiring $L({\Lambda})$ and $L({\Lambda}')$ to be orthogonal for ${\Lambda}\neq{\Lambda}'$. For $v, w\in L({\Lambda})$, ${\Lambda}\in P^+$, define a linear function $$\begin{aligned}
f_{v w}:\;{\mbox{$End$}\left(\,\bigoplus_{{\Lambda}\in P^+}\,L({\Lambda})\,
\right)}&\to & {\mathbb{F}}\end{aligned}$$ by $f_{vw}(\phi):={\left\langle \left\langle v\mid \phi w
\right\rangle \right\rangle}$, where $\phi\in {\mbox{$End$}\left(\,\bigoplus_{{\Lambda}\in P^+}\,L({\Lambda})\,
\right)}$.\
We equip the subalgebra $$\begin{aligned}
{\mbox{$gr$-$Adj$}}&:=& {\left\{\left.\;\,\phi\in{\mbox{$End$}\left(\,\bigoplus_{{\Lambda}\in P^+}\,L({\Lambda})\,
\right)}\;\;\right|\; \;\begin{array}{c} {\mbox}{The adjoint }\;\phi^* \;{\mbox}{ exists and}\\
\phi \left(L({\Lambda})\right)\subseteq L({\Lambda}),\;{\Lambda}\in P^+\;.
\end{array}\,\;\right\}} \end{aligned}$$ of the algebra of endomorphisms with the coordinate ring ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$Adj$}}]$}}$, which is generated by the linear subspace $$\begin{aligned}
span{\left\{\left.\;f_{v w}{\!\mid_{gr{\mbox}{-}Adj}}\,\;\right|\; \,v,w\in L({\Lambda})\;\,,\,\; {\Lambda}\in P^+\;\;\right\}}\end{aligned}$$ of the linear dual of ${\mbox{$gr$-$Adj$}}$. This coordinate ring is isomorphic to the symmetric algebra in this subspace.\
The multiplication map is no morphism, but the left and right multiplications with elements of ${\mbox{$gr$-$Adj$}}$, and also the adjoint map $*:{\mbox{$gr$-$Adj$}}\to{\mbox{$gr$-$Adj$}}$ are morphisms.\
For a set $M\subseteq {\mbox{$gr$-$Adj$}}$ we denote by ${\overline{M}}$ its Zariski closure in ${\mbox{$gr$-$Adj$}}$. If $M$ is a ($*$-invariant) submonoid of ${\mbox{$gr$-$Adj$}}$, then also ${\overline{M}}$ is a ($*$-invariant) submonoid. The left and right translations with elements of ${\overline{M}}$ (and the map $*:{\overline{M}}\to {\overline{M}}$) are morphisms.\
A function $f\in{\mbox{${\mathbb{F}}\,[M]$}}$ induces a function on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$, assigning $x\in {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ the value $x(f)$. In this way, ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ is equipped with a coordinate ring isomorphic to ${\mbox{${\mathbb{F}}\,[M]$}}$. Its Zariski topology coincides with the relative topology induced by the topology of the spectrum of ${\mbox{${\mathbb{F}}\,[M]$}}$.\
To investigate ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$, we introduce two new monoids ${\overline{\overline{M}}}$, ${\overline{\overline{M^*}}}$ with coordinate rings.\
Equip the subalgebra $$\begin{aligned}
{\mbox{$gr$-$End$}}&:=& {\left\{\left.\;\,\phi\in{\mbox{$End$}\left(\,\bigoplus_{{\Lambda}\in P^+}\,L({\Lambda})\,
\right)}\;\;\right|\; \;\phi \left(L({\Lambda})\right)\subseteq L({\Lambda}),\;{\Lambda}\in P^+\;.\,\;\right\}} \end{aligned}$$ of the algebra of endomorphisms with the coordinate ring ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}$ generated by the linear subspace $$\begin{aligned}
span{\left\{\left.\;f_{v w}{\!\mid_{gr{\mbox}{-}End}}\,\;\right|\; \,v,w\in L({\Lambda})\;\,,\,\; {\Lambda}\in P^+\;\;\right\}}\end{aligned}$$ of the linear dual of ${\mbox{$gr$-$End$}}$, which is isomorphic to $\bigoplus_{{\Lambda}\in P^+}L({\Lambda})\otimes L({\Lambda})$. This coordinate ring is isomorphic to the symmetric algebra in this subspace. The restriction of ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}$ to ${\mbox{$gr$-$Adj$}}$ coincides with the coordinate ring of ${\mbox{$gr$-$Adj$}}$.\
The right multiplication with an element $\phi\in {\mbox{$gr$-$End$}}$ is a morphism. We can guarantie the left multiplication $l_\phi$ to be a morphism, only if we restrict to $\phi\in{\mbox{$gr$-$Adj$}}$, because then we have $$\begin{aligned}
f_{vw}{\!\mid_{gr{\mbox}{-}End}}\,\circ\: l_\phi \;\,=\;\,f_{\phi^*v\, w}{\!\mid_{gr{\mbox}{-}End}}\;\;{\mbox}{ for all }\;\; v,w\in L({\Lambda})\;,\;{\Lambda}\in P^+.\end{aligned}$$ For a set $M\subseteq {\mbox{$gr$-$End$}}$ we denote by ${\overline{\overline{M}}}$ its Zariski closure in ${\mbox{$gr$-$End$}}$. If $M$ is a submonoid of ${\mbox{$gr$-$End$}}$, then ${\overline{\overline{M}}}$ is not necessarily a submonoid of ${\mbox{$gr$-$End$}}$. But we have:
\[Setting1\] If $M$ is a submonoid of ${\mbox{$gr$-$Adj$}}$, then ${\overline{\overline{M}}}$ is a submonoid of ${\mbox{$gr$-$End$}}$.
Also the following proposition is easy to prove:
\[Setting1b\] Let $M$ be a submonoid of ${\mbox{$gr$-$Adj$}}$. For $\phi\in {\overline{\overline{M^*}}}$ and $\psi\in{\overline{\overline{M}}}$ we get a homomorphism of algebras $\pi(\phi,\psi):{\mbox{${\mathbb{F}}\,[M]$}}\to{\mbox{${\mathbb{F}}\,[M]$}}$ by $$\begin{aligned}
\pi(\phi,\psi)\left(f_{v w}{\!\mid_{M}}\right) \;:=\; f_{\phi v\,\psi w}{\!\mid_{M}} &\quad,\quad & v,w\in L({\Lambda})\;,\;{\Lambda}\in P^+\;\;.\end{aligned}$$ $\pi$ is an action of ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ on ${\mbox{${\mathbb{F}}\,[M]$}}$.
Due to this proposition we get an action of ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ from the right by $$\begin{aligned}
{\alpha}\circ\pi(\phi,\psi) \;\,,\quad{\mbox}{ where }\;\, {\alpha}\in{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}\;\,,\,\;\phi\in {\overline{\overline{M^*}}}\;\,,\;\,\psi\in {\overline{\overline{M}}}\;\;.\end{aligned}$$ This is an action by morphisms of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$. But in general the map $({\overline{\overline{M^*}}}\times{\overline{\overline{M}}})\times{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ is no morphism. In general also the right translations of this map are no morphisms.\
The next Proposition, which is easy to prove, describes the part of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ obtained by applying ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ to the evaluation map of ${\mbox{${\mathbb{F}}\,[M]$}}$ in the unit of $M$.
\[Setting2\] Let $M$ be a submonoid of ${\mbox{$gr$-$Adj$}}$.\
1) For $\phi\in{\overline{\overline{M^*}}}$ and $\psi\in{\overline{\overline{M}}}$ we get a point ${\alpha}(\phi,\psi)\in{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ by $$\begin{aligned}
{\alpha}(\phi,\psi)(f_{vw}{\!\mid_{M}})\;\,:=\;\,{\left\langle \left\langle \phi v\mid \psi w
\right\rangle \right\rangle} \;\,{\mbox}{ for all }\;\; v,w\in L({\Lambda})\,,\; {\Lambda}\in P^+\;\;.\end{aligned}$$ If we equip ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ with the right action on itself, then the map ${\alpha}:{\overline{\overline{M^*}}}\times{\overline{\overline{M}}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ is equivariant. Furthermore we have $$\begin{aligned}
{\alpha}(\phi,z\psi)\;\,=\,\;{\alpha}(z^*\phi,\psi) &{\mbox}{ for all }& \phi\in{\overline{\overline{M^*}}}\;,\;\psi\in{\overline{\overline{M}}}\;,\;{\mbox}{ and }\;
z\in\,{\overline{M}}\;\;.\end{aligned}$$ 2) For every $\phi\in {\overline{\overline{M^*}}}$ the map ${\alpha}(\phi,\,\cdot\,):\,{\overline{\overline{M}}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$, which assigns $\psi$ the point ${\alpha}(\phi,\psi)$, is a morphism. The map ${\alpha}(1,\,\cdot\,)$ is injective.\
For every $\psi\in {\overline{\overline{M}}}$ the map ${\alpha}(\,\cdot\,,\psi):\,{\overline{\overline{M^*}}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ which assigns $\phi$ the point ${\alpha}(\phi,\psi)$, is a morphism. The map ${\alpha}(\,\cdot\,,1)$ is injective.
[**Remarks:**]{}\
1) We often say ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ maps to ${\mbox{${\mathbb{F}}\,[M]$}}$, without mentioning the map ${\alpha}$. In general this map is no morphism.\
2) The maps $\{1\}\times {\overline{\overline{M}}}\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$, and ${\overline{\overline{M^*}}}\times \{1\}\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ are injective. But if $M$ is nontrivial, then the map ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ is not injective:\
Let $\sim$ be the equivalence relation on ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$ generated by $$\begin{aligned}
(z^*x,y) \,\sim\, (x,zy) &\quad,\quad & z\,\in\, {\overline{M}} \;\;.\end{aligned}$$ Equip the quotient set, which we denote by ${\overline{\overline{M^*}}}\times_{{\overline{M}}}{\overline{\overline{M}}}$, with the induced ${\overline{\overline{M^*}}}\times{\overline{\overline{M}}}$-action from the right. Then ${\alpha}$ factors to an equivariant map $$\begin{aligned}
{\tilde}{{\alpha}}\,:\,{\overline{\overline{M^*}}}\times_{{\overline{M}}}{\overline{\overline{M}}} &\to & {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}\;\;.\end{aligned}$$ 3) The map $*:M\to M^*$ is an isomophism. Its comorphism ${\mbox}{}^*:{\mbox{${\mathbb{F}}\,[M^*]$}}\to{\mbox{${\mathbb{F}}\,[M]$}}$ is given by $$\begin{aligned}
(f_{v w}{\!\mid_{M^*}})^* & = & f_{w v}{\!\mid_{M}}\;\,{\mbox}{ for all } \;\,v,w\in L({\Lambda})\;\,, \;\,{\Lambda}\in P^+\;\;.\end{aligned}$$ We omit to state the easy compatibility conditions with the maps ${\alpha}$, $\pi$, which correspond to ${\mbox{${\mathbb{F}}\,[M]$}}$, and ${\mbox{${\mathbb{F}}\,[M^*]$}}$.\
The properties stated in the second part of the last proposition are sufficient to guarantie the irreducibility of certain orbits, belonging to the action of the product of a irreducible subgroup of ${\overline{\overline{M^*}}}$ and a irreducible subgroup of ${\overline{\overline{M}}}$.
\[SettingirreducibleOrbits\] Let $M$ be a submonoid of ${\mbox{$gr$-$Adj$}}$, and $x\in{\overline{M}}$. Let $D_1$ be an irreducible subgroup of ${\overline{\overline{M^*}}}$, let $D_2$ be an irreducible subgroup of ${\overline{\overline{M}}}$. Then the $D_1\times D_2$-orbit of the element ${\alpha}(1,x)\in {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ is irreducible.
Let $Or$ be the $D_1\times D_2$-orbit of ${\alpha}(1,x)$, i.e., $Or={\alpha}(D_1,x D_2)={\alpha}(x^* D_1, D_2)$. Let $A_1$ and $A_2$ be closed subsets of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$, such that $Or\subseteq A_1\cup A_2$. We have to show $Or\subseteq A_1$ or $Or\subseteq A_2$.\
Let $d_1\in D_1$. Due to the last proposition the map $\gamma_{d_1}:D_2\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ defined by $\gamma_{d_1}(d_2):={\alpha}(x^* d_1, d_2)$, $d_2\in D_2$, is a morphism.\
Similarly, for $d_2\in D_2$, the map $\delta_{d_2}:D_1\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[M]$}}$ defined by $\delta_{d_2}(d_1) := {\alpha}(d_1,x d_2)$, $d_1\in D_1$, is a morphism.\
Let $d_1\in D_1$. Because of $\gamma_{d_1}(D_2)\subseteq Or$, we have $\gamma_{d_1}^{-1}(A_1)\cup \gamma_{d_1}^{-1}(A_2)=D_2$. Furthermore $\gamma_{d_1}^{-1}(A_1)$ and $\gamma_{d_1}^{-1}(A_2)$ are closed. Because of the irreducibility of $D_2$ we get $\gamma_{d_1}^{-1}(A_1)=D_2$ or $\gamma_{d_1}^{-1}(A_2)=D_2$.\
Therefore the sets $$\begin{aligned}
B_1 &:=& {\left\{\left.\;d_1\in D_1\;\right|\; \gamma_{d_1}^{-1}(A_1)=D_2\;\right\}}\;\;,\\
B_2 &:=& {\left\{\left.\;d_1\in D_1\;\right|\; \gamma_{d_1}^{-1}(A_2)=D_2\;\right\}}\end{aligned}$$ satisfy $B_1\cup B_2 =D_1$.\
Note that for $d_1\in D_1$ and $d_2\in D_2$ we have $\gamma_{d_1}(d_2)=\delta_{d_2}(d_1)$. The set $B_1$ is closed, because of $$\begin{aligned}
B_1 &=& {\left\{\left.\;d_1\in D_1\;\right|\; \gamma_{d_1}(d_2)\in A_1 \;{\mbox}{ for all }\; d_2\in D_2\;\right\}}\;\;=\;\;\bigcap_{d_2\in D_2}\underbrace{\delta_{d_2}^{-1}(A_1)}_{closed}\;\;.\end{aligned}$$ Similarly, the set $B_2$ is closed. Because of the irreducibility of $D_1$ we get $B_1=D_1$ or $B_2=D_1$, which is equivalent to $Or\subseteq A_1$ or $Or\subseteq A_2$.\
[\
]{}We have ${\overline{\overline{{\mbox{$gr$-$Adj$}}}}}={\mbox{$gr$-$End$}}$. Due to Proposition \[Setting2\] we get a map ${\alpha}:{\mbox{$gr$-$End$}}\times {\mbox{$gr$-$End$}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$Adj$}}]$}}$. The following technical proposition, which follows immediately from the definition of the Zariski closures in ${\mbox{$gr$-$End$}}$, is very useful for determining such closures:
\[Setting3\] Let $M$ be a submonoid of ${\mbox{$gr$-$Adj$}}$. We have: $$\begin{aligned}
{\overline{\overline{M}}} &=& {\left\{\left.\;\phi\in{\mbox{$gr$-$End$}}\;\right|\; {\alpha}(1,\phi) {\mbox}{ factors to a homomorphism } {\mbox{${\mathbb{F}}\,[M]$}}\to{\mathbb{F}}\;\right\}}\;\;.\\
{\overline{\overline{M^*}}} &=& {\left\{\left.\;\phi\in{\mbox{$gr$-$End$}}\;\right|\; {\alpha}(\phi, 1) {\mbox}{ factors to a homomorphism } {\mbox{${\mathbb{F}}\,[M]$}}\to{\mathbb{F}}\;\right\}}\;\;.\\\end{aligned}$$
The formal Kac-Moody group $G_f$ acts faithfully on $\bigoplus_{{\Lambda}\in P^+}L({\Lambda})$. We identify $G_f$ with the corresponding subgroup of ${\mbox{$gr$-$End$}}$. Under this identification the minimal Kac-Moody group $G\subseteq G_f$ is identified with a subgroup of ${\mbox{$gr$-$Adj$}}$, invariant under taking the adjoint.\
The results of this section can be applied to the ${\mathbb{F}}$-valued points of the algebra of strongly regular functions ${\mbox{${\mathbb{F}}\,[G]$}}$, because the restriction of the coordinate ring ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$Adj$}}]$}}$ to $G$ coincides with ${\mbox{${\mathbb{F}}\,[G]$}}$.\
Also the monoid ${\mbox{$\widehat{G}$}}$ can be identified with a submonoid of ${\mbox{$gr$-$Adj$}}$. In [@M], Theorem 5.14, we showed ${\overline{G}}\,={\mbox{$\widehat{G}$}}$. In this paper we determine ${\mbox{$ {\overline{\overline{G}}} $}}$. We show the bijectivity of the map ${\tilde}{{\alpha}}\,:\,{\overline{\overline{G}}}\times_{{\overline{G}}}{\overline{\overline{G}}} \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
To prove its surjectivity, we use an induction over $|J|$, $J\subseteq I$, describing ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$. To prepare this proof, in the next two sections we first determine the ${\mathbb{F}}$-valued points of some other coordinate rings.
The ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[T_J]$}}$, ${\mbox{${\mathbb{F}}\,[T]$}}$, ${\mbox{${\mathbb{F}}\,[T_{rest}]$}}$ ($J\subseteq I$)
============================================================================================================================================================
Recall that the torus $T$ of the Kac-Moody group can be described by the following isomorphism of groups:\
$\begin{array}{ccc}
H\otimes_{\mathbb{Z}}{\mathbb{F}}^\times &\to & \;\;\:T \\
\sum_{i=1}^{2n-l}h_i\otimes s_i\;\, &\mapsto &\, \prod_{i=1}^{2n-l} t_{h_i}(s_i)
\end{array}$\
The group algebra ${\mbox{${\mathbb{F}}\,[P]$}}$ of the lattice $P$ can be identified with the classical coordinate ring on $T$, identifying $\sum c_{\lambda}e_{\lambda}\in {\mbox{${\mathbb{F}}\,[P]$}}\,$ with the function on $T$, which is defined by $$\begin{aligned}
\left(\sum_{\lambda}c_{\lambda}e_{\lambda}\right)\left(\prod_{i=1}^{2n-l}t_{h_i}(s_i)\right) \;\,:=\,\;\sum_{\lambda}\,c_{\lambda}\,
\prod_{i=1}^{2n-l}(s_i)^{{\lambda}(h_i)} &\;\,,\;\,& (s_i\in{\mathbb{F}}^\times )\;\;.\end{aligned}$$ We have similar descriptions for the tori $T_J:={\left\{\left.\;\prod_{j\in J}t_{h_j}(s_j)\;\right|\; s_j\in{\mathbb{F}}^\times \;\right\}}$, $T_{rest}$, and its classical coordinate rings, replacing the lattices $H$, $P$ by $H_J$, $P_J:={\mathbb{Z}}{\mbox}{-span}{\left\{\left.\;{\Lambda}_j\;\right|\; j\in J\;\right\}} $ or $H_{rest}$, $P_{rest}:={\mathbb{Z}}{\mbox}{-span}{\left\{\left.\;{\Lambda}_i\;\right|\; i=n+1,\ldots,2n-l\;\right\}} $, ($J\subseteq I$).\
\
In [@M], Proposition 5.1, we determined the coordinate rings ${\mbox{${\mathbb{F}}\,[T_J]$}}$, ${\mbox{${\mathbb{F}}\,[T]$}}$ and ${\mbox{${\mathbb{F}}\,[T_{rest}]$}}$. They are in general only subalgebras of the classical coordinate rings of these tori:
\[SpT1\] Let $J\subseteq I$. Let $p_J:\,P \to P_J$ be the projection defined by $p_J({\lambda}):= \sum_{j\in J} {\lambda}(h_j){\Lambda}_j$. We have:\
1) $\qquad {\mbox{${\mathbb{F}}\,[T_J]$}}\:\,=\,\:{\mbox{${\mathbb{F}}\,[\,p_J\left(X\cap P\right)]$}}\;$.\
2) $\qquad {\mbox{${\mathbb{F}}\,[T]$}}\,\:=\,\:{\mbox{${\mathbb{F}}\,[X\cap P]$}}\;$.\
3) $\qquad {\mbox{${\mathbb{F}}\,[T_{rest}]$}}\,\:=\,\:{\mbox{${\mathbb{F}}\,[P_{rest}]$}}\;$.
In [@M], Theorem 5.2, we described the relative closures of $T_J$, $T$ and $T_{rest}$ in ${\mbox{$gr$-$Adj$}}$. In the same way, but replacing ${\mbox{$gr$-$Adj$}}$ by ${\mbox{$gr$-$End$}}$, we can determine the closures of $T_J$, $T$ and $T_{rest}$. The proof of [@M], Theorem 5.2, also shows, that $\{1\}\times {\mbox{$\widehat{T}$}}_J$, $\{1\}\times {\mbox{$\widehat{T}$}}$, resp. $\{1\}\times T_{rest}$ map bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T_J]$}}$, ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T]$}}$, resp. ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T_{rest}]$}}$. Therefore we get:
\[SpT2\] Let $J\subseteq I$.\
1) We have ${\overline{\overline{T_J}}}\:=\:{\mbox{$\widehat{T}$}}_J$, and $\{1\}\times {\overline{\overline{T_J}}}$ maps bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T_J]$}}$.\
2) We have ${\mbox{$ {\overline{\overline{T}}} $}}\:=\:{\mbox{$\widehat{T}$}}$, and $\{1\}\times{\mbox{$ {\overline{\overline{T}}} $}}$ maps bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T]$}}$.\
3) $T_{rest}\,$ is closed, and $\{1\}\times T_{rest}$ maps bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[T_{rest}]$}}$.
The ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[U_J]$}}$ and ${\mbox{${\mathbb{F}}\,[U^J]$}}$ ($J\subseteq I$)\[SpU\]
=================================================================================================================================
The coordinate ring of $U$ has been described by Kac and Peterson in [@KP2], Lemma 4.3. From this Lemma follows immediately part 1) of the next theorem. Part 2) has been shown in [@M], Theorem 5.6.
\[SpU1\] Let $J\subseteq I$.\
1) Let ${\Lambda}\in F_{I\setminus J}\cap P^+$, and $v_{\Lambda}\in L({\Lambda})_{\Lambda}\setminus\{0\}\,$. Then:\
${\mbox{${\mathbb{F}}\,[U_J]$}}$ is a symmetric algebra in $\{\,f_{v_{\Lambda}yv_{\Lambda}}{\!\mid_{U_J}}\,|\,y\in {\mbox{\textbf{n}}}_J^-\,\}\,$.\
2) Let $N\in F_J\cap P^+$, and $v_N\in L(N)_N\setminus\{0\}\,$. Then:\
${\mbox{${\mathbb{F}}\,[U^J]$}}$ is as algebra generated by $\{\,f_{v_N yv_N}{\!\mid_{U^J}}\,|\,y\in ({\mbox{\textbf{n}}}^J)^-\,\}\,$.
Using these descriptions of the coordinate rings, it is possible to determine its ${\mathbb{F}}$-valued points:
\[SpU2\] Let $J\subseteq I\,$.\
1) We have ${\overline{\overline{U_J}}} = (U_f)_J$, and $\{1\}\times {\overline{\overline{U_J}}}$ maps bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U_J]$}}$.\
2) We have ${\overline{\overline{U^J}}} = (U_f)^J$, and $\{1\}\times {\overline{\overline{U^J}}}$ maps bijectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U^J]$}}$.
[**Proof of 1)**]{}: The case $J=\emptyset$ is trivial. Let $J$ be nonempty. We first show $(U_f)_J\subseteq{\overline{\overline{U_J}}}$. The coordinate ring ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}$ is a symmetric algebra in the linear span of the functions $f_{vw}{\!\mid_{gr{\mbox}{-}End}}$, $v,w\in L({\Lambda})$, ${\Lambda}\in P^+$, which is isomorphic to $\bigoplus_{{\Lambda}\in P^+}L({\Lambda})\otimes L({\Lambda})$. We get a representation of the Lie algebra $({\mbox{\textbf{n}}}_f)_J$, by assigning $x\in ({\mbox{\textbf{n}}}_f)_J$ the derivation $\delta_x: {\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}\to{\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}$, which is defined by $$\begin{aligned}
\delta_x(f_{vw}{\!\mid_{gr{\mbox}{-}End}}) \;\, := \,\; f_{v\,xw}{\!\mid_{gr{\mbox}{-}End}} &\quad,\quad & v,w\in L({\Lambda})\;\,,\,\;{\Lambda}\in P^+\;\;. \end{aligned}$$ The derivation $\delta_x$ is locally nilpotent, because $x\in({\mbox{\textbf{n}}}_f)_J$ acts locally nilpotent on $L({\Lambda})$, ${\Lambda}\in P^+$. The homomorphism of algebras $exp(\delta_x)$ satisfies $$\begin{aligned}
{\alpha}(1,1)\;\circ\; exp(\delta_x) &=& {\alpha}(1,exp(x))\;\;.\end{aligned}$$ Let $I(U_J)$ be the vanishing ideal of $U_J$ in ${\mbox{${\mathbb{F}}\,[{\mbox{$gr$-$End$}}]$}}$. Due to the last identity and Proposition \[Setting3\], it is sufficient to show $\delta_x \left( I(U_J)\right) \subseteq I(U_J)$ for all $x\in ({\mbox{\textbf{n}}}_f)_J$.\
Let $f\in I(U_J)$. Let $x\in{\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in ({\mbox{$\Delta$}}_J)^+_{re}$. For all $u\in U_J$ and $t\in{\mathbb{F}}$ we have $$\begin{aligned}
0\;=\; f(u\exp(tx)) \;=\;f(u)\,+\,t \delta_x(f)(u)\,+\,O(t^2)\;\;.\end{aligned}$$ The right side is polynomial in $t$. Due to $|\,{\mathbb{F}}\,|=\infty$ the coefficients of the powers of $t$ vanish, and we find $\delta_x(f)\in I(U_J)$.\
The elements $x\in{\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in ({\mbox{$\Delta$}}_J)^+_{re}$, generate ${\mbox{\textbf{n}}}_J$. Therefore also $\delta_x(f)\in I(U_J)$ for all $x\in{\mbox{\textbf{n}}}_J$.\
For an element $v\in L({\Lambda})$, ${\Lambda}\in P^+$, there exist only finitely many roots ${\alpha}\in{\mbox{$\Delta^+$}}$ with ${\mbox{$\bf g$}}_{\alpha}v\neq\{0\}$. Therefore for an element $x\in({\mbox{\textbf{n}}}_f)_J$ there exists an element ${\tilde}{x}\in{\mbox{\textbf{n}}}_J$, depending on $f$, such that $\delta_x(f)=\delta_{{\tilde}{x}}(f)\in I(U_J)$.\
We have $(U_f)_J\subseteq {\overline{\overline{U_J}}}$, and $\{1\}\times {\overline{\overline{U_J}}}$ maps injectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U_J]$}}$. To prove 1), it remains to show that $\{1\}\times (U_f)_J$ maps surjectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U_J]$}}$.\
Let ${\Lambda}\in F_{I\setminus J}\cap P^+$, and $v_{\Lambda}\in L({\Lambda})_{{\Lambda}}\setminus\{0\}$. Due to the description of ${\mbox{${\mathbb{F}}\,[U_J]$}}$ of the last theorem, it is sufficient to show, that for every linear map $$\begin{aligned}
l:\;{\left\{\left.\;f_{v_{\Lambda}\, yv_{{\Lambda}}}{\!\mid_{U_J}}\;\right|\; y\in {\mbox{\textbf{n}}}_J^-\;\right\}} &\to &{\mathbb{F}}\end{aligned}$$ there exists an element $\phi\in (U_f)_J$ with $l( f_{v_{\Lambda}\,yv_{\Lambda}}{\!\mid_{U_J}})={\alpha}(1,\phi)(f_{v_{\Lambda}\,y v_{\Lambda}}{\!\mid_{U_J}})$ for all $y\in{\mbox{\textbf{n}}}_J^-$.\
Choose ${\left( \;\ \mid \;\ \right)}$-dual bases of ${\mbox{\textbf{n}}}_J^+$, ${\mbox{\textbf{n}}}_J^-$, adopted to the root space decomposition: $$\begin{aligned}
\begin{array}{c}
x_{{\alpha}i}\in{\mbox{$\bf g$}}_{\alpha}\quad{\alpha}\in{\mbox{$\Delta$}}_J^+\;,\;i=1,\ldots,m_{\alpha}\\
y_{\beta i}\in{\mbox{$\bf g$}}_{-\beta}\quad\beta\in{\mbox{$\Delta$}}_J^+\;,\;i=1,\ldots,m_\beta \end{array}
&{\mbox}{ such that } & {\left(x_{{\alpha}i}\midy_{\beta j}\right)}\,=\,\delta_{\alpha \beta}\delta_{ij}\;\;.\end{aligned}$$ We have ${\left({\Lambda}\mid\beta\right)}>0$ for all $\beta\in{\mbox{$\Delta$}}_J^+$. Define recursively elements $b_{\beta j}\in{\mathbb{F}}$, $j=1,\ldots,m_\beta$, $\beta\in{\mbox{$\Delta$}}^+_J$, as follows: $$\begin{aligned}
b_{\beta 1} \;\,:=\,\; \frac{1}{(\,{\Lambda}\,|\,\beta\,){\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}}\, l(f_{v_{\Lambda}\,y_{\beta 1} v_{\Lambda}}{\!\mid_{U_J}})&{\mbox}{ for }& {\mbox{ht}}\,\beta=1\;\;.\end{aligned}$$ Let $\beta\in{\mbox{$\Delta$}}_J^+$ with ${\mbox{ht}}\,\beta>1$, and let $b_{{\alpha}i}$ be defined for all $i=1,\ldots, m_{\alpha}$, ${\alpha}\in{\mbox{$\Delta$}}_J^+$ with ${\mbox{ht}}{\alpha}<{\mbox{ht}}\beta$. Set: $$\begin{aligned}
b_{\beta j} \,\;:=\;\, \frac{1}{{\left({\Lambda}\mid\beta\right)}{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}} \left( l(f_{v_{\Lambda}\,y_{\beta j} v_{\Lambda}}{\!\mid_{U_J}}) \,-\,
{\langle \langle v_{\Lambda}\mid exp(\sum_{{\alpha},i\atop ht\,{\alpha}<ht\,\beta}b_{{\alpha}i}x_{{\alpha}i}\,)\,y_{\beta j}v_{\Lambda}\rangle \rangle} \right)\;\;.\end{aligned}$$ Next we show that we can take $\phi=\exp(\sum_{{\alpha}i}b_{{\alpha}i}x_{{\alpha}i})$. Let $\beta\in{\mbox{$\Delta$}}_J^+$. By expanding the exponential function we find for $j=1,\ldots,m_\beta$: $$\begin{aligned}
\lefteqn{ {\langle \langle v_{\Lambda}\mid exp(\sum_{{\alpha},i}b_{{\alpha}i}x_{{\alpha}i}\,)\,y_{\beta j}v_{\Lambda}\rangle \rangle}\;\,=}\\
&& {\langle \langle v_{\Lambda}\mid exp(\sum_{{\alpha},i\atop ht\,{\alpha}\,<\,ht\,\beta}b_{{\alpha}i}x_{{\alpha}i}\,)\,y_{\beta j}v_{\Lambda}\rangle \rangle} +
{\langle \langle v_{\Lambda}\mid exp(\sum_{{\alpha},i\atop ht\,{\alpha}\,=\,ht\,\beta}b_{{\alpha}i}x_{{\alpha}i}\,)\,y_{\beta j}v_{\Lambda}\rangle \rangle}\;\;.\end{aligned}$$ Denote by $\nu:{\mbox{\textbf{h}}}\to{\mbox{\textbf{h}}}^*$ the linear isomorphism induced by the invariant bilinear form ${\left( \;\ \mid \;\ \right)}$. Using $[x_{\beta i}, y_{\beta j}]=\delta_{ij}\,\nu^{-1}(\beta)$, we find that the second summand on the right side equals $b_{\beta j} {\left({\Lambda}\mid\beta\right)}{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}$. After inserting the definition of $b_{\beta j}$, the right side equals $l(f_{v_{\Lambda}\,y_{\beta j} v_{\Lambda}}{\!\mid_{U_J}})$.\
[\
]{}To prepare the proof of 2), we first show two propositions:
\[SpU3\] Let ${\Lambda}\in P^+$ and $J\subseteq I$. Then $(U_f)^J$ fixes the points of $U({\mbox{\textbf{n}}}_J^-)L({\Lambda})_{\Lambda}$.
The case $J=\emptyset$ is obvious, let $J\neq\emptyset$. If we fix an element $v\in U({\mbox{\textbf{n}}}_J^-)L({\Lambda})_{\Lambda}$, then for every element ${\tilde}{x}\in ({\mbox{\textbf{n}}}_f)^J$, there exists an element $x\in {\mbox{\textbf{n}}}^J$, such that $\exp({\tilde}{x})v=\exp(x)v$. Therefore it is sufficient to show, that every element $x\in {\mbox{\textbf{n}}}^J$ acts trivially on $U({\mbox{\textbf{n}}}_J^-)L({\Lambda})_{\Lambda}$.\
Let $v_{\Lambda}\in L({\Lambda})_{\Lambda}\setminus\{0\}$. We show by induction over $n\in{\mathbb{N}_0}$:\
$n=0\,: \quad\qquad\qquad x\, v_{\Lambda}\:=\: 0 \qquad {\mbox}{for all }\,x\in {\mbox{\textbf{n}}}^J\;$.\
$n\in{\mathbb{N}}\,:\quad\; x\, y_1\cdots y_n v_{\Lambda}\:=\: 0 \qquad {\mbox}{for all }\,x\in {\mbox{\textbf{n}}}^J,\;y_1,\,\ldots,\, y_n\in {\mbox{\textbf{n}}}_J^-\;$.\
Clearly the statement for $n=0$ is valid. The induction step from $n$ to $n+1$ follows from the equation $$\begin{aligned}
x\, y_1\,y_2\cdots y_{n+1} v_{\Lambda}&=& [x\,,\, y_1]\,y_2\cdots y_{n+1} v_{\Lambda}\;+\; y_1\,\left( x\,y_2\cdots y_{n+1} v_{\Lambda}\right)\;,\end{aligned}$$ together with $[{\mbox{\textbf{n}}}^J,{\mbox{\textbf{n}}}_J^-]\subseteq {\mbox{\textbf{n}}}^J$.\
[\
]{}
\[SpU4\] Let $J\subseteq I$, and let ${\Lambda}\in F_{I\setminus J}\cap P^+$, $N\in F_J\cap P^+$. The comorphism $m^*:{\mbox{${\mathbb{F}}\,[U]$}}\to {\mbox{${\mathbb{F}}\,[U_J]$}}\otimes{\mbox{${\mathbb{F}}\,[U^J]$}}$, dual to the multiplication map $m:U_J\times U^J\to U$, is an isomorphism of algebras. Furthermore we have: $$\begin{aligned}
m^*(f_{v_{\Lambda}\,yv_{\Lambda}}{\!\mid_{U}})\;=\; f_{v_{\Lambda}\,yv_{\Lambda}}{\!\mid_{U_J}}\,\otimes\: 1 &\quad,\quad & y\in{\mbox{\textbf{n}}}_J^- \label{U-Formel1}\;\;.\\
m^*(f_{v_N\,yv_N}{\!\mid_{U}})\;=\; 1\,\otimes\, f_{v_N\,yv_N}{\!\mid_{U^J}} &\quad,\quad & y\in({\mbox{\textbf{n}}}^J)^- \label{U-Formel2}\;\;.\end{aligned}$$
Due to [@KP2], Lemma 4.2, ${\mbox{${\mathbb{F}}\,[U]$}}$ is a Hopf algebra. Therefore for $f\in{\mbox{${\mathbb{F}}\,[U]$}}$ there exist functions $f_i,g_i\in{\mbox{${\mathbb{F}}\,[U]$}}$, $i=1,\ldots,m$, such that $f(u_1u_2)=\sum_{i=1}^m f_i(u_1)g_i(u_2)$ for all $u_1,u_2\in U$. Restricting to $u_1\in U_J$, $u_2\in U^J$ we get $f\circ m =\sum_{i=1}^{m} f_i{\!\mid_{U_J}}\otimes\: g_i{\!\mid_{U^J}}\,\in\,{\mbox{${\mathbb{F}}\,[U_J]$}}\otimes{\mbox{${\mathbb{F}}\,[U^J]$}}$.\
$m^*$ is injective, because $m$ is surjective. To show the surjectivity of $m^*$, due to Theorem \[SpU1\], it is sufficient to show the equations (\[U-Formel1\]) and (\[U-Formel2\]). Due to the last proposition we find for all $y\in {\mbox{\textbf{n}}}_J^-$, and for all $u_1\in U_J$, $u_2\in U^J\subseteq (U_f)^J$: $$\begin{aligned}
m^*(f_{v_{\Lambda}\,yv_{\Lambda}}{\!\mid_{U}})(u_1,u_2) &=& {\left\langle \left\langle v_{\Lambda}\mid u_1u_2 y v_{\Lambda}\right\rangle \right\rangle}\;\,=\;\, {\left\langle \left\langle v_{\Lambda}\mid u_1 y v_{\Lambda}\right\rangle \right\rangle} \\
&=& \left(f_{v_{\Lambda}\,yv_{\Lambda}}{\!\mid_{U_J}}\otimes\,1\,\right)(u_1,u_2)\;\;.\end{aligned}$$ For $j\in J$ the ${\alpha}_j$-string of $P(N)$ through the highest weight $N$ consists only of $N$, due to $N=\sigma_j N$. Therefore $G_J$ is contained in the stabilizer of $v_N$, and we find for $y\in ({\mbox{\textbf{n}}}^J)^-$, and for all $u_1\in U_J$, $u_2\in U^J$: $$\begin{aligned}
m^*(f_{v_N\,yv_N}{\!\mid_{U}})(u_1,u_2) &=& {\left\langle \left\langle (u_1)^*v_N\mid u_2 y v_N
\right\rangle \right\rangle}\;\,=\;\, {\left\langle \left\langle v_N\mid u_2 y v_N
\right\rangle \right\rangle} \\
&=& \left(\,1\,\otimes\, f_{v_N\,yv_N}{\!\mid_{U^J}}\right)(u_1,u_2)\;\;.\end{aligned}$$ [\
]{}[**Proof of Theorem \[SpU2\], 2):**]{} The case $J=I$ is trivial. Let $J\neq I$. We first show $(U_f)^J\subseteq {\overline{\overline{U^J}}}$: Let $\phi\in (U_f)^J$. We have $(U_f)^J\subseteq U_f= {\overline{\overline{U}}}\subseteq {\overline{\overline{G}}}$. Denote by ${\alpha}(1,\phi):{\mbox{${\mathbb{F}}\,[G]$}}\to {\mathbb{F}}$ the homomorphism of algebras corresponding to $\phi$. Denote by $I(U^J)$ the vanishing ideal of $U^J$ in ${\mbox{${\mathbb{F}}\,[G]$}}$. Because of Proposition \[Setting3\] it is sufficient to show that $I(U^J)$ is contained in the kernel of ${\alpha}(1,\phi)$.\
Let $f\in I(U^J)$. Due to the Peter and Weyl theorem $f$ is of the form $f=\sum_{i} f_{v_i w_i}{\!\mid_{G}}$. Because $\phi$ is of the form $\phi=exp(\prod_{\alpha}x_{\alpha})$, $x_{\alpha}\in ({\mbox{\textbf{n}}}^J)_{\alpha}$, ${\alpha}\in ({\mbox{$\Delta$}}^J)^+$, it is sufficient to show: $$\begin{aligned}
0 &=& \sum_i {\left\langle \left\langle v_i\mid w_i
\right\rangle \right\rangle}\;\;,\label{Nullsumme1}\\
0 &=& \sum_i {\left\langle \left\langle v_i\mid x_1 \cdots x_k w_i
\right\rangle \right\rangle} \qquad {\mbox}{ for all } x_1,\,\ldots,\, x_k\in {\mbox{\textbf{n}}}^J,\;k\in {\mathbb{N}}\;\;.\label{Nullsumme2} \end{aligned}$$ Equation (\[Nullsumme1\]) follows because of $1\in U^J$. It is sufficient to show equation (\[Nullsumme2\]) for a system of generators of ${\mbox{\textbf{n}}}^J$.\
Define recursively a multibracket for elements of ${\mbox{$\bf g$}}$: $$\begin{aligned}
[x] &:=& x \quad,\quad x \in {\mbox{$\bf g$}}\;\;,\\
{\mbox}{} [ x_{k+1},\, x_k,\,\ldots,\, x_1 ] &:=& [x_{k+1},[x_k,\,\ldots,\, x_1]] \quad,\quad x_1,\ldots ,x_k\in {\mbox{$\bf g$}},\;k\in{\mathbb{N}}\;\;.\end{aligned}$$ By an easy induction, a multibracket of the form $[y,\,x_1,\,\ldots\,x_k]$ is a linear combination of multibrackets $[x_{i_1},\ldots,x_{i_k},y]$, where $(i_1,\ldots,i_k)$ is a permutation of $(1,\ldots,k)$.\
${\mbox{\textbf{n}}}^+$ is generated by ${\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in{\mbox{$\Delta_{re}^+$}}=({\mbox{$\Delta$}}_J)^+_{re}\,\dot{\cup}\,({\mbox{$\Delta$}}^J)^+_{re}$. We have ${\mbox{\textbf{n}}}={\mbox{\textbf{n}}}_J\oplus {\mbox{\textbf{n}}}^J$ and $[{\mbox{\textbf{n}}}_J,{\mbox{\textbf{n}}}^J]\subseteq {\mbox{\textbf{n}}}^J$. All multibrackets with all elements in ${\mbox{$\bf g$}}_{\alpha}$, ${\alpha}\in({\mbox{$\Delta$}}_J)^+_{re}$ are in ${\mbox{\textbf{n}}}_J$, all multibrackets with an element in ${\mbox{$\bf g$}}_\beta$, $\beta\in ({\mbox{$\Delta$}}^J)^+_{re}$ are in ${\mbox{\textbf{n}}}^J$. Therefore ${\mbox{\textbf{n}}}^J$ is generated by all multibrackets, which contain an element in ${\mbox{$\bf g$}}_\beta$, $\beta\in ({\mbox{$\Delta$}}^J)^+_{re}$. By the remark of above it is sufficient to consider the multibrackets $$\begin{aligned}
[x_{\gamma_m},\,\cdots,\, x_{\gamma_1},\, x_{\gamma_0}] \end{aligned}$$ with $x_{\gamma_i}\in {\mbox{$\bf g$}}_{\gamma_i}$, $i=0,\ldots,m$, and $\gamma_1,\,\ldots,\,\gamma_m\in {\mbox{$\Delta_{re}^+$}}$, $\gamma_0\in ({\mbox{$\Delta$}}^J)^+_{re}$, $m\in{\mathbb{N}}$.\
$U^J$ is normal in $U$. Therefore for $t_0,\,\ldots,\, t_m\in {\mathbb{F}}$ we have $$\begin{aligned}
\lefteqn{u(t_m,\ldots,t_1,t_0)\;\,:=\;\,}\\
&& \exp(t_m x_{\gamma_m})\cdots \exp(t_1 x_{\gamma_1})exp(t_0 x_{\gamma_0})\exp(-t_1 x_{\gamma_1})\cdots \exp(-t_m x_{\gamma_m})
\;\,\in\;\,U^J\;\;. \end{aligned}$$ We consider a product $p$ of $k$ factors of such expressions in different variables. To simplify our notation, we only write down a factor in the middle of $p$: $$\begin{aligned}
p &=& \cdots\,u(t_m,\ldots,t_1,t_0) \,\cdots \quad {\mbox}{ where }\quad \ldots,\, t_m,\,\ldots,\,t_1,\,t_0,\,\ldots \in {\mathbb{F}}\;\;.\end{aligned}$$ Because of $p\in U^J$ we get $$\begin{aligned}
0 \;\,=\;\, f(p)\;\,=\;\,\sum_i {\left\langle \left\langle v_i\mid \,\cdots \,u(t_m,\,\ldots,\,t_1,\,t_0)\,\cdots \,w_i
\right\rangle \right\rangle}\;\;. \end{aligned}$$ Because the root vectors belonging to real roots act locally nilpotent, the right side is polynomial in $\cdots\,t_0,t_1,\ldots, t_m\,\cdots \,$. Because of $|{\mathbb{F}}|=\infty$ the coefficients of the monomials are zero. In particular for the monomial $\cdots\,t_m\cdots t_1 t_0\,\cdots$ we find $$\begin{aligned}
\sum_i {\left\langle \left\langle v_i\mid \,\cdots\,[x_{\gamma_m},\,\ldots,\,x_{\gamma_1},\, x_{\gamma_0}]\,\cdots\, w_i
\right\rangle \right\rangle}\;\,=\;\,0\;\;.\end{aligned}$$ Now we show $(U_f)^J\supseteq {\overline{\overline{U^J}}}$: Let $u\in {\overline{\overline{U^J}}}$. Denote by ${\alpha}$ the map ${\overline{\overline{U^-}}}\times{\overline{\overline{U}}} \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U]$}}$, and by ${\tilde}{{\alpha}}$ the maps ${\overline{\overline{U_J^-}}}\times{\overline{\overline{U_J}}} \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U_J]$}}$, ${\overline{\overline{(U^J)^-}}}\times{\overline{\overline{U^J}}} \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[U^J]$}}$. Using the last proposition and its notation, and 1), there exists an element ${\tilde}{u}\in U_f$ such that $$\begin{aligned}
\label{altiutialu}
{\alpha}(1, {\tilde}{u}) &=& \left({\tilde}{{\alpha}}(1,1)\,\otimes\,{\tilde}{{\alpha}}(1,u) \right)\,\circ\,m^*\;\;.\end{aligned}$$ Write ${\tilde}{u}$ in the form $u_J u^J$ with $u_J\in (U_f)_J$, $u^J\in (U_f)^J$. Choose elements ${\Lambda}\in F_{I\setminus J}\cap P^+$ and $v_{\Lambda}\in L({\Lambda})_{\Lambda}\setminus\{0\}$, and apply the left and right sides of the last equation to the elements $f_{v_{\Lambda}\, yv_{\Lambda}}{\!\mid_{U}}$, $y\in{\mbox{\textbf{n}}}_J^-$. Using equation (\[U-Formel1\]) of the last proposition we get $$\begin{aligned}
{\left\langle \left\langle v_{\Lambda}\mid u_J u^J y v_{\Lambda}\right\rangle \right\rangle} \;\,=\;\, {\left\langle \left\langle v_{\Lambda}\mid y v_{\Lambda}\right\rangle \right\rangle} &{\mbox}{ for all }& y\in{\mbox{\textbf{n}}}_J^-\;\;.\end{aligned}$$ Due to Proposition \[SpU3\] the left side is equal to ${\left\langle \left\langle v_{\Lambda}\mid u_J y v_{\Lambda}\right\rangle \right\rangle}$. Using Theorem \[SpU1\], 1), and 1) we conclude $u_J=1$.\
Insert ${\tilde}{u}=u^J$ in equation (\[altiutialu\]). Choose elements $N\in F_J\cap P^+$ and $v_N\in L(N)_N\setminus\{0\}$, and apply the left and right sides of (\[altiutialu\]) to the elements $f_{v_N\, yv_N}{\!\mid_{U}}$, $y\in({\mbox{\textbf{n}}}^J)^-$. Using equation (\[U-Formel2\]) of the last proposition we find $$\begin{aligned}
{\left\langle \left\langle v_N\mid u^J y v_N
\right\rangle \right\rangle} \;\,=\;\, {\left\langle \left\langle v_N\mid u y v_N
\right\rangle \right\rangle} &{\mbox}{ for all }& y\in({\mbox{\textbf{n}}}^J)^-\;\;.\end{aligned}$$ We have $1,\,u^J\in (U_f)^J\subseteq {\overline{\overline{U^J}}}$. Using Theorem \[SpU1\], 2) we conclude ${\tilde}{{\alpha}}(1,u^J)={\tilde}{{\alpha}}(1,u)$, from which follows $u=u^J\in (U_f)^J$.\
[\
]{}
The ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G]$}}$ and the Birkhoff decomposition
=================================================================================================
Let $J\subseteq I$. Define the following subsets of ${\mbox{$gr$-$End$}}$: $$\begin{aligned}
\label{Bcovering}
\widehat{G_f} &:=& {\mbox{$\widehat{G}$}}\,U_f\;\,=\;\, \bigcup_{\hat{n}\in\hat{N}} U^\pm \hat{n} U_f \;\;,\\
(\widehat{G_f})_J &:=& {\mbox{$\widehat{G}$}}_J\, (U_f)_J\;\,=\;\, \bigcup_{\hat{n}\in\hat{N}_J} U_J^\pm \hat{n} (U_f)_J \;\;.\end{aligned}$$ We call the unions on the right the Bruhat and Birkhoff coverings. During our investigation of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, we will find, that $\widehat{G_f}$ is a monoid, and the Bruhat and Birkhoff coverings are really decompositions. Similar things hold for $(\widehat{G_f})_J$.\
The monoid ${\mbox{$ {\overline{\overline{G}}} $}}$ contains $G$, as well as the closures ${\mbox{$ {\overline{\overline{T}}} $}}={\mbox{$\widehat{T}$}}$ and ${\overline{\overline{U}}}=U_f$. Therefore it also contains $\widehat{G_f}$, and due to Proposition \[Setting2\] we have: $$\begin{aligned}
\label{ESpmG}
\widehat{G_f}\times\widehat{G_f} &{\mbox}{ maps to } & {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}\;\;.\end{aligned}$$ Similarly we get: $$\begin{aligned}
\label{ESpmGJ}
(\widehat{G_f})_J\times(\widehat{G_f})_J &{\mbox}{ maps to } & {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}\;\;.\end{aligned}$$ Our first aim is to show the surjectivity of (\[ESpmG\]). The key step of the proof is an induction over $|J|$, showing the surjectivity of (\[ESpmGJ\]). The next two theorems prepare the induction step. They relate ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$ to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{J\setminus\{j\}}]$}}$, $j\in J$.\
For ${\Lambda}\in P^+$ choose a nonzero element $v_{\Lambda}\in L({\Lambda})_{\Lambda}$ and define $$\begin{aligned}
{\theta}_{\Lambda}&:=& \frac{f_{v_{\Lambda}v_{\Lambda}}{\!\mid_{G}}}{{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}} \;\,\in\;\, {\mbox{${\mathbb{F}}\,[G]$}}\;\;.\end{aligned}$$ The function ${\theta}_{\Lambda}$ is independent of the chosen element $v_{\Lambda}$, and the chosen nondegenerate contravariant symmetric bilinear form on $L({\Lambda})$. Set ${\theta}_i:={\theta}_{{\Lambda}_i}$, $i=1,\,\ldots,\, n$.\
Kac and Peterson showed in [@KP2] by checking on the dense principal open set $U^-TU^+$ of $G$: $$\begin{aligned}
\label{multgt}
{\theta}_{\Lambda}{\theta}_{{\Lambda}'}\;\,=\,\; {\theta}_{{\Lambda}+{\Lambda}'} \quad {\mbox}{ for all }\quad{\Lambda},{\Lambda}'\in P^+\;\;.\end{aligned}$$ The next theorem gives a covering of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$ by principal open sets, which are build with these functions. A variant of this covering for the full spectrum of ${\mbox{${\mathbb{F}}\,[G]$}}$ has been given by Kashiwara in [@Kas], Proposition 6.3.1.\
Denote the action of $G_J\times G_J$ on ${\mbox{${\mathbb{F}}\,[G_J]$}}$, which is induced by the action $\pi$ of $G\times G$ on ${\mbox{${\mathbb{F}}\,[G]$}}$, also by $\pi$. Write ${\theta}_{\Lambda}$ instead of ${\theta}_{\Lambda}{\!\mid_{G_J}}$ for short.
\[Ueb1\] Let $\emptyset\neq J\subseteq I$. We have $$\begin{aligned}
\lefteqn{ \bigcup_{g,h\in G_J}\; \;\bigcup_{j\in J} \;\; D_{Specm\,{\mathbb{F}}\,[G_J]}\,(\,\pi(g,h){\theta}_j\,)}\\
&=& \left\{ \begin{array}{ccl}
{\mbox{Specm\,}}\,{\mbox{${\mathbb{F}}\,[G_J]$}} &{\mbox}{ if }& J {\mbox}{ is not special }.\\
\left(\,{\mbox{Specm\,}}\,{\mbox{${\mathbb{F}}\,[G_J]$}}\,\right)\setminus\{\,{\alpha}(1,e(R(J)))\,\} &{\mbox}{ if }& J {\mbox}{ is special }.
\end{array} \right.\end{aligned}$$
Suppose there exists an element ${\alpha}\in{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$ not contained in the union on the left. Then for all $g,h\in G_J$ and $j\in J$ we have ${\alpha}(\pi(g,h){\theta}_j)=0$. Here ${\alpha}\circ\pi(g,h)$ is a homomorphism of algebras. Because of the multiplicative property (\[multgt\]) we find ${\alpha}(\,\pi(g,h){\theta}_{\Lambda}\,)=0$ for all $g,h\in G_J$ and ${\Lambda}\in P^+_J\setminus\{0\}$, where $P_J^+:=P_J\cap P^+= {\mathbb{N}_0}{\mbox}{-span}\,{\left\{\left.\;{\Lambda}_j\;\right|\; j\in J\;\right\}}$.\
Now $L_J({\Lambda}):=U({\mbox{\textbf{n}}}_J^-)L({\Lambda})_{\Lambda}$ is an irreducible $G_J$-module, ${\Lambda}\in P_J^+$. Since $G_J L({\Lambda})_{\Lambda}$ spans $L_J({\Lambda})$, we find ${\alpha}(f_{vw}{\!\mid_{G_J}})=0$ for all $v,w\in L_J({\Lambda})$, ${\Lambda}\in P_J^+\setminus\{0\}$. Due to Theorem 5.12 of [@M] this is impossible if $J$ is not special, and ${\alpha}={\alpha}(1,e(R(J)))$ if $J$ is special.\
[\
]{}The principal open subsets, which have been used in the covering of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$, can be obtained by $$\begin{aligned}
D_{Specm\,{\mathbb{F}}\,[G_J]}\,(\,\pi(g,h){\theta}_j\,) &=& \left(\,{\mbox{Specm\,}}\,{\mbox{${\mathbb{F}}\,[\, D_{G_J}(\pi(g,h){\theta}_j)\,]$}}\,\right) {\!\mid_{{\mathbb{F}}\,[G_J]}}\;\;.\end{aligned}$$ The next theorem gives a product decomposition of the principal open sets $D_{G_J}({\theta}_j)$, $j\in J$. (Set $L=J\setminus\{j\}$ and ${\Lambda}={\Lambda}_j$, $j\in J$). It can be proved in the same way as Theorem 5.11 a), and b) in [@M]. A decomposition of the coordinate rings analogous to the second part of b) has been given in [@Kas], Lemma 5.3.4 and 5.3.5.
\[Ueb2\] Let $L\subsetneqq J\subseteq I$. Set $(U^\pm_J)^L:=\bigcap_{\sigma\,\in\,{\cal W}_L}\sigma U^\pm_J\sigma^{-1}$. For ${\Lambda}\in P_J\cap F_L$ we have:\
a) $\,D_{G_J}({\theta}_{\Lambda})\:=\:(U^-_J)^L\,G_L\, T_{J\setminus L}\,(U^+_J)^L\,$.\
b) The multiplication map $$\begin{aligned}
m:\;\; (U^-_J)^L\times G_L\times T_{J\setminus L}\times (U^+_J)^L &\to & D_{G_J}({\theta}_{\Lambda}) \end{aligned}$$ is bijective, and its comorphism $$\begin{aligned}
m^*:\;\; {\mbox{${\mathbb{F}}\,[D_{G_J}({\theta}_{\Lambda})]$}} &\to &
{\mbox{${\mathbb{F}}\,[(U^-_J)^L]$}}\otimes{\mbox{${\mathbb{F}}\,[G_L]$}}\otimes {\mbox{${\mathbb{F}}\,[P_{J\setminus L}]$}}\otimes{\mbox{${\mathbb{F}}\,[(U^+_J)^L]$}}\end{aligned}$$ exists, and is an isomorphism of algebras.
\[SpG1\] Let $J\subseteq I$. Then $(\widehat{G_f})_J\times (\widehat{G_f})_J$ maps surjectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$.
We show the surjectivity by induction over $|J|$. The case $J=\emptyset$ is trivial. For $J=\{j\}$ the map $\{1\}\times G_{\{j\}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{\{j\}}]$}}$ is already surjective, because $(G_{\{j\}},{\mbox{${\mathbb{F}}\,[G_{\{j\}}]$}})$ can be identified with $(SL(2,{\mathbb{F}}),{\mbox{${\mathbb{F}}\,[SL(2,{\mathbb{F}})]$}})$.\
Now the step of the induction from $|J|\leq m$ to $|J|=m+1$, ($1<m<|I|$):\
Let $j\in J$. Due to the induction assumption we have a surjective map $$\begin{aligned}
{\tilde}{{\alpha}}:\,(\widehat{G_f})_{J\setminus\{j\}}\times(\widehat{G_f})_{J\setminus\{j\}} &\to& {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{J\setminus\{j\}}]$}}\;\;.\end{aligned}$$ Because ${\mbox{${\mathbb{F}}\,[P_{\{j\}}]$}}$ is the classical coordinate ring of the torus, we get a bijective map $$\begin{aligned}
{\tilde}{{\alpha}}:\,\{1\}\times T_{\{j\}}&\to &{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[P_{\{j\}}]$}}\;\;.\end{aligned}$$ $U_J^\pm$, ${\mbox{${\mathbb{F}}\,[U_J^\pm]$}}$, and $(U_f)_J$ can be identified with $U(A_J)^\pm$, ${\mbox{${\mathbb{F}}\,[U(A_J)^\pm]$}}$, and $U(A_J)_f$. Due to Theorem \[SpU2\], Proposition \[Setting2\], and Remark 3) after Proposition \[Setting2\], we have bijective maps $$\begin{aligned}
{\tilde}{{\alpha}} : \,\{1\}\times ((U_f)_J)^{J\setminus\{j\}} &\to & {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[(U_J)^{J\setminus\{j\}}]$}} \;\;,\\
{\tilde}{{\alpha}} : \, ((U_f)_J)^{J\setminus\{j\}}\times \{1\} &\to & {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[(U^-_J)^{J\setminus\{j\}}]$}}\;\;.\end{aligned}$$ Due to the last theorem an element $\beta\in{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[D_{G_J}({\theta}_j)]$}}$ can be written in the form $$\begin{aligned}
\beta &=& \left(\,{\tilde}{{\alpha}}(u,1)\otimes {\tilde}{{\alpha}}(x,y)\otimes {\tilde}{{\alpha}}(1,t)\otimes {\tilde}{{\alpha}}(1,{\tilde}{u})\,\right)\circ m^*\end{aligned}$$ with $u,{\tilde}{u}\in ((U_f)_J)^{J\setminus\{j\}}$, $t\in T_{\{j\}}$, and $x,y\in (\widehat{G_f})_{J\setminus\{j\}}$.\
Let $N\in P^+$. Choose ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases of $L(N)$, by choosing ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases $$\begin{aligned}
(\,a_{{\lambda}i}\,)_{\,i=1,\ldots,m_{\lambda}} &\quad,\quad & (\,c_{{\lambda}i}\,)_{\,i=1,\ldots,m_{\lambda}} \end{aligned}$$ of $L(N)_{\lambda}$ for every ${\lambda}\in P(N)$. Let $v\in L(N)_{\lambda}$, $w\in L(N)_\mu$. By applying $m^*$ to $f_{vw}{\!\mid_{G_J}}$ we find $$\begin{aligned}
\lefteqn{ m^* (f_{vw}{\!\mid_{G_J}}) }\\
&=& \sum_{({\lambda}',\,i)\,,\,{\lambda}'\geq {\lambda}\atop (\mu',\,j)\,,\,\mu'\geq \mu }
f_{v a_{{\lambda}' i}}{\!\mid_{(U_J^-)^{J\setminus\{j\}}}} \otimes\, f_{c_{{\lambda}' i} a_{\mu' j}}{\!\mid_{G_{J\setminus\{j\}}}}\otimes \,
e_{\mu'(h_j){\Lambda}_j}{\!\mid_{T_{\{j\}}}}\otimes\, f_{c_{\mu' j} w} {\!\mid_{(U_J)^{J\setminus\{j\}}}}\;\;.\end{aligned}$$ This sum is finite, due to $P(N)\subseteq N-Q_0^+$. By applying $\beta$ to $f_{vw}{\!\mid_{G_J}}$ we get $$\begin{aligned}
\beta(f_{vw}{\!\mid_{G_J}}) &=& \sum_{({\lambda}',\,i)\,,\,{\lambda}'\geq {\lambda}\atop (\mu',\,j)\,,\,\mu'\geq \mu } {\left\langle \left\langle uv\mid a_{{\lambda}' i}
\right\rangle \right\rangle}
{\left\langle \left\langle x c_{{\lambda}' i}\mid yt a_{\mu' j}
\right\rangle \right\rangle} {\left\langle \left\langle c_{\mu' j}\mid {\tilde}{u}w
\right\rangle \right\rangle} \\
&=& {\left\langle \left\langle xuv\mid yt{\tilde}{u}w
\right\rangle \right\rangle}\;\;.\end{aligned}$$ In particular for $v\in U({\mbox{\textbf{n}}}_J^-) L(N)_N \cap L(N)_{\lambda}$, $w\in U({\mbox{\textbf{n}}}_J^-) L(N)_N \cap L(N)_\mu$ we have $$\begin{aligned}
\beta(f_{vw}{\!\mid_{G_J}}) \;\,=\;\,{\left\langle \left\langle xuv\mid yt{\tilde}{u}w
\right\rangle \right\rangle}\;\,=\;\, {\alpha}(xu,yt{\tilde}{u})(f_{vw}{\!\mid_{G_J}})\;\;.\end{aligned}$$ Here ${\alpha}$ denotes the map $(\widehat{G_f})_J\times(\widehat{G_f})_J \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$ due to (\[ESpmGJ\]). Because ${\mbox{${\mathbb{F}}\,[G_J]$}}$ is already spanned by the functions $f_{vw}{\!\mid_{G_J}}$, $v,w\in U({\mbox{\textbf{n}}}_J^-)L(N)_N$, $N\in P_J^+=P_J\cap P^+$, we conclude $\beta{\!\mid_{{\mathbb{F}}\,[G_J]}}={\alpha}(xu,yt{\tilde}{u})\in{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$.\
It is easy to check that for $g,h\in G_J$ we have $$\begin{aligned}
\left(\,{\mbox{Specm\,}}\,{\mbox{${\mathbb{F}}\,[D_{G_J}(\pi(g,h){\theta}_j)]$}}\,\right){\!\mid_{{\mathbb{F}}\,[G_J]}} &=&
\left(\,{\mbox{Specm\,}}\,{\mbox{${\mathbb{F}}\,[D_{G_J}({\theta}_j)]$}}\,\right){\!\mid_{{\mathbb{F}}\,[G_J]}}\circ \:\pi(g^{-1},h^{-1})\;\;.\end{aligned}$$ Because of Theorem \[Ueb1\], and because of the equation ${\alpha}(a,b)\circ\pi(c,d)={\alpha}(ac,bd)$, we get $$\begin{aligned}
\lefteqn{ \bigcup_{g,h\in G_J} \,\bigcup_{j\in J} \,{\left\{\left.\;{\alpha}(xug^{-1},yt{\tilde}{u}h^{-1}) \;\right|\; x,y\in (\widehat{G_f})_{J\setminus\{j\}}\,,
\;u,{\tilde}{u}\in ((U_f)_J)^{J\setminus\{j\}}\,,\;t\in T_{\{j\}} \;\right\}} }\\
&=& \left\{ \begin{array}{ccl}
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}} &{\mbox}{ if }& J {\mbox}{ is not special }.\\
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}\setminus\{\,{\alpha}(1,e(R(J)))\,\} &{\mbox}{ if }& J {\mbox}{ is special }.
\end{array} \right.\qquad\qquad\qquad\qquad\qquad\qquad\end{aligned}$$ Using the Bruhat covering of $(\widehat{G_f})_J$, the Bruhat decompositions of $(G_f)_J$ and ${\mbox{$\widehat{G}$}}_J$, we find $$\begin{aligned}
(\widehat{G_f})_J &\subseteq & (\widehat{G_f})_J(G_f)_J \,\;=\,\; U_J {\mbox{$\widehat{N}$}}_J (U_f)_J\, (G_f)_J \;\,=\;\,
U_J{\mbox{$\widehat{N}$}}_J \, U_J N_J(U_f)_J\\
&=& {\mbox{$\widehat{G}$}}_J (U_f)_J\;\,=\;\,(\widehat{G_f})_J\;\;.\end{aligned}$$ From this follows $(\widehat{G_f})_{J\setminus\{j\}} T_{\{j\}}((U_f)_J)^{J\setminus\{j\}} G_J \subseteq (\widehat{G_f})_J$ for all $j\in J$. Furthermore, if $J$ is special, then $e(R(J))\in (\widehat{G_f})_J$. Therefore the map ${\alpha}: (\widehat{G_f})_J\times (\widehat{G_f})_J\to
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_J]$}}$ is surjective.\
[\
]{}
\[SpG2\] $\widehat{G_f}\times \widehat{G_f}$ maps surjectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.
Let $m: G_I\times T_{rest}\to G$ be the multiplication map, and $m^ *:{\mbox{${\mathbb{F}}\,[G]$}}\to{\mbox{${\mathbb{F}}\,[G_I]$}}\otimes {\mbox{${\mathbb{F}}\,[T_{rest}]$}}$ its comorphism. Due to the bijectivity of $m^*$, and due the last theorem and Theorem \[SpT2\], 3), the elements of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ are given by $$\begin{aligned}
\label{tensorhom}
\left({\tilde}{{\alpha}}(x,y)\otimes{\tilde}{{\alpha}}(1,t)\right)\circ m^*\;\;,\end{aligned}$$ where $x,y\in (\widehat{G_f})_I$ and $t\in T_{rest}$. Applying this expression to the matrix coefficients $f_{vw}{\!\mid_{G}}$, $v\in L({\Lambda})_{\lambda}$, $w\in L({\Lambda})_\mu$, ${\lambda},\mu\in P({\Lambda})$, ${\Lambda}\in P^+$, we find $$\begin{aligned}
\left(\,{\tilde}{{\alpha}}(x,y)\otimes{\tilde}{{\alpha}}(1,t)\,\right) (m^*(f_{vw}{\!\mid_{G}})) &=& \left({\tilde}{{\alpha}}(x,y)\otimes{\tilde}{{\alpha}}(1,t)\right)
(f_{vw}{\!\mid_{G_I}}\otimes\, e_\mu{\!\mid_{T_{rest}}})\\
&=&{\left\langle \left\langle xv\mid yw
\right\rangle \right\rangle}e_\mu(t) \;\,=\,\; {\alpha}(x,yt)(f_{vw}{\!\mid_{G}})\;\;.\end{aligned}$$ Here ${\alpha}$ denotes the map $ \widehat{G_f}\times\widehat{G_f} \to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ due to (\[ESpmG\]). Therefore ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}={\left\{\left.\;{\alpha}(x,{\tilde}{y})\;\right|\; x\in(\widehat{G_f})_I\,,\,{\tilde}{y}\in (\widehat{G_f})_I T_{rest} \;\right\}}$, in particular ${\alpha}$ is surjective.\
[\
]{}The next theorem gives one of the main results of this paper: A description of the $\widehat{G_f}\times\widehat{G_f}$-set ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.
\[SpG3\]\
1) We have ${\overline{G}}\,=\,\widehat{G}$ and ${\overline{\overline{G}}}\,=\,\widehat{G_f}$. In particular $\widehat{G_f}$ is a monoid.\
2) The map ${\mbox{$ {\overline{\overline{G}}} $}}\times_{{\overline{G}}} {\mbox{$ {\overline{\overline{G}}} $}}\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ is a ${\mbox{$ {\overline{\overline{G}}} $}}\times{\mbox{$ {\overline{\overline{G}}} $}}$-equivariant bijection.
The first equation of 1) has been shown in [@M], Theorem 5.14. To prepare the proof of the rest of the theorem, we first show the following statements a) - d):\
a) We show ${\overline{\overline{B^-}}}\subseteq \,{\overline{B^-}}$: For ${\Lambda}\in P^+$ and ${\lambda}\in P({\Lambda})$ define $$\begin{aligned}
L({\Lambda})_{{\lambda}\,\downarrow} &:=& \bigoplus_{\mu\,\in \,P({\Lambda})\cap ({\lambda}-Q_0^+)} L({\Lambda})_\mu\;\;.\end{aligned}$$ Since $L({\Lambda})_{{\lambda}\,\downarrow}$ is $B^-$-invariant, we have $$\begin{aligned}
{\left\langle \left\langle w\mid B^- v
\right\rangle \right\rangle}\;=\; 0 &{\mbox}{ for all}& v\in L({\Lambda})_{{\lambda}\,\downarrow}\;\,,\;\,w\in \left(L({\Lambda})_{{\lambda}\,\downarrow}\right)^\bot \;\;.\end{aligned}$$ These equations are also valid, if $B^-$ is replaced by its Zariski closure ${\overline{\overline{B^-}}}$, and due to the orthogonality of the weight spaces we have $\left(L({\Lambda})_{{\lambda}\,\downarrow}\right)^{\bot\, \bot} = L({\Lambda})_{{\lambda}\,\downarrow} $. Therefore $L({\Lambda})_{{\lambda}\,\downarrow}$ is also ${\overline{\overline{B^-}}}$-invariant.\
Let $b\in {\overline{\overline{B^-}}}$. Choose a pair of ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases of $L({\Lambda})$, by choosing ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases $$\begin{aligned}
(\,a_{{\lambda}i}^{\Lambda}\,)_{\,i=1,\ldots,m_{\lambda}} &\quad,\quad &
(\,c_{{\lambda}i}^{\Lambda}\,)_{\,i=1,\ldots,m_{\lambda}} \end{aligned}$$ of $L({\Lambda})_{\lambda}$ for every ${\lambda}\in P({\Lambda})$. For a fixed weight $\mu\in P({\Lambda})$ we have $$\begin{aligned}
{\langle \langle a^{\Lambda}_{\mu i}\mid b\,c^{\Lambda}_{\mu' i'}\rangle \rangle} & \neq& 0 \end{aligned}$$ at most for the finitely many weights $\mu'\in P({\Lambda})$ with $\mu'\geq \mu$. Therefore we get a well defined linear map $\psi_{\Lambda}$ by $$\begin{aligned}
\psi_{\Lambda}v \;\,:=\,\; \sum_{\mu\,i}\;\sum_{\mu'\,i'} \;a_{\mu'\, i'}^{\Lambda}{\left\langle \left\langle a^{\Lambda}_{\mu i}\mid b\,
c^{\Lambda}_{\mu' i'}
\right\rangle \right\rangle} {\left\langle \left\langle c^{\Lambda}_{\mu i}\mid v
\right\rangle \right\rangle} &\;,\;& v\in L({\Lambda})\;\;.\end{aligned}$$ These maps $\psi_{\Lambda}$, ${\Lambda}\in P^+$, define an element $\psi$ of ${\mbox{$gr$-$End$}}$. It is easy to check, that $\psi$ is the adjoint of $b$. Therefore $ b\in {\overline{\overline{B^-}}} \cap {\mbox{$gr$-$Adj$}}= {\overline{B^-}}$.\
b) Let $\psi\in{\mbox{$ {\overline{\overline{G}}} $}}$, $b\in {\overline{\overline{B}}}$ with ${\alpha}(1,\psi)={\alpha}(b,1)$. We show $$\begin{aligned}
b,\,\psi\in{\mbox{$\widehat{G}$}}&{\mbox}{and}& \psi^*\;=\;b \;\;.\end{aligned}$$ Due to the second part of Proposition \[Setting3\], the homomorphism ${\alpha}(b,1) ={\alpha}(1,\psi)$ factors to a homomorphism ${\mbox{${\mathbb{F}}\,[B^-]$}}\to{\mathbb{F}}$. Due to the first part of Proposition \[Setting3\], and part a) of above, we find $$\begin{aligned}
\psi\;\,\in\;\,{\overline{\overline{B^-}}}\;\,\subseteq\;\,{\overline{B^-}}\;\,\subseteq \;\,{\overline{G}}
\,\;= \,\;{\mbox{$\widehat{G}$}}\,\;\subseteq \,\;{\mbox{$gr$-$Adj$}}\;\;.\end{aligned}$$ By checking on the matrix coefficients, using the non-degeneracy of the contravariant bilinear forms, we find $\psi^*=b$.\
c) Let $g,h\in G_f$ and $x,y,{\tilde}{x},{\tilde}{y}\in {\mbox{$ {\overline{\overline{G}}} $}}$. By checking on the matrix coefficients we find: $$\begin{aligned}
{\alpha}(xg,yh)\;=\;{\alpha}({\tilde}{x},{\tilde}{y}) &\iff& {\alpha}(x,y)\;=\;{\alpha}({\tilde}{x}g^{-1},{\tilde}{y}h^{-1})\;\;.\end{aligned}$$ d) We show $\,{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}} ={\alpha}(U_f,{\mbox{$\widehat{N}$}}U_f)\,$: Due to the last corollary, every element of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ is of the form ${\alpha}(\psi,{\tilde}{\psi})$ with $\psi,{\tilde}{\psi}\in\widehat{G_f}$. Due to the Birkhoff covering of $\widehat{G_f}$, we can write $\psi$, ${\tilde}{\psi}$ in the form $$\begin{aligned}
\begin{array}{ccc}
\psi & = & u_-n_\sigma e(R) u_+ \\
{\tilde}{\psi} & = & {\tilde}{u}_- {\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R}) {\tilde}{u}_+
\end{array} &\;{\mbox}{ with }\; & u_-,\,{\tilde}{u}_-\in U^-\;\;,\;\; u_+,\,{\tilde}{u}_+ \in U_f\;\;,\;\; n_\sigma,\,{\tilde}{n}_{{\tilde}{\sigma}}\in N\;\;.\end{aligned}$$ From this follows: $$\begin{aligned}
(\psi,{\tilde}{\psi}) & \sim & (u_+\, ,\, e(R)n_\sigma^* u_-^*{\tilde}{u}_-{\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R}) {\tilde}{u}_+)\;\;.\end{aligned}$$ Due to the Birkhoff decomposition of ${\mbox{$\widehat{G}$}}$, we can write $e(R)n_\sigma^* u_-^*{\tilde}{u}_-{\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R})$ in the form $u'_- \hat{n}' u'_+$ with $\hat{n}'\in{\mbox{$\widehat{N}$}}$, $u'_\pm\in U^\pm$. We get $$\begin{aligned}
(\psi,{\tilde}{\psi}) & \sim & ((u_-')^*u_+\, ,\, \hat{n}' u_+'{\tilde}{u}_+)\;\;.\end{aligned}$$ Therefore ${\alpha}(\psi,{\tilde}{\psi}) = {\alpha}((u_-')^*u_+\, ,\, \hat{n}' u_+'{\tilde}{u}_+)$.\
Now we can prove the theorem:\
To 1) To show ${\overline{\overline{G}}}\,=\,\widehat{G_f}$, it is sufficient to show ${\overline{\overline{G}}}\,\subseteq\,\widehat{G_f}$. Let $\phi\in {\overline{\overline{G}}}$. Then due to d) there exist elements $u,{\tilde}{u}\in U_f$, $e(R)n_\sigma\in{\mbox{$\widehat{N}$}}$ such that $$\begin{aligned}
{\alpha}(1,\phi) & = & {\alpha}( u\, ,\, e(R)n_\sigma {\tilde}{u})\;\;.\end{aligned}$$ Using c) this is equivalent to $$\begin{aligned}
{\alpha}(1,\phi(n_\sigma{\tilde}{u})^{-1}) &=& {\alpha}(e(R)u,1)\;\;.\end{aligned}$$ The monoid ${\overline{\overline{B}}}$ contains ${\mbox{$ {\overline{\overline{T}}} $}}$ and $U_f$. Therefore $e(R)u\in {\mbox{$ {\overline{\overline{T}}} $}}U_f\subseteq {\overline{\overline{B}}}$, and due to b) we get $\phi\in {\mbox{$\widehat{G}$}}n_\sigma {\tilde}{u}\subseteq \widehat{G_f}$.\
To 2) Due to the last corollary, we only have to show the injectivity in 2). Due to the proof of d), we may start with elements $u,{\tilde}{u}\in U_f$, $g,{\tilde}{g}\in N U_f$ such that $$\begin{aligned}
{\alpha}(e(R)u,g) &=&{\alpha}(e({\tilde}{R}){\tilde}{u},{\tilde}{g}) \;\,.\end{aligned}$$ Using c) this equation is equivalent to $$\begin{aligned}
{\alpha}(e(R)u{\tilde}{u}^{-1},1) &=& {\alpha}(1,e({\tilde}{R}){\tilde}{g}g^{-1})\;\;.\end{aligned}$$ Due to b) we find $e(R)u{\tilde}{u}^{-1}\in{\mbox{$\widehat{G}$}}$ and $(e(R) u{\tilde}{u}^{-1})^* \,=\, e({\tilde}{R}){\tilde}{g}g^{-1}$. From this follows $$\begin{aligned}
(\,e(R)u\,,\, g\,) &=& (\,(e(R)u{\tilde}{u}^{-1}){\tilde}{u}\,,\, g\,)\;\,\sim\;\, (\,{\tilde}{u}\,,\, (e(R)u{\tilde}{u}^{-1})^*g\,)
\;\,=\;\,(\,{\tilde}{u}\,,\, e({\tilde}{R}){\tilde}{g}\,) \\
&\sim & (\,e({\tilde}{R}){\tilde}{u}\,,\, {\tilde}{g}\,) \;\;.\end{aligned}$$ [\
]{}The group $U_f\times U_f$ acts on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. The corresponding partition into orbits is described by the Birkhoff decomposition in the following theorem:
\
1) There are the following Bruhat- and Birkhoff decompositions of $\widehat{G_f}$: $$\begin{aligned}
\widehat{G_f} &=& \dot{\bigcup_{\hat{n}\in\hat{N}}}\; U^\pm \hat{n} U_f\;\;.\end{aligned}$$ 2) There is the following Birkhoff decomposition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$: $$\begin{aligned}
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}} &=& \dot{\bigcup_{\hat{n}\in\hat{N}} }\; {\alpha}(U_f, \hat{n} U_f)\;\;.\end{aligned}$$
Due to the Bruhat and Birkhoff coverings of $\widehat{G_f}$ and part d) of the proof of the last theorem, we only have to show that these unions are disjoint.\
a) First we do this for 2). Suppose there exist elements $u_1,u_2,{\tilde}{u}_1,{\tilde}{u}_2\in U_f$ such that $$\begin{aligned}
{\alpha}(u_1, n_\sigma e(R) u_2) &=& {\alpha}({\tilde}{u}_1,{\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R}) {\tilde}{u}_2) \;\,.\end{aligned}$$ Using part c) of the proof of the last theorem, we find for all $v,w\in L({\Lambda})$, ${\Lambda}\in P^+$: $$\begin{aligned}
\label{kBGl1}
{\left\langle \left\langle v\mid n_\sigma e(R) w
\right\rangle \right\rangle} &=& {\langle \langle {\tilde}{u}_1(u_1)^{-1}v\mid {\tilde}{n}_{{\tilde}{\sigma}}e({\tilde}{R}){\tilde}{u}_2 (u_2)^{-1}w\rangle \rangle}\;\; .\end{aligned}$$ Fix an element ${\Lambda}\in P^+$ with $P({\Lambda})\cap X\setminus R\neq\emptyset$. Fix elements $\mu\in P({\Lambda})\cap X\setminus R$ and $w_\mu\in L({\Lambda})_\mu\setminus\{0\}$. By inserting $w=w_\mu$ in the last equation we find $$\begin{aligned}
0 \;=\; {\langle \langle {\tilde}{v}\mid e({\tilde}{R}){\tilde}{u}_2 (u_2)^{-1}w_\mu\rangle \rangle} &{\mbox}{ for all }& {\tilde}{v}\in L({\Lambda})\;\;.\end{aligned}$$ Because of ${\tilde}{u}_2(u_2)^{-1}\in U_f$ we find $\mu\in X\setminus {\tilde}{R}$.\
Since $\bigcup_{{\Lambda}\in P^+}P({\Lambda})=X\cap P$ this shows $(X\setminus R)\cap P\subseteq (X\setminus{\tilde}{R})\cap P$, from which follows $R\supseteq {\tilde}{R}$. We may interchange the variables with and without $\,{\tilde}{{\mbox}{}}\,$, and get $R={\tilde}{R}$.\
Fix an element ${\Lambda}\in P^+$ with $P({\Lambda})\cap R\neq\emptyset$. Fix elements $\mu\in R\cap P({\Lambda})$ and $w_\mu\in L({\Lambda})_\mu\setminus\{0\}$. Because of ${\tilde}{u}_2 (u_2)^{-1}\in U_f$ we have $$\begin{aligned}
{\tilde}{\sigma}\mu &\in & supp({\tilde}{n}_{{\tilde}{\sigma}} e(R) {\tilde}{u}_2 (u_2)^{-1}w_\mu)\;\;.\end{aligned}$$ Choose a maximal weight $m$ of $supp({\tilde}{n}_{{\tilde}{\sigma}} e(R) {\tilde}{u}_2 (u_2)^{-1}w_\mu)$ with $m\geq {\tilde}{\sigma}\mu$. By inserting $w=w_\mu$ in equation (\[kBGl1\]), using ${\tilde}{u}_1 (u_1)^{-1}\in U_f$, we find $$\begin{aligned}
\label{kBGl3}
{\left\langle \left\langle v_m\mid n_\sigma w_\mu
\right\rangle \right\rangle} &=& {\left\langle \left\langle v_m\mid {\tilde}{n}_{{\tilde}{\sigma}}e(R){\tilde}{u}_2 (u_2)^{-1}w_\mu
\right\rangle \right\rangle}\;\;{\mbox}{ for all }\;v_m\in L({\Lambda})_m\;\;.\;\;\;\end{aligned}$$ Because ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$ is nondegenerate on $L({\Lambda})_m$, there exists an element $v_m$, such that the right side is nonzero. Therfore $\sigma \mu = m\geq {\tilde}{\sigma}\mu$. Interchanging the variables with and without $\,{\tilde}{{\mbox}{}}\,$ gives ${\tilde}{\sigma}\mu=\sigma\mu = m$.\
Inserting $m={\tilde}{\sigma}\mu$ in (\[kBGl3\]), and using ${\tilde}{u}_2 (u_2)^{-1}\in U_f$, we find $$\begin{aligned}
{\left\langle \left\langle v\mid n_\sigma w_{\mu}
\right\rangle \right\rangle} &=& {\left\langle \left\langle v\mid {\tilde}{n}_{{\tilde}{\sigma}}w_\mu
\right\rangle \right\rangle} \;\;{\mbox}{ for all }\;v\in L({\Lambda})_m \,=\, L({\Lambda})_{\sigma\mu} \,=
\,L({\Lambda})_{{\tilde}{\sigma}\mu} \;\;.\end{aligned}$$ Because of the orthogonality of the weight spaces, this equation is also valid for all $v\in L({\Lambda})$.\
This shows $$\begin{aligned}
{\left\langle \left\langle v\mid n_\sigma e(R)w
\right\rangle \right\rangle} \;=\; {\left\langle \left\langle v\mid {\tilde}{n}_{{\tilde}{\sigma}}e(R)w
\right\rangle \right\rangle} &{\mbox}{ for all }&
v, w \in L({\Lambda}) \;,\; {\Lambda}\in P^+\;\;.\end{aligned}$$ Therefore we get $n_\sigma e(R) ={\tilde}{n}_{{\tilde}{\sigma}} e(R) = {\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R})$.\
b) The Birkhoff decomposition of $\widehat{G_f}$ follows from 2), by using the embedding of $\widehat{G_f}$ in ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. Suppose there exist elements $u_1,{\tilde}{u}_1\in U^-$, $u_2,{\tilde}{u}_2\in U_f$ such that $$\begin{aligned}
(u_1)^*n_\sigma e(R) u_2 &=& ({\tilde}{u}_1)^*{\tilde}{n}_{{\tilde}{\sigma}} e({\tilde}{R}) {\tilde}{u}_2 \;\;.\end{aligned}$$ From this equation follows equation (\[kBGl1\]), and as in part a) of the proof we can deduce $R={\tilde}{R}$. Also similar as in part a), but now using a minimal weight $m$ of $supp({\tilde}{n}_{{\tilde}{\sigma}} e(R) {\tilde}{u}_2 (u_2)^{-1}w_\mu)$ with $m\leq {\tilde}{\sigma}\mu$, we find $n_\sigma e(R)= {\tilde}{n}_{{\tilde}{\sigma}}e(R)$.\
[\
]{}
The stratification of the spectrum of ${\mathbb{F}}$-valued points of ${\mbox{${\mathbb{F}}\,[G]$}}$ in $G_f\times G_f$-orbits
==============================================================================================================================
In this section, we show that the $G_f\times G_f$-orbits of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ are locally closed, irreducible, and in one to one correspondence with the finitely many special subsets of $I$. The closure relation is given by the inverse inclusion of the special sets. We give a countable covering of each orbit by big cells. We show that there exist stratified transversal slices to the orbits at any of their points.\
To cut short our notation, we denote by $x{{\,\bf \diamond\,}}y$ the image of $(x,y)\in\widehat{G_f}\times\widehat{G_f}$ under the surjective map ${\alpha}:\widehat{G_f}\times\widehat{G_f}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. We make use of this map as a parametrization of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
Recall that we have $x{{\,\bf \diamond\,}}zy = z^*x{{\,\bf \diamond\,}}y$, $x,y\in\widehat{G_f}$, $z\in {\mbox{$\widehat{G}$}}$. Recall that $\widehat{G_f}\times\widehat{G_f}$ acts on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ by morphisms from the right, i.e., $$\begin{aligned}
(x{{\,\bf \diamond\,}}y) ({\tilde}{x},{\tilde}{y}) &=& x{\tilde}{x}{{\,\bf \diamond\,}}y{\tilde}{y} \quad,\quad x,y,{\tilde}{x},{\tilde}{y}\in \widehat{G_f}\;.\end{aligned}$$ The Chevalley involution of ${\mbox{${\mathbb{F}}\,[G]$}}$ induces an involutive morphism $*$ on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, which we also call Chevalley involution. It is given by the switch map: $$\begin{aligned}
(x{{\,\bf \diamond\,}}y)^* &=& y{{\,\bf \diamond\,}}x \quad,\quad x,y\in \widehat{G_f}\;.\end{aligned}$$ In this section we also denote by $f_{vw}$ the function on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, induced by the matrix coefficient $f_{vw}{\!\mid_{G}}\,\in{\mbox{${\mathbb{F}}\,[G]$}}$. It is given by $$\begin{aligned}
f_{vw} (x{{\,\bf \diamond\,}}y) &=& {\left\langle \left\langle xv\mid yw
\right\rangle \right\rangle} \quad,\quad x,y\in \widehat{G_f}\;.\end{aligned}$$ For a set $M\subseteq {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ denote by ${\overline{\overline{\overline{M}}}}$ its Zariski closure. Note, that for a subset $A\subseteq{\mbox{$\widehat{G}$}}$ we have $1{{\,\bf \diamond\,}}{\overline{A}}\,\subseteq {\overline{\overline{\overline{1{{\,\bf \diamond\,}}A}}}}$. Similarly, for a subset $A\subseteq \widehat{G_f}$, we have $1{{\,\bf \diamond\,}}{\overline{\overline{A}}}\subseteq {\overline{\overline{\overline{1{{\,\bf \diamond\,}}A}}}}$. These formulas are useful to determine closures in ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.
\[S1\] 1) The partition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ in $G_f\times G_f$-orbits is given by $$\begin{aligned}
\label{partitionSpm}
{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}} &=& \dot{\bigcup_{\Xi\;special}} \, G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f\;\;.\end{aligned}$$ 2) Let ${\Theta}$ be special. The orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ is locally closed and irreducible. Its closure is given by $$\begin{aligned}
\label{closure}
\dot{\bigcup_{\Xi\;special\;,\;\Xi\supseteq {\Theta}}} G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f\;\;.\end{aligned}$$
a\) We first decompose the $G_f\times G_f$-orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ in a union of $U_f\times U_f$-orbits, i.e., we show $$\begin{aligned}
\label{orbit}
G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f &=& \dot{ \bigcup_{\hat{\sigma}\,\in\,{\cal W}\varepsilon(R({\Theta})){\cal W}} } U_f{{\,\bf \diamond\,}}(\hat{\sigma} T) U_f\;\;.\end{aligned}$$ By using the Birkhoff decomposition of $G_f$, and Proposition 2.14, Theorem 2.15 b) of [@M], compare also the section preliminaries, we get $$\begin{aligned}
G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f &=& U^- N U_f {{\,\bf \diamond\,}}e(R({\Theta}))G_f \;\,=\;\, U_f {{\,\bf \diamond\,}}N U e(R({\Theta})) G_f \\ &=&
\bigcup_{\sigma\in\cal W} U_f {{\,\bf \diamond\,}}e(\sigma R({\Theta})) G_f\;\;.\end{aligned}$$ Because ${\mbox{$\cal W$}}_{{\Theta}\cup{\Theta}^\bot}$ is the stabilizer of the face $R({\Theta})$ as a whole, we may restrict the last union to the minimal coset representatives ${\mbox{$\cal W$}}^{{\Theta}\cup{\Theta}^\bot}$ of ${\mbox{$\cal W$}}/{\mbox{$\cal W$}}_{{\Theta}\cup{\Theta}^\bot}$, characterized by ${\mbox{$\cal W$}}^{{\Theta}\cup{\Theta}^\bot}={\left\{\left.\;\sigma\in{\mbox{$\cal W$}}\;\right|\; \sigma{\alpha}_i\in{\mbox{$\Delta_{re}^+$}}{\mbox}{ for all } i\in {\Theta}\cup{\Theta}^\bot\;\right\}}$. Next we insert in $U_f {{\,\bf \diamond\,}}e(\sigma R({\Theta})) G_f $ the Birkhoff decomposition $G_f=(\sigma U^-\sigma^{-1})NU_f$. By using Proposition 2.14 of [@M], compare the section preliminaries, we get $$\begin{aligned}
\bigcup_{\sigma\in{\cal W}^{{\Theta}\cup{\Theta}^\bot}\,,\,\tau\in{\cal W} } U_f {{\,\bf \diamond\,}}\underbrace{\sigma U^-_{{\Theta}^\bot}\sigma^{-1}}_{\subseteq U^-} e(\sigma R({\Theta})) \tau B_f &=&
\bigcup_{\sigma\in{\cal W}^{{\Theta}\cup{\Theta}^\bot}\,,\,\tau\in{\cal W} } U_f {{\,\bf \diamond\,}}e(\sigma R({\Theta})) \tau B_f\;\;.\end{aligned}$$ Since $\bigcup_{\sigma\in{\cal W}^{{\Theta}\cup{\Theta}^\bot}\,,\,\tau\in{\cal W} } {\mbox{$\varepsilon\left(\sigma R({\Theta})\right)$}}\tau = {\mbox{$\cal W$}}\,{\mbox{$\varepsilon\left(R({\Theta})\right)$}}\,{\mbox{$\cal W$}}$, we have shown the equality in (\[orbit\]). The disjointness of the union follows from the Birkhoff decomposition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
Taking into account the the partition ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}= \dot{\bigcup}_{{\Theta}\;sp}\, {\mbox{$\cal W$}}\,{\mbox{$\varepsilon\left(R({\Theta})\right)$}} \,{\mbox{$\cal W$}}$, part 1) of the theorem follows from (\[orbit\]) the Birkhoff decomposition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
b) Next we show that the union $\bigcup_{\Xi\;sp.\;,\;\Xi\supseteq {\Theta}} G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f$ is closed, i.e., we show that it is the common zero set of the functions $$\begin{aligned}
f_{vw} \quad,\quad v,w\in L({\Lambda})\quad,\quad {\Lambda}\in P^+\setminus \overline{F_{\Theta}}\;\;.\end{aligned}$$ Due to part 1), every element of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ is of the form $g{{\,\bf \diamond\,}}e(R(\Xi))h$ with $g,h\in G_f$, $\Xi$ special. The equations $$\begin{aligned}
0\;\,=\;\, f_{vw}(g{{\,\bf \diamond\,}}e(R(\Xi))h)\;\,=\;\,{\left\langle \left\langle gv\mid e(R(\Xi))hw
\right\rangle \right\rangle} &{\mbox}{ for all }& v,w\in L({\Lambda})\,,\;{\Lambda}\in P^+\setminus
\overline{F_{\Theta}}\end{aligned}$$ are equivalent to $$\begin{aligned}
e(R(\Xi))L({\Lambda})\;\,=\;\,\{0\} &{\mbox}{ for all }& \;{\Lambda}\;\in\; P^+\setminus \overline{F_{\Theta}}\;=\;P^+\setminus R({\Theta})\;\;.\end{aligned}$$ Recall that for an element ${\Lambda}\in P^+$ we have $P({\Lambda})\cap R(\Xi)=\emptyset$ if and only if ${\Lambda}\notin R(\Xi)$. Therefore these equations are equivalent to ${\Lambda}\in P^+\setminus \overline{F_\Xi}$ for all ${\Lambda}\in P^+\setminus \overline{F_{\Theta}}$. This is equivalent to $P^+\setminus \overline{F_{\Theta}}\subseteq P^+\setminus \overline{F_\Xi}$, which in turn is equivalent to ${\Theta}\subseteq \Xi$.\
c) The closure of the $G_f\times G_f$-orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ is a union of $G_f\times G_f$-orbits. Due to b) it is contained in $\bigcup_{\Xi\;sp.\;,\;\Xi\supseteq {\Theta}} G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f$. To show equality, it is sufficient to show, that the closure contains the elements $1{{\,\bf \diamond\,}}e(R(\Xi))$, $\Xi\supseteq{\Theta}$, $\Xi$ special.\
Because left multiplications with elements of ${\mbox{$\widehat{G}$}}$ are Zariski continuous on ${\mbox{$\widehat{G}$}}$, we find $$\begin{aligned}
e(R({\Theta})){\mbox{$\widehat{G}$}}\;\,=\;\, e(R({\Theta})){\overline{G}}\;\,\subseteq\;\, {\overline{e(R({\Theta}))G}}\;\;.\end{aligned}$$ Therefore we get $$\begin{aligned}
1{{\,\bf \diamond\,}}e(R({\Theta})){\mbox{$\widehat{G}$}}\;\,\subseteq\;\, {\overline{\overline{\overline{1{{\,\bf \diamond\,}}e(R({\Theta}))G_f}}}}\;\,\subseteq\;\,{\overline{\overline{\overline{G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f}}}}\;\;.\end{aligned}$$ Now $e(R({\Theta})){\mbox{$\widehat{G}$}}$ contains for every $\Xi\supseteq {\Theta}$, $\Xi$ special, the element $e(R({\Theta}))e(R(\Xi))=
e(R({\Theta})\cap R(\Xi))=e(R(\Xi))$.\
d) If ${\Theta}$ is the biggest special set with respect to the inclusion, then the orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ is closed due to c).\
If ${\Theta}$ is not the biggest special set, then due to c) we find $$\begin{aligned}
G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f &=& {\overline{\overline{\overline{ G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f}}}}\,\setminus\,\bigcup_{\Xi\,sp.\,,\,\Xi\supsetneqq {\Theta}}{\overline{\overline{\overline{G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f}}}}\;\;.\end{aligned}$$ There are only finitely many special sets. Therefore $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ is locally closed.\
e) The algebra of strongly regular functions ${\mbox{${\mathbb{F}}\,[G]$}}$ is an integral domain. Therefore every subset of ${\overline{\overline{G}}}=\widehat{G_f}$, which contains $G$, is irreducible. In particular $G_f$ is irreducible. From Theorem \[SettingirreducibleOrbits\] follows, that the $G_f\times G_f$-orbits of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ are irreducible.\
[\
]{}Our next aim is to define and describe big cells of every $G_f\times G_f$-orbit of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. As a preparation we first determine certain principal open sets of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
Recall that ${\theta}_{\Lambda}$ denotes the function defined by ${\theta}_{\Lambda}:=\frac{1}{{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}} f_{v_{\Lambda}v_{\Lambda}}$, where $v_{\Lambda}\in L({\Lambda})_{\Lambda}\setminus\{0\}$, ${\Lambda}\in P^+$. Recall the multiplicative property ${\theta}_{\Lambda}{\theta}_{{\Lambda}'}={\theta}_{{\Lambda}+{\Lambda}'}$, ${\Lambda},{\Lambda}'\in P^+$.
\[S1b\] Let ${\Theta}$ be special. The principal open subset of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ associated with ${\theta}_{\Lambda}$, ${\Lambda}\in F_{\Theta}\cap P$, is given by $$\begin{aligned}
\label{principalopen}
\dot{ \bigcup_{\hat{\sigma}\in \widehat{\cal W}_{\Theta}} } U_f{{\,\bf \diamond\,}}(\hat{\sigma} T) U_f \;\;.\end{aligned}$$ This set, as well as its coordinate ring as a principal open set, is independent of the chosen element ${\Lambda}\in F_{\Theta}\cap P$.
We denote this principal open set by $D({\Theta})$, and its coordinate ring as a principal open set by ${\mbox{${\mathbb{F}}\,[D({\Theta})]$}}$.\
a) We first show that the principal open set of ${\theta}_{\Lambda}$ is given by (\[principalopen\]): Due to the Birkhoff decomposition, every element of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ can be written in the form $u{{\,\bf \diamond\,}}n_\sigma e(R){\tilde}{u}$, with $u,{\tilde}{u}\in U_f$, $n_\sigma\in N$, and $R$ a face of $X$. Let $v_{\Lambda}\in L({\Lambda})_{\Lambda}\setminus\{0\}$. We find $$\begin{aligned}
0\;\,\neq\;\, {\theta}_{\Lambda}(u{{\,\bf \diamond\,}}n_\sigma e(R){\tilde}{u})\;\,=\;\,\frac{ {\left\langle \left\langle uv_{\Lambda}\mid n_\sigma e(R) {\tilde}{u} v_{\Lambda}\right\rangle \right\rangle} }{{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}}
\;\,=\;\, \frac{ {\left\langle \left\langle v_{\Lambda}\mid n_\sigma e(R) v_{\Lambda}\right\rangle \right\rangle} }{{\left\langle \left\langle v_{\Lambda}\mid v_{\Lambda}\right\rangle \right\rangle}}\end{aligned}$$ if and only if ${\Lambda}\in R$ and $\sigma {\Lambda}={\Lambda}$. Because ${\Lambda}$ is an interior point of the face $R({\Theta})$, the first condition is equivalent to $R({\Theta})\subseteq R$. The second condition is equivalent to $\sigma\in{\mbox{$\cal W$}}_{\Theta}$.\
Because of $$\begin{aligned}
{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_{\Theta}&=& \dot{\bigcup_{S\;a\; face\; of\; X\atop S\supseteq R({\Theta})}} {\mbox{$\cal W$}}_{\Theta}\;{\mbox{$\varepsilon\left(S\right)$}}\;\;,\end{aligned}$$ we get $\sigma {\mbox{$\varepsilon\left(R\right)$}}\in{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_{\Theta}$. We also find that for every $\hat{\sigma}\in{\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_{\Theta}$ we have $U_f{{\,\bf \diamond\,}}\hat{\sigma}T U_f\subseteq D({\theta}_{\Lambda})$.\
b) Clearly (\[principalopen\]) does not depend on ${\Lambda}\in F_{\Theta}\cap P$. To show that the coordinate rings of the principal open sets are independent of ${\Lambda}\in F_{\Theta}\cap P$, we only have to show, that for any $N, N'\in F_{\Theta}\cap P$ there exists a function $f$ of the coordinate ring of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, and an integer $n\in{\mathbb{N}}$, such that $$\begin{aligned}
\label{fquotient}
\frac{f}{({\theta}_N)^n}{\!\mid_{D({\Theta})}}&=& \frac{1}{{\theta}_{N'}}{\!\mid_{D({\Theta})}}\;\;.\end{aligned}$$ Because $F_{\Theta}$ is open in the linear span of $F_{\Theta}$, there exists an integer $n\in{\mathbb{N}}$, such that $N-\frac{1}{n}N'\in F_{\Theta}$. Since $F_{\Theta}$ is a cone, we find $nN-N'\in F_{\Theta}\cap P$. The function $f={\theta}_{nN-N'}$ satisfies equation (\[fquotient\]).\
[\
]{}Let ${\Theta}$ be special. We call the set $BC({\Theta}):=U_f {{\,\bf \diamond\,}}e(R({\Theta}))T U_f$, as well as every translate $U_f g{{\,\bf \diamond\,}}e(R({\Theta}))T U_f h$, where $g,h\in G_f$, a big cell of the orbit $G_f {{\,\bf \diamond\,}}e(R({\Theta}))G_f$. This name is justified by the following three theorems.
\[S2\] Let ${\Theta}$ be special. The big cell $BC({\Theta})$ is principal open in the closure of $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$, i.e., $$\begin{aligned}
\label{bcint}
BC({\Theta}) &=& {\overline{\overline{\overline{G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f}}}}\cap D({\Theta})\;\;.\end{aligned}$$ The big cell $BC({\Theta})$ is dense in the closure of $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$.
a\) We first show formula (\[bcint\]). Due to the Birkhoff decomposition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, and the formulas (\[closure\]), (\[orbit\]), and (\[principalopen\]), we only have to show $$\begin{aligned}
\{\,{\mbox{$\varepsilon\left(R({\Theta})\right)$}}\,\} &=& \bigcup_{\Xi\;sp.\,,\,\Xi\supseteq {\Theta}}{\mbox{$\cal W$}}\,{\mbox{$\varepsilon\left(R(\Xi)\right)$}} \,{\mbox{$\cal W$}}\;\;\cap\;\; {\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_{\Theta}\;\;. \end{aligned}$$ Because of ${\mbox{$\widehat{{\mbox{$\cal W$}}}$}}_{\Theta}= \bigcup_{\Xi\,sp.,\,\Xi\subseteq{\Theta}} {\mbox{$\cal W$}}_{\Theta}\,{\mbox{$\varepsilon\left(R(\Xi)\right)$}}\, {\mbox{$\cal W$}}_{\Theta}$, the intersection on the right equals ${\mbox{$\cal W$}}_{\Theta}\,{\mbox{$\varepsilon\left(R({\Theta})\right)$}}\,{\mbox{$\cal W$}}_{\Theta}$. Since ${\mbox{$\cal W$}}_{\Theta}$ is the pointwise stabilizer of $R({\Theta})$, its elements fix ${\mbox{$\varepsilon\left(R({\Theta})\right)$}}$. Therefore this intersection contains only the element ${\mbox{$\varepsilon\left(R({\Theta})\right)$}}$.\
b) Due to Theorem \[S1\] the orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$ is irreducible. Therefore also its closure is irreducible. Because the big cell is open in the closure, and nonempty, it is dense.\
[\
]{}Let ${\Theta}$ be special. We equip the big cell $BC({\Theta})$ with its coordinate ring ${\mbox{${\mathbb{F}}\,[BC({\Theta})]$}}$ as a principal open set in the closure of $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$.\
Set $T^{\Theta}:=T_{I\setminus {\Theta}}T_{rest}$. Set $P^{\Theta}:={\mathbb{Z}}{\mbox}{-}span{\left\{\left.\;{\Lambda}_i\;\right|\; i=1,\ldots,2n-l,\;i\notin {\Theta}\;\right\}}$, and identify the group algebra ${\mbox{${\mathbb{F}}\,[P^{\Theta}]$}}$ with the classical coordinate ring of the torus $T^{\Theta}$.\
Note that due to Theorem \[SpU2\] the coordinate ring ${\mbox{${\mathbb{F}}\,[U_f^{\Theta}]$}}$ is isomorphic to ${\mbox{${\mathbb{F}}\,[U^{\Theta}]$}}$ by the restriction map.
\[S3\] Let ${\Theta}$ be special. We get an isomorphism $$\begin{aligned}
m:\;U_f^{\Theta}\times T^{\Theta}\times U_f^{\Theta}&\to & BC({\Theta})\end{aligned}$$ by $m(u,t,{\tilde}{u}):=u{{\,\bf \diamond\,}}e(R({\Theta})) t{\tilde}{u}$, where $u,\,{\tilde}{u}\in U_f^{\Theta}$ and $t\in T^{\Theta}$.
a\) First we show, that $m$ is surjective: Due to [@M], Proposition 2.13, compare the section preliminaries, we have $e(R({\Theta}))T_{\Theta}=e(R({\Theta}))$. Because of $T=T_{\Theta}T^{\Theta}$ we get $e(R({\Theta}))T=e(R({\Theta}))T^{\Theta}$.\
Due to the same proposition we also have $e(R({\Theta}))U_{\Theta}=e(R({\Theta}))$. Left and right multiplications with elements of ${\overline{G}}={\mbox{$\widehat{G}$}}$ are Zariski continuous on ${\overline{\overline{G}}}=\widehat{G_f}$. Therefore we get by using Theorem \[SpU2\]: $$\begin{aligned}
e(R({\Theta})) (U_f)_{\Theta}\;\,=\;\, e(R({\Theta})){\overline{\overline{U_{\Theta}}}} \;\,\subseteq \;\, {\overline{\overline{e(R({\Theta}))U_{\Theta}}}}\;\,=\;\,
{\overline{\overline{\{e(R({\Theta}))\} }}} \;\,=\;\, \{e(R({\Theta}))\}\;\;. \end{aligned}$$ Because of $U_f=(U_f)_{\Theta}\ltimes (U_f)^{\Theta}$ we find $e(R({\Theta}))U_f=e(R({\Theta}))(U_f)^{\Theta}$.\
Because of these formulas, and because ${\mbox{$\widehat{T}$}}$ is abelian, we get: $$\begin{aligned}
U_f{{\,\bf \diamond\,}}e(R({\Theta}))TU_f &=& U_f{{\,\bf \diamond\,}}e(R({\Theta})) T^{\Theta}(U_f)^{\Theta}\;\,= \;\;e(R({\Theta}))U_f{{\,\bf \diamond\,}}T^{\Theta}(U_f)^{\Theta}\\
&=& (U_f)^{\Theta}{{\,\bf \diamond\,}}e(R({\Theta})) T^{\Theta}(U_f)^{\Theta}\;\;.\end{aligned}$$ b) Next we show that the comorphism $m^*: {\mbox{${\mathbb{F}}\,[BC({\Theta})]$}} \to {\mbox{${\mathbb{F}}\,[U_f^{\Theta}]$}}\otimes {\mbox{${\mathbb{F}}\,[P^{\Theta}]$}}\otimes {\mbox{${\mathbb{F}}\,[U_f^{\Theta}]$}}$ is well defined and surjective: For ${\Lambda}\in P^+$ choose ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases of $L({\Lambda})$, by choosing ${\left\langle \left\langle \;\ \mid \;\ \right\rangle
\right\rangle}$-dual bases $$\begin{aligned}
(a_{{\lambda}i})_{i=1,\ldots,m_{\lambda}} &\quad,\quad & (b_{{\lambda}i})_{i=1,\ldots,m_{\lambda}}\end{aligned}$$ of $L({\Lambda})_{\lambda}$ for every ${\lambda}\in P({\Lambda})$. Fix elements $N\in F_{\Theta}\cap P$ and $v_N\in L(N)_N\setminus\{0\}$. For $v,w\in L({\Lambda})$, and $u,{\tilde}{u}\in U_f^{\Theta}$, $t\in T^{\Theta}$ we get $$\begin{aligned}
\frac{f_{vw}}{({\theta}_N)^k}\,(u{{\,\bf \diamond\,}}e(R({\Theta}))t{\tilde}{u})\;\,=\;\,
\sum_{{\lambda}\in P({\Lambda})\cap R({\Theta}),\;i} {\left\langle \left\langle uv\mid a_{{\lambda}i}
\right\rangle \right\rangle}e_{{\lambda}-k N}(t) {\left\langle \left\langle b_{{\lambda}i}\mid {\tilde}{u} w
\right\rangle \right\rangle}\;\;.\end{aligned}$$ This sum has only finitely many nonzero summands, because $supp(uv)$ and $supp({\tilde}{u}w)$ are finite. Denote by $p^{\Theta}:P\to P^{\Theta}$ the projection corresponding to the decomposition $P=P_{\Theta}\oplus P^{\Theta}$. Due to the last formula we have $$\begin{aligned}
\frac{f_{vw}}{({\theta}_N)^k} {\!\mid_{BC({\Theta})}}\,\circ\, m\;\,=\;\,
\sum_{{\lambda}\in P({\Lambda})\cap R({\Theta}),\;i} f_{a_{{\lambda}i} v}{\!\mid_{U^{\Theta}}}\otimes\, e_{p^{\Theta}({\lambda})-kN}\otimes f_{b_{{\lambda}i} w}{\!\mid_{U^{\Theta}}}\;\;.\end{aligned}$$ There are only finitely many nonzero summands of this sum. The function $f_{a_{{\lambda}i}v}{\!\mid_{U^{\Theta}}}$ is nonzero at most if $v\neq 0$ and if ${\lambda}$ is bigger than a weight of $supp(v)$, which is only possible for finitely many weights ${\lambda}$ in $P({\Lambda})$. Similar things hold for $f_{b_{{\lambda}i}w}{\!\mid_{U^{\Theta}}}$.\
In particular, $m^*$ is well defined. From this formula we find for $v\in L(N)$ and ${\Lambda}\in\overline{F_{\Theta}}\cap P$: $$\begin{aligned}
m^*(\frac{f_{v v_N}}{{\theta}_N}) &=& f_{v_N v}{\!\mid_{U^{\Theta}}}\otimes\, 1\otimes 1 \;\;,\\
m^*(\frac{{\theta}_{\Lambda}}{({\theta}_N)^k}) &=& 1\otimes e_{{\Lambda}-k N}\otimes 1 \;\;,\\
m^*(\frac{f_{v_N v}}{{\theta}_N}) &=& 1\otimes 1\otimes f_{v_N v}{\!\mid_{U^{\Theta}}}\;\;.\end{aligned}$$ It is easy to see, that $(\overline{F_{\Theta}}\cap P)-{\mathbb{N}_0}N= P^{\Theta}$. Taking into account Theorem \[SpU1\], 2), we have found elements of ${\mbox{${\mathbb{F}}\,[BC({\Theta})]$}}$, which are mapped onto a system of generators of ${\mbox{${\mathbb{F}}\,[U_f^{\Theta}]$}}\otimes {\mbox{${\mathbb{F}}\,[P^{\Theta}]$}}\otimes {\mbox{${\mathbb{F}}\,[U_f^{\Theta}]$}}$. Therefore $m^*$ is surjective.\
c) $m^*$ is injective, because $m$ is surjective. To show the injectivity of $m$, let $u_1,u_2,{\tilde}{u}_1,{\tilde}{u}_2\in U_f^{\Theta}$, $t_1,t_2\in T^{\Theta}$ such that $m(u_1,t_1,{\tilde}{u}_1)=m(u_2,t_2,{\tilde}{u}_2)$. Then for all $f\in {\mbox{${\mathbb{F}}\,[U^{\Theta}]$}}$ we have $$\begin{aligned}
f(u_1)\;\,=\;\, (m^*)^{-1}(f\otimes 1\otimes 1)(\underbrace{m(u_1,t_1,{\tilde}{u}_1)}_{=m(u_2,t_2,{\tilde}{u}_2)}) \;\,=\;\, f(u_2)\;\;.\end{aligned}$$ Therfore we find $u_1=u_2$. In a similar way, we get $t_1=t_2$, and ${\tilde}{u}_1={\tilde}{u}_2$.\
[\
]{}In the next theorem we give countable coverings of every $G_f\times G_f$-orbit of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ by big cells.\
Recall that ${\mbox{$\cal W$}}^J$ denotes the set of minimal coset representatives of ${\mbox{$\cal W$}}/{\mbox{$\cal W$}}_J$, and ${\mbox}{}^J{\mbox{$\cal W$}}$ denotes the set of minimal coset representatives of ${\mbox{$\cal W$}}_J\backslash {\mbox{$\cal W$}}$, $J\subseteq I$.
Let ${\Theta}$ be special. We have $$\begin{aligned}
G_f {{\,\bf \diamond\,}}e(R({\Theta}))G_f &=& \bigcup_{\sigma\in {\mbox}{}^{\Theta}{\cal W}\;,\;\tau\in {\mbox}{}^{{\Theta}\cup{\Theta}^\bot}{\cal W}} U_f
\sigma {{\,\bf \diamond\,}}e(R({\Theta}))T U_f \tau \\
&=& \bigcup_{\sigma\in {\mbox}{}^{{\Theta}\cup{\Theta}^\bot}{\cal W}\;,\;\tau\in {\mbox}{}^{\Theta}{\cal W} } U_f
\sigma {{\,\bf \diamond\,}}e(R({\Theta}))T U_f \tau \;\;.\end{aligned}$$
It is sufficient to prove the second covering of the theorem, the first follows by applying the Chevalley involution of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. Obviously the sets $U_f\sigma {{\,\bf \diamond\,}}e(R({\Theta}))T U_f \tau $ are contained in the orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f$. Therefore it is sufficient to show that the orbit $G_f{{\,\bf \diamond\,}}e(R({\Theta})) G_f$ is contained in the second union.\
By writing the elements ${\mbox{$\cal W$}}{\mbox{$\varepsilon\left(R({\Theta})\right)$}}{\mbox{$\cal W$}}$ in normal form, and inserting in (\[orbit\]), we get $$\begin{aligned}
G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f &=& \bigcup_{ \sigma\in{\cal W}^{{\Theta}\cup{\Theta}^\bot},\;\tau\in {\mbox}{}^{\Theta}{\cal W} }
U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta}))T\tau U_f\;\;.\end{aligned}$$ To transform the expression $U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta}))T\tau U_f$, we use the following decomposition of $U_f$, associated to an element $w\in{\mbox{$\cal W$}}$, proved in [@Sl1], Section 5.5:\
Set ${\mbox{$\Delta$}}^+_w:={\left\{\left.\;{\alpha}\in {\mbox{$\Delta^+$}}\;\right|\; w{\alpha}\in{\mbox{$\Delta^-$}}\;\right\}}$ and $({\mbox{$\Delta^+$}})^w:={\left\{\left.\;{\alpha}\in {\mbox{$\Delta^+$}}\;\right|\; w{\alpha}\in{\mbox{$\Delta^+$}}\;\right\}}$. Then ${\mbox{$\Delta$}}^+_w$ consists of finitely many positive real roots, and we have $$\begin{aligned}
U_f \;\,=\;\,U_w\,U_f^w &{\mbox}{ where }& U_w:=\exp(\bigoplus_{{\alpha}\in \Delta_w^+}{\mbox{$\bf g$}}_{\alpha}) \;,\;
U_f^w:=\exp(\prod_{{\alpha}\in (\Delta^+)^w}{\mbox{$\bf g$}}_{\alpha})\;\;.\end{aligned}$$ Using this decomposition, and Proposition 2.14 of [@M], compare the section preliminaries, we find $$\begin{aligned}
U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta})) T \tau U_f \;\,=\;\, U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta})) T\underbrace{\tau U_\tau \tau^{-1}}_{\subseteq U^-}
\underbrace{\tau U_f^\tau\tau^{-1}}_{\subseteq U_f}\tau\\
\subseteq\;\,U_f{{\,\bf \diamond\,}}\sigma U^-_{{\Theta}^\bot} e(R({\Theta})) T U_f\tau \;\,=\;\,
(\sigma U_{{\Theta}^\bot}^-\sigma^{-1})^* U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta})) T U_f\tau\;\;. \end{aligned}$$ Because of $\sigma\in {\mbox{$\cal W$}}^{{\Theta}\cup{\Theta}^\bot}={\left\{\left.\;\sigma\in{\mbox{$\cal W$}}\;\right|\; \sigma{\alpha}_i\in{\mbox{$\Delta_{re}^+$}}{\mbox}{ for all } i\in {\Theta}\cup{\Theta}^\bot\;\right\}}$, we have $\sigma U^-_{{\Theta}^\bot}\sigma^{-1}\subseteq U^-$, and the last expression equals $U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta}))T U_f\tau$. By doing similar transformations, we get $$\begin{aligned}
\lefteqn{ U_f{{\,\bf \diamond\,}}\sigma e(R({\Theta}))T U_f\tau \;\,=\;\,\sigma^{-1}U_f{{\,\bf \diamond\,}}e(R({\Theta}))T U_f\tau } \\
&=& \underbrace{(\sigma^{-1}U_{\sigma^{-1}}\sigma)}_{\subseteq U^-}
\underbrace{(\sigma^{-1} U_f^{\sigma^{-1}}\sigma )}_{\subseteq U_f}\sigma^{-1} {{\,\bf \diamond\,}}e(R({\Theta}))T U_f\tau
\;\,\subseteq\;\, U_f\sigma^{-1}{{\,\bf \diamond\,}}U e(R({\Theta})) T U_f\tau \\
&=& U_f\sigma^{-1}{{\,\bf \diamond\,}}e(R({\Theta}))T U_{{\Theta}^\bot}U_f\tau
\;\,=\;\, U_f\sigma^{-1}{{\,\bf \diamond\,}}e(R({\Theta}))T U_f\tau \;\;.\end{aligned}$$ Because of $({\mbox{$\cal W$}}^{{\Theta}\cup{\Theta}^\bot})^{-1}={\mbox}{}^{{\Theta}\cup{\Theta}^\bot}{\mbox{$\cal W$}}$ we have shown, that the orbit is contained in the second union.\
[\
]{}Our last aim is to show, that there exist stratified transversal slices to the $G_f\times G_f$-orbits of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ at any of their points.\
Because $G_f\times G_f$ acts by isomorphisms on ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$, it is sufficient to find stratified transversal slices at the points $1{{\,\bf \diamond\,}}e(R({\Theta}))\in G_f{{\,\bf \diamond\,}}e(R({\Theta})) G_f$, ${\Theta}$ special.\
As transversal slice at $1{{\,\bf \diamond\,}}e(R({\Theta}))$, we will use the closure ${\overline{\overline{\overline{G_{\Theta}}}}}:= {\overline{\overline{\overline{1{{\,\bf \diamond\,}}G_{\Theta}}}}}$, equipped with its coordinate ring as a closed subset of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$. We have the following description:
Let ${\Theta}$ be special. The restriction map ${\mbox{${\mathbb{F}}\,[G]$}}\to{\mbox{${\mathbb{F}}\,[G_{\Theta}]$}}$ induces a closed embedding ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{\Theta}]$}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ with image $$\begin{aligned}
{\overline{\overline{\overline{G_{\Theta}}}}} &=& (\widehat{G_f})_{\Theta}{{\,\bf \diamond\,}}(\widehat{G_f})_{\Theta}\;\,=\;\,
\dot{\bigcup_{\Xi\subseteq {\Theta},\,\Xi\,special}} (G_f)_{\Theta}{{\,\bf \diamond\,}}e(R(\Xi)) (G_f)_{\Theta}\\
&=& \dot{ \bigcup_{\hat{\sigma}\in \widehat{\cal W}_{\Theta}} } (U_f)_{\Theta}{{\,\bf \diamond\,}}(\hat{\sigma} T_{\Theta}) (U_f)_{\Theta}\;\;.\end{aligned}$$
It is not difficult to check, that the map ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{\Theta}]$}}\to{\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ is a closed embedding with image ${\overline{\overline{\overline{G_{\Theta}}}}}$. Due to Theorem \[SpG1\], $(\widehat{G_f})_{\Theta}\times (\widehat{G_f})_{\Theta}$ maps surjectively to ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{\Theta}]$}}$. To show the first equation of the theorem, we have to show, that the concatenation of the maps $$\begin{aligned}
(\widehat{G_f})_{\Theta}\times (\widehat{G_f})_{\Theta}\;\,\stackrel{{\alpha}}{\twoheadrightarrow}\;\, {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G_{\Theta}]$}} \;\,\to\;\, {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}\end{aligned}$$ coincides with the restricted map ${{\,\bf \diamond\,}}:(\widehat{G_f})_{\Theta}\times (\widehat{G_f})_{\Theta}\to {\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$.\
Let $x,y\in (\widehat{G_f})_{\Theta}$. Let $v,w\in L({\Lambda})$, ${\Lambda}\in P^+$. Choose a decomposition $L({\Lambda})=\bigoplus_{j\in J} V_j$ of $L({\Lambda})$ in an orthogonal direct sum of irreducible highest weight $({\mbox{$\bf g$}}_{\Theta}+{\mbox{\textbf{h}}})$-modules. Write $v$, $w$ as sums $v=\sum_{j\in J} v_j$, $w=\sum_{j\in J} w_j$ with $v_j,w_j\in V_j$. Then: $$\begin{aligned}
{\alpha}(x,y)(f_{vw}{\!\mid_{G_{\Theta}}}) &=& {\alpha}(x,y)(\,\sum_{j\in J}f_{v_j w_j}{\!\mid_{G_{\Theta}}}\,)\;\,=\;\,\sum_{j\in J} {\left\langle \left\langle xv_j\mid yw_j
\right\rangle \right\rangle}\\
&=&{\left\langle \left\langle xv\mid yw
\right\rangle \right\rangle}\;\,=\;\, (x{{\,\bf \diamond\,}}y) (f_{vw})\end{aligned}$$ The other equations of the theorem follow from $$\begin{aligned}
&&(\widehat{G_f})_{\Theta}{{\,\bf \diamond\,}}(\widehat{G_f})_{\Theta}\;=\; U^-_{\Theta}{\mbox{$\widehat{N}$}}_{\Theta}(U_f)_{\Theta}{{\,\bf \diamond\,}}(\widehat{G_f})_{\Theta}\;=\;
(U_f)_{\Theta}{{\,\bf \diamond\,}}(U^-_{\Theta}{\mbox{$\widehat{N}$}}_{\Theta})^* (\widehat{G_f})_{\Theta}\\
&& =\; (U_f)_{\Theta}{{\,\bf \diamond\,}}(\widehat{G_f})_{\Theta}\;=\; (U_f)_{\Theta}{{\,\bf \diamond\,}}U^-_{\Theta}{\mbox{$\widehat{N}$}}_{\Theta}(U_f)_{\Theta}\;=\; (U^-_{\Theta})^*(U_f)_{\Theta}{{\,\bf \diamond\,}}{\mbox{$\widehat{N}$}}_{\Theta}(U_f)_{\Theta}\\
&& =\;(U_f)_{\Theta}{{\,\bf \diamond\,}}{\mbox{$\widehat{N}$}}_{\Theta}(U_f)_{\Theta}\; \subseteq \;\bigcup_{\Xi\subseteq {\Theta},\,\Xi\,special} (G_f)_{\Theta}{{\,\bf \diamond\,}}e(R(\Xi)) (G_f)_{\Theta}\;\subseteq\;
(\widehat{G_f})_{\Theta}{{\,\bf \diamond\,}}(\widehat{G_f})_{\Theta}\;\;. \end{aligned}$$ The unions in the equations of the theorem are disjoint, because the unions $\bigcup_{\Xi\, special} G_f{{\,\bf \diamond\,}}e(R(\Xi)) G_f$, $\bigcup_{\hat{w}\in\hat{\cal W}} U_f{{\,\bf \diamond\,}}\hat{w}T U_f$ are disjoint.\
[\
]{}As open neighborhood of $1{{\,\bf \diamond\,}}e(R({\Theta}))$, we take the principal open set $D({\Theta})$.
Let ${\Theta}$ be special.\
1) We have $G_f{{\,\bf \diamond\,}}e(R({\Theta}))G_f\cap D({\Theta})=BC({\Theta})$. The big cell $BC({\Theta})$ is closed in the principal open set $D({\Theta})$. Its coordinate ring coincides with the coordinate ring as a closed subset of the principal open set $D({\Theta})$.\
2) We have $1{{\,\bf \diamond\,}}e(R({\Theta}))\in{\overline{\overline{\overline{G_{\Theta}}}}}\subseteq D({\Theta})$. The coordinate ring of ${\overline{\overline{\overline{G_{\Theta}}}}}$ coincides with the coordinate ring as a closed subset of the principal open set $D({\Theta})$.\
3 a) We get an isomorphism $$\begin{aligned}
\Psi:\; {\overline{\overline{\overline{G_{\Theta}}}}}\,\times \,BC({\Theta}) &\to & D({\Theta})\end{aligned}$$ by $\Psi(x{{\,\bf \diamond\,}}y, u{{\,\bf \diamond\,}}e(R({\Theta}))t{\tilde}{u}):= xu{{\,\bf \diamond\,}}yt{\tilde}{u}$, where $x,y\in(\widehat{G_f})_{\Theta}$, $u,{\tilde}{u}\in (U_f)^{\Theta}$, and $t\in T^{\Theta}$.\
b) Inserting $1{{\,\bf \diamond\,}}e(R({\Theta}))$ in the first entry (resp. second entry) of $\Psi$ induces the identity map on $BC({\Theta})$ (resp. ${\overline{\overline{\overline{G_{\Theta}}}}}$).\
c) The partition of ${\mbox{Specm\,}}{\mbox{${\mathbb{F}}\,[G]$}}$ into $G_f\times G_f$-orbits induces partitions of ${\overline{\overline{\overline{G_{\Theta}}}}}$ and $D({\Theta})$. $\Psi$ preserves the orbits, i.e., $$\begin{aligned}
\Psi(\,{\overline{\overline{\overline{G_{\Theta}}}}}\cap G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f\,,\, BC({\Theta})\,) \;\,=\;\,D({\Theta})\,\cap\, G_f{{\,\bf \diamond\,}}e(R(\Xi))G_f \quad,\quad \Xi {\mbox}{ special}\;\;.\end{aligned}$$
1\) The first part of the theorem follows immediately from the definition of the big cell and its coordinate ring.\
2) From the last theorem and the description of $D({\Theta})$ given in Theorem \[S1b\] follows $1{{\,\bf \diamond\,}}e(R({\Theta}))\in{\overline{\overline{\overline{G_{\Theta}}}}}\subseteq D({\Theta})$.\
The statement of part 2) about the coordinate rings can be seen as follows: $D({\Theta})$ is the principal open set of ${\theta}_{\Lambda}$ for an element ${\Lambda}\in F_{\Theta}\cap P$. Now $G_{\Theta}$ stabilizes every point of the highest weight space $L({\Lambda})_{\Lambda}$, because for $j\in {\Theta}$ the ${\alpha}_j$-string of $P({\Lambda})$ through ${\Lambda}$ consists only of ${\Lambda}$. Therefore ${\theta}_{\Lambda}$ takes the value 1 on $1{{\,\bf \diamond\,}}G_{\Theta}$, and also on the closure ${\overline{\overline{\overline{1{{\,\bf \diamond\,}}G_{\Theta}}}}}={\overline{\overline{\overline{G_{\Theta}}}}}$.\
3) Because of Theorem \[S3\], the statement of 3 a) is equivalent to the following statement, which we will prove: We get a bijective map $$\begin{aligned}
{\tilde}{\Psi}:\, (U_f)^{\Theta}\times {\overline{\overline{\overline{G_{\Theta}}}}}\times T^{\Theta}\times (U_f)^{\Theta}&\to & D({\Theta})\end{aligned}$$ by $\Psi(u, x{{\,\bf \diamond\,}}y, t, {\tilde}{u}):= xu{{\,\bf \diamond\,}}yt{\tilde}{u}=(x{{\,\bf \diamond\,}}y)(u,t{\tilde}{u})$, where $x,y\in(\widehat{G_f})_{\Theta}$, $u,{\tilde}{u}\in (U_f)^{\Theta}$, $t\in T^{\Theta}$. Its comorphism exists and is an isomorphism of algebras.\
From the the descriptions of $D({\Theta})$ and ${\overline{\overline{\overline{G_{\Theta}}}}}$ given in Theorem \[S1b\] and in the last theorem follows, that the image of the map ${\tilde}{\Psi}$ is $D({\Theta})$.\
Next we show that the comorphism ${\tilde}{\Psi}^*$ exists and is surjective: Choose an element ${\Lambda}\in F_{\Theta}\cap P_I$. It is easy to check, that the multiplication maps $D_{G'}({\theta}_{\Lambda})\times T_{rest}\to D_G({\theta}_{\Lambda})$, and $T_{I\setminus{\Theta}}\times T_{rest}\to T^{\Theta}$ are bijective. Furthermore their comorphisms are isomorphisms ${\mbox{${\mathbb{F}}\,[D_G({\theta}_{\Lambda})]$}}\to {\mbox{${\mathbb{F}}\,[D_{G'}({\theta}_{\Lambda})]$}}\otimes {\mbox{${\mathbb{F}}\,[P_{rest}]$}}$, and ${\mbox{${\mathbb{F}}\,[P^{\Theta}]$}}\to {\mbox{${\mathbb{F}}\,[P_{I\setminus{\Theta}}]$}}\otimes {\mbox{${\mathbb{F}}\,[P_{rest}]$}}$. Also the comorphism of the bijective map $*:U^{\Theta}\to (U^{\Theta})^-$ is an isomorphism ${\mbox{${\mathbb{F}}\,[(U^{\Theta})^-]$}}\to {\mbox{${\mathbb{F}}\,[U^{\Theta}]$}}$. Taking into account Theorem \[Ueb2\] for $J=I$ and $L={\Theta}$, we find that the map $$\begin{aligned}
{\tilde}{m}:\, U^{\Theta}\times G_{\Theta}\times T^{\Theta}\times U^{\Theta}&\to & D_G({\theta}_{\Lambda})\end{aligned}$$ given by ${\tilde}{m}(u,g,t,{\tilde}{u}):=u^* g t {\tilde}{u}$ is bijective, its comorphism exists, and is an isomorphism of algebras.\
Now identify $G$ with $1{{\,\bf \diamond\,}}G$. Then $D_G({\theta}_{\Lambda})$ is contained in $D({\Theta})$. Due to Theorem \[SpU2\] the set $U^{\Theta}$ is dense in $(U_f)^{\Theta}$. The coordinate ring ${\mbox{${\mathbb{F}}\,[(U_f)^{\Theta}]$}}$ is isomorphic to ${\mbox{${\mathbb{F}}\,[U^{\Theta}]$}}$ by the restriction map. Similar things hold for $G_{\Theta}$, ${\overline{\overline{\overline{G_{\Theta}}}}}$ and their coordinate rings.\
For a coordinate ring ${\mbox{${\mathbb{F}}\,[B]$}}$ and a nonempty subset $A\subseteq B$ denote by $res_A^B$ the restriction map ${\mbox{${\mathbb{F}}\,[B]$}}\to {\mbox{${\mathbb{F}}\,[A]$}}$. It is easy to check, that the surjective map $$\begin{aligned}
\left( res_{\,U^{\Theta}}^{(U_f)^{\Theta}}\,\otimes\, res_{G_{\Theta}}^{{\overline{\overline{\overline{G_{\Theta}}}}}} \,\otimes\,res_{T^{\Theta}}^{T^{\Theta}}\,\otimes\,res_{\,U^{\Theta}}^{(U_f)^{\Theta}}\right)^{-1}
\;\circ\;{\tilde}{m}^*\;\circ\;res_{D_G({\theta}_{\Lambda})}^{\,D({\Theta})} \end{aligned}$$ is the comorphism of ${\tilde}{\Psi}$. (Make use of ${\tilde}{\Psi}(u,g,t,{\tilde}{u})={\tilde}{m}(u,g,t,{\tilde}{u})$ for $u,{\tilde}{u}\in U^{\Theta}$, $g\in G_{\Theta}$, and $t\in T^{\Theta}$.)\
Because the maps ${\tilde}{\Psi}$ and ${\tilde}{\Psi}^*$ are surjective, they are also injective. This is shown in the same way as the injectivity of the maps $m$ and $m^*$ in the proof of Theorem \[S3\].\
3 b) follows immediately from the definition of $\Psi$. 3 c) can be checked easily by using the definition and the bijectivity of $\Psi$, and the last theorem.\
[\
]{}\
[**Acknowledgment**]{} I would like to thank the Deutsche Forschungsgemeinschaft for providing my main financial support, when most of this paper has been written. I also would like to thank the TMR-Program ERB FMRX-CT97-0100 “Algebraic Lie Theory” for providing some financial support for traveling.\
Furthermore I would like to thank the referees for their useful comments.
[AAAA1]{}
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---
abstract: 'The notion of vertex operator coalgebra is presented which corresponds to the family of correlation functions of one string propagating in space-time splitting into n strings in conformal field theory. This notion is in some sense dual to the notion of vertex operator algebra. We prove that any vertex operator algebra equipped with a non-degenerate, Virasoro preserving, bilinear form gives rise to a corresponding vertex operator coalgebra.'
author:
- Keith Hubbard
title: Constructions of vertex operator coalgebras via vertex operator algebras
---
Introduction
============
The theory of vertex operator algebras has been an ever expanding field since its inception in the 1980s when Borcherds first introduced the precise notion of a vertex algebra ([@B]). Specialized to vertex operator algebras (VOAs) by Frenkel, Lepowsky, and Meurman in [@FLM] and shown to have inherent connections to conformal field theory, modular forms, finite groups and Lie theory, vertex (operator) algebras have gained interest throughout the mathematical community. Subsequently VOAs have been interpreted as a specific type of algebra over an operad ([@HL]), and have possible applications in giving a geometric definition of elliptic cohomology ([@ST]). Quite recently, in [@K] and [@K1], the algebra induced by considering the operad structure of worldsheets swept out by closed strings propagating through space-time, has been supplemented by examining the induced coalgebra structure which gives rise to vertex operator coalgebras (VOCs).
The notion of VOC corresponds to the coalgebra of correlation functions of one string splitting into $n$ strings in space-time, whereas VOAs correspond to the algebra of correlation functions of $n$ strings combining into one string in space-time. Not only are the notions of VOC and VOA integral parts of conformal field theory but the successful integration of the two notions would provide an understanding of the correlation functions arising from any 2-dimensional worldsheet, including higher genus worldsheets.
Examples of VOAs have been quite useful, both in understanding VOA structure and also in understanding other objects (particularly the Monster finite simple group [@FLM]), but have historically been quite challenging to construct. Building a substantial pool of examples has taken decades. (See [@LL] for one recent list of constructions.) One might expect similar challenges in generating examples of VOCs. However, in this paper, via an appropriately defined bilinear form, we generate a large family of examples of VOCs by tapping into the extensive work on VOA examples. We will also explicitly calculate the family of examples corresponding to Heisenberg algebras and demonstrate the naturally adjoint nature of Heisenberg VOAs and VOCs.
Finally, dating back to the 1980s conformal field theories have included axioms about the adjointness of operators induced by manifolds of opposite orientation (cf. [@S]). The construction in this paper indicates that VOC operators satisfy this condition and are adjoint to VOA operators when an appropriate bilinear form exists
The author is indebted to his advisor, Katrina Barron, for her guidance and encouragement throughout the research, writing and revising phases of this paper which corresponds to the final chapter of his Ph.D. thesis. In addition, he thanks Stephan Stolz for numerous helpful conversations. Discussions with Bill Dwyer, Brian Hall, Hai-sheng Li and Corbett Redden were also much appreciated. The author gratefully acknowledges the financial support of the Arthur J. Schmidt Foundation.
Definitions and algebraic preliminaries
=======================================
We begin by reviewing a necessary series from the calculus of formal variables, then recall the definition of vertex operator coalgebra (cf. [@K], [@K1]) and vertex operator algebra (cf. [@FLM], [@FHL]). For later use, we will include two standard consequences of the definition of VOAs.
Delta functions
---------------
We define the “formal $\delta$-function" to be $$\delta(x)= \sum_{n \in \Z} x^n.$$
Given commuting formal variables $x_1$, $x_2$ and $n$ an integer, $(x_1 \pm x_2)^n$ will be understood to be expanded in nonnegative integral powers of $x_2$. (This is the convention throughout vertex operator algebra and coalgebra theory.) Note that the $\delta$-function applied to $\frac{x_1-x_2}{x_0}$, where $x_0$, $x_1$ and $x_2$ are commuting formal variables, is a formal power series in nonnegative integral powers of $x_2$ (cf. [@FLM], [@FHL]).
The notion of vertex operator coalgebra
---------------------------------------
The following description of a vertex operator coalgebra is the central structure of this paper. Originally motivated by the geometry of propagating strings in conformal field theory ([@K], [@K1]), VOCs may be formulated in terms of vectors spaces over an arbitrary field $\C$ using formal commuting variables $x$, $x_1$, $x_2$, $x_3$.
\[D:voc\] A *vertex operator coalgebra (over $\C$) of rank $d \in \C$* is a $\Z$-graded vector space over $\C$
$$V = \coprod_{k \in \Z} V_{(k)}$$
such that $\dim V_{(k)} < \infty$ for $k \in \Z$ and $V_{(k)} = 0$ for $k$ sufficiently small, together with linear maps
$$\begin{aligned}
\co (x) : V &\mapsto (V \otimes V)[[x,x^{-1}]] \\
v &\mapsto \co(x)v = \sum_{k\in \Z} \Delta_k(v) x^{-k-1}
\ \ \ \ \ (\text{where }\Delta_k(v) \in V \otimes V),\end{aligned}$$
$$c : V \mapsto \C,$$
$$\rho : V \mapsto \C,$$
called the *coproduct*, the *covacuum map* and the *co-Virasoro map*, respectively, satisfying the following 7 axioms:
1\. Left Counit: For all $v \in V$
$$\label{E:counit}
(c \otimes Id_V) \co(x)v=v$$
2\. Cocreation: For all $v \in V$
$$\label{E:cocreat1}
(Id_V \otimes c) \co(x)v \in V[[x]] \ \ \text{and}$$
$$\label{E:cocreat2}
\lim_{x \to 0} (Id_V \otimes c) \co(x)v=v.$$
3\. Truncation: Given $v \in V$, then $\Delta_k(v) = 0$ for $k$ sufficiently small.
4\. Jacobi Identity:
$$\begin{gathered}
\label{E:Jac}
x_0^{-1}\delta \left(\frac{x_1-x_2}{x_0} \right)
(Id_V \otimes \co(x_2)) \co(x_1)
-x_0^{-1}\delta \left(\frac{x_2-x_1}{-x_0} \right)
(T \otimes Id_V)\\
(Id_V \otimes \co(x_1)) \co(x_2)
=x_2^{-1}\delta \left(\frac{x_1-x_0}{x_2} \right)
(\co(x_0) \otimes Id_V) \co(x_2).\end{gathered}$$
5\. Virasoro Algebra: The Virasoro algebra bracket,
$$[L(j),L(k)]=(j-k)L(j+k)+\frac{1}{12}(j^3-j)\delta_{j,-k}d,$$
holds for $j, k \in \Z$, where
$$\label{E:L_def}
(\rho \otimes Id_V) \co(x) = \sum_{k \in \Z} L(k) x^{k-2}.$$
6\. Grading: For each $k \in \Z$ and $v \in V_{(k)}$
$$\label{E:VOCgrading}
L(0)v= kv.$$
7\. $L(1)$-Derivative:
$$\label{E:L(1)deriv}
\frac{d}{dx} \co(x)=
(L(1) \otimes Id_V) \co(x).$$
We denote this vertex operator coalgebra by $(V, \co, c, \rho)$ or simply by $V$ when the structure is clear.
Note that $\co$ is linear so that, for example, $(Id_V \otimes \co(x_1))$ acting on the coefficients of $\co(x_2)v \in
(V \otimes V)[[x_2,x_2^{-1}]]$ is well defined. Notice also, that when each expression is applied to any element of $V$, the coefficient of each monomial in the formal variables is a finite sum.
The definition of a VOA {#S:VOAdef}
-----------------------
Vertex operator algebras were first defined in [@FLM], but it was not until [@H2], [@H] that this definition was rigorously tied to the geometry of conformal field theory. This correspondence, along with its operadic interpretation in [@HL], helped to motivate the notion of VOC in [@K]. It is not surprising then, that the axioms of VOA and VOC would strongly resemble each other.
\[D:voa\] A *vertex operator algebra (over $\C$) of rank $d \in \C$* is a $\Z$-graded vector space over $\C$
$$V = \coprod_{k \in \Z} V_{(k)},$$
such that $\dim V_{(k)} < \infty$ for $k \in \Z$ and $V_{(k)} = 0$ for $k$ sufficiently small, together with a linear map $V \otimes V \to V[[x,x^{-1}]]$, or equivalently,
$$\begin{aligned}
Y (\cdot,x): V &\mapsto (\text{End } V) [[x,x^{-1}]] \\
v &\mapsto Y(v,x) = \sum_{k\in \Z} v_k x^{-k-1}
\ \ \ \ \text{ (where } v_n \in \text{End }V),\end{aligned}$$
and equipped with two distinguished homogeneous vectors in $V$, $\mathbf{1}$ (the *vacuum*) and $\omega$ (the *Virasoro element*), satisfying the following 7 axioms:
1\. Left unit: For all $v \in V$
$$\label{E:unit}
Y(\mathbf{1},x)v=v$$
2\. Creation: For all $v \in V$
$$\label{E:creat1}
Y(v,x) \mathbf{1} \in V[[x]] \ \ \text{and}$$
$$\label{E:creat2}
\lim_{x \to 0} Y(v,x) \mathbf{1} =v.$$
3\. Truncation: Given $v,w \in V$, then $v_k w = 0$ for $k$ sufficiently large.
4\. Jacobi Identity: For all $u,v \in V$,
$$\begin{gathered}
\label{E:Jac2}
x_0^{-1}\delta \left(\frac{x_1-x_2}{x_0} \right)
Y(u,x_1)Y(v,x_2)
-x_0^{-1}\delta \left(\frac{x_2-x_1}{-x_0} \right)
Y(v,x_2)Y(u,x_1) \\
=x_2^{-1}\delta \left(\frac{x_1-x_0}{x_2} \right)
Y(Y(u,x_0)v,x_2).\end{gathered}$$
5\. Virasoro Algebra: The Virasoro algebra bracket,
$$[L(j),L(k)]=(j-k)L(j+k)+\frac{1}{12}(j^3-j)\delta_{j,-k}d,$$
holds for $j, k \in \Z$, where
$$\label{E:L_def1}
Y(\omega,x) = \sum_{k \in \Z} L(k) x^{-k-2}.$$
6\. Grading: For each $k \in \Z$ and $v \in V_{(k)}$
$$\label{E:VOCgrading1}
L(0)v= kv.$$
7\. $L(-1)$-Derivative: Given $v \in V$,
$$\label{E:L(-1)deriv}
\frac{d}{dx} Y(v,x)=
Y(L(-1)v,x).$$
We denote a VOA either by $V$ or by the quadruple $(V, Y, \mathbf{1}, \omega)$. A vector $v \in V_{(k)}$ for some $k \in \Z$ is said to be a *homogeneous vector of weight $k$* and we write $\text{wt }v=k$. A pair of basic properties of VOAs will be necessary for our discussion (cf. [@FHL], [@LL]):
$$\begin{aligned}
Y(v,x) \mathbf{1} &=e^{xL(-1)}v \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ for } v \in V, \\
\text{wt } v_k w&=r + s-k-1 \ \ \ \ \ \ \ \ \text{ for } v \in V_{(r)}, \ w \in V_{(s)}\end{aligned}$$
A family of examples of VOCs
============================
One of the most natural questions to ask about vertex operator coalgebras is “what do they look like?", or even, “do any exist?". The main purpose of this paper is to answer the latter question in the affirmative and to provide concrete insight into the former question.
A family of examples of VOCs {#S:family_examples}
----------------------------
Let the vector space $V = \coprod_{k \in \Z} V_{(k)}$ be a module over the Virasoro algebra, $\mathcal{V} = \oplus_{j \in \Z} \C L(j) \oplus \C d$, such that for all homogeneous vectors $L(0) \cdot v= \text{wt }(v) v$. We will say that a bilinear form $(\cdot,\cdot)$ on $V$ is *Virasoro preserving* if it satisfies the condition
$$\label{E:L-antisym}
(L(k)v_1,v_2)=(v_1,L(-k)v_2)$$
for all $k \in \Z$, $v_1,v_2 \in V$. In particular, $k=0$ in Property (\[E:L-antisym\]) indicates that all Virasoro preserving bilinear forms are graded, i.e.
$$(V_{(k)},V_{(\ell)})=0$$
for $k \neq \ell$. If $V$ is a VOA, the bilinear form is said to be *invariant* if, for all $u,v,w \in V$,
$$\label{E:VOAinvar}
(Y(u,x)v,w)=(u,Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})w).$$
Any invariant bilinear form on V is Virasoro preserving ((2.31) in [@L]).
Note that there is a natural extension of $(\cdot,\cdot):V^{\otimes 2} \to \C$ to $(\cdot,\cdot):V^{\otimes 4} \to \C$ given by $(u_1 \otimes u_2,v_1 \otimes v_2)=(u_1,v_1) (u_2,v_2)$, for $u_1,u_2,v_1,v_2 \in V$.
\[T:examples\] Let $(V,Y,\mathbf{1},\omega)$ be a vertex operator algebra equipped with a nondegenerate and Virasoro preserving bilinear form $(\cdot,\cdot)$. Given the linear operators
$$\begin{aligned}
c : V & \to \C \\
v &\mapsto (v,\mathbf{1}),\end{aligned}$$
$$\begin{aligned}
\rho : V & \to \C \\
v &\mapsto (v,\omega),\end{aligned}$$
and
$$\begin{aligned}
\co : V &\to (V \otimes V)[[x,x^{-1}]] \\
v &\mapsto \co(x)v = \sum_{k\in \Z} \Delta_k(v) x^{-k-1},\end{aligned}$$
defined by
$$(\co(x)u, v \otimes w) = (u, Y(v, x)w),$$
the quadruple $(V,\co,c, \rho)$ is a vertex operator coalgebra.
We will show that all 8 axioms for VOCs are satisfied.
1\. Positive energy: Trivially satisfied.
2\. Left Counit: Given $u \in V$, for all $v \in V$
$$\begin{aligned}
((c \otimes Id_V) \co(x)u,v)&=(\co(x)u, \mathbf{1} \otimes v)\\
&=(u, Y(\mathbf{1},x)v) \\
&=(u,v).\end{aligned}$$
Thus, by nondegeneracy, $(c \otimes Id_V) \co(x)u=u$.
3\. Cocreation: Given $u \in V$, then for all $v \in V$
$$\begin{aligned}
((Id_V \otimes c) \co(x)u,v)&= (\co(x)u, v \otimes \mathbf{1}) \\
&= (u, Y(v,x)\mathbf{1}) \\
&= (u, e^{x L(-1)}v) \in \C[x]\end{aligned}$$
and
$$\lim_{x \to 0} ( u, e^{x L(-1)}v)= (u,v).$$
4\. Truncation: Pick $N \in \Z$ such that $V_{(n)}=0$ for all $n \leq N$. Given $u \in V_{(r)}$, let $v \in V_{(s)}$ and $w \in V_{(t)}$.
$$\begin{aligned}
( \co(x)u,v \otimes w)
&= ( u,Y(v,x) w)\\
&= \sum_{k \in \Z} (u,v_k w) x^{-k-1}\end{aligned}$$
For $(u,(v)_k w)$ to be nonzero, we must have $\text{wt } u=
\text{wt } v + \text{wt } w -k -1$, i.e. $r=s+t-k-1$; but $s,t > N$ so we must have $r > 2N-k-1$ or $r-2N > -k-1$. Hence, $( \co(x)u,v \otimes w) \in \C[[x^{-1}]]x^{r-2N-1}$ for any $s,t \in \Z$.
5\. Jacobi Identity: Given $u \in V$, then for all $v_1,v_2,v_3 \in V$
$$\begin{aligned}
\label{E:511}
((Id_V \otimes \co(x_2)) \co(x_1)u,v_1 \otimes v_2 \otimes v_3)
&= (\co(x_1)u,v_1 \otimes Y(v_2,x_2) v_3) \\
&= (u,Y(v_1,x_1) Y(v_2,x_2) v_3), \notag\end{aligned}$$
$$\begin{aligned}
\label{E:512}
((T \otimes Id_V)
(Id_V \otimes \co(x_1)) \co(x_2)u, & v_1 \otimes v_2 \otimes v_3) \\
&= ((Id_V \otimes \co(x_1)) \co(x_2)u,v_2 \otimes v_1 \otimes v_3) \notag \\
&= (\co(x_2)u,v_2 \otimes Y(v_1,x_1) v_3) \notag \\
&= (u,Y(v_2,x_2) Y(v_1,x_1) v_3), \notag\end{aligned}$$
$$\begin{aligned}
\label{E:513}
((\co(x_0) \otimes Id_V) \co(x_2)u,v_1 \otimes v_2 \otimes v_3)
&= ( \co(x_2)u, Y(v_1,x_0) v_2 \otimes v_3) \\
&= (u,Y(Y(v_1,x_0)v_2,x_2)v_3). \notag\end{aligned}$$
Equations (\[E:511\]), (\[E:512\]) and (\[E:513\]) make it clear that the VOA Jacobi identity (\[E:Jac2\]) is equivalent to the VOC Jacobi identity (\[E:Jac\]).
6\. Virasoro Algebra: Given $u \in V$, for all $v \in V$
$$\begin{aligned}
\label{E:519}
((\rho \otimes Id_V) \co(x)u,v)
&=(\co(x)u, \omega \otimes v) \\
&=(u, Y(\omega,x)v) \notag \\
&=\sum_{k \in \Z} (u, L(k)v) x^{-k-2} \notag \\
&=\sum_{j \in \Z} (L(j)u,v) x^{j-2}. \notag\end{aligned}$$
Note that in the last equality we have used Virasoro preservation, (\[E:L-antisym\]).
Equation (\[E:519\]) shows that the Virasoro algebra bracket,
$$[L(j),L(k)]=(j-k)L(j+k)+\frac{1}{12}(j^3-j)\delta_{j,-k}d,$$
follows from the Virasoro bracket relation on VOAs.
7\. Grading: Equation (\[E:519\]) shows that $L(0)=Res_x x Y(\omega,x)$ so grading follows from VOAs.
8\. $L(1)$-Derivative: Given $u \in V$, then for all $v,w \in V$
$$\begin{aligned}
((L(1) \otimes Id_V) \co(x)u,v \otimes w)
&=(\co(x)u,L(-1)v \otimes w) \\
&= (u, Y(L(-1)v, x)w) \\
&= \frac{d}{dx}(u, Y(v, x)w) \\
&= \frac{d}{dx}(\co(x)u,v \otimes w) \end{aligned}$$
Here the first equality uses Virasoro preservation.
Li showed in [@L] that if a simple VOA satisfies the condition $L(1)V_{(1)}=0$ then there exists a nondegenerate, invariant bilinear form on V. Thus we are guaranteed a family of VOAs equipped with the type of form required for Theorem \[T:examples\]. Additionally, Heisenberg VOAs may be explicitly equipped with an appropriate bilinear form and they will be the focus of more concrete discussion in the next section.
Vertex operator algebras and coalgebras associated with Heisenberg algebras
---------------------------------------------------------------------------
While the construction in the last section does describe a family of VOCs, it is not extremely explicit in nature. In this section we will explicitly construct VOCs from Heisenberg algebras. We begin with the construction of the vector space for the Heisenberg VOA following [@D].
Let $\mathbf{h}$ be a finite dimensional vector space equipped with a symmetric, nondegenerate bilinear form $\langle \cdot, \cdot \rangle$. Since $\mathbf{h}$ may be considered as an abelian Lie algebra, let $\hat{\mathbf{h}}$ be the corresponding affine Lie algebra, i.e., let
$$\hat{\mathbf{h}}=\mathbf{h} \otimes \C [t,t^{-1}] \oplus \C c,$$
where $c$ is nonzero, with the Lie bracket defined by
$$\begin{aligned}
[\alpha \otimes t^m,\beta \otimes t^n] &= \langle \alpha, \beta \rangle m \delta_{m,-n}c \\
[\hat{\mathbf{h}},c] &=0 \end{aligned}$$
for $\alpha, \beta \in \mathbf{h}$, $m,n \in \Z$. There is a natural $\Z$-grading on $\hat{\mathbf{h}}$ under which $\alpha \otimes t^m$ has weight $-m$ for all $\alpha \in \mathbf{h}$, and $m \in \Z$, and $c$ has weight 0. The element $\alpha \otimes t^m$ of $\hat{\mathbf{h}}$ is usually denoted $\alpha(m)$. Three graded subalgebras are of interest:
$$\hat{\mathbf{h}}^+=\mathbf{h} \otimes t \C [t],$$
$$\hat{\mathbf{h}}^-=\mathbf{h} \otimes t^{-1} \C [t^{-1}],$$
$$\hat{\mathbf{h}}_{\Z}=\hat{\mathbf{h}}^+ \oplus \hat{\mathbf{h}}^- \oplus \C c.$$
The subalgebra $\hat{\mathbf{h}}_{\Z}$ is a *Heisenberg algebra*, by which we mean that its center is one-dimensional and is equal to its commutator subalgebra. Note that $\hat{\mathbf{h}}^+$ and $\hat{\mathbf{h}}^-$ are abelian, but that $\hat{\mathbf{h}}_{\Z}$ is necessarily non-abelian.
We consider the induced $\hat{\mathbf{h}}_{\Z}$-module
$$M(1)=U(\hat{\mathbf{h}}_{\Z}) \otimes_{U(\hat{\mathbf{h}}^+ \oplus \C c)} \C$$
where $U$ indicates the universal enveloping algebra and $\C$ is viewed as a $\Z$-graded $(\hat{\mathbf{h}}^+ \oplus \C c)$-module by
$$\begin{aligned}
c \cdot 1 &= 1, \\
\hat{\mathbf{h}}^+ \cdot 1 &= 0, \\
\text{deg } 1 &=0.\end{aligned}$$
The module $M(1)$ may be generalized (cf. [@L] and [@FLM]). $M(1)$ is linearly isomorphic to $S(\hat{\mathbf{h}}^-)$ in a way that preserves grading. Thus we often write basis elements of $M(1)$ as
$$v=\alpha_1(-n_1) \cdots \alpha_r(-n_r)$$
for $\alpha_i \in \mathbf{h}$, $n_i \in \Z_+$, $i=1, \ldots, r$, and observe that $v$ has weight $n_1+ \cdots+n_r$. (Tensor products are suppressed in this notation.) Note that the $\alpha_i(-n_i)$’s all commute so their order is irrelevant. Using the form $\langle \cdot, \cdot \rangle$ on $\mathbf{h}$, we may choose $\{ \gamma_i \}_{i=1}^{d}$ to be an orthonormal basis and we lose no generality by considering only $\alpha_i$ from this set of basis elements. Thus, we will typically prove results for the set of generating elements
$$\text{gen}M = \left\{ \alpha_1(-n_1) \cdots \alpha_r(-n_r) |
r \in \N, \ \alpha_j \in \{ \gamma_i \}_{i=1}^{d}, \ n_j \in \Z_+, \ j=1, \ldots, r \right\}$$
and then extend linearly to all of $M(1)$.
Next we define a bilinear form on $M(1)$ which we will use throughout the rest of this section.
There is a unique bilinear form $(\cdot,\cdot)$ on $M(1)$ satisfying
$$\begin{aligned}
(\alpha(m) \cdot u, v)&= (u, \alpha(-m) \cdot v), \label{E:311} \\
(1,1)&=1. \label{E:3181}\end{aligned}$$
for all $u,v \in M(1)$, $\alpha \in \mathbf{h}$, $m \in \Z \smallsetminus \{0 \}$.
More precisely, given $v=\alpha_1(-n_1) \cdots \alpha_r(-n_r)$ let
$$p(v) = \alpha_r(n_r) \cdots \alpha_1(n_1) \alpha_1(-n_1) \cdots \alpha_r(-n_r) \in \Z_+ \subset M(1).$$
The unique bilinear form $(\cdot, \cdot)$ on $M(1)$ satisfying (\[E:311\]) and (\[E:3181\]) is defined on basis elements $u, v \in \text{gen}M$ by
$$\begin{aligned}
\label{E:form}
(u,v)= \left\{ \begin{array}{ll}
p(u) & \text{if } u=v \\
0 & \text{otherwise}
\end{array} \right. \end{aligned}$$
Further, this form is nondegenerate, graded and symmetric.
We will construct the form in (\[E:form\]) from (\[E:311\]) and (\[E:3181\]), thus showing the form is unique. Consider $u,v \in \text{gen}M$ such that $u \neq v$. Then there is an element $\alpha(-n) \in \hat{\mathbf{h}}$ and a positive integer $t$ such that $\alpha(-n)^t$ is contained in $u$ or $v$ but not in the other. We may assume $\alpha(-n)^t$ is in $u$, say $u=\alpha(-n)^t \alpha_1(-n_1) \cdots \alpha_r(-n_r)$. But then
$$\begin{aligned}
(u,v)
&=& ( \alpha(-n)^t \alpha_1(-n_1) \cdots \alpha_r(-n_r),v) \label{E:3311}\\
&=& ( \alpha_1(-n_1) \cdots \alpha_r(-n_r), \alpha(n)^t v) \nonumber \\
&=& (\alpha_1(-n_1) \cdots \alpha_r(-n_r), 0) \nonumber \\
&=&0 \nonumber .\end{aligned}$$
(As a biproduct, (\[E:3311\]) shows the form is graded and symmetric.) Now we need only examine the form applied to a single basis element. Let $u=\alpha_1(-n_1) \cdots \alpha_r(-n_r)$. Then
$$\begin{aligned}
(u,u)
&=&(\alpha_1(-n_1) \cdots \alpha_r(-n_r), \alpha_1(-n_1) \cdots \alpha_r(-n_r)) \label{E:3312}\\
&=&(\alpha_r(n_r) \cdots \alpha_1(n_1) \alpha_1(-n_1) \cdots \alpha_r(-n_r), 1) \nonumber \\
&=& (p(u),1) \nonumber \\
&=& p(u), \nonumber \end{aligned}$$
thus proving that (\[E:form\]) is the unique form satisfying (\[E:311\]) and (\[E:3181\]). Nondegeneracy is immediate from (\[E:3312\]).
Given $\alpha \in \mathbf{h}$, we will need the following three series in formal variable $x$ with coefficients in $\hat{\mathbf{h}}_{\Z}$:
$$\begin{aligned}
\alpha^+(x) &= \sum_{k \in \Z_+} \alpha(k) x^{-k-1} \\
\alpha^-(x) &= \sum_{k \in \Z_+} \alpha(-k) x^{k-1} \\
\alpha(x) &= \alpha^-(x) + \alpha^+(x).\end{aligned}$$
Via Equation (\[E:311\]) we see that
$$\begin{aligned}
(\alpha^-(x) v_1, v_2)&= (v_1, \alpha^+(x) v_2), \label{E:313} \\
(\alpha(x) v_1, v_2)&= (v_1, \alpha(x) v_2). \label{E:315}\end{aligned}$$
The vertex operator algebra associated to a Heisenberg algebra is described using a normal ordering procedure, indicated by open colons $\no \ \ \no$, which reorders the enclosed expression so that all operators $\alpha_1(-m)$ are placed to the left (multiplicatively) of all operators $\alpha_2(n)$, for $\alpha_1, \alpha_2 \in \mathbf{h}$, $m, n \in \Z_+$. For example,
$$\begin{aligned}
\no \alpha_1(x) \alpha_2(x) \no v
&= \ \no (\alpha_1^-(x) + \alpha_1^+(x))
\alpha_2(x) \no v \\
&= \alpha_1^-(x) \alpha_2(x) v +
\alpha_2(x) \alpha_1^+(x) v.\end{aligned}$$
This normal ordering moves degree-lowering operators to the right. Notice that $\alpha^+(x)$ will only produce elements of lower weight than the basis element of $M(1)$ to which it is applied, while $\alpha^-(x)$ will only produce elements of higher weight. Therefore applying all the degree-lowering operators before all the degree-raising operators guarantees that no single weight-space has infinitely many summands in it.
We now have the notation to describe the VOA associated to a Heisenberg algebra. Define a linear map $Y ( \cdot,x): M(1) \to (\text{End } M(1))[[x,x^{-1}]]$ by
$$Y(v, x) = \ \no
\left( \frac{1}{(n_1-1)!} \left(\frac{d}{dz}\right)^{n_1-1} \alpha_1(x) \right) \cdots
\left( \frac{1}{(n_r-1)!} \left(\frac{d}{dz}\right)^{n_r-1} \alpha_r(x) \right)
\ \no$$
for $v=\alpha_1(-n_1) \cdots \alpha_r(-n_r)$. We also define two distinguished elements of $M(1)$, $\mathbf{1}=1$ and $\omega = \frac{1}{2} \sum_{i=1}^{d} \gamma_i(-1)^2$, where $\{ \gamma_i \}_{i=1}^{d}$ is the orthonormal basis of $\mathbf{h}$ as above.
The quadruple $(M(1), Y, \mathbf{1}, \omega)$ as defined above is a vertex operator algebra.
Our treatment here largely mirrors [@D], with the above proposition being Proposition 3.1 in [@D]. A proof may be found in [@G]. For our purposes, however, the nondegenerate bilinear form on $M(1)$, described in (\[E:form\]), is equally relevant. We will now prove an additional property of that bilinear form.
The bilinear form on $M(1)$ defined in Equation (\[E:form\]) is Virasoro preserving.
First, we will explicitly calculate the $L(k)$ operators and then show that $(L(k)v,w)=(v,L(-k)w)$. Symmetry of the form allows us to only consider $k \in \N$.
Using the definition of the $L(k)$ operators we see that $\sum_{k \in \Z} L(k) x^{-k-2} = Y(\frac{1}{2}\sum_{i=1}^{d} \gamma_i(-1)^2, x)$. Employing the definition of $Y$, for $k \in \Z_+$ we have
$$\begin{aligned}
L(k) &= \sum_{i=1}^{d} \left( \frac{1}{2} \sum_{j =1}^{k-1} \gamma_i(j) \gamma_i(k-j)
+ \sum_{j \in \Z_+} \gamma_i(-j) \gamma_i(k+j) \right) \\
L(-k) &= \sum_{i=1}^{d} \left( \frac{1}{2} \sum_{j =1}^{k-1} \gamma_i(-j) \gamma_i(-k+j)
+ \sum_{j \in \Z_+} \gamma_i(-k-j) \gamma_i(j) \right).\end{aligned}$$
Given $u, v \in \text{gen}M$,
$$(L(0)u, v)=\text{wt }(u) \ (u,v) = \text{wt }(v) \ (u,v) = (u,L(0)v)$$
since either $(u,v)=0$, or $u=v$ implying $\text{wt }(u)=\text{wt }(v)$. For $k \in \Z_+$ we make use of (\[E:311\]) to observe that,
$$\begin{aligned}
(L(k)u,v)
&= \sum_{i=1}^{d} \left( \frac{1}{2} \sum_{j =1}^{k-1} (\gamma_i(j) \gamma_i(k-j)u,v)
+ \sum_{j \in \Z_+} (\gamma_i(-j) \gamma_i(k+j)u,v) \right) \\
&= \sum_{i=1}^{d} \left( \frac{1}{2} \sum_{j =1}^{k-1} (u,\gamma_i(-k+j) \gamma_i(-j)v)
+ \sum_{j \in \Z_+} (u,\gamma_i(-k-j) \gamma_i(j)v) \right) \\
&= (u,L(-k)v).\end{aligned}$$
Using the Heisenberg VOA $M(1)$ along with the nondegenerate, Virasoro preserving bilinear form defined in (\[E:form\]), we will follow the construction of a VOC in Section \[S:family\_examples\]. First, we define a linear map $c : V \to \C$ by
$$c(1) =1$$
$$c(\alpha_1(-n_1) \cdots \alpha_r(-n_r)) = 0$$
where $r \geq 1$ and $\alpha_1(-n_1) \cdots \alpha_r(-n_r) \in \text{gen}M$. It is clear that $c(v)=(v,\mathbf{1})$ for all $v \in M(1)$. Next, we define a linear map $\rho : V \to \C$ by
$$\begin{aligned}
\rho(\gamma_i(-1)^2)= 1\end{aligned}$$
for each basis element $\gamma_i$ of $\mathbf{h}$ and $\rho(v)=0$ for $v$ any other basis element of $M(1)$. Again it is clear that $\rho(v)=(v,\omega)$ for all $v \in M(1)$. Finally, we need to define a linear map $\co (x): V \to (V \otimes V)[[x,x^{-1}]]$ such that
$$(\co(x)u, v \otimes w) = (u, Y(v, x)w).$$
For notational simplicity, given $\alpha \in \mathbf{h}$, $n \in \Z_+$ and $x$ a formal variable, let
$$\begin{aligned}
\alpha^+(n,x) &= \frac{1}{(n-1)!} \left(\frac{d}{dz}\right)^{n-1}
\sum_{k \in \Z_+} \alpha(k) x^{-k-1} \\
\alpha^-(n,x) &= \frac{1}{(n-1)!} \left(\frac{d}{dz}\right)^{n-1}
\sum_{k \in \Z_+} \alpha(-k) x^{k-1} \\
\alpha(n,x) &= \alpha^-(n,x) + \alpha^+(n,x).\end{aligned}$$
Let $v$ denote the basis element $\beta_1(-m_1) \cdots \beta_s(-m_s)$ and define $\co (x): V \to (V \otimes V)[[x,x^{-1}]]$ as
$$\co(x)u=\sum_{v \in \text{gen}M}
\frac{1}{p(v)} v \otimes \ \no
\beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no u.$$
For all $u,v,w \in M(1)$, $(\co(x)u, v \otimes w) = (u, Y(v, x)w)$.
First, note that our definition of $\co$ is equivalent to
$$(\co(x)u, v \otimes w) =
(\frac{1}{p(v)} v \otimes \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no u, v \otimes w)$$
for all $u = \alpha_1(-n_1) \cdots \alpha_r(-n_r)$, $v = \beta_1(-m_1) \cdots \beta_s(-m_s)$, $w = \mu_1(-\ell_1) \cdots \mu_t(-\ell_t)$. Given these basis elements we use induction on $s$ to show that
$$\label{E:34}
(u, \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no w)
=(\ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no u, w).$$
For $s=0$, this is trivial. If we assume that (\[E:34\]) is true for $s-1$ and appeal to (\[E:313\]), we see that
$$(u, \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no w)$$
$$\begin{gathered}
= (u, \beta_s^-(m_s,x) \ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x) \ \no w) \\
+ (u, \ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x) \ \no \ \beta_s^+(m_s,x) w) \end{gathered}$$
$$\begin{gathered}
= ( \beta_s^+(m_s,x) u, \ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x) \ \no w) \\
+ (\ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x)\ \no u, \beta_s^+(m_s,x) w)\end{gathered}$$
$$\begin{gathered}
= ( \ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x) \ \no \beta_s^+(m_s,x) u, w) \\
+ \beta_s^-(m_s,x) \ \no \beta_1(m_1,x) \cdots \beta_s(m_{s-1},x) \ \no u, w) \end{gathered}$$
$$=(\ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no u, w).$$
Finally, using Equation (\[E:34\]) we see that
$$\begin{aligned}
(\co(x)u, v \otimes w)
&=(\frac{1}{p(v)} v \otimes \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x)
\ \no u, v \otimes w) \\
&=\frac{(v, v)}{p(v)} ( \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no u, w) \\
&=(u, \ \no \beta_1(m_1,x) \cdots \beta_s(m_s,x) \ \no w) \\
&=(u, Y(v, x)w).\end{aligned}$$
Theorem \[T:examples\] proves that the quadruple $(M(1),\co,c,\rho)$ associated to the Heisenberg algebra $\hat{\mathbf{h}}_{\Z}$ is a vertex operator coalgebra.
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|
[Theory of decoherence in Bose-Einstein condensate interferometry]{}
**B J Dalton**
ARC Centre for Quantum-Atom Optics
and Centre for Atom Optics and Ultrafast Spectroscopy
Swinburne University of Technology
Melbourne, Victoria 3122, Australia
Email: [email protected]
**Abstract.** A full treatment of decoherence and dephasing effects in BEC interferometry has been developed based on using quantum correlation functions for treating interferometric effects. The BEC is described via a phase space distribution functional of the Wigner type for the condensate modes and the positive P type for the non-condensate modes. Ito equations for stochastic condensate and non-condensate field functions replace the functional Fokker-Planck equation for the distribution functional and stochastic averages of field function products determine the quantum correlation functions.
Introduction
============
Bose-Einstein condensates (BEC) in cold atomic gases are an example of a quantum system on a macroscopic scale. Well below the transition temperature essentially all bosons occupy small number of single particle states (or modes) – in simple situations only one mode. Interferometry based on BECs (such as by splitting a BEC in a single trap into two traps and then allowing the BECs to recombine) offers possible improvements in precision over single atom interferometry by a factor given by square root of the boson number $N$ \[1, 2\]. Such BEC interferometry is based on having almost all bosons in one (or perhaps two) modes.
A typical double-well interferometry experiment involves starting with a BEC in a single well trap, then changing the trap to a possibly asymmetric double-well and then back to a single well. Asymmetry could lead to excitation of bosons to higher energy states of final trap, or to changes to spatial interference patterns. The process of exciting one boson from the ground to the first excited state for a single well trap via two quantum pathways is shown, both pathways involving an intermediate double well trap. The near degeneracy of energy levels for the intermediate asymmetric double well facilitates the boson transfer to the excited state. The two non-observed quantum pathways involve the boson transfer occurring in different halves of the process, and superposition of the two quantum transition amplitudes can lead to interference effects in the excitation probability.
Interferometry experiments involving measurements of boson positions are best described in terms of quantum correlation functions, which are expectation values of products of bosonic field operators specified at different spatial points, and are related to many-boson position measurements \[3\]. If boson-boson interactions were absent and the BEC isolated from the environment, idealised forms of quantum correlation functions would result, with interferometric effects clearly visible. Interactions of the BEC with the external environment (such as fluctuating trap fields) and internal boson-boson interactions tend to degrade the interference pattern. Boson-boson interactions can result in dephasing (associated with interactions within condensate modes) and decoherence effects (associated with interactions causing transitions from condensate modes) even when external environmental effects are absent.
Recently a simple theory of double-well BEC interferometry has been developed \[4\] based on a two mode approximation and allowing for possible fragmentations of the original BEC into two modes - which may be localised in each well. Many previous theories ignore fragmentation, with all bosons occupying a single condensate wave function which satisfies the standard Gross-Pitaevskii equation (see \[5\] and references therein). The two-mode theory was developed from the quantum principle of least action and gives self consistent coupled equations for the two mode functions (generalised Gross-Pitaevskii equations) and for the amplitudes (matrix equations) describing fragmentation of the BEC. Self consistency results in the mode functions depending on the relative importance of various ways the BEC can fragment, whilst the fragmentation amplitudes depend on the mode functions. Numerical studies are planned. However, only transitions within the two condensate modes are allowed for, and although some dephasing processes are included, decoherence processes associated with elementary collective excitations (Bogoliubov) and single boson excitations (thermal modes) are not. The theory is also restricted to small $N$.
A full treatment of decoherence and dephasing effects on quantum correlation functions is required, and this is the subject of the present paper. Several possible approaches to developing such a theory may be identified. These all have desirable features such as exploiting the physics of large occupancy differences for condensate and non-condensate modes, not being restricted to small boson numbers and avoiding the explicit consideration of large numbers of modes. Collective and single boson excitations from condensate modes, and possible fragmentation effects can be allowed for, though the presence of these processes may not be explicit.
One such approach is a master equation method \[6\], in which a condensate density operator is defined for which a master equation is derived allowing for interactions with non-condensate modes, which constitute a reservoir. The difficulty with this method is that it is hard to evaluate the non-condensate contributions to quantum correlation functions. A second approach is based on a Heisenberg equation method that has been applied in numerous many-body theory cases. Heisenberg equations for field operators and products of field operators are derived, and taking the expectation values with the initial density operator results in a heirarchy of coupled equations for quantum correlation functions. An ansatz (such as assuming that a suitable high order correlation function factorises) produces a truncated set of coupled equations from which correlation functions of the required order can be calculated. The problem with this method is that it is hard to confirm the validity of the ansatz.
The present approach is a generalised phase space method, but involving a distribution functional rather than a distribution function \[7\]. The field operator for the bosonic system is written as a sum of condensate and non-condensate mode contributions \[8\]. The BEC state is described by a density operator which satisfies the Liouville-von Neumann equation (LVN), and which is mapped onto a phase space distribution functional. The latter has the feature that the highly occupied condensate modes are described via a generalised Wigner representation (since the bosons in condensate modes behave like a classical mean field), whilst the basically unoccupied non-condensate modes are described via a positive P representation (these bosons should exhibit quantum effects). The LVN equation is replaced by a functional Fokker-Planck equation (FFPE) for the distribution functional, which is based on the truncated Wigner approximation \[7\] that can be applied when large condensate mode occupancy occurs. The FFPE are finally replaced by coupled Ito stochastic equations (c-number Langevin equations) for condensate and non-condensate field functions, where the Ito equations contain deterministic and random noise terms - identifiable from the FFPE. Stochastic averages of the field functions then give the quantum correlation functions. There are no obvious difficulties with this approach, though further development will be needed to explicitly incorporate Bogoliubov collective excitations.
Generalised phase space functional theory
=========================================
The Hamiltonian in terms of bosonic field operators $\widehat{\Psi }(\mathbf{r}),\widehat{\Psi }(\mathbf{r})^{\dag }$ is$$\begin{aligned}
\widehat{H} &=&\dint d\mathbf{r(}\frac{\hbar ^{2}}{2m}\nabla \widehat{\Psi }(\mathbf{r})^{\dag }\cdot \nabla \widehat{\Psi }(\mathbf{r})+\widehat{\Psi }(\mathbf{r})^{\dag }V\widehat{\Psi }(\mathbf{r})) \nonumber \\
&&+\dint d\mathbf{r}\frac{g}{2}\widehat{\Psi }(\mathbf{r})^{\dag }\widehat{\Psi }(\mathbf{r})^{\dag }\widehat{\Psi }(\mathbf{r})\widehat{\Psi }(\mathbf{r}), \label{Eq. Hamiltonian}\end{aligned}$$where the boson mass is $m$, the trap potential is $V(\mathbf{r},t)$ and $g$ specifies boson-boson interactions in the zero range approximation.
The field operator $\widehat{\Psi }(\mathbf{r})$ is written as sum of a condensate term $\widehat{\Psi }_{C}(\mathbf{r})$ and a non-condensate term $\widehat{\Psi }_{NC}(\mathbf{r})$ \[6\]$$\widehat{\Psi }_{C}(\mathbf{r})=\widehat{c}_{1}\phi _{1}(\mathbf{r})+\widehat{c}_{2}\phi _{2}(\mathbf{r})\qquad \widehat{\Psi }_{NC}(\mathbf{r})=\dsum\limits_{k\neq 1,2}^{n}\widehat{c}_{k}\phi _{k}(\mathbf{r}),
\label{Eq.Cond&NonCondField}$$which are defined in terms of $n$ orthonormal mode functions $\phi _{k}(\mathbf{r})$ and bosonic mode annihilation operators $\widehat{c}_{k}$. Condensate and non-condensate contributions to the field operators commute.
Replacing $\widehat{\Psi }(\mathbf{r})$ by $\widehat{\Psi }_{C}(\mathbf{r})+\widehat{\Psi }_{NC}(\mathbf{r})$ the Hamiltonian is sum of three terms \[6\], Hamiltonians $\widehat{H}_{C}$ and $\widehat{H}_{NC}$ for the condensate and non-condensate - which are of the same form as in Equation (\[Eq. Hamiltonian\]), and the interaction $\widehat{V}$ between condensate and non-condensate. This is the sum of three contributions, which are linear, quadratic and cubic in the condensate operators.
Spatial coherence effects for interference experiments in BECs may be described via quantum correlation functions, which depend on the density operator $\widehat{\rho }$ for the bosonic system and are defined by$$\begin{aligned}
&&G^{N}(\mathbf{r}_{1}\mathbf{,r}_{2}\mathbf{,..,r}_{N};\mathbf{s}_{N}\mathbf{,..,s}_{2}\mathbf{,s}_{1}) \nonumber \\
&=&Tr(\widehat{\rho }(t)\,\widehat{\Psi }\,(\mathbf{r}_{1})^{\dag }\,..\widehat{\Psi }\,(\mathbf{r}_{N})^{\dag }\,\widehat{\Psi }\,(\mathbf{s}_{N})\,..\,\widehat{\Psi }\,(\mathbf{s}_{1})). \label{Eq.QuantumCorrFns}\end{aligned}$$The quantum correlation function with $r_{i}=s_{i}\,(i=1,...,N)$ determines the simultaneous probability of detecting one boson at $r_{1}$, .., the $N$th at $r_{N}$, see \[3\]. It is evident that the quantum correlation functions will contain condensate terms (describing the main interference effects), non-condensate terms and mixed terms involving both condensate and non-condensate operators (describing effects degrading the interference patterns).
In the phase space functional method the density operator $\widehat{\rho }$ is first mapped uniquely onto a characteristic functional $\chi \lbrack \xi
_{C}(\mathbf{r}),\xi _{C}^{+}(\mathbf{r}),\xi _{NC}(\mathbf{r}),\xi
_{NC}^{+}(\mathbf{r})]$ of the four functions $\xi _{C}^{+}(\mathbf{r}),\xi
_{C}(\mathbf{r}),\xi _{NC}^{+}(\mathbf{r})$ and $\xi _{NC}(\mathbf{r})$$$\begin{aligned}
&&\chi \lbrack \xi _{C},\xi _{C}^{+},\xi _{NC},\xi _{NC}^{+}] \nonumber \\
&=&Tr(\widehat{\rho }\,\exp i\dint d\mathbf{r}\{\xi _{C}(\mathbf{r})\widehat{\Psi }_{C}^{\dag }(\mathbf{r})+\widehat{\Psi }_{C}(\mathbf{r})\xi _{C}^{+}(\mathbf{r})\}\, \nonumber \\
&&\times \exp i\dint d\mathbf{r}\{\xi _{NC}(\mathbf{r})\widehat{\Psi }_{NC}^{\dag }(\mathbf{r})\}\,\exp i\dint d\mathbf{r}\{\widehat{\Psi }_{NC}(\mathbf{r})\xi _{NC}^{+}(\mathbf{r})\}) \label{Eq.CharFnal}\end{aligned}$$The characteristic functional is of the Wigner $(W)$ type for condensate modes and the positive P* *$(P^{+})$ type for the non-condensate modes.
The (quasi) distribution functional $P[\psi _{C}(\mathbf{r}),\psi _{C}^{+}(\mathbf{r}),\psi _{NC}(\mathbf{r}),\psi _{NC}^{+}(\mathbf{r})]$ involves four field functions $\psi _{C}(\mathbf{r}),\psi _{C}^{+}(\mathbf{r}),\psi
_{NC}(\mathbf{r}),\psi _{NC}^{+}(\mathbf{r})$ corresponding to field operators $\widehat{\Psi }_{C}(\mathbf{r}),\widehat{\Psi }_{C}(\mathbf{r})^{\dag },\widehat{\Psi }_{NC}(\mathbf{r})$ and $\widehat{\Psi }_{NC}(\mathbf{r})^{\dag }$. Although non-unique and possibly negative (and hence not interpretable as a probability distribution), it is required to determine the characteristic functional $\chi \lbrack \xi _{C}(\mathbf{r}),\xi _{C}^{+}(\mathbf{r}),\xi _{NC}(\mathbf{r}),\xi _{NC}^{+}(\mathbf{r})]$ via a functional integration process with weight function $w(\psi _{1},\psi
_{1}^{+},..,\psi _{i},\psi _{i}^{+},..,\psi _{n},\psi _{n}^{+},)$ given by $\dprod\limits_{i}(\Delta \mathbf{r}_{i})$ $$\begin{aligned}
&&\chi \lbrack \xi _{C}(\mathbf{r}),\xi _{C}^{+}(\mathbf{r}),\xi _{NC}(\mathbf{r}),\xi _{NC}^{+}(\mathbf{r})] \nonumber \\
&=&\diiiint D^{2}\psi _{C}\,D^{2}\psi _{C}^{+}\,D^{2}\psi _{NC}\,D^{2}\psi
_{NC}^{+}\,\,P[\psi _{C}(\mathbf{r}),\psi _{C}^{+}(\mathbf{r}),\psi _{NC}(\mathbf{r}),\psi _{NC}^{+}(\mathbf{r})] \nonumber \\
&&\times \exp i\dint d\mathbf{r\,\{}\xi _{C}(\mathbf{r})\psi _{C}^{+}(\mathbf{r})+\psi _{C}(\mathbf{r})\xi _{C}^{+}(\mathbf{r})\}\,
\label{Eq.DistribnFnal} \\
&&\times \exp i\dint d\mathbf{r\,\{}\xi _{NC}(\mathbf{r})\psi _{NC}^{+}(\mathbf{r})\}\exp i\dint d\mathbf{r\,\{}\psi _{NC}(\mathbf{r})\xi _{NC}^{+}(\mathbf{r})\}. \nonumber\end{aligned}$$
Quantum averages of symmetrically ordered products of condensate field operators $\{\widehat{\Psi }_{C}^{\dag }(\mathbf{r}_{1})....\widehat{\Psi }_{C}^{\dag }(\mathbf{r}_{p})\widehat{\Psi }_{C}(\mathbf{s}_{q})..\widehat{\Psi }_{C}(\mathbf{s}_{1})\}$ and normally ordered products of non-condensate field operators $\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{1})\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{2})....\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{r})\widehat{\Psi }_{NC}(\mathbf{v}_{s})..\widehat{\Psi }_{NC}(\mathbf{v}_{1})$ are given by functional integrals of the distribution functional $P[\psi _{C},\psi _{C}^{+},\psi _{NC},\psi _{NC}^{+}]$ with products of field functions,.where the condensate field operator $\widehat{\Psi }_{C}(\mathbf{r}_{i})^{\dag }$ is replaced by $\psi _{C}^{+}(\mathbf{r}_{i})$, $\widehat{\Psi }_{C}(\mathbf{s}_{j})$ is replaced by $\psi (\mathbf{s}_{j})$ and with analogous replacements for the non-condensate field operators.$$\begin{aligned}
&&Tr[\widehat{\rho }\,\{\widehat{\Psi }_{C}^{\dag }(\mathbf{r}_{1})...\widehat{\Psi }_{C}^{\dag }(\mathbf{r}_{p})\widehat{\Psi }_{C}(\mathbf{s}_{q})..\widehat{\Psi }_{C}(\mathbf{s}_{1})\}\, \label{Eq.QuantumAverages} \\
&&\times \widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{1})...\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{r})\widehat{\Psi }_{NC}(\mathbf{v}_{s})..\widehat{\Psi }_{NC}(\mathbf{v}_{1})] \nonumber \\
&=&\diiiint D^{2}\psi _{C}\,D^{2}\psi _{C}^{+}\,D^{2}\psi _{NC}\,D^{2}\psi
_{NC}^{+}\,\,P[\psi _{C}(\mathbf{r}),\psi _{C}^{+}(\mathbf{r}),\psi _{NC}(\mathbf{r}),\psi _{NC}^{+}(\mathbf{r})] \nonumber \\
&&\times \psi _{C}^{+}(\mathbf{r}_{1})\,..\psi _{C}^{+}(\mathbf{r}_{p})\,\psi _{C}(\mathbf{s}_{q})\,...\psi _{C}(\mathbf{s}_{1})\,\psi
_{NC}^{+}(\mathbf{u}_{1})\,..\psi _{NC}^{+}(\mathbf{u}_{r})\,\psi _{NC}(\mathbf{v}_{s})...\psi _{NC}(\mathbf{v}_{1}). \nonumber\end{aligned}$$Symmetric ordering is defined as the average over all $N(p,q)=(p+q)!$ permutations of the factors $\widehat{\Psi }^{\dag }(\mathbf{r}_{1})..\widehat{\Psi }^{\dag }(\mathbf{r}_{p})\widehat{\Psi }(\mathbf{s}_{q})..\widehat{\Psi }(\mathbf{s}_{1})$. These results plus equal time commutation rules give the quantum correlation functions.
For example, the first order quantum correlation function that is used to exhibit macroscopic spatial coherence in BECs is given by$$\begin{aligned}
&&G^{1}(\mathbf{r}_{1};\mathbf{s}_{1})=\left\langle \,\widehat{\Psi }(\mathbf{r}_{1})^{\dag }\,\widehat{\Psi }(\mathbf{s}_{1})\right\rangle
\nonumber \\
&=&-\frac{1}{2}\delta (\mathbf{r}_{1}-\mathbf{s}_{1})
\label{Eq.FirstOrderCorrFn} \\
&&+\diiiint D^{2}\psi _{C}\,D^{2}\psi _{C}^{+}\,D^{2}\psi _{NC}\,D^{2}\psi
_{NC}^{+}\,\,P[\psi _{C}(\mathbf{r}),\psi _{C}^{+}(\mathbf{r}),\psi _{NC}(\mathbf{r}),\psi _{NC}^{+}(\mathbf{r})] \nonumber \\
&&\times (\psi _{C}^{+}(\mathbf{r}_{1})+\,\psi _{NC}^{+}(\mathbf{r}_{1}))(\psi _{C}(\mathbf{s}_{1})+\,\psi _{NC}(\mathbf{s}_{1})). \nonumber\end{aligned}$$The result includes pure condensate terms, pure non-condensate terms and mixed terms. The delta function term arises from the difference between normal and symmetric ordering for the condensate terms.
The Liouville-von Neumann equation for the density operator replaced by the functional Fokker-Planck equation for the distribution functional by using the correspondence rules involving functional derivatives, such as $\widehat{\Psi }_{C}(\mathbf{s})\widehat{\rho }\leftrightarrow \left( \psi _{C}(\mathbf{s})+\frac{1}{2}\delta /\delta \psi _{C}^{+}(\mathbf{s})\right)
P[\psi _{C},..,\psi _{NC}^{+}]$ and $\widehat{\Psi }_{NC}(\mathbf{s})\widehat{\rho }\leftrightarrow \left( \psi _{NC}(\mathbf{s})\right) P[\psi
_{C},..,\psi _{NC}^{+}]$. The density operator is multiplied by various field operators in the LVN equation and the overall effect on the distribution functional is obtained by applying such rules in succession. The general form for the FFPE after applying the truncated Wigner approximation to remove the third order functional derivatives is$$\begin{aligned}
&&\frac{\partial }{\partial t}P[\psi _{C},\psi _{C}^{+},\psi _{NC},\psi
_{NC}^{+}] \nonumber \\
&=&\dint d\mathbf{r\{}\dsum\limits_{\alpha =1}^{4}-\frac{\delta }{\delta
\phi _{\alpha }}A_{\alpha }(\psi _{C},\psi _{C}^{+},\psi _{NC},\psi
_{NC}^{+}) \nonumber \\
&&+\frac{1}{2}\dsum\limits_{\alpha ,\beta =1}^{4}\frac{\delta ^{2}}{\delta
\phi _{\alpha }\delta \phi _{\beta }}D_{\alpha \beta }(\psi _{C},\psi
_{C}^{+},\psi _{NC},\psi _{NC}^{+})\} \nonumber \\
&&\times P[\psi _{C},\psi _{C}^{+},\psi _{NC},\psi _{NC}^{+}],
\label{Eq.FFPE}\end{aligned}$$where $\phi _{1}\equiv \psi _{C},\phi _{2}\equiv \psi _{C}^{+},\phi
_{3}\equiv \psi _{NC},\phi _{4}\equiv \psi _{NC}^{+}$. The drift vector $A_{\alpha }$ involves the fields $\psi _{C},\psi _{C}^{+},\psi _{NC},\psi
_{NC}^{+}$ and their spatial derivatives. The positive definite diffusion matrix $D_{\alpha \beta }$ also involves these fields.
The functional Fokker-Planck equation for the distribution functional is then replaced by Ito stochastic field equations for $\psi _{C},\psi
_{C}^{+},\psi _{NC},\psi _{NC}^{+}$, which are now regarded as time dependent stochastic fields. The general form for the stochastic field equations is$$\frac{\partial }{\partial t}\phi _{\alpha }=A_{\alpha }(\psi _{C},\psi
_{C}^{+},\psi _{NC},\psi _{NC}^{+})+\dsum\limits_{\beta }d_{\alpha \beta
}(\psi _{C},\psi _{C}^{+},\psi _{NC},\psi _{NC}^{+})\Gamma _{\beta }(\mathbf{r},t). \label{Eq.ItoStochastic}$$The first term is deterministic and is associated with the drift vector. The second term is a random noise term and is associated with diffusion matrix, the matrix $d$ being related to the diffusion matrix via $D=d\,d^{T}$. The $\Gamma _{\beta }(\mathbf{r},t)$ are Gaussian-Markov random noise terms$$\overline{\Gamma _{\alpha }(\mathbf{r}_{1},t_{1})}=0\qquad \overline{\Gamma
_{\alpha }(\mathbf{r}_{1},t_{1})\Gamma _{\beta }(\mathbf{r}_{2},t_{2})}=\delta _{\alpha \beta }\delta (t_{1}-t_{2})\delta (\mathbf{r}_{1}-\mathbf{r}_{2}) \label{Eq.GauusianMarkoff}$$where the bar denotes a stochastic average.
Quantum averages of the field operators are now given by stochastic averages, which are equivalent to the functional integrals in Equation ([Eq.QuantumAverages]{}) involving the distribution functional.
$$\begin{aligned}
&&Tr[\widehat{\rho }\,\{\widehat{\Psi }^{\dag }(\mathbf{r}_{1})..\widehat{\Psi }^{\dag }(\mathbf{r}_{p})\widehat{\Psi }(\mathbf{s}_{q})..\widehat{\Psi
}(\mathbf{s}_{1})\}\,\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{1})..\widehat{\Psi }_{NC}^{\dag }(\mathbf{u}_{r})\widehat{\Psi }_{NC}(\mathbf{v}_{s})..\widehat{\Psi }_{NC}(\mathbf{v}_{1})] \nonumber \\
&=&\overline{\psi _{C}^{+}(\mathbf{r}_{1})\,..\psi _{C}^{+}(\mathbf{r}_{p})\,\psi _{C}(\mathbf{s}_{q})\,..\psi _{C}(\mathbf{s}_{1})\,\psi
_{NC}^{+}(\mathbf{u}_{1})\,..\psi _{NC}^{+}(\mathbf{u}_{r})\,\psi _{NC}(\mathbf{v}_{s})..\psi _{NC}(\mathbf{v}_{1})} \nonumber \\
&& \label{Eq.StochAver}\end{aligned}$$
Conclusion
==========
It is shown how the quantum correlation functions required for describing interferometry using BECs can obtained via stochastic averages of products of field functions, which satisfy Ito equations derived from the functional Fokker-Planck equation for the phase space distribution functional that represents the quantum density operator. The phase space distribution functional is of the Wigner type for the condensate modes and the positive P type for the non-condensate modes. Decoherence and dephasing effects in BEC interferometry with large boson numbers are fully treated, unlike previous theories in which only condensate modes are considered or the theory is restricted to small boson numbers.
Acknowledgements
================
The author is grateful for discussions with M K Olsen and P D Drummond on applying Wigner and positive P phase space distributions to BECinterferometry.
References {#references .unnumbered}
==========
[9]{} Kasevich M A 2002 *Science* **298** 1363
Dunningham J A, Burnett K and Barnett S M 2002 *Phys. Rev. Letts.* **89** 150401
Bach R and Rzazewski K 2004 *Phys. Rev. A* **70** 063622
Dalton B J 2006 Two-mode theory of BEC interferometry *Preprint* Q-ph/0601012
Ananakian D and Bergeman T 2006 *Phys. Rev. A* **73** 013604
Gardiner C W and Zoller P 2000 *Phys. Rev. A* **61** 033601
Steele M J, Olsen M K, Plimak L I, Drummond P D, Tan S M, Collett M J, Walls D F and Graham R 1998 *Phys. Rev. A* **58** 4824
Gardiner C W and Zoller P 1998 *Phys. Rev. A* **58** 536
|
---
abstract: 'It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$, admit real uniformly discrete uniqueness sets ${\Lambda}$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having “periodic gaps”. The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets ${\Lambda}$ satisfy a strong non-uniqueness property: Every discrete function $c(\lambda)\in l^2({\Lambda})$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.'
author:
- Alexander Olevskii and Alexander Ulanovskii
title: Discrete Uniqueness Sets for Functions with Spectral Gaps
---
A.O.: School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Israel\
E-mail: [email protected]
A.U.: Stavanger University, 4036 Stavanger, Norway\
E-mail: [email protected]
Introduction
============
[**Paley-Wiener Space**]{}. We will use the standard form of the Fourier transform: $$F(t)=\hat f(t):=\int_{{\mathbb R}}e^{-2\pi i tx}f(x)\,dx.$$
Given a measurable set $S$, the Paley-Wiener space $PW_S$ consists of the inverse Fourier transforms of all square-integrable functions $F$ which vanish a.e. outside $S$. The set $S$ is called the spectrum of the space $PW_S$. Clearly, if the measure of $S$ is finite, then $F\in L^1({{\mathbb R}})$, and so every function $f \in PW_S$ is continuous. If $S$ is bounded, then every $f \in PW_S$ is an entire function of exponential type.
[**Uniformly discrete sets**]{}. A set ${\Lambda}\subset{{\mathbb R}}$ is called [*uniformly discrete*]{} (u.d.) if $\delta({\Lambda})>0$, where $$\label{sep}\delta({\Lambda}):=\inf_{\lambda,\lambda'\in{\Lambda},\lambda\ne\lambda'} |\lambda-\lambda'|$$ is the infimal distance between different elements of ${\Lambda}$.
A u.d. set ${\Lambda}$ is said to have the [*uniform density*]{} $D({\Lambda})$, if ${\Lambda}$ is regularly distributed in the following sense:$$\mbox{Card}\left(
{\Lambda}\cap(x,x+r)\right)=rD({\Lambda})+o(r) \mbox{ uniformly on } x \mbox{ as } r\to \infty.$$
[**Uniqueness Problem**]{}. Let $F$ be a space of continuous functions on the real line ${{\mathbb R}}$. A set ${\Lambda}\subset{{\mathbb R}}$ is called a [*uniqueness set*]{} for $F$ if $$f \in F, f|_{\Lambda}=0 \Rightarrow f =0.$$ Otherwise, ${\Lambda}$ is called a [*non-uniqueness set*]{}.
[**Problem**]{}. [*Which spaces of continuous functions on ${{\mathbb R}}$ admit u.d. uniqueness sets?*]{}
We will consider this problem for spaces of function whose spectrum belongs to a fixed set $S$. It is natural to distinguish between the following cases: $S$ is a bounded set, an unbounded set of finite measure, and a set of infinite measure.
In the present paper we focus on spaces of continuous functions whose spectrum lies in a set $S$ of infinite measure. In Sec. 3–5 we establish that wide spaces of such functions admit u.d. uniqueness sets, provided $S$ has periodic gaps. The periodicity condition is important. In particular, in Sec. 6, for sets $S$ with randomly distributed gaps we show that every u.d. set ${\Lambda}$ satisfies some strong non-uniqueness property.
We start with a short survey of known results on the first two cases. A detailed discussion of these and related results can be found in [@ou1]. For simplicity of presentation, we focus on the one-dimensional case.
Spectra of Finite Measure
=========================
[**Bounded Spectra**]{}. The classical case is when $S = [a,b]$ an interval. Then the elements of $PW_S$ are entire functions of exponential type. The distribution of zeros of such functions is very well studied, see [@levin]. In particular, if the density $D({\Lambda})$ exists, then the condition $D({\Lambda})\geq |S|$ is necessary while the condition $D({\Lambda}) >|S|$ is sufficient for ${\Lambda}$ to be a uniqueness set for $PW_S$, where $|S|=b-a$ denotes the measure of $S$. This can be shown by standard complex variable techniques. A classical result of Beurling and Malliavin [@BM] states that the same is true for irregular sets ${\Lambda}$, provided the uniform density is replaced with a certain exterior one (the Beurling–Malliavin density).
In the case of disconnected spectra $S$, the uniqueness property of u.d. sets cannot be expressed in terms of their density: Some “dense” (relatively to the measure of $S$) u.d. sets ${\Lambda}$ may be non-uniqueness sets for $PW_S$. For example, one can easily check that ${\Lambda}={{\mathbb Z}}$ is a non-uniqueness set for $PW_S$, where $S = [0,\epsilon]\cup[1,1+\epsilon],0< \epsilon<1$.
On the other hand, some “sparse” u.d. sets ${\Lambda}$ may be uniqueness sets for $PW_S$ with a “large” spectrum $S$. This phenomenon was discovered by Landau [@L], who proved that certain perturbations of ${{\mathbb Z}}$ produce uniqueness sets for $PW_S$ whenever $S$ is a finite union of intervals $[k+a,k+1-a],$ where $0<a<1/2$ is any fixed number. The uniqueness sets ${\Lambda}$ constructed by Landau have a complicated structure.
A more general result is proved in [@ou2]:
The set $${\Lambda}:=\{n+2^{-|n|}, n\in{{\mathbb Z}}\}$$ is a uniqueness set for $PW_S$, for every bounded set $S$ satisfying $|S|<1.$
This theorem remains true for the bounded sets $S$ of arbitrarily large measure satisfying $|S_1|<1$, where we denote by $$\label{pr}
S_a:=(S+a{{\mathbb Z}})\cap[0,a]$$ the “projection” of $S$ onto $[0,a]$.
Moreover, the result holds true also for the unbounded sets of finite measure which have a “moderate accumulation” at infinity, see [@ou2].
Using re-scaling, one may formulate a corresponding result for any bounded set $S$.
[**Unbounded Spectra of Finite Measure**]{}. It was shown in [@ou] (see also [@ou1], Lec. 10) that for every (bounded or unbounded) set $S$ in ${{\mathbb R}}$ of finite measure, the space $PW_S$ possesses a u. d. uniqueness set:
For every set $S$ of finite measure, there is a u.d. set ${\Lambda}$ satisfying $D({\Lambda})=|S|$, which is a uniqueness set for $PW_S$.
By the discussion above, the density condition $D({\Lambda})=|S|$ is optimal, since one cannot get a smaller density when $S$ is an interval.
Sobolev Spaces with Periodic Spectral Gaps
==========================================
Periodic Spectral Gaps
----------------------
We say that $S$ has periodic “strong” gaps if there exists $a>0$ such that $$\label{2}
|\overline{S_a}|<a,$$ where $\overline{S_a}$ denotes the closure of $S_a$, and the set $S_a$ is defined in (\[pr\]). Condition (\[2\]) means that there is a non-empty interval $I\subset[0,a]$ such that $S\cap (I+a{{\mathbb Z}})=\emptyset.$
We say that $S$ has periodic “weak” gaps if $$\label{3}
|S_a|<a.$$Condition (\[3\]) means that there is a set of positive measure $Q\subset[0,a]$ such that $S\cap (Q+a{{\mathbb Z}})=\emptyset.$
Observe that [*every set $S$ of finite measure has periodic weak gaps*]{}, since we have $|S_a|<a$, for every $a>|S|.$
Uniqueness Sets for Sobolev Spaces
----------------------------------
Given any u.d. set ${\Lambda}$, it is obvious that there is a non-trivial smooth function $f$ which vanishes on ${\Lambda}$. However, this is no longer so if the spectrum of $f$ has weak periodic gaps. We will state the result for Sobolev spaces.
For every number $\alpha>1/2$, we denote by $W^{(\alpha)}$ the Sobolev space of functions $f$ such that the Fourier transform $F=\hat f$ satisfies $$\label{4}
\|F\|_\alpha^2:=\int_{{\mathbb R}}(1+|t|^{2\alpha})|F(t)|^2\,dt<\infty.$$ It is clear that the functions $F$ satisfying (\[4\]) belong to $L^1({{\mathbb R}})$, and so $W^{(\alpha)}$ consists of continuous functions. We denote by $W^{(\alpha)}_S$ the subspace of $W^{(\alpha)}$ of functions $f$ with spectrum in $S$, i.e. $F=0$ a.e. outside $S$.
\[t1\] Suppose a set $S$ satisfies $|S_a|<a$, for some $a>0$. Then there is a u.d. set ${\Lambda}$ of density $D({\Lambda})=a$, which is a uniqueness set for $W^{(\alpha)}_{S}, \alpha>1/2$.
Decomposition of ${{\mathbb Z}}$
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Let $A \subset[0,1], |A| < 1$. There exist pairwise disjoint sets $Z_j \subset{{\mathbb Z}}, j\in{{\mathbb N}}$, such that every exponential system $$\label{exps}\{e^{-i 2\pi n t},n\in Z_j\}$$ is complete in $L^2(A)$.
1\. Observe that the exponential family $$\label{sys}\{e^{-i2\pi nt}, |n| > N\}$$is complete in $L^2(A)$, for every natural $N$. Indeed, assume there exists a non-trivial function $F\in L^2(A)$ orthogonal to the system (\[sys\]). Extend $F$ by zero to $[0,1]\setminus A$. Then $F$ is orthogonal to the system (\[sys\]) in $L^2(0,1)$. Since the trigonometrical system forms an orthonormal basis in $L^2(0,1)$, we conclude that $F$ is a trigonometric polynomial: $$F(t)=\sum_{|n|\leq N}c_je^{-i2\pi n t}.$$Clearly, $F$ cannot vanish on the set of positive measure $[0,1]\setminus A$, which is a contradiction.
2\. Fix a sequence $\epsilon_k, k\in{{\mathbb N}}$, satisfying $\epsilon_k\to0, k\to\infty.$ We will now construct a sequence of disjoint finite symmetric sets $\Gamma_k\subset{{\mathbb Z}}, k\in {{\mathbb N}},$ with the following property: For every $|m|\leq k,$ there is a trigonometric polynomial $P_{k,m}$ whose frequencies belong to $\Gamma_k$, such that $$\label{s}
\|e^{i2\pi m t}-P_{k,m}(t)\|_{L^2(A)}<\epsilon_k.$$
Set $\Gamma_1:=\{-1,0,1\}$. Clearly, (\[s\]) holds with $m=0,-1,1$. Then set$$\Gamma_k:=\{n:n_{k-1}<|n|\leq n_k\},$$where $n_1=1$, and we choose $n_j, j>1,$ inductively as follows: By Step 1, there exists $n_2$ so large that for every $|m|\leq 2$ there is a polynomial $P_{2,m}$ satisfying (\[s\]) with $k=2$ and whose frequencies belong to the set $\Gamma_2$, and so on. On the $k$-th step, we choose an integer $n_k$ so large that for every $|m|\leq k$ there is a polynomial $P_{k,m}$ satisfying (\[s\]) and whose frequencies belong to the set $\Gamma_k$.
3\. Now, take a partition of ${{\mathbb N}}$ into disjoint infinite subsets $\Delta_j$ and set $$Z_j:=\bigcup_{k\in \Delta_j}\Gamma_k.$$ It follows from the construction above that every exponential system (\[exps\]) is complete in $L^2(A)$.
It is easy to see that one may construct the sets $Z_j$ so that $$D(\cup_{j=1}^\infty Z_j)=1.$$
Periodization and Fourier Transform
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For an integrable function $H$ on the circle group ${{\mathbb T}}:={{\mathbb R}}/{{\mathbb Z}}$, we denote by$$c_n(H) := \int_{{\mathbb T}}H(t) e^{2\pi i nt}\, dt,\quad n\in{{\mathbb Z}},$$ the Fourier coefficients of $H$.
Given $F\in L^1 ({{\mathbb R}})$, consider its “periodization” $$H(u):= \sum_{k\in{{\mathbb Z}}} F(u+k), \quad u\in [0,1].$$ Clearly, $H$ is defined a.e. and belongs to $L^1({{\mathbb T}})$. Direct calculation shows that its Fourier coefficients satisfy $ c_n (H) = f(n),$ where $f$ is the inverse Fourier transform of $F$.
Similarly, for the periodization $H_v$ of the function $$F_v(t):= e^{2\pi i vt} F(t),$$ we have $$\label{ol}
c_n (H_v)= f(n+v), \quad n\in{{\mathbb Z}}.$$
It is easy to check that the periodization of an $L^2$-function does not always belong to $L^2({{\mathbb T}})$. However, the following is true:
Assume $F$ satisfies $\|F\|_\alpha<\infty.$ Then $$\int_0^1 |H(t)|^2\,dt<\infty.$$
Indeed, we have $$|H(t)|^2=\left|\sum_{n\in{{\mathbb Z}}}F(t+na)\right|^2\leq \sum_{n\in{{\mathbb Z}}}\frac{1}{(1+|n|^\alpha)^2}\sum_{n\in{{\mathbb Z}}}|F(t+na)|^2(1+|n|^\alpha)^2,$$and the lemma easily follows from the definition of $\|F\|_\alpha$ in (\[4\]).
Proof of Theorem 3
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By re-scaling we can assume that $a=1$.
Using Lemma 1 with $A=S_1$, write ${{\mathbb Z}}= \bigcup_{j=1}^\infty Z_j$, where each exponential system (\[exps\]) is complete in $L^2(S_1)$. It means that each $Z_j$ is a uniqueness set for the space $PW_{S_1}$.
Fix a sequence $\{\alpha_j\}$ dense in $[0,1],$ and set $$\label{ol2}
{\Lambda}:= \bigcup_{j=1}^\infty (Z_j + \alpha_j).$$We may assume that ${\Lambda}$ is u.d. and $D({\Lambda})=1.$
Now we will prove that ${\Lambda}$ is a uniqueness set for the space $W_S^{(\alpha)}$. We have to show that every function $f\in W_S^{(\alpha)}$ satisfying $$\label{ol3}
f|_{\Lambda}= 0$$ must vanish on ${{\mathbb R}}$.
Let $F:=\hat f$. Fix $j\in{{\mathbb N}}$ and consider the function $$F_j(t):= e^{2\pi \alpha_j t} F(t),$$ and its periodization $H_j$. Recall that $F$ vanishes a.e. outside $S$. Since $S\subset S_1+{{\mathbb Z}}$, we have $$H_j = 0 \quad \mbox{a.e. on } {{\mathbb T}}\setminus S_1.$$ Also, by Lemma 2, $H_j \in L^2 ({{\mathbb T}})$.
By (\[ol\]), (\[ol2\]) and (\[ol3\]), $$c_n (H_j)= f(\alpha_j +n) = 0,\quad n\in Z_j.$$Since $Z_j$ is a uniqueness set for $PW_{S_1}$, we have $H_j=0$ a.e. By (\[ol\]), this means that $$f(n+\alpha_j) = 0,\quad n\in {{\mathbb Z}}.$$ Since this equality is true for all $j $, $f$ is continuous and the sequence $\{\alpha_j\}$ is dense in $[0,1]$, we conclude that $ f= 0.$
Uniqueness Sets for Fast Decreasing Functions
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Theorem \[t1\] shows that certain classes of smooth functions $f$ having periodic weak spectral gaps admit u.d. uniqueness sets. In this section we show that a similar result holds for functions $f$ whose Fourier transforms $F$ are smooth functions.
Let us denote by $Y$ the space of continuous functions $f$ satisfying $$\label{dec}
\sup_{x\in{{\mathbb R}}}(1+x^2)|f(x)|<\infty.$$ Denote by $Y_S$ the subspace of $Y$ of functions $f$ such that $F=\hat f=0$ outside $S$.
\[t2\] Suppose a set $S$ satisfies $|S_a|<a$, for some $a>0$. Then there is a u.d. set ${\Lambda}$ of density $D({\Lambda})=a$, which is a uniqueness set for $Y_{S}$.
The proof below shows that in Theorem \[t2\] condition (\[dec\]) in the definition of $Y$ can be somewhat relaxed. However, the result is no longer true if no decay condition is imposed, see Theorem 6 below.
Proof of Theorem \[t2\]
-----------------------
The proof follows the same idea used in the proof of Theorem 3. However, the periodization of $F$ cannot be defined pointwisely.
We will use the following corollary of the classical Poisson summation formula:
\[l3\] Assume a continuous function $f$ satisfies (\[dec\]) and $\hat f(t)=0, t\in Q+{{\mathbb Z}}$, for some set $Q\subset[0,1], |Q|>0$. Then for every $x\in [0,1]$ we have $$\label{pp1}
\sum_{n\in{{\mathbb Z}}}f(x+n)e^{-i2\pi nt}=0, \quad t\in Q.$$
If $F=\hat f$ is also fast decreasing, then this claim follows directly from the Poisson formula.
Otherwise, apply the Poisson formula to the convolution $(f\ast h_\epsilon)(x)$, where $$h_\epsilon:=\left(\frac{1}{2\epsilon}{\bf 1_{(-\epsilon,\epsilon)}}\right)^{2\ast}$$ and ${\bf 1_{(-\epsilon,\epsilon)}}$ is the indicator function of $(-\epsilon,\epsilon)$: $$\sum_{n\in{{\mathbb Z}}} (f*h_\epsilon)(x+n)e^{-i2\pi nx} = 0, \quad x\in [0,1),\ t\in Q.$$
Now, we claim: $$\sum_{n\in{{\mathbb Z}}} (f\ast h_\epsilon)(x+n)e^{-i2\pi nt} \to \sum_{n\in{{\mathbb Z}}} f(x+n)e^{-i2\pi nt}$$ as $\epsilon\to 0$, which proves the lemma. Indeed, fix $\delta>0$ and decompose the left side into two sums: $
\sum_{|n|< N} + \sum_{ |n|\geq N}.$ One can chose $N= N(\delta)$ so that modulus of the second summand is $< \delta$, for every $x,t\in[0,1]$ and $0<\epsilon<1$. Clearly, each term of the first summand goes to $f(x)e^{-i2\pi nt}$ as $\epsilon\to 0,$ due to the continuity of $f$.
Now, we can finish the proof of Theorem 4. By re-scaling, we may assume that $a=1$, so that $|S_1|<1$.
Following the proof of Theorem 3, we may find pairwise disjoint sets $Z_j\subset{{\mathbb Z}}, j\in{{\mathbb N}},$ such that for every $j$ the system $$E(Z_l):=\{e^{-2\pi i kt}, k\in Z_l\}$$ is complete in $L^2(S_1)$.
Set $${\Lambda}:=\cup_{j\in{{\mathbb N}}}(Z_j+\alpha_j),$$where $\{\alpha_l, l\in{{\mathbb N}}\}$ is dense in $(0,1)$. It remains to check that ${\Lambda}$ is a uniqueness set for $Y_S$.
Assume $f|_{\Lambda}= 0$, for some $f\in Y_S$, i.e. we have $$f|_{Z_j + \alpha_j} = 0, \quad j=1,2,\dots$$
Fix $j$ and consider a $1$-periodic function $$g_j (x) :=\sum_{n\in {{\mathbb Z}}} f(n + \alpha_j) e^{-i 2\pi nx}.$$ Clearly, $g\in L^2(0,1)$ and is orthogonal in $L^2(0,1)$ to all the exponential functions in $E(Z_j)$. On the other hand, due to Lemma 3, $$g_j(x) = 0,\quad t\in Q:=[0,1]\setminus S_1.$$ The completeness of $E(Z_j)$ in $L^2(S_1)$ implies that $g_j = 0$ a.e. Hence, $$f(n+\alpha_j) = 0,\quad n\in{{\mathbb Z}}.$$ This is true for every $j$. Recalling that $\{\alpha_j\}$ is dense on $[0,1]$ and $f \in C({{\mathbb R}}) $, we conclude that $f=0$ on ${{\mathbb R}}$.
Distributions with Periodic Spectral Gaps
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Strong Gaps
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If $S$ has periodic strong gaps, then the results above can be extended to wider function spaces.
Denote by $X$ the space of continuous functions that have at most polynomial growth on ${{\mathbb R}}$. Every element $f\in X$ is a Schwartz distribution. Its spectrum is the minimal closed set $S$ such that for every test function $\varphi$ satisfying $\hat \varphi=0$ in a neighbourhood of $S$, we have $$\int_{{\mathbb R}}f(t)\varphi(t)\,dt=0.$$
Given closed set $S$, we denote by $X_S$ the subspace of $X$ consisting of functions with spectrum in $S$.
Without loss of generality, we may assume the spectral gaps are $[0,\delta] + {{\mathbb Z}}.$
\[t3\] There is a u.d. set ${\Lambda}, D({\Lambda}) =1 $, which is a uniqueness set for $X_S, S = [0, 1-\delta]+{{\mathbb Z}}$, for every $0<\delta<1.$
Consider $Z_j$ as in Lemma 1 (this can be done independently on $\delta$). Choose ${\Lambda}$ as in the proof of Theorem 4. Given $f\in X_S,$ consider the function $ g:= f\cdot\varphi,$ where $\hat\varphi$ is a Schwartz function supported by $[0 ,\delta/2]$.
It is easy to see that $g$ satisfies the assumptions of Theorem 4 with $a=1 ,S= [0,1-\delta/2]$. If $f|_{\Lambda}= 0$, then the same is true for $g$. So, Theorem 4 implies $f= 0.$
Weak Gaps
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Here we show that Theorem 5 is no longer true for the weak spectral gaps. This is a direct corollary of a result from [@ou09].
We need the following
Given a closed (not necessarily bounded) set $S$, the Bernstein space $B_S$ is the set of continuous bounded functions $f$ on ${{\mathbb R}}$ whose spectrum (in distributional sense) lies in $S$.
[([@ou09])]{} There is a closed set $S$ of Lebesque measure zero such that every bounded function $c(\lambda)$ defined on a u.d. set ${\Lambda}$ can be interpolated by a function $f\in B_S$.
It is obvious that every set of measure zero has weak periodic gaps with an arbitrary period $a$. However, no u.d. set ${\Lambda}$ is a uniqueness set for $B_S$.
A few words about the proof of Theorem 6. It is based on a classical result of D.E. Menshov (1916) (see [@b]): [*There is a probability measure $\mu$ on ${{\mathbb R}}$ supported by a compact set $K$ of measure zero, and such that its Fourier transform $$\hat\mu (x)=\int_K e^{-2\pi i t}\,d\mu(t)$$ vanishes at infinity*]{}.
Here is a short sketch of the proof (see details in [@ou1], Lec. 10).
1\. Given $\delta$, by re-scaling one can get a probability measure $\mu_\delta$ supported by a compact $K$ of Lebesgue measure zero, such that $$\hat\mu_\delta(0)=1, |\hat\mu_\delta(x)|<\delta,\quad |x|>\delta.$$
2\. Using this, one can construct a family of compact sets $K_j$ of measure zero, which goes to infinity, and functions $g_j\in B_{K_j}, j\in{{\mathbb N}},$ satisfying $$\|g_j\|_\infty=g_j(0)=1, \ |g_j(t)|<e^{-j}, \quad |t|>e^{-j}.$$
3\. Set $$S:=\cup_{j=1}^\infty K_j.$$It is a closed (non-compact) set of measure zero.
Fix any $\delta>0$ and any u.d. set ${\Lambda}$. Using appropriate translates of the functions $g_j$, one can define functions $f_\lambda\in B_S$ satisfying $$\|f_\lambda\|_\infty=f_j(\lambda)=1, |f(\lambda')|<e^{-2|\lambda'-\lambda|/\delta},\quad \lambda_j\in{\Lambda},\lambda_l\ne\lambda_j.$$
4\. Consider the linear operator $T:l^\infty({\Lambda}) \rightarrow l^\infty({\Lambda}) $ defined by $$(Tc)_\lambda:=\sum_{\lambda'\in{\Lambda},\lambda'\ne \lambda}f_{\lambda'}(\lambda)c_{\lambda'}, \quad \lambda\in{\Lambda}, \quad c=\{c_{\lambda'}, \lambda'\in{\Lambda}\}\in l^\infty({\Lambda}).$$ Clearly, $\|T\|<1$. Hence, the operator $ T+I$ is surjective. Therefore, for every datum $c=\{c_\lambda\}\in l^\infty({\Lambda})$ there is a sequence $ b=\{b_\lambda\}\in l^\infty({\Lambda})$ satisfying $(I+T)b=c$. Hence, the function $$f(x):= \sum_{\lambda\in{\Lambda}} b_\lambda f_\lambda (x).$$ belongs to $B_S$ and solves the interpolation problem $ f|_{\Lambda}= c$.
Non-Periodic Spectral Gaps
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Here we show that the periodicity of spectral gaps is crucial for existence of discrete uniqueness sets.
Let us consider spectra $S$ which are unions of disjoint intervals of a given length. For simplicity, we assume that each interval has length one: $$\label{gam}
S=\cup_{j=1}^\infty [\gamma_j,\gamma_j+1].$$We also assume that the distances $\xi_j$ between the intervals belong to a fixed interval, say $[2,3]$: $$\label{xi}
\quad \xi_j:= \gamma_{j+1}-\gamma_j-1\in [2,3], \quad j\in{{\mathbb N}}.$$ Clearly, $S$ belongs to a half-line $[\gamma_1,\infty)$ and admits a representation $$\label{g}
S=\Gamma+[0,1], \ \Gamma:=\cup_{j=1}^\infty\{\gamma_j\},$$where $\Gamma$ satisfies $\delta(\Gamma)\geq 3$. Here $\delta(\Gamma)$ is the separation constant defined in (1).
Now we introduce a certain property of u.d. sets. We say that u.d. set $\Gamma$ satisfies property (C) if it contains arbitrary long arithmetic progressions with rationally independent steps. More precisely, we assume
\(C) For every $m\in{{\mathbb N}}$ there are rationally independent numbers $q_1,...,q_m$, such that for every $N\in{{\mathbb N}}$ the set $\Gamma$ contains arithmetic progressions of length $N$ with differences $q_1,...,q_m$. The latter means that there exist $a_1,...,a_m$ such that $$\bigcup_{j=1}^m\{a_j+q_j, a_j+2q_j,\dots,a_j+Nq_j\}\subset\Gamma.$$
\[int\] Assume $S$ is given in (\[gam\])–(\[g\]), where $\Gamma$ satisfies property [(C)]{}. Then no u.d. set ${\Lambda}$ is a uniqueness set for the Sobolev space $W_S^{\alpha}$.
Below we prove this result in a stronger form.
One may also check that, under the assumptions of Theorem \[int\], no u.d. set ${\Lambda}$ is a uniqueness set for the space $Y_S$.
Interpolation Sets
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A set ${\Lambda}$ is called an [*interpolation set*]{} for the Paley-Wiener space $PW_S$, if for every sequence $\{c_\lambda,\lambda\in{\Lambda}\}\in l^2({\Lambda})$ there exists $f\in PW_S$ satisfying$$f(\lambda)=c_\lambda,\quad \lambda\in{\Lambda}.$$
The following criteria is well-known (see e.g. [@ou1], Lec. 4):
Let $S$ be a bounded set and ${\Lambda}$ a u.d. set. Then ${\Lambda}$ is a set of interpolation for $PW_S$ if and only if there is a constant $C>0$ such that the inequality $$\int_S \left|\sum_{\lambda\in{\Lambda}}c_\lambda e^{i2\pi\lambda t}\right|^2\,dt\geq C \sum_{\lambda\in{\Lambda}}|c_\lambda|^2$$holds for every finite sequence $c_\lambda$.
Theorem 7 is a direct corollary of the following
[**Main Lemma**]{}. [*Assume $S$ is a set from Theorem 7. Then for every $\delta>0$ there is a bounded subset $S(\delta)\subset S$ such that every u.d. set ${\Lambda}$ satisfying $\delta({\Lambda})\geq\delta$ is a set of interpolation for $PW_{S(\delta)}$.*]{}
Indeed, consider a set ${\Lambda}\cup\{c\}$ for some point $c\not\in{\Lambda}$. By the Main Lemma, it is a set of interpolation for $PW_{S(\delta)}$, for some bounded subset $S(\delta)\subset S$. Then there exists $f\in PW_{S(\delta)}$ satisfying $f(c)=1$ and $$f(\lambda)=0,\quad \lambda\in{\Lambda}.$$ It remains to observe that $PW_{S(\delta)}\subset W_S^{\alpha}$.
Proof of Main Lemma
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Suppose $\Gamma$ satisfies property [(C)]{}. Then for every $0<\epsilon<1$ there exist $N\in{{\mathbb N}}$ and $\eta_{j}\in\Gamma, j=1,...,N,$ such that the exponential polynomial $$P(t)=\frac{1}{N}\sum_{j=1}^N e^{i \eta_{j}t}$$ satisfies $$\label{p}
|P(t)|\leq\epsilon , \quad \epsilon < |t|<1/\epsilon.$$
1\. Fix any integer $m>1/\epsilon$. Then fix numbers $q_1,...,q_m$ in the definition of property (C). Since $q_j$ are rationally independent, the set of points $$\label{set}\{kq_j\in(-1/\epsilon,1/\epsilon)\}$$ is separated, where $ k\in{{\mathbb Z}},k\ne0, j=1,\dots, m$. Hence, the distance between any two points in this set exceeds some positive number $\rho$. We may assume that $\rho<\epsilon$.
2\. For $n\in{{\mathbb N}}$ and $q\geq 2$, consider the $(1/q)$-periodic exponential polynomial $$P_{n,q}(t):=\frac{1}{n}\sum_{j=0}^{n-1}e^{i 2\pi j q t}=\frac{1}{n}\frac{e^ {i2\pi nq t}-1}{e^{i2\pi q t}-1}.$$ From the properties of Dirichlet kernel, it is well-known that it satisfies $$\label{rho}
|P_{n,q}(t)|\leq\rho, \quad \mbox{dist}(t, (1/q){{\mathbb Z}})\geq\rho,$$provided $n$ is large enough.
3\. Choose $n$ so large that (\[rho\]) holds with $q=q_j$, $j=1,\dots,m$. Then, since the set (\[set\]) is $\rho$-separated, for every $t$ satisfying $\epsilon<|t|<1/\epsilon,$ the inequality $$|P_{n, q_j}(t)|<\epsilon$$holds for all but at most one value of $j\in\{1,\dots,m\}$.
4\. By the definition of property (C), there exist $a_j$ such that $a_j+kq_j\in\Gamma, k=0,\dots, n-1$. Set $$P(t)=\frac{1}{m}\sum_{j=1}^m e^{i2\pi a_j t}P_{n, q_j}(t).$$ By Step 3, we see that $$|P(t)|<\frac{1+(m-1)\epsilon}{m}<\epsilon, \quad \epsilon<|t|<1/\epsilon,$$which completes the proof.
Fix $\delta>0$ and assume that a u.d. set ${\Lambda}$ satisfies $\delta({\Lambda})\geq\delta$.
By Lemma 5, for every $0<\epsilon<1$ there is an exponential polynomial $P$ with frequencies in $\Gamma$ satisfying (\[p\]). We denote the set of its frequencies by $\Gamma_P\subset\Gamma$, and set $$S(\delta):=\Gamma_P+[0,1].$$Clearly, $S(\delta)$ is a bounded subset of $S$.
Now we fix any positive smooth function $\Phi$ which vanishes outside $[0,1]$ such that its Fourier transform $\varphi=\hat \Phi$ satisfies $\varphi(0)=1$ and $$\label{ph}
\sup_{x\in{{\mathbb R}}}(1+x^4)|\varphi (x)|<\infty.$$
Set $$H(t):=(\Phi\ast \sum_{\gamma\in \Gamma_P}\delta_\gamma)(t)= \sum_{\gamma\in \Gamma_P} \Phi(t-\gamma).$$Then the support of $H$ belongs to $S(\delta)$ and its Fourier transform is given by $$h(x):=\hat H(x)=P(x)\varphi(x).$$Clealry, $h(0)=1.$
When $\epsilon$ is sufficiently small, from (\[p\]) and (\[ph\]) we get $$|h(x)|<\frac{\epsilon}{1+x^2}, \mbox{ for all } |x|>\delta,$$where $\delta$ is the separation constant of ${\Lambda}$. Using this estimate and assuming that $\epsilon $ is sufficiently small, for every $\lambda\in{\Lambda}$ we get the estimate $$\sum_{\mu\in{\Lambda},\mu\ne\lambda}|h(\mu-\lambda)|<\sum_{\mu\in{\Lambda},\mu\ne\lambda}\frac{\epsilon}{1+(\mu-\lambda)^2}
< 2\sum_{n\in{{\mathbb N}}}\frac{\epsilon}{1+(\delta n)^2}<\frac{1}{2}.$$
Set $$M:=\max_{t\in [0,1]}|\Phi(t)|.$$ Then $$\int_{S(\delta)}\left|\sum_{\lambda\in{\Lambda}}c_\lambda e^{i\lambda t}\right|^2\,dt \geq \frac{1}{M}\int_{S(\delta)}\left|\sum_{\lambda\in{\Lambda}}c_\lambda e^{i\lambda t}\right|^2 H(t)\,dt$$ $$=\frac{1}{M}\left(\sum_{\lambda\in{\Lambda}}|c_\lambda|^2+\sum_{\lambda,\mu\in{\Lambda},\lambda\ne\mu}c_\lambda\bar c_\mu h(\lambda-\mu)\right)\geq$$$$\frac{1}{M}\left(\ \sum_{\lambda\in{\Lambda}}|c_\lambda|^2-\sum_{\lambda,\mu\in{\Lambda},\lambda\ne\mu}\frac{|c_\lambda|^2+|c_\mu|^2}{2}|h(\lambda-\mu)|\right)\geq$$ $$=\frac{1}{M}\left(\sum_{\lambda\in{\Lambda}}|c_\lambda|^2-\sum_{\lambda\in{\Lambda}}|c_\lambda|^2\sum_{\lambda,\mu\in{\Lambda},\mu\ne\lambda}|h(\lambda-\mu)|\right)>
\frac{1}{2M}\sum_{\lambda\in{\Lambda}}|c_\lambda|^2.$$By Lemma 4, this completes the proof.
Random Spectra do not Admit u.d. Uniqueness Sets
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Here we consider the situation when $S$ is a countable union of unite intervals, the distances between the intervals being randomly distributed. More precisely, below we assume that $S$ and $\Gamma$ are defined in (\[gam\]) and (\[g\]), and that the $\xi_j:=\gamma_{j+1}-\gamma_j-1$ are independent random variables uniformly distributed over the interval $[2,3]$. With these assumptions, we have
\[tr\] With probably one no u.d. set ${\Lambda}$ is a uniqueness set for $W_{S}^{(\alpha)}$.
Theorem 8 follows from the following claim, which is an analogue of the Main Lemma in Sec. 6.1: [*With probability one, for every fixed $\delta>0$ there is a bounded subset $S(\delta)\subset S$ such that every u.d. set ${\Lambda}, \delta({\Lambda})\geq\delta,$ is a set of interpolation for $PW_{S(\delta)}$.*]{}
Recall that $$S=\cup_{j=1}^\infty \{\gamma_j\}+[0,1], \quad \gamma_{j+1}-\gamma_j\in[3,4], \quad j\in{{\mathbb N}}.$$ It is easy to see that given any integers $k\geq 1, N\geq 2$ and number $q\in (3,4)$, the set $$\{\gamma_k, \gamma_k+q,\dots,\gamma_k+Nq\}+[1/4,3/4]$$belongs to $S$ whenever $$|\gamma_{k+j}-(\gamma_k+jq)|<\frac{1}{4},\quad j=1,2,\dots, N.$$ Recall also that $\gamma_{j+1}-\gamma_{j}$ is uniformly distributed over $[3,4]$. So, the probability that the latter inequalities hold true is positive and independent on $k$.
Now, fix any $m\in{{\mathbb N}}$ and $q_1,\dots,q_m\in(3,4)$. By the the Borel-Cantelli lemma, one can see that with probability one there are integers $k_1,\dots,k_m$ such that the finite sequence $$\Gamma^\ast:= \bigcup_{j=1}^{m} \{\gamma_{k_j}, \gamma_{k_j}+q_j,\dots, \gamma_{k_j}+Nq_j\}$$satisfies $$S(\delta):=\Gamma^\ast+[1/4,3/4]\subset S.$$ Now, choosing $m$ and $N$ sufficiently large, the claim above follows exactly the same way as in the proof of the Main Lemma.
Remarks
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Multi-dimensional Extensions
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All our one-dimensional results above admit multi-dimensional extensions. Here we give a very brief account of these extensions.
The definitions in Sec. 1 can be extended to the multi-dimensional situation. In particular, given a set $S\subset{{\mathbb R}}^p$, the Paley-Wiener space $PW_S$ consists of the ($p$-dimensional) inverse Fourier transforms of the $L^2({{\mathbb R}}^p)$-functions which vanish a.e. outside $S$. A set ${\Lambda}\subset{{\mathbb R}}^p$ is uniformly discrete (u.d.), if the infimal distance between its different elements is positive. A u.d. set ${\Lambda}$ possesses a uniform density $D({\Lambda})$ if $$\mbox{Card} ({\Lambda}\cap ([0,r]^p+s))=r^p D({\Lambda})+o(r^p)\mbox{ uniformly on } s \mbox{ as } r\to\infty.$$Here $s=(s_1,\dots,s_p)\in {{\mathbb R}}^p$ and $$[0,r]^p+s=\{x=(x_1,\dots,x_p)\in{{\mathbb R}}^p: s_j\leq x_j\leq s_j+r, j=1,\dots,p\}.$$
We will denote by $|S|$ the $p$-dimensional measure of a set $S\subset{{\mathbb R}}^p$.
Denote by $S_a$, where $a$ is a positive number, the “projection” of a set $S\subset{{\mathbb R}}^p$ onto the cube $[0,a]^p$: $$S_a:=(S+a{{\mathbb Z}}^p)\cap [0,a]^p.$$
We will now formulate a multi-dimensional analogue of Theorem 1: [*Suppose that $s_1,\dots,s_p$ are real numbers linearly independent over the set of integers. Then the set $${\Lambda}:=\{ m_1+s_12^{-|m_1|-...-|m_p|},\dots, m_p+s_p2^{-|m_1|-...-|m_p|}, (m_1,\dots,m_p)\in{{\mathbb Z}}^p\}$$ is a uniqueness set for $PW_S$, for every bounded set $S\subset{{\mathbb R}}^n$ satisfying $|S_1|<1$.*]{}
The proof of this result goes on the same lines as the proof of Theorem 2 in [@u].
Choosing the numbers $s_j$ small, one can make the set ${\Lambda}$ in the above result an arbitrarily small perturbation of the lattice ${{\mathbb Z}}^p$.
Also Theorem 2, as noted in [@ou], admits an extension to several dimensions: [*For every set $S\subset{{\mathbb R}}^p$ of finite measure there is a u.d. set ${\Lambda}\subset{{\mathbb R}}^p, D({\Lambda})= |S|,$ which is a uniqueness set for $PW_S$.* ]{}
In order to get multi-dimensional versions of Theorems 3 and 4, one may introduce multi-dimensional analogues of the spaces $W^{(\alpha)}_S$ and $Y_S$ as follows: The space $W^{(\alpha)}({{\mathbb R}}^p), \alpha>p/2,$ consists of functions $f$ defined on ${{\mathbb R}}^p$ which are the Fourier transform of functions $F$ vanishing outside $S$ and satisfying $$\|F\|^2:=\int_{{{\mathbb R}}^p}(1+|t|^{2\alpha})|F(t)|^2\,dt<\infty,$$where $|t|^2=t_1^2+\dots+t_p^2$ and $dt=dt_1\cdot\dots\cdot dt_p.$
The space $Y_S({{\mathbb R}}^p)$ consists of continuous functions $f$ satisfying $$\sup_{x\in{{\mathbb R}}^p}(1 + |x|^{2p})|f(x)| < \infty, \quad |x|^2:=x_1^2+\dots+x_p^2,$$ and such that the Fourier transform $\hat f$ vanishes outside $S$. One may check that both Lemmas 1 and 2 above admit multi-dimensional extensions. This allows to get multi-dimensional analogues of Theorems 3 and 4: [*Assume $S\subset{{\mathbb R}}^p$ is such that $|S_a|<a^p$, for some $a>0$. Then the spaces $W^{(\alpha)}_S({{\mathbb R}}^p), \alpha>p/2,$ and $Y_S({{\mathbb R}}^p)$ admit a u.d. uniqueness set*]{}.
One may also check that the $p$-dimensional versions of Theorems 5–8 hold true.
Questions
---------
We leave open several problems which might be of certain interest.
1\. In connection with Theorems 1 and 2, one may ask: [*Does there exist a u.d. set ${\Lambda}, D({\Lambda})=1,$ which is a uniqueness set for $PW_S$, for every set $S\subset{{\mathbb R}},|S| < 1$?*]{}
2\. The following question arises in connection with Theorems 3 and 4: [*Let $S\subset{{\mathbb R}}$ be a set with periodic weak gaps. Does the space $PW_S\cap C({{\mathbb R}})$ admit a u.d. uniqueness set?* ]{}
3\. It also seems an interesting question if Theorem 2 remains true for the Fourier transforms of integrable functions: [*Let $S\subset{{\mathbb R}}$ be a set of finite measure. Is it true that the space $$\widehat{L_S} := \{f=\hat F: F\in L^1({{\mathbb R}}), F=0 \mbox{ a.e. outside } S\}$$ admits a u.d. uniqueness set?*]{}
Theorem 5 implies that the answer is “yes” whenever $S_a$ is not dense on $[0,a]$, for some $a.$
The first author thanks the Israel Science Foundation for partial support.
We thank also the CIRM center at Trento University for the kind hospitality during our two weeks RiP stay.
[99]{} Bari, N.K. *Trigonometric Series*, Fizmatgiz, Moscow, 1961 (in Russian). English translation: Bary, N. K. *A Treatise on Trigonometric Series. Vols. I, II*, The Macmillan Co., New York, 1964.
Beurling, A., Malliavin, P. *On the closure of characters and the zeros of entire functions.* Acta Math., **118** (1967), 79–93.
Landau, H. J. *A sparse regular sequence of exponentials closed on large sets.* Bull. Amer. Math. Soc. **70** (1964), 566–569.
Levin, B. Ya. *Lectures on entire functions.* **150** AMS, Providence, RI, 1996.
Olevskii, A., Ulanovskii, A. *Universal sampling and interpolation of band-limited signals.* Geom. Funct. Anal. (GAFA) **18** (2008), 1029–1052.
Olevskii, A., Ulanovskii, A. *Approximation of discrete functions and size of spectrum.* Algebra i Analiz **21** (2009), no. 6, 227–240; reprinted in St. Petersburg Math. J. **21** (2010), no. 6, 1015–1025
Olevskii, A., Ulanovskii, A. *Uniqueness sets for unbounded spectra.* C. R. Acad. Sci. Paris, Sér. I **349** (2011), 679–681.
Olevskii, A., Ulanovskii, A. *Functions with Disconnected Spectrum: Sampling, Interpolation, Translates.* AMS, University Lecture Series, **65**, 2016.
Ulanovskii, A. *On Landau’s phenomenon in ${{\mathbb R}}^n$.* Math. Scand. **88** (2001), no. 1, 72–78.
|
---
abstract: |
The geometry of a floating bridge on a drumhead soundboard produces string stretching that is first order in the amplitude of the bridge motion. This stretching modulates the string tension and consequently modulates string frequencies at acoustic frequencies. Early work in electronic sound synthesis identified such modulation as a source of bell-like and metallic timbre. And increasing string stretching by adjusting banjo string-tailpiece-head geometry is known to enhance characteristic banjo tone. Hence, this mechanism is likely a significant source of the ring, ping, clang, and plunk common to the family of instruments that share floating-bridge/drumhead construction. Incorporating this mechanism into a full, realistic model calculation remains an open challenge.
\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
PACS: 43.75.006: Gh
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[**key words:**]{} banjo, frequency modulation, floating bridge, tailpiece
[**contact info:**]{} [email protected], (626) 395-4252, FAX: (626) 568-8473; 452-48 Caltech, Pasadena CA 91125
[**separate figures:**]{} FIG1-politzer.eps, FIG2-politzer.eps, FIG3-politzer.eps
author:
- David Politzer
title: Banjo timbre from string stretching and frequency modulation
---
[**1. What is a Banjo?**]{}
A banjo is a drum with strings mounted on a neck. With minor caveats, that is what makes it a banjo. So that is what must be responsible for its characteristic sound. Actually, the banjo is the American instrument fitting that description.[@dickey],[@rae],[@moore] Cultures around the world have their own versions. While there is great variation among their voices, they are acoustically identifiable as belonging to the banjo family. Among the many are the akonting and kora of west Africa, the sarod of India and its neighbors, the dramyin of Tibet, the dashpuluur of Tuva, the sanxian of China, and the shamisen of Japan. And banjos in America today come in several readily identifiable and acoustically distinguishable varieties.
A reasonable question is: what is it in the mechanics of sound production by drum and strings that distinguishes the sound of banjos as a class from that of other stringed instruments? While it may not be easy to quantify the defining characteristics of that sound, “Ring the banjo" is a phrase used and commonly understood in America since before the mid-19$^{\text{th}}$ Century, an era when banjos had no metal parts.
[**2. Geometry of Break Angle and String Stretch**]{}
A possible answer lies in the geometry, common to all members of the banjo family, of how the strings are attached, how they go over the bridge, and how the bridge moves.
The ideal, textbook string with fixed ends must stretch as it vibrates. However, the amount of stretch is second order in the amplitude of vibration. The typical textbook analysis ignores this stretching and arrives at a description of normal modes and frequencies that gives a very satisfactory account for most musical situations. Of course, it is possible to pluck a string with such ferocity that the initial sound is, in fact, manifestly distorted by the stretching. Even under normal conditions, second order stretching certainly contributes to the characteristic timbre of plucked strings. Such timbre distinctions are generally very sensitive to non-linearities (e.g., as produced by stretching) and non-harmonic frequency ratios (e.g., as produced by inherent string stiffness). However, second-order stretching and string stiffness are features common to all plucked instruments. So they are not likely candidates for distinguishing the sound of one instrument from another, e.g., banjo from guitar. And this would be true even if the floating bridge effects described here are in some sense smaller than the non-ideal string features common to all plucked instruments.
The floating bridge on a drumhead produces a different behavior with respect to stretch. “Floating" refers to the bridge’s relation to the strings. Specifically, the floating bridge goes up and down relative to the ends of the strings, which are fixed to the rim and the neck. That is to be contrasted, for example, with a bridge and saddle, as on a flat-top guitar, where the bridge end of the string goes up and down with the bridge.
“Break angle" is the angle the strings make going over the bridge. It is determined by the bridge height and tailpiece geometry, as roughly illustrated in FIG. 1. (Here and in what follows, American banjo terminology is used to describe the various parts and motions. However, all instruments in this world-wide family have analogous parts, e.g., some way to do the same job as the tailpiece to anchor the string ends to the edge of the drum.)
String tension is determined by scale length, string gauge, and chosen pitch of the open string. With a non-zero break angle, the string tension produces a downward force on the bridge. When the bridge is at rest, this is canceled by the upward force of the distorted head.
That there must be some string stretch somewhere is suggested by the following very simple, heuristic consideration. In FIG. 2, $L$ is the scale length (bridge to nut), $l$ is the bridge to tailpiece distance, and $\theta_o$ is the equilibrium break angle. The equilibrium length of the string from nut to tailpiece is
$S_o = L + l$ / cos $\theta_o$ .
If the bridge moves up a distance $x$, the total string must stretch a length
$\Delta S = \sqrt{L^2 + x^2} - L + \sqrt{ l^2 + (l \text{ tan} \theta_o + x)^2} - \sqrt{l^2 + (l \text{ tan} \theta_o )^2}$ .
In practice $x$ is much smaller than $l$. For example, $x$ could be 0.1 mm and $l$ could be 4 cm. Using $x \ll l$:
$\Delta S \simeq x \text{ sin }\theta_o $ .
As $\theta_o \to 0$ (and $x \ll l$) there remains a stretch proportional to $x^2$, i.e., yet smaller by a factor of $x/l$.
A more realistic calculation is presented in the Appendix, which takes account of the fact that, in practice, friction prevents the strings from sliding through the bridge notches for the small motions associated with actual playing. The stretch $\Delta L$ of the long string segment ($L$ in FIG. 2 and 3), is still first order in the vertical bridge displacement, with the bridge necessarily rocking back along the string direction in response to the vertical motion of its base. In the limit of large stretching modulus, the stretched equilibrium condition is particularly simple. String stretching on both sides of the bridge produces additional horizontal forces on the top of the bridge that must balance. The balance due just to that stretching yields
$\Delta L \simeq x$ [$\left\{ {\text{ sin }\theta_o \text{ cos }\theta_o \over 1 + \text{cos}^2 \theta_o}\right\}$]{}.
[**3. From Stretch to Frequency Modulation**]{}
Localized stretch and changes in tension propagate along a string at the longitudinal speed of sound in the material. For steel strings, that is roughly 20 times greater than the speed of transverse waves in normally tuned strings. Hence, it is reasonable to approximate the stretch as producing an instantaneous increase in tension. If a given stretch were applied once and for all, there would be a corresponding rise in pitch. If the stretching happened very slowly, one could still think of the stretch as a change in pitch, i.e., an adiabatic change.
Strings of different materials have different stretching moduli. In particular, steel strings are much stiffer (longitudinally) than gut, nylon, or other synthetics. Since it is the drumhead that moves air, the sound volume is a function of the magnitude of the bridge motion. So, for a given sound volume, steel strings experience greater changes in tension than non-metallic strings. In the early $20^{\text{th}}$ Century, most banjo players embraced metal strings — for producing a sound that was more satisfyingly banjo-like (although there have always been individuals who prefer the older and more mellow sound). And this is a potential clue: longitudinal string stiffness is a likely contributor to banjo timbre.
Strings of different gauges mounted on a particular banjo will experience different changes in tension for a given bridge motion. However, the fractional pitch changes will be about the same for all strings of the same material because the tuned pitches are proportional to the square root of the ratio of tension to density.
If tension changes while a string is vibrating, although the tension change is a linear response to the small length change, the string vibration is inherently non-linear. Some care is then required when thinking in terms of Fourier components. In particular, it is the entire bridge motion that modulates a given string’s tension. For a typical pluck, that bridge motion is not sinusoidal or even periodic.
The important picture to take from this discussion is that each string’s tension is modulated by the motion of the bridge, and that motion is roughly periodic with the period of the lowest notes being played but, in fact, mirrors the full sound of the instrument. The thus modulated tensions manifest acoustically because each string’s frequencies, harmonics, and partials are proportional to the square root of its modulated tension.
[**4. The Sound of Frequency Modulation**]{}
Slow frequency modulation gives a familiar form of tremolo. In 1973, Chowning found that, when the frequency of the modulation is increased and itself enters the audio range, the tremolo warble disappears, and it is the timbre of the note that is effected.[@chowning] The originally dull, sinusoidal, signal-generator sound becomes brighter, more metallic, and bell-like when subjected to audio range frequency modulation. The abstract mathematics is the same as for FM radio signals.[@FM] In its simplest form, the modulation induces frequency sidebands along with the original signal, spaced on the order of the modulation frequency. From ref. , “As the index \[i.e., the relative frequency range of the modulation\] sweeps upward, energy is swept gradually outward into higher order side bands; this is the originally exciting, now extremely annoying ‘FM sweep’. The important thing to get from these Bessel functions is that the higher the index, the more dispersed the spectral energy — normally a brighter sound."
One might wonder whether string stretch from bridge motion can actually alter the sound appreciably, thinking that it cannot introduce frequencies that were not already present in its absence. The concern is the following. If a particular string’s motion is [*exactly*]{} periodic and bridge motion is caused [*only*]{} by that string, then the frequencies of all partials are integer multiples of the fundamental frequency, with or without stretching. Of course, on a real instrument, plucked string motion is not exactly periodic. But, more importantly, the timbre is not just the list of frequencies present but also their relative strengths. And the mechanism described can redistribute those strengths — because it is non-linear. If one imagines deconstructing the sound and then synthesizing it with an independently variable frequency modulation, nothing special happens when the modulation passes through an exact integer divisor of the frequency in question.
[**5. Observational Support**]{}
The proposed mechanism is inherently non-linear. So a necessarily but not sufficient corollary is that its effects be amplitude dependent. Indeed, banjos sound more banjo-like played loud than soft, even when the soft is put through a linear amplifier. The clearest difference comes in the early part of the note, i.e., when the amplitude is greatest, both in the discerned sound and in the analyzed waveform and spectrum. A careful study of the early part of each pluck of loud versus soft could confirm a non-linear origin of the banjo timbre. However, that does not distinguish between various possible non-linear mechanisms.
Conversely, it is possible that characteristic banjo timbre results from a particularly strong linear effect that produces dramatic inharmonicity. Strong string-drumhead coupling has been suggested independently by several people. However, a quick comparison of banjo versus acoustic guitar using 0.010$''$ steel strings showed that the standard deviation of the first fifteen harmonic frequencies from pure integer ratios were both about 0.10% of the average values. The banjo was about 0.101%, and the guitar about 0.09%. Perhaps more precision is needed here.
A linear mechanism which is clearly stronger on the banjo than on other plucked string instruments is the sympathetic vibration of one string with another of their unison harmonics. Typically, one of the strongest is the third harmonic of one string with the second of another, tuned a fifth higher in pitch. While there does not yet exist much documented scientific literature on the American banjo, the effect is a standard element in the literature on the shamisen.[@shamisen] This effect certainly contributes to a bright, quick sound for drumhead instruments, where bridge motion enhances that inter-string coupling.
These and other mechanisms deserve further study.
Three kinds of readily available observations seem to support specifically the string stretching and frequency modulation proposal. First, with modern software, you can construct functions of time and then listen to them. In particular, you can listen to the sound of sinusoidal modulations of sinusoidal functions and even add an amplitude envelope typical of plucked string sound.[@risset] Of course, it will not sound like a banjo. A huge number of details are missing. But the extra ring and brightness of tone stand out.
A second demonstration requires the facility to record and speed up the recording. Play a low note on a banjo and push down periodically (perhaps 6 to 12 times per second) on the head near a foot of the bridge. That will produce an audible frequency tremolo. Speed up the recording until the modulation frequency is well above 20 Hz. (36 Hz should do.) The sound will have acquired a definite metallic plink, akin to banging on sheet metal. This is most dramatic if the original note was quite low and the original break angle as small as possible. (Adjustment of tailpiece or choice of tailpiece can accomplish the latter.) This demonstration could also be performed with any low note frequency tremolo on any instrument — except that it would miss the connection to bridge motion.
The third category of support (and most relevant to the specific, proposed mechanism) comes from very well-established, universally agreed upon lore among banjo players. Without any agreement on why or how, experienced banjo players and builders know that break angle is an important issue. Some tailpieces are adjustable over a range with the turn of a screw, while others produce a fixed break angle, whose value depends on the geometry of the tailpiece and the banjo on which it is mounted. The range on current, popular instruments is roughly $6^{\text{o}}$ to $15^{\text{o}}$.
Often, the tailpiece advice comes with the observation that a steeper angle produces greater down-pressure of the strings on the bridge. However, at equilibrium, that force is canceled by an upward force of the head. Furthermore, the string-head system acting on the bridge supplies the same return force as a function of bridge displacement as with a shallower angle — at least over the relevant range of angles and assuming the head force on the bridge is linear with displacement over the range of bridge motion. (Further discussion of head linearity is given below.) It is essential to remember that the strings are retuned to their original tensions after the tailpiece is adjusted. If stretching were ignored, the fluctuating component of the forces on the bridge would be independent of break angle, and the value of the break angle would have no sonic impact.
So, even if the mechanism of tailpiece alteration is not widely understood, the consequence is: increasing the break angle makes the sound more banjo-like. Words that are often used to describe the sound of larger angles are: “sharper," “snappier," or“brighter," while smaller break angles produce “mellow," “warm," or “round tone."[@siminoff] It is not that gut strings with gentle break angles are not banjo-like. It is just that steel strings with sharp break angles are more so.
And the most apparent consequence of break angle on the mechanics of sound production is through the mechanism proposed in this note.
[**6. Contrast with Other Stringed Instruments**]{}
There are other acoustic, stringed instruments with floating bridges, where the bridge moves relative to the fixed ends of the string. These include the violin family, mandolins, and arch-top guitars. However, their bridges, riding on wooden soundboards, do not move nearly as much as the bridge on a banjo for the same sort of pluck. For example, the violin, with a soundboard that has around 94% the area of a typical banjo, produces a far quieter sound when plucked. Also, the quintessential banjo features disappear if the skin on a banjo is replaced with wood. Such instruments exist, made by instrument manufacturers, individual luthiers, and hobbyists. They may be called banjos if they are strung and played like banjos, but their sound is quieter, and sustain is longer. More significantly, it is widely acknowledged that they sound distinctly like dulcimers and not at all like banjos.
[**7. Non-linearities from the Drumhead Itself**]{}
Drumheads were likely initially chosen for soundboards because of the sound volume produced. They are inordinately efficient transducers of a varying, localized force into sound. (Just tap or gently rub a drumhead and listen.) And banjo players generally opt for as low a mass bridge as is structurally sound. The combined effect is that, in comparison to other stringed instruments, the banjo is relatively loud, with a short sustain. This is certainly an essential aspect of its characteristic sound, amplitude envelope being an important part of distinguishing different sounds.[@risset] However, the timbre corresponding to the banjo’s “ring" is something beyond that.
Banjo drumheads also have their own characteristic sound. Some of that comes from their interaction with string tension [*via*]{} the bridge (as discussed above). But there may well be other non-linearities inherent in the use of a drumhead that contribute to that sound, as well. This deserves further study, but some basic issues are clear.
The dynamics of the head are relevant both for how the whole head vibrates in response to driving by the bridge and how the head pushes back on the bridge. The non-trivial stress tensor of a banjo head, even at equilibrium, is apparent to the player, particularly in the vicinity of the bridge. In addition, it is also possible that typical motions of the head in the vicinity of the bridge go beyond the range in which the relation of stress to strain can be linearized. This is an additional possible mechanism for the brightening of sound by the generic drum/string system. But modeling or even just picturing a non-linear stress/strain relation as it impacts bridge motion is far more challenging than the one-dimensional analog presented by the string.
[**8. Further Work**]{}
Motion of the bridge of a stringed instrument has long been a subject of study.[@raman],[@morse] The bridge end of the string must move to transfer energy. However, incorporating the concomitant stretching required by the floating bridge is something that has not as yet been done. This letter simply highlights the issue and identifies an obvious consequence. One way to proceed further would be to incorporate into a model the stretching string, the various forces on the bridge, and the dynamics of the head. The resulting predicted sounds could be compared to real instruments, with particular attention to the sonic consequence of each part varied separately. There are certainly other aspects that contribute to the characteristic sounds of different banjo designs. But of particular interest here is the identification of what they all have in common but is unique to their family.
[**9. Summary**]{}
Any specific instrument in the banjo family has features responsible for its characteristic timbre, and these vary considerably. Also, banjo players are known for their penchant for adjusting and swapping parts in a quest for their own notion of ideal sound. There is no agreed-upon ideal. However, all these instruments have a pluck which identifies them as banjo-like and distinguishes them from anything else. First-order string stretching and the consequent frequency modulation are proposed as a key contributor to that distinctive sound.
[**Acknowledgement**]{}
I would like to thank Frank Rice of Caltech for constructive observations.
[**Appendix: String-Stuck-to-Bridge Geometry**]{}
In practice, the friction from down pressure of the strings on the bridge prevents them from sliding over the bridge as it goes up and down. (Players often notice this sticking when tuning.) Similarly, the base of the bridge is fixed by friction relative to the head. So the frictional forces are forces of constraint, and there is actually a range of angles over which the bridge can be set relative to the head and strings. What happens when the bridge is in motion is a complex, dynamical question that depends on the bridge mass and geometry and on the head elastic moduli, and it couples the motions of all the strings. However, a more realistic estimate of the string stretching than the one presented in Section 2 can be made for a vertical bridge base displacement $x$, assuming that the situation is static. For simplicity, also assume that the elastic modulus of the head is so much higher than that of the string that only bridge base motion perpendicular to the head is allowed. (This is the standard picture of vibrating diaphragms and strings.) If the base of the bridge is raised, the top of the bridge rocks back toward the tailpiece, but the total torque of the string segments about the bridge base must remain zero in equilibrium. A possible geometry is sketched in FIG. 3, where the bridge is initially perpendicular to the long part of the string. Note that for the string to give zero net torque about the bridge base (point $a$) in its initial position, the initial tension in the tailstring must be higher than in the long part by a factor 1/cos$\theta_o$.
Using the parameters defined in FIG. 3, for small vertical bridge base motion $x$, the string stretch $\Delta L$ generically has a term linear in $x$ whose coefficient is a function of the break angle $\theta_o$, the lengths $L$ and $l$, the bridge height $h$, and the string stretching elastic constant $k$. $k$ is proportional to the Young’s modulus of the string and is the proportionality constant in $T_L = k \thinspace \Delta_o L$, where $T_L$ is the initial, tuned tension in the long string segment of stretched length $L$, and $\Delta_o L$ is the amount it had to be stretched to reach that tension. The natural hierarchy of length scales is $\Delta L < x \ll \Delta L_o \ll h < l \ll L$.
To lowest order in $x$, the balance of torques at equilibrium implies
$\Delta L = x$ [${\text{sin} \theta_o \text{cos} \theta_o \over 1 + \text{cos}^2 \theta_o} \left\{ {1 \ - \ {\Delta L_o \over l} \over 1 \ - \ { \text{tan} \theta_o \over 1 + \text{cos}^2 \theta_o} \ {\Delta L_o \over h} (1 \ - \ \text{sin} \theta_o \text{cos} \theta_o \ h/l)} \right\}$]{} .
The leading term for $\Delta L_o \to 0$ (which is equivalent to $k \to \infty$) represents the balance of the additional horizontal forces on the top of the bridge due to the new stretching that accompanies $x$. The terms that are down by $\Delta L_o$ (or $1/k$) arise because of the additional need to balance the torques on the bridge due to the original ($x=0$) tensions when the top point then moves in response to $x$.
To lowest order in $x$, the vertical motion of the top of the bridge is the same as the motion perpendicular to the long string segment, and both are equal to $x$. So this aspect is no different from any stringed instrument.
[99]{}
J. Dickey, [*The structural dynamics of the American five-string banjo*]{}, J. Acoust. Soc. Am., [**114**]{} (5), 2958 (2003).
J. Rae, [*Banjo*]{}, in [*The Science of String Instruments*]{}, T. Rossing, ed., ch. 5 (Springer, New York, 2010). L. Stephey & T. Moore, [*Experimental investigation of an American five-string banjo*]{}, J. Acoust. Soc. Am. [**124**]{} (5), 3276 (2008). J. Chowning, [*The Synthesis of Complex Audio Spectra by Means of Frequency Modulation*]{}, J. Audio Eng. Soc. [**21**]{} (7), 526 (1973). B. Schottstaedt, [*An Introduction to FM*]{}, https://ccrma.stanford.edu/software/snd/snd/fm.html. e.g., reviewed by S. Yoshikawa in [*The Science of String Instruments*]{}, T. Rossing, ed., ch. 11 (Springer, New York, 2010)
J.-C. Risset and M. Matthews, [*Analysis of Musical Instrument Tones*]{}, Physics Today [**22**]{} (2), 23 (1969). R. Siminoff, [*How to Set Up the Best Sounding Banjo*]{}, (Hal Leonard, Milwaukee, 1999). C.V. Raman, [*The small motion at the nodes of a vibrating string*]{}, Nature [**82**]{} 9 (1909).
P.M. Morse, [*Vibration and Sound*]{}, ch. III §13, p.133, (McGraw-Hill, New York, 1936).
[**Figure Captions**]{}
FIG 1: schematic of break angle, tailpiece, string, bridge, and head
FIG 2: break angle $\theta_o$ and bridge motion $x$ determine stretch
FIG 3: bridge base motion $x$, with string stuck to bridge of height $h$
{width="4.4in"}
{width="3.7in"}
{width="4.3in"}
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abstract: 'Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (1) a fluid phase, and (2) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically-consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching implications. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.'
author:
- |
K. Garikipati[^1], E. M. Arruda[^2], K. Grosh[^3], H. Narayanan[^4], S. Calve[^5]\
University of Michigan, Ann Arbor, Michigan 48109, USA
bibliography:
- 'mybib.bib'
title: 'A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics'
---
Background {#sect1}
==========
Development of biological tissue, when described in the biomechanics literature, is generally broken down into the distinct processes of *growth*, *remodelling* and *morphogenesis*. Growth, or conversely, resorption, involves the addition or loss of mass. Remodelling results from a change in microstructure, which could manifest itself as an evolution of macroscopic quantities such as state of internal stress, stiffness or material symmetry. It also appears sometimes as fibrosis or hypertrophy. Morphogenesis involves both growth and remodelling, as well as more complex changes in form. A classical example of morphogenesis is the development of an embryo from a fertilized egg. These terms are based on the definitions developed by [@Taber:95], and will be followed in this work. In the present communication we will focus exclusively on growth, its continuum formulation, and the implications that the process holds for the standard machinery of continuum mechanics. Remodelling is treated in an accompanying paper [@remodelpaper]. The larger and more complex problem of morphogenesis will not be treated in this body of work.
The ideas here are applicable to soft (e.g., tendon, muscle) and hard (e.g., bone) tissue. In this paper, growth of biological tissue will be treated at a macroscopic scale. The continuum formulation (e.g., constitutive laws) at this scale may be motivated by cellular, sub-cellular or molecular processes. However, we will not explicitly model processes at this fine a scale. The formulation can be applied with a specific tissue e.g., muscle, as the body of interest. Our experiments, described separately, are on self-organizing tendon [@Calveetal:2003] and cardiac muscle constructs [@Baaretal:2003], engineered [*in vitro*]{}.
The principal notion to be borne in mind while developing a continuum formulation for growth is that one is presented with a system that is open with respect to mass. Scalar mass sources and sinks, and vectorial mass fluxes must be considered. A mass source was first introduced in the context of biological growth by @CowinHegedus:76. The mass flux is a more recent addition of @EpsteinMaugin:2000, who, however, did not elaborate on the specific nature of the transported species. @KuhlSteinmann:02 also incorporated the mass flux and specified a Fickean diffusive constitutive law for it. In their paper the diffusing species is the material of the tissue itself. The approach to mass transport that is followed in our paper is outlined in the next two paragraphs.
In order to be precise about the physiological relevance of our formulation, we have found it appropriate to adopt a different approach from the papers in the preceding paragraph in regard to mass transport. We do not consider mass transport of the material making up any tissue during growth. Instead, it is the nutrients, enzymes and amino acids necessary for growth of tissue, byproducts of metabolic reactions, and the tissue’s fluid phase [@Swartzetal:99] that undergo diffusion[^6] in our treatment. There do exist certain physiological processes in which cells or the surrounding matrix migrate within a tissue. One such process is observed when leukocytes (white blood cells) such as neutrophils and monocytes are signalled to pass through a capillary wall and are induced, by specific chemical attractors, to migrate to a site of infection. This is the process of chemotaxis [@GuytonHall:1996; @Vander:2003]. The migrant cells or matrix then participate in some form of cell proliferation or death. Fibroblasts also migrate within the extra cellular matrix during wound healing. A third example is the migration of stem cells to different locations during the embryonic development of an animal. These processes involve very *short range* diffusion, and can be treated by the approach described in this paper. We have chosen to focus upon homeostasis, defined by @Vander:2003 as “…a state of reasonably stable balance between the physiological variables…”[^7]. Since, to the best of our knowledge, processes of the type just described are not observed during homeostatic tissue growth, we will ignore transport of the solid phase of the tissue.
The processes of cell proliferation and death, hypertrophy and atrophy, are complex and involve several cascades of biochemical reactions. We will treat them in an elementary fashion, using source/sink terms that govern inter-conversion of species, and the mass fluxes that supply reactants and remove byproducts. The treatment will be mathematical; specific constituents will not be identified with any greater detail than to say that they are either the tissue’s solid phase, the interstitial fluid phase, precursors of the solid phase (these would include amino acids, nutrients and enzymes), or byproducts of reactions. We will return to explicitly incorporate biochemical and cellular processes within our description of mass transport in a subsequent paper.
Virtually all biological tissue consists of a solid and fluid phase and can be treated in the context of mixture theory [@TruesdellToupin:60; @TruesdellNoll:65; @BedfordDrumheller:1983]. When growth is of interest, additional species (reactants and byproducts) must be considered as outlined above. The solid phase is an anisotropic composite that is inhomogeneous at microscopic and macroscopic scales. The fluid, being mostly water, may be modelled as incompressible, or compressible with a very large bulk modulus. This level of complexity will be maintained throughout our treatment. The use of mixture theory leads to difficulties associated with partitioning the boundary traction into portions corresponding to each species. [@RajagopalWineman:1990], suggested a resolution to this problem that holds in the case of saturated media—a condition that is applicable to soft biological tissue. An alternative is to apply the theory of porous media that grew out of the classical work of Fick and Darcy in the 1800s [@Terzaghi:1943; @deBoer:2000]. In this approach fluxes are introduced for each species. Since a species that diffuses must do so within some medium, one may think of the various constituents diffusing through the solid phase[^8]. This strategy has been adopted in the present work.
Recent work
-----------
In a simplification that avoided the complexity of mixture theory or porous media, [@CowinHegedus:76] accounted for the fluid phase via irreversible sources and fluxes of momentum, energy and entropy. This approach was also followed by [@EpsteinMaugin:2000] and [@KuhlSteinmann:02]. While the approach followed in the present paper, i.e. derivation of a mass balance law with mass source and flux, and postulating sources and fluxes for momentum, energy and entropy, has been attempted recently by [@EpsteinMaugin:2000], and [@KuhlSteinmann:02], there are important differences between those works and our paper. Epstein and Maugin conclude that the mass flux vanishes unless the internal energy depends upon strain gradient terms (a second-order theory). This view ignores Fickean diffusion (where the flux is linearly dependent upon concentration of the relevant species). Our treatment also results in the dependence of mass flux upon strain gradient, but without the requirement of a strain gradient dependence of the internal energy. Instead, the dissipation inequality motivates a constitutive relation for the mass flux of each transported species. When properly formulated in a thermodynamic setting, the mass flux can be constrained to depend upon the strain/stress gradient *and* the gradient in concentration (mass per unit system volume) of the corresponding species. The latter term is the Fickean contribution to the mass flux. The form obtained is essentially identical to [@DeGrootMazur:1984].
While modelling hard biological tissue, it is common to assume a Fickean flux, and a mass source that depends upon the strain energy density [@HarriganHamilton:1993], and therefore upon the strain. This introduces coupling between mass transport and mechanics. One possible difficulty with this approach is that one could conceive of a mass source that satisfies other requirements, such as the dissipation inequality, but does not depend upon any mechanical quantities. Strain- or stress-mediated mass transport would then be absent in the boundary value problems solved with such a formulation.
The present paper is aimed at a complete treatment of mass transport, coupled with mechanics, for the growth problem. Initial sections (Sections \[sect2\]–\[sect3bis\]) treat the balance of mass, balance of linear and angular momenta, the forms of the First and Second Laws for this problem, and kinematics of growth, respectively. The Clausius-Duhem inequality and its implications for constitutive relations are the subject of Section \[sect5\]. Examples are provided as appropriate to illustrate the important results. A preliminary numerical example appears in Section \[sect6\]. A discussion and conclusion are provided in Section \[sect7\].
Balance of mass for an open system {#sect2}
==================================
The body of interest, $\sB$, occupies the open region $\Omega_0\subset\mathbb{R}^3$ in the reference configuration. Points in $\sB$ are parameterized by their reference positions, $\bX$. The deformation of $\sB$ is a point-to-point map, $\Bvarphi (\bX,t)\in\mathbb{R}^3$, of $\Omega_0$, carrying the point at $\bX$ to its current position $\bx = \Bvarphi(\bX,t)$, at time $t\in [0,T]$. In its current configuration, $\sB$ occupies the open region $\Omega_t =
\Bvarphi_t(\Omega_0),\; \Omega_t\subset\mathbb{R}^3$ (see Figure \[potato\]). The tangent map of $\Bvarphi$ is the deformation gradient $\bF :=
\partial\Bvarphi/\partial\bX$.
The body consists of several species, of which the solid phase of the tissue is denoted $\mathrm{s}$, and the fluid phase is $\mathrm{f}$. The remaining species, $\alpha,\dots,\omega$, are precursors of the tissue or byproducts of its breakdown in chemical reactions. The index $\iota$ will be used to indicate an arbitrary species. (Where appropriate, we will use the term “system” to refer to the species collectively. Where the presence of several species is an unimportant fact, we will use the term “body”. ) The system is open with respect to mass. Species $\alpha,\dots,\omega$ have sources/sinks, $\Pi^\alpha,\dots,\Pi^\omega$, and mass fluxes, $\bM^\alpha,\dots,\bM^\omega$, respectively. The sources specify mass production rates per unit volume of the body in its reference configuration, $\Omega_0$. The fluxes specify mass flow rates per unit cross-sectional area in $\Omega_0$. Importantly, *these flux vectors are defined relative to the solid phase*. As discussed in Section \[sect1\], the solid tissue phase has only a mass source/sink associated with it, and no flux. The fluid tissue phase has only a mass flux, and no source/sink[^9].
Since the solid phase, $\mathrm{s}$, does not undergo transport, its motion is specified entirely by $\Bvarphi(\bX,t)$. We describe the remaining species $\mathrm{f},\alpha,\dots,\omega$ as convecting with the solid phase and diffusing with respect to it. They therefore have a velocity relative to $\mathrm{s}$. Since the remaining species convect with $\mathrm{s}$, it implies a local homogenization of deformation. The modelling assumption is made that at each point, $\bX$, the individual phases undergo the same deformation.
We define concentrations of the species $\rho_0^\iota=\bar{\rho}_0^\iota f^\iota$ as masses per unit volume in $\Omega_0$. The intrinsic species density is $\bar{\rho}_0^\iota$, and $f^\iota$ is the volume fraction of $\iota$, for $\iota = \mathrm{s,f},\alpha,\dots,\omega$. In an experiment it is far easier to measure the concentration, $\rho_0^\iota$, rather than the intrinsic species density, $\bar{\rho}_0^\iota$[^10]. The concentrations also have the property $\sum\limits_{\iota}\rho_0^\iota = \rho_0$, the total material density of the tissue, with the sum being over all species $\mathrm{s,f},\alpha,\dots,\omega$. The concentrations, $\rho_0^\iota$, change as a result of mass transport and inter-conversion of species, implying that the total density in the reference configuration, $\rho_0$, changes with time. They are parameterized as $\rho_0^\iota(\bX,t)$.
Balance of mass in the reference configuration {#sect2.1}
----------------------------------------------
Recall that $\Omega_0$ is a fixed volume. The statement of balance of mass of the solid phase of the tissue, written in integral form over $\Omega_0$, is
$$\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0}
\rho_0^\mathrm{s} (\bX,t)\mathrm{d}V = \int\limits_{\Omega_0}
\Pi^\mathrm{s} (\bX,t)\mathrm{d}V. \label{massbalintA}$$
Since the solid phase of the tissue does not undergo mass transport, there is no associated flux. Localizing the result gives
$$\frac{\partial\rho_0^\mathrm{s}}{\partial t} = \Pi^\mathrm{s},
\label{massballocA}$$
where the explicit dependence upon position and time has been suppressed.
The fluid phase of the tissue, $\mathrm{f}$, may be thought of as the interstitial or lymphatic fluid that perfuses the tissue. As explained above, we do not consider sources of fluid in the region of interest. The fluid therefore enters and leaves $\Omega_0$ as a flux, $\bM^\mathrm{f}$. The balance of mass in integral form is $$\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0}
\rho_0^\mathrm{f} (\bX,t)\mathrm{d}V =
-\int_{\partial\Omega_0}\bM^\mathrm{f}(\bX,t)\cdot\bN \mathrm{d}A,
\label{massbalintB}$$
where $\bN$ is the unit outward normal to the boundary, $\partial\Omega_0$. Applying the Divergence Theorem to the surface integral and localizing the result gives $$\frac{\partial\rho_0^\mathrm{f}}{\partial t} = -
\Bnabla\cdot\bM^\mathrm{f}, \label{massballocB}$$
where $\Bnabla(\bullet)$ is the gradient operator defined on $\Omega_0$, and $\Bnabla\cdot(\bullet)$ denotes the divergence of a vector or tensor argument on $\Omega_0$.
For the precursor and byproduct species, $\iota=\alpha,\dots,\omega$, the balance of mass in integral form is $$\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0} \rho_0^\iota
(\bX,t)\mathrm{d}V = \int\limits_{\Omega_0} \Pi^\iota
(\bX,t)\mathrm{d}V
-\int_{\partial\Omega_0}\bM^\iota(\bX,t)\cdot\bN \mathrm{d}A.
\label{massbalintI}$$
In local form it is $$\frac{\partial\rho_0^\iota}{\partial t} = \Pi^\iota -
\Bnabla\cdot\bM^\iota,\;\forall\,\iota=\alpha,\dots,\omega.
\label{massballocI}$$
Of course, this last equation is the general form of mass balance for any species $\iota$, recalling that in particular, $\bM^\mathrm{s} = \bzero$ and $\Pi^\mathrm{f} = 0$. This form will be used in the development that follows.
The fluxes, $\bM^\iota,\;\forall\,\iota=\mathrm{f},\alpha,\dots,\omega$, represent mass transport of the fluid, of precursors to the reaction site, and of byproducts from sites of tissue breakdown. The sources, $\Pi^\iota,\;\forall\iota=\mathrm{s},\alpha,\dots,\omega$, arise from inter-conversion of species. The sources/sinks in (\[massballocA\]) and (\[massballocI\]) are therefore related, as tissue and byproducts are formed by consuming precursors (amino acids and nutrients, for instance). To maintain a degree of simplicity in this initial exposition, we will restrict our description of tissue breakdown to the reverse of this reaction.
The sources, $\Pi^\iota$ for various species, satisfy a relation that is arrived as follows: Summing (\[massballocI\]) over all species leads to the law of mass balance for the system, $$\sum\limits_\iota\frac{\partial\rho_0^\iota}{\partial t} =
\sum\limits_\iota\left(\Pi^\iota - \Bnabla\cdot\bM^\iota\right),
\label{massballocItot}$$
where the sum runs over all species, with $\Pi^\mathrm{f} = 0$ and $\bM^\mathrm{s} = \bzero$. Alternatively, in writing the mass balance equation for the system, the interconversion terms (sources/sinks) play no role, and only the fluxes at the boundaries need be accounted for. In integral form we have $$\frac{\mathrm{d}}{\mathrm{d}t}\sum\limits_\iota\int\limits_{\Omega_0}\rho_0^\iota
\mathrm{d}V =
-\sum\limits_\iota\int\limits_{\partial\Omega_0}\bM^\iota\cdot\bN
\mathrm{d}A.$$
Applying the Divergence Theorem and localizing leads to $$\sum\limits_\iota \frac{\partial\rho_0^\iota}{\partial t} =
\sum\limits_\iota(-\Bnabla\bM^\iota). \label{massballoctot}$$
Comparing the equivalent forms (\[massballocItot\]) and (\[massballoctot\]) it emerges that the sources and sinks satisfy $$\sum\limits_\iota\Pi^\iota = 0, \label{sourcebalance}$$
a conclusion that is consistent with classical mixture theory [@TruesdellNoll:65]. The section that follows contains an example in which the Law of Mass Action is invoked to describe a set of inter-related sources, $\Pi^\iota$.
### Sources, sinks and stoichiometry: An example based upon the Law of Mass Action {#sect2.1.1}
The conversion of precursors to tissue and the reverse process of its breakdown are governed by a series of chemical reactions. The stoichiometry of these reactions varies in a limited range. Continuing in the simple vein adopted above, it is assumed that the formation of tissue and byproducts from precursors, and the breakdown of tissue, are governed by the forward and reverse directions of a single reaction: $$\sum\limits_{\iota=\alpha}^{\omega} n_\iota[\iota] \longrightarrow
[\mathrm{s}]. \label{chemreac}$$
Here, $n_\iota$ is the (possibly fractional) number of moles of species $\iota$ in the reaction. For a tissue precursor, $n_\iota > 0$, and for a byproduct, $n_\iota < 0$. By the Law of Mass Action for this reaction, the rate of the forward reaction (number of moles of $\mathrm{s}$ produced per unit time, per unit volume in $\Omega_0$) is $k_\mathrm{f}\prod\limits_{\iota=\alpha}^\omega
[\rho_0^\iota]^{n_\iota}$, where $\prod$ on the right hand-side denotes a product, not to be confused with the source, $\Pi$. The rate of the reverse reaction (number of moles of $\mathrm{s}$ consumed per unit time, per unit volume in $\Omega_0$) is $k_\mathrm{r}[\rho_0^\mathrm{s}]$, where $k_f$ and $k_r$ are the corresponding reaction rates. Assuming, for the purpose of this example, that the solid phase is a single compound, let the molecular weight of $\mathrm{s}$ be $\sM_\mathrm{s}$. From the above arguments the source term for $\mathrm{s}$ is $$\Pi^\mathrm{s} =
\left(k_\mathrm{f}\prod\limits_{\iota=\alpha}^\omega
[\rho_0^\iota]^{n_\iota} -
k_\mathrm{r}[\rho_0^\mathrm{s}]\right)\sM_\mathrm{s},
\label{sourceA}$$
Since the formation of one mole of $\mathrm{s}$ requires consumption of $n_\iota$ moles of $\iota$, we have $$\Pi^\iota =
-\left(k_\mathrm{f}\prod\limits_{\vartheta=\alpha}^\omega
[\rho_0^\vartheta]^{n_\vartheta} -
k_\mathrm{r}[\mathrm{s}]\right)n_\iota\sM_\iota, \label{sourceI}$$
where $\sM_\iota$ is the molecular weight of species $\iota$. Since, due to conservation of mass, $\sM_\mathrm{s} =
\sum\limits_{\iota=\alpha}^\omega n^\iota\sM_\iota$ the sources satisfy $\sum\limits_{\iota=\mathrm{s},\alpha}^\omega \Pi^\iota =
0$.
Balance of mass in the current configuration {#sect2.2}
--------------------------------------------
In the current configuration, $\Omega_t$, the concentration, source and mass flux of species $\iota$ are $\rho^\iota(\bx,t),\,\pi^\iota\mathrm(\bx,t)$, and $\bm^\iota(\bx,t)$ respectively. The boundary is $\partial\Omega_t$, and has outward normal $\bn$ (Figure \[potato\]). Since the deformation, $\Bvarphi(\bX,t)$, is applied to the body (the system), standard arguments yield $\rho^\iota = \rho_0^\iota(\mathrm{det}\bF)^{-1}$, and $\pi^\iota
= \Pi^\iota(\mathrm{det}\bF)^{-1}$. By Nanson’s formula, the normals satisfy $\bn\mathrm{d}a =
\bF^{-\mathrm{T}}\bN\mathrm{d}A$, where $\mathrm{d}A$ and $\mathrm{d}a$ are the area elements on the boundaries $\partial\Omega_0$ and $\partial\Omega_t$ respectively. The Piola transform then gives $\bm^\iota =
(\mathrm{det}\bF)^{-1}\bF\bM^\iota$.
The balance of mass in integral form for the solid tissue phase is $$\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_t}\rho^\mathrm{s}(\bx,t)\mathrm{d}v =
\int\limits_{\Omega_t}\pi^\mathrm{s}(\bx,t)\mathrm{d}v.
\label{massbalcurr1}$$ Applying Reynolds’ Transport Theorem to the left hand-side, localizing the result and employing the product rule gives, $$\frac{\partial\rho^\mathrm{s}}{\partial t} +
\bv\cdot\Bnabla_x\rho^\mathrm{s} + \rho^\mathrm{s}
\Bnabla_x\cdot\bv = \pi^\mathrm{s},\nonumber$$ or, $$\frac{\mathrm{d}\rho^\mathrm{s}}{\mathrm{d}t} = \pi^\mathrm{s} -
\rho^\mathrm{s} \Bnabla_x\cdot\bv, \label{massbalcurr2}$$ where $\bv(\bx,t) = \bV(\bX,t)\circ\Bvarphi^{-1}(\bx,t)$ is the spatial velocity, $\Bnabla_x(\bullet)$ is the spatial gradient operator, and $\Bnabla_x\cdot(\bullet)$ is the spatial divergence. The time derivative on the left hand-side in (\[massbalcurr2\]) is the material time derivative, that may be written explicitly as $(\partial\rho^\mathrm{s}(\bx,t)/\partial
t)_X$, implying that the reference position is held fixed.
For the fluid tissue phase, $\mathrm{f}$, the integral form is, $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_t}\rho^\mathrm{f}(\bx,t)\mathrm{d}v & = & -
\int\limits_{\partial\Omega_t}\bm^\mathrm{f}(\bx,t)\cdot\bn\mathrm{d}a.
\label{massbalcurr1B}\end{aligned}$$
Invoking the Divergence Theorem in addition to the arguments used above for the solid phase now gives $$\frac{\mathrm{d}\rho^\mathrm{f}}{\mathrm{d}t} = -
\Bnabla_x\cdot\bm^\mathrm{f} - \rho^\mathrm{f} \Bnabla_x\cdot\bv.
\label{massbalcurr2B}$$
Finally, for the precursor and byproduct species, $\iota =
\alpha,\dots,\omega$, proceeding as above gives the local form in $\Omega_t$: $$\frac{\mathrm{d}\rho^\iota}{\mathrm{d}t} = \pi^\iota-
\Bnabla_x\cdot\bm^\iota - \rho^\iota
\Bnabla_x\cdot\bv,\;\forall\,\iota = \alpha,\dots,\omega.
\label{massbalcurr2I}$$
Balance of linear and angular momenta {#sect3}
=====================================
Balance of linear momentum {#sect3.1}
--------------------------
We first write the balance of linear momentum in the reference configuration, $\Omega_0$. The body, $\sB$, is subject to surface traction, $\bT$, and body force per unit mass, $\bg$. The natural boundary condition then implies that $\bT =
\sum_{\iota}\bP^\iota\bN$ on $\partial\Omega_0$, where $\bP^\iota$ is the partial first Piola-Kirchhoff stress tensor corresponding to species $\iota$, and the index runs over all species. Thus, $\bP^\iota\bN$ is the corresponding partial traction. The mass fluxes, $\bM^\iota,\;(\iota = \mathrm{f},\alpha,\dots,\omega)$, and mass sources, $\Pi^\iota,\;(\iota =
\mathrm{s},\alpha,\dots,\omega)$ make important contributions to the balance of linear momentum, as shown below.
The body undergoes deformation, $\Bvarphi(\bX,t)$, and has a material velocity field $\bV(\bX,t) =
\partial\Bvarphi(\bX,t)/\partial t$. In discussing momentum and energy, it proves convenient to define a material velocity of species $\iota$ relative to the solid phase as $\bV^\iota = (1/\rho_0^\iota)\bF\bM^\iota$. Recall (from Section \[sect2\]) that the remaining species are described as deforming with the solid phase and diffusing relative to it. Therefore $\bF$ is common to all species. The spatial velocity corresponding to $\bV^\iota$ is $\bv^\iota =
(1/\rho^\iota)\bm^\iota = \bV^\iota$, by the Piola transform. Since fluxes are defined relative to the solid tissue phase, which does not diffuse, the total material velocity of the solid phase is $\bV$, and for each of the remaining species it is $\bV +
\bV^\iota$, $\iota = \mathrm{f},\alpha,\dots,\omega$. Formally, we can write the material velocity as $\bV + \bV^\iota$, $\iota =
\mathrm{s},\mathrm{f},\alpha,\dots,\omega$ with the understanding that $\bV^\mathrm{s} = \bzero$. Likewise, $\Pi^\mathrm{f} = 0$. This convention has been adopted in the remainder of the paper.
The balance of linear momentum of species $\iota$ written in integral form over $\Omega_0$ is, $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0}
\rho_0^\iota(\bV+\bV^\iota) \mathrm{d}V &=& \int\limits_{\Omega_0}
\rho^\iota_0\bg \mathrm{d}V + \int\limits_{\Omega_0}
\rho^\iota_0\bq^\iota \mathrm{d}V + \int\limits_{\Omega_0}
\Pi^\iota(\bV+\bV^\iota) \mathrm{d}V \nonumber\\
& &+ \int\limits_{\partial \Omega_0}\bP^\iota\bN \mathrm{d}A -
\int\limits_{\partial \Omega_0}(\bV+\bV^\iota)\bM^\iota\cdot\bN
\mathrm{d}A, \label{linmombalI1}\end{aligned}$$ where $\bg$ is the body force per unit mass, and $\bq^\iota$ is the force per unit mass exerted upon $\iota$ by the other species present. Attention is drawn to the fact that the mass source distributed through the volume, and the influx over the boundary affect the rate of change of momentum in (\[linmombalI1\]). Summing over all species, the balance of linear momentum for the system is obtained: $$\begin{aligned}
\sum\limits_{\iota}\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_0} \rho_0^\iota(\bV+\bV^\iota) \mathrm{d}V &=&
\sum\limits_{\iota}\int\limits_{\Omega_0} \rho^\iota_0\bg
\mathrm{d}V + \sum\limits_{\iota}\int\limits_{\Omega_0}
\rho^\iota_0\bq^\iota \mathrm{d}V\nonumber\\& & +
\sum\limits_{\iota}\int\limits_{\Omega_0} \Pi^\iota(\bV+\bV^\iota)
\mathrm{d}V + \sum\limits_{\iota} \int\limits_{\partial
\Omega_0}\bP^\iota\bN \mathrm{d}A\nonumber\\
& & - \sum\limits_{\iota}\int\limits_{\partial
\Omega_0}(\bV+\bV^\iota)\bM^\iota\cdot\bN \mathrm{d}A,
\label{linmombalall}\end{aligned}$$
The interaction forces, $\rho_0^\iota\bq^\iota$, satisfy a relation with the mass sources, $\Pi^\iota$, that is elucidated by the following argument: The rate of change of momentum of the entire system is affected by external agents only, and is independent of internal interactions of any nature ($\bq^\iota$ and $\Pi^\iota$). This observation leads to the following equivalent expression for the rate of change of linear momentum of the system:
$$\begin{aligned}
\sum\limits_{\iota}\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_0} \rho_0^\iota(\bV+\bV^\iota) \mathrm{d}V &=&
\int\limits_{\Omega_0}\rho_0\bg \mathrm{d}V +
\int\limits_{\partial \Omega_0}\bP\bN \mathrm{d}A \nonumber\\
& &- \sum\limits_{\iota}\int\limits_{\partial
\Omega_0}(\bV+\bV^\iota)\bM^\iota\cdot\bN \mathrm{d}A.
\label{linmombalsys}\end{aligned}$$
Here, $\bP = \sum\limits_{\iota}\bP^\iota$ and $\rho_0=\sum\limits_{\iota}\rho_0^\iota$. Since both (\[linmombalall\]) and (\[linmombalsys\]) represent the balance of linear momentum of the system, it follows that,
$$\begin{aligned}
\sum\limits_{\iota}\int\limits_{\Omega_0} \rho^\iota_0\bq^\iota
\mathrm{d}V + \sum\limits_{\iota}\int\limits_{\Omega_0}
\Pi^\iota(\bV+\bV^\iota) \mathrm{d}V = 0\end{aligned}$$
Recalling the relation between the sources (\[sourcebalance\]), and localizing leads to
$$\begin{aligned}
\sum\limits_{\iota}\left(\rho^\iota_0\bq^\iota +
\Pi^\iota\bV^\iota\right)=0, \label{interforcebalance}\end{aligned}$$
a result that is also consistent with classical mixture theory [@TruesdellNoll:65].
Having established (\[interforcebalance\]) we return to the balance of linear momentum for a single species (\[linmombalI1\]) in order to simplify it. Writing $(\bV+\bV^\iota)[\bM^\iota\cdot\bN]$ as $((\bV+\bV^\iota)\otimes\bM^\iota)\bN$, and using the Divergence Theorem, $$\begin{aligned}
\int\limits_{\Omega_0} \left(\frac{\partial\rho_0^\iota}{\partial
t}\left(\bV+\bV^\iota\right) +
\rho_0^\iota\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right)\right) \mathrm{d}V =
\int\limits_{\Omega_0}\rho^\iota_0\left(\bg+\bq^\iota\right)\mathrm{d}V\qquad& &\nonumber\\
+\int\limits_{\Omega_0}\left(\Pi^\iota\left(\bV+\bV^\iota\right) +
\Bnabla\cdot\bP^\iota\right)
\mathrm{d}V& & \nonumber\\
-
\int\limits_{\Omega_0}\Bnabla\cdot\left(\left(\bV+\bV^\iota\right)\otimes\bM^\iota\right)\mathrm{d}V
& &\end{aligned}$$
Using the mass balance equation (\[massballocI\]), and applying the product rule to the last term gives $$\begin{aligned}
\int\limits_{\Omega_0} \rho_0^\iota\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right) \mathrm{d}V &=&
\int\limits_{\Omega_0}\rho^\iota_0\left(\bg+\bq^\iota\right)\mathrm{d}V\nonumber\\
& &+
\int\limits_{\Omega_0}\left(\Bnabla\cdot\bP^\iota-\left(\Bnabla\left(\bV+\bV^\iota\right)\right)\bM^\iota\right)\mathrm{d}V\end{aligned}$$
Localizing this result gives the balance of linear momentum for a single species in the reference configuration:
$$\rho_0^\iota\frac{\partial}{\partial t}\left(\bV+\bV^\iota\right)
= \rho^\iota_0\left(\bg+\bq^\iota\right) +
\Bnabla\cdot\bP^\iota-\left(\Bnabla\left(\bV+\bV^\iota\right)\right)\bM^\iota
\label{ballinmomrefI}$$
The balance of linear momentum for a single species in the current configuration, $\Omega_t$, is obtained via similar arguments and the Reynolds Transport Theorem: $$\begin{aligned}
\rho^\iota\frac{\partial}{\partial t}\left(\bv+\bv^\iota\right)
&=& \rho^\iota\left(\bg+\bq^\iota\right) +
\Bnabla_x\cdot\Bsigma^\iota\nonumber\\
& & - \left(\Bnabla_x\left(\bv+\bv^\iota\right)\right)\bm^\iota -
\rho^\iota\left(\Bnabla_x\left(\bv+\bv^\iota\right)\right)\bv,
\label{ballinmomcurrI}\end{aligned}$$
where $\Bsigma^\iota =
(\mathrm{det}\bF)^{-1}\bP^\iota\bF^\mathrm{T}$ is the partial Cauchy stress of species $\iota$.
Angular Momentum {#sect3.2}
----------------
For the purely mechanical theory, balance of angular momentum implies that the Cauchy stress is symmetric: $\Bsigma^\iota = \Bsigma^{\iota^\mathrm{T}}$. We now re-examine this result in the presence of mass transport. At the outset one might expect a statement on the symmetry of some stress or stress-like quantity. We derive this result for any species, $\iota$, beginning with the integral form of balance of angular momentum written over $\Omega_0$.
$$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0} \Bvarphi
\times \rho^\iota_0(\bV+\bV^\iota)\mathrm{d}V &=&
\int\limits_{\Omega_0}\Bvarphi\times\left[\rho^\iota_0\left(\bg+\bq^\iota\right)+\Pi^\iota\left(\bV+\bV^\iota\right)\right]\mathrm{d}V\nonumber\\
& &+
\int\limits_{\partial\Omega_0}\Bvarphi\times\left(\bP^\iota-\left(\bV+\bV^\iota\right)\otimes\bM^\iota\right)\bN
\mathrm{d}A\end{aligned}$$
Applying properties of the cross product, the Divergence Theorem and product rule gives $$\begin{aligned}
\int\limits_{\Omega_0}\bV\times\rho^\iota_0\bV^\iota +
\Bvarphi\times \left(\frac{\partial\rho^\iota_0}{\partial
t}\left(\bV+\bV^\iota\right)+ \rho^\iota_0\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right)\right)\mathrm{d}V =\qquad\quad& &\nonumber\\
\int\limits_{\Omega_0}\Bvarphi\times\rho^\iota_0\left(\bg+\bq^\iota+\Pi^\iota\left(\bV+\bV^\iota\right)\right)\mathrm{d}V& &\nonumber\\
+
\int\limits_{\Omega_0}\left(\Bvarphi\times\Bnabla\cdot\bP^\iota-\Bvarphi\times\left(\Bnabla\left(\bV+\bV^\iota\right)\bM^\iota\right)\right)\mathrm{d}V
& &\nonumber\\
\int\limits_{\Omega_0}\left(-\Bvarphi\times\left(\bV+\bV^\iota\right)\Bnabla\cdot\left(\bM^\iota\right)\right)
\mathrm{d}V & &\nonumber\\
-
\int\limits_{\Omega_0}\Bepsilon\colon\left(\left(\bP^\iota-\left(\bV+\bV^\iota\right)\otimes\bM^\iota\right)\bF^\mathrm{T}\right)\mathrm{d}V,\end{aligned}$$
where $\Bepsilon$ is the permutation symbol, and $\Bepsilon\colon\bA$ is written as $\epsilon_{ijk}A_{jk}$ in indicial form, for any second-order tensor $\bA$. Using the mass balance equation (\[massballocI\]), and balance of linear momentum (\[ballinmomrefI\]), we have $$\int\limits_{\Omega_0}\bV\times\rho^\iota_0\bV^\iota\mathrm{d}V =
-\int\limits_{\Omega_0}\Bepsilon\colon\left(\left(\bP^\iota-\left(\bV+\bV^\iota\right)\otimes\underbrace{\bM^\iota}_{\rho_0^\iota\bF^{-1}\bV^\iota}\right)\bF^\mathrm{T}\right)\mathrm{d}V.$$
Recalling the relation of the permutation symbol to the cross product, and the indicated relation between $\bM^\iota$ and $\bV^\iota$ leads to $$\bzero =
-\int\limits_{\Omega_0}\Bepsilon\colon\left(\left(\bP^\iota-\bV^\iota\otimes\rho_0^\iota\bF^{-1}\bV^\iota\right)\bF^\mathrm{T}\right)\mathrm{d}V.$$
Localizing this result and again applying the properties of the permutation symbol we are led to the symmetry condition, $$\left(\bP^\iota-\bV^\iota\otimes\rho^\iota_0\bF^{-1}\bV^\iota\right)\bF^\mathrm{T}
=
\bF\left(\bP^\iota-\bV^\iota\otimes\rho^\iota_0\bF^{-1}\bV^\iota\right)^\mathrm{T}.$$
But, $(\bV^\iota\otimes\bF^{-1}\bV^\iota)\bF^\mathrm{T}
= \bV^\iota\otimes\bV^\iota$. Thus, the symmetry $\bP^\iota\bF^\mathrm{T} = \bF(\bP^\iota)^\mathrm{T}$ that results from conservation of angular momentum for the purely mechanical theory, is retained in this case. The partial Cauchy stresses are therefore symmetric: $\Bsigma^\iota = \Bsigma^{\iota^\mathrm{T}}$. This is in contrast with the non-symmetric Cauchy stress arrived at by @EpsteinMaugin:2000. The origin of this difference lies in the fact that these authors use a single species with $\bV
=
\partial\Bvarphi/\partial t$ as the material velocity, rather than multiple species with material velocities $\bV + \bV^\iota$.
Balance of energy and the entropy inequality {#sect4}
============================================
Balance of energy {#sect4.1}
-----------------
Since mass is undergoing transport with respect to $\sB$, and inter-conversion between species $\iota =
\mathrm{s},\alpha,\dots,\omega$, it is appropriate to work with energy and energy-like quantities per unit mass. In addition to the terms introduced in previous sections, the internal energy per unit mass of species $\iota$ is denoted $e^\iota$; the heat supply to species $\iota$ per unit mass of that species is $r^\iota$; and the partial heat flux vector of $\iota$ is $\bQ^\iota$, defined on $\Omega_0$. An interaction energy appears between species: The energy transferred to $\iota$ by all other species is $\tilde{e}^\iota$, per unit mass of $\iota$. In the arguments to follow in this section we will use the fluxes $\bM^\iota$ *and* the associated velocities, $\bV^\iota$. Working in $\Omega_0$, we relate the rate of change of internal and kinetic energies of species $\iota$ to the work done on $\iota$ by mechanical loads, processes of mass production and transport, heating and energy transfer:
$$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{\Omega_0}\rho_0^\iota
\left ( e^\iota + \frac{1}{2} \Vert\bV+\bV^\iota\Vert^2 \right )
\mathrm{d}V = \int\limits_{\Omega_0} \left(\rho_0^\iota\bg
\cdot\left(\bV+\bV^\iota\right) + \rho_0^\iota r^\iota
\right)\mathrm{d}V\qquad& &\nonumber\\
+\int\limits_{\Omega_0}\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)\mathrm{d}V&
&\nonumber\\
+ \int\limits_{\Omega_0} \left(\Pi^\iota\left(e^\iota
+ \frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)+\rho^\iota_0\tilde{e}^\iota\right)\mathrm{d}V & & \nonumber \\
+\int\limits_{\partial
\Omega_0}\left(\left(\bV+\bV^\iota\right)\cdot\bP^\iota -
\bM^\iota\left(e^\iota +\frac{1}{2}
\Vert\bV+\bV^\iota\Vert^2\right) -
\bQ^\iota\right)\cdot\bN\mathrm{d}A. \label{energyspecies1}\end{aligned}$$
Summing over all species, the rate of change of energy of the system is,
$$\begin{aligned}
\sum\limits_{\iota}\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_0}\rho_0^\iota \left ( e^\iota + \frac{1}{2}
\Vert\bV+\bV^\iota\Vert^2 \right ) \mathrm{d}V =
\sum\limits_{\iota}\int\limits_{\Omega_0} \left(\rho_0^\iota\bg
\cdot\left(\bV+\bV^\iota\right) + \rho_0^\iota r^\iota \right)\mathrm{d}V& &\nonumber\\
+\sum\limits_{\iota}\int\limits_{\Omega_0}\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)\mathrm{d}V&
&\nonumber\\
+ \sum\limits_{\iota}\int\limits_{\Omega_0}
\left(\Pi^\iota\left(e^\iota
+ \frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)+\rho^\iota_0\tilde{e}^\iota\right)\mathrm{d}V & & \nonumber \\
+\sum\limits_{\iota}\int\limits_{\partial
\Omega_0}\left(\left(\bV+\bV^\iota\right)\cdot\bP^\iota -
\bM^\iota\left(e^\iota +\frac{1}{2}
\Vert\bV+\bV^\iota\Vert^2\right) -
\bQ^\iota\right)\cdot\bN\mathrm{d}A.\quad& & \label{energysum}\end{aligned}$$
The inter-species energy transfers are related to interaction forces and mass sources. To demonstrate this, we proceed as follows: The rate of change of energy of the system can also be expressed by considering the system interacting with its environment, in which case the internal interactions between species (interaction forces, mass interconversion and inter-species energy transfers) play no role. This viewpoint gives,
$$\begin{aligned}
\sum\limits_{\iota}\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_0}\rho_0^\iota \left ( e^\iota + \frac{1}{2}
\Vert\bV+\bV^\iota\Vert^2 \right ) \mathrm{d}V =
\sum\limits_{\iota}\int\limits_{\Omega_0} \left(\rho_0^\iota\bg
\cdot\left(\bV+\bV^\iota\right) + \rho_0^\iota r^\iota
\right)\mathrm{d}V &
&\nonumber\\
+\sum\limits_{\iota}\int\limits_{\partial
\Omega_0}\left(\left(\bV+\bV^\iota\right)\cdot\bP^\iota -
\bM^\iota\left(e^\iota +\frac{1}{2}
\Vert\bV+\bV^\iota\Vert^2\right) -
\bQ^\iota\right)\cdot\bN\mathrm{d}A.\quad& & \label{energysys}\end{aligned}$$
Since (\[energysum\]) and (\[energysys\]) are equivalent, it follows that,
$$\begin{aligned}
\sum\limits_{\iota}\left(\int\limits_{\Omega_0}\left(\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)
+ \Pi^\iota\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)+\rho^\iota_0\tilde{e}^\iota\right)\mathrm{d}V
\right)= 0,\end{aligned}$$
and on localizing this result,
$$\begin{aligned}
\sum\limits_{\iota}\left(\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)
+ \Pi^\iota\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)+\rho^\iota_0\tilde{e}^\iota\right)=
0. \label{energycond1}\end{aligned}$$
This result relating the interaction energies to interaction forces between species, their sources and relative velocities, is identical to that obtained from classical mixture theory [@TruesdellNoll:65]. Together with (\[sourcebalance\]) and (\[interforcebalance\]) it demonstrates that the present formulation is consistent with mixture theory.
Equation (\[energyspecies1\]) for the rate of change of energy of a single species can be further simplified by applying the Divergence Theorem and product rule, giving first,
$$\begin{aligned}
\int\limits_{\Omega_0}\left(\frac{\partial\rho_0^\iota}{\partial
t} \left(e^\iota + \frac{1}{2}\Vert\bV+\bV^\iota\Vert^2 \right) +
\rho^\iota_0\frac{\partial}{\partial t}\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2 \right)\right) \mathrm{d}V
=\qquad
& &\nonumber\\
\int\limits_{\Omega_0} \left(\rho_0^\iota\bg
\cdot\left(\bV+\bV^\iota\right) + \rho_0^\iota r^\iota +
\Pi^\iota\left(e^\iota
+ \frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)+\rho^\iota_0\tilde{e}^\iota\right)\mathrm{d}V& & \nonumber \\
+\int\limits_{\Omega_0}\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)
\mathrm{d}V&
&\nonumber\\
+\int\limits_{\Omega_0}\left(\left(\bV+\bV^\iota\right)\cdot\Bnabla\cdot\bP^\iota
+
\bP^\iota\colon\Bnabla\left(\bV+\bV^\iota\right)\right)\mathrm{d}V& &\nonumber\\
-
\int\limits_{\Omega_0}\left(\Bnabla\cdot\left(\bM^\iota\right)\left(e^\iota
+\frac{1}{2} \Vert\bV+\bV^\iota\Vert^2\right)\right)\mathrm{d}V & &\nonumber\\
-\int\limits_{\Omega_0}\left( \left(\Bnabla e^\iota+
\left(\bV+\bV^\iota\right)\cdot\Bnabla\left(\bV+\bV^\iota\right)\right)\cdot\left(\bM^\iota\right)
- \Bnabla\cdot\bQ^\iota\right)\mathrm{d}V.\end{aligned}$$
Using the balance of mass (\[massballocI\]), balance of linear momentum (\[ballinmomrefI\]), and localizing the result, we have,
$$\begin{aligned}
\rho^\iota_0\frac{\partial e^\iota}{\partial t} &=&
\bP^\iota\colon\Bnabla\left(\bV+\bV^\iota\right)-
\Bnabla\cdot\bQ^\iota + \rho_0^\iota r^\iota
+\rho^\iota_0\tilde{e}^\iota - \Bnabla e^\iota\cdot\bM^\iota
\label{energybalI}\end{aligned}$$
Summing over $\iota$ gives,
$$\begin{aligned}
& &\sum\limits_{\iota}\rho^\iota_0\frac{\partial e^\iota}{\partial
t} =\nonumber\\
& &\qquad\sum\limits_{\iota}\left(
\bP^\iota\colon\dot{\bF}+\bP^\iota\colon\Bnabla\bV^\iota -
\Bnabla\cdot\bQ^\iota +\rho_0^\iota r^\iota +
\rho^\iota_0\tilde{e}^\iota - \Bnabla e^\iota\cdot\bM^\iota\right)
\label{energybaltot}\end{aligned}$$
Substituting for $\sum\limits_\iota
\rho^\iota_0\tilde{e}^\iota$ from (\[energycond1\]),
$$\begin{aligned}
\sum\limits_{\iota}\rho^\iota_0\frac{\partial e^\iota}{\partial t}
&=& \sum\limits_{\iota}\left(
\bP^\iota\colon\dot{\bF}+\bP^\iota\colon\Bnabla\bV^\iota -
\Bnabla\cdot\bQ^\iota +\rho_0^\iota r^\iota- \Bnabla
e^\iota\cdot\bM^\iota\right)\nonumber\\
&&-
\sum\limits_{\iota}\left(\rho_0^\iota\bq^\iota\cdot(\bV+\bV^\iota)
- \Pi^\iota\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)\right).
\label{energybaltot1}\end{aligned}$$
This form of the balance of energy is most convenient for combining with the entropy inequality leading to the Clausius-Duhem form of the dissipation inequality.
The entropy inequality: Clausius-Duhem form {#sect4.2}
-------------------------------------------
Let $\eta^\iota$ be the entropy per unit mass of species $\iota$, and $\theta$ the absolute temperature. The entropy production inequality holds for the system as a whole. Accordingly, we write
$$\begin{aligned}
\sum\limits_{\iota}\frac{\mathrm{d}}{\mathrm{d}t}
\int\limits_{\Omega_0} \rho_0^\iota \eta^\iota \mathrm{d}V &\geq&
\sum\limits_{\iota}\int\limits_{\Omega_0}\left(
\Pi^\iota \eta^\iota + \frac{\rho_0^\iota r^\iota}{\theta}\right) \mathrm{d}V\nonumber\\
& & - \sum\limits_{\iota}\int\limits_{\partial \Omega_0}
\left(\bM^\iota \cdot \bN\eta^\iota + \frac{\bQ^\iota}{\theta}
\cdot \bN \right)\mathrm{d}A.\end{aligned}$$
Applying the Divergence Theorem, using the mass balance equation (\[massballocI\]), and localizing the result, we have the entropy inequality,
$$\sum\limits_{\iota}\rho_0^\iota\frac{\partial\eta^\iota}{\partial
t} \geq \sum\limits_{\iota}\left(\frac{\rho_0^\iota
r^\iota}{\theta} -\Bnabla\eta^\iota\cdot\bM^\iota -
\frac{\Bnabla\cdot\bQ^\iota}{\theta} +
\frac{\Bnabla\theta\cdot\bQ^\iota}{\theta^2}\right).
\label{entropyineq}$$
Now, multiplying Equation (\[entropyineq\]) by $\theta$, subtracting it from Equation (\[energybaltot1\]) and using (\[ballinmomrefI\]) for $\rho_0^\iota\bq^\iota$ gives,
$$\begin{aligned}
& &\sum\limits_{\iota}\rho^\iota_0\left(\frac{\partial
e^\iota}{\partial t} -\theta\frac{\partial\eta^\iota}{\partial
t}\right) +\sum\limits_{\iota}\Pi^\iota\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right) +
\frac{\Bnabla\theta\cdot\bQ^\iota}{\theta}
\nonumber\\
& &+\sum\limits_{\iota} \left(\rho^\iota_0\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right) - \rho_0^\iota\bg -
\Bnabla\cdot\bP^\iota +
\Bnabla\left(\bV+\bV^\iota\right)\bM^\iota\right)\cdot\left(\bV+\bV^\iota\right)
\nonumber\\
& &-\sum\limits_{\iota}\left(\bP^\iota\colon\dot{\bF} -
\bP^\iota\colon\Bnabla\bV^\iota + \left(\Bnabla e^\iota -
\theta\Bnabla\eta^\iota\right) \cdot\bM^\iota\right)\leq 0
\label{redentropyineqfin}\end{aligned}$$
Equation (\[redentropyineqfin\]) is the reduced entropy inequality—also referred to as the Clausius-Duhem inequality—for growth processes.
The kinematics of growth {#sect3bis}
========================
The formulation up to this point has introduced some elements of coupling between mass transport, mechanics and thermodynamics. Mass transport and mechanics are further coupled due to the kinematics of growth. Local volumetric changes take place as species concentrations evolve. As concentration increases, the material of a species swells, and conversely, shrinks as concentration decreases. This observation has led to an active field of study within the literature on biological growth [@Skalak:81; @SkalakHoger:96; @TaberHumphrey:2001; @LubardaHoger:02; @AmbrosiMollica:2002]. Our treatment follows in the same vein.
The elasto-growth decomposition {#sect3bis.1}
-------------------------------
Finite strain kinematics treats the total deformation gradient as arising from a geometrically-necessary elastic deformation accompanying growth, as well as a separate elastic deformation due to an external stress. The deformation gradient is subject to a split reminiscent of the classical decomposition of multiplicative plasticity [@Bilbyetal:1957; @Lee:1969]
At a continuum point the reference concentration of each species admits the notion of an “original” state in which the concentration of a species is $\rho_\mathrm{org}^\iota(\bX)$. This is a state that may never be attained in a physical system. However, if attained, the corresponding species would be stress-free in the absence of deformation. Neglecting other possible kinematics (such as plasticity) and microstructural details, the set of quantities $\{\rho_0^\mathrm{s},\dots,\rho_0^\omega\}$, and the temperature, $\theta$, fully specify the reference state of the material at a point. As mass transport alters the reference density to its value $\rho_0^\iota(\bX,t)$, the species swells if $\rho_0^\iota >
\rho_\mathrm{org}^\iota$, and shrinks if $\rho_0^\iota <
\rho_\mathrm{org}^\iota$. Assuming that these volume changes are isotropic leads to the following growth kinematics: For each species, one can define a “growth deformation gradient tensor”, $\bF^{\mathrm{g}^\iota} :=
\frac{\rho_0^\iota}{\rho_\mathrm{org}^\iota}{\bf 1}$, where ${\bf
1}$ is the second-order isotropic tensor. The tensor $\bF^{\mathrm{g}^\iota}$ is analogous to the plastic deformation gradient of multiplicative plasticity. As the ratio $\rho_0^\iota/\rho_\mathrm{org}^\iota$ is a local quantity, $\bF^{\mathrm{g}^\iota}$ varies pointwise and adjacent neighborhoods will, in general, be incompatible due to the action of $\bF^{\mathrm{g}^\iota}$ alone. However, further elastic deformation, $\tilde{\bF}^{\mathrm{e}^\iota}$ occurs to ensure compatibility, leading to an internal stress, that can in general be different for each species. The action of these kinematic tangent maps can be conceived of in the absence of external stress. With an external stress, there is further elastic deformation, $\bar{\bF}^\mathrm{e}$, common to all species. This sequence of maps is pictured in Figure \[growthkinematicsfig\]. The kinematic relations are: $$\bF =
\bar{\bF}^\mathrm{e}\tilde{\bF}^{\mathrm{e}^\iota}\bF^{\mathrm{g}^\iota},\quad
\bF^{\mathrm{g}^\iota} =
\frac{\rho_0^\iota}{\rho_\mathrm{org}^\iota}{\bf 1}.
\label{growthkinematicseq}$$
Clearly, the elastic deformation gradients can be combined to write $\bF^{\mathrm{e}^\iota} =
\bar{\bF}^\mathrm{e}\tilde{\bF}^{\mathrm{e}^\iota}$, the “total” elastic deformation gradient of species $\iota$.
Restrictions on constitutive relations from the Clausius-Duhem inequality {#sect5}
=========================================================================
As is the practice in field theories of continuum physics, we use the Clausius-Duhem inequality (\[redentropyineqfin\]) to obtain restrictions on the constitutive relations. We begin with very general assumptions on the specific internal energy of each species: $ e^\iota = \hat{e}^\iota(\bF^{\mathrm{e}^\iota},
\eta^\iota, \rho_0^\iota)$. Then, applying the chain rule and regrouping some terms, (\[redentropyineqfin\]) becomes $$\begin{aligned}
& &\sum\limits_{\iota}\left(\frac{\partial
e^\iota}{\partial\bF^{\mathrm{e}^\iota}}-\bP^\iota\bF^{\mathrm{g}^{\iota\mathrm{T}}}\right)\colon\dot{\bF}^{\mathrm{e}^\iota}
+ \sum\limits_{\iota}\left(\frac{\partial e^\iota}{\partial
\eta^\iota}-\theta\right)\frac{\partial\eta^\iota}{\partial t}\nonumber\\
&+&\sum\limits_{\iota} \left(\rho^\iota_0\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right) - \rho_0^\iota\bg -
\Bnabla\cdot\bP^\iota +
\Bnabla\left(\bV+\bV^\iota\right)\bM^\iota\right)\cdot(\bV^\iota+\bV)\nonumber\\
&+&\sum\limits_{\iota}\left(\rho^\iota_0\bF^{-\mathrm{T}}\left(\Bnabla
e^\iota -
\theta\Bnabla\eta^\iota\right)\right)\cdot\bV^\iota\nonumber\\
&+&\sum\limits_{\iota}\Pi^\iota\left(e^\iota +
\frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right) +
\frac{\Bnabla\theta\cdot\bQ^\iota}{\theta}\nonumber\\
&+& \sum\limits_{\iota}\rho^\iota_0\frac{\partial
e^\iota}{\partial\rho^\iota_0}\frac{\partial
\rho^\iota_0}{\partial t} -
\sum\limits_{\iota}\bP^\iota\colon(\Bnabla\bV^\iota +
\bF^{\mathrm{e}^\iota}\dot{\bF}^{\mathrm{g}^\iota}) \leq 0.
\label{redentropyineq1}\end{aligned}$$
Inequality (\[redentropyineq1\]) represents a fundamental restriction upon the physical processes during biological growth. Any constitutive relations that are prescribed must satisfy this restriction, as is well-known [@TruesdellNoll:65]. Guided by (\[redentropyineq1\]), we prescribe the following constitutive relations in classical form: $$\begin{aligned}
&&\bP^\iota\bF^{\mathrm{g}^\iota\mathrm{T}} = \rho_0^\iota
\frac{\partial e^\iota}{\partial\bF^{\mathrm{e}^\iota}}
\label{stress-constrelI}\\
\nonumber\\
&&\theta = \frac{\partial e^\iota}{\partial \eta^\iota},\;\forall\,\iota\label{temp-constrelI}\\
\nonumber\\
&&\rho^\iota_0\bV^\iota =\nonumber\\
& &
-\frac{\tilde{\bD}^\iota}{\rho^\iota_0}\left(\rho^\iota_0\frac{\partial\bV}{\partial
t} - \rho_0^\iota\bg - \Bnabla\cdot\bP^\iota +
\left(\Bnabla\bV\right)\bM^\iota+\rho^\iota_0\bF^{-\mathrm{T}}\left(\Bnabla
e^\iota -
\theta\Bnabla\eta^\iota\right)\right),\label{VIconstrel1}\\
&&\mathrm{where}\;\bw\cdot\tilde{\bD}^\iota\bw
\ge 0,\;\;\forall\, \bw \in\mathbb{R}^3\nonumber\\
\nonumber\\
&&\bQ^\iota = -{\bK}^\iota\Bnabla\theta,\;\bw\cdot{\bK}^\iota\bw
\ge 0,\;\;\forall\, \bw \in\mathbb{R}^3\label{qconstrelI}\end{aligned}$$
With the constitutive relations (\[stress-constrelI\]–\[qconstrelI\]) ensuring the non-positiveness of certain terms the entropy inequality is further reduced to $$\begin{aligned}
& & \sum\limits_{\iota}\left(\rho^\iota_0\frac{\partial
e^\iota}{\partial\rho^\iota_0}\frac{\partial
\rho^\iota_0}{\partial t} - \bP^\iota\colon(\Bnabla\bV^\iota +
\bF^{\mathrm{e}^\iota}\dot{\bF}^{\mathrm{g}^\iota})\right)\nonumber\\
&&+\sum\limits_{\iota}\left(
\rho^\iota_0\bV^\iota\cdot\left(\frac{\partial\bV^\iota}{\partial
t} +
\left(\Bnabla\bV^\iota\right)\bF^{-1}\bV^\iota\right)+\Pi^\iota\left(e^\iota
+ \frac{1}{2}\Vert\bV+\bV^\iota\Vert^2\right)\right)\nonumber\\
&&+\sum\limits_{\iota} \left(\rho^\iota_0\frac{\partial}{\partial
t}\left(\bV+\bV^\iota\right) - \rho_0^\iota\bg -
\Bnabla\cdot\bP^\iota +
\Bnabla\left(\bV+\bV^\iota\right)\bM^\iota\right)\cdot\bV \le 0.
\label{dissipation1}\end{aligned}$$
The left hand-side of (\[dissipation1\]) is the dissipation, $\sD$, a quantity that we will return to below.
Equation (\[stress-constrelI\]) specifies a constitutive relation for $\bP^\iota\bF^{\mathrm{g}^{\iota\mathrm{T}}}$, which is a truly elastic stress. Equation (\[temp-constrelI\]) implies a uniform temperature in each species, and corresponds with the definition usually employed for temperature in thermal physics. The heat flux in species $\iota$ is given by the product of a positive semi-definite conductivity tensor, ${\bK}^\iota$ and the temperature gradient, which is the Fourier Law of heat conduction. Equation (\[VIconstrel1\]) requires more detailed discussion which appears below.
Constitutive relations for fluxes {#sect5.1}
---------------------------------
Turning to (\[VIconstrel1\]), we first point out that since $\bM^\iota = \rho_0^\iota\bF^{-1}\bV^\iota$, this is an implicit relation for $\bV^\iota$. Rewriting it as an explicit one for $\rho^\iota_0\bV^\iota$ we have, $$\begin{aligned}
\rho^\iota_0\bV^\iota =& &\left(\bone +
\frac{\tilde{\bD}^\iota\Bnabla\bV\bF^{-1}}{\rho^\iota_0}\right)^{-1}\frac{\tilde{\bD}^\iota}{\rho^\iota_0}\nonumber\\
& &\cdot\left\{-\left(\rho^\iota_0\frac{\partial\bV}{\partial t} -
\rho_0^\iota\bg - \Bnabla\cdot\bP^\iota +
\rho^\iota_0\bF^{-\mathrm{T}}\left(\Bnabla e^\iota -
\theta\Bnabla\eta^\iota\right)\right)\right\}. \label{VIconstrel2}\end{aligned}$$
The constitutive relation for flux, $\bM^\iota =
\rho^\iota_0\bF^{-1}\bV^\iota$ is then obtained: $$\begin{aligned}
\bM^\iota & &=\underbrace{\bF^{-1}\left(\bone +
\frac{\tilde{\bD}^\iota\Bnabla\bV\bF^{-1}}{\rho^\iota_0}\right)^{-1}\frac{\tilde{\bD}^\iota}{\rho^\iota_0}\bF^{-\mathrm{T}}}_{\bD^\iota}\nonumber\\
&\cdot&\left\{\underbrace{-\left(\rho^\iota_0\bF^\mathrm{T}\frac{\partial\bV}{\partial
t} - \rho_0^\iota\bF^\mathrm{T}\bg -
\bF^\mathrm{T}\Bnabla\cdot\bP^\iota + \rho^\iota_0\left(\Bnabla
e^\iota -
\theta\Bnabla\eta^\iota\right)\right)}_{\boldmath{\sF}^\iota}\right\}.
\label{MIconstrel}\end{aligned}$$
As is common in descriptions of mass transport, the tensor delineated as $\bD^\iota$ will be referred to as the mobility tensor of species $\iota$. Recall that in (\[VIconstrel1\]) we have taken $\tilde{\bD}^\iota$ to be positive semi-definite[^11]. The flux is thus written as the product of a mobility tensor and a thermodynamic driving force, $\boldmath{\sF}^\iota$. This is the Nernst-Einstein relation. We proceed now to examine the four separate terms in the thermodynamic driving force: $$\boldmath{\sF}^\iota =
-\rho^\iota_0\bF^\mathrm{T}\frac{\partial\bV}{\partial t} +
\rho_0^\iota\bF^\mathrm{T}\bg +
\bF^\mathrm{T}\Bnabla\cdot\bP^\iota - \rho^\iota_0\left(\Bnabla
e^\iota - \theta\Bnabla\eta^\iota\right) \label{drivingforceI}$$
The first two terms respectively represent the influences of inertia and body force. Thus, the inertial effect is to drive species $\iota$ in the opposite direction to the body’s acceleration. The body force’s influence is directed along itself. The third term represents the stress divergence effect. In the case of a non-uniform partial stress, $\bP^\iota$, there exists a thermodynamic driving force for transport along $\bP^\iota$. We demonstrate this effect for the case of the fluid species in Section \[sect5.2\] below, for which it translates to the more intuitive notion of transport along a fluid pressure gradient.
The fourth term in $\boldmath{\sF}^\iota$ admits the following interpretation: The Legendre transformation $\psi^\iota = e^\iota
- \theta\eta^\iota$ allows one to rewrite $\Bnabla e^\iota -
\theta\Bnabla\eta^\iota$ as $\Bnabla\psi^\iota\vert_\theta$ (at uniform temperature), where $\psi^\iota$ is the mass-specific Helmholtz free energy. An assumption inherent in the development that began in Section \[sect2\] is that any mass entering or leaving $\Omega_0$ at a point $\bX$ on the boundary, $\partial\Omega_0$, has the field values $\rho_0^\iota,e^\iota,\eta^\iota,\theta$, and $\psi^\iota$ corresponding to $\bX$. Likewise, the incremental mass of species $\iota$ created or absorbed via the source/sink $\Pi^\iota$ at $\bX$ has the field values of that point. Consider a sufficiently small neighborhood of a point, say $\mathsf{N}(\bX)\subset\Omega_0$. Changing the mass of species $\iota$ in $\mathsf{N}(\bX)$ by $\delta\mathsf{m}^\iota$ units causes a change in the Helmholtz free energy of $\iota$ in $\mathsf{N}(\bX)$ by $\delta\Psi^\iota =
\psi^\iota\delta\mathsf{m}^\iota$. By definition therefore, $\psi^\iota =
\partial\Psi^\iota/\partial\mathsf{m}^\iota$. This derivative gives the *chemical potential*, $\mu^\iota$, of the transported species, $\iota$. Thus, we have $\mu^\iota = e^\iota - \theta\eta^\iota$, and $\Bnabla e^\iota -\theta\Bnabla\eta^\iota =
\Bnabla\mu^\iota\vert_\theta$. This last term in ${\boldmath{\sF}}^\iota$ thus represents the thermodynamic driving force due to a chemical potential gradient.
It has recently come to our attention that the constitutive relation for flux (\[MIconstrel\]) is precisely the result arrived at by @DeGrootMazur:1984, including the identification of the chemical potential gradient term. However, their approach involves a slightly different application of the Second Law, and a less detailed treatment of the mechanics.
The gradient of internal energy in (\[drivingforceI\]) leads to a strain gradient-dependent term. A concentration gradient-driven term arises from the gradient of mixing entropy. Together with the other terms that were remarked upon above, they represent a complete thermodynamic formulation of coupled mass transport and mechanics. This is the central result of our paper.
Transport of the fluid species: The example of an ideal fluid {#sect5.2}
-------------------------------------------------------------
Consider the stress divergence term $\bF^\mathrm{T}\Bnabla\cdot\bP^\iota$. An elementary calculation gives $$\bF^\mathrm{T}\Bnabla\cdot\bP^\iota =
\Bnabla\cdot\left(\bF^\mathrm{T}\bP^\iota\right) -
\Bnabla\bF^\mathrm{T}\colon\bP^\iota. \label{stressdivI}$$
In indicial form, where lower/upper case indices are for components of quantities in the current/reference configuration respectively, this relation is $$F_{iK}P^\iota_{iJ,J} = \left(F_{iK}P^\iota_{iJ}\right)_{,J} -
F_{iK,J}P^\iota_{iJ}.$$
For an ideal fluid, supporting only an isotropic Cauchy stress, $p\bone$, we have $\bP^\mathrm{f} =
\mathrm{det}(\bF)p\bF^{-\mathrm{T}}$, where $p$ is positive in tension. The arguments that follow assume this case. (The more general case of a non-ideal, viscous fluid will merely have additional terms from the viscous Cauchy stress.) The stress divergence term is $$\bF^\mathrm{T}\Bnabla\cdot\bP^\mathrm{f} =
\Bnabla\left(\mathrm{det}(\bF)p\right) -
\Bnabla\bF^\mathrm{T}\colon\bF^{-\mathrm{T}}\mathrm{det}(\bF)p,$$
demonstrating the appearance of a hydrostatic stress-driven contribution to ${\boldmath{\sF}}^\mathrm{f}$. This is Darcy’s Law for transport of a fluid down a pressure gradient.
For the special case of a compressible, ideal fluid we have $e^\mathrm{f} =
\bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f})$; i.e., the fluid stores strain energy as a function of its *current, intrinsic* density. Fluid saturation conditions hold in biological tissue, for which case the fluid volume fraction, $f^\mathrm{f}$, is simply the pore volume fraction. Recall from Section \[sect2\] that the individual species deform with the common deformation gradient $\bF$. Therefore the pores deform *homogeneously* with the surrounding solid phase. Physically this corresponds to the pore size being smaller than the scale at which the homogenization assumption of a continuum theory holds. Momentarily ignoring changes in reference concentration of the fluid, we have $\bF^{\mathrm{e}^\mathrm{f}} = \bF$. Then, since $\rho^\mathrm{f}_0 = \bar{\rho}^\mathrm{f}_0 f^\mathrm{f}$, we can write $\hat{e}^\mathrm{f}(\bF,\eta^\mathrm{f},\rho^\mathrm{f}_0) =
\hat{e}^\mathrm{f}(\bF,\eta^\mathrm{f},\bar{\rho}^\mathrm{f}_0
f^\mathrm{f}) =
\bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f}_0/\mathrm{det}\bF)=
\bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f})$. In this case a simple calculation shows that the hydrostatic pressure is $$p =
-\frac{\bar{\rho}^\mathrm{f}}{\mathrm{det}(\bF)}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}},$$
and the stress divergence term is $$\bF^\mathrm{T}\Bnabla\cdot\bP^\mathrm{f} =
-\Bnabla\left(\bar{\rho}^\mathrm{f}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}}\right)
+
\Bnabla\bF^\mathrm{T}\colon\bF^{-\mathrm{T}}\bar{\rho}^\mathrm{f}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}}.$$
The Eshelby stress as a thermodynamic driving force {#sect5.3}
---------------------------------------------------
Combining the stress divergence and chemical potential gradient contributions to the driving force for any species, and using the mass-specific Helmholtz free energy, $\psi^\iota$, we write,
$$\bF^\mathrm{T}\Bnabla\cdot\bP^\iota - \rho^\iota_0\left(\Bnabla
e^\iota - \theta\Bnabla\eta^\iota\right) =
\Bnabla\cdot\left(\bF^\mathrm{T}\bP^\iota\right) -
\Bnabla\bF^\mathrm{T}\colon\bP^\iota
- \rho^\iota_0\Bnabla\psi^\iota\vert_\theta.$$
Regrouping terms this expression is $$-\Bnabla\cdot\underbrace{\left(\rho^\iota_0\psi^\iota\vert_\theta\bone
- \bF^\mathrm{T}\bP^\iota\right)}_{\mbox{Eshelby
stress},\;\BXi^\iota} +
\left(\Bnabla\rho^\iota_0\right)\psi^\iota_0\vert_\theta -
\Bnabla\bF^\mathrm{T}\colon\bP^\iota.$$
Thus, the divergence of the well-known Eshelby stress tensor is also among the driving forces for mass transport. Also observe the presence of a strain gradient-dependent driving force, $-\Bnabla\bF^\mathrm{T}\colon\bP^\iota$ in the developments of Sections \[sect5.2\] and \[sect5.3\], independent of the pressure gradient term for the fluid species.
[**Remark 1**]{}: The final version of the dissipation inequality (\[dissipation1\]), and the mass balance equation can be manipulated to restrict the mathematical form of the mass source. It is common to make the mass source depend upon the strain energy density [@HarriganHamilton:1993] while respecting the restriction imposed by the dissipation inequality. This form is often used while modelling hard tissue. Such an approach leads to strain-mediated mass transport. However, with a strain-independent source, strain-mediated (or stress-mediated) mass transport would not be obtained with such a formulation.
[**Remark 2**]{}: We expect that evaluation of the dissipation, $\sD$, using (\[dissipation1\]) from field quantities in a boundary value problem will provide a test of soundness, and if necessary indications for improvement, of our constitutive models.
[**Remark 3**]{}: Since soft biological tissues usually demonstrate rate-dependent response, it has been common to employ a solid viscoelastic constitutive model for them. This approach fits within our framework, with a modification of the internal energy to include its dependence upon internal variables that represent the viscoelastic stress-like parameters. However, a more physiologically-valid model may be one with a purely hyperelastic solid phase, and a viscous fluid. In such a composite model the rate-dependent behavior would arise from the fluid.
[**Remark 4**]{}: The constitutive relations (\[stress-constrelI\]) and (\[MIconstrel\]) respectively specify the partial stress, $\bP^\iota$, and flux, $\bM^\iota$, of a species. The flux also implies the relative velocity, $\bV^\iota$. The velocity of the solid phase, $\bV$ is obtained from the local form of the balance of linear momentum for the system (\[linmombalsys\]). With all these quantities known, the individual interaction forces between species, $\rho^\iota_0\bq^\iota$, can be obtained from (\[ballinmomrefI\]). They are, however, not needed while solving for the balance of linear momentum of the system.
A numerical example {#sect6}
===================
The theory developed in Sections \[sect2\]–\[sect5\] has been implemented in a computational formulation, retaining much of the complexity of the coupled balance laws and constitutive relations. For realistic soft tissue material parameters, the contribution of the fluxes and interaction forces between species to the balance of linear momentum of the composite tissue is negligible. This simplification has been used. As a preliminary demonstration of the theory[^12], we present a computation of the coupled physics in the early stages of uniaxial extension of a cylindrical soft tissue specimen. The motivation for this model problem comes from our experimental model of engineered, functional tendon constructs grown *in vitro*, having the same cylindrical geometry. The experimental aspects of our broad-based project on soft tissue growth are described elsewhere [@Calveetal:2003]. In addition to engineering scaffold-less tendon constructs from neonatal rat fibroblast cells, we have the ability to impose a range of mechanical, chemical, nutritional and electrical stimuli on them and study the tissue’s response. Besides modelling these experiments, the mathematical formulation described here presents researchers with a vehicle for testing scenarios and framing hypotheses that can be experimentally-validated in our laboratory, thereby driving the experimental studies.
Material models and parameters {#sect6.1}
------------------------------
The engineered tendon construct is $12$ mm in length and $1\;\mathrm{mm}^2$ in area. In this paper an internal energy density for the solid phase based upon the worm-like chain model is used. The reader is directed to @Riefetal:97 and [@Bustamanteetal:2003] where the one-dimensional version of this model has been applied to long chain molecules. It has been described and implemented into an anisotropic representative volume element by @Bischoffetal:2002, and is summarized here. The internal energy density of a single constituent chain of an eight-chain model (Figure \[eightchain\]) is, $$\begin{aligned}
\bar{\rho}_0^\mathrm{s}\hat{e}^\mathrm{s}(\bF^{\mathrm{e}^\mathrm{s}},\rho_0^\mathrm{s},\eta^\mathrm{s})
&=& \frac{N k \theta}{4 A}\left(\frac{r^2}{2L} +
\frac{L}{4(1-r/L)} -
\frac{r}{4}\right)\nonumber\\
&-&\frac{N k \theta}{4\sqrt{2L/A}}\left(\sqrt{\frac{2A}{L}} +
\frac{1}{4(1 - \sqrt{2A/L})} -\frac{1}{4} \right)\log(\lambda_1^{a^2}\lambda_2^{b^2}\lambda_3^{c^2})\nonumber\\
&+& \frac{\gamma}{\beta}({J^{\mathrm{e}^\mathrm{s}}}^{-2\beta} -1)
+ 2\gamma{\bf 1}\colon\bE^{\mathrm{e}^\mathrm{s}} \label{wlcmeq}\end{aligned}$$ Here, $N$ is the density of chains, $k$ is the Boltzmann constant, $r$ is the end-to-end length of a chain, $L$ is the fully-extended length, and $A$ is the persistence length that measures the degree to which the chain departs from a straight line. The preferred orientation of tendon collagen is described by an anisotropic unit cell with sides $a,b$ and $c$—see Figure \[eightchain\]. All lengths in this model have been rendered non-dimensional (Table \[mattab\]) by dividing by the link length in a chain.
.
The elastic stretches along the unit cell axes are respectively denoted $\lambda^\mathrm{e}_1,\lambda^\mathrm{e}_2$ and $\lambda^\mathrm{e}_3$, and $\bE^{\mathrm{e}^\mathrm{s}} =
\frac{1}{2}(\bC^{\mathrm{e}^\mathrm{s}} - {\bf 1})$ is the elastic Lagrange strain. The factors $\gamma$ and $\beta$ control bulk compressibility. The end-to-end length is given by
$$r =
\frac{1}{2}\sqrt{a^2\lambda_1^{\mathrm{e}^2}+b^2\lambda_2^{\mathrm{e}^2}+c^2\lambda_3^{\mathrm{e}^2}},\quad
\lambda_I^{\mathrm{e}} = \sqrt{\bN_I\cdot\bC^{\mathrm{e}}\bN_I}
\label{rwlcm}$$
Preliminary mechanical tests of the engineered tendon have been carried out in our laboratory but, at this stage, the worm-like chain model has not been calibrated to these tests. Instead, published data for the worm-like chain, obtained by calibrating against rat cardiac tissue [@Bischoffetal:2002], has been employed.
The fluid phase was modelled as an ideal, nearly-incompressible fluid: $$\bar{\rho}^\mathrm{f}_0\hat{e}^\mathrm{f}(\bF^{\mathrm{e}^\mathrm{f}},\rho_0^\mathrm{f},\eta^\mathrm{f})
=
\frac{1}{2}\kappa(\mathrm{det}(\bF^{\mathrm{e}^\mathrm{f}})-1)^2,$$
where $\kappa$ is the fluid bulk modulus.
Only a solid and a fluid phase were included for the tissue. Low values were chosen for the mobilities of the fluid [@Swartzetal:99] with respect to the solid phase (see Table \[mattab\]). In order to demonstrate growth, the solid phase must have a source term, $\Pi^\mathrm{s}$ (Section \[sect2\]), and the only other phase, the fluid, must have $\Pi^\mathrm{f} =
-\Pi^\mathrm{s}$. Therefore, contrary to the case made in Section \[sect2\], a non-zero value of the fluid source, $\Pi^\mathrm{f}$, was assumed. A form motivated by first-order reactions was used: $$\Pi^\mathrm{f} = -k^\mathrm{f}(\rho_0^\mathrm{f} -
\rho_{0_\mathrm{ini}}^\mathrm{f}),\quad \Pi^\mathrm{s} =
-\Pi^\mathrm{f}, \label{piform}$$
where $k^\mathrm{f}$ is the reaction rate, and $\rho_{0_\mathrm{ini}}^\mathrm{f}$ is the initial fluid concentration. This term acts as a source for the solid when $\rho_0^\mathrm{f}
> \rho_{0_\mathrm{ini}}^\mathrm{f}$, and a sink when $\rho_0^\mathrm{f} < \rho_{0_\mathrm{ini}}^\mathrm{f}$.
In a very simple approximation, the fluid’s mixing entropy was written as $$\eta^\mathrm{f}_\mathrm{mix} =
-\frac{k}{\sM^\mathrm{f}}\log\frac{\rho_0^\mathrm{f}}{\rho_0}.
\label{mixentropy}$$
Recall that in the notation of Section \[sect2\], $\sM^\mathrm{f}$ is the fluid’s molecular weight.
Symbol Value Units
---------------------------------- ---------------------------- ------------------------------------------------------- --------------------------------
Chain density $N$ $7\times 10^{21}$ $\mathrm{m}^{-3}$
Temperature $\theta$ $310.0$ K
Persistence length $A$ $1.3775$ –
Fully-stretched length $L$ $25.277$ –
Unit cell axes $a,\;b,\;,c$ $9.2981,\;12.398,\;6.1968$ –
Bulk compressibility factors $\gamma,\;\beta$ $1000,\; 4.5$ –
Fluid bulk modulus $\kappa$ $1$ GPa
Fluid mobility tensor components $D_{11},\;D_{22},\;D_{33}$ $1\times 10^{-8},\;1\times 10^{-8},\;1\times 10^{-8}$ $\mathrm{m}^{-2}\mathrm{sec}$
Fluid conversion reaction rate $k^\mathrm{f}$ $-1.\times 10^{-7}$ $\mathrm{sec}^{-1}$
Gravitational acceleration $\bg$ $9.81$ $\mathrm{m}.\mathrm{sec}^{-2}$
Molecular weight of fluid $\sM^\mathrm{f}$ $2.9885\times 10^{-23}$ $\mathrm{kg}$
: Material parameters used in the analysis[]{data-label="mattab"}
Boundary and initial conditions; coupled solution method {#sect6.2}
--------------------------------------------------------
Boundary conditions for mass transport consisted of the specified fluid concentration at all external surfaces of the cylinder. This value was fixed at $500\,\mathrm{kg.m}^{-3}$. With these boundary conditions the fluid flux normal to surfaces of the specimen is determined by solving the initial and boundary value problem. The bottom planar surface was fixed in the $\be_3$ direction and a displacement was applied at the top surface, also in the $\be_3$ direction, to give a nominal strain rate of $0.05\,\mathrm{sec}^{-1}$ in the $\be_3$ direction. This is the only mechanical load on the problem. Initial conditions were $\rho_0^\mathrm{f}(\bX,0) =
500\,\mathrm{kg.m}^{-3},\;\rho_0^\mathrm{s}(\bX,0) =
500\,\mathrm{kg.m}^{-3}$, and for the mechanical problem, $\bu(\bX,0) = \bzero,\,\bV(\bX,0) = \bzero$.
The coupled problem was solved by a staggered scheme based upon operator splits [@Armero:99; @Garikipatietal:01]. The details will be presented in a future communication that will focus upon computational aspects and numerical examples. Here we only mention that the staggered scheme consists of identifying the displacement and species concentrations as primitive variables associated with the mechanical and mass transport problems. The mechanical problem is solved holding the concentrations fixed. The resulting displacement field is then held constant to solve the mass transport problem. The transient solution is obtained for mechanics using energy-momentum conserving schemes [@SimoTarnow:1992b; @SimoTarnow:1992a; @Gonzalezphd:1996], and for mass transport using the Backward Euler Method. Hexahedral elements are employed, combined with nonlinear projection methods [@SimoTaylorPister:85] to treat the near-incompressibility imposed by the fluid. The numerical formulation has been implemented within the nonlinear finite element program, FEAP [@FEAPmanual].
Evolution of stress and flux {#sect6.3}
----------------------------
The following contour plots represent the stress, and various contributions to the total flux in the early stages of loading of the model problem. Symmetry has been employed to model a single quadrant of the cylinder.
The longitudinal stress, $\sigma_{33}$ in Figure \[stressfig\] arises from the stretch and the evolution in concentration.
[![Longitudinal Cauchy stress, $\sigma_{33}$ (Pa) at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading.[]{data-label="stressfig"}](S33-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Longitudinal Cauchy stress, $\sigma_{33}$ (Pa) at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading.[]{data-label="stressfig"}](S33-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Stress gradient-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux corresponding to a tensile $\sigma_{33}$ wave travelling downwards.[]{data-label="M1fig"}](M1-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Stress gradient-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux corresponding to a tensile $\sigma_{33}$ wave travelling downwards.[]{data-label="M1fig"}](M1-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Internal energy gradient-driven flux, ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux. This corresponds to a lower energy near the top of the cylinder as the tensile stress ($\sigma_{33}$) wave travels downward and relaxes some of the strain energy of contraction.[]{data-label="M2fig"}](M2-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Internal energy gradient-driven flux, ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux. This corresponds to a lower energy near the top of the cylinder as the tensile stress ($\sigma_{33}$) wave travels downward and relaxes some of the strain energy of contraction.[]{data-label="M2fig"}](M2-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Gravity-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The negative values indicate a downward flux, due to the action of gravity.[]{data-label="M3fig"}](M3-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Gravity-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The negative values indicate a downward flux, due to the action of gravity.[]{data-label="M3fig"}](M3-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Inertia-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The negative values indicate a downward flux as the tissue accelerates upward.[]{data-label="M4fig"}](M4-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Inertia-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The negative values indicate a downward flux as the tissue accelerates upward.[]{data-label="M4fig"}](M4-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Concentration gradient-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. Note that the maximum and minimum values are many orders of magnitude lower than for the other flux contributions reported above. This is a demonstration of mechanics influences dominating diffusion over the classical concentration gradient contribution.[]{data-label="M5fig"}](M5-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Concentration gradient-driven flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. Note that the maximum and minimum values are many orders of magnitude lower than for the other flux contributions reported above. This is a demonstration of mechanics influences dominating diffusion over the classical concentration gradient contribution.[]{data-label="M5fig"}](M5-100.eps "fig:"){width="7.5cm"}]{} (b)
[![Total flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux, dominated by the stress gradient driven contribution.[]{data-label="Mfig"}](M-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Total flux ($\mathrm{kg.m}^{-2}\mathrm{sec}$) in the $\be_3$ direction at $1 \,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate an upward flux, dominated by the stress gradient driven contribution.[]{data-label="Mfig"}](M-100.eps "fig:"){width="7.5cm"}]{} (b)
The flux contributions in Figures \[M1fig\]–\[Mfig\] can be summarized as follows: The fluid flux is dominated by the contribution from the stress gradient in the $\be_3$ direction. The latter arises as the stress ($\sigma_{33}$) wave of tension travels down the cylinder in the first few microseconds after application of the load (the time taken to travel the length of the cylinder is $12 \,\mu\mathrm{sec}$). Additionally, as the fluid concentration changes due to the flux, it causes a further change in the stress (Section \[sect5\]). Other flux terms are qualitatively sensible; i.e., their directions are consistent with the physics of the problem, as argued in each of the figure captions[^13]. There is some loss of axial symmetry in a few of the plots due to the coarseness of the finite element mesh for this example. It appears that spatial oscillations in the solution lead to a further loss of symmetry in Figures \[M2fig\]b–\[M3fig\]b and \[Mfig\]a. These oscillations arise due to large and dominant advective terms, and need to be remedied by stabilized finite element methods. Here, we only aim to demonstrate that various driving forces for mass transport are in agreement with their theoretical underpinnings in the paper. The resorption of the solid phase is shown indirectly in Figure \[Pifig\]. A positive fluid source, $\Pi^\mathrm{f}$, means that $\Pi^\mathrm{s} < 0$. Since $\Pi^\mathrm{s}$ is the only term balancing $\partial\rho_0^\mathrm{s}/\partial t$ \[see (\[massballocA\])\], it follows that $\partial\rho_0^\mathrm{s}/\partial t < 0$.
[![Rate of fluid production, $\Pi^\mathrm{f}$ ($\mathrm{kg.m}^{-3}.\mathrm{sec}^{-1}$), at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate that the local fluid concentrations have fallen below their initial values.[]{data-label="Pifig"}](Pi-1.eps "fig:"){width="7.5cm"}]{} (a)
[![Rate of fluid production, $\Pi^\mathrm{f}$ ($\mathrm{kg.m}^{-3}.\mathrm{sec}^{-1}$), at $1
\,\mathrm{nanosec.}$ and $100\,\mathrm{nanosec.}$ after the beginning of loading. The positive values indicate that the local fluid concentrations have fallen below their initial values.[]{data-label="Pifig"}](Pi-100.eps "fig:"){width="7.5cm"}]{} (b)
Discussion and conclusions {#sect7}
==========================
A general framework for growth of biological tissue has been presented in this paper. While simplified models were used for source terms, $(\Pi^\iota)$, they can be formulated on the basis of the kinetics of chemical reactions in order to develop more realistic growth laws. This approach, we believe, is fundamental to a proper treatment of mass transport in tissue. Results are obtained that differ fundamentally from the classical setting of continuum mechanics. Most notable among these differences are the mass fluxes driven by gradients in stress, strain, energy and entropy, in addition to body force and inertia. Importantly, though our treatment differs from classical mixture theory, the two are fully consistent as established at several points in this paper. The balance laws in Section \[sect2\] and \[sect3\] introduce a degree of coupling between the phenomena of mass transport and mechanics. This is visible most transparently in the balance of linear momentum (\[ballinmomrefI\]) that includes mass fluxes, $\bM^\iota$. The balance of mass, described by Equations (\[massballocA\]) and (\[massballocI\]), is also dependent upon the mechanics as the discussion in Section \[sect5.1\] makes clear. Notably, this ensures mechanics-mediated mass transport even with a mass source that is independent of strain/stress, as the discussion at the end of Section \[sect5.3\] establishes. The discussion in Sections \[sect5.1\]–\[sect5.3\] provides many insights into the nature of this coupling. The mechanics problem also has a constitutive dependence upon mass concentration, via (\[stress-constrelI\]) and the fact that the growth deformation gradient tensor, $\bF^{\mathrm{g}^\iota}$ is determined by the concentration. The viscoelastic nature of the composite tissue would emerge naturally from a model incorporating a hyperelastic solid and viscous fluid.
We have formally allowed all species to be load bearing and develop a stress. At the scales that are of interest in a tissue, the only relevant load bearing species are the solid and fluid phases. Nevertheless, inasmuch as a transported species such as a nutrient has a molecular structure that can be subject to loads at the scale of pico-newtons, it is not inconsistent to speak of the partial stress of this species. Since the constitutive relation (\[stress-constrelI\]) indicates that the partial stresses are scaled by concentrations, the contribution to total stress from any species besides the solid and fluid phases will be negligible.
We have chosen to leave remodelling out of our formulation in this communication, to focus upon the above issues. Remodelling includes any evolution in properties, state of stress, material symmetry, volume or shape brought about by microstructural changes. In the development of biological tissue, growth and remodelling occur simultaneously. As density changes due to growth, the material also remodels by microstructural evolution within the neighborhood of each point. A rigorous treatment of this phenomenon has been presented in the continuum mechanical setting in a companion paper [@remodelpaper].
[^1]: Asst. Professor, Department of Mechanical Engineering, [[email protected]]{}
[^2]: Assoc. Professor, Department of Mechanical Engineering and Program in Macromolecular Science and Engineering
[^3]: Assoc. Professor, Departments of Mechanical Engineering, and Biomedical Engineering
[^4]: Graduate research assistant, Department of Mechanical Engineering
[^5]: Graduate research assistant, Program in Macromolecular Science and Engineering
[^6]: We use the terms “mass transport” and “diffusion” interchangeably.
[^7]: @Vander:2003 go on to say: “This simple definition cannot give a complete appreciation of what homeostasis truly entails, however. There probably is no such thing as a physiological variable that is constant over long periods of time. In fact, some variables undergo fairly dramatic swings about an average value during the course of a day, yet may still be considered ‘in balance’. That is because homeostasis is a *dynamic process*, not a static one.”
[^8]: A more sophisticated, and physiologically-correct, description is that the interstitial fluid diffuses relative to the solid phase, while precursors and byproducts of reactants diffuse with respect to the fluid.
[^9]: Considering the case of the lymphatic fluid, this implies that lymph glands are assumed not to be present.
[^10]: In our experiments we have measured the mass concentration $\rho_0^\iota$ of collagen in engineered tendons grown *in vitro*. These results will be presented elsewhere [@Calveetal:2004].
[^11]: If $\Vert\tilde{\bD}^\iota\Bnabla\bV\bF^{-1}/\rho^\iota_0\Vert << 1$, we have $\bD^\iota \approx \tilde{\bD}^\iota$.
[^12]: This numerical section has been included mainly for completeness of this theoretical paper. A separate paper, currently in preparation, will present the computational formulation and contain a detailed examination of a number of initial and boundary value problems for growth.
[^13]: In order to compare the flux contributions, they have all been plotted on the same scale: $-1\times
10^{-4}--1\times 10^{-4} \,\mathrm{kg.m}^{-2}\mathrm{sec}$. However the plots also show the maximum and minimum field values at the top and bottom of the legend bars. These values represent a better comparison of the relative flux magnitudes.
|
---
abstract: 'We review our recent proposal of a method to extend the quantization of *spherically symmetric* isolated horizons, a seminal result of loop quantum gravity, to a phase space containing horizons of *arbitrary* geometry. Although the details of the quantization remain formally unchanged, the physical interpretation of the results can be quite different. We highlight several such differences, with particular emphasis on the physical interpretation of black hole entropy in loop quantum gravity.'
address: 'Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA'
author:
- C Beetle and J Engle
title: Entropy of generic quantum isolated horizons
---
The confirmation of the Bekenstein–Hawking entropy formula by Ashtekar, Baez, Corichi and Krasnov [@abck1997; @dl2004; @meissner2004] (ABCK) is one of the triumphs of loop quantum gravity. The ABCK approach begins by quantizing a classical phase space whose points correspond to spacetimes with inner boundary at a *spherically symmetric* isolated horizon [@ak2004; @abf1998; @ack1999; @ashtekar_etal2000]. The calculation relies on spherical symmetry (just of the intrinsic geometry of the horizon itself) to make the symplectic structure on that phase space, and thus the ensuing quantization, well-defined.
This paper is concerned with an apparent inconsistency in the roles played by spherical symmetry before and after quantization in the ABCK approach. Imposing spherical symmetry classically restricts the allowed *bulk* fields such that they induce a round metric on the horizon. After quantization, however, the bulk spin network states in ABCK are (virtually) generic at the horizon, no different from those allowed on an *arbitrary* 2-surface in loop quantum gravity. In this sense, the *only* place that spherical symmetry is used at all in deriving the *quantum* theory of ABCK is in making the symplectic structure of the *classical* theory well-defined.
Here, we review a new way [@be2010] to justify the ABCK symplectic structure classically, including its crucial Chern–Simons surface term. Most importantly, our proposed scheme does not rely on spherical symmetry, or indeed on any restriction of the intrinsic horizon geometry, to make the symplectic structure well-defined. Rather, it allows one simply to quantize the phase space of *all* isolated horizons of a given total area ${\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}$. The intrinsic geometry of the horizon, its shape, *is not fixed a priori*. This approach renders moot the question of how to impose spherical symmetry at the horizon quantum mechanically, which is the key missing ingredient that would make ABCK a consistent quantization of a spherically symmetric horizon. Rather, our approach renders the ABCK quantization entirely consistent in another way, namely, by broadening the *classical* picture to eliminate the requirement of symmetry *ab initio*.
ABCK Phase Space and Quantization
=================================
ABCK use a covariant phase space, each point of which corresponds a space*time* $\mathcal{M}$ of the form shown in Figure \[st.sn.tf\](a) in which the classical (Einstein) equations of motion hold. It is bounded to the future and past by partial Cauchy slices $M_\pm$, which extend to spatial infinity $i^0$, and has an inner boundary at a null surface $\Delta$ diffeomorphic to $S^2 \times {\mathbb{R}}$. Boundary conditions at $\Delta$ make it is a spherically symmetric [[horizon]{}]{}. In a precise sense [@ak2004], this means that $\Delta$ models the surface of a quiescent black hole in perfect equilibrium with its immediate surroundings.
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![(a) The space-time arena considered both here and in ABCK. (b) Bulk spin networks fix charges for the quantum Chern–Simons theory via the quantum horizon boundary condition. (c) Classical horizon shape is probed by transverse, not pullback, fluxes.[]{data-label="st.sn.tf"}](qhbc.pdf "fig:"){width="4.5cm"}
(a) (b) (c)
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Under the above conditions (and one or two additional technical assumptions), the integral $$\label{symInt}
\Omega(\delta_1, \delta_2)
:= {{}^\beta\mspace{-1mu}\Omega}_{\mathrm{B}}(\delta_1, \delta_2) + {{}^\beta\mspace{-1mu}\Omega}_{\mathrm{S}}(\delta_1, \delta_2)
:= \frac{1}{4\pi G {\beta}} \int_M \delta_{[1} \Sigma^i \wedge \delta_{2]} {{}^\beta\mspace{-6mu}A}_i
+ \frac{1}{2\pi} \frac{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}}{4\pi G {\beta}} \oint_{S} \delta_1 V \wedge \delta_2 V$$ takes the same value over *any* partial Cauchy slice $M$ through $\mathcal{M}$ with inner boundary $S$. Here, $\delta_{1, 2}$ represent a pair of tangent vectors to the ABCK phase space, *i.e.*, solutions of the linearized equations of motion on the given background that preserve the total area and spherical symmetry of $S$. The bulk term ${{}^\beta\mspace{-1mu}\Omega}_{\mathrm{B}}(\delta_1, \delta_2)$ in (\[symInt\]) is the standard symplectic structure of loop quantum gravity. The surface term ${{}^\beta\mspace{-1mu}\Omega}_{\mathrm{S}}(\delta_1, \delta_2)$ has a form familiar from a Chern–Simons theory for the $U(1)$ [[connection]{}]{} $V_a$ induced on $S$ by the bulk geometry. The curvature of $V_a$ is $$\label{edV}
{\mathrm{d}}V = - \tfrac{1}{4} {\mathcal{R}}\, \epsilon
\qquad\leadsto\qquad {\mathrm{d}}V = -\tfrac{2\pi}{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}}\, {\mathpalette{\smallunderarrow@{\relax}\Leftarrowfill@}}{\Sigma}_i\, r^i
\quad \text{if ${\mathcal{R}}$ is \textit{constant},}$$ where ${\mathcal{R}}$ is the the intrinsic scalar curvature of $S$, $\epsilon$ is its 2-from area element, and $r^i$ is a gauge-fixed internal radial direction. It is the latter equation here, which holds only in spherical symmetry, that makes (\[symInt\]) independent of $M$, and thus a well-defined symplectic structure.
The sum of bulk and surface terms in the symplectic structure (\[symInt\]) suggests that the Hilbert space of the quantum theory should be a tensor product ${\ifx^pre\relax\mathcal{H}\else\HilScr^pre\relax\fi} = {\ifx_B\relax\mathcal{H}\else\HilScr_B\relax\fi} \otimes {\ifx_S\relax\mathcal{H}\else\HilScr_S\relax\fi}$ of bulk and surface factors. This is indeed what happens at first in the ABCK quantization. The bulk Hilbert space ${\ifx_B\relax\mathcal{H}\else\HilScr_B\relax\fi}$ is the standard one of loop quantum gravity, spanned by spin network states whose underlying graphs may include edges that end at one of a finite set $\mathcal{P}$ of [[points]{}]{} on the horizon. Such a spin network is shown in Figure \[st.sn.tf\](b). The surface Hilbert space ${\ifx_S\relax\mathcal{H}\else\HilScr_S\relax\fi}$ is a direct limit [@abck1997] of Hilbert spaces for a quantum Chern–Simons theory, where the limit runs over such sets $\mathcal{P}$ of punctures, ordered by inclusion. The two Hilbert space factors at this stage are entirely independent of one another and, in particular, the quantum Chern–Simons connection on the horizon is totally unrelated to the geometric degrees of freedom in the bulk. We denote that connection by $X$, rather than $V$, to emphasize the absence of such a relation.
The initial Hilbert space ${\ifx^pre\relax\mathcal{H}\else\HilScr^pre\relax\fi}$ of the ABCK quantization is reduced to the true, physical Hilbert space ${\ifx^phys\relax\mathcal{H}\else\HilScr^phys\relax\fi}$ of the model in a series of steps. The first reasserts the physical relationship between the Chern–Simons connection $X$ and the bulk variables by restricting to states $| \psi \rangle$ in ${\ifx^pre\relax\mathcal{H}\else\HilScr^pre\relax\fi}$ that satisfy the [[horizon boundary condition]{}]{} $$\label{qhbc}
\biggl[ \hat I_{\mathrm{B}} \otimes \hat U[X, C] \biggr] | \psi \rangle
= \biggl[ \exp \biggl( - \frac{2 \pi {\mathrm{i}}}{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}}\, \hat\Sigma[\operatorname{int}C, r] \biggr)
\otimes \hat I_{\mathrm{S}} \biggr] | \psi \rangle.$$ Here, $\hat U[X, C]$ is the Chern–Simons holonomy operator around an arbitrary loop $C$ in $S$, while $\hat\Sigma[\operatorname{int}C, r]$ is the canonical flux of loop quantum gravity, in the gauge-fixed internal direction $r^i$, through the interior of $C$ within $S$. The quantum horizon boundary condition (\[qhbc\]) is modeled on the classical relation (\[edV\]), under the assumption that ${\mathcal{R}}$ is *constant*. The kinematical Hilbert space ${\ifx^kin\relax\mathcal{H}\else\HilScr^kin\relax\fi}$ of states $| \psi \rangle$ obeying (\[qhbc\]) is then further reduced to the physical Hilbert space ${\ifx^phys\relax\mathcal{H}\else\HilScr^phys\relax\fi}$ of the ABCK model by imposing the diffeomorphism and Hamiltonian constraints.
Area Connection and Quantization of Generic Horizons
====================================================
Our proposed application [@be2010] of the ABCK quantization to the phase space of *all* isolated horizons with total area ${\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}$ hinges on the definition $$\label{accp}
\mathring V_a := V_a + \tfrac{1}{4} ({\mathord{\ast}}{\mathrm{d}}\psi)_a
\qquad\text{with}\qquad
{\ifx\relax\relax \triangle \else \triangle_{\mathrm{\relax}} \fi}\psi := {\mathord{\ast}}{\mathrm{d}}{\mathord{\ast}}{\mathrm{d}}\psi := {\mathcal{R}}- {\ifx\relax\relax\langle\else\mathopen\relax\langle\fi {\mathcal{R}}\ifx\relax\relax\rangle\else\mathclose\relax\rangle\fi}$$ of a new $U(1)$ connection on the horizon in the classical theory. Here, ${\ifx\relax\relax \triangle \else \triangle_{\mathrm{\relax}} \fi}$ is the scalar Laplacian and ${\mathord{\ast}}$ is the Hodge dual operation, both intrinsic to $S$, while ${\ifx\relax\relax\langle\else\mathopen\relax\langle\fi {\mathcal{R}}\ifx\relax\relax\rangle\else\mathclose\relax\rangle\fi} := \oint_S {\mathcal{R}}\epsilon / {\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}$ denotes the *average* value of the curvature of $S$. We call the solution $\psi$ of this Poisson equation the [[potential]{}]{} of $S$ and the $U(1)$ connection $\mathring V_a$, the [[connection]{}]{} because $$\label{edV0}
{\mathrm{d}}\mathring V = - \tfrac{1}{4} {\ifx\relax\relax\langle\else\mathopen\relax\langle\fi {\mathcal{R}}\ifx\relax\relax\rangle\else\mathclose\relax\rangle\fi}\, \epsilon = -\tfrac{2\pi}{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}}\, {\mathpalette{\smallunderarrow@{\relax}\Leftarrowfill@}}{\Sigma}_i\, r^i
\qquad\leadsto\qquad
\exp \oint_C {\mathrm{i}}\mathring V = \exp \biggl( - \frac{2 \pi {\mathrm{i}}}{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}} \int_{\operatorname{int}C} \Sigma_i r^i \biggr)$$ for *any* geometry. That is, in close analogy to (\[qhbc\]), the *classical* holonomy of the area connection about any closed loop $C \subset S$ depends *solely* on the area of $S$ interior to $C$.
Using the area connection $\mathring V_a$ from (\[edV0\]), we show [@be2010] that the symplectic structure integral $$\label{rsymInt}
\Omega(\delta_1, \delta_2)
:= {{}^\beta\mspace{-1mu}\Omega}_{\mathrm{B}}(\delta_1, \delta_2) + {{}^\beta\mspace{-1mu}\mathring\Omega}_{\mathrm{S}}(\delta_1, \delta_2)
:= {{}^\beta\mspace{-1mu}\Omega}_{\mathrm{B}}(\delta_1, \delta_2)
+ \frac{1}{2\pi} \frac{{\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}}{4\pi G {\beta}}
\oint_{S} \delta_1 \mathring V \wedge \delta_2 \mathring V$$ is independent of $M$ throughout the entire phase space of *all* isolated horizons of area ${\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}$. Remarkably, (\[rsymInt\]) differs from (\[symInt\]) only in that the surface term involves the area connection rather than the spin connection, and yet is well-defined on a much larger phase space. Immediately after quantization, *i.e.*, at the level of ${\ifx^pre\relax\mathcal{H}\else\HilScr^pre\relax\fi}$, this classical distinction is irrelevant because the quantum Chern–Simons connection $X$ on $S$ has no physical relation to the bulk variables. When that relation is restored by the quantum horizon boundary condition (\[qhbc\]), the meaning given to $X$ is exactly analogous to that of the classical area connection exhibited in (\[edV0\]).
Quantum Horizon Shape and Entropy
=================================
We now consider the situation from a purely quantum mechanical perspectve. The mere existence of a classical connection $\mathring V_a$ with holomies (\[edV0\]) on a generic isolated horizon dispels the notion that just imposing the quantum horizon boundary condition (\[qhbc\]) makes a quantum horizon spherically symmetric. Moreover, since (\[qhbc\]) *does* determine the boundary state uniquely in terms of the bulk, we must look to the bulk to see whether symmetry is actually present.
The geometric content of bulk loop quantum gravity states is probed [@al2004] by flux operators $$\label{sfo}
\frac{1}{8 \pi G {\beta}}\, \hat\Sigma[T, f] := \frac{\hbar}{2} \sum\nolimits_{x \in T} f^i(x)
\sum\nolimits_{e \text{ at } x} \kappa(T, e)\, \hat J_i^{(x, e)},$$ where the sums are over all analytic curves $e$ extending from each point $x$ of a transversely oriented 2-surface $T$ in $M$, $f^i(x)$ is a Lie-algebra valued smearing function, $\kappa(T, e) = \pm 1$ (or $0$) according to the orientation of $e$ relative to $T$, and $\hat J_i^{(x, e)}$ is a generator of internal $SU(2)$ gauge along $e$. We distinguish between [[fluxes]{}]{}, wherein $T$ is a subset of $S$, and [[fluxes]{}]{}, wherein $T$ intersects $S$ in a curve. The two cases are illustrated in Figure \[st.sn.tf\](c).
The quantum horizon boundary condition (\[qhbc\]) involves only pullback fluxes. Classically, however, information about the intrinsic dyad induced on $S$ inheres in the transverse fluxes. Because the ABCK quantization yields no restriction on the transverse fluxes, we argue that it is properly viewed as the *consistent* quantization of an isolated horizon of fixed total area ${\relax{a}_{\scriptscriptstyle\mspace{-2mu}\Delta}}$, but *arbitrary* shape.
In the months since Loops ’11, an invitation from Abhay Ashtekar to visit him at Penn State resulted in some very helpful discussions that revealed an important subtlety in our proposed scheme. Namely, even though we argue that the final Hilbert space contains quantum states corresponding to all possible classical shapes of the horizon, and all of these enter into the statistical ensemble underlying the ABCK entropy, it is *not* these shapes that are counted in determining the entropy. Rather, what one counts are possible states of the “quantum area element,” a quantity that classically is *pure gauge.* Due to the distributional nature of quantum geometry, it is no longer pure gauge in the quantum theory.
Shortly after this work was finished, a different strategy for allowing arbitrary shapes in the calculation of quantum entropy was published by Perez and Pranzetti [@pp2010], which, in contrast to the present work, is *$SU(2)$-covariant*. Let us remark on the relative strengths and weaknesses of their work as compared to the present one. A strength of both is that quantum black holes *with arbitrary horizon shape* are described, and an advantage of Perez-Pranzetti over the present approach is that it is $SU(2)$ covariant, something necessary to obtain the correct coefficient in the next to leading order term in the entropy [@gm2004; @gour2002; @carlip2000; @enp2009; @abbdv2009]. However, a disadvantage of [@pp2010] is that one has to perform a separate quantization for each horizon shape (i.e., diffeomorphism equivalence class of horizon geometries) and then combine them, a procedure which neglects information in the Poisson algebra of observables describing the horizon shape. As in ABCK, it is also not clear that this fixing of the horizon shape is actually reflected in each elementary quantization.
We thank the organizers of the conference for the opportunity to present these results. We also are indebted to Abhay Ashtekar, who invited us to Penn State to discuss this work.
References {#references .unnumbered}
==========
[10]{} url \#1[[\#1]{}]{}urlprefix\[2\]\[\][[\#2](#2)]{} Ashtekar A, Baez J, Corichi A and Krasnov K 1998 *Phys. Rev. Lett.* **80** 904–907 Domagala M and Lewandowski J 2004 *Class. Quantum Grav.* **12** 5233–5244
Meissner K 2004 *Class. Quantum Grav.* **21** 5245–5251
Ashtekar A and Krishnan B 2004 *Living Rev. Rel.* **7** 10 Ashtekar A, Beetle C and Fairhurst S 1999 *Class. Quantum Grav.* **16** L1–L7 Ashtekar A, Corichi A and Krasnov K 1999 *Adv. Theor. Math. Phys.* **3** 419–478 Ashtekar A *et al* 2000 *Phys. Rev. Lett.* **85** 3564–3567 Beetle C and Engle J 2010 *Class. Quantum Grav.* **27** 235024
Ahtekar A and Lewandowski J 2004 *Class. Quantum Grav.* **21** R53–R152
Perez A and Pranzetti D 2011 *Entropy* **13** 744–777 Engle J, Noui K and Perez A 2010 *Phys. Rev. Lett.* **105** 031302
Agullo I, Barbero J F, Borja E, Diaz-Polo J and Villasenor E 2009 *Phys. Rev. D* **80** 084006 Ghosh A and Mitra P 2005 *Phys. Rev. D* **71** 027502 Gour G 2002 *Phys. Rev. D* **66** 104022 Carlip S 2000 *Class. Quantum Grav.* **17** 4175–4186
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---
abstract: 'Personalized interventions in social services, education, and healthcare leverage individual-level causal effect predictions in order to give the best treatment to each individual or to prioritize program interventions for the individuals most likely to benefit. While the sensitivity of these domains compels us to evaluate the fairness of such policies, we show that actually auditing their disparate impacts per standard observational metrics, such as true positive rates, is impossible since ground truths are unknown. Whether our data is experimental or observational, an individual’s actual outcome under an intervention different than that received can never be known, only predicted based on features. We prove how we can nonetheless point-identify these quantities under the additional assumption of monotone treatment response, which may be reasonable in many applications. We further provide a sensitivity analysis for this assumption by means of sharp partial-identification bounds under violations of monotonicity of varying strengths. We show how to use our results to audit personalized interventions using partially-identified ROC and xROC curves and demonstrate this in a case study of a French job training dataset.'
author:
- |
Nathan Kallus\
Cornell University\
`[email protected]`
- |
Angela Zhou\
Cornell University\
`[email protected]`
bibliography:
- 'beyondfairness.bib'
- 'sensitivity.bib'
title: 'Assessing Disparate Impacts of Personalized Interventions: Identifiability and Bounds'
---
Introduction {#sec:intro}
============
The expanding use of predictive algorithms in the public sector for risk assessment has sparked recent concern and study of fairness considerations [@propublica; @bs14; @barocas-hardt-narayanan]. One critique of the use of predictive risk assessment argues that the discussion should be reframed to instead focus on the role of *positive interventions* in distributing beneficial resources, such as directing pre-trial services to prevent recidivism, rather than in meting out pre-trial detention based on a risk prediction [@bdivz17]; or using risk assessment in child welfare services to provide families with additional childcare resources rather than to inform the allocation of harmful suspicion [@shroff2017predictive; @eubanks2018automating]. However, due to limited resources, interventions are necessarily targeted. Recent research specifically investigates the use of models that predict an intervention’s benefit in order to efficiently target their allocation, such as in developing triage tools to target homeless youth [@triage-rice-13; @kube-das-19]. Both ethics and law compel such personalized interventions to be fair and to avoid disparities in how they impact different groups defined by certain protected attributes, such as race, age, or gender.
The delivery of interventions to better target those individuals deemed most likely to respond well, even if a prediction or policy allocation rule does not have access to the protected attribute, might still result in disparate impact (with regards to social welfare) for the same reasons that these disparities occur in machine learning classification models [@chen-js18]. (See \[apx-substantive\] for an expanded discussion on our use of the term “disparate impact.”) However, in the problem of personalized interventions, the “fundamental problem of causal inference,” that outcomes are not observed for interventions not administered, poses a fundamental challenge for evaluating the fairness of any intervention allocation rule, as the true “labels” of intervention efficacy of any individual are never observed in the dataset. Metrics commonly assessed in the study of fairness in machine learning, such as group true positive and false positive rates, are therefore conditional on potential outcomes which are not observed in the data and therefore cannot be computed as in standard classification problems.
The problem of personalized policy learning has surfaced in econometrics and computer science [@manski05; @kt15], gaining renewed attention alongside recent advances in causal inference and machine learning [@athey17; @wager17; @dell2014]. In particular, [@bd12] analyze optimal treatment allocations for malaria bednets with nonparametric plug-in estimates of conditional average treatment effects, accounting for budget restrictions; [@davis-heller] use the generalized random forests method of [@wager2017estimation] to evaluate heterogeneity of causal effects in a program matching at-risk youth in Chicago with summer jobs on outcomes and crime; and [@kube-das-19] use BART [@hill2011bayesian] to analyze heterogeneity of treatment effect for allocation of homeless youth to different interventions, remarking that studying fairness considerations for algorithmically-guided interventions is necessary.
In this paper, we address the challenges of assessing the disparate impact of such personalized intervention rules in the face of unknown ground truth labels. We show that we can actually obtain point identification of common observational fairness metrics under the assumption of *monotone treatment response*. We motivate this assumption and discuss why it might be natural in settings where interventions only either help or do nothing. Recognizing nonetheless that this assumption is not actually testable, we show how to conduct sensitivity analyses for fairness metrics. In particular, we show how to obtain sharp partial identification bounds on the metrics of interest as we vary the strength of violation of the assumption. We then show to use these tools to visualize disparities using partially identified ROC and xROC curves. We illustrate all of this in a case study of personalized job training based on a dataset from a French field experiment.
Problem Setup
=============
We suppose we have data on individuals $(X,A,T,Y)$ consisting of:
- Prognostic features $X \in \mathcal{X}$, upon which interventions are personalized;
- Sensitive attribute $A\in\mathcal A$, against which disparate impact will be measured;
- Binary treatment indicator $T\in \{0,1\}$, indicating intervention exposure; and
- Binary response outcome $Y\in\{0,1\}$, indicating the benefit to the individual.
Our convention is to identify $T=1$ with an active intervention, such as job training or a homeless prevention program, and $T=0$ with lack thereof. Similarly, we assume that a positive outcome, $Y = 1$, is associated with a beneficial event for the individual, e.g., successful employment or non-recidivation. Using the Neyman-Rubin potential outcome framework [@ir15], we let $Y(0), Y(1) \in \{0,1\}$ denote the potential outcomes of each treatment. We let the observed outcome be the potential outcome of the assigned treatment, $Y=Y(T)$, encapsulating non-interference and consistency assumptions, also known as SUTVA [@rubin1980randomization]. Importantly, for any one individual, we never *simultaneously* observe $Y(0)$ and $Y(1)$. This is sometimes termed the fundamental problem of causal inference. We assume our data either came from a randomized controlled trial (the most common case) or an unconfounded observational study so that the treatment assignment is ignorable, that is, $Y(1), Y(0) \indep T \mid X,A$. When both treatment and potential outcomes are binary, we can exhaustively enumerate the four possible realizations of potential outcomes as $(Y(0),Y(1))\in\{0,1\}^2$. We call units with $(Y(0),Y(1))=(0,1)$ responders, $(Y(0),Y(1))=(1,0)$ anti-responders, and $Y(0)=Y(1)$ non-responders. Such a decomposition is also common in instrumental variable analysis [@angrist96] where the binary outcome is take-up of treatment with the analogous nomenclature of compliers, never-takers, always-takers, and defiers. In the context of talking about an actual outcome, following [@manski1997monotone], we replace this nomenclature with the notion of response rather than compliance. We remind the reader that due to the fundamental problem of causal inference, response type is *unobserved*.
We denote the conditional probabilities of each response type by $$p_{ij}=p_{ij}(X,A)=\mathbb P(Y(0)=i,Y(1)=j\mid X,A).$$ By exhaustiveness of these types, $p_{00}+p_{01}+p_{10}+p_{11}=1$. (Note $p_{ij}$ are *random variables*.)
We consider evaluating the fairness of a personalized intervention policy $Z=Z(X,A)\in\{0,1\}$, which assigns interventions based on observable features $X,A$ (potentially just $X$). Note that by definition, the intervention has zero effect on non-responders, negative effect on anti-responders, and a positive effect only on responders. Therefore, in seeking to benefit individuals with limited resources, the personalized intervention policy should seek to target only the responders. Naturally, response type is unobserved and the policy can only mete out interventions based on observables.
In classification settings, minimum-error classifiers on the efficient frontier of type-I and -II errors are given by Bayes classifiers that threshold the probability of a positive label. In personalized interventions, policies that are on the efficient frontier of social welfare (fraction of positive outcomes, $\Prb{Y(Z)=1}$) and program cost (fraction intervened on, $\Prb{Z=1}$) are given by thresholding ($Z=\indic{\tau\geq\theta}$) the *conditional average treatment effect* (CATE): $$\begin{aligned}
\tau=\tau(X,A)&=\Efb{Y(1)-Y(0)\mid X,A} =p_{01}-p_{10}\\&= \mathbb P(Y=1 \mid T=1, X, A) - \mathbb P(Y=1 \mid T=0, X, A ),\end{aligned}$$ where the latter equality follows by the assumed ignorable treatment assignment. Estimating $\tau$ from unconfounded data using flexible models has been the subject of much recent work [@wager2017estimation; @shalit-johansson-sontag-17; @hill2011bayesian].
We consider observational fairness metrics in analogy to the classification setting, where the “true label” of an individual is their *responder status*, $R = \indic{Y(1)>Y(0)}$. We define the analogous true positive rate and true negative rate for the intervention assignment $Z$, conditional on the (unobserved) events of an individual being a responder or non-responder, respectively: $$\label{eq:tprdef}\begin{aligned}
{\op{TPR}}_a&=\mathbb P(Z=1\mid A=a,Y(1)>Y(0)),\\
{\op{TNR}}_a&=\mathbb P(Z=0\mid A=a,Y(1)\leq Y(0)).
\end{aligned}$$
Interpreting Disparities for Personalized Interventions
-------------------------------------------------------
The use of predictive models to deliver interventions can induce disparate impact if responding (respectively, non-responding) individuals of different groups receive the intervention at disproportionate rates under the treatment policy. This can occur even with efficient policies that threshold the true CATE $\tau$ and can arise from the disparate predictiveness of $X,A$ of response type (i.e., how far $p_{ij}$ are from $0$ and $1$). This is problematic because the choice of features $X$ is usually made by the intervening agent (e.g., government agency, etc.).
We discuss one possible interpretation of TPR or TNR disparities in this setting when the intervention is the bestowal of a benefit, like access to job training or case management. From the point of view of the intervening agent, there are specific program goals, such as employment of the target individual within 6 months. Therefore, false positives are costly due to program cost and false negatives are missed opportunities. But outcomes also affect the individual’s utility. Discrepancies in TPR across values of $A$ are of concern since they suggest that the needs of those who could actually benefit from intervention (responders) in one group are not being met at the same rates as in other groups. Arguably, for benefit-bestowing interventions, TPR discrepancies are of greater concern. Nonetheless, from the point of view of the individual, the intervention may always grant some positive resource (e.g., from the point of view of well-being), regardless of responder status, since it corresponds to access to a good (and the individual can gain other benefits from job training that may not necessarily align with the intervener’s program goals, such as employment in 1 year or personal enrichment). If so, then TNR discrepancies across values of $A$ imply a “disparate benefit of the doubt” such that the policy disparately over-benefits one group over another using the limited public resource without the cover of advancing the public program’s goal, which may raise fairness and envy concerns, especially since this “waste” is at the cost of more slots for responders.
Beyond assessing disparities in TPR and TNR for one fixed policy, we will also use our ability to assess these over varying CATE thresholds in order to compute xAUC metrics [@kz-19] in \[sec-group-disparities\]. These give the disparity between the probabilities that a non-responder from group $a$ is ranked above a responder from group $b$ and vice-versa. Thus, they measure the disproportionate access one group gets relative to another in *any* allocation of resources that is non-decreasing in CATE.
We emphasize that the identification arguments and bounds that we present on fairness metrics are primarily intended to facilitate the *assessment* of disparities, which may require further inquiry as to their morality and legality, not necessarily to promote statistical parity via adjustments such as group-specific thresholds, though that is also possible using our tools. We defer a more detailed discussion to \[sec-discussion\] and re-emphasize that assessing the distribution of outcome-conditional model errors are of central importance both in machine learning [@hardt2016equality; @barocas-hardt-narayanan; @mitchell-potash-barocas] and in the economic efficiency of targeting resources [@pmt-2016; @pmt-2016-brown; @berger].
Related Work
============
[@madras-creager-pz19] consider estimating joint treatment effects of race and treatment under a deep latent variable model to reconstruct unobserved confounding. For evaluating fairness of policies derived from estimated effects, they consider the gap in population accuracy $\mathrm{Acc}_a = \Prb{Z = Z^*\mid A=a}$, where $Z^*= \mathbb{I}[\tau(X) > 0]$ is the (identifiable) optimal policy. In contrast, we highlight the unfairness of even optimal policies and focus on outcome-conditional error rates (TPR, TNR), where the non-identifiability of responder status introduces challenges regarding identifiability.
The issue of model evaluation under the censoring problem of selective labels has been discussed in situations such as pretrial detention, where detention censors outcomes [@lkllm17; @kz18-2]. Sensitivity analysis is used in [@jsfg18] to account for possible unmeasured confounders. The distinction is that we focus on the targeted delivery of interventions with unknown (but estimated) causal effects, rather than considering classifications that induce one-sided censoring but have definitionally known effects.
Our emphasis is distinct from other work discussing fairness and causality that uses graphical causal models to decompose predictive models along causal pathways and assessing the normative validity of path-specific effects [@klrs17; @hardtscholkopf-disc17], such as the effect of [probabilistic]{} hypothetical interventions on race variables or other potentially immutable protected attributes. When discussing treatments, we here consider interventions corresponding to allocation of concrete resources (e.g., give job training), which are in fact physically manipulable by an intervening agent. The correlation of the intervention’s *conditional average* treatment effects by, say, race and its implications for downstream resource allocation are our primary concern. There is extensive literature on partial identification in econometrics, e.g. [@manski2003partial]. In contrast to previous work that analyzes partial identification of average treatment effects when data is confounded and using monotonicity to improve precision [@manski2003partial; @balke1997bounds; @beresteanu2012partial], we focus on unconfounded (e.g., RCT) data and achieve full identification by assuming monotonicity and consider sensitivity analysis bounds for *nonlinear* functionals of partially identified sets, namely, true positive and false positive rates.
Identifiability of Disparate Impact Metrics {#sec-fairness-id}
===========================================
Since the definitions of the disparate impact metrics in \[eq:tprdef\] are conditioned on an unobserved event, such as the response event $Y(1) > Y(0)$, they actually cannot be identified from the data, even under ignorable treatment. That is, the values of ${\op{TPR}}_a,{\op{TNR}}_a$ can vary even when the joint distribution of $(X,A,T,Y)$ remains the same, meaning the data we see cannot possibly tell us about the specific value of ${\op{TPR}}_a,{\op{TNR}}_a$.
\[lemma:unidentifiable\] ${\op{TPR}}_a,{\op{TNR}}_a$ (or discrepancies therein over groups) are generally not identifiable.
Essentially, \[lemma:unidentifiable\] follows because the data only identifies the marginals $p_{10}+p_{11},\,p_{01}+p_{11}$ while ${\op{TPR}}_a,{\op{TNR}}_a$ depend on the joint via $p_{01}$, which can vary even while marginals are fixed. Since this can vary independently across values of $A$, discrepancies are not identifiable either.
Identification under Monotonicity
---------------------------------
We next show identifiability if we impose the additional assumption of monotone treatment response.
\[asm:monotone\] $Y(1) \geq Y(0)$. (Equivalently, $p_{10}=0$.)
says that anti-responders do not exist. In other words, the treatment either does nothing (e.g., an individual would have gotten a job or not gotten a job, regardless of receiving job training) or it benefits the individual (would get a job if and only if receive job training), but it never harms the individual. This assumption is reasonable for positive interventions. As [@k19-monotonicity] points out, policy learning in this setting is equivalent to the binary classification problem of predicting responder status.
\[prop-id\] Under \[asm:monotone\], $$\label{eq:id}
\begin{aligned}
{\op{TPR}}_a&=\frac{ \Eb{\tau \mid A=a, Z=1} \Prb{Z=1\mid A=a} }{\Eb{\tau \mid A=a} }, \\
{\op{TNR}}_a&=\frac{\Eb{ (1 - \tau) \mid A=a, Z=0} \Prb{Z=0\mid A=a} }{ \Eb{(1-\tau) \mid A=a} }.
\end{aligned}$$
Since the quantities on the right hand sides in \[eq:id\] are in terms of identified quantities (functions of the distribution of $(X,A,T,Y)$), this proves identifiability. Given a sample and an estimate of $\tau$, it also provides a simple recipe for estimation by replacing each average or probability by a sample version, since both $A$ and $Z$ are discrete.
Thus, \[prop-id\] provides a novel means of assessing disparate impact of personalized interventions under monotone response. This is relevant because monotonicity is a defensible assumption in the case of many interventions that bestow an additional benefit, good, or resource, such as the ones mentioned in \[sec:intro\]. Nonetheless, the validity of \[asm:monotone\] is itself not identifiable. Therefore, should it fail even slightly, it is not immediately clear whether these disparity estimates can be relied upon. We therefore next study a sensitivity analysis by means of constructing partial identification bounds for ${\op{TPR}}_a,{\op{TNR}}_a$.
Partial Identification Bounds for SensitivityAnalysis {#sec-pi-bounds}
=====================================================
We next study the partial identification of disparate impact metrics when \[asm:monotone\] fails, i.e., $p_{10}\neq0$. We first state a more general version of \[prop-id\]. For any $\eta=\eta(X,A)$, let $$\begin{aligned}
\rho^{{\op{TPR}}}_a(\eta) &\coloneqq \frac{ \Eb{ \tau + \eta \mid {A=a, Z=1} } \Prb{Z=1\mid A=a} }{ \Eb{\tau+ \eta \mid A=a} },\\
\rho^{{\op{TNR}}}_a(\eta) &\coloneqq \frac{ \Eb{ 1-(\tau + \eta) \mid {A=a,Z=0} } \Prb{Z=0\mid A=a} }{ \Eb{1-(\tau + \eta) \mid A=a} }.\end{aligned}$$
\[prop-id2\] ${\op{TPR}}_a=\rho^{{\op{TPR}}}_a(p_{10}),\,{\op{TNR}}_a=\rho^{{\op{TNR}}}_a(p_{10})$.
Since the anti-responder probability $p_{10}$ is unknown, we cannot use \[prop-id2\] to identify ${\op{TPR}}_a, {\op{TNR}}_a$. We instead use \[prop-id2\] to compute bounds on them by restricting $p_{10}$ to be in an uncertainty set. Formally, given an uncertainty set $\mathcal U$ for $p_{10}$ (i.e., a set of functions of $x,a$), we define the simultaneous identification region of the TPR and TNR for all groups $a\in\mathcal A$ as: $$\Theta=\braces{\prns{\rho^{{\op{TPR}}}_a(\eta),\rho^{{\op{TNR}}}_a(\eta)}_{a\in\mathcal A}\;:\:\eta\in\mathcal U}\subseteq\R{2\times\abs{\mathcal A}}.$$ For brevity, we will let $\rho_a(\eta)=\prns{\rho^{{\op{TPR}}}_a(\eta),\rho^{{\op{TNR}}}_a(\eta)}$ and $\rho(\eta)=(\rho_a(\eta))_{a\in\mathcal A}$.
The set $\Theta$ describes all possible simultaneous values of the group-conditional true positive and true negative rates. As long as $\forall\eta\in\mathcal U$ we have $0\leq\eta(X,A)\leq \min\prns{\Prb{Y=1\mid T=0,X,A},\Prb{Y=0\mid T=1,X,A}}$ (which is identified from the data) by \[prop-id2\] this set is necessarily sharp [@manski2003partial] given only the restriction that $p_{10}\in\mathcal U$. (In particular, this bound on $\eta$ can be achieved by just point-wise clipping $\mathcal U$ with this identifiable bound as necessary.) That is, given a joint on $(X,A,T,Y)$, on the one hand, every $\rho\in\Theta$ is realized by some full joint distribution on $(X,A,T,Y(0),Y(1))$ with $p_{10}\in\mathcal U$, and on the other hand, every such joint gives rise to a $\rho\in\Theta$. In other words, $\Theta$ is an *exact* characterization of the in-fact possible simultaneous values of the group-conditional TPRs and TNRs.
Therefore, if, for example, we are interested in the minimal and maximal possible values for the true (unknown) TPR discrepancy between groups $a$ and $b$, we should seek to compute $
\inf_{\rho\in\Theta}\
\rho^{{\op{TPR}}}_a-\rho^{{\op{TPR}}}_b$ and $\sup_{\rho\in\Theta}\
\rho^{{\op{TPR}}}_a-\rho^{{\op{TPR}}}_b.
$ More generally, for any $\mu\in\R{2\times\abs{\mathcal A}}$, we may wish to compute $$\label{eq:supportfn}h_{ \Theta } ({\mu}) \coloneqq
\sup_{\rho \in \Theta} {\mu}^\top \rho .$$ Note that this, for example, covers the above example since for any $\mu$ we can also take $-\mu$. The function $h_{ \Theta }$ is known as the *support function* of ${ \Theta }$ [@rockafellar2015convex]. Not only does the support function provide the maximal and minimal contrasts in a set, it also exactly characterizes its convex hull. That is, $\op{Conv}\prns{ \Theta }=\braces{\rho:\mu^\top\rho\leq h_{\Theta}(\mu)\ \forall\mu}$. So computing $h_{ \Theta }$ allows us to compute $\op{Conv}\prns{ \Theta }$.
Our next result gives an explicit program to compute the support function when $\mathcal U$ has a product form of within-group uncertainty sets: $$\label{eq:productU}\mathcal U=\braces{\eta:\eta(\;\cdot\;,a)\in\mathcal{U}_{a}\ \forall a\in\mathcal A},$$ which leads to $\Theta=\prod_{a\in\mathcal A}\Theta_a$ where $\Theta_a=\braces{\rho_a(\eta_a):\eta_a\in\mathcal U_a}$.
\[prop-supp-fn-rep\] Let $r_a^z \coloneqq \Prb{Z=z\mid A=a}$ and $\tau_a^z\coloneqq\Eb{\tau\mid A=a,Z=z}$. Suppose $\mathcal U$ is as in . Then \[eq:supportfn\] can be reformulated as: $$\begin{aligned}
h_{\Theta} ({\mu})=~&\omit{\rlap{$\sum_{a\in\mathcal A}h_{\Theta_a} ({\mu_a})$}}\\
h_{\Theta_a} ({\mu_a})=~&\sup_{\omega_a, t_a} && \mu_a^{{\op{TPR}}} r_a^1 \; \prns{t_a\tau_a^1+\Eb{\omega_a(X)\mid {A =a, Z=1}}}
\\&&&+ \frac{\mu_a^{{\op{TNR}}} r_a^0 }{{t_a}-1} (t_a \;
\prns{1-\tau_a^0}
+ \Eb{\omega_a(X) \mid {A=a, Z=0}} )
\\
&\quad\mathrm{s.t. } &&
\omega_a(\cdot)\in t_a\;\mathcal U_a,
~~ t_a\prns{r_a^0\tau_a^0+r_a^1\tau_a^1}+\Eb{\omega_a \mid A=a}=1. \end{aligned}$$
For a fixed value of $t_a$, the above program is a linear program, given that $\mathcal U_a$ is linearly representable. Therefore a solution may be found by grid search on the univariate $t_a$. Moreover, if $\mu_a^{{\op{TPR}}}=0$ or $\mu_a^{{\op{TNR}}}=0$, the above remains a linear program even with $t_a$ as a variable [@charnes1962programming]. With this, we are able to express group-level disparities through assessing the support function at specific contrast vectors $\mu$.
Partial Identification under Relaxed Monotone Treatment Response
----------------------------------------------------------------
We next consider the implications of the above for the following relaxation of the monotone treatment response assumption:
\[asm:monotone-B\] $p_{10}\leq B$.
Note that \[asm:monotone-B\] with $B=0$ recovers \[asm:monotone\] and \[asm:monotone-B\] with $B=1$ is a vacuous assumption. In between these two extremes we can consider milder or stronger violations of monotone response and the partial identification bounds they corresponds to. This provides us with a means of sensitivity analysis of the disparities we measure, recognizing that monotone response may not hold exactly and that disparities may not be exactly identifiable. For the rest of the paper, we focus solely on partial identification under \[asm:monotone-B\]. Note that \[asm:monotone-B\] corresponds exactly to the uncertainty set $\mathcal U_B=\braces{\eta:0\leq\eta(X,A)\leq \min\prns{B,\Prb{Y=1\mid T=0,X,A},\Prb{Y=0\mid T=1,X,A}}}$.We define $\Theta_B=\prod_{a\in\mathcal A}\Theta_{B,a}$ to be the corresponding identification region.
Under \[asm:monotone-B\], our bounds take on a particularly simple form. Let $$\begin{aligned}
\mathcal B_a^z(B)=\E\bigl[\min\bigl(B,\,&\Prb{Y=1\mid T=0,X,A},\\&\Prb{Y=0\mid T=1,X,A}\bigr)\mid A=a,Z=z\bigr]\end{aligned}$$ and define $$\begin{aligned}
\overline{\rho}^{{\op{TPR}}}_a(B) &= \frac{ (\tau_a^1+\mathcal B_a^1(B)) r_a^1 }{
\tau_a^0 r_a^0+(\tau_a^1+\mathcal B_a^1(B)) r_a^1},\\
\underline{\rho}^{{\op{TPR}}}_a(B) &=\frac{ \tau_a^1 r_a^1 }{
(\tau_a^0+\mathcal B_a^0(B)) r_a^0+\tau_a^1 r_a^1},\\
\overline{\rho}^{{\op{TNR}}}_a(B) &=
\frac{ (1-\tau_a^0) r_a^0 }{
(1-\tau_a^0) r_a^0+(1-\tau_a^1-\mathcal B_a^1(B)) r_a^1}
,\\
\underline{\rho}^{{\op{TNR}}}_a(B) &=
\frac{ (1-\tau_a^0-\mathcal B_a^0(B)) r_a^0 }{
(1-\tau_a^0-\mathcal B_a^0(B)) r_a^0+(1-\tau_a^1) r_a^1}
. \end{aligned}$$
\[prop-flp-1class\] Suppose \[asm:monotone-B\] holds. Then $[\underline{\rho}^{{\op{TPR}}}_a(B),\overline{\rho}^{{\op{TPR}}}_a(B)]$ and $[\underline{\rho}^{{\op{TNR}}}_a(B),\overline{\rho}^{{\op{TNR}}}_a(B)]$ are the sharp identification intervals for ${\op{TPR}}_a$ and ${\op{TNR}}_a$, respectively. Moreover, $(\underline{\rho}^{{\op{TPR}}}_a(B),\underline{\rho}^{{\op{TNR}}}_a(B))\in\Theta_{B,a}$ and $(\overline{\rho}^{{\op{TPR}}}_a(B),\overline{\rho}^{{\op{TNR}}}_a(B))\in\Theta_{B,a}$, i.e., the two extremes are simultaneously achievable.
Partial Identification of Group Disparities and ROC and xROC Curves {#sec-group-disparities}
===================================================================
We discuss diagnostics to summarize possible impact disparities across a [range]{} of possible policies.
#### TPR and TNR disparity.
Discrepancies in model errors (TPR or TNR) are of interest when auditing classification performance on different groups with a given, fixed policy $Z$. Under \[asm:monotone\], they are identified by \[prop-id\]. Under violations of \[asm:monotone\], we can consider their partial identification bounds. If the *minimal* disparity remains nonzero, that provides strong evidence of disparity. Similarly, if the *maximal* disparity is large, a responsible decision maker should be concerned about the possibility of a disparity.
Under \[asm:monotone-B\], \[prop-flp-1class\] provides that the sharp identification intervals of ${\op{TPR}}_a-{\op{TPR}}_b$ and ${\op{TNR}}_a-{\op{TNR}}_b$ are, respectively, given by $$\label{eq-tpr-tnr-disp}\begin{aligned}
&[\underline \rho_a^{{\op{TPR}}}(B)-\overline \rho_b^{{\op{TPR}}}(B),\ \overline \rho_a^{{\op{TPR}}}(B)-\underline \rho_b^{{\op{TPR}}}(B)],\\&[\underline \rho_a^{{\op{TNR}}}(B)-\overline \rho_b^{{\op{TNR}}}(B),\ \overline \rho_a^{{\op{TNR}}}(B)-\underline \rho_b^{{\op{TNR}}}(B)].
\end{aligned}$$ Given effect scores $\tau$, we can then use this to plot *disparity curves* by plotting the endpoints of \[eq-tpr-tnr-disp\] for policies $Z=\mathbb{I}[\tau \geq \theta]$ for varying thresholds $\theta$.
#### Robust ROC Curves
We first define the analogous *group-conditional* *ROC curve* corresponding to a CATE function $\tau$. These are the parametric curves traced out by the pairs $(1-{\op{TNR}}_a,{\op{TPR}}_a)$ of policies that threshold the CATE for varying thresholds. To make explicit that we are now computing metrics for different policies, we use the notation ${\rho}(\eta; \tau \geq \theta)$ to refer to the metrics of the policy $Z=\indic{\tau\geq\theta}$. Under \[asm:monotone\], \[prop-id\] provides point identification of the group-conditional ROC curve: $$\begin{aligned}
&{\op{ROC}}_a(\tau) \coloneqq \{ (1- {\rho}_a^{{\op{TNR}}}(0; \tau \geq \theta),{\rho}_a^{{\op{TPR}}}(0; \tau \geq \theta) ) : \theta \in \Rl \}\end{aligned}$$ When \[asm:monotone\] fails, we cannot point identify ${\op{TPR}}_a,{\op{TNR}}_a$ and correspondingly we cannot identify ${\op{ROC}}_a(\tau)$. We instead define the *robust ROC* curve as the union of all partially identified ROC curves. Specifically: $$\begin{aligned}
&{\Theta^{\op{ROC}}_a}(\tau) \coloneqq \{ (1- {\rho}^{{\op{TNR}}}_a(\eta_a; \tau \geq \theta),{\rho}^{{\op{TPR}}}_a(\eta_a; \tau \geq \theta) ) \colon \theta \in \Rl, \eta_a \in \mathcal{U}_a \}.\end{aligned}$$ Plotted, this set provides a visual representation of the region that the true ROC curve can lie in. We next prove that under \[asm:monotone-B\], we can easily compute this set as the area between two curves.
Let $\mathcal{U}=\mathcal{U}_{B}$. Then ${\Theta^{\op{ROC}}_a}(\tau)$ is given as the area between the two parametric curves $\underline{\op{ROC}}_a(\tau)\coloneqq\{ (1- {\underline\rho}_a^{{\op{TNR}}}(B; \tau \geq \theta),{\underline\rho}_a^{{\op{TPR}}}(B; \tau \geq \theta) ) \colon \theta \in \Rl \}$ and $\overline{\op{ROC}}_a(\tau)\coloneqq\{ (1- {\overline\rho}_a^{{\op{TNR}}}(B; \tau \geq \theta),{\overline\rho}_a^{{\op{TPR}}}(B; \tau \geq \theta) ) \colon \theta \in \Rl \}$.
This follows because the extremes are simultaneously achievable as noted in \[prop-flp-1class\]. We highlight, however, that the lower (resp., upper) ROC curve may not be simultaneously realizable as an ROC curve of any single policy.
#### Robust xROC Curves
Comparison of group-conditional ROC curves may not necessarily show impact disparities as, even in standard classification settings ROC curves can overlap despite disparate impacts [@kz-19; @hardt2016equality]. At the same time, comparing disparities for fixed policies $Z$ with fixed thresholds may not accurately capture the impact of using $\tau$ for rankings. [@kz-19] develop the $\op{xAUC}$ metric for assessing the *bipartite ranking* quality of risk scores, as well as the analogous notion of a $\op{xROC}$ curve which parametrically plots the TPR of one group vs. the FPR of *another group*, at any fixed threshold. This is relevant if effect scores $\tau$ are used for downstream decisions by different facilities with different budget constraints or if the score is intended to be used by a “human-in-the-loop” exercising additional judgment, e.g., individual caseworkers as in the encouragement design of [@behncke-evaluation]. Under \[asm:monotone\], we can point identify ${\op{TPR}}_a,{\op{TNR}}_a$, so, following [@kz-19], we can define the point-identified xROC curve as $$\begin{aligned}
&\op{xROC}_{a,b}(\tau)= \{ (1- {\rho}_b^{{\op{TNR}}}(0; \tau \geq \theta),{\rho}_a^{{\op{TPR}}}(0; \tau \geq \theta) ) :\theta\in\Rl\}.
\end{aligned}$$ Without \[asm:monotone\], we analogously define the *robust xROC* curve as the union of all partially identified xROC curves: $$\begin{aligned}
&\Theta^{\op{xROC}}_{a,b}(\tau)= \{ (1- {\rho}_b^{{\op{TNR}}}(\eta_a; \tau \geq \theta),{\rho}_a^{{\op{TPR}}}(\eta_a; \tau \geq \theta) ) :\theta\in\Rl,\eta_a\in\mathcal U_a\}.
\end{aligned}$$
Let $\mathcal{U}=\mathcal{U}_{B}$. Then ${\Theta^{\op{xROC}}_{a,b}}(\tau)$ is given as the area between the two parametric curves $\underline{\op{xROC}}_{a,b}(\tau)\coloneqq\{ (1- {\underline\rho}_b^{{\op{TNR}}}(B; \tau \geq \theta),{\underline\rho}_a^{{\op{TPR}}}(B; \tau \geq \theta) ) \colon \theta \in \Rl \}$ and $\overline{\op{xROC}}_{a,b}(\tau)\coloneqq\{ (1- {\overline\rho}_b^{{\op{TNR}}}(B; \tau \geq \theta),{\overline\rho}_a^{{\op{TPR}}}(B; \tau \geq \theta) ) \colon \theta \in \Rl \}$.
This follows because $\mathcal U_B$ takes the form of a product set over $a\in\mathcal A$.
Case Study: Personalized Job Training(Behaghel et al.) {#sec-behaghel}
======================================================
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We consider a case study from a three-armed large randomized controlled trial that randomly assigned job-seekers in France to a control-group, a job training program managed by a public vendor, and an out-sourced program managed by a private vendor [@behaghel-2014]. While the original experiment was interested in the design of contracts for program service delivery, we consider a task of heterogeneous causal effect estimation, motivated by interest in personalizing different types of counseling or active labor market programs that would be beneficial for the individual. Recent work in policy learning has also considered personalized job training assignment [@wager17; @kt15] and suggested excluding sensitive attributes from the input to the decision rule for fairness considerations, but without consideration of fairness in the causal effect estimation itself and how significant impact disparities may still remain after excising sensitive attributes because of it.
We focus on the public program vs. control arm, which enrolled about 7950 participants in total, with $n_1 = 3385$ participants in the public program. The treatment arm, $T=1$, corresponds to assignment to the public program. The original analysis suggests a small but statistically significant positive treatment effect of the public program, with an ATE of $0.023$. We omit further details on the data processing to \[apx-crepon-data\]. We consider the group indicators: *nationality* ($0,1$ denoting French nationals vs. non-French, respectively), *gender* (denoting woman vs. non-woman), and *age* (below the age of 26 vs. above). (Figures for gender appear in \[apx-crepon-data\].)
In \[fig-disparity-curves\], we plot the identified “disparity curves” of \[eq-tpr-tnr-disp\] corresponding to the maximal and minimal sensitivity bounds on TPR and TNR disparity between groups. Levels of shading correspond to different values of $B$, with color legend at right. We learn $\tau$ by the Generalized Random Forests method of [@wager2017estimation; @athey2019generalized] and use sample splitting, learning $\tau$ on half the data and using our methods to assess bounds on $\rho^{{\op{TPR}}}, \rho^{{\op{TNR}}}$ and other quantities with out-of-sample estimates on the other half of the data. We bootstrap over 50 sampled splits and average disparity curves to reduce sample uncertainty.
In general, the small probability of being a responder leads to increased sensitivity of TPR estimates (wide identification bands). The curves and sensitivity bounds suggest that with respect to nationality and gender, there is small or no disparity in true positive rates but the true negative rates for nationality, gender, and age may differ significantly across groups, such that non-women would have a higher chance of being bestowed job-training benefits when they are in fact not responders. However, TPR disparity by age appears to hold with as much as -0.1 difference, with older actually-responding individuals being less likely to be given job training than younger individuals. Overall, this suggests that differences in heterogeneous treatment effects across age categories could lead to significant adverse impact on older individuals.
This is similarly reflected in the robust ROC, xROC curves (\[fig-xroc\]). Despite possibly small differences in ROCs, the xROCs indicate strong disparities: the sensitivity analysis suggests that the likelihood of ranking a non-responding young individual above a responding old individual (xAUC [@kz-19]) is clearly larger than the symmetric error, meaning that older individuals who benefit from the treatment may be disproportionately shut out of it as seats are instead given to non-responding younger individuals.
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Discussion and Conclusion {#sec-discussion}
=========================
We presented identification results and bounds for assessing disparate model errors of causal-effect maximizing treatment policies, which can lead disparities in access to those who stand to benefit from treatment across groups. Whether this is “unfair” would naturally rely on one’s normative assumptions. One such is “claims across outcomes,” that individuals have a claim to the public intervention if they stand to benefit, which can be understood within [@adler]’s axiomatic justification of fair distribution. There may also be other justice-based considerations, e.g. minimax fairness. We discuss this more extensively in \[apx-substantive\]. With the new ability to *assess* disparities using our results, a second natural question is whether these disparities warrant adjustment, which is easy to do given our tools combined with the approach of [@hardt2016equality]. This question again is dependent both on one’s viewpoint and ultimately on the problem context, and we discuss it further in \[apx-substantive\]. Regardless of normative viewpoints, auditing allocative disparities that would arise from the implementation of a personalized rule must be a crucial step of a responsible and convincing program evaluation. We presented fundamental identification limits to such assessments but provided sensitivity analyses that can support reliable auditing.
Omitted proofs
==============
To prove this we exhibit a simple example satisfying ignorability where both ${\op{TPR}}_a,{\op{TNR}}_a$ and differences therein varies while the joint distribution of $(X,A,T,Y)$ does not. Let $\mathcal X=\{0,1\}$, $Z=\indic{X=1}$, $\Prb{T=t,X=x\mid A=a}=\frac14$, $\Prb{A=a}=1/\abs{\mathcal A}$. To specify a joint distribution of $(X,A,T,Z,Y(1),Y(0))$ that satisfies ignorable treatment, it only remains to specify $p_{ij}$.
Note that in this case $${\op{TPR}}_a=\frac{p_{01}(1,a)}{p_{01}(0,a)+p_{01}(1,a)},~
{\op{TPR}}_a=\frac{1-p_{01}(0,a)}{2-p_{01}(0,a)-p_{01}(1,a)}.$$
The result follows by noting that where the corresponding joint distribution of $(X,A,T,Z,Y)$ is completely specified by $p_{01}+p_{11},\ p_{10}+p_{11}$, while $p_{01}$ could vary as long as these sums are neither 0 nor 1. Since we can vary this independently across values of $A$, differences are not identifiable either.
$$\begin{aligned}
& \mathbb P(Z=1\mid A=a,Y(1)>Y(0)) \\
=& \frac{\mathbb P(Y(1)>Y(0)\mid A =a, Z=1) \mathbb P(Z=1\mid A=a) }{ \mathbb P(Y(1)>Y(0)\mid A=a) } \\
=& \frac{ \mathbb E [ \E[ Y(1)=1 \mid \substack{T=1\\A =a,X=x} ] - \E[ Y(0)=1 \mid \substack{T=0\\A =a,X=x }]\mid \substack{Z=1\\A=a} ] \mathbb P(Z=1\mid A=a) }{\mathbb E [ \mathbb P(Y(1)=1 \mid \substack{T=1\\A =a,X=x}) - \mathbb P(Y(0)=1 \mid \substack{T=0\\A =a,X=x })\mid A=a]} \; \\
=& \frac{\mathbb E [ \tau(X,A) \mid \substack{Z=1 \\A =a}] \mathbb P(Z=1\mid A=a) }{ \E[ \tau(X,A) \mid A =a ] } \; \end{aligned}$$
where the first equality holds by Bayes’ rule, the second by iterating expectations on $X$ and \[asm:monotone\], and the third by unconfoundedness and consistency of potential outcomes. The proof for identification of ${\op{TNR}}$ is identical for the quantity $\mathbb P(Z=0\mid A=a,Y(1)\leq Y(0)) $.
Recalling that CATE identifies, under violations of \[asm:monotone\] $$\tau=\Efb{Y(1)-Y(0)\mid X,A} =p_{01}-p_{10},$$ $$\begin{aligned}
=\;& \frac{\mathbb E [ \tau + \eta \mid \substack{Z=1 \\A =a}] \mathbb P(Z=1\mid A=a) }{ \E[ \tau + \eta \mid A =a ] } =\frac{ (p_{01}-p_{10} + p_{10}) \mathbb P(Z=1\mid A=a) }{\mathbb E [ (p_{01}-p_{10} + p_{10})\mid A=a]} \;\\
=\;& \mathbb P(Z=1\mid A=a,Y(1)>Y(0))\end{aligned}$$
\[proof-supp-fn-rep\] The support function evaluated at $\mu$ is: $$\begin{aligned}
\max_{\eta} & \sum_{a \in \mathcal{A}} \mu^{{\op{TPR}}}_a \frac{ \Eb{ \tau + \eta \mid {A=a, Z=1} } r_a^1 }{ \Eb{\tau+ \eta \mid A=a} } + \mu^{{\op{TNR}}}_a \frac{ \Eb{ 1-(\tau + \eta) \mid {A=a,Z=0} } r^a_0 }{ \Eb{1-(\tau + \eta) \mid A=a} } \\
\mathrm{s.t. } \;\;& 0 \leq \eta(x,a) \leq \mathcal B_a^z(B), \quad \forall x \in \mathcal{X}, \; \forall a \in \mathcal{A}
\end{aligned}$$ We apply the Charnes-Cooper transformation [@charnes1962programming]with the bijection $t_a = \frac{1}{\Eb{\tau + \eta \mid A =a } },\; \omega_a = \eta t_a$. The denominator of the second term under this bijection is equivalently $$\Eb{1-(\tau + \eta) \mid A=a} = 1 - \frac{1}{t_a}$$ such that we can rewrite the second term as $$\begin{aligned}
&{\mu_a^{{\op{TNR}}} r_a^0 } \left( \frac{1}{1 - \nicefrac{1}{t_a}} \Eb{ 1-\tau \mid \substack{A=a\\ Z=0} } + \frac{\nicefrac{1}{t_a}}{1 - \nicefrac{1}{t_a}} \Eb{{\omega_a} \mid \substack{A=a\\ Z=0}} \right) \\&= \frac{\mu_a^{{\op{TNR}}} r_a^0 }{{t_a}-1} (t \;\Eb{ 1-\tau \mid \substack{A=a\\ Z=0} } + \Eb{\omega_a \mid \substack{A=a\\ Z=0}} )
\end{aligned}$$ and the objective function overall as: $$\begin{aligned}
\max_{\eta} & \sum_{a \in \mathcal{A}} (\mu^{{\op{TPR}}}_a r_a^1) (t_a \tau_a^1 +\Eb{\omega_a \mid \substack{A=a\\ Z=1}} ) + \frac{\mu_a^{{\op{TNR}}} r_a^0 }{{t_a}-1} (t_a \;(1-\tau_a^0)+ \Eb{\omega_a \mid \substack{A=a\\ Z=0}} )
\end{aligned}$$
The new constraint set (including the constraint yielding the definition of $t_a$) is: $$\begin{aligned}
\mathcal{U} = \{ \Eb{\tau t_a + \omega_a \mid A =a } = 1, \;\;
\omega_a(x,a) \leq t_a \mathcal B_a^z(B), \quad \forall x \in \mathcal{X}, \; \forall a \in \mathcal{A} \} \end{aligned}$$
\[proof-flp\] We first consider the case of maximizing or minimizing the TPR. We leverage the invariance in the objective function under the surjection on $\eta(x,a)$ to its marginal expectation over a $Z=z, A=a$ partition. $$w(x,a) = \begin{cases}
\Eb{ \eta \mid Z=1, A=a } & \text{ if }Z = 1 \\
\Eb{ \eta \mid Z=0, A=a } & \text{ if } Z= 0
\end{cases}$$ Therefore we can reparametrize the program as optimizing over coefficients $x,y$ of the optimal solution, $w(x,y) = x \mathbb{I}[Z = 0] + y \mathbb{I}[Z = 1] $. Define the fractional objective
$$\begin{aligned}
g(\alpha,\beta) &= \frac{ \Eb{ \tau + x \mathbb{I}[Z = 0] +y \mathbb{I}[Z = 1] \mid {A=a, Z=1} } \Prb{Z=1\mid A=a} }{ \Eb{\tau+ x \mathbb{I}[Z = 0] + y \mathbb{I}[Z = 1] \mid A=a} } \\
&= \frac{ (\Eb{ \tau \mid {A=a, Z=1} }+y) r_a^1 }{ \Eb{\tau \mid A=a} + x r_a^0 + yr_a^1 } \end{aligned}$$
First note that without loss of generality that when maximizing, we can set $x = 0$ since this decreases the objective regardless of the value of $y$. We can consider the constrained problem $\max_{y \leq B} h(y)$ where $h(y) = g(0, y)$. Then we have the first and second derivatives, $$\begin{aligned}
\frac{\partial h }{\partial y} &= \frac{r_a^1 (\E[\tau\mid A=a] - \E[\tau \mid Z=1, A=a] }{(y r_a^1 +\ \E[\tau \mid A=a] )^2},\\\frac{\partial^2 h }{\partial \beta^2} &= \frac{(r_a^1)^2 ( \E[\tau \mid A=a] - \E[\tau \mid Z=1,A=a] ) }{(y r_a^1 +\ \E[\tau \mid A=a] )^3}\end{aligned}$$
By inspection, since $y\geq 0$ we have that $\frac{\partial^2 h }{\partial y^2} \geq 0$ so the function is convex. So when maximizing $h$ on the constraints for $y$, it attains optimal value at the boundary (since $h$ is increasing). When minimizing, note that the derivative is not vanishing anywhere on the constraint set so it suffices to check the endpoints, where the minimum is achieved at $\beta = 0$.
We now consider the case of minimizing or maximizing the TNR.
Now consider a generic $ f(x) = \frac{a - b x}{c - bx - dy}$ which represents the TNR sensitivity bound with $\omega = x \mathbb{I}[Z=0] + y \mathbb{I}[Z=1] $, and the constants $$\begin{aligned}
a &= r_a^0-\E[ \tau \mid Z=0, A=a], \quad c = 1 - \E[\tau \mid A=a]\\
b &= r_a^0 , \quad d = r_a^1
\end{aligned}$$ Without loss of generality we know that we can set $y$ to its upper bound $B$ when maximizing as we are only increasing the objective value; then $c' = c - B r_a^1$. We verify that the second derivative is negative, so that the function is concave: $$\frac{\partial^2 f}{\partial x^2} = \frac{2 b^2 (a - c')}{(c' -bx)^3} = \frac{2 (r_a^0)^2 ( r_a^0-\E[ \tau \mid Z=0, A=a] - (1 - \E[\tau \mid A=a]- B r_a^1 ) ) }{(1 - \E[\tau \mid A=a]- B r_a^1 ) - xr_a^0 }$$
Checking the sign of the numerator simplifies to checking the sign of $$a - c' = (-r_a^1 + \E[ \tau + B\mid Z=1, A=a]) )$$ which is negative. The denominator is lower bounded by $1 - \E[\tau \mid A=a] - B$ which is always positive: therefore the problem is concave. The first derivative $ \frac{\partial f}{\partial x} = \frac{b(a-c')}{(c'-bx)^2}$ is negative on the domain; therefore the maximum is achieved at $x = 0$. Therefore, when maximizing, $\omega = B \mathbb{I}[Z=1] $. For minimizing the TPR, we take a similar approach: analogously, we can set $y$ to its lower bound without loss of generality. Following the same analysis, the function is still concave $\frac{\partial^2 f}{\partial x^2} = \frac{2 b^2 (a - c')}{(c' -bx)^3}$ since $- r_a^1 -\E[ \tau \mid A=a]<0$ and decreasing with nonzero first-derivative; so the minimum is achieved at $\omega = \mathbb{I}[Z=0]B$.
Behaghel et al. Job Training {#apx-crepon-data}
============================
We processed the data using replication files available with the AEJ: Applied Economics journal electronic supplement. For the sake of simplicity, we analyze the trial as if it were a randomized controlled trial (without accounting for noncompliance or different randomization probabilities that differ by region). Thus, we consider intention-to-treat effects (as intention to treat is ultimately the policy lever available). We further restricted some covariates, omitting some where personalized allocation based on these covariates seemed unilkely for fairness reasons. The covariates we retain include: length of previous employment, salary, education level, reason for unemployment, region, years of experience at previous job, statistical risk level, job search type (full-time or non-full time), wage target, time of first unemployment spell, job type, and number of children.
An interacted linear model indicates potential heterogeneity of treatment effect with significance on college education, economic layoff, those seeking work due to fixed term contracts or those with previous layoffs.
![Diagnostics for gender protected attribute for \[sec-behaghel\] (not-woman vs. woman)](figs/tpr_disp_curve_gender.pdf "fig:"){width="0.4\linewidth"}![Diagnostics for gender protected attribute for \[sec-behaghel\] (not-woman vs. woman)](figs/tnr_disp_curve_gender.pdf "fig:"){width="0.4\linewidth"} ![Diagnostics for gender protected attribute for \[sec-behaghel\] (not-woman vs. woman)](figs/roc-gender-smoothed.pdf "fig:"){width="0.4\linewidth"}![Diagnostics for gender protected attribute for \[sec-behaghel\] (not-woman vs. woman)](figs/xroc-gender-smoothed.pdf "fig:"){width="0.4\linewidth"}
![ROC curves under \[asm:monotone\] for \[sec-behaghel\] ](figs/roc-nationality-nom.pdf "fig:"){width="0.33\linewidth"}![ROC curves under \[asm:monotone\] for \[sec-behaghel\] ](figs/roc-age-nom.pdf "fig:"){width="0.33\linewidth"}![ROC curves under \[asm:monotone\] for \[sec-behaghel\] ](figs/roc-gender-nom.pdf "fig:"){width="0.33\linewidth"}
Substantive Discussion: Fairness vs. Justice {#apx-substantive}
============================================
We first caveat our use of “disparate impact”: while our selection of protected attibutes parallels choices of protected attributes that appear elsewhere in the literature on fair machine learning, for the case of interventions, there may not be precedent from discrimination case law, nonetheless assessing fairness with respect to these social groups may be of concern. We view disparate impact in this domain as assessing fairness of outcome rates under a personalization model.
#### Should true positive rates be adjusted for?
Our presentation of an identification strategy of fairness metrics for allocating interventions with unknown causal effects begs the question: should disparities in TPR and FNR be adjusted for in the interventional welfare setting? Is responder-accuracy parity a meaningful prescriptive notion of fairness?
One critique of outcome-conditional fair classification metrics recognizes the dependence of false positive rates on the underlying *base rate*, ${\mathbb{P}}(Y=1\mid A=a)$, [@cdg-18; @c16]. The equivalent situation occurs when the within-group ATE varies by the protected attribute, e.g. $\E[\tau \mid A=a]$ differs. Ultimately, external domain knowledge is required to adjudicate whether group-wide disparities in ATE should be adjusted for, or to decide which normative notion of distributive justice or fairness is appropriate. For example, consider the case of job training. From an economic perspective, multiple mechanisms could explain heterogeneity in CATE by race. Active labor market programs (see [@labor-cb16]) may be less effective for one group vs. another group due to the presence of labor-market discrimination. Alternatively, they could be less effective due to correlation of group status and efficacy that is mediated by occupation choice: one group may be more interested in labor markets where the primary benefits of job search counseling, in reducing search frictions, are not barriers to employment in the first place relative to other factors such as skills gaps. Intuitively, the former mechanism of ATE variation by group reflects a notion of “disparity” which remains problematic, while the latter may seem to reflect an unproblematic causal mechanism. While mediation analysis and fairness defined in terms of path-specific effects could further decompose the treatment effect along these stated mechanisms, in policy settings, collecting all of the relevant information can be burdensome, and deciding on a causal graph can be difficult.
**Claims Across Outcomes** We first outline different frameworks for thinking about fairness/equity of algorithms and interventions. Analogous to the proposals arising from metrics proposed in fairness in machine learning, one might view the decision-maker’s concern to be of ensuring *accuracy* parity, that the decisions meted out are overall beneficial to individual. We view a theory of fairness that assesses disparities in outcome-conditional error rates in the context of a theory of normative claims arising from “claims across outcomes”. [@adler] develops a “claims across outcomes” framework of fairness and social welfare, in the context of an overall welfarist theory of justice.
On the one hand, fair classification from the point of view of assessing or equalizing TPR or TNR disparities may be interpreted in a claims context as: for an individual with “true outcome” $Y$ and covariates $X$, an individual with the true label $Y=1$ as having a comparative claim for $\hat{Y}=1$, if the predictor $\hat{Y}$ is an allocation tool. We can map the setting of personalized interventions to the “claims across outcomes” setting: the potential outcomes framework posits for each individual the random variables of outcomes $Y(0), Y(1)$. In the responder setting, the true label is responder status $Y(1) > Y(0)$. However, since these are *jointly unobservable*, in situations where heterogeneous treatment effects are plausible, the best guess is an individual-level treatment effect conditional on covariates, $\mathbb{E}[Y(0)\mid X=x], \E[Y(1)\mid X=x]$. In this interventional setting, one can think of individuals having claims in favor of favorable outcomes, e.g. a claim in favor of $Y(1)$ if $Y(1) > Y(0)$.
For the case of interventions, classification decisions $Z$ are allocative of real interventions, and we argue that implicitly, the consideration of social welfare (balancing efficiency and program costs) is an important factor in the original design of social programs or personalized interventions. This is in sharp contrast to the literature on fair classification which considers settings such as lending in finance, or risk prediction in the criminal justice system, where overriding concerns are primarily those of *vendor* utility.
On the other side of the spectrum, we can recall axiomatically justified social welfare functions that apply to the case of *deterministic* resource allocation, where outcomes are generally known. A decision-maker might also be concerned with equity considerations, adopting a min-max welfare criterion, appealing to Rawlsian justice frameworks. Another approach is simply assessing the population cardinal welfare of the allocation, e.g. the policy value $\E[Y(\pi(X))]$ or a social-welfare transformation thereof, $\E[g(Y(\pi(X)))]$. The literature on policy learning addresses welfare functionals that are linear functionals of potential outcomes, see [@kt15]. Cardinal welfare constraints such as those studied in [@heidari2018fairness] can be applied with an imputed CATE function.
#### Comparison to other work on fair classification and welfare.
[@liu-drsh18] study the implications of classifier-based decisions, as well as proposals for statistical parity, on group welfare. Their work addresses selection rules that have known marginal impacts by group. [@hu2019fair] studies the welfare weights implied by classification parity metrics and shows that enforcing classification parity metrics are not Pareto-improving. Rather than studying the welfare implications of classification parity, we are concerned with assessing non-identifiable model errors in the causal-effect personalized intervention setting. Since in the personalized intervention setting, welfare is a primary objective for the Planner (e.g. social services, or social protection more broadly), modulo cost considerations, combining the distributional information from identification of classification errors with other social welfare objectives is of possible interest.
We next aim to provide concrete examples of discussions regarding the distributional impacts of interventions, in order to provide additional context on different contexts wherein different notions of “fairness” from the fairness in machine learning literature map onto welfare or justice concerns, as stated in discussions on interventional outcomes.
**Lexicographic fairness or maximin (Rawlsian) fairness.**
In a large multi-site graduation trial on testing an intensive, composite intervention targeted at the “ultra-poor”, which comprised wraparound services including coaching and revenue-generating resources, still the poorest seemed to benefit least from the intervention in terms of sustained revenue [@banerjee2015multifaceted]. In this setting, concerns about maximin fairness (Rawlsian justice) might override considerations of efficiency insofar as one might be willing to invest resources to help the worst-off on humanitarian grounds.
**Universalism.**
Criticisms of targeted policies in general note practical difficulties introduced by imposing and enforcing eligibility guidelines. [@m05]. Although discussion of resource constraints may be used to justify a targeting scheme, critics of targeting argue that the most efficient targeting is not as welfare-improving as simply advocating for greater resources [@eubanks2018automating].
**Additional distributional preferences on $ Y(Z)$ with respect to equitable or redistributive aims of the policy.**
[@berger] consider profiling based on covariates as a means of allocating government services, in the example of allocating predicting unemployment duration to allocate reemployment services. They outline competing equity vs. efficiency concerns, in the case that unemployment duration is correlated with treatment efficacy (e.g efficacy of reemployment services), and conclude that “ tradeoffs between alternative social goals in designing profiling systems are likely to be empirically important... the form and extent of these tradeoffs may depend on empirical relationships between the impacts of the program being allocated and the equity-related characteristics of potential participants." While outcome-conditional true positive rates or true negative rates compare model performance across binary protected attributes, program designers may remain concerned regarding the distribution of benefits. [@carneiro2002removing] consider “removing the veil of ignorance” under the simplifying of constant treatment response to consider distributional (quantile) treatment effects, as a relaxation of the anonymity axiom of cardinal social welfare. Distributional preferences are relevant when program designers are concerned about model performance at finer-grained levels than discrete protected attribute.
|
---
abstract: 'We prove that a crossing change along a double point circle on a $2$-knot is realized by ribbon-moves for a knotted torus obtained from the $2$-knot by attaching a $1$-handle. It follows that any $2$-knots for which the crossing change is an unknotting operation, such as ribbon $2$-knots and twist-spun knots, have trivial Khovanov-Jacobsson number.'
address:
- 'Department of Mathematics, University of South Alabama, Mobile, AL 36688, U.S.A.'
- 'Department of Mathematics, University of South Florida, Tampa, FL 33620, U.S.A.'
- 'Graduate School of Science and Technology, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan (Department of Mathematics, University of South Florida, April 2003–March 2005)'
author:
- 'J. Scott Carter'
- Masahico Saito
- Shin Satoh
title: |
Ribbon-moves for 2-knots with 1-handles attached\
and Khovanov-Jacobsson numbers
---
A [*surface-knot*]{} or [*-link*]{} is a closed surface embedded in $4$-space ${\mathbb{R}}^4$ locally flatly. Throughout this note, we always assume that all surface-knots are oriented. A [*ribbon-move*]{} (cf. [@Og]) is a local operation for (a diagram of) a surface-knot as shown in Figure \[fig01\]. We say that surface-knots $F$ and $F'$ are [*ribbon-move equivalent*]{}, denoted by $F\sim F'$, if $F'$ is obtained from $F$ by a finite sequence of ribbon-moves.
![[]{data-label="fig01"}](fig01.eps)
The ribbon-move is a special case of the [*crossing change*]{}: Assume that a surface-knot $F$ has a double point circle $L$ in a diagram such that (i) $L$ has no self-intersection, and (ii) at every triple point on $L$, the sheet transverse to $L$ is either top or bottom (not middle). The condition (i) means that $L$ does not go through the same triple point twice. When $L$ satisfies these conditions, we can perform a crossing change along $L$ by exchanging the roles of over- and under-sheets as indicated in Figure \[fig02\] (cf. [@Yas]). See [@CSbook] for details on diagrams of surface-knots.
![[]{data-label="fig02"}](fig02.eps)
For a $2$-knot $K$ (a knotted sphere in ${\mathbb{R}}^4$), a crossing change is not necessarily realized by ribbon-moves; indeed, a ribbon-move does not change the Farber-Levine pairing of $K$ but a crossing change might (cf. [@Og]). On the other hand, when we consider the ${\mathbb{T}}^2$-knot (knotted torus in ${\mathbb{R}}^4$) $K+h$ obtained from $K$ by attaching a $1$-handle $h$ on $K$, we obtain the following.
\[thm1\] Let $K$ and $K'$ be $2$-knots such that $K'$ is obtained from $K$ by a crossing change. Then for any $1$-handles $h$ and $h'$ on $K$ and $K'$, respectively, the ${\mathbb{T}}^2$-knot $K+h$ is ribbon-move equivalent to $K'+h'$.
Along the double point circle $L$ for which we perform the crossing change, there is a neighborhood $N$ identified with $(B^3,t)\times S^1$, where $(B^3,t)$ is a tangle with two strings as shown in the left of Figure \[fig03\]. In the figure, the orientations of tangles are induced from that of $K$, and all bands are attached in an orientation-compatible manner. For an interval $I$ in $S^1$, we take a $1$-handle $h_1=b_1\times I$ on $K$, where $b_1$ is a band as indicated in the figure.
We observe that $K+h_1$ is ambient isotopic to $(K'\cup T)+h_2$ (cf. [@Sa2]), where $T=m \times S^1$ is a ${\mathbb{T}}^2$-knot linking with $K'$, and the $1$-handle $h_2=b_2\times I$ connects between $K'$ and $T$. See the center of Figure \[fig03\].
Consider a $1$-handle $h_3=b_3\times I$ on $K'\cup T$. Since both of $h_2$ and $h_3$ connect between $K'$ and $T$, the ${\mathbb{T}}^2$-knot $(K'\cup T)+h_2$ is ribbon-move equivalent to $(K'\cup T)+h_3$.
Finally we see that $(K'\cup T)+h_3$ is ambient isotopic to $K'+h_4$, where $h_4=b_4\times I$ is the $1$-handle on $K'$ as shown in the right of the figure. Thus we obtain $$K+h\sim K+h_1=
(K'\cup T)+h_2\sim (K'\cup T)+h_3
=K'+h_4\sim K'+h'.$$ This completes the proof.
![[]{data-label="fig03"}](fig03.eps)
We say that the crossing change is an [*unknotting operation*]{} for a surface-knot $F$ if the trivial surface-knot is obtained from $F$ by a finite sequence of crossing changes. It is still unknown whether the crossing change is an unknotting operation for [*any*]{} surface-knot.
Khovanov [@Kh] introduced a categorification of the Jones polynomial, that is, a chain complex for a given classical knot diagram such that its graded Euler characteristic is the Jones polynomial. Khovanov [@Kh4] and Jacobsson [@Ja] proved that it defines an invariant for cobordisms (relative to boundary diagrams). Specifically, a cobordism between two knot diagrams gives rise to a chain map (we call it a Khovanov-Jacobsson homomorphism) between corresponding chain complexes, that is invariant under equivalence of cobordisms of diagrams. See also [@Dror]. In particular, a diagram of a ${\mathbb{T}}^2$-knot is a cobordism between empty diagrams, giving rise to a homomorphism ${\mathbb{Z}}\rightarrow {\mathbb{Z}}$ defined up to sign, a multiplication by a constant. We call this constant the [*Khovanov-Jacobsson number*]{}.
\[thm2\] Let $K$ be a $2$-knot for which the crossing change is an unknotting operation. Then for any $1$-handle $h$ on $K$, the ${\mathbb{T}}^2$-knot $K+h$ has the trivial Khovanov-Jacobsson number.
Let $K_0$ be the trivial $2$-knot and $h_0$ the trivial $1$-handle on $K_0$. By assumption and Theorem \[thm1\], the ${\mathbb{T}}^2$-knot $K+h$ is ribbon-move equivalent to $K_0+h_0$, which is the trivial ${\mathbb{T}}^2$-knot.
Consider two movies as shown in Figure \[fig04\]. It is seen from the definitions [@Dror; @Ja] that the corresponding Khovanov-Jacobsson homomorphisms $H^*(|\bigcirc)\rightarrow H^*(\bigcirc |)$ are the same for these movies. This implies that a ribbon-move does not change the Khovanov-Jacobsson number. Hence the ${\mathbb{T}}^2$-knot $K+h$ has the same number as that of the trivial ${\mathbb{T}}^2$-knot $K_0+h_0$.
![[]{data-label="fig04"}](fig04.eps)
By Theorem \[thm2\], if there is a $2$-knot $K$ such that the Khovanov-Jacobsson number of $K+h$ is non-trivial, then the crossing change is not an unknotting operation for $K$. However, we have no such examples at present.
\[cor3\] Let $K$ be a ribbon $2$-knot or twist-spun knot. Then for any $1$-handle $h$ on $K$, the ${\mathbb{T}}^2$-knot $K+h$ has trivial Khovanov-Jacobsson number.
This follows from Theorem \[thm2\] and the fact that the crossing change is an unknotting operation for every ribbon $2$-knot or twist-spun knot (cf. [@AS; @Sa1]).
We say that a surface-knot is [*pseudo-ribbon*]{} [@Ka] if it has a diagram without triple points. The notions of ribbon and pseudo-ribbon $2$-knots are the same [@Ya] (see also [@KS]). On the other hand, for ${\mathbb{T}}^2$-knots, they are not coincident in the sense that the family of pseudo-ribbon ${\mathbb{T}}^2$-knots properly contains that of ribbon ${\mathbb{T}}^2$-knots.
\[prop4\] Any pseudo-ribbon ${\mathbb{T}}^2$-knot has trivial Khovanov-Jacobsson number.
By the results of Teragaito [@Te] and Shima [@Sh], every pseudo-ribbon ${\mathbb{T}}^2$-knot $T$ is (i) a ribbon ${\mathbb{T}}^2$-knot, or (ii) a ${\mathbb{T}}^2$-knot obtained from a split union of a Boyle’s turned ${\mathbb{T}}^2$-knot $T'$ [@Bo] and a trivial $2$-link $U=U_1\cup U_2\cup\dots\cup U_n$ by surgery along $1$-handles $h_1,h_2,\dots,h_n$ for some $n\geq 0$, where each $h_i$ connects between $T'$ and $h_i$ $(i=1,2,\dots,n)$.
For the case (i), there is a ribbon $2$-knot $K$ and a $1$-handle $h$ such that $T=K+h$. Hence the conclusion follows from Corollary \[cor3\].
For the case (ii), we see that $T=(T'\cup U)+(\bigcup_{i=1}^n h_i)$ is ribbon-move equivalent to $T'$. We consider two movies for a classical knot diagram $D$ in a plane, one of which keep $D$ still and the other twists $D$ by a $2\pi$-rotation of the plane. Then it follows from the definitions [@Dror; @Ja; @Kh4] that the corresponding Khovanov-Jacobsson homomorphisms $H^*(D)\rightarrow H^*(D)$ are the same for these movies. This implies that $T'$ has the same Khovanov-Jacobsson number as that of a non-turned (that is, just spun) ${\mathbb{T}}^2$-knot, which is ribbon. Hence this case reduces to (i).
Achknowledgments {#achknowledgments .unnumbered}
================
The first, second, and third authors are partially supported by NSF Grant DMS $\#0301095$, NSF Grant DMS $\#0301089$, and JSPS Postdoctoral Fellowships for Research Abroad, respectively. The third author expresses his gratitude for the hospitality of the University of South Florida and the University of South Alabama.
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[http://www.math.toronto.edu/\~drorbn/papers/Cobordism/]{}
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, [*An invariant of tangle cobordisms*]{}, preprint available at:
[http://xxx.lanl.gov/abs/math.GT/0207264]{}
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, [*A note on unknotting numbers of twist-spun knots*]{}, preprint.
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|
---
abstract: 'We extend the learning from demonstration paradigm by providing a method for learning unknown constraints shared across tasks, using demonstrations of the tasks, their cost functions, and knowledge of the system dynamics and control constraints. Given safe demonstrations, our method uses hit-and-run sampling to obtain lower cost, and thus unsafe, trajectories. Both safe and unsafe trajectories are used to obtain a consistent representation of the unsafe set via solving an integer program. Our method generalizes across system dynamics and learns a guaranteed subset of the constraint. We also provide theoretical analysis on what subset of the constraint can be learnable from safe demonstrations. We demonstrate our method on linear and nonlinear system dynamics, show that it can be modified to work with suboptimal demonstrations, and that it can also be used to learn constraints in a feature space.'
author:
- 'Glen Chou, Dmitry Berenson, Necmiye Ozay'
bibliography:
- 'refs.bib'
title: Learning Constraints from Demonstrations
---
Introduction
============
Related Work
============
Preliminaries and Problem Statement
===================================
Conclusion
==========
In this paper we propose an algorithm that learns constraints from demonstrations, which acts as a complementary method to IOC/IRL algorithms. We analyze the properties of our algorithm as well as the theoretical limits of what subset of an unsafe set can be learned from safe demonstrations. The method works well on a variety of system dynamics and can be adapted to work with suboptimal demonstrations. We further show that our method can also learn constraints in a feature space. The largest shortcoming of our method is the constraint space gridding, which yields a complex constraint representation and causes the method to scale poorly to higher dimensional constraints. We aim to remedy this issue in future work by developing a grid-free counterpart of our method for convex unsafe sets, which can directly describe standard pose constraints like task space regions [@tsr].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by a Rackham first-year graduate fellowship, ONR grants N00014-18-1-2501 and N00014-17-1-2050, and NSF grants CNS-1446298, ECCS-1553873, and IIS-1750489.
|
---
abstract: 'Approximating the strong force on matter in the flat background space of the early post-inflationary universe as an effective strong gravity indicates about 83% of all matter in the universe is now comprised of small closed impenetrable systems, each with 4.4 times the proton mass and radius $3\times10^{^{-14}}$cm.'
author:
- 'T. R. Mongan'
title: 'Cold dark matter from “strong gravity”'
---
84 Marin Avenue, Sausalito, CA 94965 USA
[email protected]
Dark Matter {#dark-matter .unnumbered}
-----------
Many cosmological models assume all four forces governing the universe were unified very early in the history of the universe. Taking a cue from force unification and approximating the strong force in the flat background space of the post-inflationary early universe as an effective strong gravity acting only on matter, the analysis below indicates dark matter is largely composed of impenetrable spheres with radius $3\times10^{-14}$ cm and mass 4.4 times the proton mass $m_{p}$.
After the initial force symmetry broke, the gravitational structure constant $\frac{Gm_{p}^{2}}{\hbar c}=5.9\times10^{-39}$, with $\hbar=1.05\times10^{-27}$ g cm$^{2}$/sec, $c=3\times10^{10}$cm/sec and $m_{p}=1.67\times10^{-24}g$, is the ratio of the strength of gravity and the strong force after inflation. In the flat homogeneous and isotropic space of the post-inflationary universe with matter energy density $\rho$, approximate the strong force as strong gravity acting only on matter with strength $G_{S}=\left(\frac{M_{P}}{m_{p}}\right)^{2}G=1.7\times10^{38}G$, where the gravitational constant $G=6.67\times10^{-8}$cm$^{3}$/g sec$^{2}$ and the Planck mass $M_{P}=\sqrt{\frac{\hbar c}{G}}=2.18\times10^{-5}$ g. Then, the strong gravity Friedmann equation $\left(\frac{dR}{dt}\right)^{2}-\frac{8\pi G_{S}\rho R^{2}}{3}=-c^{2}$ describes the local curvature of spaces defining closed massive systems bound by the effective strong gravity. Because a strong force at short distances is involved, a quantum mechanical analysis of such systems is necessary. The Schrodinger equation resulting from Elbaz-Novello quantization of the Friedmann equation [@key-1] for a closed massive system bound by the effective strong gravity is $-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dr^{2}}\psi-\frac{2G_{S}\mu M}{3\pi r}\psi=-\frac{\mu c^{2}}{2}\psi$, where $M=2\pi^{2}\rho r^{3}$ is the conserved mass of the closed system with radius $r$ and $\mu$ is an effective mass. This Schrodinger equation is identical in mathematical form (but not interpretation) to the Schrodinger equation for the hydrogen atom and can be solved immediately. The ground state curvature energy $-\frac{\mu}{2\hbar^{2}}\left(\frac{2G_{S}\mu M}{3\pi}\right)^{2}$ of this Schrodinger equation must equal $\frac{\mu c^{2}}{2}$ for consistency with the corresponding Friedmann equation, so the effective mass $\mu=\frac{3\pi\hbar c}{2G_{S}M}$ . The ground state solution of this Schrodinger equation describes a stable closed system bound by the effective strong gravity, with zero orbital angular momentum and radius $<r>=\frac{G_{S}M}{\pi c^{2}}$. Setting the radius $r=\frac{G_{S}M}{\pi c^{2}}=\frac{\hbar M}{\pi cm_{p}^{2}}$ of the small closed system equal to the Compton wavelength $\frac{\hbar}{Mc}$ yields $M=\sqrt{2\pi^{2}\frac{\hbar c}{G_{S}}}=m_{p}\pi\sqrt{2}=4.4m_{p}$ and radius $r=\frac{\hbar M}{\pi cm_{p}^{2}}=\frac{\hbar\sqrt{2}}{cm_{p}}=2.9\times10^{-14}$ cm. The density of dark matter particles is $\frac{3\left(4.4m_{p}\right)}{4\pi}\left(\frac{cm_{p}}{\hbar\sqrt{2}}\right)^{3}=6.7\times10^{16}$ g/cm$^{3}$.
Geodesic paths inside the stable ground state closed systems created by the effective strong gravity are all circles with radius $<r>=\frac{G_{S}M}{\pi c^{2}}$, so matter within these closed systems is permanently confined within a sphere of radius $<r>$. No particle can enter or leave these small closed systems after they form, to increase or decrease the amount of matter in those closed systems. These small closed systems act like rigid impenetrable spheres interacting only gravitationally and constituting the majority of dark matter.
Astrophysical consequences {#astrophysical-consequences .unnumbered}
--------------------------
As the universe expanded after inflation, the matter density of the universe steadily dropped. When the matter density in the early universe fell to $6.7\times10^{16}$ g/cm$^{3}$, most matter in the universe coalesced into dark matter - small closed spherically symmetric systems with zero orbital angular momentum. This resulted in a universe packed with small invisible and impenetrable systems interacting only gravitationally. Assuming uniform matter density in the universe and instantaneous coalescence with maximum packing fraction, 74% of the matter in the universe is small closed impenetrable systems constituting the bulk of dark matter. With non-uniform density, coalescence would occur first in lower density volumes, and expansion of the universe might allow slightly more than 74% of matter to coalesce into small closed systems.
The matter fraction of the energy density in the universe today is about 0.3. If the hadronic matter fraction is about 0.05, dark matter is about (0.25/0.3) = 83% of all matter. If 74% of all matter is small closed systems, those small bound systems account for $0.74\times0.3=0.22$ of the energy density in the universe today, or about $\left(0.22/0.25\right)=88\%$ of the dark matter. Using today’s scale factor of the universe $R_{0}\approx10^{28}$ cm and today’s matter density $2.4\times10^{-30}$ g/cm$^{3}$, the matter density $6.7\times10^{16}$ g/cm$^{3}$ required for coalescence into small closed systems with mass $4.4m_{p}$ occurred when the scale factor of the universe was $R_{c}\approx3\times10^{12}$ cm.
Assuming velocity-independent rigid sphere scattering [@key-2], an approximate lower bound on the self-interaction collision cross-section/mass ratio for these dark matter particles is $\frac{\sigma}{M}=\frac{4\pi\left(2r\right)^{2}}{M}=16\sqrt{2}\left(\frac{G_{s}}{c^{2}}\right)^{2}m_{p}=0.006$ cm$^{2}$/g. This is about a third of Hennawi and Ostriker’s [@key-3] best fit value of 0.02 cm$^{2}$/g based on growth of black holes in the center of galaxies. This lower bound self-interaction collision cross-section/mass ratio for dark matter particles is consistent with upper bounds on $\frac{\sigma}{M}$ based on evaporation of galactic halos [@key-4], optical and X-ray observations of the colliding galactic cluster 1E0657-56 [@key-5], and X-ray observations of the merging galaxy cluster MACSJ0025.4-1222 [@key-6]. As noted by Hennawi and Ostriker [@key-3], values of $\frac{\sigma}{M}$ in this range indicate observed density profiles in galaxies occur by early and efficient growth of massive black holes at the center of galaxies, followed by successive galactic mergers. The mass of the dark matter particles is below the Lopes/Silk 10 GeV threshold [@key-7] that would allow capture by the solar gravitational field and accumulation in the center of the Sun.
The impenetrable spheres of dark matter are the ultimate defense against gravitational collapse. The radius of a close-packed sphere of $n$ dark matter particles is $R_{n}=\sqrt[3]{n}\sqrt{2}\left(\frac{\hbar}{cm_{p}}\right)=\sqrt[3]{n}(2.9\times10^{-14}$cm). The Schwarschild radius of that sphere, $R_{S}=\frac{8.8Gnm_{p}}{c^{2}}=1.1n\times10^{-51}$cm, is smaller than the physical radius of the sphere until $\sqrt[3]{n}(2.9\times10^{-14}$cm) $=1.1n\times10^{-51}$cm, or $n=1.4\times10^{56}$. This suggests a minimum mass for an accretionary black hole of $6.2\times10^{56}m_{p}=1.0\times10^{33}$ g, or about half the solar mass, and a minimum Schwarschild radius of 1.5 km. This limit is below the Tolman-Oppenheimer-Volkoff limit of 0.7 solar masses for the mass of neutron stars that can be supported against gravitational collapse by neutron degeneracy pressure, as well as the maximum neutron star mass of 1.5 to 3 solar masses calculated by Bombaci [@key-8]. The limit prohibits black holes with mass around $10^{20}$ g proposed as anti-matter factories by Bambi et al [@key-9].
The surface temperature of a Schwarzschild black hole of mass $M$ is $T=\frac{\hbar c^{3}}{8\pi GMk}=\frac{\hbar c}{4\pi R_{S}k},$ where the Boltzmann constant $k=1.38\times10^{-16}$(g cm$^{2}$/sec$^{2})$/$^{o}K$. So, the maximum surface temperature of an accretionary black hole cannot exceed that of a minimum mass Schwarzschild black hole with mass $1\times10^{33}$ g and surface temperature $1.2\times10^{-7}$ $^{o}K$. The microwave background radiation temperature will not drop below the surface temperature of a minimum mass Schwarzschild black hole until the scale factor of the universe is $2.3\times10^{6}$times greater than today, so accretionary black holes will not begin to evaporate by Hawking radiation until that time.
Consistency with DAMA/LIBRA results {#consistency-with-damalibra-results .unnumbered}
-----------------------------------
Dark matter particles with mass $4.4m_{p}$ are consistent with DAMA/LIBRA nuclear recoil results [@key-10]. DAMA/LIBRA results and astronomical observations of the colliding galaxies 1E0657-56 [@key-5] both indicate a very low cross section for scattering of dark matter on ordinary matter. In a parton model of the struck nuclei at DAMA/LIBRA, detectable nuclear recoils only occur in very low probability events when an incoming dark matter particle collides almost head-on with a parton carrying most of the center-of-mass momentum of the struck nucleus.
Accelerator production {#accelerator-production .unnumbered}
----------------------
The impenetrable spheres of dark matter will not be created in particle accelerators because of Lorentz contraction of the accelerated particles. In the colliding particle center of mass system, the energy of colliding particles is dumped into a thin Lorentz-contracted disk. This does not create the uniform matter distribution in a sphere of radius $3\times10^{-14}$ cm necessary to reproduce early universe conditions resulting in coalescence of impenetrable spheres of dark matter.
[10]{} Elbaz, E., Novello, M., Salim, J. M., Motta da Silva, M. C., & Klippert, R., Gen. Rel. Grav. 29, 481, 1997; Novello, M., Salim, J. M., Motta da Silva, M. C., & Klippert, R., Phys. Rev. D54, 6202, 1996
Schiff, L. I., Quantum Mechanics (2nd ed.), McGraw-Hill Book Company, Inc., New York, 1955
Hennawi, J.F. & Ostriker, J. P., ApJ 572, 41, 2002
Gnedin, O.Y. & Ostriker, J. P., ApJ 561, 61, 2001
Markevitch, M., Gonzalez, A.H., Clowe, D., Vikhilin, A., Forman, W., Jones, C., Murray, S., & Tucker, W., ApJ. 606, 819, 2004
Bradac, M., et al, arxiv:0806.2320
Lopes, I.P. & Silk, J., Phys. Rev. Lett. 88, 151303, 2002
Bombaci, I. Astron. Astrophysics, 305, 871-877, 1996
Bambi, C., Dolgov, A., and Petrov, A., arXiv:0806:3440
Petriello, F. & Zurek, K, arXiv:0806.3989
|
---
abstract: 'We present a comprehensive analysis on the MSSM Higgs sector CP violation at photon colliders including the chargino contributions as well as the contributions of other charged particles. The chargino loop contributions can be important for the would-be CP odd Higgs production at photon colliders. Polarization asymmetries are indispensable in determining the CP properties of neutral Higgs bosons.'
address: |
$^1$Center for Gifted Students and $^2$Department of Physics,\
Korea Advanced Institute of Science and Technology,\
Daejeon 305-701, Korea\
author:
- 'Saebyok ${\rm Bae^1}$[^1], Byungchul ${\rm Chung}^2$[^2] and P. ${\rm Ko^2}$[^3]'
title: ' MSSM Higgs sector CP violation at photon colliders: Revisited '
---
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Introduction {#sec:intro}
============
The discovery of Higgs boson(s) at the current/future colliders is one of the most important goals of high energy particle physics experiments. Its (non)discovery would be crucial for testing our present understanding of the origin of electroweak symmetry breaking (EWSB) and the subsequent generation of masses of electroweak (EW) gauge bosons and chiral fermions in the Standard Model (SM). This would be also true of the Minimal Supersymmetric Standard Model (MSSM), which is the most popular candidate for the new physics beyond the SM.
The Higgs sector in the MSSM possesses three neutral Higgs particles: two CP-even neutral scalars ($h$ and $H$), one CP-odd neutral scalar ($A$), and a pair of charged Higgs scalars ($H^\pm$) [@Gunion]. The tree-level MSSM Higgs potential does not allow spontaneous CP violation unlike general two-Higgs doublet model. Even if one include the one-loop corrected effective potential for the Higgs sector, the spontaneous CP violation [@Maekawa] can not be realistic, because the resulting lightest neutral Higgs boson should be far less than the current lower limit on the Higgs boson [@Pomarol]. Still, there are many new explicitly CP violating complex parameters in the soft supersymmetry (SUSY) breaking sector of the MSSM Lagrangian, and some of them can have large phases (without conflict with the electron/neutron electric dipole moment (EDM) constraints), and thus can lead to some observable consequences in various CP violating phenomena in $K$ and $B$ decays [@Ko] and electroweak baryogenesis [@Carena], etc. Especially, the complex phases of the stop and sbottom trilinear couplings $A_{t,b}$ and the Higgsino mass parameter $\mu$ can cause the mixing between CP-odd and CP-even neutral Higgs bosons in the neutral Higgs sector via loop corrections in the MSSM, namely, the Higgs sector CP violation [@PiWa].
In most phenomenological studies of the MSSM, the large SUSY CP violating (CPV) phases were usually neglected, since they may lead to large EDMs of electron and neutron, or $\epsilon_K$, depending on whether they are flavor preserving (FP) or flavor changing (FC). The SUSY CPV phases are assumed to be very small, so that the only source of CP violation would be the single Kobayashi-Maskawa (KM) phase in the CKM mixing matrix in the charged weak current of down type quarks. In this case, the SUSY effects on $K$ and $B$ phenomenology are minimal in the sense that deviations from the SM predictions are quite small. However one can consider large FP SUSY CPV phases, since one can avoid the EDM constraints in basically three different ways:
- Decoupling (Effective SUSY Model): The 1st/2nd generation sfermions are heavy (and degenerate to some extent) enough, so that the SUSY CP and $\epsilon_K$ problems are evaded. Only third generation sfermions and gauginos have to be lighter than $O(1)$ TeV in order that one solves the gauge hierarchy problem by SUSY [@Dimopoulos]. In this case, the SUSY CPV phases need not be zero, and they can lead to substantial deviations from the SM cases, especially for the third generation. In this scenario, $B$ factories may be able to probe the SUSY CPV phases from direct asymmetry in $B \rightarrow X_s \gamma$ and the lepton forward-backward asymmetry in $B \rightarrow X_s l^+ l^-$, etc.
- Cancellation: Various contributions to electron/neutron EDMs may cancel one another, leading to the net results which are consistent with experimental lower bounds [@Ibrahim]. In this case, many of the SUSY CPV phases can be $O(1)$ as in the decoupling scenario. However this scenario is tightly constrained when the data on the mercury ($^{199}$Hg) atom EDM is included [@mercury].
- Non-universal Scenario: $|A_e|, |A_{u,c}|, |A_{d,s}| \lesssim 10^{-3} |\mu|$ to evade $e/n$ EDM’s, but $A_t , A_b , A_{\tau}$ can have large CP violating phases [@Chang]. However there is a strong two-loop Barr-Zee type constraint for large $\tan\beta$. Therefore large CPV phases can be allowed in this scenario and decoupling scenario only for $\tan \beta \lesssim 20-30$.
The reliable determinations of the neutral Higgs sector CP-violation in the MSSM can be achieved by observing the CP-properties of all the three neutral Higgs particles directly. Higgs bosons can be produced in $\gamma \gamma$ collisions via one-loop diagrams in which all the possible charged particles participate. The $s$-channel resonance productions of neutral Higgs bosons in $\gamma \gamma$ collisions have been considered as crucial tools of studying the CP properties of Higgs particles . Because the polarizations of the colliding photons can strongly govern both the $\gamma \gamma$ luminosity spectrum and the cross sections, obtaining the highly polarizated photon beams is important to Higgs boson detections. This is possible by Compton backscattering of laser photons off the linear collider electron and positron beams which can produce high luminosity $\gamma \gamma$ collisions with a wide spectrum of $\gamma \gamma$ center of mass energy [@Exp_pp].
In particular, one can observe CP violating effects through the $s$-channel resonance for CP-odd neutral Higgs particle production in the linear collider. Due to the mixing effect between the CP-odd and CP-even neutral Higgs bosons, there are the additional loop contributions of charged scalars and vectors to the would-be CP-odd neutral Higgs $H_2$ production in $\gamma \gamma$ collision, resulting in the enhanced production cross section. In Ref. [@SYChoi], the CP violation of the neutral Higgs sector at a photon collider was studied using the $s$-channel resonance production cross sections and the polarization asymmetries of Higgs particles for $3 \le \tan \beta \le 10$. In the loop diagrams relevant to $\gamma \gamma
\rightarrow$ neutral Higgs bosons, the contributions of charginos were neglected by assuming that they were heavy enough to be decoupled from the productions of the Higgs bosons. However, charginos are not much heavier than the lighter stop in many SUSY breaking scenarios, and their effects should be included in a realistic analysis. The current lower limit on the lighter chargino mass from LEP II experiment is only $M_{\tilde{\chi}^-_1}
> 103~ (83.6)$ GeV for $m_{\tilde{\nu}} > (<) \, 300$ GeV in the minimal supergravity scenario [@Diaz]. It is even less stringent in the AMSB scenario: $M_{\tilde{\chi}^-_1} > 45$ GeV. Therefore we include the chargino contributions to $\gamma\gamma\rightarrow H_{k=1,2,3}$, and investigated their effects when other parameters are fixed as Ref. [@SYChoi].
In this work, we investigate the neutral Higgs productions at $\gamma\gamma$ collisions, including the chargino loop contributions as well as other charged particles in the MSSM, and study the CP properties of the MSSM Higgs sector. This paper is organized as follows. In Section II, we review briefly the loop-induced CP violation and the mixing of CP-even and CP-odd Higgs bosons in the neutral Higgs sector of the MSSM. In Section III, we derive the cross sections for the Higgs productions in $\gamma\gamma$ collisions and the polarization asymmetries in terms of two form factors appearing in the $\gamma\gamma \rightarrow H_{k=1,2,3}$ amplitudes. In Section IV, we present detailed numerical analyses and discuss the potential importance of chargino loop contributions to the CP violation in $\gamma\gamma \rightarrow H_k$. The formulae for the chargino and stop mass matrices, their eigenvalues and the corresponding mixing matrices are given in Appendix A. The interaction Lagrangians relevant to $\gamma\gamma
\rightarrow H_k$ are recapitulated for both convenience and completeness.
The Neutral Higgs Sector in the MSSM
====================================
The MSSM Lagrangian possesses many new CP violating phases in the soft SUSY breaking terms in addition to the KM phase in the CKM matrix element. Using Peccei-Quinn $U(1)_{\rm PQ}$ and $U(1)_{\rm
R}$ symmetries, we can redefine some parameters to be real. We will work in the basis where $B\mu$ and the wino mass parameter ${M}_2$ are real. In the MSSM, the Higgs potential is CP-conserving at the tree level and only the soft terms (and the usual CKM mixing matrix) can have CP violating phases. However, CPV phases in soft terms can induce CP violation in the effective potential of Higgs bosons through quantum corrections involving squarks and other SUSY particles in the loop. The effective potential of the Higgs fields at the one-loop level[^4] can be written as \[VHiggs\] [V]{}\_ [Higgs]{}\^[eff]{} &=& \_[1]{}\^2 \_[1]{}\^\_[1]{}+\_[2]{}\^2 \_[2]{}\^\_[2]{} +(m\_[12]{}\^2 \_[1]{}\^ \_[2]{} + [h.c.]{})\
&&+ \_[1]{}(\_[1]{}\^\_[1]{})\^2+\_[2]{}(\_[2]{}\^ \_[2]{})\^2 +\_[3]{}(\_[1]{}\^ \_[1]{})(\_[2]{}\^ \_[2]{}) +\_[4]{}(\_[1]{}\^ \_[2]{})(\_[2]{}\^ \_[1]{})\
&&+ \_[5]{}(\_[1]{}\^\_[2]{})\^2+\_[5]{}\^\* (\_[2]{}\^ \_[1]{})\^2 +\_[6]{}(\_[1]{}\^ \_[1]{})(\_[1]{}\^ \_[2]{}) +\_[6]{}\^\*(\_[1]{}\^ \_[1]{})(\_[2]{}\^ \_[1]{})\
&&+ \_[7]{}(\_[2]{}\^ \_[2]{})(\_[1]{}\^ \_[2]{}) +\_[7]{}\^\*(\_[2]{}\^ \_[2]{})(\_[2]{}\^ \_[1]{}). The fields $\Phi_i$ $(i=1,2)$ are the scalar components of the Higgs superfields, with $\Phi_2 (\Phi_1)$ giving masses to the up-type (down-type) fermions. In the MSSM, one has $\lambda_i =0$ ($i=5,6,7$) at tree level so that there is not Higgs sector CP violations in the MSSM. But these couplings are generated at one loop level and can be complex if $\mu, A_t$ possess CPV phases. Also the Higgs bilinear couplings $m_{12}^2$ [@Pil] can be complex by quantum corrections. For small $\tan \beta \sim O(1)$, where the stop contributions are dominant over sbottom or chargino contributions to the Higgs sector CP violations [@PiWa; @Pil; @IbNa], one has, for example [@Bae], \[m12\] m\_[12]{}\^2 && B+ h\_t\^2 A\_t , where the top Yukawa coupling $h_t=\frac{\sqrt{2} m_t(\bar{m_t})}{ v \sin\beta }$, and $m_{{\tilde t}_i}$ $(i=1, 2)$ are the masses of the lighter and heavier stops. The contributions of the 1st and 2nd generation squarks are negligible because of their small Yukawa couplings. The mixing of two CP-even Higgs bosons is denoted by the real parameter $B \mu$, whereas the $h_t^2 A_t \mu$ term with the complex $A_t$ trilinear coupling generates the mixings among all two CP-even and one CP-odd neutral Higgs bosons. Therefore, the quadratic term of the Higgs fields with the coefficient $m_{12}^2$ plays an important role in the Higgs mixing. If $h_t^2 A_t \mu$ terms are much less than $B \mu$, and $\mu_1^2 \sim \mu_2^2 \sim B \mu$, we can expect that the scalar-scalar mixing is much larger than the scalar-pseudoscalar mixing. For large $\tan \beta \gtrsim 30$, the contribution of the chargino sector can dominate those of the stop and sbottom sectors in the mixing between the CP-even and CP-odd Higgs bosons [@IbNa]. The same is true of other quartic couplings $\lambda_{5,6,7}$, whose imaginary parts vanish in the CP conserving limit (or at tree level) in the MSSM. One has to keep in mind that there is a strong constraint from two-loop Barr-Zee type $e/n$ EDM constraints for large $\tan \beta$ ($40 \lesssim \tan \beta \lesssim 60$). Therefore, we will choose rather low $\tan \beta \lesssim 20$ and allow maximal CPV phase in the $\mu$ and $A_t$ parameters.
Since the electroweak gauge symmetry is broken spontaneously into $U(1)_{\rm em}$, two Higgs doublets can be written as \[phi1\] \_[1]{} = (
[c]{} \_[1]{}\^[+]{}\
(v\_[1]{}+\_[1]{}+ia\_[1]{})/
), \_[2]{} = e\^[i ]{} (
[c]{} \_[2]{}\^[+]{}\
(v\_[2]{}+\_[2]{}+ia\_[2]{})/
), where the VEVs $v_i$ are real. The relative phase $\xi$, which is renormalization-scheme dependent,[^5] is determined from the minimum energy conditions of the Higgs potential [@PiWa], i.e., the vanishing tadpole conditions $T_{\phi}={\partial{\cal
V}_{\rm Higgs}^{eff}}/{\partial \phi}=0$. It turns out $\xi$ is very small in the $\overline{MS}$ scheme, and will be ignored in the numerical analysis. Because the electroweak symmetry is spontaneously broken to $U(1)_{\rm em}$, three Goldstone bosons are eaten by $W^{\pm}, Z^0$ gauge bosons, and one ends up with two charged Higgs and three neutral Higgs bosons. The $3 \times 3$ $({\rm mass})^2$ mass matrix ${\cal M}_N^2$ for three neutral Higgs bosons is a real symmetric matrix, and is diagonalized by a $3 \times 3$ orthogonal matrix $O$: \[Oij\] O\^[T]{} [M]{}\^2\_[N]{} O= (M\_[H\_[1]{}]{}\^2, M\_[H\_[2]{}]{}\^2, M\_[H\_[3]{}]{}\^2 ), where $M_{H_{3}} \geq M_{H_{2}} \geq
M_{H_{1}}$. The corresponding mass eigenstates, $H_{i}$ $(i=1,2,3)
\,$, are defined from the weak eigenstates as \[s-psmixing\] (a, \_1, \_2)\^T =O(H\_1, H\_2, H\_3)\^T.
Neutral Higgs Boson Productions at photon colliders {#sec:nh}
====================================================
Both within the SM and the MSSM, the neutral Higgs decays into two gluons ($gg$) or two photons ($\gamma\gamma$) have been interesting subjects. The inverse of the former process is a main production mechanism for the neutral Higgs bosons at hadron colliders if the Higgs bosons have intermediate masses. The latter is an important mode for tagging the neutral Higgs bosons at hadron colliders. Its inverse process is the mechanism for neutral Higgs productions in the $\gamma \gamma$ collision which can be run at next linear colliders (NLC).
The reactions $g g \rightarrow H_k$ ($k=1,2,3$) are generated by the (s)quark loops, and have been already discussed by two groups in the presence of the MSSM Higgs sector CP violation [@ggh]. We have calculated these processes and confirmed their results, although we do not reproduce them here. The case for $\gamma\gamma \rightarrow H_k$ is more complicated than the previous case ($g g \rightarrow H_k $), since one has to include all the charged particle ($W^{\pm}$, $H^{\pm}$ and charginos) contributions as well as the (s)quark loop contributions. It is straightforward to perform the loop integrations. The only thing to take into account is the various mixing components for charginos and neutral Higgs bosons. We present the chargino mass matrix ${\cal M_C}$, its mass eigenvalues $M_{\tilde{\chi}^-_1}$, $M_{\tilde{\chi}^-_2}$ and two mixing matrices $U$ and $V$: $U^* {\cal M_C} V^{\dagger} = {\rm diag}( M_{\tilde{\chi}^-_1},
M_{\tilde{\chi}^-_2})$ (see Appendix A for explicit expressions).
The interaction Lagrangian between the charginos and three neutral Higgs bosons is ( H\_j \^+\_k \^-\_l ) = H\_j \^-\_l, (with $j=1,2,3$ and $k,l=1,2$) where $$\kappa^j_{kl} = - \frac{g}{\sqrt{2}} \left[ e^{+i\xi} U_{k1} V_{l2}
(O_{3, j} +i\cos\beta \, O_{1, j})
+ U_{k2} V_{l1}(O_{2, j} +i \sin\beta \, O_{1, j}) \right].$$ In this work, it suffices to keep $\kappa_{kk}^j$ only, since we consider the chargino loop contribution to $\gamma\gamma
\rightarrow H_i$. Note that there are two CP violating phases ($\xi$ and $\theta_\mu = $ arg($\mu$)) in the couplings $\kappa^j_{kk}$. Also note that the $H_j -\tilde{\chi}^+_k
-\tilde{\chi}^-_k$ couplings arise from the Higgs-gaugino-Higgsino couplings in the current basis. Thus the chargino loop effects will be maximized if the wino-Higgsino mixing is large. This requires $\mu \approx M_2$. In our study, however, we are interested in large $\mu$ parameter (which we fix to $\mu = 1.2$ TeV) in order to have large CP mixing between CP-even and CP-odd Higgs bosons from the stop loop. Then the charginos become too heavy to be relevant to $\gamma\gamma \rightarrow H_i$. For a smaller wino mass parameter $M_2 = 150$ GeV, the wino-Higgsino mixing becomes smaller, but the lighter chargino mass becomes also very light, and the loop function will be enhanced. The net result turns out that the light chargino loop effects are important for the reaction $\gamma\gamma \rightarrow H_i$ even if the lighter chargino is dominantly a wino state ($M_2 \ll |\mu|$).
The amplitudes for $\gamma ( k_1, \epsilon_1 ) + \gamma ( k_2,
\epsilon_2 ) \rightarrow H_i (q)$ (with $i=1,2,3$) can be defined in terms of two form factors $A_i (s)$ and $B_i (s)$ as follows in a model independent way (we closely follow the convention of Ref. [@SYChoi] in the following): (H\_i) = M\_[H\_i]{} { A\_i(s) - B\_i(s) \_ \_1\^\_2\^k\_1\^k\_2\^}, where $s \equiv ( k_1 + k_2
)^2 = M_{H_k}^2$. Including the chargino loop contributions, the CP-even form factors $A_i$ at $s=M_{H_i}^2$ are A\_i(s=M\_[H\_i]{}\^2) = \_[f=t,b]{}A\_i\^f + \_[\_j=\_[1,2]{}, \_[1,2]{}]{} A\_i\^[\_j]{} +A\_i\^[H\^]{} + A\_i\^[W\^]{} + \_[j=1,2]{} A\_i\^[\^\_j]{}, The CP-even functions $A_i^f$, $A_i^{\tilde{f}_j}$, $A_i^{H^{\pm}}$, and $A_i^{W^{\pm}}$ are given in Ref. [@SYChoi]. We confirmed their results and reproduced them and the related form factor loop functions in Tables 1 and 2 for completeness. The chargino contribution to $A$ form factor is A\_i\^[\^\_j]{}= 2 [Re]{} ( \^i\_[jj]{} ) F\_[sf]{}(\_[i \^\_j]{}), where $\tau_{i X}=
M_{H_i}^2/ 4 M_{X}^2$. The form factor $F_{sf}(\tau)=\tau^{-1}
\left[ 1+ (1-\tau^{-1}) f(\tau) \right]$ (and other loop functions defined in Table II) depends on the scaling function $f(\tau)$ [@Gunion]: \[f\] f() =- \_0\^1 ={\^2() & [for]{} 1\
- \^2 & [for]{} 1. . On the other hand, the CP-odd form factor $B_i$ have contributions only from the fermion loops and not from the boson loops: B\_i(s=M\_[H\_i]{}\^2) = \_[f=t, b]{} B\_i\^f + \_[j=1,2]{} B\_i\^[\^\_j]{}, where $B_i^f$ are given in [@SYChoi] (see also Tables 1 and 2), and the chargino contributions are \[B\_i\] B\_i\^[\^\_j]{} = -2 [Im]{} ( \^i\_[jj]{} ) F\_[pf]{}(\_[i \^\_j]{}), where $F_{pf}(\tau)=\tau^{-1}f(\tau)$. Therefore, when a CP-odd Higgs boson $A$ is produced in $\gamma \gamma$ collision in the CP-conserving limit, only fermion loops (not boson loops) contributes to the production reaction.
It is also convenient to define two helicity amplitudes ${\cal M}_{\pm\pm}$ by \_[\_1 \_2]{} = - M\_[H\_k]{} { A\_k (s) \_[\_1 \_2]{} + i \_1 B\_k (s) \_[\_1 \_2]{} }, where $\lambda_{1,2}=\pm $ are photon helicities. Then, in the narrow-width approximation, the partonic cross sections of the $s$-channel Higgs productions [@SYChoi] are (H\_i) = ( |[M]{}\_[++]{} |\^2 + | [M]{}\_[–]{} |\^2 ) (1-M\_[H\_i]{}\^2/s) \_0(H\_i) (1-M\_[H\_i]{}\^2/s). By using the amplitudes of $\gamma \gamma \rightarrow H_i$ at $s=M_{H_i}^2$, we can also obtain the unpolarized decay rates of the neutral Higgs bosons into two photons, (H\_i ) = M\_[H\_i]{} ( | A\_i(s=M\_[H\_i]{}\^2) |\^2 + | B\_i(s=M\_[H\_i]{}\^2) |\^2 ).
The Higgs sector CP violation can be measured in the following three polarization asymmetries ${\cal A}_a$ ($a=1,2,3$) [@Grza] which are defined in terms of two independent helicity amplitudes: \[asy1\] [A]{}\_1 &=& = ,\
\[asy2\] [A]{}\_2 &=& = ,\
\[asy3\] [A]{}\_3 &=& = , In the CP-conserving limit, one of the form factors $A_i$ and $B_i$ must vanish, so that ${\cal A}_1={\cal A}_2=0$, and ${\cal
A}_3=+1(-1)$ for a pure CP-even (CP-odd) Higgs scalar. >From the definition of the function $f(\tau)$ in Eq. (\[f\]), we find that the form factors $A_i$ and $B_i$ may be complex, when the Higgs masses $M_{H_i}$ are two times larger than the particle mass in the loop. This will induce rich structures in the polarization asymmetries ${\cal A}_a$ as functions of Higgs masses and other SUSY parameters in the presence of Higgs sector CP violation.
Numerical Analyses
==================
The CP violation in the neutral Higgs sector through the stop loop with the complex $A_t$ parameter always appear in the combination of ${\rm arg}(A_t \mu)$. In the following numerical analyses, we assume that the $\mu$ parameter is real and positive, in order to simplify the discussions. For the complex $\mu$ parameter, the chargino mass marix will contain CPV phase, thereby there would be additional CP violating effects in the chargino loop contributions to $\gamma\gamma \rightarrow H_i$. However, this CP violating effect is independent of the CP violation in the neutral Higgs sector through the mixing between the CP-even and the CP-odd Higgs bosons. Since our focus in this work is to examine the reaction $\gamma\gamma \rightarrow H_i$ in the presence of Higgs sector CP violation through the mixing, we ignore complex phase in the chargino sector. We also assume $A_t = A_b$ for simplicity even if these couplings are independent in general. The CP violating phase ${\rm arg}(A_t)$ is varied between $0$ and $2\pi$. Also we choose the same parameters as Ref. [@SYChoi] (except for the wino mass parameter $M_2$) in our numerical analyses in order to investigate the chargino contributions more clearly; |A\_t|=|A\_b| = 0.4 [TeV]{}, = 1.2 [TeV]{}, M\_2 = 150 [GeV]{}, M\_[SUSY]{} = 0.5 [TeV]{}. \[param\] Using these parameter set, we investigate in detail $\hat \sigma_0
(\gamma \gamma \rightarrow H_i)$ and ${\cal A}_a (H_i)$ for two different values of $ \tan\beta =3$ and $\tan \beta = 10$ as functions of each Higgs boson mass ($M_{H_k}$) and CP violating phase ${\rm arg}(A_t)$ with/without chargino loop contributions. As discussed in Section II, we do not consider a very large $\tan
\beta$ case, since the $A_t$ phase is strongly constrained by the two-loop Barr-Zee type contributions to the EDMs of electron and neutron. Note that the chargino contributions to the Higgs mixing are negligible [@IbNa] for our choice of $\tan\beta = 3$ and $\tan\beta = 10$.
It turns out the Higgs sector CP violation is most prominent in the would-be CP-odd Higgs boson $H_2$ production at photon colliders. Therefore we first discuss the production of the would-be CP-odd Higgs scalar. In Fig. \[ccc\], we show the production cross section for $\gamma \gamma \rightarrow H_2$ as a function of $M_2$ in the CP conserving limit (${\rm
arg}(A_t)=0^\circ$) for $\tan\beta = 3$ (on the left side) and $\tan\beta = 10$ (on the right side), respectively. In both cases, we assumed $\mu = 1.2$ TeV, and we set $M_{H^+} = 300$ GeV so that $M_{H_2} = 291~(290)$ GeV for $\tan\beta = 3 ~(10)$, respectively. The solid (dashed) curve represents the case with (without) chargino contributions. For ${\rm arg}(A_t)=0^\circ$ (thick solid curve), $H_2$ will be the pure CP-odd state ($A$) for our parameter set (\[param\]), since we can neglect the effects of charginos on the Higgs mixing due to $\tan \beta \lesssim 20$ [@IbNa]. In this case, $\hat \sigma_0 (\gamma \gamma
\rightarrow H_2)$ has only the fermion loop contributions, since the couplings of $H_2$ to the sfermion pairs, the charged Higgs-boson and $W$-boson pairs vanish in the CP conserving limit. The cross section for $\gamma \gamma \rightarrow H_2$ without chargino loop contributions is independent of $M_2$ (the horizontal dash-dotted lines), and are quite small ($\lesssim 1$ fb). The bottom-quark contribution is negligible compared to the top-quark contribution for two reasons: (i) the small $b$ quark mass and (ii) the smaller electric charge of $b$ quark (note that the $\gamma \gamma \rightarrow H_2$ amplitude depends on $e_q^2$). For our choice of parameters, the bottom quark contribution turns out to get significant only for $\tan\beta \geq 10$, and can be safely neglected for $\tan\beta \lesssim 10$. On the other hand, the cross section for $\gamma \gamma \rightarrow H_2$ is enhanced almost by an order of magnitude when the chargino loop contributions are included. The chargino loop contributions to $\gamma \gamma \rightarrow H_2$ can not be ignored at all, if charginos are not very heavy. This is true even if we set $M_2
\ll |\mu|$ so that the wino-Higgsino mixing is not large. Still the lighter charginos are light enough ($M_2 = 150$ GeV for our parameter set) and the loop contribution is important. Also because of the $1/\tan\beta$ suppression factor for the top loop, the chargino loop contribution becomes more important for larger $\tan\beta$. Finally, as the $M_2$ increases, the lighter chargino becomes heavier and the chargino loop contribution decreases rather quickly due to the decoupling theorem. Since the chargino mass arises dominantly from SUSY breaking rather than from electroweak symmetry breaking, the decoupling of the chargino loop contribution is more effective than the top loop contribution. Also, the couplings ${\rm Im}(\kappa^2_{jj})$ decrease more quickly as functions of $\tan\beta$ compared to the loop functions as $M_2$ increases. Therefore, the difference between the cross sections for $\tan\beta=3$ and $\tan\beta = 10$ increases as $M_2$ increases.
In Fig. \[ddd\], we show the cross section for $\gamma \gamma
\rightarrow H_2$ as a function of ${\rm arg}(A_t)$ for $\tan\beta =3$ (on the left side) and $\tan\beta = 10$ (on the rigth side), respectively. The solid (the dash-dotted) curves represents the case with (without) the chargino loop contributions. For ${\rm arg}(A_t)=0^\circ$ (or $180^\circ$), the cross section is strongly enhanced by the chargino loop contributions as discussed in the previous paragraph. As ${\rm arg}(A_t)$ is turned on, the cross section is significantly enhanced even without the chargino loop contributions. This is because all the charged particles including bosons begin to contribute in the presence of CP violation in the Higgs sector. The dash-dotted curves strongly depend on ${\rm arg}(A_t)$ for the following reasons. First of all, the stop masses and the mixing angles depend on ${\rm arg}(A_t)$ very sensitively. Note that the stop masses have the $LR$ mixing term $m^2_{\tilde{t} LR}= m_t(A_t^* e^{-i\xi} - \mu/\tan\beta)$, as shown in Eq. (\[mLR\]) of the Appendix A. Since the mixing between CP-even and CP-odd neutral Higgs bosons arises from the stop loop \[see Eq. (\[m12\])\], the $A_t$ phase affects the CP mixing through Im ($A_t \mu$) and the stop masses in Eq. (\[m12\]). Also once CP is broken in the Higgs sector, all the charged particles including bosons as well as fermions contribute to $\gamma\gamma \rightarrow H_2$. Therefore the stop loop contribution will depend on ${\rm arg}(A_t)$. Still the dominant contribution comes from the chargino loops (see the solid curves in Fig. 2). The net result depends on ${\rm arg}(A_t)$ rather mildly, mainly through the ${\rm arg}(A_t)$ of the CP-odd and CP-even Higgs mixing.
Also note that the sensitivity of the cross section $\hat \sigma_0 (\gamma \gamma \rightarrow H_2)$ to ${\rm arg}(A_t)$ decreases as $\tan \beta$ increases. This tendency can be understood by the strong phase dependences of stop masses, since stop loops contribute to (i) the mixing of the CP-even and the CP-odd Higgs bosons, and (ii) the loop diagrams. The scalar-pseudoscalar mixing is typically characterized by $${\rm Im}(m_{12}^2) \propto h_t^2 {\rm arg}(A_t \mu), ~~~~$$ whose $\tan\beta$ dependence is negligible for $3\leq \tan\beta
\leq 10$ \[see Eq. (\[m12\])\]. Also the stop mass eigenvalues are sensitive to the CP phase ${\rm arg}(A_t)$ when $|A_t|= |\mu|
/\tan\beta$ due to the $LR$ mixing ($\tan \beta = 3$ for our parameter set $|A_t|=|\mu|/3=0.4$ TeV). The CP mixing would be a decreasing function of $\tan\beta$ for $\tan\beta \ge 3$, and the stop masses are less sensitive to CP phase ${\rm arg}(A_t)$ for the larger $\tan\beta=10$. Therefore, the phase dependence of the mixing would be a decreasing function of $\tan\beta$ for $\tan\beta \ge 3$. Another dependence of the cross section on ${\rm arg}(A_t)$ originates from the stop masses in the loop, which is sensitive to the phase ${\rm arg}(A_t)$ in our choice of SUSY parameter set. In other words, $\tan\beta$-dependence of the phase sensitivity comes dominantly from the stop masses as in the CP-even and CP-odd Higgs mixing. Therefore, the cross section depends on the phase ${\rm arg}(A_t)$ less sensitively when $\tan
\beta$ becomes larger for our parameter set. Finally, the heavier Higgs boson ($H$) is also strongly affected by the CP mixing, since it can have a large mixing with the CP-odd scalar $A$. The discussions for $H$ will be similar to those for $A$, and will not be repeated.
In Figs. \[aaa\] and \[bbb\], we show that the cross sections $\hat \sigma_0 (H_i)$ ($i=1,2,3$) in units of fb for five different $A_t$ phases; ${\rm arg}(A_t) = 0^\circ$ (thick solid curve), $40^\circ$ (dash-dotted curve), $80^\circ $ (dashed curve), $120^\circ$ (dotted curve) and $160^\circ$ (solid curve) for $\tan\beta = 3$ (Fig. \[aaa\]) and $\tan\beta = 10$ (Fig. \[bbb\]). We present two different cases: without the chargino loop contributions (the left column) as in Ref. [@SYChoi] and with the chargino loop contributions (the right column) with $M_2 = 150$ GeV for which $M_{\tilde{\chi}^-_1} = 146~(148.2)$ GeV for $\tan\beta =3 ~(10)$. The chargino contributions to $\gamma\gamma \rightarrow H_1$ is negligible, since $M_{H_1}$ is far below the chargino pair threshold $2 M_{\tilde{\chi}^-_1}$ for our parameter set. On the other hand, two heavier Higgs productions are affected by chargino loops by significant amounts, and we can observe rich structures in the production cross sections due to the interference of all the charged particles’ contributions. For example, the production cross sections for $\gamma \gamma \rightarrow H_2$ without the chargino loop contributions (the left columns of Figs. \[aaa\] and \[bbb\]) have only a single peak at the point $M_{H_2} = 2m_t$ for ${\rm arg}(A_t)=0^\circ$. If the chargino loop contributions are included (the right columns), the production cross sections have two comparable peaks at the point $M_{H_2} = 2M_{\tilde{\chi}^-_1}$ (lighter chargino) and $M_{H_2} = 2m_t$ in the CP-conserving limit. As the CP violating phase arg($A_t$) increases, the cross section $\hat \sigma_0 (\gamma \gamma \rightarrow H_2)$ starts to get extra contributions from the charged boson loops (involving sfermions, the charged Higgs-boson and the $W$-boson pairs) due to the mixing between the CP-odd and the CP-even neutral Higgs bosons.
For a larger $\tan\beta = 10$ (Fig. 4), there appear three qualitative differences compared to the lower $\tan\beta=3$; the effect of bottom quark loop contribution, the dominant chargino loop contributions, and the interchange of the CP-properties of the neutral Higgs bosons.
- Since the bottom quark Yukawa coupling (to the CP even Higgs boson) is proportional to $1/\cos\beta$, the bottom quark contribution can be significant in the region of large $\tan\beta$. For ${\rm arg}(A_t)=0^\circ$, the CP-odd Higgs boson $H_2$ has pseudoscalar couplings to top and bottom quarks, where the coupling of $H_2$ to top (bottom) quark is proportional to $\cot\beta$ ($\tan\beta$) \[see Eqs. (\[hff\]) and (\[R\]) of the Appendix B\]. Furthermore, there are additional differences from different electric charges of top and bottom quarks, since the $\gamma\gamma \rightarrow H_i$ amplitudes depend on $e_q^2$, which are $(2/3)^2$ vs. $(-1/3)^2$ for (s)top and (s)bottom, respectively. On the other hand, the loop functions have weaker $\tan\beta$-dependences. For our parameter set, it turns out that the bottom quark contribution begins to dominate the top quark contribution when $\tan\beta \sim 10$, and can be neglected for $\tan\beta < 10$.
- In the CP conserving limit, the chargino contribution to $\gamma\gamma \rightarrow A$ is dominant over the top quark contribution, since the latter is suppressed by $1/\tan\beta$ relative to the former, even if we assume the mixing angles in the chargino sector are $O(0.1)$. This is the reason why the top quark contribution decreases more quickly than the lighter chargino contributions as $\tan \beta$ increases in Figs. \[aaa\] and \[bbb\].
- The final point is the interchange of the CP properties of the heavier Higgs bosons $H_2$ and $H_3$ for large ${\rm arg}(A_t)$ and large $\tan\beta=10$. Since there are only fermion contributions to the CP-odd Higgs production, i.e., two peaks at $M_{H_i}=2M_{\tilde{\chi}^-_1}$ and $2m_t$, we can find from Fig. \[bbb\] that $H_3$ for ${\rm
arg}(A_t)=160^\circ$ has the same CP-odd property as $H_2$ for ${\rm arg}(A_t)=0^\circ$. This can be checked even more easily by using the polarization asymmetry ${\cal A}_3$ which is +1($-1$) for a CP-even (CP-odd) Higgs boson, as discussed below in relation with polarization asymmetries (Figs. 5–7).
The importance of chargino loop contributions for $H_3$ production is also similar to the case of $H_2$ production as discussed above, and we will not repeat it again.
The number of events is determined by the combination of the luminosity and the cross section for $\gamma \gamma \rightarrow H_2$. Although the photon beam luminosity depends on many parameters, if one only consider the high energy part of the generated photons, the 0.3 conversion factor and the comparable photon spot size to electron beam, the approximate luminosity of $\gamma \gamma$ collider [@ggcol] is \^ 0.3\^2[L]{}\^[e e]{}\_[geom]{} 0.1[L]{}\^[e e]{}\_[geom]{}, where ${\cal L}^{e e}_{\rm geom}$ is the luminosity of $e^+ e^-$ collider. Taking $100~{\rm fb}^{-1}$ as a nominal integrated luminosity in the $\gamma\gamma$ mode, we can infer from Figs. \[aaa\] and \[bbb\] that the maximum number of events for the CP-odd Higgs boson is approximately 100 (10) per a year for $\tan\beta=3$ ($\tan\beta=10$), when the unpolarized cross section does not contain chargino-loop contributions. However, the chargino-loop contributions enhance the maximum number of events as approximately 880 (710) for $\tan\beta=3$ ($\tan\beta=10$). Hence, the chargino loop contributions for the production of the would-be CP-odd Higgs boson can be significant at the $\gamma \gamma$ collider for larger $\tan\beta$.
In Fig. \[eee\], we show three polarization asymmetries of $H_2$ as functions of ${\rm arg}(A_t)$ for $\tan\beta = 3$ (the left column) and $\tan\beta = 10$ (the right column). As in Figs. 1 and 2, we set $M_{H^+} =
300$ GeV so that $M_{H_2} = 291~(290)$ GeV for $\tan\beta = 3 ~(10)$, respectively. The case with (without) the chargino loop is represented by solid (dash-dotted) curves. We have fixed $M_2 = 150$ GeV as before. The polarization asymmetries ${\cal A}_i (\Phi)$’s satisfy the following relations: \_[1,2]{}() = - [A]{}\_[1,2]{}(360\^-), [A]{}\_[3]{}() = + [A]{}\_[3]{}(360\^-), where $\Phi={\rm arg}(A_t \mu) + \xi$ with $\xi=0$. Namely, ${\cal A}_{1,2}$ are CP-odd observables (antisymmetric about $\Phi=180^\circ$) and ${\cal A}_3$ is a CP-even observable (symmetric about $\Phi=180^\circ$). Note that the chargino loops not only enhance the cross section but also affect the polarization asymmetries by significant amounts.
In Fig. \[fff\], we show the polarization asymmetries ${\cal A}_a (H_j)$ as functions of the neutral Higgs masses for ${\rm arg}(A_t) = 0^{\circ}$, $40^{\circ}$, $80^{\circ}$, $120^{\circ}$ and $160^{\circ}$ with $\tan\beta=3$, including all the charged particles in the loops. The lightest Higgs boson $H_1$ still behaves like a CP-even scalar, since $-0.03\% \lesssim {\cal A}_1 \le 0$, $0 \le {\cal A}_2 \lesssim 0.4\%$, and ${\cal A}_3 \simeq 1$. On the other hand, the heavier $H_2$ and $H_3$ are generically admixtures of CP-even and CP-odd states if the phase of $A_t$ does not vanish. For $H_2$ and $H_3$, chargino, top and stop loops give main contributions to the asymmetries above the chargino-pair threshold, but the chargino and $W^\pm$ loop contributions affect them below the chargino-pair threshold.
In Fig. \[ggg\], we present the polarization asymmetries for $\tan\beta=10$. Again, the lightest Higgs boson $H_1$ behaves like a CP-even scalar for the larger $\tan\beta$, since $-0.1\% \lesssim {\cal A}_1 \le 0$, $0 \le {\cal A}_2 \lesssim 0.3\%$, and ${\cal A}_3 \simeq 1$. If $\tan\beta$ becomes larger, the top (stop) loop contribution is accompanied by the bottom (sbottom) contribution to the polarization asymmetries of the heavier Higgs bosons $H_2$ and $H_3$. Fig. \[ggg\] indicates that as the CP violating phase ${\rm arg}(A_t)$ increases for the case of large $\tan\beta$, the value of the asymmetry ${\cal A}_3$ of $H_2$ approaches that of $H_3$ at ${\rm arg}(A_t)=0^\circ$ and vice versa, i.e., the CP-properties of the heavier Higgs bosons $H_2$ and $H_3$ are interchanged.
From Figs. \[fff\] and \[ggg\], the polarization asymmetry ${\cal A}_2 (H_1)$ is the most sensitive CP observable in detecting the CP violation of the lightest Higgs boson for both small and large $\tan\beta$, when the chargino contributions are included. This result is different from the first paper of Ref. [@SYChoi], where charginos are neglected by assuming they are very heavy, and thus ${\cal A}_2$ (${\cal A}_1$) is the most powerful CP observable for $\tan\beta=3~(\tan\beta=10)$. Unfortunately the asymmetry itself is very small so that it would not be easy to find nonzero ${\cal A}_2 (H_1)$. Still asymmetries for heavier neutral Higgs bosons can be sizable and thus be used as the probes of Higgs sector CP violation if they can be produced with high statistics at NLCs. Therefore, we need to prepare the colliding photon beams with large linear polarizations as well as high center of mass energy $\sqrt{s_{\gamma\gamma}}$ in order to produce neutral Higgs bosons and determine their CP properties in a model independent manner.
Conclusions
===========
In this work, we presented a comprehensive analysis of the neutral Higgs boson productions through $\gamma \gamma \rightarrow
H_{i=1,2,3}$ in the presence of the Higgs sector CP-violation of the MSSM. In particular, we have included the chargino loop contributions as well as the contributions from squarks, $W^\pm$ and charged Higgs particles. In many scenarios of SUSY breaking, charginos are not too heavy that their effects are generically important. First of all, the production of the would-be CP-odd $H_2$ boson is enhanced by an order of magnitude when chargino loop contributions are included even without the Higgs sector CP violation. If the phase of the $A_t$ parameter is turned on, CP violation in the Higgs sector become very rich in the structures. This is also true of the case of the heaviest Higgs boson $H_3$. Also the polarization asymmetries are affected by the Higgs sector CP violation.
If the $A_t$ parameter has a large CP violating phase, its effects can appear in various physical observables: the Higgs sector CP-violation as discussed in this work, and also the direct CP violation in $B\rightarrow X_s \gamma$ [@baek], for example. Since the latter is an indirect signature, it is important to probe SUSY CP violation in a direct way. Thus it is important to probe CP violation from the soft SUSY breaking sector such as arg($A_t$) in the Higgs sector CP violation by using $\gamma\gamma$ colliders as discussed in this work. In this regards, the $\gamma\gamma$ mode at NLC with high $\sqrt{s_{\gamma\gamma}}$ and luminosity, and high quality beam polarizations will be indispensable for this purpose by measuring the cross sections of $\gamma \gamma \rightarrow H_i$ ($i=1,2,3$) and three asymmetries ${\cal A}_a (H_j)$ ($a,j=1,2,3$) in the MSSM.
We are grateful to S.Y. Choi and Jae Sik Lee for useful communications and to W.Y. Song for discussions. This work was supported in part by BK21 Haeksim program of MOE, and by KOSEF through CHEP at Kyungpook National University.
Charginos and scalar tops
=========================
The chargino mass matrix in the $(\tilde{W}^+, \tilde{H}^+)$ basis [@Haber] is $${\cal M_C} =
\left(
\begin{array}{cc}
{M}_2 & \sqrt{2} e^{-i \xi} m_W \cos \beta
\\
\sqrt{2} m_W \sin \beta & \mu
\end{array}
\right),$$ where ${M}_2 > 0$ and $\mu$ are gaugino and Higgsino masses, and $e^{+i \xi}$ is the the phase of the up-type Higgs VEV [@PiWa]. Since the mass matrix $X$ is a general complex matrix, it is diagonalized by a biunitary transformation: $$\label{M_D}
U^* X V^{-1} \equiv {\rm diag}(M_{\tilde{\chi}^-_1}, M_{\tilde{\chi}^-_2}),$$ with $M_{\tilde{\chi}^-_2} \ge M_{\tilde{\chi}^-_1} \ge 0$. In order for $M_{\tilde{\chi}^-_{i=1,2}}$ to be positive, we define the unitary matrix $U$ as a product of two unitary matrices $$\label{H}
U \equiv HU'.$$ The angles $\theta_1$ and $\phi_1$ of the unitary matrix U’ = (
[cc]{} & e\^[+i\_1]{}\
- e\^[-i\_1]{} &
), are given by \_1 &=& ,\
\_1 &=& , where $\theta_\mu={\rm arg}(\mu)$. The unitary mixing matrix $V$ is $$V =
\left(
\begin{array}{cc}
\cos \frac{\theta_2}{2} & \sin \frac{\theta_2}{2} e^{-i\phi_2} \\
-\sin \frac{\theta_2}{2} e^{+i\phi_2} & \cos \frac{\theta_2}{2}
\end{array}
\right),$$ where $$\begin{aligned}
\tan \theta_2 &=& \frac{2 \sqrt{2}m_W \left[ M_2^2 \sin^2\beta
+ |\mu|^2\cos^2\beta + M_2 |\mu| \sin2\beta \cos(\theta_\mu
+ \xi) \right]^{1/2}}{M_2^2 - |\mu|^2 + 2 m_W^2 \cos2\beta},
\\
\tan\phi_2 &=& \frac{M_2 \sin\xi \sin\beta
- |\mu|\sin \theta_\mu \cos\beta}{M_2 \cos\xi \sin\beta
+ |\mu|\cos \theta_\mu \cos\beta}.\end{aligned}$$ By using the unitary matrix $H = {\rm diag}(e^{i\gamma_1}, e^{i \gamma_2})$, where $\gamma_{1,2}$ are the phases of the diagonal elements of $U'^* X V^{-1}$, we finally obtain $$U^*XV^{-1} = {\rm diag}(M_{\tilde{\chi}^-_1}, M_{\tilde{\chi}^-_2}).$$ And the mass eigenvalues of charginos are M\_[\^-\_1,\^-\_2]{}\^2 &=& ( M\_2\^2 + ||\^2 + 2m\_W\^2 ) \^[1/2]{}. Note that the mass eigenvalues and the mixing angles depend on the CP violating phases $\xi$ and $\theta_\mu$.
The stop $(\rm{mass})^2$ matrix ${\cal M}_{\tilde t}^2$ [@Bae] is written as \[stop mass\] [L]{}\_[mass]{}\^[eff]{} &=& - ( \^\*\_L \^\*\_R ) [M]{}\^2\_ (
[c]{} \_L\
\_R
)\
&=& - ( \^\*\_L \^\*\_R ) (
[cc]{} m\_[L]{}\^2 &m\_[LR]{}\^2\
m\_[LR]{}\^[2\*]{} &m\_[R]{}\^2
) (
[c]{} \_L\
\_R
), where \[mL\] m\_[L]{}\^2 &=& M\_[\_L]{}\^2 + m\_t\^2 + m\_Z\^2 (-\^2\_W),\
\[mR\] m\_[R]{}\^2 &=& M\_[\_R]{}\^2 + m\_t\^2 + m\_Z\^2 \^2\_W,\
\[mLR\] m\_[LR]{}\^2 &=& m\_t (A\_t\^\* e\^[-i ]{}-). The stop mixing angle $\theta_{\tilde t}$ is \[theta t\] \_[t]{} = ( ). The relations between the mass and the weak eigenstates of stops are given by \[mw transf1\] &=& \_L \_[t]{} +\_R e\^[-i\_[t]{}]{} \_[t]{},\
\[mw transf2\] [\_2]{} &=& -\_L e\^[i\_[t]{}]{} \_[t]{} +\_R \_[t]{}, where $\beta_{\tilde t}=-\arg (m_{\tilde{t} LR}^2)$. The mass eigenvalues of the lighter and heavier stops are \[mt1,2\] m\^2\_[[[t]{}\_1]{},[[t]{}\_2]{}]{} = . Note that $m_{{{\tilde t}_1},{{\tilde t}_2}}^2$ is dependent on the CP violating phases, $\arg( A_t )$ and $\arg( \mu )$ due to $m_{{\tilde t}LR}^2$ in Eq. (\[mLR\]).
Relevant couplings
==================
In this sections, we list the couplings relevant to $\gamma\gamma \rightarrow H_i$ that appear in Table 1.
- Higgs-fermion-fermion couplings: \[hff\] [L]{}\_[H|[f]{}f]{} = - [g m\_f 2 m\_W]{} |[f]{} f H\_i , where $$\begin{aligned}
\label{R}
R_\beta^d & = & \bar{R}_\beta^u = \cos\beta \equiv c_\beta,
%\nonumber \\
~~~~R_\beta^u = \bar{R}_\beta^d = \sin\beta \equiv s_\beta,
\nonumber \\
v_f^d & = & O_{2,i},~~~~v_f^u = O_{3,i},
%\nonumber \\
~~~~a_f^d = a_f^u = O_{1,i}.\end{aligned}$$ Here the matrix $O$ diagonalizes the Higgs mass matrix as in Eq. (5). In the presence of Higgs sector CP violation, the Higgs bosons couple with both CP-even and CP-odd bilinears, $\bar{f} f$ and $\bar{f} \gamma_5 f$, simultaneously.
- The Higgs-$W$-$W$ couplings are determined by the gauge couplings: \_[HW\^+ W\^-]{} = g m\_W ( c\_O\_[2,i]{} + s\_O\_[3,i]{} ) H\_i W\_\^+ W\^[-]{} .
- The Higgs-sfermion-sfermion couplings: \_[H\_i \_j \_k]{} = g\^i\_[\_j \_k]{} \_j\^\* \_k H\_i , with $$g^i_{\tilde{f}_j \tilde{f}_k} = \tilde{C}^f_{\alpha;\beta\gamma}
O_{\alpha , i} ( U_f )^*_{\beta j} ( U_f )_{\gamma k} .$$ The matrix $U_f$ diagonalize the sfermion mass matrix: $$U_f^{\dagger} M_{\tilde{f}}^2 U_f = {\rm diag}( m_{\tilde{f}_1}^2,
m_{\tilde{f}_2}^2 )$$ with $m_{\tilde{f}_1} \le m_{\tilde{f}_2}$. The indices $\alpha$ and $\{\beta, \gamma\}$ label the three neutral Higgs bosons $(a, \phi_1, \phi_2 )$ and the sfermion chiralities $\{L, R\}$, respectively. The explicit expressions for $\tilde{C}^f_{\alpha;\beta\gamma} $ can be found in Ref. [@JSLee].
- The $H_i - H^+ - H^-$ couplings are determined by the Higgs potential. If we define \_[H\_i H\^+ H\^- ]{} = v C\_i H\_i H\^+ H\^- , then the couplings $C_i$ are given by [@SYChoi] $$C_i = \sum_{\alpha=1,2,3} O_{\alpha,i} c_\alpha$$ with $$\begin{aligned}
c_1 & = & 2 s_\beta c_\beta {\rm Im}( \lambda_5 e^{2 i \xi} )
- s_\beta^2 {\rm Im}( \lambda_6 e^{i \xi} )
- c_\beta^2 {\rm Im}( \lambda_7 e^{i \xi} ),
\nonumber \\
c_2 & = & 2 s_\beta^2 c_\beta \lambda_1 + c_\beta^3 \lambda_3 - s_\beta^2
c_\beta \lambda_4 - 2 s_\beta^2 c_\beta {\rm Re} ( \lambda_5 e^{2 i \xi} )
\nonumber \\
&&+ s_\beta ( s_\beta^2 - 2 c_\beta^2 ) {\rm Re} ( \lambda_6 e^{i \xi} )
+ s_\beta c_\beta^2 {\rm Re} ( \lambda_7 e^{i \xi} ),
\nonumber \\
c_3 & = & 2 c_\beta^2 s_\beta \lambda_2 + s_\beta^3 \lambda_3 - c_\beta^2
s_\beta \lambda_4 - 2 c_\beta^2 s_\beta {\rm Re} ( \lambda_5 e^{2 i \xi} )
\nonumber \\
&&+ c_\beta s_\beta^2 {\rm Re} ( \lambda_6 e^{i \xi} )
+ c_\beta ( c_\beta^2 - 2 s_\beta^2 ) {\rm Re} ( \lambda_7 e^{i \xi} ).\end{aligned}$$
J.F. Gunion, H.E. Haber, G. Kane, and S. Dawson, [*The Higgs Hunter’s Guide*]{} (Addison-Wesley Publishing Company, 1990).
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.
=7.0cm =7.0cm
=7.0cm =7.0cm
=7.0cm =7.0cm
---------------------------- --------------------------------------------------------------------
$A$’s and $B$’s Expressions
$A_i^f$ $- 2 ( \sqrt{2} G_F )^{1/2} M_{H_i} N_c e_f^2 \left(
{v_f^i \over R_\beta^f} \right) F_{sf} ( \tau_{if} )$
$A_i^{\tilde{f}_j}$ $ {M_{H_i} N_c e_f^2 g^i_{\tilde{f}_j
\tilde{f}_j } \over 2 m_{\tilde{f}_j}^2} F_0 ( \tau_{i \tilde{f} }
)$
$A_i^{W^\pm}$ $( \sqrt{2} G_F )^{1/2} M_{H_i} \left( c_\beta
O_{2,i} + s_\beta O_{3,i} \right) F_1 ( \tau_{iW} )$
$A_i^{H^\pm}$ ${M_{H_i} v C_i \over 2 m_{H^\pm}^2} F_0 ( \tau_{i
H} )$
$A_i^{\tilde{\chi}^\pm_j}$ $2 {\rm Re}( \kappa_{jj}^i )
{M_{H_i} \over M_{\tilde{\chi}^-_j}} F_{sf} ( \tau_{i
\tilde{\chi}^\pm_j} ) $
$B_i^f$ $ 2 ( \sqrt{2} G_F )^{1/2} M_{H_i} N_c e_f^2 \left( {
\overline{R_\beta^f} a_f^i \over R_\beta^f} \right) F_{pf} (
\tau_{if} )$
$B_i^{\tilde{\chi}^\pm_j}$ $- 2 {\rm Im} ( \kappa_{jj}^i )
{M_{H_i} \over M_{\tilde{\chi}^-_j} }F_{pf}( \tau_{i
\tilde{\chi}^\pm_j} )$
---------------------------- --------------------------------------------------------------------
: The amplitudes $A_i^X$’s and $B_i^X$’s, where $i$ labels three neutral Higgs bosons, and $X$ labels the species of charged particles in the triangle loop (with $\tau_{iX} \equiv
M_{H_i}^2/4m_{X}^2$).[]{data-label="table1"}
------------------- ------------------------------------------------------
$F$’s Definitions
$F_{sf} ( \tau )$ $\tau^{-1} \left[ 1 + ( 1 - \tau^{-1} ) f ( \tau )
\right]$
$F_{pf} ( \tau )$ $\tau^{-1} f ( \tau )$
$F_0 ( \tau )$ $\tau^{-1} \left[ -1 + \tau^{-1} f ( \tau ) \right]$
$F_1 ( \tau )$ $2 + 3 \tau^{-1} + 3 \tau^{-1} ( 2 - \tau^{-1} )
f ( \tau )$
------------------- ------------------------------------------------------
: Form factor loop functions $F$’s in terms of the scaling function $f(\tau)$ defined in Eq. (12).[]{data-label="table2"}
[^1]: Former address: Department of Physics, KAIST, Daejeon, Korea ([email protected])
[^2]: E-mail address: [email protected]
[^3]: E-mail address: [email protected]
[^4]: We follow the notations of the recent third paper of Ref. [@PiWa].
[^5]: Refer to the third paper of Ref. [@PiWa].
|
---
abstract: 'During the last 70 years, the quantum theory of angular momentum has been successfully applied to describing the properties of nuclei, atoms, and molecules, their interactions with each other as well as with external fields. Due to the properties of quantum rotations, the angular momentum algebra can be of tremendous complexity even for a few interacting particles, such as valence electrons of an atom, not to mention larger many-particle systems. In this work, we study an example of the latter: a rotating quantum impurity coupled to a many-body bosonic bath. In the regime of strong impurity-bath couplings the problem involves addition of an infinite number of angular momenta which renders it intractable using currently available techniques. Here, we introduce a novel canonical transformation which allows to eliminate the complex angular momentum algebra from such a class of many-body problems. In addition, the transformation exposes the problem’s constants of motion, and renders it solvable exactly in the limit of a slowly-rotating impurity. We exemplify the technique by showing that there exists a critical rotational speed at which the impurity suddenly acquires one quantum of angular momentum from the many-particle bath. Such an instability is accompanied by the deformation of the phonon density in the frame rotating along with the impurity.'
author:
- Richard Schmidt
- Mikhail Lemeshko
title: 'Deformation of a quantum many-particle system by a rotating impurity'
---
Introduction
============
An important part of modern condensed matter physics deals with so-called ‘impurity problems’, aiming to understand the behavior of individual quantum particles coupled to a complex many-body environment. The interest in quantum impurities goes back to the classic works of Landau, Pekar, Fröhlich, and Feynman, who showed that propagation of electrons in crystals is largely affected by the quantum field of lattice excitations and can be rationalized by introducing the quasiparticle concept of the polaron [@LandauPolaron; @LandauPekarJETP48; @FrohlichAdvPhys54; @FeynmanPR55]. In turn, the properties of a quantum many-body system can be drastically modified by the presence of impurities. The most known examples are the Kondo effect [@KondoPTP64] – suppression of electron transport due to magnetic impurities in metals – and the Anderson orthogonality catastrophe which leads to the edge singularities in the X-ray absorption spectra of metals [@AndersonPRL67].
In many instances, the impurities – even those possessing an internal structure – can be accurately described as point-like particles. The latter is justified by the separation of the energy scales inherent to the impurity and the surrounding bath. A well-known example is that of Bose- and Fermi-polarons realized in cold atomic gases by a number of groups [@ChikkaturPRL00; @SchirotzekPRL09; @PalzerPRL09; @KohstallNature12; @KoschorreckNature12; @SpethmannPRL12; @FukuharaNatPhys13; @ScellePRL13; @Cetina15; @MassignanRPP14]. There, the spherically symmetric ground state of an alkali atom lies hundreds of THz lower than any of its electronically excited states. Given ultracold collision energies, such an energy gap renders all the processes happening inside of an atom irrelevant.
More complex systems, such as molecules, are extended objects and therefore possess a number of fundamentally different types of internal motion. The latter stem from the relative motion of the nuclei, such as rotation and vibration, which couple to each other as well as to the electronic spin and orbital degrees of freedom [@LemKreDoyKais13; @KreStwFrieColdMol; @BernathBook; @LevebvreBrionField2]. This results in a rich low-energy dynamics which is highly susceptible to external perturbations. Moreover, in many experimental realizations molecular rotation is coupled to a phononic bath pertaining to the surrounding medium such as superfluid helium [@ToenniesAngChem04], rare-gas matrix [@MatrixIsolationBook], or a Coulomb crystal formed in an ion trap [@WillitschIRPC12], which needs to be properly accounted for by a microscopic theory.
The concept of orbital angular momentum, however, goes far beyond physically rotating systems and is being used to describe e.g. the excited-state electrons in solids, whose motion is perturbed by lattice vibrations [@Mahan90], or Rydberg atoms immersed into a Bose-Einstein condensate [@Greene2000; @BalewskiNature13]. Despite the ubiquitous use of the angular momentum concept in various branches of physics, a versatile theory describing the redistribution of orbital angular momentum in quantum many-body systems has not yet been developed.
Recently, we have undertaken the first step towards such a theory by deriving a generic Hamiltonian which describes the coupling of an $SO(3)$-symmetric impurity – a quantum rotor – with a bath of harmonic oscillators [@SchmidtLem15]. We have shown that the problem can be approached most naturally by introducing the quasiparticle concept of the *‘angulon’* – a quantum rotor dressed by a quantum field. The angulon is an eigenstate of the total angular momentum of the system, which remains a conserved quantity in the presence of the impurity-bath interactions. It was found that even single-phonon excitations of the bath alone are capable of drastically modifying the rotational spectrum of the impurity, which manifests itself in the emerging Many-Body-Induced Fine Structure [@SchmidtLem15].
Here we demonstrate that rotation of an anisotropic impurity can, in turn, substantially alter the collective state of a many-particle system. The effects are most significant in the regime of strong correlations, which however requires adding an infinite number of angular momentum vectors pertaining to possible many-body states. The resulting angular momentum algebra involves Wigner $3nj$-symbols [@VarshalovichAngMom] of an arbitrarily high order and is therefore intractable using standard techniques. In order to overcome this problem, here we introduce a canonical transformation, which, to our knowledge, has never appeared in the literature before. The transformation renders the Hamiltonian independent of the impurity coordinates, thereby eliminating the complex angular momentum algebra from the many-body problem. Furthermore, the transformation singles out the conserved quantities of the many-body problem and renders it solvable exactly in the limit of a slowly rotating impurity.
The transformation makes it apparent that there exists a critical rotational speed which leads to an instability, accompanied by a discontinuity in the many-particle spectrum. Unlike in the vortex instability, originating from rotation of a condensate around a given axis [@Pitaevskii2003], the instability we uncover here corresponds to the finite transfer of three-dimensional angular momentum between the impurity and the bath. It exists solely due to the discrete energy spectrum inherent to quantum rotation. We demonstrate that the emerging instability is ushered by a macroscopic deformation of the surrounding bath, i.e. the phonon density modulation in the frame co-rotating with the impurity.
The canonical transformation
============================
We start from the general Hamiltonian of the angulon problem, as defined in Ref. [@SchmidtLem15]: $$\begin{gathered}
\label{Hamil}
\hat H= B \mathbf{\hat{J}^2} + \sum_{k \lambda \mu} \omega_k {\hat b^\dagger}_{k\lambda \mu} {\hat b}_{k\lambda \mu} \\+ \sum_{k \lambda \mu} U_\lambda(k) \left[ Y^\ast_{\lambda \mu} (\hat \theta,\hat \phi) {\hat b^\dagger}_{k \lambda \mu}+ Y_{\lambda \mu} (\hat \theta,\hat \phi) {\hat b}_{k \lambda \mu} \right],\end{gathered}$$ where $Y_{\lambda \mu} (\hat \theta,\hat \phi)$ are the spherical harmonics [@VarshalovichAngMom] depending on the molecular angle *operators* $\hat \theta$ and $\hat \phi$, $\sum_k\equiv\int dk$, and $\hbar \equiv 1$.
The first term of Eq. (\[Hamil\]) corresponds to the kinetic energy of the translationally-localized linear-rotor impurity, with $B$ the rotational constant and $\mathbf{\hat{J}}$ the angular momentum operator. In the absence of an external bath, the impurity eigenstates, $\vert j, m \rangle$, are labeled by the angular momentum, $j$, and its projection, $m$, onto the laboratory-frame $z$-axis. Unperturbed rotational states form $(2j+1)$-fold degenerate multiplets with energies $E_j = B j(j+1)$ [@LevebvreBrionField2; @BernathBook; @LemKreDoyKais13].
![\[transf\] Action of the canonical transformation, Eq. (\[Transformation\]), on the many-body system. Left: in the laboratory frame, $(x, y, z)$, the molecular angular momentum, $\mathbf{J}$, combines with the bath angular momentum, $\boldsymbol{\Lambda}$, to form the total angular momentum of the system, $\mathbf{L}$. Right: after the transformation, the bath degrees of freedom are transferred to the rotating frame of the molecule, $(x', y', z')$. As a result, the molecular angular momentum in the transformed space coincides with the total angular momentum of the system in the laboratory frame.](transformation.pdf){width="\linewidth"}
The second term of Eq. (\[Hamil\]) represents the kinetic energy of the bosonic bath, where the corresponding creation and annihilation operators, ${\hat b^\dagger}_\mathbf{k}$ and ${\hat b}_\mathbf{k}$, are expressed in the spherical basis, ${\hat b^\dagger}_{k\lambda \mu}$ and ${\hat b}_{k\lambda \mu}$. Here $k=|\mathbf{k}|$, while $\lambda$ and $\mu$ define, respectively, the boson angular momentum and its projection onto the laboratory $z$-axis, see Appendix \[sec:Bklm\] for details.
The last term of Eq. (\[Hamil\]) describes the interaction between the impurity and the bath. The angular-momentum-dependent coupling strength, $U_\lambda(k)$, depends on the microscopic details of the two-body interaction between the impurity and the bosons. For example, in Ref. [@SchmidtLem15] we showed that for a linear rotor immersed into a Bose gas, the couplings are given by $$\label{Ulamk}
U_\lambda(k) = u_\lambda \left[\frac{8 k^2\epsilon_k \rho }{\omega_k(2\lambda+1)}\right]^{1/2} \int dr r^2 f_\lambda(r) j_\lambda (kr).$$ This assumes that in the impurity frame, the interaction between the rotor and a bosonic atom is expanded as $$\label{VimpBos}
V_\text{imp-bos} (\mathbf{r'}) = \sum_\lambda u_\lambda f_\lambda (r') Y_{\lambda 0} ( \Theta', \Phi' ),$$ with $u_\lambda$ and $f_\lambda(r')$ giving the strength and shape of the potential in the corresponding angular momentum channel. The prefactor of Eq. (\[Ulamk\]) depends on the bath density, $\rho$, the kinetic energy of the bare atoms, $\epsilon_k$, and the dispersion relation of the bosonic quasiparticles, $\omega_k$. Since the angulon Hamiltonian (\[Hamil\]) describes the interactions between a quantum rotor and a bosonic bath of, in principle, any kind, we will approach it from an entirely general perspective, exemplifying the couplings by the ones of Eq. (\[Ulamk\]).
Many-body problems such as given by the Hamiltonian are typically hard to solve. The conventional approaches to tackle them include, when applicable, perturbation theory, renormalization group, or in principle uncontrolled methods such as those based on the selective diagram resummations, as well as purely numerical techniques. An alternative, actively used since the development of classical mechanics, involves canonical transformations of the underlying Hamiltonian [@WagnerUnitaryBook; @LLI]. Here the idea is to partially diagonalize the Hamiltonian and/or to expose the constants of motion, which allows to reveal some of the eigenstates’ properties exactly. In the context of impurity problems, typical approaches employ the collective bath variables as a generator of the symmetry transformations, as it has been used e.g. in the polaron theory [@Lee1953; @GirardeauPF61; @Devreese13].
In the angulon problem discussed in this paper, the total angular momentum is a good quantum number. However, due to the coupling of bath degrees of freedom with the impurity coordinates, as given by the third term of Eq. , this conservation law is not apparent. Here we introduce a canonical transformation which makes this constant of motion explicit and allows to achieve several other goals listed below. The corresponding operator $\hat S$ uses the composite angular momentum of the bath as a generator of rotation, which transfers the environment degrees of freedom into the frame co-rotating along with the quantum rotor. The transformation is given by: $$\label{Transformation}
\fbox{$ \hat{S} = e^{- i \hat\phi \otimes \hat \Lambda_z} e^{- i \hat\theta \otimes \hat\Lambda_y} e^{- i \hat\gamma \otimes\hat \Lambda_z} \\$}$$ The angle operators, $(\hat\phi, \hat\theta, \hat\gamma)$, act in the Hilbert space of the rotor, and $$\label{Lambda}
\hat {{\mathbf{\Lambda}}}=\sum_{k\lambda\mu\nu}{\hat b^\dagger}_{k\lambda\mu}\boldsymbol\sigma^{\lambda}_{\mu\nu}{\hat b}_{k\lambda \nu}$$ is the collective angular momentum operator of the many-body bath, acting in the Hilbert space of the bosons. Here $\boldsymbol\sigma^{\lambda}$ denotes the vector of matrices fulfilling the angular momentum algebra in the representation of angular momentum $\lambda$.
The transformation brings the Hamiltonian (\[Hamil\]) into the following form: $$\begin{gathered}
\label{transH}
\hat{\mathcal{H}} \equiv \hat S^{-1} \hat H \hat S= B (\hat{\mathbf{J}}' - \hat{\mathbf{\Lambda}})^2 \\ + \sum_{k\lambda\mu}\omega_k {\hat b^\dagger}_{k\lambda\mu}{\hat b}_{k\lambda\mu} + \sum_{k\lambda} V_\lambda(k) \left[{\hat b^\dagger}_{k\lambda0}+{\hat b}_{k\lambda0}\right]\end{gathered}$$ Here $V_\lambda(k)=U_\lambda(k) \sqrt{(2\lambda+1)/(4\pi)}$ and $\hat{\mathbf{J}}'$ is the ‘anomalous’ angular momentum operator acting in the rotating frame of the impurity. Since the components of $\hat{\mathbf{J}}'$ act in the body-fixed frame, they obey anomalous commutation relations [@BiedenharnAngMom; @LevebvreBrionField2] as opposed to the ‘ordinary’ angular momentum operator, $\hat{\mathbf{J}}$ of Eq. (\[Hamil\]), which acts in the laboratory frame. The details of the derivation, as well as the properties of the $\hat{\mathbf{J}}'$ operator are presented in Appendix \[app:transfo\].
Let us now discuss the physical meaning of the transformation $\hat S$. In order to describe the composite system, it is natural to introduce two coordinate frames, as schematically shown in Fig. \[transf\]. The laboratory frame, $(x,y,z)$, is singled out by the collective state of the bosons, while the rotating impurity frame, $(x',y',z')$, is defined by the instantaneous orientation of the molecular axes. The relative orientation of the two frames is given by the eigenvalues of the Euler angle operators, $(\hat\phi, \hat\theta, \hat\gamma)$, acting in the impurity Hilbert space. The $\hat S$ operator transforms the many-body state of the bosons into the rotating molecular frame, using $\hat{{\mathbf{\Lambda}}}$ as a generator of quantum rotations. In turn, as we show below, the molecular state in the transformed frame becomes an eigenstate of the total angular momentum of the system, which is a constant of motion.
Introducing the body-fixed coordinate frame bound to the impurity makes explicit an additional quantum number, $n$, which gives the projection of the angular momentum onto the rotor axis $z'$. The angular momentum basis states, $\vert j, m, n \rangle$, are therefore the eigenstates of the $\hat{\mathbf{J}}^2$, $\hat{J}_z$, and $\hat{J}'_z$ operators, as given by Eqs. (\[J2eig\])–(\[JPrzeig\]) of the Appendix.
For a linear-rotor molecule in the absence of a bath the total angular momentum, $\hat{\mathbf{L}} = \hat{\mathbf{J}} + \hat{\mathbf{\Lambda}}$, coincides with $\hat{\mathbf{J}}$. Therefore, $\hat{\mathbf{L}}$ is perpendicular to the molecular axis $z'$, resulting in $n=0$. With the bosons present, the total angular momentum is no longer perpendicular to $z'$, providing the molecular state with nonzero $n$ in the transformed frame. In other words, the transformation (\[Transformation\]) converts a linear-rotor molecule into an effective ‘symmetric top’ [@LevebvreBrionField2] by dressing it with a boson field.
Compared to the original Hamiltonian, Eq. (\[Hamil\]), the transformed Hamiltonian, Eq. (\[transH\]), possesses the following properties:
1. *$\hat{\mathcal{H}}$ is explicitly expressed through the total angular momentum, which is a constant of motion.* Due to the isotropy of space, the eigenstates of the original Hamiltonian, $\hat H$, are simultaneous eigenstates of the total angular momentum operators, $\hat{\mathbf{L}}^2$ and $\hat{\mathbf{L}}_z$, and thus can be labeled as $\vert L, M \rangle$. The transformed states, $\hat S^{-1} \vert L, M \rangle$, are hence the eigenstates of the transformed Hamiltonian, $\hat{\mathcal{H}}$. As detailed in Appendix \[app:moltrans\], these transformed states are also eigenstates of the $\hat{\mathbf{J}}'^2$ operator with the eigenvalues $L(L+1)$, corresponding to the total angular momentum. Consequently, the $\hat{\mathbf{J}}'^2$ operator in Eq. (\[transH\]) can be replaced by the classical number $L(L+1)$.
2. \[2lab\] *$\hat{\mathcal{H}}$ does not contain the impurity coordinates $(\hat \theta,\hat \phi)$, which allows to bypass the intractable angular momentum algebra, arising from the impurity-bath coupling.* The angle operators of the original Hamiltonian, Eq. (\[Hamil\]), couple the impurity states with every single boson excitation, which results in the problem of adding an infinite number of angular momenta in three dimensions. The latter involves working with Wigner $3nj$-symbols of an arbitrarily large order. In the transformed Hamiltonian, on the other hand, the problem is reduced to adding the angular-momentum projections of the impurity and the bath. There, the impurity-bath coupling, $\hat{\mathbf{J}}' \cdot \hat{\mathbf{\Lambda}}$, has the form of spin-orbit interaction and does not lead to an involved angular momentum algebra.
3. *$\hat{\mathcal{H}}$ can be solved exactly in the limit of a slowly rotating impurity, $B\to 0$*, see Sec. \[sec:deform\].
4. *$\hat{\mathcal{H}}$ allows to find the eigenstates containing an infinite number of phonon excitations, which is crucial e.g. to account for the macroscopic deformation of the condensate.* This follows directly from sub. \[2lab\], and is detailed in Sec. \[sec:deform\].
5. *$\hat{\mathcal{H}}$ contains information about the deformation of the condensate in the rotating impurity frame.* Compared to the laboratory frame, where the deformation of the bath is averaged over the angles, this provides an additional insight into the nature of the many-body state and, consequently, into the origin of the angulon instability, discussed in Sec. \[sec:deform\].
![\[energies\] Change of the angulon spectral function, $A_L (\omega)$, where $\omega=E-B L(L+1)$, with the rotational constant $B$, for three lowest total angular momentum states. The $L>0$ states show an instability in the spectrum. The red dashed line shows the deformation energy, Eq. (\[defEnergy\]), which is independent of $L$. The circles indicate the points for which the phonon density modulation is shown in Fig. \[PhonDens\].](SpecFunc){width="\linewidth"}
Macroscopic deformation of the bath and the emerging instability {#sec:deform}
================================================================
In the limit of a slowly-rotating impurity, $B \to 0$, the Hamiltonian (\[transH\]) can be solved exactly by means of an additional canonical transformation: $$\label{Htilde}
\hat{\mathscr{H}} = \hat{U}^{-1} \hat{\mathcal{H}} \hat{U}$$ where $$\label{Utransf}
\hat{U} = \exp \left[ \sum_{k \lambda} \frac{V_\lambda (k)}{W_{k\lambda}} \left( {\hat b}_{k \lambda 0} - {\hat b^\dagger}_{k \lambda 0} \right) \right]$$ with $W_{k\lambda}=\omega_k+B\lambda(\lambda+1)$. This transformation removes the terms linear in the bosonic operators, replacing them by the deformation energy of the bath, $$\label{defEnergy}
E_\text{def} = - \sum_{k\lambda} V_\lambda(k)^2/W_{k\lambda}$$ As a consequence, in the limit of $B=0$, the vacuum of phonon excitations, ${\, | 0 \rangle}$, becomes the exact ground state of Eq. (\[Htilde\]). On the other hand, such a coherent shift transformation corresponds to a macroscopic deformation of the bath, and could not be easily performed on the original Hamiltonian (\[Hamil\]) where the impurity coordinates are strongly coupled with the bath degrees of freedom.
Here we are interested in the effect of a slowly-rotating impurity on the many-body state of the environment. Therefore we introduce a variational ansatz based on single-phonon excitations on top of the bosonic state macroscopically deformed by the operator $\hat U$: $$\label{PsiChevy}
\vert \psi \rangle = g_{LM} \vert 0 \rangle \vert L M 0 \rangle + \sum_{k \lambda n} \alpha_{k \lambda n} {\hat b^\dagger}_{k \lambda n} \vert 0 \rangle \vert L M n \rangle$$ The states of an isolated symmetric-top molecule are characterised by three quantum numbers: the angular momentum, $L$, its projection, $M$, onto the laboratory-frame $z$-axis, and its projection, $n$, onto the molecular symmetry axis, $z'$. For a linear rotor molecule, the angular momentum vector is always perpendicular to the molecular axis and therefore $n$ is identically zero. The transformation (\[Transformation\]), however, transfers the bosons to the molecular frame, thereby creating an effective ‘many-body symmetric-top’ state. The latter consists of a linear rotor impurity dressed by the field of bosons carrying finite angular momentum. As a result, the total angular momentum of such a symmetric-top is no longer perpendicular to the linear rotor axis and provides the finite values of the projection $n$. See Appendix \[app:moltrans\] for more details.
It is worth emphasizing that the non-transformed many-body wavefunction corresponding to Eq. (\[PsiChevy\]) is given by ${\, | \phi \rangle}= \hat S \cdot \hat U{\, | \psi \rangle}$. Therefore it is a highly-involved object with an infinite number of degrees of freedom entangled with each other. The simple ansatz of Eq. (\[PsiChevy\]) was made possible by the consecutive canonical transformations, Eqs. (\[Transformation\]) and (\[Utransf\]). Furthermore, it is straightforward to extend Eq. (\[PsiChevy\]) to bath excitations of higher order, since this does not generate any complexities related to the angular momentum algebra.
Performing the variational solution for the energy, $E = \langle \psi \vert \hat{\mathscr{H}} \vert \psi \rangle/ \langle \psi \vert \psi \rangle$, we obtain the condition $$\label{Dyson}
-E+BL(L+1)-\Sigma_L(E)=0$$ which has the form of a Dyson equation with self-energy $\Sigma_L(E)$ [@Rath2013], as given by Eq. (\[selfenergyapp\]), see Appendix \[sec:variational\] for a detailed derivation. Eq. (\[Dyson\]) can be rewritten in terms of the angulon Green’s function as $\left[G_L(E)\right]^{-1}=0$, where $$\label{GFct}
\left[G_L(E)\right]^{-1}= [G^0_L(E)]^{-1} - \Sigma_L(E) ,$$ with $[G^0_L(E)]^{-1} =-E+B L(L+1)$.
![\[PhonDens\] Phonon density in the impurity frame for selected values of $\log [B/u_0]$, which are specified in the right-top corner of the panels and (a)–(d) as labeled in Fig. \[energies\]; (e) far to the right from the instability, at $\log[B/u_0] = 3.5$ for $L=1$ and at $\log[B/u_0] = 3.0$ for $L=2$. The coordinates $(x,z)$ are in units of $(m u_0)^{-1/2}$.](PhonDensity.pdf){width="\linewidth"}
The ground- and excited-state properties of the system are contained in the spectral function, $\mathcal A_L (E)= \text{Im}[G_L(E + i 0^+)]$. Without restricting the generality of what follows, we assume potentials whose angular momentum expansion, Eq. (\[VimpBos\]), is given by the Gaussian form-factors, $f_\lambda(r)=(2\pi)^{-3/2}e^{-r^2/(2r_\lambda^2)}$, and nonzero magnitudes, $u_0$ and $u_1$, in two lowest angular momentum channels. We assume an anisotropy ratio of $u_1/u_0=5$, a range $r_0 = r_1 = 15~(m u_0)^{-1/2}$, and set the interactions with $\lambda>1$ to zero. Furthermore, we use a Bogoliubov-type dispersion relation, $\omega_k = \sqrt{\epsilon_k (\epsilon_k + 2 g_\text{bb} n)}$, where $\epsilon_k = k^2/2 m$ with $m$ the mass of a boson. We choose the boson-boson interaction $g_\text{bb} = 418 (m^3 u_0)^{-1/2}$ and density $n = 0.014 (m u_0)^{3/2}$. This choice of parameters reproduces the speed of sound in superfluid $^4$He for $u_0 = 2\pi \times100$ GHz [@DonnellyHe98]. Fig. \[energies\] shows the dependence of the spectral function on the rotational constant $B$ for the three lowest rotational states. The width of the lines reflects the lifetimes of the corresponding levels. In Ref. [@SchmidtLem15], we studied the non-transformed Hamiltonian using a variational ansatz based on single-phonon excitations. Using this ansatz we found that the angulon states become stable after crossing the phonon threshold at zero energy. Here this is no longer the case, since the transformation $\hat U$ of eq. introduces an infinite number of phonon excitations into the variational ansatz. This leads to a energetic renormalization of the phonon emission threshold providing all the excited angulon states with decay channels for phonon emission. This, in turn, leads to a finite lifetime for any magnitude of the impurity-bath coupling.
In the limit of $B \to 0$ the molecule is not rotating and is inducing an anisotropic deformation of the bath, corresponding to the mean-field-like deformation energy, Eq. (\[defEnergy\]). The magnitude of the deformation energy decreases with $B$ monotonously and determines the general shape of the spectrum. Apart from the deformation energy which is identical for all $L$’s, the energy of the angulon acquires an additional contribution due to phonon excitations in the surrounding medium. The latter corresponds to the rotational Lamb shift discussed in Ref. [@SchmidtLem15], which has been observed as the renormalization of the rotational spectrum for molecules in superfluid helium nanodroplets [@ToenniesAngChem04]. Most importantly, we find that for the excited states with $L>0$ there exists a critical rotational constant, where a discontinuity in the rotational spectrum occurs. This effect corresponds to a transfer of one quantum of angular momentum *from the bath to the impurity*. One can see that the faster the rotation (i.e. the larger $L$), the earlier this instability occurs. Such an instability has been briefly discussed in Ref. [@SchmidtLem15], where it was referred to as Many-Body-Induced Fine Structure of the second kind.
While the instability can be detected using spectroscopy in the laboratory frame, an insight into its origin can be gained by making use of the canonical transformation, Eq. (\[Transformation\]). Namely, in the frame co-rotating with the impurity, the instability manifests itself as a change of the phonon density, $\langle {\hat b^\dagger}_\mathbf{r} {\hat b}_\mathbf{r} \rangle$; for analytic expressions see Appendix \[sec:deformation\]. Fig. \[PhonDens\] shows the phonon density for $L=1$ and $2$ at five different values of the impurity rotational constant. Darker shade corresponds to higher density. Far to the left of the instability, panels (a), the impurity is rotating slowly and the bosons are able to adiabatically follow its motion. As a result, the surrounding bath becomes polarized, which manifests itself in a highly-asymmetric phonon density. The shape of the density modulation is given by the first spherical harmonic which arises due to the $\lambda=1$ term in the impurity-boson potential included in our model. Closer to the instability, panels (b), the phonon density increases, signaling the onset of the resonant phonon excitations. At the right edge of the instability, panels (c), the phonon density drops drastically. Further away from the instability, the density distribution becomes the more symmetric the faster the impurity rotates, as illustrated in panels (d), (e). In other words, when the rotational constant exceeds the critical value given by the instability, it becomes energetically unfavorable for the bosons to follow the motion of the impurity. As a consequence, the bosonic bath does not possess finite angular momentum, which results in the spherically-symmetric density distribution. Thus, the phonon density in the transformed frame can serve as a fingerprint of the angular momentum transfer from the bath to the impurity which takes place at the instability point.
It is important to note that the ‘angulon instability’ discussed here is fundamentally different from the vortex instability [@Pitaevskii2003], also associated with rotation. The comparison between the two is summarized in Table \[tab:inst\]. First, the rotation of the impurity is inherently three-dimensional and does not involve any specific rotation axis. This is different for a vortex line which singles out a particular direction in space. Second, the formation of a vortex requires a transfer of one unit of angular momentum per particle in the bath. In the angulon instability, on the other hand, a finite (small) number of rotational quanta is shared between the impurity and the collective state of the many-particle environment. Finally, the vortex instability leads to a finite circulation around the vortex line, which is absent for the angulon instability.
Angulon Vortex
----------------------------------- --------------------------------- -------------------------------
Corresponding rotation Spherical, $\hat{\mathbf{L}}^2$ Planar, $\hat{\mathbf{L}}_z$
\[2pt\] Angular momentum transfer $\hbar$ $\hbar$ per particle
\[2pt\] Circulation zero integer$\times 2 \pi \hbar/m$
\[2pt\]
: \[tab:inst\] Comparison of the angulon instability with the vortex instability.
Experimental implementation
===========================
![\[PA\] Detection of the angulon self-energy $\Sigma_L$ using (a) photoassociation spectroscopy [@UlmanisCR12] and (b) shift of $p$- and $d$-wave Feshbach resonances [@KohlerRMP06].](PAimp){width="\linewidth"}
The described effects can be observed experimentally both with molecules trapped in strongly-interacting superfluids, such as helium droplets [@ToenniesAngChem04], or molecular impurities immersed in weakly-interacting Bose-Einstein condensates [@Pitaevskii2003]. The dependence of the angulon self-energy, $\Sigma_L$ of Eq. , on the many-body parameters can be revealed by measuring the relative shift between the rotational states of a diatomic molecule. Since the effects will be most pronounced for the molecular states possessing a small rotational constant $B$, experiments involving molecules in highly-excited vibrational states provide the most natural setup. In the context of ultracold gases, the latter include photoassociation spectroscopy [@UlmanisCR12] and measuring nonzero angular momentum Feshbach resonances [@KohlerRMP06]. In both cases, the shifts of the spectroscopic lines will be proportional to the angulon self-energy, as schematically illustrated in Fig. \[PA\]. An alternative possibility is measuring $\Sigma_L$ as a shift of the microwave lines in the spectra of weakly bound molecules [@MarkPRA07], prepared using one of these techniques. In frequency domain, at sufficiently low temperatures the width of the lines will correspond to the angulon lifetime. The instability shown in Fig. \[energies\] corresponds to the vanishing quasiparticle weight with a related emergence of a broad incoherent background and therefore can be detected as a line broadening with increasing impurity-bath interactions. In the time domain, on the other hand, the angulon Green’s function can be detected using Ramsey and spin-echo techniques [@KnapPRX12; @Schmidt15]. In such measurement, the angulon instability will leads to dephasing dynamics with a related pronounced decay of the Ramsey and spin-echo contrast [@KnapPRX12; @Schmidt15].
While in superfluid helium the interactions cannot be tuned as easily as in ultracold gases, the range of chemical species amenable to trapping is essentially unlimited [@ToenniesAngChem04]. The latter, combined with the advances in the theory of molecule-helium interactions [@SzalewiczIRPC08] paves the way to studying angulon physics in a broad range of parameters.
Conclusions
===========
In this paper, we studied the redistribution of orbital angular momentum between a quantum impurity and a many-particle environment. We introduced a technique which allows to drastically simplify the problem of adding an infinite number of angular momenta which occurs in the regime of strong interactions. The essence of the method – a novel canonical transformation – paves the way to eliminating the complex angular momentum algebra from the problem, as well as to exposing the problem’s constants of motion. We exemplified the technique’s capacity by studying an instability which occurs in the spectrum of the many-particle system due the interaction between the bath and the rotating impurity. Such an instability should be detectable with molecules in superfluid helium droplets [@ToenniesAngChem04] and might be responsible for the long timescales emerging in molecular rotation dynamics in the presence of an environment [@PentlehnerPRL13], which presently lacks even a qualitative explanation. Moreover, the rotating impurities can be prepared experimentally in perfectly controllable settings, based on ultracold molecules immersed into a Bose or Fermi gas [@JinYeCRev12; @KreStwFrieColdMol; @LemKreDoyKais13] and cold molecular ions inside Coulomb crystals [@WillitschIRPC12]. It is important to note that the transformation, as defined by Eq. , is quite general, and can be applied to extended Fröhlich Hamiltonians [@Rath2013], to impurities with complex rotational structure [@LevebvreBrionField2], Rydberg molecules [@Greene2000; @bendkowsky2009; @BellosPRL13; @KruppPRL14], as well as to the case of a Fermionic bath [@ChevyPRA06].
The ultimate goal of our approach is to find a series of canonical transformations that would lead to exact solutions to the many-body Hamiltonians of the same class as Eq. (\[Hamil\]). This resonates with Wegner’s idea of the continuous unitary transformations [@WegnerAnnPhys94], which underlies one of the Hamiltonian formulations of the renormalization group approach [@KehreinRG].
Finally, the impurity problem considered here can be used as a building block of a general theory describing the redistribution of orbital angular momentum in quantum many-particle systems. This opens up a perspective of applying the techniques of this article to the several problems in condensed matter [@Mahan90] and chemical [@EncyclChemPhys] physics.
Acknowledgements
================
We are grateful to Eugene Demler, Jan Kaczmarczyk, Laleh Safari, and Hendrik Weimer for insightful discussions. The work was supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.
The angular momentum representation {#sec:Bklm}
===================================
The creation and annihilation operators of Eq. (\[Hamil\]) are expressed in the angular momentum representation, which is related to the Cartesian representation as: $$\label{AklmAk}
{\hat b^\dagger}_{k\lambda \mu} =\frac{k}{(2\pi)^{3/2}} \int d\Phi_k d\Theta_k~\sin\Theta_k~{\hat b^\dagger}_\mathbf{k} ~i^\lambda~ Y^*_{\lambda \mu} (\Theta_k, \Phi_k)$$ $$\label{AkAklm}
{\hat b^\dagger}_\mathbf{k} = \frac{(2\pi)^{3/2}}{k} \sum_{\lambda \mu} {\hat b^\dagger}_{k\lambda \mu}~i^{-\lambda}~Y_{\lambda \mu} (\Theta_k, \Phi_k),$$ The quantum numbers $\lambda$ and $\mu$ define, respectively, the angular momentum of the bosonic excitation and its projection onto the laboratory-frame $z$-axis. Eqs. (\[AklmAk\]) and (\[AkAklm\]) correspond to the following commutation relations: $$\label{AkComm}
[{\hat b}_\mathbf{k}, {\hat b^\dagger}_\mathbf{k'}] = (2\pi)^3\delta^{(3)}(\mathbf{k-k'})$$ $$\label{AklmComm}
[{\hat b}_{k\lambda \mu}, {\hat b^\dagger}_{k'\lambda' \mu'}] = \delta(k-k') \delta_{\lambda \lambda'} \delta_{\mu \mu'}$$
In the coordinate space, the transformation between the representations is defined as: $$\label{ArlmAr}
{\hat b^\dagger}_{r\lambda \mu} = r \int d\Phi_r d\Theta_r~\sin\Theta_r ~{\hat b^\dagger}_\mathbf{r}~i^\lambda~Y^*_{\lambda \mu} (\Theta_r,\Phi_r)$$ $$\label{ArArlm}
{\hat b^\dagger}_\mathbf{r} = \frac{1}{r} \sum_{\lambda \mu} {\hat b^\dagger}_{r\lambda \mu} ~i^{-\lambda}~Y_{\lambda \mu} (\Theta_r,\Phi_r)$$ with the corresponding commutation relations: $$\label{ArComm}
[{\hat b}_\mathbf{r}, {\hat b^\dagger}_\mathbf{r'}] = \delta^{(3)}(\mathbf{r-r'})$$ $$\label{ArlmComm}
[{\hat b}_{r \lambda \mu}, {\hat b^\dagger}_{r' \lambda' \mu'}] = \delta(r-r') \delta_{\lambda \lambda'} \delta_{\mu \mu'}$$
The operators in the coordinate and momentum space are related through the Fourier transform, $$\label{brViabk}
{\hat b}^\dagger_\mathbf{r} = \int \frac{d^3 k}{(2\pi)^3} {\hat b}^\dagger_\mathbf{k} e^{i \mathbf{k \cdot r}},$$ from which one can obtain the corresponding relation for the angular momentum components $$\label{brlmViabklm}
{\hat b}^\dagger_{r \lambda \mu} = i^\lambda \sqrt{\frac{2}{\pi}} r \int k dk~j_\lambda (kr)~{\hat b}^\dagger_{k\lambda\mu}$$ with $j_\lambda (kr)$ the spherical Bessel function [@AbramowitzStegun].
The canonical transformation {#app:transfo}
============================
Here we provide details on the derivation of the transformed Hamiltonian, Eq. (\[transH\]).
In the angular momentum representation, the boson creation and annihilation operators, ${\hat b^\dagger}_{k\lambda \mu}$ and ${\hat b}_{k\lambda \mu}$, are defined as irreducible tensors of rank $\lambda$ [@VarshalovichAngMom]. Consequently, they are transformed by the $\hat{S}$-operator of Eq. (\[Transformation\]) in the following way: $$\begin{aligned}
\label{bDagRot}
\hat{S}^{-1} {\hat b^\dagger}_{k\lambda \mu} \hat{S} &= \sum_\nu D_{\mu \nu}^{\lambda \ast} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) {\hat b^\dagger}_{k\lambda \nu}\\
\label{bRot}
\hat{S}^{-1} {\hat b}_{k\lambda \mu} \hat{S} &= \sum_\nu D_{\mu \nu}^{\lambda} ( \hat{\phi}, \hat{\theta}, \hat{\gamma}) {\hat b}_{k\lambda \nu}\end{aligned}$$ Here $D_{\mu \nu}^{\lambda} ( \hat{\phi}, \hat{\theta}, \hat{\gamma})$ are Wigner $D$-matrices [@VarshalovichAngMom] whose arguments are the angle operators defining the relative orientation of the impurity frame with respect to the laboratory frame. These expressions can also be derived using the explicit expression for the angular momentum of the bosons, Eq. (\[Lambda\]).
The Wigner rotation matrix appearing in Eq. (\[bDagRot\]) is complex conjugate with respect to the one of Eq. (\[bRot\]) and therefore corresponds to an inverse rotation. As a result, $$\label{bDagbRot}
\hat{S}^{-1} \Bigl( \sum_\mu {\hat b^\dagger}_{k\lambda \mu} {\hat b}_{k\lambda \mu} \Bigr) \hat{S} = \sum_\mu {\hat b^\dagger}_{k\lambda \mu} {\hat b}_{k\lambda \mu} ,$$ and the second term of Eq. (\[Hamil\]) does not change under the transformation.
Similarly, in the last term of Eq. (\[Hamil\]) we use that $Y_{\lambda \mu} (\hat{\theta}, \hat{\phi}) = \sqrt{\frac{2 \lambda +1}{ 4 \pi}} D^{\lambda \ast}_{\mu 0} (\hat{\phi}, \hat{\theta}, 0)$, which leads to cancellation of the Wigner $D$-matrices. In such a way, the transformation $\hat{S}$ eliminates the molecular angle variables from the Hamiltonian.
The transformation of the molecular rotational Hamiltonian, $B \mathbf{\hat{J}^2}$, turns out to be slightly more cumbersome. In the laboratory frame, the angular momentum vector is defined by its spherical components, $\hat{\mathbf{J}} = \{\hat{J}_{-1}, \hat{J}_{0}, \hat{J}_{+1} \}$, where: $$\begin{aligned}
\label{J0}
\hat{J}_0 &= \hat{J}_z \\
\label{Jplus}
\hat{J}_{+1} &= -\frac{1}{\sqrt{2}} \left(\hat{J}_x + i\hat{J}_y \right)\\
\label{Jminus}
\hat{J}_{-1} &= \frac{1}{\sqrt{2}} \left(\hat{J}_x - i\hat{J}_y \right)\end{aligned}$$ see Refs. [@BiedenharnAngMom; @VarshalovichAngMom]. We use the analogous notation for the components of the total angular momentum of the bosons $\hat{\boldsymbol{\Lambda}} = \{\hat{\Lambda}_{-1}, \hat{\Lambda}_{0}, \hat{\Lambda}_{+1} \}$, Eq. (\[Lambda\]). The operators (\[J0\])–(\[Jminus\]) obey the following commutation relations with each other: $$\label{Jicommute}
\left[\hat{J}_i, \hat{J}_k \right] = - \sqrt{2} C_{1, i; 1, k}^{1, i+k} \hat{J}_{i+k},$$ where $i,k = \{-1, 0, +1 \}$, and with the rotation operators: $$\begin{gathered}
\label{JiComm}
\left[\hat{J}_k, D^{\lambda}_{\mu \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \right] \\= (-1)^{k+1} \sqrt{\lambda(\lambda+1)} C_{\lambda, \mu; 1,-k}^{\lambda, \mu-k} D^{\lambda}_{\mu-k, \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma})\end{gathered}$$ $$\begin{gathered}
\label{JiCommAst}
\left[\hat{J}_k, D^{\lambda \ast}_{\mu \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \right] \\= \sqrt{\lambda(\lambda+1)} C_{\lambda, \mu; 1, k}^{\lambda, \mu+k} D^{\lambda \ast}_{\mu+k, \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma})\end{gathered}$$ Here $C_{l_1, m_1; l_2, m_2}^{l_3, m_3}$ are the Clebsch-Gordan coefficients [@VarshalovichAngMom].
By using the latter property, one can show that the operators (\[J0\])–(\[Jminus\]) transform under Eq. (\[Transformation\]) in the following way: $$\label{JiTransformed}
\hat{\mathcal{J}}_{i} \equiv \hat{S}^{-1} \hat{J}_{i} \hat{S} = \hat{J}_{i} - \sum_{k=-1,0,1} D^{1 \ast}_{i k} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \hat{\Lambda}_{k}$$
After some angular momentum algebra, we obtain the following expression for the square of the angular momentum in the transformed frame: $$\label{J2transformed}
\hat{S}^{-1} \mathbf{\hat{J}^2} \hat{S} \equiv \hat{\mathcal{J}}_{0}^2 - \hat{\mathcal{J}}_{+1} \hat{\mathcal{J}}_{-1} - \hat{\mathcal{J}}_{-1} \hat{\mathcal{J}}_{+1} =(\mathbf{\hat{J}'} - \mathbf{\hat{\Lambda}})^2$$
Here $\mathbf{\hat{J}'}$ is the angular momentum operator in the rotating molecular (i.e. body-fixed) coordinate frame [@LevebvreBrionField2; @BiedenharnAngMom], which can be expressed via the laboratory-frame components as: $$\label{JiPrimeviaJi}
\hat{J}'_{i} = \sum_k D^{1}_{k, i} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \hat{J}_{k}$$ The spherical components of $\mathbf{\hat{J}'}$ are expressed through the Cartesian components using the relations analogous to Eqs. (\[J0\])-(\[Jminus\]). Note that this makes the $\mathbf{\hat{J}'}$ operators different from the so-called contravariant angular momentum components used by Varshalovich [@VarshalovichAngMom].
The molecular-frame angular momentum operators obey the anomalous commutation relations with one another [@BernathBook; @BiedenharnAngMom], $$\label{JiPrimecommute}
\left[\hat{J}'_i, \hat{J}'_k \right] = \sqrt{2} C_{1, i; 1, k}^{1, i+k} \hat{J}'_{i+k}$$ and the following commutation relations with the rotation matrices: $$\begin{gathered}
\label{JiPrimeComm}
\left[\hat{J}'_k, D^{\lambda}_{\mu \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \right] \\= - \sqrt{\lambda(\lambda+1)} C_{\lambda, \nu; 1, k}^{\lambda, \nu+k} D^{\lambda}_{\mu, \nu + k} (\hat{\phi}, \hat{\theta}, \hat{\gamma})\end{gathered}$$ $$\begin{gathered}
\label{JiPrimeCommAst}
\left[\hat{J}'_k, D^{\lambda \ast}_{\mu \nu} (\hat{\phi}, \hat{\theta}, \hat{\gamma}) \right] \\= (-1)^k \sqrt{\lambda(\lambda+1)} C_{\lambda, \nu; 1, - k}^{\lambda, \nu-k} D^{\lambda}_{\mu, \nu - k} (\hat{\phi}, \hat{\theta}, \hat{\gamma})\end{gathered}$$
It is worth noting that in the case of a linear-rotor molecule, the molecule-boson interaction does not depend on the third Euler angle, $\hat{\gamma}$. However, this angle must be preserved in Eq. (\[Transformation\]), as well as in all the derivations described above, in order to keep the transformation unitary.
Molecular states in the transformed space {#app:moltrans}
=========================================
In the main text and Fig. \[transf\] we have introduced two coordinate frames: the laboratory one, $(x,y,z)$, and the molecular one, $(x',y',z')$. A general molecular state, therefore, can be characterised by three quantum numbers: the magnitude of angular momentum, $j$; its projection, $m$, onto the laboratory-frame $z$-axis; and its projection, $n$, onto the molecular-frame $z'$-axis: $$\begin{aligned}
\label{J2eig}
\hat{\mathbf{J}}^2 \vert j, m, n \rangle &= j(j+1) \vert j, m, n \rangle \\
\label{Jzeig}
\hat{J}_z \vert j, m, n \rangle &= m \vert j, m, n \rangle \\
\label{JPrzeig}
\hat{J}'_z \vert j, m, n \rangle &= n \vert j, m, n \rangle\end{aligned}$$
In the angular representation, the corresponding wave functions are given by [@LevebvreBrionField2]: $$\label{JMNwf}
\langle \phi, \theta, \gamma \vert j, m, n \rangle = \sqrt{\frac{2j+1}{8 \pi^2}} D^{j \ast}_{mn} (\phi, \theta, \gamma)$$
The action of the space-fixed and molecule-fixed components of angular momentum is given by the general formula [@BernathBook; @BiedenharnAngMom]: $$\begin{aligned}
\label{JiKet}
\hat{J}_k \vert j, m, n \rangle &= \sqrt{j(j+1)} C_{j, m; 1, k}^{j, m+k} \vert j, m+k, n \rangle\\
\label{JiPrimeKet}
\hat{J}'_k \vert j, m, n \rangle &= (-1)^{k} \sqrt{j(j+1)} C_{j, n; 1, -k}^{j, n-k} \vert j, m, n-k \rangle\end{aligned}$$ where $k = \{-1, 0, +1\}$. Thus, in the molecular frame the raising operators lower the projection quantum number $n$ and the lowering operators raise it.
Unlike for nonlinear polyatomic molecules [@LevebvreBrionField2], the angular momentum of a linear rotor is always perpendicular to the internuclear axis (defining $z'$), and therefore $n$ is identically zero. However, this is the case only before the transformation $\hat{S}$ is applied. Let us consider the most general many-body state in the non-transformed frame, $$\label{LM}
\vert L, M \rangle = \sum_{\underset{j m; i}{k \lambda \mu}} a_{k \lambda j}^i C_{j, m; \lambda, \mu}^{L, M} \vert j m 0 \rangle \otimes \vert k \lambda \mu \rangle_i$$ The molecular states $\vert j m 0 \rangle$ are the eigenstates of the molecular angular momentum operator, as given by Eqs. (\[J2eig\])–(\[Jzeig\]). The same relations are fulfilled for the collective bosonic states: $\hat{\mathbf{\Lambda}}^2 \vert k \lambda \mu \rangle = \lambda(\lambda+1) \vert k \lambda \mu \rangle$ and $\hat{\mathbf{\Lambda}}_z \vert k \lambda \mu \rangle =\mu \vert k \lambda \mu \rangle$, where $\hat{\mathbf{\Lambda}}$ is defined by Eq. (\[Lambda\]), and $k$ is the linear momentum. The index $i$ labels all the possible boson configurations resulting in a collective state $\vert k \lambda \mu \rangle$, spanning the complete many-body Hilbert space of the bosonic bath.
It is straightforward to show that the state (\[LM\]) is an eigenstate of the total angular momentum operator, $\hat{\mathbf{L}} = \hat{\mathbf{J}}+\hat{\mathbf{\Lambda}}$: $$\begin{aligned}
\label{L2eig}
\hat{\mathbf{L}}^2\vert L, M \rangle &= L(L+1) \vert L, M \rangle \\
\label{Lzeig}
\hat{L}_z \vert L, M \rangle &= M \vert L, M \rangle\end{aligned}$$
By acting on $ \vert L, M \rangle$ with $\hat{S}^{-1}$, after some angular momentum algebra, we obtain the state in the transformed frame: $$\label{LMtil}
\hat{S}^{-1} \vert L, M \rangle = \sum_{k \lambda n i} f^i_{k \lambda n} \vert L M n \rangle \otimes \vert k \lambda n \rangle_i$$ where the coefficients are given by $$\label{fn}
f^i_{k \lambda n} = (-1)^{\lambda+n} \sum_{j} a_{k \lambda j}^i C_{L, -n; \lambda, n}^{j, 0}$$
We see that the transformation effectively transferred the angular momentum of the bosons to the molecular frame. This is reflected by the fact that the transformed state, $\hat{S}^{-1} \vert L, M \rangle$, becomes an eigenstate of the body-fixed angular momentum operator, $\hat{\mathbf{J}}'^2$, with the eigenvalues of the total angular momentum operator, $\hat{\mathbf{L}}^2$, i.e. $$\label{J2Slm}
\hat{\mathbf{J}}'^2 \left( \hat{S}^{-1} \vert L, M \rangle \right)= L(L+1) \left( \hat{S}^{-1} \vert L, M \rangle \right)$$
Each state $\vert L M n \rangle$ in the superposition of Eq. (\[LMtil\]) is an effective symmetric-top state [@LevebvreBrionField2], with the projection of total angular momentum on the molecular axis entirely determined by the boson field.
Derivation of the Dyson equation from the variational principle {#sec:variational}
===============================================================
We minimize the energy obtained from the expectation value of Eq. with respect to the variational state: $$\label{PsiChevy2}
\vert \psi \rangle = g_{LM} \vert 0 \rangle \vert L M 0 \rangle + \sum_{k \lambda n} \alpha_{k \lambda n} {\hat b^\dagger}_{k \lambda n} \vert 0 \rangle \vert L M n \rangle$$ Minimization with respect to $\alpha^*_{k\lambda n}$ and $g_{LM}^*$ yields the following equations: $$\label{varequation1}
\left[-E+BL(L+1)\right]g_{LM} + B\sqrt{L(L+1)}\sum_{k\lambda}\xi_{k\lambda}\alpha_{k\lambda n}=0$$ and $$\begin{aligned}
\label{varequation2}
&&\left[-E+BL(L+1)+W_{k\lambda}\right]\alpha_{k\lambda n}-2 B \sum_{\nu} \boldsymbol{\sigma}^\lambda_{ n\nu} \boldsymbol{\eta}^L_{ n\nu}\alpha_{k\lambda\nu}\nonumber\\
&&+B\delta_{ n,\pm1}\xi_{k\lambda}\sum_{k'\lambda'}\xi_{k'\lambda'}\alpha_{k'\lambda' n}\nonumber\\
&&=-B\sqrt{L(L+1)}\xi_{k\lambda}g_{LM}\delta_{ n,\pm1}\end{aligned}$$ where we defined $\delta_{ n,\pm 1}=\delta_{ n,1}+\delta_{ n,-1}$, $\xi_{k\lambda}=\sqrt{\lambda(\lambda+1)}V_\lambda(k)/W_{k\lambda}$, $W_{k\lambda}=\omega_k + B \lambda (\lambda+1)$, and $ \boldsymbol{\eta}^L_{ n\nu}={\langle LM n | \,}\hat {{\mathbf{J}}}'{\, | LM\nu \rangle}$. In what follows, we show that Eqs. and can be solved in closed form.
First, the angular-momentum coupling term of Eq. (\[varequation2\]) is given by: $$\begin{aligned}
\boldsymbol{\sigma}^\lambda_{ n\nu} \boldsymbol{\eta}^L_{ n\nu}&=& n^2\delta_{ n\nu}+\frac{1}{2}\sqrt{\lambda (\lambda+1)-\nu(\nu+1)}\nonumber\\
&\times&\sqrt{L (L+1)-\nu(\nu+1)}\delta_{ n,\nu+1}\nonumber\\
&+&\frac{1}{2}\sqrt{\lambda (\lambda+1)-\nu(\nu-1)}\nonumber\\
&\times&\sqrt{L (L+1)-\nu(\nu-1)}\delta_{ n,\nu-1}\end{aligned}$$ Assuming that $V_\lambda(k)\neq 0 $ for $\lambda=0,1$ only, we obtain that Eqs. and are solved by $\alpha_{k\lambda n}=0$ for $\lambda = 0$. Consequently, $g_{LM}$, $\alpha_{k1\pm1}$, and $\alpha_{k 1 0}$ are the only variational parameters.\
For $\alpha_{k 1 0}$ we obtain $$\begin{aligned}
\label{alphaeq1}
&&\left[-E+BL(L+1)+\omega_k+2B\right]\alpha_{k10}\nonumber\\&& - B\sqrt{2L (L+1) }(\alpha_{k11}+\alpha_{k1-1})=0\end{aligned}$$
For the $\alpha_{k1\pm1}$ components we find two identical equations $$\begin{aligned}
\label{alphaeq2}
&&\left[-E+BL(L+1)+\omega_k\right]\alpha_{k1,\pm1}\nonumber\\&& - B\sqrt{2L (L+1) }\alpha_{k10}+B\xi_{k1}\sum_{k'}\xi_{k' 1}\alpha_{k'1,\pm1}\nonumber\\
&=&-B\sqrt{L(L+1)} \xi_{k1} g_{LM}\end{aligned}$$ By symmetry we expect $|\alpha_{k11}|=|\alpha_{k1-1}|$, however, if $\alpha_{k11}=-\alpha_{k1-1}$ were true, Eq. would imply $\alpha_{k10}=0$. This in turn would lead to a contradiction in Eq. which shows that $\alpha_{k11}=\alpha_{k1-1}$.
Thus, from Eq. we obtain $$\alpha_{k10}=\frac{2B \sqrt{2L(L+1)}}{-E+\omega_k +B L (L+1) +2B} \alpha_{k11}$$
Let us now define the inverse propagator $$P_E(k)=BL(L+1)-E + \omega_k - \frac{4B^2 L(L+1)}{-E+\omega_k +B L (L+1) +2B}$$ and rewrite Eq. as: $$\label{alphaxx}
\alpha_{k11}=-\frac{B \xi_{k 1}}{P_E(k)} \sum_{k'}\xi_{k' 1}\alpha_{k' 1 1}-\frac{B\sqrt{L(L+1)}\xi_{k1}}{P_E(k)}g_{LM}$$ In addition it is convenient to introduce the variable $\chi$ as $$g_{LM}\chi=\sum_{k}\xi_{k 1}\alpha_{k11}$$ After multiplying Eq. with $\xi_{k1}$ and integration over $k$ we find $$\chi=-B\sqrt{L(L+1)}\frac{\int_0^\infty dk\, \xi_{k1}^2/P_E(k)}{1+B \int_0^\infty dk\, \xi_{k1}^2/P_E(k)}$$\
This finally yields the Dyson equation $$-E+BL(L+1)-\Sigma_L(E)=0$$ where the self-energy is given by $$\label{selfenergyapp}
\Sigma_L(E)=B^2 L (L+1) \frac{\int_0^\infty dk\, \xi_{k1}^2/P_E(k)}{1+B \int_0^\infty dk\, \xi_{k1}^2/P_E(k)}$$ and $$\xi_{k1}=\sqrt{2}\frac{V_1(k)}{\omega_k+2B}$$ Furthermore, we absorbed the deformation energy $E_\text{def}$, Eq. (\[defEnergy\]), which is identical for all the $L$-levels, into the definition of $E$. Note, that if $B/u_\lambda \gg 1$, the self energy $\Sigma_L\to BL(L+1)$ and the Dyson equation is solved by $E=0$. This means that for weak interactions the impurity levels are shifted by the mean-field deformation energy only.\
The self-energy of Eq. can be partially evaluated analytically. It is convenient to define $$\omega=E-BL(L+1)$$ and to rewrite the retarded self-energy, $\Sigma_L^\text{ret}(\omega) \equiv \Sigma_L(\omega+i0^+)$, as $$\Sigma_L^{\text{ret}}(\omega)=2 B^2 L (L+1) \frac{\chi_L(\omega)}{1+2 \chi_L(\omega)}$$ where $$\label{eqchi}
\chi_L(\omega)=\int_0^\infty dk\, \frac{V_1(k)^2}{[\omega_k+2B]^2}\frac{1}{P_{\omega+i 0^+}(k)}.$$ The integrand of $\chi_L(\omega)$ possesses poles at the momenta $k_0$ satisfying $\omega_{k_0}=\omega$ for $L=0$ and at the momenta $k_{1,2}$ satisfying $\omega_{k_{1}}=\omega+2 B L $ and $\omega_{k_{2}}= \omega-2B L(L+1) $ for states with $L>0$. Using the relation $1/(x+i0^+)=\mathcal{P}(1/x)-i\pi\delta(x)$ this reveals the onset of the scattering continua in the spectral function.
For $L=0$ one finds $$\text{Im}\chi_{L=0}(\omega)=\pi\theta(\omega)\zeta_0$$ where $$\zeta_0=\frac{V_1(k_0)^2}{[\omega+2 B]^2}\left.\left[\frac{\partial \omega_k}{\partial k}\right]^{-1}\right|_{k=k_0}$$ while for $L>0$ one has $$\begin{aligned}
\text{Im}\chi_{L>0}(\omega)&=&\frac{\pi}{2}\theta(\omega_{k1})\left[1-\frac{1}{\sqrt{1+4 L (L+1)}}\right]\zeta_1\nonumber\\&+&
\frac{\pi}{2}\theta(\omega_{k2})\left[1+\frac{1}{\sqrt{1+4 L (L+1)}}\right]\zeta_2\nonumber\\\end{aligned}$$ where $$\zeta_{1,2}=\frac{V_1(k_0)^2}{[\omega+2 B]^2}\left.\left[\frac{\partial \omega_k}{\partial k}\right]^{-1}\right|_{k=k_{1,2}}$$ Finally, the real part of $\chi_L (\omega)$ follows from the principal value integration.
Deformation of the phonon density {#sec:deformation}
=================================
From Eq. (\[ArArlm\]) we obtain the expression for the phonon density in the rotating impurity frame: $$\begin{gathered}
\label{PhDens}
n(\mathbf{r}) \equiv \langle {\hat b^\dagger}_\mathbf{r} {\hat b}_\mathbf{r} \rangle \\= \frac{1}{r^2} \sum_{\underset{\lambda' \mu'}{\lambda \mu} } ~i^{-\lambda+\lambda'}~Y_{\lambda \mu} (\Theta_r,\Phi_r) Y^\ast_{\lambda' \mu'} (\Theta_r,\Phi_r) \langle {\hat b^\dagger}_{r\lambda \mu} {\hat b}_{r\lambda' \mu'} \rangle\end{gathered}$$
Using Eq. (\[brlmViabklm\]), we evaluate the partial-wave contributions: $$\begin{gathered}
\label{brlmViabklmAver}
\langle {\hat b}^\dagger_{r \lambda \mu} {\hat b}_{r \lambda' \mu'} \rangle \\= i^{\lambda-\lambda'} \frac{2}{\pi} r^2 \int k dk \int k' dk'~j_\lambda (kr) j_{\lambda'} (k'r)~\langle {\hat b}^\dagger_{k\lambda\mu} {\hat b}_{k'\lambda' \mu'} \rangle\end{gathered}$$
We calculate the expectation values, $\langle \dots \rangle$, with respect to the states in the transformed frame, $\vert \phi \rangle = \hat U \vert \psi \rangle$, where $\hat U$ and $\vert \psi \rangle$ are given by Eqs. (\[Utransf\]) and (\[PsiChevy\]) of the main text. Finally, the expectation values of $\langle {\hat b}^\dagger_{k\lambda\mu} {\hat b}_{k'\lambda' \mu'} \rangle$ are given by: $$\begin{gathered}
\label{blkmAver}
\langle {\hat b}^\dagger_{k\lambda\mu} {\hat b}_{k'\lambda' \mu'} \rangle = \delta_{\mu 0} \delta_{\mu' 0} \Biggl [ 3 \frac{V_\lambda(k)}{W_{k \lambda}} \frac{V_{\lambda'} (k')}{W_{k' \lambda'}} - g^{\ast}_{LM} \alpha_{k' \lambda' 0} \frac{V_\lambda(k)}{W_{k \lambda}} \\- g_{LM} \alpha^\ast_{k \lambda 0} \frac{V_{\lambda'} (k')}{W_{k' \lambda'}}\frac{V_\lambda(k)}{W_{k \lambda}} + \vert \alpha_{k \lambda \mu}\vert^2 \Biggr ]\end{gathered}$$
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abstract: 'The temperature dependence of the resistance $R(T)$ of ultrathin quench-condensed films of Ag, Bi, Pb and Pd has been investigated. In the most resistive films films, $R(T)=R_{0}\exp \left( T_{0}/T\right) ^{x}$, where $x=0.75\pm 0.05$. Surprisingly, the exponent $x$ was found to be constant for a wide range of $R_{0}$ and $T_{0}$ in all four materials, possibly implying a consistent underlying conduction mechanism. The results are discussed in terms of several different models of hopping conduction.'
address: |
School of Physics and Astronomy, University of Minnesota, Minneapolis,\
MN 55455, USA
author:
- 'N. Marković,\* C. Christiansen, D. E. Grupp,$^{\S }$ A. M. Mack,$^{\P }$ G. Martinez-Arizala, and A. M. Goldman'
date: 'December 18, 1999'
title: Anomalous Hopping Exponents of Ultrathin Films of Metals
---
INTRODUCTION
============
In highly disordered materials, electrical conduction occurs by the hopping of electrons between localized sites. This results in a thermally activated electrical resistance of the form:
$$R(T)=R_{0}\exp \left( \frac{T_{0}}{T}\right) ^{x} \label{RT}$$
where $T$ is temperature, and $R_{0}$, $T_{0}$ and $x$ are constants which depend on the disorder, the details of the interactions and the dimensionality of the system. Simple activated hopping over a constant barrier results in the Arrhenius form with $x=1$. For noninteracting electrons, when the average hopping distance depends on temperature due to the compromise between hopping to sites which are close in energy, but farther away, Mott variable range hopping [@Mott] is expected, with $%
x=1/(d+1)$, where $d$ is the dimension. Efros and Shklovskii (ES) showed that including Coulomb interactions between electrons results in a soft gap in the density of states at the Fermi energy, which changes the variable range hopping exponent to $x=1/2$ in all dimensions [@Efros].
Hopping conduction has been investigated in a wide variety of materials, such as doped semiconductors [@Dai; @Zabrodski], semiconducting heterostructures [@VanKeuls], amorphous metals [@Lee; @Shlimak; @Liu; @Hsu; @Adkins], magnetic materials [@Ioselevich] and superconductors [@Dekker]. Both the Mott and the ES forms of variable range hopping have been observed, as well as a crossover between the two regimes [@VanKeuls; @Lee]. It should be emphasized, however, that it is often hard to distinguish between Mott and ES hopping, particularly in experiments in which the resistance changes only by one or two orders of magnitude. The unambiguous identification of the Mott or ES hopping can be further complicated by factors which are usually neglected, such as the granularity of the system, possible temperature dependence of $R_{0}$, or correlations between electron hops.
While investigating the transport properties of ultrathin quench-condensed films over the course of many years, we have often found that the resistance of the thinnest films was thermally activated with $x\simeq 0.75$. A similar hopping exponent has been reported by other authors [@VanKeuls; @Adkins; @Gershenson; @vanderPutten], but has rarely been discussed [*per se*]{}. Since there is no theory which predicts this value of the hopping exponent, its origin has been left an open question. Here we report a detailed study of the temperature dependence of the resistance in very disordered films of four different materials: Ag, Bi, Pb and Pd. The films were grown in separate runs over several years, in different cryostats and on different substrates, yet they all show the same thermally activated resistance with an almost identical exponent. A careful analysis of the data points to a new conduction mechanism in this regime, or perhaps calls for a modification of the conventional picture. We compare our results with those of other experiments and available theoretical calculations, and suggest that the model that may possibly explain the anomalous hopping exponent is the [*collective*]{} variable range hopping model of Fisher [*et al*]{}. [@Fisher]. Developed to describe vortices in superconductors, this approach has not been considered before in the context of charge transport in disordered electronic systems.
In Section II we survey the various models that have been considered in the discussion of transport in disordered films. We also exhibit the mapping of the model for collective vortex hopping onto the problem of charge transport in disordered systems and estimate the value of the hopping exponent. Experimental details of film growth and resistance measurements are given in Sec. III. In Sec. IV, we analyze the temperature dependence of the film resistance using several different methods to show that the exponent obtained is really a property of the system, rather than a consequence of an improper fit. The results are discussed and compared with other experimental and theoretical work in Sec. V.
SURVEY OF HOPPING MODELS
========================
In recent years, extensions of the basic variable range hopping model to include percolation effects and correlations between electron hops have been developed. Deutscher [*et al*]{}. [@Deutscher] proposed a hopping mechanism which leads to a thermally activated resistance with $x$ close to $%
1/2$ without considering Coulomb interactions. The mechanism was based on the superlocalization property of wavefunctions on incipient percolation clusters [@Levy], and may be relevant for atomically disordered systems as well as for granular percolative structures. The detailed microstructure of ultrathin quench-condensed films is still not known, and although these films are usually considered to be homogeneous, they may actually contain small grains or clusters. It is then possible that the electrons are restricted to move on a sublattice which is fractal over some range of length scales, and that their wavefunctions decay faster then exponentially with distance. Based on this assumption, a hopping conductivity law has been derived [@Deutscher] near the percolation threshold, which has the form of Eq \[RT\] with $x=3/7$. This is experimentally almost indistinguishable from the Efros-Shklovskii law with $x=1/2$ if only the temperature dependence of the resistance is studied, but can be identified through the behavior of the parameter $T_{0}$ and the nature of the crossover to the conventional Mott regime [@Deutscher].
Generally speaking, there is no reason to assume that the prefactor $R_{0}$ in Eq. \[RT\] is independent of temperature. Van Keuls [*et al*]{}. [@VanKeuls] studied the resistivity in a gated $GaAs/Al_{x}Ga_{1-x}As$ heterostructures as a function of temperature, electron density and magnetic field. Assuming the prefactor of the form:
$$R_{0}=bT^{m} \label{R0}$$
where $b$ and $m$ are constants, these workers fit Eq. \[RT\] to their data with $x=1/3$ in low magnetic fields and with $x=1/2$ in high magnetic fields. The same crossover was observed as a function of electron density and temperature, and it scaled with the separation between the electron layer and the nearby screening gate, as predicted by Aleiner and Shklovskii [@Aleiner]. In addition to introducing a temperature dependent prefactor, this experiment also raised the issue of the importance of correlations between the electron hops.
In an electron glass, where the screening length is long [@Monroe] and the interactions long-ranged, electron hopping may be correlated [@Pollak]. Excitations can leave the system far from equilibrium [@Ovadyahu] and relaxation occurs through the rearrangement of charge. The energy of a single electron hop may then be significantly reduced by the motion of the surrounding charges. At sufficiently low temperatures, such collective hopping might be the dominant conduction mechanism.
In the analysis of their data, Van Keuls [*et al*]{}. [@VanKeuls] assumed that the number of configurations of occupied states reached by the correlated hopping of a number of electrons is proportional to the single-particle density of states. In that case, the qualitative behavior of the resistance remains unchanged, and the effects of correlations enter through the constants which determine the parameter $T_{0}$ in different regimes.
Yet another issue which can be relevant in extremely thin films is the possibility that the electrons might interact logarithmically rather than as 1/r. As shown by Keldysh [@Keldysh], the range of the logarithmic interaction is given by:
$$r_{\log }=\frac{\kappa }{\kappa _{s}+1}d \label{r}$$
where $\kappa $ and $\kappa _{s}$ are the dielectric constants of the film and the substrate, respectively, and $d$ is the film thickness. One possible consequence of the logarithmic electron-electron interactions is that the Coulomb gap in the density of states (linear in 2D) may change to an exponential form [@Shklovskii]. This leads to a modified variable range hopping law with a temperature dependent exponent.
Alternatively, the behavior of the logarithmically interacting electrons might be similar to that of vortices in 2D superconductors, which are known to interact logarithmically. Collective variable range hopping of vortices in disordered thin-film superconductors was studied by Fisher [*et al*]{}. [@Fisher]. Including the effect of correlations, these authors found that multivortex hopping results in a lower energy than single vortex hopping. They also suggested that such multiparticle hopping might dominate single particle hopping even in the case of inverse power law interactions.
This approach may then be mapped onto a disordered two-dimensional system of charges. Following the arguments of Fisher [*et al*]{}. [@Fisher], the energy $U(r)$ of a multi-particle excitation of length $r$, can be estimated to be:
$$U(r)\approx K(\frac{l}{r})^{1/2} \label{Ur}$$
where $l$ is the distance between charges, and $K$ is the bare single-particle excitation energy. The latter is equal to $e^{2}/\kappa a$, where $a$ is the localization length and $\kappa $ is the dielectric constant. The simultaneous hopping of many charges may then result in a lower energy than the hop of a single charge.
The electrical resistance is a product of the probability for an electron to tunnel a distance $r$, $\Gamma _{r}$, and the probability for an excitation with the energy $U(r)$ to occur, $\Gamma _{U}$:
$$R\propto 1/\Gamma _{r}\Gamma _{U} \label{R}$$
A lower bound for the multihop rate can be estimated as follows: if all of the $(r/l)^{2}$ electrons in the $r$ by $r$ region hop a distance comparable to the spacing $l$, then the rate should be proportional to the single-hop rate , $exp(-l/a)$, raised to the power of the number of electrons, resulting in:
$$\Gamma _{r}\propto e^{-(l/a)(r/l)^{2}} \label{gammar}$$
The probability for an excitation of energy $U(r)$ is proportional to $%
exp(-U(r)/T)$, or using Eq. \[Ur\]:
$$\Gamma _{U}\propto e^{-(K/T)(l/r)^{1/2}} \label{gammaU}$$
The minimum resistance is obtained when the hopping distance $r_{hop}$ is:
$$r_{hop}^{2}\approx al(\frac{T_{0}}{T})^{4/5} \label{rhop}$$
where
$$T_{0}=K(l/a)^{1/4} \label{T0}$$
Substituting Eqs. \[gammar\] through \[T0\] back into Eq. \[RT\] results in a hopping form such as that of Eq. \[RT\] with $x=4/5$.
In the other limit, the minimum number of electrons participating in a collective hop could be taken as $(r/d)$, which leads to $x=2/3$. The collective variable range hopping mechanism may therefore result in a resistance of the form of Eq. \[RT\], where the range of exponents is $%
2/3<x<4/5$, depending on the fraction of electrons participating in the process.
EXPERIMENTAL METHODS
====================
The temperature dependence of the resistance has been studied in ultrathin quench-condensed films of Ag, Bi, Pd [@Liu] and Pb [@Martinez1; @Martinez2]. The films were deposited on liquid helium cooled substrates and resistance measurements were performed [*in situ*]{} at temperatures down to $0.15K$. Ultra-high vacuum conditions and temperatures below $20K$ were sustained throughout each run, in order to avoid contamination or crystallization. The substrates were glazed alumina (for Bi and Pd films) or $SrTiO_{3}$ (100) (for Ag and Pb films). The $SrTiO_{3}$ (100) substrates were $0.75mm$ thick and had a $100nm$ thick Au gate on the back. Such a gate can be used to study the response of the film to a perpendicular electric field and was used to establish glass-like behavior in the most resistive films [@Martinez1; @Martinez2].
Films were deposited in thickness increments between $0.05$ and $0.5\AA $ on top of a thin germanium layer $(5-10\AA )$. (The Pd films were the exception to this as they were deposited directly onto glazed alumina substrates where they became connected at monolayer coverage.) Films grown on amorphous Ge are believed to be homogeneous, since they become connected at an average thickness of about one monolayer [@Strongin]. The thicknesses of the films studied ranged from $5\AA $ up to $15\AA $. These nominal values of the film thickness are calculated from the deposition rate, which was measured using a quartz crystal monitor placed in the vicinity of the substrate. The first low-temperature scanning tunneling microscopy studies of the morphology of films grown in a similar manner indicate that the thinnest films may indeed be homogenous, while the thicker ones contain small clusters [@Ekinci].
Resistance measurements were carried out using a standard dc four-probe technique. Very low bias currents $(<10nA)$ were used to avoid Joule heating of the sample and to make sure that the voltage across the sample was a linear function of the applied current. When measuring very resistive films ($10^{4}-10^{8}$ $\Omega )$, because of the long time constants of the circuit, it was necessary to monitor the voltage as a function of time after the current was changed, and allow adequate time for the voltage to stabilize. To avoid the voltage offset errors due to thermal EMFs, both polarities of the current were used to determine the resistance.
The resistance of a series of Ag films was also studied in a magnetic field. Magnetic fields of up to 20kG (12 kG) were applied in direction parallel (perpendicular) to the plane of the substrate using a split-coil superconducting magnet. In a regime where the anomalous hopping exponent is observed, the resistance was found to be independent of magnetic field.
RESULTS AND ANALYSIS
====================
The temperature dependence of the sheet resistance (resistance per square) for series of Ag, Bi, Pb and Pd films is shown in Fig. \[logR\]. The logarithm of the sheet resistance, plotted as a function of $T^{-0.75}$ follows a straight line for each film, indicating that the resistance is thermally activated with $x\approx 0.75$. Using other values for $x$, such as $1$, $1/2$ or $1/3$ yielded considerably larger deviations from the data.
Since the prefactor in Eq. \[RT\] is generally expected to be temperature-dependent $(m\neq 0$ in Eq. \[R0\]$)$, we attempted to fit the data using different combinations of $m$ and $x$. As shown in Fig. \[xm\], using values of $m$ greater than zero actually increased the error of the fit. The maximum deviation in the fit of the combinations of Eqs. 1 and 2 to the data became much larger than the noise in $R$ as $m$ was increased. Furthermore if values of $m$ were chosen to force either Mott or ES hopping exponents of $x=1/3$ or $1/2$, respectively, the quality of fits as measured by chi squared would be significantly worse than that with $m=0$, in contrast with the findings of Van Keuls [*et al*]{}. [@VanKeuls].
Assuming that the hopping exponent is $x\approx 0.75$, the activation energy $T_{0}$ can be extracted from the fit to Eq. \[RT\]. The values of the parameter $T_{0}$ for different films of all four materials are shown in Fig. \[T0vsR\] as a function of $R_{14K}$, which is the sheet resistance measured at $14K$. This quantity is inversely proportional to the film thickness, so by using $R_{14K}$ instead of the thickness, one can avoid systematic errors in the nominal thicknesses of the films of different materials. It is apparent in Fig. \[T0vsR\] that $T_{0}$ changes greatly as $R_{14K}$ (and therefore also the film thickness) changes, ranging from around $100K$ for the thinnest films, to around $10K$ for the thickest films. The same qualitative and quantitative behavior was found for all four materials.
A more direct method of determining the exponent $x$ (which is exact under the condition $m<<\left( T_{0}/T\right) ^{x}$) has been developed by Zabrodskii and Zinov’eva [@Zabrodski]. The method is based on defining the function $w=-d(\log R)/d(\log T)$. If $R$ is given by Eq. \[RT\], then $\log w\propto -x\log T$. By plotting $\log w$ as a function of $\log T$, $x$ can be easily extracted from the slope of the resulting straight line. The benefit of the Zabrodskii-Zinov’eva approach is the simplicity of fitting a line rather than a complicated function with up to four adjustable parameters($b,$ $m$, $T_{0\text{ }}$and $x$). Once it has been determined that $m=0$, this method eliminates the danger of finding a local minimum instead of the best fit. The results of determining $x$ this way are shown in Fig. \[zab\] . For very resistive films, plotting $\log w$ vs. $\log T$ indeed yielded straight lines. The values of $x$ varied slightly between different materials, from $x\approx 0.7$ for Ag to $x\approx 0.8$ for Bi. Remarkably, the value of $x$ did not change between different films in the same series over a significant range of sheet resistances, as shown in Fig. \[xR\].
Even though $R_{14K}$ and $T_{0}$ change from film to film, as more material is added to increase the average thickness and decrease the sheet resistance of the film, $x$ stays constant over three orders of magnitude in $R_{14K}$. For thicker films (smaller $R_{14K}$), $x$ drops rather abruptly to a value between $1/3$ and $1/2$. In this regime, the Zabrodskii-Zinov’eva plots no longer produce straight lines, indicating that the hopping exponent changes as a function of temperature. Further increase of the film thickness leads to another crossover to a weakly localized regime where the temperature dependence of the resistance is logarithmic (not shown in Fig. \[xR\]).
DISCUSSION
==========
An activated temperature dependence of the resistance with an anomalous hopping exponent $x\approx 0.75$ has been observed in disordered films of four different materials, grown on different substrates and measured in different cryostats. This strongly suggests that the exponent $x\approx 0.75$ is a general property of ultrathin films of metals in the very strongly localized regime.
The same exponent has been reported by Adkins and Astrakharchik [@Adkins] in ultrathin quench-condensed films of Bi with a Ge underlayer. In that experiment, the temperature dependence of the resistance changed to simply activated (with $x=1$) when the Coulomb interaction was screened in the presence of a nearby metallic gate. This behavior was ascribed to the fixed range hopping of dipoles in screened films, but no details were given on the origin of $x\approx 0.75$ in unscreened films. The authors suggest that the films may be in the crossover regime between the variable range hopping and the fixed range hopping regime. Such a crossover can occur when the optimal hopping distance $r_{hop}$ becomes comparable with the localization length $%
\xi $.
In our experiment, $x$ stays constant over several orders of magnitude in sheet resistance, and then drops abruptly as the sheet resistance decreases further. If our films were merely at the crossover between $x\approx 1/2$ (or $1/3$) and $x\approx 1$, the change in $x$ would be expected to be gradual. The observed constancy of $x$ implies that a consistent mechanism may be governing the conduction in this regime, a mechanism different from Mott or Efros-Shklovskii variable range hopping which is usually observed in less resistive films.
Furthermore, it was not possible to obtain a satisfactory fit to the data using a temperature-dependent prefactor, as in the work of Van Keuls [*et al*]{}. It is interesting that these workers obtain $x\approx 0.75$ in all magnetic fields if $m$ is taken to be zero. However, the activation energies obtained from such fits are reported to be unacceptably small [@VanKeuls].
There are several other mechanisms which may be relevant in a very disordered 2D system. For example, Dai et al.[@Dai] observed an exponent $x{{}\sim }1$ in Si:B, which changed to $x{{}\sim }1/2$ when a magnetic field was applied. They suggested that the $x{{}\sim }1$ was due to the exchange interaction between the electron spins, which is destroyed in a magnetic field. It must be noted that the Ag did not show any magnetoresistance up to the highest field available, 20 kG, so the exchange interaction is most likely not the origin of the anomalous hopping exponent observed in these films.
If we allow a possibility that our films are granular on a very small scale (which we cannot unequivocally rule out), than we must consider the superlocalization mechanism of Deutscher [*et al.* ]{}[@Deutscher] as a possible candidate to explain our data. Without Coulomb interactions, this model predicts $x\approx 0.43,$ which obviously cannot account for our results. Including the Coulomb interactions may lead to a higher exponent, as proposed by van der Putten [*et al*]{} [@vanderPutten]. These authors studied the hopping conductivity of percolating carbon-black-polymer compounds and found $x\approx 0.66,$ which is much closer to our result, although still too low. They interpret their results as evidence of superlocalization on a fractal network with Coulomb dominated hopping. The activation energies were found to be independent of the electron concentration, as predicted by Deutscher [*et al*]{}. In contrast, the activation energies found in our experiment depend strongly on the film thickness, which is closely related to the electron concentration. Furthermore, if the Coulomb interactions were screened, one might expect the exponent to decrease towards $0.43$, rather than to increase towards $1$, as observed in screened Bi films by Adkins and Astrakharchik [@Adkins].
Another possibility is that the anomalous exponent is a consequence of the exponential gap in the density of states, which can arise if the electrons interact logarithmically [@Shklovskii]. In that case, the hopping exponent would be something close to, but smaller than $1$ at higher temperatures, and then cross over smoothly to $1/2$ at low temperatures. Forcing a fit of Eq. \[RT\] to the data would result in an exponent which changes continuously with temperature. In the less resistive regime where we observe a temperature-dependent exponent, a closer inspection shows the opposite trend: the exponent is close to $1/2$ at higher temperatures, and increases with decreasing temperature. On the other hand, we cannot rule out the possibility that we might observe a smooth crossover to $1/2$ in the most resistive films if we could measure at much lower temperatures.
Finally, we consider the collective variable range hopping mechanism, proposed in the context of vortices in disordered superconductors by Fisher [*et al*]{}. [@Fisher]. The mapping of this model onto a 2D electron system may actually be exact, if the electron-electron interactions are logarithmic over relevant length scales, but the authors suggest that collective hopping may dominate over the single-particle hopping even in the case of a conventional Coulomb interaction. The range of exponents predicted by the collective hopping model is $2/3<x<4/5$, depending on the ratio of electrons which participate in the process. The exponent found in our experiment, as well as the exponents found by other groups [@Adkins; @VanKeuls; @Gershenson; @vanderPutten], are well within that range. The activation energies are expected to depend on the localization length, which in turn depends on the film thickness, as shown in Fig. \[T0vsR\].
In conclusion, we have addressed the issue of the anomalous hopping exponent $x\approx 0.75$ observed in ultrathin films of metals and related 2D systems. We argue that this hopping exponent is a general property of very strongly disordered systems, rather than a result of an improper fit or a signature of some sort of a crossover behavior. The usual models of hopping conduction do not explain this result. Our data can be explained by a [*collective*]{} variable range hopping mechanism, but our work by no means provides a proof of such a mechanism. Future experimental and theoretical studies will be needed to shed more light on this matter.
We gratefully acknowledge useful discussions with Boris Shklovskii and Leonid Glazman. This work was supported in part by the National Science Foundation under Grant No. NSF/DMR-987681.
[\*]{}
: Present Address: Department of Applied Physics, Technical University of Delft, the Netherlands.
<!-- -->
[$^{\S }$]{}
: Present Address: Center for Integrated Systems, Stanford University, Stanford, CA, USA.
<!-- -->
[$^{\P }$]{}
: Present Address: Seagate Technology, Bloomington, MN, USA.
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|
---
abstract: 'Tseliakhovich & Hirata recently discovered that higher-order corrections to the cosmological linear-perturbation theory lead to supersonic coherent baryonic flows just after recombination (i.e. $z \approx 1020$), with rms velocities of $\sim$30 km/s relative to the underlying dark-matter distribution, on comoving scales of $\la 3$ Mpc$h^{-1}$. To study the impact of these coherent flows we performed high-resolution N-body plus SPH simulations in boxes of 5.0 and 0.7 Mpc$h^{-1}$, for bulk-flow velocities of 0 (as reference), 30 and 60 km/s. The simulations follow the evolution of cosmic structures by taking into account detailed, primordial, non-equilibrium gas chemistry (i.e. H, He, H$_2$, HD, HeH, etc.), cooling, star formation, and feedback effects from stellar evolution. We find that these bulk flows suppress star formation in low-mass haloes (i.e. $M_{\rm vir} \la 10^8$M$_{\odot}$ until $z\sim 13$), lower the abundance of the first objects by $\sim 1\%-20\%$, and as consequence delay cosmic star formation history by $\sim 2\times 10^7\,\rm yr$. The gas fractions in individual objects can change up to a factor of two at very early times. Coherent bulk flow therefore has implications for (i) the star-formation in the lowest-mass haloes (e.g. dSphs), (ii) the start of reionization by suppressing it in some patches of the Universe, and (iii) the heating (i.e. spin temperature) of neutral hydrogen. We speculate that the patchy nature of reionization and heating on several Mpc scales could lead to enhanced differences in the HI spin-temperature, giving rise to stronger variations in the HI brightness temperatures during the late dark ages.'
author:
- |
Umberto Maio$^{1}$[^1], Leon V. E. Koopmans$^{2}$, and Benedetta Ciardi$^{3}$\
\
${}^1$ Max Planck Institute for Extraterrestrial Physics, Giessenbachstra[ß]{}e 1, 85741 D-Garching, Germany\
${}^2$ Kapteyn Astronomical Institute, University of Groningen, P.O.Box 800, 9700AV, Groningen, the Netherlands\
${}^3$ Max Planck Institute for Astrophysics, Karl-Schwarzschild-Stra[ß]{}e 1, 85741 D-Garching,Germany
bibliography:
- 'bibl.bib'
title: 'The impact of primordial supersonic flows on early structure formation, reionization and the lowest-mass dwarf galaxies'
---
\[firstpage\]
Cosmology:theory – early Universe
Introduction {#Sect:introduction}
============
Cosmic structure-formation models are based on Jeans theory [@Jeans1902] applied to primordial matter fluctuations in an expanding Universe [e.g. @wmap7_2010]. The matter density perturbations are linearly expanded to first order, which is supposed to work well as long as the density contrast is much smaller than unity [e.g. @Peebles1974; @ColesLucchin2002; @CiardiFerrara2005], and the equations that govern the evolution of the dark and baryonic matter become linear. To study highly non-linear evolution when the density contrast approaches and far exceeds unity, however, fully numerical simulations are required. According to the concordance cosmological model [e.g. @wmap7_2010] the Universe is flat and has an expansion rate of $H_0\simeq 71\,\rm km/s/Mpc$. The cosmological constant (or dark energy) represents the main energy-density content, $\ol\simeq 0.7$, whereas dark matter contributes $\om\simeq 0.26$ and baryonic matter is only a small fraction of $\ob\simeq 0.04$. Within this context, baryonic structures arise from in-fall and condensation of gas into growing dark-matter potential wells [e.g. @GunnGott1972; @Peebles1974; @WhiteRees1978]. Both in-fall and condensation of gas only take place [*after*]{} the time of recombination ($z\simeq 1020$), when baryons and photons decouple [@wmap7_2010]. On the other hand, dark matter perturbations can already start growing when their density exceeds that of radiation (i.e. at $z_{\rm eq}\sim 3100-3200$). Recently @TseliakhovichHirata2010 for the first time included quadratic terms in the cosmological perturbation theory to account for the advection of small-scale perturbations by large-scale velocity flows. They find that, at the redshift of decoupling $z\simeq 1020$, coherent supersonic flows of the baryons relative to the underlying dark-matter distribution are formed on scales of a few Mpc or less, with typical velocities on those scales with a rms value of $\sim 30\,\rm km/s$, sourced by density perturbations up to scales of $\sim$100Mpc. These gas bulk velocities could actually suppress the formation of the very early and low-mass structures, induce higher baryon acoustic oscillations [@Dalal_et_al_2010], and have further implications on high-redshift galaxy clustering, 21-cm studies, and reionization. In fact, as we find, they might impact the formation of the lowest-mass galaxies (e.g. dSph) on scales as large as the local group, depending of the magnitude of the bulk flow at decoupling. The effect can thus cause a spatially varying bias in baryonic structure formation, whereas small-scale baryonic structures are suppressed more in regions where the bulk-velocity is larger. Detailed numerical simulations considering this non-linear effect are not yet available and the very high dynamic range required is not foreseen in the near future. It therefore looks very challenging to quantitatively assess these effects in the highly non-linear regime discussed by [@TseliakhovichHirata2010]. Indeed, one would need to resolve, at the same time, large scales (of the order of the horizon) and extremely small scales (of the order of the Jeans length). Given these extreme difficulties, we adopt the following approach: we focus on small scales where the effect is expected to be largest, at least properly dealing with all the detailed physics and chemistry, and we assume [*a constant bulk velocity shift in the initial conditions*]{} as determined by [@TseliakhovichHirata2010] on the relevant scales. Even though this approximation is still crude, we believe it is a reasonable assumption to study the lowest-mass structures in the box (which have virial radii much smaller than the box size), we expect it to be able to provide the first quantitative results in the highly non-linear regime and assess its effects on early structure formation. In this paper, we make a first step towards understanding the effects of coherent supersonic bulk flow on the formation of the smallest baryonic structures, in particular focusing on the suppression of star formation and the delay of reionization. In Sect. \[sect:sims\], we describe our simulations, and in Sect. \[sect:results\] we show our results. We summarize and conclude in Sect. \[sect:conclusion\].
Simulations {#sect:sims}
===========
We perform numerical simulations by using the parallel tree/SPH P-Gadget2 code [@Springel2005], which implements gravity, hydrodynamics, molecular and atomic evolution, cooling, population III and population II star formation, supernova/wind feedback, and a full chemistry network involving e$^-$, H, H$^+$, H$^-$, He, He$^+$, He$^{++}$, H$_2$, H$_2^+$, D, D$^+$, HD, and HeH$^+$ [for more details see @Maio2006; @Maio2007; @Maio2010; @MaioPhD; @Maio_et_al_2010b]. Particularly important for our purposes is the capability of the code of following gas collapse down to the catastrophic molecular cooling branch, by resolving the Jeans scales of primordial structures with $\sim 10^2$ particles [@Maio2009], and completely fulfilling the correct resolution requirements for SPH simulations [see @BateBurkert1997; @Maio2009 for details]. We generate the initial conditions by using the N-GenIC code and add bulk velocity shifts for the gas in the $x$-direction $\vbx = 0, 30, 60 \,\rm km/s$ (referred to as HR-Shift00, HR-Shift30, and HR-Shift60, respectively) at recombination epoch and properly scaled down [according to @TseliakhovichHirata2010] to the simulation initial redshift of $z_{in}=100$. The value $\vbx\sim 60\,\rm km/s$ has been considered to account for statistical variations from the expected rms value of $\sim 30\,\rm km/s$, but it should be considered as an upper limit which can be found in less than 1% of the volume of the Universe. Even though we expect that the presence of bulk flows at higher redshift does not affect the structure formation process, we nevertheless run additional simulations with $z_{in}=1020$, with $\vbx=0,60\,\rm km/s$ (HR-Shift00Rec, HR-Shift60Rec). We should caution the reader that the Gadget code implements N-body/SPH structure formation, cooling, star formation, and feedback effects in matter-dominated universes, while contributions from radiation or from the primordial hot plasma in the cosmological dynamics are neglected. Therefore, its use at $z >> 200$ might be questionable. In any case, the results are very similar to the simulations with $z_{in}=100$, and thus we don’t discuss them any further. Our reference simulations have a box side of $0.7\,\rm Mpc/{\it h}\simeq 1~Mpc$, and are run in the frame of the standard $\Lambda$CDM cosmological model ($\ol=0.7$, $\om=0.26$, $\ob=0.04$, $h=0.7$, $\sigma_8=0.9$, $n=1$). Matter and gas fields are sampled with 320$^3$ particles, respectively, allowing a resolution of $\sim 10^2\msun/h$ for the gas component and $\sim 7.5\times 10^2 \msun/h$ for the dark-matter component. The chemical abundances are initialized as in @Maio2007 [@Maio2010]. To check if the effects of primordial supersonic gas flows on structure formation could be seen in bigger but lower-resolution simulations, we also run a set of three larger boxes (LR-Shift00, LR-Shift30, and LR-Shift60) with a size of 5 Mpc/[*h*]{}, $\vbx=0, 30, 60,\rm km/s$, and with the same parameters described before. This box is comparable to the smallest scale where the bulk flows are expected according to [@TseliakhovichHirata2010]. The corresponding mass resolution is $\sim 4\times 10^4\msun/{\it h}$ and $\sim 3\times 10^5\msun/{\it h}$, for gas and dark-matter species. In this case, we did not find any relevant differences among the three runs, implying that the effect is observable [*only*]{} if the numerical resolution is high enough to resolve halo masses of $\sim 10^4\msun/{\it h}$. This highlights the difficulty of current large-scale simulations to address these effects.\
In Table \[tab:sims\] we list the main features of the numerical set-ups.
--------------- ----------------- ----------------- ------------------------ -------------------------------- ------------------------------- ---------- ---------------------
Model box side number of mean inter-particle $M_{gas}\rm [M_\odot/{\it h}]$ $M_{dm}\rm [M_\odot/{\it h}]$ initial bulk shift \[km/s\]
\[Mpc/[*h*]{}\] particles separation \[kpc/$h$\] redshift at $z=1020$
HR-Shift00 0.7 $2\times 320^3$ 2.187 $1.16\times 10^2$ $7.55\times 10^2 $ 100 0
HR-Shift30 0.7 $2\times 320^3$ 2.187 $1.16\times 10^2$ $7.55\times 10^2 $ 100 30
HR-Shift60 0.7 $2\times 320^3$ 2.187 $1.16\times 10^2$ $7.55\times 10^2 $ 100 60
HR-Shift00Rec 0.7 $2\times 320^3$ 2.187 $1.16\times 10^2$ $7.55\times 10^2 $ 1020 0
HR-Shift60Rec 0.7 $2\times 320^3$ 2.187 $1.16\times 10^2$ $7.55\times 10^2 $ 1020 60
LR-Shift00 5.0 $2\times 320^3$ 15.62 $4.23\times 10^4$ $ 2.75\times 10^5$ 100 0
LR-Shift30 5.0 $2\times 320^3$ 15.62 $4.23\times 10^4$ $ 2.75\times 10^5$ 100 30
LR-Shift60 5.0 $2\times 320^3$ 15.62 $4.23\times 10^4$ $ 2.75\times 10^5$ 100 60
\[tab:sims\]
--------------- ----------------- ----------------- ------------------------ -------------------------------- ------------------------------- ---------- ---------------------
Results {#sect:results}
=======
We start by looking at the overall evolution of the star formation rate density of the Universe, and then we consider the main statistical quantities involved in the cosmological evolution process, i.e. the distributions of dark-matter haloes and gas clouds.
Star formation rate density
---------------------------
The behavior of the star formation rate densities is plotted in Fig. \[fig:sfr\], for all the cases considered before. What emerges is a systematic shift in the onset of star formation, with a delay increasing for higher values of $\vbx$. For the $\vbx=0\,\rm km/s$ case, the onset happens at $z\sim 16.3$, when the Universe is roughly $2.5\times 10^8\,\rm yr$ old, while in the $\vbx=30\,\rm km/s$ and $60\,\rm km/s$ cases, star formation sets in at $z\sim 16.1$ and 15.6, respectively, corresponding to a delay of $\sim 10^7\,\rm yr$ and $\sim 2\times 10^7\,\rm yr$ compared to the case without velocity shift. Differences at very early times are almost one order of magnitude and at later times a factor of a few, persisting down to $z\sim13$, when the global trends rapidly converge to similar values. The reason why this happens is due to the higher kinetic energy given to the gas at the recombination epoch: because of that, first dark-matter haloes can retain less material, gas condenses more slowly hindering molecule formation, the corresponding cooling is less efficient, and the resulting star formation is delayed. In particular, this is very relevant for small primordial haloes, with masses of $\sim 10^4\msun - 10^8\msun$ [see also @TseliakhovichHirata2010], whose dimensions are comparable to or smaller than the baryon Jeans length. In this case in fact, the gas cannot fragment within the halo and partially or entirely flows out of it, since larger dark-matter potential wells are needed to retain the gas. Therefore, this effect on small scales influences the growth of larger objects, as well. From this we conclude that the inflow and condensation of gas, and thus also star formation, is already strongly suppressed in haloes in the mass-range $\la 10^8\msun$ [*before reionization*]{}. Those low-mass halos surviving to the present day, or even those that merge into larger masses, would therefore have a lower baryonic and stellar mass fraction. We further note that the effect of suppressing star formation is similar to that of reionization, but it has a very different physical origin and occurs well before reionization itself.
![Total star formation rates for bulk velocity shifts of $0\,\rm km/s$ (black solid line), $30\,\rm km/s$ along the x-axis (blue dotted line), $60\,\rm km/s$ along the x-axis (red dashed line) are shown for the simulation box of side $\sim 1$ Mpc. []{data-label="fig:sfr"}](figure/sfr_07Mpch_vshift.ps){width="40.00000%"}
Dark-matter haloes and gas clouds
---------------------------------
In each simulation, we identify cosmic structures with the use of a friend-of-friend (FoF) algorithm. Each object is represented by all those particles (dark-matter, gas and star) closer than a limiting linking length of $20\%$ the mean inter-particle separation. The dark-matter halo and gas cloud distributions are shown in Fig. \[fig:haloes07\], at different redshift as indicated by the legends, for the runs starting at $z_{in}=100$. In general, at very high redshift we cannot give solid conclusions, since small-number statistic dominates the sample. At later times (i.e. $z{\la}23$), some systematic behavior clearly emerges. The mass distributions of dark matter haloes range between $\sim 10^4\msun/h$ and $10^7\msun/h$. The three cases of $\vbx = 0, 30, 60\,\rm km/s$ show a slight decrement with increasing $\vbx$, but only at a few percent level. Hence, as expected, the effect of the bulk-flow of gas on the overall growth of the dark-matter dominated haloes is minimal.
{width="22.00000%"} {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} {width="22.00000%"} {width="22.00000%"}
More clear differences arise from the analyses of the gas clouds. In Fig.\[fig:haloes07\], we also show the mass distribution of the gaseous component of the objects and we clearly see decrements up to tens of percent, mostly evident in the $\vbx = 60\,\rm km/s$ case. Interestingly, there is a cascade effect from smaller to larger masses (expected from the simple analyses of the star formation rates), which delays and slightly hinders the accumulation of gas in the dark-matter haloes (i.e. suppression of gas inflow in lower-mass haloes also suppresses the gas accumulation in high-mass haloes through accretion of low-mass haloes). As expected, the average gas cloud mass decreases for larger $\vbx$. At $z\simeq 23$, it drops from $\sim 1.70\times 10^3\msun/{\it h}$ ($\vbx=0\,\rm km/s$) down to $\sim 1.62\times 10^3\msun/{\it h}$ ($\vbx=30\,\rm km/s$), and $\sim 1.39\times 10^3\msun/{\it h}$ ($\vbx=60\,\rm km/s$). While at $z\simeq 15$, it is $\sim 4.41\times 10^3\msun/{\it h}$ ($\vbx=0\,\rm km/s$), $\sim 4.26\times 10^3\msun/{\it h}$ ($\vbx=30\,\rm km/s$), and $\sim 4.06\times 10^3\msun/{\it h}$ ($\vbx=60\,\rm km/s$). The differences correspond to decrements with respect to the reference run of $\sim 6\%-3\%$ ($\vbx=30\,\rm km/s$) and $\sim 20\%-10\%$ ($\vbx=60\,\rm km/s$) at $z\simeq 23-15$, respectively. Also the total abundance of gas clouds shows a similar trend, with mass-weighted number counts dropping, with respect to the reference run, by $\sim 5\%-3\%$ ($\vbx=30\,\rm km/s$) and $\sim 20\%-50\%$ ($\vbx=60\,\rm km/s$) at $z\simeq 23-15$. We have verified that the excess small clumps observed in the low-mass tail of the distributions (at masses ${\la}500\msun/{\it h}$) for $\vbx>0$ are due to residual gas which, differently from the reference run, is not trapped by the dark-matter haloes and, later collapses to form such clumps. It should be noted though that statistical errors in this mass regime are substantial, due to the small number of particles contained in a halo.
![Evolution of the gas content (top) and of the gas fraction (bottom) for the first (and most massive) halo formed in the simulations. Data refer to velocity shifts of $0\,\rm km/s$ (solid lines, black crosses), $30\,\rm km/s$ (dotted lines, blue rhombi), $60\,\rm km/s$ (dashed lines, red triangles). []{data-label="fig:evolution07"}](figure/Mpch07/halo_evolution_1Mpc_gasonly_compare.ps){width="40.00000%"}
As a more specific example, in Fig. \[fig:evolution07\] we show the evolution of the gas content (top panel) and gas fraction (bottom panel) of the first (and most massive) halo formed in the simulations. The gas mass ranges between $\sim 10^3\msun/h$ and $10^7\msun/h$, and the corresponding dark-matter mass is about one order of magnitude larger. The gas mass variations due to the different values of $\vbx$ are significant at $z{\ga}16$, while at smaller redshift the total amount converges to $\sim 2-6\times 10^6\msun/h$. In this respect, the gas fractions in the halo are more informative, and suggest a depletion of gas for higher $\vbx$ up to a factor of 2. Indeed, at $z{\ga}24$, the gas fraction for the $\vbx=0\,\rm km/s$ case is $\sim 0.04$, while in the $\vbx=30\,\rm km/s$ and $\vbx=60\,\rm km/s$ cases it drops to $\sim 0.03$ and below $0.02$, respectively. Similar strong variations are observed also at later times (e.g. at $z\sim 23$), but the general trend converges at $z{\la}16$, when the discrepancies among the cases become $< 10\%$ and all the gas fractions reach values of $\sim 0.10$. The same effects are observed in the gas mass and density profiles as function of the (physical) radius – see Fig. \[fig:profiles07\] and compare to Fig. 2 in [@TseliakhovichHirata2010]. At early times (left panels) the amount of gas falling in the halo depends on $\vbx$, and the larger $\vbx$, the more gas is depleted in the distribution tails. At larger radii, the mass depletion is up to $\sim 50\%$ and smoothly decreases towards the core, where the gas is even denser and more abundant. This can be seen as a consequence of the fact that there is less gas falling in, and it sinks more easily in the potential wells of the halo. Of course, given the low number of SPH particles reaching those regimes, the statistical errors might be quite significant and ad hoc simulations are needed to address this issue in more detail.
Summary and conclusions {#sect:conclusion}
=======================
{width="45.00000%"} {width="45.00000%"}
Stimulated by very recent analytical work [@TseliakhovichHirata2010], we have run cosmological, high-resolution numerical N-body/SPH simulations including detailed, non-equilibrium chemistry evolution, cooling, star formation and feedback effects [see e.g. @Maio2007; @Maio_et_al_2010b], which consistently follow the gas behavior down to the catastrophic cooling regime, and correctly resolve the small, primordial Jeans scales with $\sim 10^2$ SPH particles. To take into account very early, supersonic, coherent, Mpc-scale flows, generated at recombination time from the advection of small-scale perturbations by large-scale velocity flows, we set initial gas velocity shifts along the $x$-direction, $\vbx=0,30,60\,\rm km/s$ based on analytic estimates of the rms differences in dark-matter and baryonic gas velocities at $z=1020$. We find that in the simulations with larger $\vbx$ the onset of star formation is delayed by some $\sim 10^7\,\rm yr$, and it is smaller by a factor of a few in comparison to the $\vbx=0\,\rm km/s$ case, until redshift $z\sim 13$. As a consequence, the beginning of the reionization process is delayed by the same amount and initially driven by more massive objects. More ad hoc calculations should be done to quantify the differences in terms of evolution, topology and photon budget. This might also have consequences in terms of detection of the 21cm signal from neutral hydrogen, because the building up of a Ly$\alpha$ background [see e.g. @2010Natur.468...49R for a recent review] necessary for the observability of the line [e.g. @2006PhR...433..181F] would be delayed as well. We speculate that this could lead to more patchy reionization and HI heating, with larger HI brightness-temperature fluctuations. More research is needed to quantify these effects. Because we find that the amount of gas in the first haloes is expected to drop significantly (up to a factor of 2 at early times, i.e. $\sim 50\%$), there could also be interesting consequences in terms of feedback effects, as e.g. less gas in small mass objects should make it easier to further remove gas via photoevaporation, winds or SN explosions. It will be interesting to investigate in more details the interplay between bulk-flow induced feedback effects. Slight differences (at $\sim 1\%$ level) in the dark-matter statistical distributions also arise, but stronger differences (up above $\sim 10\%$ level) are seen in the gas clouds. A cascade effect transfers the inability of primordial, small haloes to completely retain the gas to larger, lower-redshift haloes, whose gaseous components are systematically smaller for $\vbx > 0\,\rm km/s$. In this way, the original hindering for the very first objects is transferred to the structure which form later on, affecting their growth history. Numerical studies performed on lower-resolution simulations show no detectable differences among the different cases, and this means that the lack of numerical resolution can be a strong limitation to probe higher-order corrections to linear perturbation theory, and SPH particle masses of $\sim 10^2\msun/{\it h}$ are required. The precise initial redshift does not alter significantly our findings either.\
Our general conclusions from this albeit preliminary, but [*first*]{} fully non-linear, study of the effects of bulk gas flows on non-linear cosmic structure growth are that it: (i) delays early star formation, which affects both reionization (by delaying it to lower redshifts) and the heating of gas of higher redshift, possible affecting total/global emission and absorption features of HI against the CMB by making these effects more patchy; (ii) suppresses star formation in the lowest-mass haloes over all redshifts which has a cascading effect over nearly 300 million years on higher-mass haloes which are their merger-products. Because these small early objects, if they survive till the present day, are expected to be the parent population of e.g. dSph galaxies, the effct of the bulk flows can also have an impact on the stellar/gas content of the lowest-mass dwarf satellites around more massive galaxies (e.g. the MW) and be highly spatially dependent (i.e. the effect is only present where there were large bulk flows at decoupling). This could result in a strongly spatially varying bias as suggested by @TseliakhovichHirata2010 and @Dalal_et_al_2010. In addition, suppression of gas infall and condensation on these small scales could play an important role in the “missing satellite problem” [see e.g. @Kravtsov2010 and references therein].
acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the referee C. Hirata, for his swift and postitive feeback. We acknowledge useful discussions with J. Bolton, S. D. M. White and U. Pen. UM acknowledges S. Khochfar and the tmox group. LVEK acknowledges the generous support by an ERC-ST Grant.
\[lastpage\]
[^1]: [email protected]
|
---
abstract: 'Let $\Phi^\infty(d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\le 4$. In this article we determine $\Phi^\infty(5)$ and $\Phi^\infty(6)$.'
author:
- 'Maarten Derickx and Andrew V. Sutherland'
title: |
Torsion subgroups of elliptic curves over\
quintic and sextic number fields
---
Introduction
============
Let $E$ be an elliptic curve over a number field $K$. By the Mordell-Weil theorem [@Weil29], the set of $K$-rational points on $E$ forms a finitely generated abelian group $E(K)$. In particular, its torsion subgroup $E(K)_{\rm tors}$ is finite, and it is well known that it can be generated by two elements [@Silverman09]. There thus exist integers $m,n\ge 1$ such that $$E(K)_{\rm tors}\simeq {{\mathbb{Z}}}/m{{\mathbb{Z}}}\oplus {{\mathbb{Z}}}/mn{{\mathbb{Z}}}.$$ The uniform boundedness conjecture states that for every number field $K$ there exists a bound $B$ such that $\#E(K)_{\rm tors}\le B$ for every elliptic curve $E$ over $K$. This conjecture is now a theorem due to Merel [@Merel96], who actually proved the strong version of this conjecture in which the bound $B$ depends only on the degree $d\coloneqq[K\!:{{\mathbb{Q}}}]$. It follows that for every positive integer $d$, there is a finite set $\Phi(d)$ of isomorphism classes of torsion subgroups that arise for elliptic curves over number fields of degree $d$; we may identify elements of $\Phi(d)$ by pairs of positive integers $(m,mn)$.
The set $\Phi(1)$ was famously determined by Mazur [@Mazur77a], who proved that $$\Phi(1)=\{(1,n):1\le n\le 12,\ n\ne 11\}\ \cup\ \{(2,2n):1\le n\le 4\}.$$ The set $\Phi(2)$ was determined in a series of papers by Kenku, Momose, and Kamienny, culminating in [@KM88; @Kamienny92], that yield the result $$\Phi(2)=\{(1,n):1\le n\le 18,\ n\ne 17\}\ \cup\ \{(2,2n):1\le n\le 6\}\ \cup\ \{(3,3),(3,6),(4,4)\}.$$ For $d>2$ the sets $\Phi(d)$ have yet to be completely determined. However, if we distinguish the subset $\Phi^\infty(d)\subseteq \Phi(d)$ of torsion subgroups that arise for infinitely many ${\overline{{{\mathbb{Q}}}}}$-isomorphism classes of elliptic curves defined over number fields of degree $d$, we can say more.
We have $\Phi^\infty(1)=\Phi(1)$ and $\Phi^\infty(2)=\Phi(2)$. In [@JKS04] Jeon, Kim, and Schweizer found $$\Phi^\infty(3) =\, \{(1,n):1\le n\le 20, \ n\ne 17,19\}\ \cup\ \{(2,2n):1\le n\le 7\},$$ and in [@JKP06] Jeon, Kim and Park obtained $$\begin{aligned}
\Phi^\infty(4) =\ &\{(1,n):1\le n\le 24, \ n\ne 19,23\}\ \cup\ \{(2,2n):1\le n\le 9\}\\
&\cup\ \{(3,3n):1\le n\le 3\}\ \cup \{(4,4),(4,8),(5,5),(6,6)\}.\end{aligned}$$ In this article we determine the sets $\Phi^\infty(5)$ and $\Phi^\infty(6)$.
\[thm:main\] Let $\Phi^\infty(d)$ denote the set of pairs $(m,mn)$ for which $E(K)_{\rm{tors}}\simeq {{\mathbb{Z}}}/m{{\mathbb{Z}}}\oplus{{\mathbb{Z}}}/mn{{\mathbb{Z}}}$ for infinitely many non-isomorphic elliptic curves $E$ over number fields $K$ of degree $d$. Then $$\Phi^\infty(5)=\,\{(1,n):1\le n\le 25, \ n\ne 23\}\ \cup\ \{(2,2n):1\le n \le 8\},$$ and $$\begin{aligned}
\Phi^\infty(6) =\ &\{(1,n):1\le n\le 30,\ n\ne 23,25,29\}\ \cup\ \{(2,2n):1\le n\le 10\}\\
&\cup\ \{(3,3n):1\le n\le 4\}\ \cup \{(4,4),(4,8),(6,6)\}.\end{aligned}$$
For $d=5,6,7,8$, the elements $(1,n)\in \Phi^\infty(d)$ were determined in [@DvH13] using a strategy that we generalize here. The key steps involve computing (or at least bounding) the gonalities of certain modular curves, and determining whether their Jacobians have rank zero or not. To obtain gonality bounds we require explicit models for the modular curves $X_1(m,mn)$ that parametrize triples $(E,P,Q)$, where $E$ is an elliptic curve with independent points $P$ of order $m$ and $Q$ of order $mn$; for our approach to be computationally feasible, it is important that these models have low degree and reasonably small coefficients. Optimized models for $X_1(n)=X(1,n)$ for $n\le 50$ were constructed in [@Sutherland12]. Here we extend the approach of [@Sutherland12] to construct optimized models for $X_1(m,mn)$ for $m^2n\le 120$, as described in §\[sec:models\]; these can be found at [@models]. These models are necessarily defined only over the cyclotomic field ${{\mathbb{Q}}}(\zeta_m)$. The need to work over ${{\mathbb{Q}}}(\zeta_m)$ requires us to develop some new techniques for determining when the Jacobian of $X_1(m,mn)$ has rank zero over ${{\mathbb{Q}}}(\zeta_m)$; these are described in §\[sec:Jrank\]. The case $X_1(2,30)$ proved to be computationally challenging, so we used an alternative strategy based on ideas in [@DKM16], as explained in §\[sec:proofs\].
In principle our methods can also determine $\Phi^\infty(7)$; we have computed explicit models for all of the relevant modular curves and proved that their Jacobians have rank zero. We can prove $(2,2n)\in \Phi^\infty(7)$ for $1\le n\le 10$ and not for $n>15$; it remains only to determine for $11\le n\le 15$ whether the gonality of $X_1(2,2n)$ is greater than $7$ or not (we expect the answer is yes in each case). Similar comments apply to $\Phi^{\infty}(8)$ but not $\Phi^{\infty}(9)$, which requires new techniques, as explained in .
The source code for computations on which results of this article depend is available at [@mdmagma].
Background {#sec:background}
==========
In this section we briefly recall background material and introduce some notation. Let $K$ be a field and let $X/K$ be a *nice curve*, by which we mean that $X/K$ is of dimension 1, smooth, projective, and geometrically integral. The *gonality* $\gamma(X)$ of $X$ is the minimal degree of a finite $K$-morphism $X\to {{\mathbb{P}}}^1_K$. If $L/K$ is a field extension and $X_L:=X\times_K L$ is the base change of $X$ to $L$, we necessarily have $\gamma(X_L)\le \gamma(X)$, and we call $\gamma(X_L)$ the $L$-*gonality* of $X$. If $K$ is a number field, ${\mathfrak p}$ is a prime of $K$ of good reduction for $X$, and $X_{{{\mathbb{F}}}_{\mathfrak p}}$ is the reduction of $X$ to the residue field ${{\mathbb{F}}}_{\mathfrak p}$ of ${\mathfrak p}$, then $\gamma(X_{{{\mathbb{F}}}_{\mathfrak p}})\le \gamma(X)$, and we call $\gamma(X_{{{\mathbb{F}}}_{\mathfrak p}})$ the ${{\mathbb{F}}}_{\mathfrak p}$-*gonality* of $X$.
\[thm:Abramovich\] Let $\Gamma\subseteq \operatorname{PSL}_2({{\mathbb{Z}}})$ be a congruence subgroup. The ${{\mathbb{C}}}$-gonality of the modular curve $X_\Gamma$ is at least $(\lambda_1/24)[\operatorname{PSL}_2({{\mathbb{Z}}}):\Gamma]$, where $\lambda_1\ge 975/4096$.
See [@Abramovich96 Thm.0.1] for the first statement and [@KS03 p.176] for the lower bound on $\lambda_1$.
Let $d(X)$ denote the least integer for which the set $\{a\in X(\overline{K}):[K(a):K]=d\}$ of *points of degree $d$* on $X$ is infinite. We have the following result of Frey [@Frey94], which can be viewed as a corollary of Faltings’ proof of Lang’s conjecture [@Faltings94].
\[thm:Frey\] Let $X$ be nice curve over a number field. Then $d(X)\le \gamma(X)\le 2d(X)$.
If $f\in K(X)$ is a function of degree $d$, then by Hilbert irreducibility there are infinitely many points of degree $d$ over $K$ among the roots of $f-c$ as $c$ varies over $K$; this proves the first inequality, and the second is [@Frey94 Prop.1].
There is one situation in which the lower bound of is known to be tight.
\[lem:rank0\] Let $X/K$ be a nice curve whose Jacobian has rank $0$. Then $d(X)=\gamma(X)$.
Let $d < \gamma(X)$ be a positive integer. The map $\pi\colon X^{(d)}\to{\operatorname{Jac}}(X)$ from the $d$th symmetric power of $X$ to its Jacobian is injective, since otherwise we could construct a function $f\in K(X)$ of degree $d<\gamma(X)$ from the difference of two linearly equivalent divisors of degree $d$ and view $f$ as a map $X\to{{\mathbb{P}}}^1_K$. If ${\operatorname{rk}}( {\operatorname{Jac}}(X)(K))=0$ then $\pi(X^{(d)}(K))$ is finite, and so is $X^{(d)}(K)$; it follows that $X$ has only finitely many points of degree $d$.
The proofs of together imply the following corollary.
\[cor:rank0\] Let $X/K$ be a nice curve over a number field. If $K(X)$ contains a function of degree $d$ then $X$ has infinitely many points of degree $d$. When ${\operatorname{rk}}({\operatorname{Jac}}(X)(K))=0$ the converse also holds.
For positive integers $m$ and $n$ we use $Y_1(m,mn)$ to denote the modular curve that parameterizes triples $(E,P,Q)$, where $E$ is an elliptic curve with independent points $P$ of order $m$ and $Q$ of order $mn$, and $X_1(m,mn)$ is its projectivization obtained by adding cusps. We view $X_1(m,mn)$ as a ${{\mathbb{Z}}}[\frac{1}{mn}]$-scheme that is isomorphic to the coarse moduli space for the corresponding algebraic stack $\mathcal{X}_1(m,mn)$ in the sense of [@DR73]; all the cases of interest to us have $mn \ge 5$ and are also fine moduli spaces. By fixing a primitive $m$th root of unity $\zeta_m$ in the function field of $X_1(m,mn)$, we may also view $X_1(m,mn)$ as a nice curve over ${{\mathbb{Q}}}(\zeta_m)$ that has good reduction at all primes not dividing $mn$, and we write $J_1(m,mn)$ for the Jacobian of $X_1(m,mn)$ as a curve over ${{\mathbb{Q}}}(\zeta_m)$. There is an associated congruence subgroup $$\Gamma_1(m,mn):=\left\{\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in \operatorname{SL}_2({{\mathbb{Z}}}):a\equiv d\equiv 1\bmod mn,\ c\equiv 0\bmod mn,\ b\equiv 0\bmod m\right\},$$ and after fixing an embedding ${{\mathbb{Q}}}(\zeta_m)\hookrightarrow {{\mathbb{C}}}$, the quotient ${{\mathbb{H}}}^*/\Gamma_1(m,mn)$ of the extended upper half-plane by the action of $\Gamma_1(m,mn)$ is a compact Riemann surface isomorphic to $X_1(m,mn)({{\mathbb{C}}})$. The image of $\Gamma_1(m,mn)$ in $\operatorname{PSL}_2({{\mathbb{Z}}})$ has index $\frac{m^3n^2}{2}\prod_{p|mn}(1-\frac{1}{p^2})\ge m^2n-1$.
Let $X_{0,1}(m,mn)$ be the projectivisation of the modular curve that parametrizes triples $(E,G,Q)$, where $E$ is an elliptic curve, $G$ is a cyclic subgroup of order $m$ (or equivalently, a cyclic isogeny of degree $m$) and $Q$ is an independent point of order $mn$ (so $G\cap \langle Q\rangle=\{0\}$). The curve $X_{0,1}(m,mn)_{{{\mathbb{Q}}}(\zeta_m)}$ is isomorphic to $X_{1,1}(m,mn)$ (as can be seen by considering the corresponding congruence subgroups, or by writing down a natural transformation between the two functors on schemes over ${{\mathbb{Z}}}/nm{{\mathbb{Z}}}$, where we view $X_{0,1}(m,mn)_{{{\mathbb{Q}}}(\zeta_m)}$ as parametrizing quadruples $(E,G,Q,\zeta_m)$ with $E,G,Q$ as above and $\zeta_m$ a chosen primitive $m$th root of unity). Unlike $X_1(m,mn)$, the curve $X_{0,1}(m,mn)$ has the advantage of always being defined over ${{\mathbb{Q}}}$. Now let $X_1(m^2n):=X_1(1,m^2n)$ parameterize pairs $(E,Q)$ in which $E$ is an elliptic curve with a point $Q$ of order $m^2n$, and consider the map $$\varphi\colon X_1(m^2n)\to X_{0,1}(m,mn)$$ that sends the pair $(E,Q)$ to the triple $(E/\langle mnQ\rangle,\,E[m]/\langle mnQ\rangle,\, Q \bmod \langle mnQ\rangle)$. The group $E[m]/\langle mnQ\rangle$ and the point $Q \bmod \langle mnQ\rangle $ are independent because $Q \bmod \langle mnQ\rangle $ has the same order $mn$ as $Q \bmod E[m]$. The map $\varphi$ is defined over ${{\mathbb{Q}}}$ and has degree $m$. The group $({{\mathbb{Z}}}/m^2n{{\mathbb{Z}}})^\times$ acts on $X_1(m^2n)$ via the diamond operators $\langle a\rangle\colon (E,P)\mapsto (E,aP)$. We have $a\equiv 1\bmod mn$ precisely when $\langle a\rangle$ stabilizes $\varphi$, meaning $\varphi=\varphi\circ\langle a\rangle$, and the quotient of $X_1(m^2n)$ by this automorphism subgroup is isomorphic to $X_{0,1}(m,mn)$.
Throughout this article ${\overline{{{\mathbb{Q}}}}}$ denotes a fixed algebraic closure of ${{\mathbb{Q}}}$ that contains all number fields $K$ under consideration, and we identify $\overline{K} = {\overline{{{\mathbb{Q}}}}}$. For a modular curve $X$ defined over a number field $K$, we define the *degree over ${{\mathbb{Q}}}$* of a point $a\in X(\overline{{{\mathbb{Q}}}})$ to be the absolute degree $[L:{{\mathbb{Q}}}]$ of the minimal extension $L/K$ for which $a\in X(L)$ and let ${{\mathbb{Q}}}(a)$ denote the field $L$ (which contains $K$). For the sake of clarity we may refer to $[L:K]$ as the *degree of $a$ over $K$*.
Finally, we use $\phi(m)\coloneqq[{{\mathbb{Q}}}(\zeta_m):{{\mathbb{Q}}}]=\#({{\mathbb{Z}}}/m{{\mathbb{Z}}})^\times$ throughout to denote the Euler function.
Constructing models of X1(m,mn) {#sec:models}
===============================
Our method for constructing explicit methods of $X_1(m,mn)$ is a generalization of the technique used in [@Sutherland12] to construct models for $X_1(n)\coloneqq X_1(1,n)$, which we now briefly recall. Given $n>3$ one begins as in [@Reichert86] with the universal family of elliptic curves $$E(b,c):=y^2+(1-c)y-by=x^3-bx^2$$ in Tate normal form with rational point $P=(0,0)$ and imposes the constraint $$\left\lceil\frac{n+1}{2}\right\rceil P + \left\lfloor\frac{n-1}{2}\right\rfloor P = 0$$ by requiring the $x$-coordinates of the two summands to coincide (the $y$-coordinates of the summands cannot coincide because $\lceil (n+1)/2\rceil\ne \lfloor(n-1)/2\rfloor$, so the points must sum to zero). After clearing denominators and removing spurious factors corresponding to torsion points whose order properly divides $n$, one obtains a (singular) affine plane curve $C/{{\mathbb{Q}}}$ with the same function field as $X_1(n)$. Each non-singular point $(b_0,c_0)$ on this curve determines an elliptic curve $E(b_0,c_0)$ on which $P=(0,0)$ is a point of order $n$. The equations obtained by this method are typically much larger than necessary, but the algorithm in [@Sutherland12] can be used to obtain models with lower degrees, fewer terms, and smaller coefficients.
Constructing models using elliptic surfaces {#sec:specialmodels}
-------------------------------------------
We use a similar approach to construct equations for $X_1(m,mn)$. For $m=2$ we use the parameterized family of elliptic curves constructed by Jain in [@Jain10], in which the elliptic curve $$E_2(q,t):\quad y^2 = x^3+(t^2-qt-2)x^2-(t^2-1)(qt+1)^2x,$$ has the rational point $P_2:=(0,0)$ of order 2 and the rational point $$Q_2(q,t) := \bigl((t+1)(qt+1)\,,\,t(qt+1)(t+1)\bigr)$$ of infinite order; see [@Jain10 Thm.1.ii]. As suggested to us by Noam Elkies, for any $n>1$, setting $$\left\lceil\frac{2n+1}{2}\right\rceil Q_2(q,t) + \left\lfloor\frac{2n-1}{2}\right\rfloor Q_2(q,t)=0,$$ allows us to construct a model for $X_1(2,2n)$; as above, it is enough to equate the $x$-coordinates of the two summands, and this yields a polynomial equation in $q$ and $t$. With $n=7$, for example, after clearing denominators and removing spurious factors we obtain the equation $$\begin{aligned}
7q^{12}t^4 &+ 56q^{11}t^3 + 70q^{10}t^4 + 112q^{10}t^2 + 208q^9t^3 + 64q^9t - 111q^8t^4 + 144q^8t^2\\
&- 624q^7t^3 - 156q^6t^4 - 1104q^6t^2 - 512q^5t^3 - 832q^5t - 55q^4t^4 - 592q^4t^2\\
&- 256q^4 - 136q^3t^3 - 256q^3t - 10q^2t^4 - 96q^2t^2 - 16qt^3 - t^4 = 0.\end{aligned}$$ This equation is not as compact as we might wish, and its degree in both $q$ and $t$ is greater than the gonality of $X_1(2,14)$, which is $3$. However, after applying the optimizations described in [@Sutherland12] we obtain the equation $$(u^2 + u)v^3 + (u^3 + 2u^2 - u - 1)v^2 + (u^3 - u^2 - 4u - 1)v - u^2 - u = 0,$$ whose degree in $u$ and $v$ matches the gonality of $X_1(2,14)$. The relation between the $(u,v)$ coordinates and the $(q,t)$ coordinates is given by $$q = \frac{u+v}{v-u},\qquad\qquad t = \frac{(u-v)(u+v)(u+v+2)}{u^3 + u^2v + 2u^2 + uv^2 + 2uv + v^3 + 2v^2}.$$ The reader may wish to compare this model for $X_1(2,14)$ with the one given in [@JKL11a p.589].
A similar approach can be used to obtain equations for $X_1(3,3n)$. From [@Jain10 Thm.4.ii] we have the parameterized family of elliptic curves $$E_3(q,t):\quad y^2 + (qt-q+t+2)xy+(qt^2-qt+t)y=x^3,$$ with the rational point $P_3:=(0,0)$ of order 3 and the rational point $$Q_3(q,t):=(-t,t^2)$$ of infinite order. For $n\ge 1$, equating the $x$-coordinates of $\left\lceil\frac{3n+1}{2}\right\rceil Q_3(q,t)$ and $\left\lfloor\frac{3n-1}{2}\right\rfloor Q_3(q,t)$ yields an equation $F(q,t)=0$ that we may use to construct a model for $X_1(3,3n)$. In order to obtain a geometrically integral curve we must factor $F$ over ${{\mathbb{Q}}}(\zeta_3)$ rather than ${{\mathbb{Q}}}$ (if we only factor over ${{\mathbb{Q}}}$, over ${{\mathbb{Q}}}(\zeta_3)$ we will have the union of two curves that are both birationally equivalent to $X_1(3,3n)$).
To obtain equations for $X_1(4,4n)$ we use a parameterization due to Kumar and Shioda; see the example following [@KS13 Rem. 10]. We have the family $$E_4(q,t):\quad y^2 + xy + (1/16)(q^2-1)(t^2-1)y = x^3 + (1/16)(q^2-1)(t^2-1)x^2,$$ with the rational point $P_4:=(0,0)$ of order 4 and the rational point $$Q_4(q,t):= \bigl((q+1)(t^2-1)/8\,,\,(q+1)^2(t-1)^2(t+1)/32\bigr)$$ of infinite order. For any $n\ge 1$, equating the $x$-coordinates of $\left\lceil\frac{4n+1}{2}\right\rceil Q_4(q,t)$ and $\left\lfloor\frac{4n-1}{2}\right\rfloor Q_4(q,t)$ yields an equation $F(q,t)=0$ that we may use to construct a model for $X_1(4,4n)$ after factoring $F(q,t)$ over ${{\mathbb{Q}}}(\zeta_4)={{\mathbb{Q}}}(i)$.
\[rem:verify\] It is usually obvious which factor of $F(q,t)$ is the correct choice (the biggest one), but one can verify the correct choice by checking that it yields a curve of the same genus as $X_1(m,mn)$. The other non-conjugate factors of $F(q,t)$ correspond to modular curves $X$ that admit a non-constant map from $X_1(m,mn)$ of degree greater than $1$; provided $X_1(m,mn)$ has genus $g>1$, the Riemann-Hurwitz formula implies $g > g(X)$. For $g\le 1$ one can instead prove that none of the non-conjugate factors yield a model for $X_1(m,mn)$ by finding a non-singular point that does not yield a triple $E(E,P,Q)$ with $P$ and $Q$ of the correct order (always possible).
A general method {#sec:generalmodels}
----------------
The methods in the previous section for $m=2,3,4$ rely on parameterizations obtained from elliptic surfaces that do not exist in general. We now sketch a general method that works for any $m>3$. Rather than constructing a model for the curve $X_1(m,n)$, we will construct a model for its quotient by the involution $(E,P,Q)\to(E,-P,Q)$, which we denote $X_1(m,n)^+$. The curve $X_1(m,n)^+$ is defined over ${{\mathbb{Q}}}(\zeta_m)^+$, the maximal real subfield of ${{\mathbb{Q}}}(\zeta_m)$, and its base change from ${{\mathbb{Q}}}(\zeta_m)^+$ to ${{\mathbb{Q}}}(\zeta_m)$ is isomorphic to $X_1(m,n)$; for $m=1,2,3,4,6$ we have ${{\mathbb{Q}}}(\zeta_m)^+={{\mathbb{Q}}}$ and $X_1(m,n)^+\simeq X_{0,1}(m,n)$. For the sake of brevity we give details only for the cases in which $X_1(m)$ has genus 0, which suffices for our purposes.
Let us fix $m>3$ and $n\ge 1$. We may view $E(b,c)$ as the universal elliptic curve with rational points $P=(0,0)$ and $Q=(x,y)$; let $e(b,c,x,y)=0$ be the equation defining $E(b,c)$. Let $f(b,c)=0$ be an equation for $X_1(m)$ constructed as in [@Sutherland12], and let $h(b,c,x)$ denote the irreducible polynomial obtained by equating the $x$-coordinates of $\lceil(mn+1)/2\rceil Q$ and $\lfloor(mn-1)/2\rfloor Q$ and removing spurious factors as above. The equations $e(b,c,x,y)=f(b,c)=h(b,c,x)=0$ define a curve in $\mathbb{A}^4[b,c,x,y]$; this curve is not reduced (because there are two choices for the unconstrained $y$-coordinate), but it contains the curve we seek.
Let us now assume $m\in \{4,5,6,7,8,9,10,12\}$ so that $g(X_1(m))=0$, in which case we may write $b=b(t)$ and $c=c(t)$ as functions of a single rational parameter $t$, as in [@Kubert76 Table3]. In this case there is no $f(b,c)$ to compute, our curve equation becomes $e(t,x,y)=0$, and we compute $h(t,x)$ by equating the $x$-coordinates of $\lceil(mn+1)/2\rceil Q$ and $\lfloor(mn-1)/2\rfloor Q$ and removing spurious factors as above. Let $H$ be the resultant of $e$ and $h$ with respect to the variable $y$; the polynomial $H(t,x)$ is the square of a polynomial $F\in{{\mathbb{Z}}}[t,x]$ that is irreducible over ${{\mathbb{Q}}}$ but splits into $\phi(m)/2$ geometrically irreducible factors over ${{\mathbb{Q}}}(\zeta_m)^+$ that are $\operatorname{Gal}({{\mathbb{Q}}}(\zeta_m)^+/{{\mathbb{Q}}})$-conjugates. We may take any of these factors as our model for $X_1(m,n)^+$. The base change of this model from ${{\mathbb{Q}}}(\zeta_m)^+$ to ${{\mathbb{Q}}}(\zeta_m)$ is then a model for $X_1(m,mn)$.
By combining this method with §\[sec:specialmodels\] and applying the algorithm in [@Sutherland12], we have constructed optimized models for $X_1(m,mn)$ for $m^2n\le 120$ and verified them as explained in ; these models are available in electronic form at [@models].
Modular Jacobians of rank zero over cyclotomic fields {#sec:Jrank}
=====================================================
In [@CES03], Conrad, Edixhoven, and Stein give an explicit method for computing $L$-ratios for modular forms on $X_1(n)$. By applying the proven parts of the Birch and Swinnerton-Dyer conjecture (see [@Kato04 Cor.14.3] or [@KL89]) they are then able to prove that the rank of $J_1(p)\coloneqq{\operatorname{Jac}}(X_1(p))$ is zero over ${{\mathbb{Q}}}$ for all primes $p < 73$ except for $p=37, 43,53,61, 67$; see [@CES03 §6.1.3, §6.2.2]. The software developed by Stein for performing these computations is available in the computer algebra system Magma [@magma] which can compute provably correct bounds on $L$-ratios as exact rational numbers; in particular, one can unconditionally determine when the $L$-ratio is nonzero, in which case the rank is provably zero.
Here we adapt this method to prove that $J_1(m,mn)$ has rank $0$ over ${{\mathbb{Q}}}(\zeta_m)$ for suitable values of $m$ and $n$. For the sake of brevity we focus on the cases $m=2,3,4,6$ in which ${{\mathbb{Q}}}(\zeta_m)$ has degree at most 2, which suffices for our application; the method generalizes to arbitrary $m$ and abelian extensions $K/{{\mathbb{Q}}}$ that contain an $m$th root of unity. This includes $m=5$, which we list below for reference but do not need to prove our main result.
\[thm:rank0\] The rank of $J_1(m,mn)$ is zero over ${{\mathbb{Q}}}(\zeta_m)$ if any of the following hold:
[2]{}
- $m=1$ and $n \leq 36$;
- $m=2$ and $n \leq 21$;
- $m=3$ and $n \leq 10$;
- $m=4$ and $n \leq 6$;
- $m=5$ and $n \leq 4$;
- $m=6$ and $n \leq 5$.
All computations referred to in the proof below were performed using the `IsX1mnRankZero` function implemented in [@mdmagma], which uses the `LRatio` function in Magma [@magma] to determine when the $L$-ratio is nonzero.
Let $J$ be the Jacobian of the quotient of the curve $X_1(m^2n)$ by the subgroup of diamond operators that stabilize the map $\varphi\colon X_1(m^2n)\to X_{0,1}(m,mn)$, as described in §\[sec:background\]. We then have $J_1(m,mn) \simeq J_{{{\mathbb{Q}}}(\zeta_m)}$, and it suffices to prove that the rank of $J_{{{\mathbb{Q}}}(\zeta_m)}({{\mathbb{Q}}}(\zeta_m)) = J({{\mathbb{Q}}}(\zeta_m))$ is zero.
For $m\le 2$ we have ${{\mathbb{Q}}}(\zeta_m)={{\mathbb{Q}}}$ and the strategy of [@CES03] can be applied directly; the desired result follows from a computation that finds the $L$-ratios to be nonzero for all modular forms $f$ corresponding to simple isogeny factors of $J$, for $m\le 2$ and $n$ as in the theorem.
We now assume $m\in \{3,4,6\}$ so that ${{\mathbb{Q}}}(\zeta_m)$ is a quadratic extension of ${{\mathbb{Q}}}$. Let $f$ be a newform of level dividing $m^2n$ such that the abelian variety $A_f$ associated to $f$ is an isogeny factor of $J$. Proving that ${\operatorname{rk}}(J({{\mathbb{Q}}}(\zeta_m)))=0$ is equivalent to proving that ${\operatorname{rk}}(A_f({{\mathbb{Q}}}(\zeta_m)))=0$ for all such $f$.
Let $A$ be the Weil descent of $A_{f,{{\mathbb{Q}}}(\zeta_m)}$ down to ${{\mathbb{Q}}}$. From the definition of the Weil descent we have $A({{\mathbb{Q}}}) = A_f({{\mathbb{Q}}}(\zeta_m))$, and the identity map $A_{f,{{\mathbb{Q}}}(\zeta_m)} \to A_{f,{{\mathbb{Q}}}(\zeta_m)}$ over ${{\mathbb{Q}}}(\zeta_m)$ induces a morphism $A_f \to A$ over ${{\mathbb{Q}}}$; it follows that $A$ is isogenous to $A_f \oplus A/A_f$. Let $\chi_m : ({{\mathbb{Z}}}/m{{\mathbb{Z}}})^\times \to {{\mathbb{Q}}}^\times$ denote the quadratic character of ${{\mathbb{Q}}}(\zeta_m)$. One sees that $A/A_f$ is isogenous to $A_{f_{\chi_m}}$ by comparing traces of Frobenius on their Tate modules. In particular, ${\operatorname{rk}}(A_{f}({{\mathbb{Q}}}(\zeta_m)))=0$ if and only if both ${\operatorname{rk}}(A_f({{\mathbb{Q}}}))=0$ and ${\operatorname{rk}}(A_{f_{\chi_m}}({{\mathbb{Q}}}))=0$ hold.
The theorem now follows from a computation; we find that the $L$-ratios of $f$ and ${f_{\chi_m}}$ are nonzero for $m=3,4,6$ and $n$ as in the theorem and all newforms $f$ of level dividing $m^2n$ such that the abelian variety $A_f$ associated to $f$ is an isogeny factor of $J$.
In fact (as kindly pointed out to us by the referee), it suffices to compute the $L$-ratios of the forms $f$; the twists $f_{\chi_m}$ already arise among the untwisted $f$ under consideration. Indeed, it follows from [@AtkinLi78 Prop 3.1] that for any Dirichlet character ${\varepsilon}$ of conductor dividing $mn$, if $f$ is a modular form in $S_2(\Gamma_0(m^2n),{\varepsilon})$, then so is $f_{\chi_m}$, since $\chi_m^2=1$.
We also computed the relevant $L$-ratios for $m\le 6$ and $n$ just past the range listed in ; for each of these $(m,n)$ we found that at least one relevant $L$-ratio was zero.
Proof of the main theorem {#sec:proofs}
=========================
The first step in our proof is to show for $d=5,6$ that ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ contains a function of degree $d/\phi(m)$ if and only if $(m,mn)$ is one of the pairs for $\Phi^\infty(d)$ appearing in the theorem. The following lemma illustrates why the forward implication is useful.
\[lem:suff\] If ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ contains a function of degree $\frac{d}{\phi(m)}$ then $(m,mn)\in \Phi^\infty(d)$.
We first note that for $mn \le 4$, either $\phi(m)=1$ and $(m,mn)\in \Phi(1)=\Phi^\infty(1)\subseteq\Phi^\infty(d)$, or $\phi(m)=2$ divides $d$ and $(m,mn)\in\Phi(2)=\Phi^\infty(2)\subseteq\Phi^\infty(d)$; in both cases the lemma holds.
We now assume $mn\ge 5$, in which case $X_1(m,mn)$ is a fine moduli space. By Merel’s proof of the uniform boundedness conjecture [@Merel96] there is a positive integer $B$ such that for all number fields $K$ of degree $d$ and elliptic curves $E/K$ we have $E(K)_{\rm tors} \subseteq E[B]$. The integer $B$ is necessarily divisible by $mn$, since $X_1(m,mn)$ has points of degree $d$ over ${{\mathbb{Q}}}$.
Now let $f\in {{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ be a function of degree $d/\phi(m)$. For each $a \in {{\mathbb{Q}}}(\zeta_m)$ the points in $f^{-1}(a)$ have degree at most $d/\phi(m)$ over ${{\mathbb{Q}}}(\zeta_m)$, hence degree at most $d$ over ${{\mathbb{Q}}}$. By Hilbert irreducibility, there are infinitely many $a$ for which the points in $f^{-1}(a)$ have degree exactly $d$, but we also need to show that there are infinitely many $a$ for which the torsion subgroups of the elliptic curves corresponding to the points in $f^{-1}(a)$ are actually isomorphic to ${{\mathbb{Z}}}/m{{\mathbb{Z}}}\times {{\mathbb{Z}}}/mn{{\mathbb{Z}}}$ and not any larger. In order to show this we consider the maps $$X_1(B,B) \overset{\pi}{\longrightarrow} X_1(m,mn)\overset{f}{\longrightarrow} {{\mathbb{P}}}^1,$$ where $\pi$ sends $(E,P,Q)$ to $(E,(B/m)P, B/(mn)Q)$, and let $\varphi\coloneqq f\circ\pi$. Let $A \subseteq {{\mathbb{P}}}^1({{\mathbb{Q}}}(\zeta_m))$ be the set of $a$ for which every $b\in \varphi^{-1}(a)$ has degree $d\deg\pi$ over ${{\mathbb{Q}}}$. The set $A$ is infinite, by Hilbert irreducibility, and for $a\in A$ every $c \in f^{-1}(a)$ has degree $d$ over ${{\mathbb{Q}}}$. We claim that for all such $c=(E,P,Q)$ we have $E({{\mathbb{Q}}}(c))_{\rm tors}=\langle P,Q\rangle$, which implies $(m,mn)\in \Phi^\infty(d)$ as desired.
Suppose not. Then we can construct a point $c'=(E,P',Q')$ on $X_1(m',m'n')$ of degree $d$ over ${{\mathbb{Q}}}$ with $P\in\langle P'\rangle,\, Q\in\langle Q'\rangle$, and $\langle P,Q\rangle\subsetneq \langle P',Q'\rangle$, and we have maps $$X_1(B,B) \overset{\pi_1}{\longrightarrow} X_1(m',m'n')\overset{\pi_2}{\longrightarrow}X_1(m,mn)\overset{f}{\longrightarrow} {{\mathbb{P}}}^1,$$ in which $\pi=\pi_2\circ\pi_1$, with $\deg\pi_1 < \deg\pi$, and $c=\pi_2(c')$. If we now consider $b\in\pi_1^{-1}(c')$, then $b\in\varphi^{-1}(a)$ has degree $d\deg \pi_1<d\deg\pi$ over ${{\mathbb{Q}}}$, a contradiction.
We now outline the strategy of the proof. Let $T$ be the set of pairs $(m,mn)$ identifying torsion subgroups that we wish to prove is equal to $\Phi^\infty(d)$. To prove $\Phi^\infty(d)=T$ we proceed as follows.
1. Prove that ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ contains a function of degree $d/\phi(m)$ for all $(m,mn)\in T$.
2. Compute the set $T_1\coloneqq \{(m,mn): \phi(m)|d\text{ and }B(m,mn)\le 2d\}$ where $B(m,mn)$ is the lower bound on $\gamma(X_1(m,mn))$ given by (we have $\Phi^\infty(d)\subseteq T_1$).
3. Verify that ${\operatorname{rk}}(J_1(m,mn)({{\mathbb{Q}}}(\zeta_m)))=0$ for all $(m,mn)\in T_1$ via .
4. Compute the set $T_2\coloneqq \{(m,mn): \phi(m)|d\text{ and }B(m,mn)\le d\}\subseteq T_1$ (now $\Phi^\infty(d)\subseteq T_2$).
5. For $(m,mn)\in T_2-T$ prove that ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ has no functions of degree $d/\phi(m)$.
In step (2) the restriction on $m$ follows from the Weil pairing, and the restriction on $n$ is from ; the tighter restriction on $n$ in step (4) is . If the verification in step (3) succeeds, then and together imply that each pair $(m,mn)\in T_2$ lies in $\Phi^\infty(d)$ if and only if ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ contains a function of degree $d/\phi(m)$.
\[remark:higher\_d\] We have completed steps (1)-(4) for $d=5,6,7,8$. This strategy cannot be applied with $d = 9$ because (3) fails; as proved in [@DvH13], we have $\gamma(X_1(37))=18$ and ${\operatorname{rk}}(J_1(37)({{\mathbb{Q}}}))\ne 0$.
We now prove , beginning with the case $d=5$, following the strategy above.
\[prop:d5\] $\Phi^\infty(5) = \{(1,n):1\le n\le 25, n\ne 23\}\cup \{(2,2n):1\le n\le 8\}$.
The existence of the Weil pairing implies that $(m,mn)\in\Phi^\infty(5)$ only if $\phi(m)$ divides $5$; we thus have $m\le 2$. The elements $(1,n)\in \Phi^\infty(5)$ are determined in [@DvH13 Thm.3], so we only need to consider $m=2$.
For $1\le n\le 6$ the genus of $X_1(2,2n)$ is either 0 or 1 and $X_1(2,2n)({{\mathbb{Q}}})\ne\emptyset$; it follows that ${{\mathbb{Q}}}(X_1(2,2n))$ contains functions of every degree $d\ge 2$, including $d=5$.
For $n=7$ we used the method of §\[sec:specialmodels\] and a modified version of the algorithm in [@Sutherland12] to construct the model $$X_1(2,14): v^3 - (u^3 + u^2 + u - 1)v^2 - (u^5 + 3u^4 + 3u^3 + u^2 + u)v + u^5 + u^4 = 0,$$ with maps $$q=\frac{v + 1}{v-2u + 1},\qquad t=\frac{(v+1)(2u-v+1)(2u(u+1)+v+1)}{v^3 + (2u^2 + 1)v^2 - (2u^3 - 2u^2 - 2u - 1)v + u^4 + (u+1)^4}$$ that give points $(E_2(q,t),P_2(q,2),Q_2(q,t))$ on $X_1(2,14)$ as defined in §\[sec:specialmodels\]. We may then take $u$ as a function of degree 5 in ${{\mathbb{Q}}}(X_1(2,14))$.
For $n=8$ we similarly constructed $$X_1(2,16): v^4 + (u^3 - 2u)v^3 - (2u^4 + 2)v^2 + (u^5 + u^3 + 2u)v + 1 = 0,$$ which also has $u$ as a function of degree 5 in ${{\mathbb{Q}}}(X_1(2,16))$ (we omit the maps $q(u,v)$ and $t(u,v)$ for reasons of space). This completes step (1) of our proof strategy.
Proceeding to steps (2)–(4), from we find that $\gamma(X_1(2,2n))>10$ for $n>18$, and $\gamma(X_1(2,2n))>5$ for $n>13$. By we have ${\operatorname{rk}}(J_1(2,2n)({{\mathbb{Q}}}))=0$ for $n\le 18$, thus by it suffices to prove that ${{\mathbb{Q}}}(X_1(2,2n))$ contains no functions of degree 5 for $9\le n\le 13$.
For step (5) we begin by proving that $\gamma(X_1(2,2n)_{{{\mathbb{F}}}_3})>5$ for $n=10,11,13$, and that $\gamma(X_1(2,24)_{{{\mathbb{F}}}_5})>5$ for $n=12$, using methods similar to those in [@DvH13]. This involves exhaustively searching the Reimann–Roch spaces of a suitable set divisors for functions of degree $d\le 5$; the Magma code we used to perform these computations can be found in [@mdmagma].
For $n=9$ we actually have $\gamma(X_1(2,18))=4$, so in this case we need to show that ${{\mathbb{Q}}}(X_1(2,18))$ contains no functions of degree exactly equal to 5; this is addressed by below.
\[prop:d5m2n9\] ${{\mathbb{Q}}}(X_1(2,18))$ does not contain a function of degree $5$
We proceed by verifying conditions 1–5 of [@DvH13 Prop,7] for $p=d=5$. For $k={{\mathbb{Q}}},{{\mathbb{F}}}_5$, let $W_d^r(k)$ denote the closed subscheme of $\operatorname{Pic}^d(X_1(2,18)_k(k))$ corresponding to line bundles of degree $d$ whose global sections form a $k$-vector space of dimension strictly greater than $r$.
- The map $J_1(2,18)({{\mathbb{Q}}})\to J_1(2,18)_{{{\mathbb{F}}}_5}({{\mathbb{F}}}_5)$ is injective because $J_1(2,18)({{\mathbb{Q}}})$ is finite.
- Using Magma, a brute force search of the Riemann–Roch space of all effective divisors of degree $5$ on $X_1(2,18)_{{{\mathbb{F}}}_5}$ finds that ${{\mathbb{F}}}_5(X_1(2,18)_{{{\mathbb{F}}}_5})$ contains no functions of degree 5.
- A similar brute force computation shows that $W_5^2({{\mathbb{F}}}_5)=\emptyset$.
- A brute force computations finds that $5-\gamma(X_1(2,18)_{{{\mathbb{F}}}_5})=5-4=1$ and $\#W_4^1({{\mathbb{F}}}_5)=3$. The surjectivity of the map $W_4^1({{\mathbb{Q}}}) \to W_4^1({{\mathbb{F}}}_5)$ is verified in [@mdmagma] by finding three modular units of degree $4$ on $X_1(2,18)$ with linearly independent pole divisors.
- The reduction map $X_1(2,18)({{\mathbb{Q}}})\to X_1(2,18)_{{{\mathbb{F}}}_5}({{\mathbb{F}}}_5)$ is surjective because the 9 elements of $X_1(2,18)_{{{\mathbb{F}}}_5}({{\mathbb{F}}}_5)$ are precisely the reductions of the 9 cusps in $X_1(2,18)({{\mathbb{Q}}})$.
It follows from [@DvH13 Prop.7] that ${{\mathbb{Q}}}(X_1(2,18))$ contains no functions of degree 5.
We now address $d=6$ using the same proof strategy.
\[prop:d6\] We have $$\begin{aligned}
\Phi^\infty(6)= &\{(1,n):1\le n\le 30, n\ne 23,25,29\}\ \cup\ \{(2,2n):1\le n\le 10\}\\
&\cup\ \{(3,3n):1\le n\le 4\}\ \cup\ \{(4,4),(4,8),(6,6)\}.\end{aligned}$$
We have $\phi(m)$ dividing $6$ for $m=1,2,3,4,6,7,9,14,18$. The elements $(1,n)\in \Phi^\infty(6)$ are determined in [@DvH13 Thm.3], and we can immediately rule out $m=7,9,14,18$, since for these $m$ we have $\phi(m)=6$ and $d/\phi(m)=1$, but $g(X_1(m,mn))>1$ for all $m>6$ and $n\ge 1$. Faltings’ theorem implies that $X_1(m,mn)({{\mathbb{Q}}}(\zeta_m))$ is finite for $m>6$. This leaves only $m=2,3,4,6$.
We have $\Phi^\infty(2),\Phi^\infty(3)\subseteq \Phi^\infty(6)$ (given a degree 2 or 3 point $(E,P,Q)$ on $X_1(m,mn)$ we can always base change $E$ to a number field of degree 6). It thus suffices to show that ${{\mathbb{Q}}}(\zeta_m)(X_1(m,mn))$ contains a function of degree 6 for the pairs $(2,16),(2,18),(2,20)$, and a function of degree 3 for the pairs $(3,9),(3,12),(4,8),(6,6)$.
The map $\pi\colon X_1(2,2n)\to X_1(2n)$ given by $(E,P,Q)\mapsto (E,Q)$ has degree 2. For $n=8,9,10$ we have a function $f$ of degree 3 in ${{\mathbb{Q}}}(X_1(2n))$, because $(1,2n)\in \Phi^\infty(3)$ and ${\operatorname{rk}}(J_1(2n)({{\mathbb{Q}}}))=0$ (by ), and $\pi\circ f$ is then a function of degree 6 in ${{\mathbb{Q}}}(X_1(2,2n))$.
The curves $X_1(3,9)$, $X_1(4,8)$, $X_1(6,6)$ all have genus 1 and a ${{\mathbb{Q}}}(\zeta_m)$-rational point (take a rational cusp), hence they are isomorphic to elliptic curves and admit functions of every degree $d\ge 2$. The map $\colon X_1(3,12)\to X_1(12)_{{{\mathbb{Q}}}(\zeta_3)}$ given by $(E,P,Q)\mapsto (E,Q)$ has degree 3 and $X_1(12)_{{{\mathbb{Q}}}(\zeta_3)}$ has genus 0 and a ${{\mathbb{Q}}}(\zeta_3)$-rational point, hence it is isomorphic to ${{\mathbb{P}}}^1_{{{\mathbb{Q}}}(\zeta_m)}$; it follows that ${{\mathbb{Q}}}(\zeta_3)(X_1(3,12))$ contains a function of degree 3.
This completes step (1) of our proof strategy. We now proceed to steps (2)–(5) for each $m=2,3,4,6$ in turn.
We begin with $m=2$. From we find that $\gamma(X_1(2,2n))>12$ for $n>21$, and $\gamma(X_1(2,2n))>6$ for $n>15$. By we have ${\operatorname{rk}}(J_1(2,2n)({{\mathbb{Q}}}))=0$ for $n\le 21$, thus by it suffices to prove that ${{\mathbb{Q}}}(X_1(2,2n))$ contains no functions of degree 6 for $11\le n\le 15$. For $11\le n\le 14$ we proceed as in to prove $\gamma(X_1(2,2n)_{{{\mathbb{F}}}_p})>6$ using $p=3$ for $n=11,13,14$ and $p=5$ for $n=12$, which was done with a computation in Magma. The code written for these and the subsequent computations can be found in [@mdmagma]. For $n=15$ we instead apply and .
We next consider $m=3$. We have $\gamma(X_1(3,3n))>6$ for $n>8$ and $\gamma(X_1(3,3n))>3$ for $n>5$, and ${\operatorname{rk}}(J_1(3,3n)({{\mathbb{Q}}}(\zeta_3))=0$ for $n\le 8$. It suffices to show that ${{\mathbb{Q}}}(\zeta_3)(X_1(3,15))$ contains no functions of degree 3, which again follows from a Magma computation.
The cases $m=4,6$ are similar. For $m=4$ we have $\gamma(X_1(4,4n))>6$ for $n>6$ and $\gamma(X_1(4,4n))>3$ for $n>3$, and ${\operatorname{rk}}(J_1(4,4n)({{\mathbb{Q}}}(\zeta_4))=0$ for $n\le 6$; a computation shows that $\gamma(X_1(4,12)_{{{\mathbb{F}}}_5})>3$. For $m=6$ we have $\gamma(X_1(6,6n))>6$ for $n>2$ and ${\operatorname{rk}}(J_1(6,6n)({{\mathbb{Q}}}(\zeta_6))=0$ for $n\le 2$; a Magma computation shows that $\gamma(X_1(6,12)_{{{\mathbb{F}}}_5})>3$.
$X_1(2,30)$ has no non-cuspidal points of degree less than $6$. \[lem:d6m2n15\]
$X_1(2,30)$ has a degree two map to $X_1(30)$ given by $(E,P,Q)\mapsto (E,Q)$, so we start by considering the points of degree less than 6 on $X_1(30)$.
Theorem 3 of [@DvH13] states that $X_1(30)$ has only finitely many points of degree less than $6$, and these points are explicitly determined in [@DvHnp]. The non-cuspidal points on $X_1(30)$ of degree less than 6 all have degree 5 and arise from two Galois conjugacy classes of elliptic curves that we now describe.
Let $x_1, x_2 \in \overline {{\mathbb{Q}}}$ be zeros of $x^5+x^4-3x^3+3x+1$ and $x^5+x^4-7x^3+x^2+12x+3$, respectively, and define $$y_1 := 2x_1^4 + x_1^3 - 6x_1^2 + 4x_1 +4, \quad y_2 := \frac {3x_2^4+7x_2^3+6x_2^2+11x_2-73} {53}.$$ Let $E_{x,y}$ denote the curve $E(b,c)$ in Tate normal form with $b=rs(r-1)$ and $c=s(r-1)$, where $r=(x^2y-xy+y-1)/(x^2y-x)$ and $s:=(xy-y+1)/(xy)$.
The point $P_0 := (0,0)$ is a point of order 30 on both $E^{}_{x_1,y_1}$ and $E^{}_{x_2,y_2}$. Moreover, every non-cuspidal point $(E,P) \in X_1(\overline {{\mathbb{Q}}})$ of degree less than 6 over ${{\mathbb{Q}}}$ can be obtained as either $(E_{\sigma(x_1),\sigma(y_1)},dP_0)$ or $(E_{\sigma(x_2),\sigma(y_2)},dP_0)$ for some $\sigma \in \operatorname{Gal}({\overline{{{\mathbb{Q}}}}}/{{\mathbb{Q}}})$ and $d \in ({{\mathbb{Z}}}/30{{\mathbb{Z}}})^\times$; thus every such point has degree 5.
The 2-division polynomials of $E^{}_{x_1,y_1}$ and $E^{}_{x_2,y_2}$ each have just one root in ${{\mathbb{Q}}}(x_1)$ and ${{\mathbb{Q}}}(x_2)$, respectively, corresponding to $x(15P_0)$; it follows that $E^{}_{x_1,y_1}({{\mathbb{Q}}}(x_1))$ and $E^{}_{x_2,y_2}({{\mathbb{Q}}}(x_2))$ contain no other points of order 2. Thus every non-cuspidal point in $X_1(2,30)(\overline {{\mathbb{Q}}})$ that maps to a point of degree less than 6 in $X_1(30)(\overline {{\mathbb{Q}}})$ has degree at least $2\cdot 5 = 10$. The lemma follows.
\[prop:d6m2n15\] $X_1(2,30)$ has only finitely many points of degree $6$.
The proof below is based on ideas presented in [@DKM16 §4].
We have $d(X_1(2,30))=\gamma(X_1(2,30))$ by , since ${\operatorname{rk}}(J_1(2,30)({{\mathbb{Q}}}))=0$, by , and $d(X_1(2,30))=\gamma(X_1(2,30))\ge 6$, by . It thus suffices to prove there are no functions of degree $6$ in ${{\mathbb{Q}}}(X_1(2,30))$.
We first show that if ${{\mathbb{Q}}}(X_1(2,30))$ contains a function of degree 6 then it contains a function whose pole and zero divisors consist entirely of cusps. Let $f \in {{\mathbb{Q}}}(X_1(2,30))$ be a function of degree $6$ and let $c \in X_1(2,30)({{\mathbb{Q}}})$ be any of its 12 rational cusps. We may assume $f$ has a pole at $c$ by replacing $f$ with $1/(f -f(c))$ if necessary. The pole divisor of $f$ can then be written as $c + D$, where $D$ is a divisor of degree $5$. Now let $c'\ne c$ be a rational cusp not in the support of $D$, and define $f' := f -f(c')$. The function $f'$ has the same poles as $f$, and its zero divisor can be written as $c'+D'$ with $D'$ a divisor of degree $5$. The divisors $D$ and $D'$ must consist entirely of cusps, by , and the claim follows.
An enumeration of all $f \in {{\mathbb{Q}}}(X_1(2,30))$ of degree at most 6 with $\operatorname{div}(f)$ supported on cusps computed as in [@DvH13 Footnote 7] finds none with degree exactly 6; the proposition follows.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the referee for several helpful remarks on an earlier draft of this paper.
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---
abstract: 'Our purpose is to investigate the local boundedness, the upper semicontinuity, and the stability of the solution map of tensor complementarity problems. To do this, we focus on the set of R$_0$–tensors and show that this set plays an important role in the investigation. Furthermore, by using a technique in semi-algebraic geometry, we obtain some results on the finite-valuedness and the lower semicontinuity of the solution map.'
author:
- Vu Trung Hieu
date: 'Received: date / Accepted: date'
title: 'On the R$_0$–tensors and the solution map of tensor complementarity problems'
---
Introduction
============
The tensor complementarity problems were firstly introduced by Song and Qi in [@SongQi2014; @SongQi2015]. These problems have attracted a lot of attention of researchers (see [@QCC2018] and the references given there). It is well-known that Huang and Qi have presented an explicit relationship between $n$-person noncooperative games and tensor complementarity problems [@HuangQi17]. Structured tensors and different properties of their solution sets have been intensively investigated [@LQX2017; @SongQi2016; @SongYu2016; @ZW2018]. The global uniqueness and solvability for tensor complementarity problems have been discussed in [@BHW2016] and [@LLW2017]. Methods and algorithms to solve a tensor complementarity problem have been interested by several authors [@DZ2018; @LLW2017; @XLX2017].
The involved function in a tensor complementarity problem is the sum of a homogeneous polynomial function and an arbitrary given vector. Thus, the tensor complementarity problem is a special case of the homogeneous complementarity problem which was mentioned in a work of Oettli and Yen [@OettliYen95]. Besides, the tensor complementarity problems also is a subclass of the polynomial complementarity problems which have been recently introduced by Gowda [@Gowda16]. The tensor complementarity problem is a natural extension of the linear complementarity problem [@CPS1992], hence, several properties of both problems are similar. The boundedness, the continuity and the stability of the solution map of linear complementarity problems have been deeply investigated (see, e.g., [@CPS1992; @Gowda1992; @OettliYen95; @Phung2002; @Robinson79]). We will investigate these properties of the solution map of tensor complementarity problems and simultaneously show that the set of R$_0$–tensors plays an important role in the investigation.
In the paper, firstly, we prove that the set of R$_0$–tensors is open. Accordingly, the local boundedness of the solution map is shown. Secondly, since the involved function in a tensor complementarity problem is polynomial, by using tools in semi-algebraic geometry, we obtain the generic finite-valuedness of the solution map of tensor complementarity problems. Consequently, a necessary condition for the lower semicontinuity of the solution map is obtained. Furthermore, we show that the set of R$_0$–tensors is generic semi-algebraic in the set of real tensors. A lower bound for the dimension of the complement of R$_0$–tensors is obtained. Finally, the paper shows a closed relation between the upper semicontinuity of the solution map and the R$_0$ property of the involved tensors. A result on the stability of the solution map is introduced.
The organization of the paper is as follows. Section 2 gives a brief introduction to tensor complementarity problems and semi-algebraic geometry. Main results are presented in the next four sections. Section 3 investigates the openness of the set of R$_0$–tensors and the local boundedness of the solution map. Besides, this section proves the semi-algebraicity and the genericity of the set R$_0$–tensors. The finite-valuedness and the lower semicontinuity are discussed in Section 4. The last two sections give results on the upper semicontinuity and the stability of the solution map.
Preliminaries
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In this section, we will recall some definitions, notations, and auxiliary results on tensor complementarity problems and semi-algebraic geometry.
Tensor complementarity problems
-------------------------------
The scalar product of two vectors $x, y$ in the Euclidean space $\operatorname{\mathbb{R}}^n$ is denoted by $\langle x,y\rangle$. Let $F: \operatorname{\mathbb{R}}^n\to\operatorname{\mathbb{R}}^n$ be a vector-valued function. The *nonlinear complementarity problems* defined by $F$ is the problem $$\operatorname{CP}(F) \qquad {\text{Find}}\ \, x\in \operatorname{\mathbb{R}}^n \ \; \text{such that}\ \, \ x\geq 0, \; F(x) \geq 0, \; \langle x,F(x)\rangle=0.$$ The solution set is denoted by $\operatorname{Sol}(F)$.
The following remark shows that a solution of a complementarity problem can be characterized by using some Lagrange multipliers.
\[KKT\] A vector $x$ solves $\operatorname{CP}(F)$ if and only if there exists a vector $\lambda\in \operatorname{\mathbb{R}}^n$ such that the following system is satisfied $$\label{KKT_formula}
\left\lbrace \begin{array}{r}
F(x)-\lambda=0,\\
\langle \lambda,x \rangle=0,\\ \lambda\geq 0,\; x \geq 0.
\end{array}\right.$$
To find the solution set of a complementarity problem, we will find the solutions on each pseudo-face of $\operatorname{\mathbb{R}}^n_+$. For every index set $\alpha\subset [n]=\{1,\dots,n\}$, we associate this set with the following *pseudo-face* $$K_{\alpha}=\left\lbrace x\in \operatorname{\mathbb{R}}_+^n:x_i=0, \forall i\in\alpha; \; x_i >0, \forall i\in [n]\setminus\alpha\right\rbrace.$$ The subsets $K_{\alpha},\alpha\subset [n],$ is a finite disjoint decomposition of $\operatorname{\mathbb{R}}^n_+$. Therefore, we have $$\label{quasiface}
\operatorname{Sol}(F)=\bigcup_{\alpha\subset [n]}\left[ \operatorname{Sol}(F)\cap K_{\alpha}\right].$$
Throughout this paper, we assume that $m$ and $n$ are given integers, and $m,n\geq2$. An $m$-th order $n$-dimensional *tensor* $\operatorname{\mathcal{A}}= (a_{i_1\cdots i_m})$ is a multi-array of real entries $a_{i_1\cdots i_m}\in\operatorname{\mathbb{R}}$, where $i_j\in[n]$ and $j\in[m]$. The set of all real $m$-th order $n$-dimensional tensors is denoted by $\operatorname{\mathbb{R}}^{[m,n]}$. For any tensor $\operatorname{\mathcal{A}}= (a_{i_1\cdots i_m})$, the Frobenius norm of $\operatorname{\mathcal{A}}$ is defined and denoted as $$\|\operatorname{\mathcal{A}}\|:=\sqrt{\sum_{i_1,i_2,...,i_m=1}^{n}a^2_{i_1i_2\cdots i_m}}.$$ This norm can also be considered as a vector norm. So, the norm of $(\operatorname{\mathcal{A}},a)$ in $\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^{n}$ can be defined as follows $$\|(\operatorname{\mathcal{A}},a)\|:=\sqrt{\|\operatorname{\mathcal{A}}\|^2+\sum_{i=1}^{n}a^2_i}.$$ Clearly, $\operatorname{\mathbb{R}}^{[m,n]}$ is a real vector space of dimension $n^m$. Here, each tensor $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[m,n]}$ is a real vector having ${n^m}$ components. If $m=2$ then $\operatorname{\mathbb{R}}^{[2,n]}$ is the space of $n\times n$–matrices which is isomorphic to $\operatorname{\mathbb{R}}^{n\times n}$. Note that if $\operatorname{\mathcal{A}}= (a_{i_1\cdots i_m})$ and $\operatorname{\mathcal{B}}= (b_{i_1\cdots i_m})$ are tensors in $\operatorname{\mathbb{R}}^{[m,n]}$ then the sum of them is defined by $$\operatorname{\mathcal{A}}+\operatorname{\mathcal{B}}=(a_{i_1\cdots i_m}+b_{i_1\cdots i_m}).$$
For any $x = (x_1,..., x_n)^T\in \operatorname{\mathbb{R}}^n$, $\operatorname{\mathcal{A}}x^{m-1}$ is a vector whose $i$-th component defined by $$\label{Axm_1}(\operatorname{\mathcal{A}}x^{m-1})_i:=\sum_{i_2,...,i_m=1}^{n}a_{ii_2\cdots i_m}x_{i_2}\cdots x_{i_m}, \ \forall i\in[n],$$ and $\operatorname{\mathcal{A}}x^m$ is a polynomial of degree $m$, defined by $$\operatorname{\mathcal{A}}x^{m}:=\langle x,Ax^{m-1} \rangle=\sum_{i_1,i_2,...,i_m=1}^{n}a_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}.$$ The polynomials $(Ax^{m-1})_i$ and $\operatorname{\mathcal{A}}x^{m}$ are homogeneous of degree respectively $m-1$ and $m$, that is $\operatorname{\mathcal{A}}(tx)^{m-1}=t^{m-1}( \operatorname{\mathcal{A}}x^{m-1}) $ and $\operatorname{\mathcal{A}}(tx)^{m}=t^m(\operatorname{\mathcal{A}}x^m)$ for all $t\geq 0$ and $x\in\operatorname{\mathbb{R}}^n$.
\[norm\_beta\] By the continuity of the polynomial function $\operatorname{\mathcal{A}}x^{m-1}$, if $U$ is an bounded set in $\operatorname{\mathbb{R}}^n$ then there exists $\beta>0$ such that $\|\operatorname{\mathcal{A}}x^{m-1}\|\leq \beta\|\operatorname{\mathcal{A}}\|$ for all $x\in U$.
Let $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[m,n]}$ and $a \in\operatorname{\mathbb{R}}^n$ be given. If $F(x)=\operatorname{\mathcal{A}}x^{m-1}+ a$ then one says that $\operatorname{CP}(F)$ is a *tensor complementarity problem* which defined by $\operatorname{\mathcal{A}}$ and $a$. This problem and its solution set are denoted respectively by $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ and $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$. By definition, $x$ solves $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ if and only if $$\label{TCP}
x\geq 0, \ \operatorname{\mathcal{A}}x^{m-1}+a\geq 0, \ \operatorname{\mathcal{A}}x^{m}+\left\langle x,a \right\rangle =0.$$ Clearly, the vector $0$ solves $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ for all $a\in\operatorname{\mathbb{R}}^n_+$. The solution map of tensor complementarity problems is denoted and defined by $$\label{Sol}
\operatorname{Sol}:\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^{n}\rightrightarrows \operatorname{\mathbb{R}}^n, \ (\operatorname{\mathcal{A}},a)\mapsto\operatorname{Sol}(\operatorname{\mathcal{A}},a).$$
A subset $K\subset \operatorname{\mathbb{R}}^n$ is called a cone [@Bernstein p. 89] if $\lambda >0$ and $x\in K$ then $\lambda x\in K$. The cone $K$ is bounded if and only if $K=\{0\}$.
The solution set of $\operatorname{TCP}(\operatorname{\mathcal{A}},0)$ is a closed cone. Indeed, suppose that $x$ is a solution of $\operatorname{TCP}(\operatorname{\mathcal{A}},0)$. For each $t\geq0$, from , one has $$tx\geq 0, \ \; \operatorname{\mathcal{A}}(tx)^{m-1}=t^{m-1}(\operatorname{\mathcal{A}}x^{m-1}) \geq 0, \ \; \operatorname{\mathcal{A}}(tx)^{m}=t^{m}(\operatorname{\mathcal{A}}x^m)=0.$$ By definition, $tx$ solves $\operatorname{TCP}(\operatorname{\mathcal{A}},0)$. This shows that $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is a cone. The closedness of $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is implied from the continuity of $F(x)=\operatorname{\mathcal{A}}x^{m-1}$.
Let us recall that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor (sometimes, we say that $\operatorname{\mathcal{A}}$ is R$_0$ or $\operatorname{\mathcal{A}}$ has R$_0$ property) if $\operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$. We denote $\operatorname{\mathcal{R}_0}$ to be the set all $m$-th order $n$-dimensional R$_0$–tensors and $\mathcal{O}\in \operatorname{\mathbb{R}}^{[m,n]}$ to be the zero tensor. The complement of $\operatorname{\mathcal{R}_0}$ is denoted and defined by $$C(\operatorname{\mathcal{R}_0})=\operatorname{\mathbb{R}}^{[m,n]}\setminus\operatorname{\mathcal{R}_0}.$$ Clearly, $\mathcal{O}$ belongs to $C(\operatorname{\mathcal{R}_0})$ since $\operatorname{Sol}(\mathcal{O},0)=\operatorname{\mathbb{R}}^n_+$.
Semi-algebraic geometry
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Recall a subset in $\operatorname{\mathbb{R}}^n$ is *semi-algebraic* if it is the union of finitely many subsets of the form $$\label{basicsemi}
\big\{x\in \operatorname{\mathbb{R}}^n\,:\,f_1(x)=...=f_\ell(x)=0,\ g_{\ell+1}(x)<0,\dots,g_m(x)<0\big\},$$ where $\ell,m$ are natural numbers, and $f_1,\dots, f_\ell, g_{\ell+1},\dots,g_m$ are polynomials with real coefficients. The semi-algebraic property is preserved by taking finitely union, intersection, minus and taking closure of semi-algebraic sets. The well-known Tarski-Seidenberg Theorem states that the image of a semi-algebraic set under a linear projection is a semi-algebraic set.
There are some ways to define the dimension of a semi-algebraic set. Here, we choose the geometric approach which is represented in [@BCF98 Corollary 2.8.9]. If $S\subset \operatorname{\mathbb{R}}^n$ is a semi-algebraic set, then there exists a decomposition of $S$ into a disjoint union of semi-algebraic subsets [@BCF98 Theorem 2.3.6] $$S=\bigcup_{i=1}^sS_i,$$ where each $S_i$ is semi-algebraically diffeomorphic to $(0,1)^{d_i}$. Here, let $(0,1)^{0}$ be a point, $(0,1)^{d_i}\subset \operatorname{\mathbb{R}}^{d_i}$ is the set of points $x=(x_1,\dots,x_{d_i})$ such that $x_j\in (0,1)$ for all $j=1,\dots,d_i$. The *dimension* of $S$ is, by definition, $$\dim(S):=\max\{d_1,...,d_s\}.$$ The dimension is well-defined and not depends on the decomposition of $S$. We would like remind that if the dimension of a nonempty semi-algebraic set $S$ is zero then $S$ has finitely many points.
\[dense\] For a subset $S$ of $\operatorname{\mathbb{R}}^n$, the following notions are different: $S$ is full measure, meaning its complement $\operatorname{\mathbb{R}}^n\setminus S$ has measure zero, and $S$ is topologically generic, meaning it contains a countable intersection of dense and open sets. However, for semi-algebraic sets, the following properties are equivalent (see, e.g., [@DHP16 Lemma 2.3]): $S$ is dense in $\operatorname{\mathbb{R}}^n$; $\operatorname{\mathbb{R}}^n\setminus S$ has measure zero; and $\dim(\operatorname{\mathbb{R}}^n\setminus S)<n$. If the semi-algebraic set $S$ satisfies one of three previous properties, then one says this set is generic.
We will use the Tarski-Seidenberg Theorem in the third form in the next section. To present the theorem, we have to describe semi-algebraic sets via the language of first-order formulas. A *first-order formula* (with parameters in $\operatorname{\mathbb{R}}$) is obtained by the induction rules [@Coste02]:
1. If $p \in\operatorname{\mathbb{R}}[X_1, . . . , X_n]$, then $p > 0$ and $p= 0$ are first-order formulas;
2. If $P,Q$ are first-order formulas, then “$P$ *and* $Q$”, “$P$ *or* $Q$”, “*not* $Q$”, which are denoted respectively by $P \wedge Q$, $P \vee Q$, and $\neg Q$, are first-order formulas;
3. If $Q$ is a first-order formula, then $\exists X\, Q$ and $\forall X\, Q$, where $X$ is a variable ranging over $\operatorname{\mathbb{R}}$, are first-order formulas.
Formulas obtained by using only the rules (i) and (ii) are called *quantifier-free formulas*. A subset $S \subset \operatorname{\mathbb{R}}^n$ is semi-algebraic if and only if there is a quantifier-free formula $Q_S(X_1,...,X_n)$ such that $$(x_1,...,x_n) \in S\ \; \text{if and only if }\; Q_S(x_1,..., x_n).$$ In this case, $Q_S(X_1,...,X_n)$ is said to be a *quantifier-free formula defining $S$.*
The Tarski-Seidenberg Theorem in the third form [@Coste02 Theorem 2.6], says that if $Q(X_1,...,X_n)$ is a first-order formula, then the set $$S=\big\lbrace (x_1,...,x_n)\in\operatorname{\mathbb{R}}^n: \ Q(x_1,...,x_n) \big\rbrace$$ is a semi-algebraic set.
The set of R$_0$–tensors
========================
In this section, we prove that the set $\operatorname{\mathcal{R}_0}$ of R$_0$–tensors is open in $\operatorname{\mathbb{R}}^{[m,n]}$. Accordingly, the local boundedness of the solution map is shown. Moreover, we show that $\operatorname{\mathcal{R}_0}$ is generic semi-algebraic. The dimension of the complement $C(\operatorname{\mathcal{R}_0})$ is discussed in the last subsection.
Local boundedness of the solution map
-------------------------------------
\[open\_cone\] The set $\operatorname{\mathcal{R}_0}$ of all R$_0$–tensors is open in $\operatorname{\mathbb{R}}^{[m,n]}$.
If the complement $C(\operatorname{\mathcal{R}_0})$ is closed, then the set $\operatorname{\mathcal{R}_0}$ is open. So, we need only prove the closedness of $C(\operatorname{\mathcal{R}_0})$. Let $\{\operatorname{\mathcal{A}}^k\}\subset C(\operatorname{\mathcal{R}_0})$ be a convergent sequence with $\operatorname{\mathcal{A}}^k\to A$. On the contrary, we suppose that $\operatorname{\mathcal{A}}\in\operatorname{\mathcal{R}_0}$. For each $k$, $\operatorname{Sol}(\operatorname{\mathcal{A}}^k,0)$ is unbounded. There exists an unbounded sequence $\{x^k\}$ such that $x^k\in\operatorname{Sol}(\operatorname{\mathcal{A}}^k,0)$ and $x^k\neq 0$ for each $k$. Without loss of generality we can assume that $\|x^k\|^{-1}x^k\to\bar x$ and $\|\bar x\|=1.$ By definition, one has $$\operatorname{\mathcal{A}}^k(x^k)^{m-1}\geq 0, \ \operatorname{\mathcal{A}}^k(x^k)^m=0.$$ Dividing these ones by $\|x^k\|^{m-1}$ and $\|x^k\|^{m}$, respectively, and taking $k\to+\infty$, we obtain $\operatorname{\mathcal{A}}(\bar x)^{m-1}\geq0$ and $\operatorname{\mathcal{A}}(\bar x)^m=0.$ It follows that $\bar x \in \operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$. This contradicts to $\|\bar x\|=1$. Therefore, $\operatorname{\mathcal{A}}$ must be in $\operatorname{\mathcal{C}}(\operatorname{\mathcal{R}_0})$. Hence, $C(\operatorname{\mathcal{R}_0})$ is closed. The proof is complete.
The set $\operatorname{\mathcal{R}_0}$ is a cone in $\operatorname{\mathbb{R}}^{[m,n]}$. Indeed, for any $t>0$, one has $$(t\operatorname{\mathcal{A}}) x^{m-1}=t^{m-1} (\operatorname{\mathcal{A}}x^{m-1}), \ (t\operatorname{\mathcal{A}})x^{m}=t^{m} (\operatorname{\mathcal{A}}x^{m}).$$ This leads to $ \operatorname{Sol}(t\operatorname{\mathcal{A}},0)=\operatorname{Sol}(\operatorname{\mathcal{A}},0)$. Hence, $\operatorname{\mathcal{A}}\in \operatorname{\mathcal{R}_0}$ if and only if $t\operatorname{\mathcal{A}}\in \operatorname{\mathcal{R}_0}$. This implies that $\operatorname{\mathcal{R}_0}$ is a cone.
The boundedness of solution sets of tensor complementarity problems and polynomial complementarity problems under the R$_0$ condition is mentioned in [@SongYu2016] and [@Gowda16]. Based on the openness of the set $\operatorname{\mathcal{R}_0}$, we show that the the solution map is locally bounded.
Here, $\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)$ stands for the closed ball in $\operatorname{\mathbb{R}}^{[m,n]}$ centered at $\operatorname{\mathcal{O}}$ with radius $\varepsilon$. Similarly, $\operatorname{\mathbb{B}}(0,\delta)$ is the closed ball in $\operatorname{\mathbb{R}}^n$ centered at $0$ with radius $\delta$.
\[bounded1\] The following two statements are equivalent:
The tensor $\operatorname{\mathcal{A}}$ is R$_0$;
There exists $\varepsilon>0$ such that the following set $$S(\varepsilon,\delta):=\bigcup_{(\operatorname{\mathcal{B}},b)\in \operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)\times \operatorname{\mathbb{B}}(0,\delta)} \operatorname{Sol}(\operatorname{\mathcal{A}}+\operatorname{\mathcal{B}}, a+b),$$ is bounded, for any $\delta>0$ and $a\in\operatorname{\mathbb{R}}^n$.
$\rm(a) \Rightarrow (b)$ Let $\operatorname{\mathcal{A}}$ be an R$_0$–tensor. Since the set $\operatorname{\mathcal{R}_0}$ is open in $\operatorname{\mathbb{R}}^{[m,n]}$, due to Proposition \[open\_cone\], there exists $\varepsilon>0$ such that $\operatorname{\mathcal{A}}+\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)\subset\operatorname{\mathcal{R}_0}$. We suppose that there exists $a\in\operatorname{\mathbb{R}}^n$ and $\delta>0$ such that the set $S(\varepsilon,\delta)$ is unbounded. Let $\{x^k\}$ be an unbounded sequence and $\{(\operatorname{\mathcal{B}}^k,b^k)\}$ be a sequence in $\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)\times \operatorname{\mathbb{B}}(0,\delta)$ satisfying $$\label{xk_sol}
x^k\neq 0, \ \|x^k\|^{-1}x^k\to\bar x,\ x^k\in\operatorname{Sol}( \operatorname{\mathcal{A}}+\operatorname{\mathcal{B}}^k,a+b^k).$$ By the compactness of the sets $\operatorname{\mathcal{A}}+\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)$ and $a+\operatorname{\mathbb{B}}(0,\delta)$, we can assume that $$\operatorname{\mathcal{A}}+\operatorname{\mathcal{B}}^k\to\bar\operatorname{\mathcal{A}}\in \operatorname{\mathcal{A}}+\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon), \ a+b^k\to \bar a\in a+\operatorname{\mathbb{B}}(0,\delta).$$ From , it is easy to check that $\bar x$ solves $\operatorname{TCP}(\bar\operatorname{\mathcal{A}},0)$. Since $\|\bar x\|=1$, $\bar\operatorname{\mathcal{A}}$ is not R$_0$. This contradicts to the fact that $\bar\operatorname{\mathcal{A}}$ belongs to $\operatorname{\mathcal{A}}+\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)\subset\operatorname{\mathcal{R}_0}$.
$\rm(b) \Rightarrow (a)$ Suppose that there exists $\varepsilon>0$ such that $S(\varepsilon,\delta)$ is bounded for any $a\in\operatorname{\mathbb{R}}^n$ and $\delta>0$. Clearly, one has $\operatorname{Sol}(\operatorname{\mathcal{A}},0)\subset S(\varepsilon,\delta)$. This implies that $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is bounded, namely, $\operatorname{\mathcal{A}}$ is an R$_0$–tensor. The proof is complete.
The tensor $\operatorname{\mathcal{A}}$ is R$_0$ if and only if $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is bounded for every $a\in\operatorname{\mathbb{R}}^n$ (see [@SongYu2016 Theorem 3.2]). Moreover, $\operatorname{\mathcal{A}}$ is an R$_0$–tensor if and only if the set $$\bigcup_{b\in \operatorname{\mathbb{B}}(0,\delta)} \operatorname{Sol}(\operatorname{\mathcal{A}},a+b)$$ is bounded, for every $a\in\operatorname{\mathbb{R}}^n$ and $\delta>0$ [@Gowda16 Proposition 2.1].
Recall that the set-valued map $\Psi:\operatorname{\mathbb{R}}^m\rightrightarrows\operatorname{\mathbb{R}}^n$ is *locally bounded* at $\bar x$ if there exists an open neighborhood $U$ of $\bar x$ such that the set $\cup_{x\in U} \Psi(x)$ is bounded.
The following two statements are equivalent:
The tensor $\operatorname{\mathcal{A}}$ is R$_0$;
The solution map $\operatorname{Sol}$ is locally bounded at $(\operatorname{\mathcal{A}},a)$, for every $a\in\operatorname{\mathbb{R}}^n$.
$\rm(a) \Rightarrow (b)$ Suppose that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor. By Theorem \[bounded1\], there exists $\varepsilon>0$ such that, for every $a\in\operatorname{\mathbb{R}}^n$, the following set is bounded $$S(\varepsilon,\varepsilon)=\bigcup_{(\operatorname{\mathcal{B}},b)\in U}\operatorname{Sol}(\operatorname{\mathcal{B}},b),$$ where $U=(\operatorname{\mathcal{A}},a)+\operatorname{\mathbb{B}}(\operatorname{\mathcal{O}},\varepsilon)\times \operatorname{\mathbb{B}}(0,\varepsilon)$ is an open neighborhood of $(\operatorname{\mathcal{A}},a)$. This means that the map $\operatorname{Sol}$ is locally bounded at $(\operatorname{\mathcal{A}},a)$.
$\rm(b) \Rightarrow (a)$ Suppose that the assertion in $\rm(b)$ is true. Taking $a=0$, there exists an open neighborhood $U$ of $(\operatorname{\mathcal{A}},0)$ such that $$\operatorname{Sol}(\operatorname{\mathcal{A}},0)\subset \bigcup_{(\operatorname{\mathcal{B}},b)\in U} \operatorname{Sol}(\operatorname{\mathcal{B}},b)$$ is bounded. It follows that $\operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$. The proof is complete.
Semi-algebraicity and genericity of $\operatorname{\mathcal{R}_0}$
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The set $\operatorname{\mathcal{R}_0}$ is semi-algebraic in $\operatorname{\mathbb{R}}^{[m,n]}$.
Remind that $\operatorname{\mathcal{A}}\in\operatorname{\mathcal{R}_0}$ if $\operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$. Since $0$ always belongs to $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$, the set $\operatorname{\mathcal{R}_0}$ can be described as follows: $$\label{R0}
\begin{array}{cl}
\operatorname{\mathcal{R}_0}&=\left\lbrace\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[m,n]}: \nexists x\in \operatorname{\mathbb{R}}^n_+\setminus\{0\} \left( \left[\operatorname{\mathcal{A}}x^{m-1}\geq 0\right]\wedge \left[ \operatorname{\mathcal{A}}x^{m}=0 \right]\right) \right\rbrace \medskip\\
&=\left\lbrace\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[m,n]}: \forall x\in \operatorname{\mathbb{R}}^n_+\setminus\{0\}\left( \left[\operatorname{\mathcal{A}}x^{m-1}\ngeq 0\right]\vee \left[ \operatorname{\mathcal{A}}x^{m}\neq 0 \right] \right) \right\rbrace.
\end{array}$$ Because $\operatorname{\mathbb{R}}^n_+$ and $\{0\}$ are semi-algebraic, the set $K=\operatorname{\mathbb{R}}^n_+\setminus\{0\}$ is also a semi-algebraic set in $\operatorname{\mathbb{R}}^n$. Let $Q_K(x)$ be the quantifier-free formula defining $K$. Since $\left( \operatorname{\mathcal{A}}x^{m-1}\right)_i$, where $i=1,...,n,$ and $\operatorname{\mathcal{A}}x^{m}$ are polynomials, the following formulas $$Q_1(\operatorname{\mathcal{A}},x)=\bigvee_{i=1}^{m}\left[ \left( \operatorname{\mathcal{A}}x^{m-1}\right)_i<0\right]$$ and $$Q_2(\operatorname{\mathcal{A}},x)=\left[ \operatorname{\mathcal{A}}x^{m}>0\right] \vee\left[ \operatorname{\mathcal{A}}x^{m}<0\right]$$ are quantifier-free. From the last equation in , $\operatorname{\mathcal{A}}\in \operatorname{\mathcal{R}_0}$ if and only if $Q(\operatorname{\mathcal{A}})$, where $Q(\operatorname{\mathcal{A}})$ is the following first-order formula $$Q(\operatorname{\mathcal{A}}):=\forall x \left( Q_K(x) \wedge\left[Q_1(\operatorname{\mathcal{A}},x) \vee Q_2(\operatorname{\mathcal{A}},x) \right] \right) .$$ According to the Tarski-Seidenberg Theorem in the third form, $\operatorname{\mathcal{R}_0}$ is a semi-algebraic set in $\operatorname{\mathbb{R}}^{[m,n]}$.
Let $\Phi:X \to Y$ be a differentiable map between manifolds, where $X\subset \operatorname{\mathbb{R}}^m$ and $Y\subset \operatorname{\mathbb{R}}^n$. A point $y \in Y$ is called a *regular value* for $\Phi$ if either the level set $\Phi^{-1}(y)=\emptyset$ or the derivative map $$D\Phi(x): T_xX\to T_yY$$ is surjective at every point $x \in \Phi^{-1}(y)$, where $T_xX$ and $T_yY$ denote respectively the tangent spaces of $X$ at $x$ and of $Y$ at $y$. So $y$ is a regular value of $f$ if and only if $\operatorname{rank}D\Phi(x)=n$ for all $x\in \Phi^{-1}(y)$.
\[RegularLevel\] Consider the differentiable semi-algebraic map $\Phi:X \to \operatorname{\mathbb{R}}^n$ where $X\subset \operatorname{\mathbb{R}}^n $. Assume that $y \in Y$ is a regular value of $\Phi$ and $\Phi^{-1}(y)$ is nonempty. According to the Regular Level Set Theorem [@Loring_2010 Theorem 9.9], one has $\dim\Phi^{-1}(y)=0$. It follows that $\Phi^{-1}(y)$ has finite points.
\[Sard\_parameter\] Let $\Phi : \operatorname{\mathbb{R}}^p\times X\to \operatorname{\mathbb{R}}^n$ be a differentiable semi-algebraic map, where $X\subset \operatorname{\mathbb{R}}^n $. Assume that $y \in \operatorname{\mathbb{R}}^n$ is a regular value of $\Phi$. According to the Sard Theorem with parameter [@DHP16 Theorem 2.4], there exists a generic semi-algebraic set $\operatorname{\mathbb{S}}\subset\operatorname{\mathbb{R}}^p$ such that, for every $p\in \operatorname{\mathbb{S}}$, $y$ is a regular value of the map $\Phi_p:X\to Y$ with $\Phi_p(x)=\Phi(p,x)$.
\[generic\_1\] The set $\operatorname{\mathcal{R}_0}$ of all R$_0$–tensors is generic in $\operatorname{\mathbb{R}}^{[m,n]}$.
We will show that there exists a generic semi-algebraic set $\operatorname{\mathbb{S}}\subset\operatorname{\mathbb{R}}^{[m,n]}$ such that $\operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$ for all $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{S}}$. Indeed, let $K_{\alpha}\neq\{0\}$ be a given pseudo-face of $\operatorname{\mathbb{R}}^n_+$. To avoid confusion, we only consider the case $\alpha=\{1,...,\ell\}$, where $\ell<n$, because other cases can be treated similarly. Then, if $x\in K_{\alpha}$ then $x_{\ell+1}\neq 0$. We consider the function $$\Phi_\alpha:\operatorname{\mathbb{R}}^{[m,n]}\times K_{\alpha}\times \operatorname{\mathbb{R}}^{\ell} \ \to \ \operatorname{\mathbb{R}}^{n+\ell}$$ which is defined by $$\label{Phi_alpha}
\Phi_\alpha(\operatorname{\mathcal{A}},x,\lambda_\alpha)=\left( \operatorname{\mathcal{A}}x^{m-1}-\lambda, x_{\alpha}\right)^T,$$ where $x_{\alpha}=(x_{1},...,x_{\ell})$, $\lambda_{\alpha}=(\lambda_{1},...,\lambda_{\ell})$ and $\lambda=(\lambda_{1},...,\lambda_{\ell},0,...,0)\in\operatorname{\mathbb{R}}^n$. The Jacobian matrix of $\Phi_\alpha$ is determined as follows $$D\Phi_\alpha=\left[ \begin{array}{c|c|c}
\ D_{\operatorname{\mathcal{A}}}(\operatorname{\mathcal{A}}x^{m-1}-\lambda) \ & \ D_{x}(\operatorname{\mathcal{A}}x^{m-1}-\lambda) \ & \ D_{\lambda_{\alpha}}(\operatorname{\mathcal{A}}x^{m-1}-\lambda) \\
\hline
D_{\operatorname{\mathcal{A}}}(x_{\alpha}) & D_{x}(x_{\alpha}) & D_{\lambda_{\alpha}}(x_{\alpha}) \\
\end{array}\right].$$
We claim that the rank of $D\Phi_\alpha$ is $n+\ell$ for all $x\in K_\alpha$. Indeed, it is easy to check that the rank of $D_{x}(x_{\alpha})$ is $\ell$. Therefore, if we prove that the rank of $D_{\operatorname{\mathcal{A}}}(\operatorname{\mathcal{A}}x^{m-1}-\lambda)$ is $n$ then the claim follows. Clearly, one has $$D_{\operatorname{\mathcal{A}}}(\operatorname{\mathcal{A}}x^{m-1}-\lambda)=\begin{bmatrix}
Q_{1} &0_{1\times n} & \cdots &0_{1\times n} \\
0_{1\times n} &Q_{2} & \cdots &0_{1\times n} \\
& & \ddots & \\
0_{1\times n} &0_{1\times n} & \cdots & Q_{n}
\end{bmatrix},$$ $0_{1\times n}$ is the zero $1\times n$–matrix, $Q_{i}$ is an $1\times n$–matrix. From and , for each $i\in[n]$, we conclude that $Q_{i}$ is a nonzero matrix because the $(\ell+1)$–th entry of $Q_{i}$ is given by $$\dfrac{\partial(\operatorname{\mathcal{A}}x^{m-1}-\lambda)_i}{\partial a_{i\ell\cdots \ell}} =x_{\ell}^{m-1}\neq 0.$$ This shows that $\operatorname{rank}D_{\operatorname{\mathcal{A}}}(\operatorname{\mathcal{A}}x^{m-1}-\lambda)=n$.
Therefore, $0\in \operatorname{\mathbb{R}}^{n+\ell}$ is a regular value of $\Phi_\alpha$. According to Remark \[Sard\_parameter\], there exists a generic semi-algebraic set $\operatorname{\mathbb{S}}_{\alpha}\subset \operatorname{\mathbb{R}}^{[m,n]}$, such that if $\operatorname{\mathcal{A}}\in \operatorname{\mathbb{S}}_{\alpha}$ then $0$ is a regular value of the map $$\Phi_{\alpha,\operatorname{\mathcal{A}}}:K_{\alpha}\times \operatorname{\mathbb{R}}^{\ell} \to \operatorname{\mathbb{R}}^{n+\ell}, \ \Phi_{\alpha,\operatorname{\mathcal{A}}}(x,\lambda_\alpha) =\Phi_\alpha(\operatorname{\mathcal{A}},x,\lambda_\alpha).$$ Remark \[RegularLevel\] claims that if the set $\Omega(\alpha,\operatorname{\mathcal{A}}):=\Phi^{-1}_{\alpha,\operatorname{\mathcal{A}}}(0)$ is nonempty then it is a zero dimensional semi-algebraic set. Hence, $\Omega(\alpha,\operatorname{\mathcal{A}})$ is a finite set. Moreover, from and Remark \[KKT\], one has $$\operatorname{Sol}(\operatorname{\mathcal{A}},0)\cap K_{\alpha}=\pi(\Omega(\alpha,\operatorname{\mathcal{A}})),$$ where $\pi$ is the projection $\operatorname{\mathbb{R}}^{n+\ell} \to \operatorname{\mathbb{R}}^n$ which is defined by $\pi(x,\lambda_{\alpha}) = x$. Thus, the cardinality of $\operatorname{Sol}(\operatorname{\mathcal{A}},0)\cap K_{\alpha}$ is finite.
If $K_{\alpha}=\{0\}$, i.e. $\alpha=[n]$, then $\operatorname{Sol}(\operatorname{\mathcal{A}},0)\cap K_{\alpha}=\{0\}$. By the finite decomposition in , $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is a finite set.
By setting $\operatorname{\mathbb{S}}:=\cap_{\alpha\subset [n]}\operatorname{\mathbb{S}}_{\alpha}$, for any $\operatorname{\mathcal{A}}$ in $\operatorname{\mathbb{S}}$, the cardinality of $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is finite. Since $\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ is a cone, one has $\operatorname{Sol}(\operatorname{\mathcal{A}},0)=\{0\}$. This follows that $\operatorname{\mathbb{S}}\subset \operatorname{\mathcal{R}_0}$, consequently, $\operatorname{\mathcal{R}_0}$ is generic in $\operatorname{\mathbb{R}}^{[m,n]}$. The proof is completed.
Theorem 6 in [@OettliYen95] asserts that the set of all R$_0$–matrices is dense in $\operatorname{\mathbb{R}}^{n\times n}$. This is a special case of Theorem \[generic\_1\] when $m=2$.
The dimension of $C(\operatorname{\mathcal{R}_0})$
--------------------------------------------------
From Remark \[dense\] and Theorem \[generic\_1\], the complement $C(\operatorname{\mathcal{R}_0})$ is thin in the set of real $m$-th order $n$-dimensional tensors. A natural question is: *How $C(\operatorname{\mathcal{R}_0})$ is thin in $\operatorname{\mathbb{R}}^{[m,n]}$*? The dimension of $C(\operatorname{\mathcal{R}_0})$ tell us about the thinness of this set. The following theorem gives a rough lower estimate for $\dim C(\operatorname{\mathcal{R}_0})$.
The dimension of the semi-algebraic set $C(\operatorname{\mathcal{R}_0})$ satisfies the following inequalities $$(n-1)^{m}\leq\dim C(\operatorname{\mathcal{R}_0})\leq n^m-1.$$
The second inequality immediately follows from Theorem \[generic\_1\] and Remark \[dense\]. To prove the first inequality, let $\alpha\subset [n]$ be given with $\alpha\neq[n]$, we consider the set $$\operatorname{\mathbb{S}}_{\alpha}=\left\lbrace \operatorname{\mathcal{A}}=(a_{i_1i_2\cdots i_m})\in\operatorname{\mathbb{R}}^{[m,n]}:a_{i_1i_2\cdots i_m}=0, \ \forall i_j\in[n]\setminus\alpha \right\rbrace.$$ It follows that $\operatorname{\mathbb{S}}_{\alpha}$ is a subspace of $\operatorname{\mathbb{R}}^{[m,n]}$ with the dimension $|\alpha|^{m}$. Hence, $\operatorname{\mathbb{S}}_{\alpha}$ is semi-algebraic. Let us denote by $\bar K_{\alpha}$ the face $$\bar K_{\alpha}=\left\lbrace x\in \operatorname{\mathbb{R}}_+^n:x_i=0, \forall i\in\alpha; \; x_i \geq 0, \forall i\in [n]\setminus\alpha\right\rbrace.$$ A trivial verification shows that $\bar K_{\alpha}\subset\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ for all $\operatorname{\mathcal{A}}\in \operatorname{\mathbb{S}}_{\alpha}$. We conclude that the subspace $\operatorname{\mathbb{S}}_{\alpha}$ is a subset of $C(\operatorname{\mathcal{R}_0})$. Thus, one has $$|\alpha|^{m}=\dim\operatorname{\mathbb{S}}_{\alpha}\leq\dim C(\operatorname{\mathcal{R}_0}).$$ Take $\alpha=\{2,...,n\}$, one has $|\alpha|=n-1$. The first inequality is obtained.
Lower semicontinuity of solution maps
=====================================
We will prove that the solution map of tensor complementarity problems is finite-valued on a generic semi-algebraic set in the parametric space. Consequently, a necessary condition for the lower semicontinuity of the solution map is given.
Now we recall some notions in set-valued analysis. The set-valued map $\Psi:\operatorname{\mathbb{R}}^m\rightrightarrows \operatorname{\mathbb{R}}^n$ is *finite-valued* on $S\subset \operatorname{\mathbb{R}}^m$ if the cardinality of the image $\Psi(x)$ is finite, namely $|\Psi(x)|<+\infty$, for all $x\in S$. The map $\Psi$ is *upper semicontinuous* at $x\in \operatorname{\mathbb{R}}^m$ iff for any open set $V\subset Y$ such that $\Psi(x)\subset V$ there exists a neighborhood $U$ of $x$ such that $\Psi(x')\subset V$ for all $x'\in U$. If $\Psi$ is upper semicontinuous at every $x\in \operatorname{\mathbb{R}}^m$ then $\Psi$ is said that to be upper semicontinuous on $\operatorname{\mathbb{R}}^m$. The map $\Psi$ is *lower semicontinuous* at $\bar x$ (see [@FaPa03 Proposition 2.1.17]) if $\Psi(\bar x)=\liminf_{x\to \bar x}\Psi(x)$, where $$\liminf_{x\to \bar x}\Psi(x)=\left\lbrace u\in\operatorname{\mathbb{R}}^n\;: \;\forall x^k\to \bar x, \; \exists u^k\to u \; \text{ with } \; u^k\in \Psi(x^k)\right\rbrace.$$ If $\Psi$ is lower semicontinuous at every $x\in X$ then $\Psi$ is said that to be lower semicontinuous on $X$.
\[Coste413\] The number of connected components of $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ does not excess $ \chi=d(2d-1)^{5n}$, where $d=\max\left\{2, m-1\right\}.$ Indeed, let $\Omega$ be the set of all $(x,\lambda)\in \operatorname{\mathbb{R}}^n\times\operatorname{\mathbb{R}}^n$ such that the following conditions are satisfied $$\left\lbrace \begin{array}{r}
\operatorname{\mathcal{A}}x^{m-1}+q-\lambda=0,\\
\langle \lambda,x \rangle=0,\\ \lambda\geq 0,x \geq 0.
\end{array}\right.$$Clearly, $\Omega$ is a semi-algebraic set determined by $3n+1$ polynomial equations and inequalities in $2n$ variables, whose degrees do not exceed the number $d$. According to [@Coste02 Proposition 4.13], the number of connected components of $\Omega$ does not excess the number $\chi$. By the definition of $\Omega$, one has $\operatorname{Sol}(\operatorname{\mathcal{A}},a)=\pi(\Omega),$ where $\pi$ is the projection $$\operatorname{\mathbb{R}}^{n+n} \to \operatorname{\mathbb{R}}^n, \ \pi(x,\lambda) = x.$$ By the continuity of $\pi$, the number of connected components of $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ also does not excess $\chi$.
In the following proposition, we consider the finite-valuedness of two solution maps of the tensor complementarity problems $\operatorname{Sol}$ given by and $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ defined by $$\label{Sol_A}
\operatorname{Sol}_{\operatorname{\mathcal{A}}}:\operatorname{\mathbb{R}}^{n}\rightrightarrows \operatorname{\mathbb{R}}^n, \ a\mapsto\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)=\operatorname{Sol}(\operatorname{\mathcal{A}},a),$$ where $\operatorname{\mathcal{A}}$ is given.
\[generic\_3\] There exists a generic semi-algebraic set $\operatorname{\mathbb{S}}\subset\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^{n}$ such that the map $\operatorname{Sol}$ is finite-valued on $\operatorname{\mathbb{S}}$.
To prove the assertion, we apply the argument in the proof of Theorem \[generic\_1\] again, the only difference being in the analysis of the function $$\Phi_\alpha:\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^n\times K_{\alpha}\times \operatorname{\mathbb{R}}^{\ell} \ \to \ \operatorname{\mathbb{R}}^{n+\ell}$$ which defined by $$\Phi_\alpha(\operatorname{\mathcal{A}},a,x,\lambda_\alpha)=\left( \operatorname{\mathcal{A}}x^{m-1}+a-\lambda, x_{\alpha}\right) ^T.$$ Note that, since $D_{\operatorname{\mathcal{A}}} \Phi_\alpha$ has rank $n$, the rank of $D\Phi_\alpha$ is $n+\ell$ for $x\in K_\alpha\neq\{0\}$, and the proof is complete.
\[generic\_5\] Let $\operatorname{\mathcal{A}}$ be given. There exists a generic semi-algebraic set $\operatorname{\mathbb{S}}_{\operatorname{\mathcal{A}}}\subset\operatorname{\mathbb{R}}^{n}$ such that the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is finite-valued on $\operatorname{\mathbb{S}}_{\operatorname{\mathcal{A}}}$. This property is implied from [@LLP2018 Theorem 3.2] with the note that $\operatorname{\mathbb{R}}^n_+$ is a semi-algebraic set satisfying the linearly independent constraint qualification.
\[generic\_4\] If the solution map $\operatorname{Sol}$ is lower semicontinuous at $(\operatorname{\mathcal{A}},a)$, then $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ has finite elements. Hence, if $\dim\operatorname{Sol}(\operatorname{\mathcal{A}},a)\geq 1$, then $\operatorname{Sol}$ is not lower semicontinuous at $(\operatorname{\mathcal{A}},a)$.
According to Proposition \[generic\_3\], there exists a generic set $\operatorname{\mathbb{S}}$ in $\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^n$ such that $\operatorname{Sol}$ is finite-valued on $\operatorname{\mathbb{S}}$. By the density of $\operatorname{\mathbb{S}}$, there exists a sequence $\{(\operatorname{\mathcal{A}}^k,a^k)\}\subset \operatorname{\mathbb{S}}$ such that $(\operatorname{\mathcal{A}}^k,a^k)\to (\operatorname{\mathcal{A}},a)$. From Remark \[Coste413\], $\operatorname{Sol}(\operatorname{\mathcal{A}}^k,a^k)$ has finitely points and $|\operatorname{Sol}(\operatorname{\mathcal{A}}^k,a^k)|\leq\chi$. Since $\operatorname{Sol}$ is lower semicontinuous, one has $$\operatorname{Sol}(\operatorname{\mathcal{A}},a)=\liminf_{k\to+\infty}\operatorname{Sol}(\operatorname{\mathcal{A}}^k,a^k).$$ It follows that $|\operatorname{Sol}(\operatorname{\mathcal{A}},a)|\leq\chi$. The first assertion is proved. The second assertion follows the first one.
\[example\_1\] Consider the tensor complementarity problem $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ where $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[3,2]}$ given by $a_{111}=a_{122}=-1$, $a_{211}=a_{222}=-1$ and all other $a_{i_1i_2i_3}=0$. Obviously, one has $$\operatorname{\mathcal{A}}x^{m-1}+q=\begin{bmatrix}
-x_1^2-x_2^2\\
-x_1^2-x_2^2
\end{bmatrix}+\begin{bmatrix}
a_1\\
a_2
\end{bmatrix},$$ where the parameters $a_1,a_2\in\operatorname{\mathbb{R}}$. From Remark \[KKT\], $x\in\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ if and only if there exists $\lambda\in\operatorname{\mathbb{R}}^2$ such that $$\begin{bmatrix}
-x_1^2-x_2^2\\
-x_1^2-x_2^2
\end{bmatrix}+\begin{bmatrix}
a_1\\
a_2
\end{bmatrix}-\begin{bmatrix}
\lambda_1\\
\lambda_2
\end{bmatrix}=
\begin{bmatrix}
0\\
0
\end{bmatrix}.$$ An easy computation shows that $$\operatorname{Sol}(\operatorname{\mathcal{A}},a)=\left\{\begin{array}{cl}
\left\lbrace(0,0),(0,\sqrt{a_2})\right\rbrace & \text{ if } 0\leq a_2<a_1, \\
\left\lbrace(0,0),(\sqrt{a_1},0)\right\rbrace & \text{ if } 0\leq a_1< a_2, \\
\left\lbrace(0,0)\right\rbrace \cup S_{a_1}& \text{ if } 0\leq a_1=a_2, \\
\emptyset & \text{ if otherwise},\\
\end{array}\right.$$ where $$S_{a_1}=\{(x_1,x_2):x_1^2+x_2^2=a_1,x_1\geq 0, x_2\geq 0\}.$$ Clearly, $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)$ is finite-valued for all $a\in\operatorname{\mathbb{S}}$, where $$\operatorname{\mathbb{S}}=\operatorname{\mathbb{R}}^2\setminus \{a\in\operatorname{\mathbb{R}}^2: 0 < a_1=a_2\}.$$ The set $\operatorname{\mathbb{S}}$ is generic semi-algebraic set in $\operatorname{\mathbb{R}}^2$. Moreover, since $\dim S_{a_1}=1$ with $a_1>0$, according to Theorem \[generic\_4\], the map $\operatorname{Sol}$ is not lower semicontinuous at $(A,a)$ where $a\in\operatorname{\mathbb{R}}^2$ with $0 < a_1=a_2$.
Upper semicontinuity of the solution map
========================================
This section establishes a closed relationship between the R$_0$ property and the upper semicontinuity of the solution map of tensor complementarity problems. Furthermore, two results on the single-valued continuity of the solution map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ are obtained.
Necessary and sufficient conditions
-----------------------------------
\[usc\_1\] If $\operatorname{\mathcal{A}}$ is an R$_0$–tensor, then the map $\operatorname{Sol}$ is upper semicontinuous at $(A,a)$, for every $a\in\operatorname{\mathbb{R}}^n$ satisfying $\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq\emptyset$.
Suppose that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor but there is $a\in\operatorname{\mathbb{R}}^n$ such that $\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq \emptyset$ and $\operatorname{Sol}$ is not upper semicontinuous at $(\operatorname{\mathcal{A}},a)$. There exists a nonempty open set $V$ containing $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$, a sequence $\{(\operatorname{\mathcal{A}}^k,a^k)\}$, and a sequence $\{x^k\}$ satisfying $(\operatorname{\mathcal{A}}^k,a^k)\to (\operatorname{\mathcal{A}},a)$ and $$\label{V_open0}
x^k\in\operatorname{Sol}(\operatorname{\mathcal{A}}^k,a^k)\setminus V.$$ According to Theorem \[bounded1\], there exists $k_0$ such that $\{x^k, k\geq k_0\}$ is a bounded subsequence. So, the sequence $\{x^k\}$ is bounded. Without loss of generality we can assume that $x^k\to \bar x$. It is easy to check that $\bar x$ solves $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$. It follows that $\bar x\in V$. By the openness of $V$ and , one has $\bar x\notin V$. We obtain a contradiction. Therefore, the map $\operatorname{Sol}$ is upper semicontinuous at $(\operatorname{\mathcal{A}},a)$.
\[SolA\_usc\] If $\operatorname{\mathcal{A}}$ is an R$_0$–tensor, then the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous at $a$, for every $a\in\operatorname{\mathbb{R}}^n$ satisfying $\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq\emptyset$.
Suppose that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor and $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is nonempty. From Proposition \[usc\_1\], the map $\operatorname{Sol}$ is upper semicontinuous at $(A,a)$. For any open neighborhood $V$ of $\operatorname{Sol}(A,a)$, there exists an open neighborhood $U$ of $(A,a)$ such that $(\operatorname{\mathcal{B}},b)\in U$ then $\operatorname{Sol}(\operatorname{\mathcal{B}},b)\subset V$. Consider the map $$\varphi:\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^n\rightrightarrows \operatorname{\mathbb{R}}^n, \ (\operatorname{\mathcal{B}},b)\mapsto b.$$ Clearly, $\varphi$ is surjective, continuous and linear. According to the Theorem Open Mapping [@Rudin91 Theorem 2.11], $\varphi(U)$ is an open neighborhood of $a$. By definition, $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous at $a$.
Consider the tensor complementarity problem $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ given in Example \[example\_1\]. One has $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(0)=\{0\}$, so $\operatorname{\mathcal{A}}$ is an R$_0$–tensor. From Corollary \[SolA\_usc\], the solution map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous on $\operatorname{\mathbb{R}}^2_+$.
The inverse assertion in Proposition \[SolA\_usc\] is not true. Indeed, choose $\operatorname{\mathcal{A}}=\operatorname{\mathcal{O}}\in\operatorname{\mathbb{R}}^{[3,2]}$, one has $$\label{SolO}
\operatorname{Sol}_{\operatorname{\mathcal{O}}}(a_1,a_2)=\left\{\begin{array}{ccc}
\operatorname{\mathbb{R}}^2_+ & \text{ if } &a_1=0,a_2=0, \\
\operatorname{\mathbb{R}}_+\times\{0\}& \text{ if } & a_1=0,a_2>0, \\
\{0\}\times\operatorname{\mathbb{R}}_+ & \text{ if } & a_1>0,a_2=0, \\
\{(0,0)\} & \text{ if } & a_1>0,a_2>0,\\
\emptyset & \text{ if } & \text{ otherwise.}
\end{array}\right.$$ It is easy to check that $\operatorname{Sol}_{\operatorname{\mathcal{O}}}$ is upper semicontinuous on $\operatorname{dom}\operatorname{Sol}_{\operatorname{\mathcal{O}}}=\operatorname{\mathbb{R}}^2_+$, but $\operatorname{\mathcal{O}}$ has not R$_0$ property.
\[usc\_2\] Assume that $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is nonempty and bounded. If the map $\operatorname{Sol}$ is upper semicontinuous at $(\operatorname{\mathcal{A}},a)$, then $\operatorname{\mathcal{A}}$ is a R$_0$–tensor.
Suppose that $\operatorname{Sol}(\operatorname{\mathcal{A}},0)\neq\{0\}$ and $y\in\operatorname{Sol}(\operatorname{\mathcal{A}},0)$ with $y\neq 0$. According to Remark \[KKT\], there exists $\lambda\in\operatorname{\mathbb{R}}^n$ such that $$\label{KKT_H0}
\left\lbrace \begin{array}{r}
\operatorname{\mathcal{A}}y^{m-1}-\lambda=0,\\
\langle \lambda, y \rangle=0,\\ \lambda\geq 0, \; y \geq 0.
\end{array}\right.$$ For each $t\in(0,1)$, we take $y_t=t^{-1}y$ and $ \lambda_t=t^{-(m-1)}\lambda$. We will show that for every $t$ there exists $\operatorname{\mathcal{A}}_t\in\operatorname{\mathbb{R}}^{[m,n]}$ with $\operatorname{\mathcal{A}}_t\to \operatorname{\mathcal{A}}$ when $t\to 0$ and the following system is satisfied $$\label{KKT_Ht}
\left\lbrace \begin{array}{r}
\operatorname{\mathcal{A}}_t (y_t)^{m-1}+{q}-\lambda_t=0,\\
\langle \lambda_t,y_t \rangle=0,\\ \lambda_t\geq 0, \; y_t \geq 0.
\end{array}\right.$$
Since $y=(y_1,...,y_n)\neq 0$, there exists $y_{\ell}\neq 0$, so one has $y^{m-1}_{\ell}\neq 0$. Taking $\operatorname{\mathcal{Q}}\in\operatorname{\mathbb{R}}^{[m,n]}$ such that $$\operatorname{\mathcal{Q}}x^{m-1}=\left( q_1x_i^{m-1},...,q_nx_i^{m-1}\right),$$ where $q_j=-a_j/y_{\ell}^{m-1}$ for $j=1,...,n$. It is clear that $\operatorname{\mathcal{Q}}y^{m-1}+ a=0$. We take $\operatorname{\mathcal{A}}_t= \operatorname{\mathcal{A}}+t\operatorname{\mathcal{Q}}$ and prove that the system is true. Indeed, the last two inequalities in are obvious. Consider the left-hand side of the first equation in , from we have $$\begin{array}{cl}
\operatorname{\mathcal{A}}_t(y_t)^{m-1}+ a-\lambda_t&=(\operatorname{\mathcal{A}}_t=\operatorname{\mathcal{A}}+t\operatorname{\mathcal{Q}})(t^{-1}y)^{m-1} +a- t^{-(m-1)}\lambda\\
&=t^{-(m-1)}(\operatorname{\mathcal{A}}y^{m-1}-\lambda)+(\operatorname{\mathcal{Q}}y^{m-1}+a) \\
&= 0.
\end{array}$$ The second equation in is obtained by $$\langle \lambda_t,y_t \rangle=\langle t^{-(m-1)}\lambda,t^{-1}y \rangle=t^{-m}\langle \lambda,y \rangle=0.$$ According to Remark \[KKT\], the system leads to $y_t \in \operatorname{Sol}(\operatorname{\mathcal{A}}_t,a)$. Remind that this assertion is true for all $t\in (0,1)$.
Since $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is nonempty bounded, let $V$ be a nonempty bounded open set containing $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$. By the upper semicontinuity of $\operatorname{Sol}$ at $(\operatorname{\mathcal{A}},a)$, there exists $\delta>0$ such that $\operatorname{Sol}(\operatorname{\mathcal{B}},b)\subset V$ for all $(\operatorname{\mathcal{B}},b)\in\operatorname{\mathbb{R}}^{[m,n]}\times\operatorname{\mathbb{R}}^n$ satisfying $\|(\operatorname{\mathcal{B}},b)-(\operatorname{\mathcal{A}},a)\|<\delta$. Taking $t$ small enough such that $\|(\operatorname{\mathcal{A}}_t,a)-(\operatorname{\mathcal{A}},a)\|<\delta$, we have $\operatorname{Sol}(\operatorname{\mathcal{A}}_t,a)\subset V$. So, $y_t\in V$ for every $t>0$ sufficiently small. This is impossible, because $V$ is bounded and $y_t =t^{-1}y \to \infty$ as $t\to 0$. The proof is complete.
The main result of this section is shown in the following theorem.
The following two statements are equivalent:
The tensor $\operatorname{\mathcal{A}}$ is R$_0$;
The map $\operatorname{Sol}$ is upper semicontinuous at $(\operatorname{\mathcal{A}},a)$, for every $a\in\operatorname{\mathbb{R}}^n$ satisfying $\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq\emptyset$.
Proposition \[usc\_1\] shows that $\rm(b)$ follows $\rm(a)$. Hence, we need only prove the direction $\rm(b) \Rightarrow (a)$. Note that $0\in\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq\emptyset$ for every $a\in\operatorname{\mathbb{R}}^n_+$. According to Remark \[generic\_5\], then there exists $q\in\operatorname{\mathbb{R}}^n_+$ such that $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is bounded. By assumptions, the map $\operatorname{Sol}$ is upper semicontinuous at $(\operatorname{\mathcal{A}},a)$. Proposition \[usc\_2\] says that $\operatorname{\mathcal{A}}$ is R$_0$. The proof is complete.
Single-valued continuity
------------------------
Recall that $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ is said to have the *GUS-property* if $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ has a unique solution for every $a\in\operatorname{\mathbb{R}}^n$. Some special structured tensors which have GUS-property are shown in [@BHW2016; @LLW2017]. A new property of the GUS-property of tensor complementarity problems is given in the following theorem.
If $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ has the GUS-property, then the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued continuous on $\operatorname{\mathbb{R}}^n$.
By assumptions, $\operatorname{TCP}(\operatorname{\mathcal{A}},0)$ has a unique solution. This implies that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor. Corollary \[SolA\_usc\] shows that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous on $\operatorname{\mathbb{R}}^n$. Therefore, $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued continuous on $\operatorname{\mathbb{R}}^n$.
Consider the tensor complementarity problem $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ where $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[3,2]}$ given by $a_{111}=a_{222}=1$ and all other $a_{i_1i_2i_3}=0$. Obviously, one has $$\operatorname{\mathcal{A}}x^{m-1}+q=\begin{bmatrix}
x_1^2\\
x_2^2
\end{bmatrix}+\begin{bmatrix}
a_1\\
a_2
\end{bmatrix},$$ where the parameters $a_1,a_2\in\operatorname{\mathbb{R}}$. An easy computation shows that $$\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a_1,a_2)=\left\{\begin{array}{cl}
\left\lbrace(\sqrt{-a_1},\sqrt{-a_2})\right\rbrace & \text{ if } a_1<0, a_2<0 \\
\left\lbrace (0,\sqrt{-a_2} )\right\rbrace & \text{ if } a_1\geq 0, a_2<0 \\
\left\lbrace (\sqrt{-a_1},0)\right\rbrace & \text{ if } a_1< 0, a_2\geq0 \\
\left\lbrace (0,0)\right\rbrace & \text{ if } a_1\geq 0, a_2\geq0 \\
\end{array}\right.$$ The problem $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ has the GUS-property, the domain of $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is $\operatorname{\mathbb{R}}^2$ and $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued and continuous on $\operatorname{\mathbb{R}}^2$.
Recall that a tensor $\operatorname{\mathcal{A}}$ is *copositive* if $\operatorname{\mathcal{A}}x^m\geq 0$ for all $x \geq 0$. A function $F:\operatorname{\mathbb{R}}^n\to \operatorname{\mathbb{R}}^n$ is *monotone* on $X\subset\operatorname{\mathbb{R}}^n$ if for all $x,y\in X$ the following inequality is satisfied $$\label{condition_monotone}
\left\langle F(y)-F(x),y-x\right\rangle \geq0.$$ If $F(x)=\operatorname{\mathcal{A}}x^{m-1}$ is monotone on $\operatorname{\mathbb{R}}^n_+$ then $\operatorname{\mathcal{A}}$ is copositive. Indeed, by taking $y=0$ in , $\operatorname{\mathcal{A}}$ satisfies the copositive condition.
If the R$_0$–tensor $\operatorname{\mathcal{A}}$ is copositive, then $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is nonempty for every $q\in\operatorname{\mathbb{R}}^n$ [@Gowda16 Corollary 7.2].
Assume that $\operatorname{\mathcal{A}}$ is an R$_0$–tensor. If $F(x)=\operatorname{\mathcal{A}}x^{m-1}$ is monotone on $\operatorname{\mathbb{R}}^n_+$, then the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued continuous on a generic semi-algebraic set in $ \operatorname{\mathbb{R}}^{n}$.
By the copositity and the R$_0$ property of $\operatorname{\mathcal{A}}$, according to Corollary 7.2 in [@Gowda16], one has $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)\neq\emptyset$ for all $a\in\operatorname{\mathbb{R}}^n$. From Theorem \[generic\_4\], there exists a generic semi-algebraic set $\operatorname{\mathbb{S}}\subset \operatorname{\mathbb{R}}^{n}$ such that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is finite-valued on $\operatorname{\mathbb{S}}$.
For every $a\in\operatorname{\mathbb{R}}^n$, by the monotonicity of $F$, $F+a$ also is monotone. It follows that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)$ is convex [@FaPa03 Theorem 2.3.5]. Since $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)$ is nonempty and has finite points, it has an unique point. This implies that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued on $\operatorname{\mathbb{S}}$. Moreover, $\operatorname{\mathcal{A}}$ is an R$_0$–tensor, Remark \[SolA\_usc\] implies that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous on $\operatorname{\mathbb{S}}$. From what has already been shown, $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued continuous on $\operatorname{\mathbb{S}}$.
Consider the tensor complementarity problem $\operatorname{TCP}(\operatorname{\mathcal{A}},a)$ where $\operatorname{\mathcal{A}}\in\operatorname{\mathbb{R}}^{[3,2]}$ given by $a_{111}=a_{122}=1$, $a_{211}=a_{222}=1$ and all other $a_{i_1i_2i_3}=0$. Obviously, one has $$F(x)=\operatorname{\mathcal{A}}x^{m-1}+a=\begin{bmatrix}
x_1^2+x_2^2\\
x_1^2+x_2^2
\end{bmatrix}+\begin{bmatrix}
a_1\\
a_2
\end{bmatrix},$$ where the parameters $a_1,a_2\in\operatorname{\mathbb{R}}$. The Jacobian matrix of $F$ is positive semidefinite on $\operatorname{\mathbb{R}}^2_+$. Hence, $F$ is monotone on $\operatorname{\mathbb{R}}^2_+$. From Remark \[KKT\], an easy computation shows that $$\operatorname{Sol}_{\operatorname{\mathcal{A}}}(a_1,a_2)=\left\{\begin{array}{cl}
\left\lbrace(0,\sqrt{-a_2})\right\rbrace & \text{ if } a_2<0, a_2\leq a_1, \\
\left\lbrace(\sqrt{-a_1},0)\right\rbrace & \text{ if } a_1<0, a_1\leq a_2, \\
\left\lbrace(0,0)\right\rbrace& \text{ if } 0\leq a_1,0\leq a_2,\\
S_{-a_1}& \text{ if } a_1=a_2<0, \\
\end{array}\right.$$ where $$S_{-a_1}=\{(x_1,x_2):x_1^2+x_2^2=a_1,x_1\geq 0, x_2\geq 0\}, \ a_1<0.$$ The tensor $\operatorname{\mathcal{A}}$ is R$_0$ since $\operatorname{Sol}_{\operatorname{\mathcal{A}}}(0,0)=\{(0,0)\}$. The map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is single-valued continuous on the generic semi-algebraic set $\operatorname{\mathbb{S}}$, where $$\operatorname{\mathbb{S}}=\operatorname{\mathbb{R}}^2\setminus \{q\in\operatorname{\mathbb{R}}^2: a_1=a_2<0\}.$$
Stability of the solution map
=============================
This section discusses on the stability of the solution map of tensor complementarity problems. We will show that the map $\operatorname{Sol}$ is locally upper-Hölder when the involved tensor is R$_0$. In addition, if the tensor is copositive then one obtains a result on the stability of the solution map.
Recall that the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ defined in is said to be *locally upper-Hölder* at ${a}$ if there exist $\gamma>0,c>0$ and $\varepsilon>0$ such that $$\operatorname{Sol}_{\operatorname{\mathcal{A}}}(b)\subset \operatorname{Sol}_{\operatorname{\mathcal{A}}}(a)+\gamma\|b-a\|^{c}\operatorname{\mathbb{B}}(0,1)$$ for all $a$ satisfying $\|b- a\|< \varepsilon$, where $\operatorname{\mathbb{B}}(0,1)$ is the closed unit ball in $\operatorname{\mathbb{R}}^n$.
If $\operatorname{\mathcal{A}}$ is R$_0$ and $\operatorname{Sol}(\operatorname{\mathcal{A}},a)\neq\emptyset$, then the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is locally upper-Hölder at $a$.
By assumptions, Corollary \[SolA\_usc\] claims that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous at $a$. According to [@LLP2018 Theorem 4.1], the upper semicontinuity and the local upper-Hölder stability of $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ at $a$ are equivalent. Hence, the assertion is proved.
Let $C$ be a nonempty closed cone. Here $\operatorname{int}C^+$ stands for the interior of the dual cone $C^+$ of $C$. Note that $q\in\operatorname{int}C^+$ if and only if $\left\langle v,q \right\rangle >0$ for all $v\in C$ and $v\neq 0$ [@LTY2005 Lemma 6.4].
\[Holder2\] If $\operatorname{\mathcal{A}}$ is copositive and $a\in\operatorname{int}(\operatorname{Sol}(\operatorname{\mathcal{A}},0)^+)$, then the map $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is locally upper-Hölder at ${a}$.
Since the upper semicontinuity and the local upper-Hölder stability of $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ at $a$ are equivalent [@LLP2018 Theorem 4.1], we need only to prove that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous at $a$.
Since $\operatorname{\mathcal{A}}$ is copositive and $a\in\operatorname{int}(\operatorname{Sol}(\operatorname{\mathcal{A}},0)^+)$, $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ is nonempty compact [@Gowda16 Corollary 7.3]. We suppose that $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is not upper semicontinuous at $a$. There is a nonempty open set $V$ containing $\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ such that, for every integer number $k\geq 1$, there exists $q^k\in \operatorname{\mathbb{R}}^n$ satisfying $q^k\to q$ and $$\label{V_open1}
x^k\in\operatorname{Sol}(K,P+q^k)\setminus V.$$ By repeating the argument in the proof of Proposition \[usc\_1\], we can prove that the sequence $\{x^k\}$ is bounded. So, without loss of generality we assume that $x^k\to \bar x$. It easy to check that $\bar x\in\operatorname{Sol}(\operatorname{\mathcal{A}},a)$. This leads to $\bar x\in V$. Besides, since $V$ is an open nonempty set, the relation implies that $\bar x\notin V$. This is a contradiction. Therefore, $\operatorname{Sol}_{\operatorname{\mathcal{A}}}$ is upper semicontinuous at $a$.
Theorem 7.5.1 in [@CPS1992] shown an interesting result on the stability of the solution map of linear complementarity problems under the copositive condition. Here, we obtain an analogical one for the solution map of tensor complementarity problems.
If $a\in\operatorname{int}(\operatorname{Sol}(\operatorname{\mathcal{A}},0)^+)$, then there exist constants $\varepsilon>0,\gamma>0$ and $c>0$ such that, if $\operatorname{\mathcal{B}}\in\operatorname{\mathbb{R}}^{[m,n]}$ and $b\in\operatorname{\mathbb{R}}^n$ satisfying $$\label{epsilon}
\max\{\|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\|,\|b- a\|\}<\varepsilon,$$ and $\operatorname{\mathcal{B}}$ is copositive, then the following statements hold:
The set $\operatorname{Sol}(\operatorname{\mathcal{B}},a)$ is nonempty and bounded;
One has $$\label{ell}\operatorname{Sol}(\operatorname{\mathcal{B}},b)\subset \operatorname{Sol}(\operatorname{\mathcal{A}},a)+\gamma(\|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\|+\|b-a\|)^{c}\operatorname{\mathbb{B}}(0,1).$$
$\rm (a)$ We prove that there exists $\varepsilon_1>0$ such that if $\operatorname{\mathcal{B}}$ is copositive and $ b\in\operatorname{\mathbb{R}}^n$ satisfying $$\label{eps_1}
\max\{\|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\|,\|b- a\|\}<\varepsilon_1$$ then $\operatorname{Sol}(\operatorname{\mathcal{B}},b)$ is nonempty and bounded. Suppose that the assertion is false. There exists a sequence $(\operatorname{\mathcal{B}}^k,b^k)$, where $(\operatorname{\mathcal{B}}^k,b^k)\to (\operatorname{\mathcal{A}},a)$ and $\operatorname{\mathcal{B}}^k$ is copositive, such that $\operatorname{Sol}(\operatorname{\mathcal{B}}^k,b^k)$ is empty or unbounded, for each $k\in\mathbb{N}$. From [@Gowda16 Corollary 7.3], we conclude that $b^k\notin\operatorname{int}(\operatorname{Sol}(\operatorname{\mathcal{B}}^k,0)^+)$. This implies that there exists $x^k\in\operatorname{Sol}(\operatorname{\mathcal{B}}^k,0)$ such that $x^k\neq 0$ and $\left\langle x^k,b^k \right\rangle\leq 0$. We can assume that $\|x^k\|^{-1}x^k\to\bar x\in\operatorname{\mathbb{R}}^n_+$ with $\|\bar x\|=1$.
Clearly, since $\left\langle x^k,b^k \right\rangle\leq 0$ for each $k\in\mathbb{N}$, one has $\left\langle \bar x,a \right\rangle\leq 0$. If we prove that $\bar x$ solves $\operatorname{TCP}(\operatorname{\mathcal{A}},0)$, then this contradicts to the assumption that $a\in\operatorname{int}(\operatorname{Sol}(\operatorname{\mathcal{A}},0)^+)$, and $\rm(a)$ is proved. Thus, we only need to show that $\bar x\in \operatorname{Sol}(\operatorname{\mathcal{A}},0)$. Because $x^k$ belongs to $\operatorname{Sol}(\operatorname{\mathcal{B}}^k,0)$, one has $$\label{x_k}
\operatorname{\mathcal{B}}^k (x^k)^{m-1}\geq 0, \ \operatorname{\mathcal{B}}^k (x^k)^{m} =0.$$ By dividing the inequality and the equation in by $\|x^k\|^{m-1}$ and $\|x^k\|^{m}$, respectively, and taking $k\to+\infty$, we obtain $$\operatorname{\mathcal{A}}\bar x^{m-1}\geq 0, \ \operatorname{\mathcal{A}}\bar x^{m}= 0.$$ This implies that $\bar x\in \operatorname{Sol}(\operatorname{\mathcal{A}},0)$.
$\rm (b)$ We prove the inclusion . According to Proposition \[Holder2\], there exist $\gamma_0>0,c>0$ and $\varepsilon$ such that $$\label{ell0}
\operatorname{Sol}(\operatorname{\mathcal{A}},b)\subset \operatorname{Sol}(\operatorname{\mathcal{A}},a)+\gamma_0\|b-a\|^{c}\operatorname{\mathbb{B}}(0,1)$$ for every $b$ satisfying $\|b-a\|< \varepsilon$. Suppose that $\operatorname{\mathcal{B}}$ is copositive and $b\in\operatorname{\mathbb{R}}^n$ satisfying . For each $z\in \operatorname{Sol}(\operatorname{\mathcal{B}},b)$, by setting $$\label{q_hat} \hat b=b+\left(\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\right) z^{m-1},$$ we have $$A z^{m-1}+ \hat b=\operatorname{\mathcal{B}}z^{m-1}+b\geq 0, \ \; \left\langle z, \operatorname{\mathcal{A}}z^{m-1}+ \hat b\right\rangle =\left\langle z,\operatorname{\mathcal{B}}z^{m-1}+b\right\rangle = 0.$$ These show that $z\in\operatorname{Sol}(\operatorname{\mathcal{A}},\hat b)$. Besides, since $\operatorname{Sol}(\operatorname{\mathcal{B}},b)$ is bounded and nonempty, Remark \[norm\_beta\] states that there exists $\beta>0$ such that $$\label{norm_ineq}
\|(\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}) z^{m-1}\|\leq \beta\|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\|$$ for any $z\in\operatorname{Sol}(\operatorname{\mathcal{B}},b)$. From , , and , one has $$\begin{array}{ll}
\|\hat b-a\| &\leq \; \|b-a\|+\|(\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}})z^{m-1}\| \smallskip \\
&\leq \; \|b-a\|+\beta \|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\| \smallskip \\
&\leq \; (1+\beta)\varepsilon_1.
\end{array}$$ Choosing $\varepsilon_1$ small enough such that $\|\hat b-a\|<\varepsilon$, if necessary. From and , there exists $x\in\operatorname{Sol}(\operatorname{\mathcal{A}},a)$ such that $$\begin{array}{rl}
\|z-x\| \; & \leq \; \gamma_0\|\hat b-a\|^{c} \smallskip\\
&\leq \; \gamma_0\left(\|b-a\|+\beta \|\operatorname{\mathcal{B}}- \operatorname{\mathcal{A}}\| \right)^{c} \smallskip \\
& \leq \; \gamma \left(\|b-a\|+\|\operatorname{\mathcal{B}}-\operatorname{\mathcal{A}}\| \right)^{c},
\end{array}$$ where $\gamma=\max\left\lbrace \gamma_0^{c},\gamma_0^{c}\beta\right\rbrace $. Then the inclusion is obtained.
Conclusions
===========
In this paper, we have proved that the set $\operatorname{\mathcal{R}_0}$ of all R$_0$–tensors is an open generic semi-algebraic cone . Upper and lower estimates for the dimension of the complement $C(\operatorname{\mathcal{R}_0})$ are shown. Several results on local boundedness, upper semicontinuity and stability of the solution map have been obtained. In our further research, we intend to develop these results for polynomial complementarity problems and semi-algebraic variational inequalities.
The author would like to express his deep gratitude to Professor Nguyen Dong Yen for the enthusiastic encouragement.
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abstract: 'Aim of this paper is to study the non-existence of global solutions of the fractional differential problem involving generalized Katugampola derivative. We utilize the test function method and fractional integration by parts formula to obtain the result. An illustrative example is also given.'
address: 'Sandeep P Bhairat Department of Mathematics, Institute of Chemical Technology,Mumbai – 400 019, (M.S) India.'
author:
- Sandeep P Bhairat
title: 'Non-existence of global solutions for a generalized fractional differential problem'
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Definition]{} \[theorem\][Example]{}
Introduction
============
In the past thirty years, the interest to fractional differential equations paid more attention of many researchers in several areas such as bioengineering, physics, mechanics and applied sciences, [@hr2; @ve]. For recent development and historical arguments, see the monographs [@kst; @skm]. The existence of solutions for various class of fractional differential equations are studied extensively with number of fractional derivatives in [@as]-[@kmf],[@mdk; @ke; @oo]. Whereas for the non-existence of solutions, one can see the recent papers [@neh; @nehh; @newnec; @newne; @oldnex].
Recently, the existence and uniqueness of solution of initial value problem $$\label{p}
\begin{cases}
& ^{\rho}D_{a+}^{\alpha,\beta}x(t)=f(t,x(t)),\quad 0<\alpha<1,0\leq\beta\leq1,
\rho>0, \, t>a>0,\\
& ^{\rho}I_{a+}^{1-\gamma }x(t) \big|_{t=a}=\phi,\quad \qquad\phi\in \mathbb{R},\gamma=\alpha-\beta-\alpha\beta,
\end{cases}$$ for the generalized fractional differential problem is studied in [@oo]. The operators $^{\rho}I_{a+}^{\sigma }$ and $^{\rho}D_{a+}^{\sigma,\xi }$ are the Katugampola fractional integral [@ki] and generalized Katugampola fractional derivative [@oo], respectively. In this paper we consider the generalized fractional differential problem of type $$\label{pex}
\begin{cases}
& ^{\rho}D_{a+}^{\alpha,\beta}x(t) \geq {\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\mu}{|x(t)|}^{m},\quad \rho>0,\, t>a>0,\,\mu\in\mathbb{R},\,\,m>1,\\
& ^{\rho}I_{a+}^{1-\gamma }x(t) \big|_{t=a}=x_a,\qquad\qquad x_a\in \mathbb{R},\,\gamma=\alpha+\beta-\alpha\beta,
\end{cases}$$ where $0<\alpha<1,0\leq \beta\leq1$. We prove that no solutions can exist for all time for certain values of $\mu$ and $m$ in an appropriate weighted space of continuous functions.
Preliminaries
=============
In this section, we list some definitions and lemmas useful throughout the paper.
\[d1\][@kst] The space $X_{c}^{p}(a,b)\,(c\in\mathbb{R},p\geq1)$ consists of those real-valued Lebesgue measurable functions $g$ on $(a,b)$ for which ${\|g\|}_{X_{c}^{p}}<\infty,$ where $$\begin{gathered}
{\|g\|}_{X_{c}^{p}}={\bigg(\int_{a}^{b}{|t^cg(t)|}^{p}\frac{dt}{t}\bigg)}^{\frac{1}{p}},\quad p\geq1,\,\,c\in\mathbb{R},\\
{\|g\|}_{X_{c}^{p=\infty}}=\text{ess sup}_{a\leq t\leq b}|t^cg(t)|,\quad c\in\mathbb{R}.\end{gathered}$$ In particular, when $c=\frac{1}{p},$ we see that $X_{{1}/{p}}^{c}(a,b)=L_p(a,b).$
\[d2\][@oo] Let $\Omega=[a,b]\,\,(0<a<b<\infty)$ be a finite interval on $\mathbb{R}^{+}$ and $\rho>0.$ Denote by $C[a,b]$ a space of continuous functions $g$ on $\Omega$ with the norm $$\begin{gathered}
{\|g\|}_{C}=\max_{t\in\Omega}|g(t)|.\end{gathered}$$ The weighted space $C_{\gamma,\rho}[a,b]$ of functions $g$ on $(a,b]$ is defined by $$\begin{gathered}
\label{space}
C_{\gamma,\rho}[a,b]=\{g:(a,b]\to\mathbb{R}:{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\gamma}g(t)\in{C[a,b]}\},\quad0\leq\gamma<1\end{gathered}$$ with the norm $$\begin{gathered}
{\|g\|}_{C_{\gamma,\rho}}={\bigg\|{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\gamma}g(t)\bigg\|}_{C}=\max_{t\in\Omega}\bigg|{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\gamma}g(t)\bigg|,\end{gathered}$$ and $C_{0,\rho}[a,b]=C[a,b].$
\[d3\][@oo] Let $\delta_\rho=\big(t^{\rho-1}\frac{d}{dt}\big),\,\Omega=[a,b]\,(0<a<b<\infty),\rho>0$ and $0\leq\gamma<1.$ Denote $C_{\delta_\rho,\gamma}^{n}[a,b]$ the Banach space of functions $g$ which are continuously differentiable, with $\delta_\rho,$ on $[a,b]$ upto $(n-1)$ order and have the derivative $\delta_\rho^ng$ on $(a,b]$ such that $\delta_\rho^ng\in{C_{\gamma,\rho}[a,b]}:$ $$\begin{gathered}
C_{\delta_\rho,\gamma}^{n}[a,b]=\big\{\delta_\rho^kg\in{C[a,b]}, k=0,1,\cdots,n-1,\,\,\delta_\rho^ng\in{C_{\gamma,\rho}[a,b]}\big\},\quad n\in\mathbb{N},\end{gathered}$$ with the norm $$\begin{gathered}
{\|g\|}_{C_{\delta_\rho,\gamma}^{n}}=\sum_{k=0}^{n-1}{\|\delta_\rho^kg\|}_{C}+{\|\delta_\rho^ng\|}_{C_{\gamma,\rho}},\quad
{\|g\|}_{C_{\delta_\rho}^{n}}=\sum_{k=0}^{n}\max_{t\in\Omega}|\delta_\rho^kg(t)|.\end{gathered}$$ Note that, for $n=0,$ we have $C_{\delta_\rho,\gamma}^{0}[a,b]=C_{\gamma,\rho}[a,b].$
\[d4\][@ki] Let $\alpha>0$ and $h\in{X_{c}^{p}(a,b)},$ where $X_{c}^{p}$ is as in Definition \[d1\]. The left-sided Katugampola fractional integral $^{\rho}I_{a+}^{\alpha}$ of order $\alpha$ is defined by $$\begin{gathered}
\label{kil}
^{\rho}I_{a+}^{\alpha}h(t)=\int_{a}^{t}s^{\rho-1}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{\alpha-1}\frac{h(s)}{\Gamma(\alpha)}ds,\quad t>a.\end{gathered}$$
\[d5\][@ki] Let $\alpha>0$ and $h\in{X_{c}^{p}(a,b)},$ where $X_{c}^{p}$ is as in Definition \[d1\]. The right-sided Katugampola fractional integral $^{\rho}I_{b-}^{\alpha}$ of order $\alpha$ is defined by $$\begin{gathered}
\label{kir}
^{\rho}I_{b-}^{\alpha}h(t)=\int_{t}^{b}s^{\rho-1}{\bigg(\frac{s^\rho-t^\rho}{\rho}\bigg)}^{\alpha-1}\frac{h(s)}{\Gamma(\alpha)}ds,\quad t<b.\end{gathered}$$
\[d6\][@kd] Let $\alpha\in{\mathbb{R}^{+}{\setminus}\mathbb{N}}$ and $n=[\alpha]+1,$ where $[\alpha]$ is integer part of $\alpha$ and $\rho>0.$ The left-sided Katugampola fractional derivative $^{\rho}D_{a+}^{\alpha}$ is defined by $$\begin{aligned}
\label{kdl}
^{\rho}D_{a+}^{\alpha}h(t)={\bigg(t^{\rho-1}\frac{d}{dt}\bigg)}^{n}\int_{a}^{t}s^{\rho-1}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{n-\alpha-1}\frac{h(s)}{\Gamma(n-\alpha)}ds.\end{aligned}$$
\[d7\][@kd] Let $\alpha\in{\mathbb{R}^{+}\setminus\mathbb{N}}$ and $n=[\alpha]+1,$ where $[\alpha]$ is integer part of $\alpha$ and $\rho>0.$ The right-sided Katugampola fractional derivative $^{\rho}D_{b-}^{\alpha}$ is defined by $$\begin{aligned}
\label{kdr}
^{\rho}D_{b-}^{\alpha}h(t)={\bigg(-t^{\rho-1}\frac{d}{dt}\bigg)}^{n}\int_{t}^{b}s^{\rho-1}{\bigg(\frac{s^\rho-t^\rho}{\rho}\bigg)}^{n-\alpha-1}\frac{h(s)}{\Gamma(n-\alpha)}ds.\end{aligned}$$
\[l1\][@ki] Suppose that $\alpha>0,\beta>0,p\geq1,0<a<b<\infty$ and $\rho,c\in\mathbb{R}$ such that $\rho\geq{c}.$ Then, for $h\in{X_{c}^{p}(a,b)},$ the semigroup property of Katugampola integral is valid. This is $$\begin{gathered}
\label{ski}
^{\rho}I_{a+}^{\alpha}{^{\rho}I_{a+}^{\beta}h(t)}={^{\rho}I_{a+}^{\alpha+\beta}h(t)}.\end{gathered}$$
A similar property for right-sided operator also holds.
\[l2\][@kd] Suppose that ${^{\rho}I_{a+}^{\alpha}}$ and $^{\rho}D_{a+}^{\alpha}$ are as defined in Definitions \[d4\] and \[d6\], respectively. Then,
(i)
: ${^{\rho}I_{a+}^{\alpha}}{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\sigma-1}=\frac{\Gamma(\sigma)}{\Gamma(\sigma+\alpha)}{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\alpha+\sigma-1},\qquad\alpha\geq0,\sigma>0, t>a.$
(ii)
: ${^{\rho}D_{a+}^{\alpha}}{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\alpha-1}=0,\qquad0<\alpha<1, t>a.$
\[l3\][@oo] Let $0<a<b<\infty,\alpha>0$ and $0\leq\gamma<1.$\
(i) If $0<\alpha<\gamma,$ then $^{\rho}I_{a+}^{\alpha}$ is bounded from $C_{\gamma,\rho}[a,b]$ into $C_{\gamma,\rho}[a,b].$\
(ii) If $\gamma\leq\alpha$, then $^{\rho}I_{a+}^{\alpha}$ is bounded from $C_{\gamma,\rho}[a,b]$ into $C[a,b].$\
In particular, $^{\rho}I_{a+}^{\alpha}$ is bounded in $C_{\gamma,\rho}[a,b].$
\[l4\] Let $0\leq\alpha<1$ and $0\leq\gamma<1.$ If $h\in{C_{\gamma,\rho}^1[a,b]},$ then the fractional derivatives $^{\rho}D_{a+}^{\alpha}$ and $^{\rho}D_{b-}^{\alpha}$ exist on $(a,b]$ and $[a,b),$ respectively $(a>0)$ and are represented in the forms: $$\begin{gathered}
\label{kd1}
^{\rho}D_{a+}^{\alpha}h(t)=\frac{h(a)}{\Gamma(1-\alpha)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{-\alpha}+\int_{a}^{t}s^{\rho-1}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{-\alpha}\frac{h^{'}(s)}{\Gamma(1-\alpha)}ds,\\
^{\rho}D_{b-}^{\alpha}h(t)=\frac{h(b)}{\Gamma(1-\alpha)}{\bigg(\frac{b^\rho-t^\rho}{\rho}\bigg)}^{-\alpha}-\int_{t}^{b}s^{\rho-1}{\bigg(\frac{s^\rho-t^\rho}{\rho}\bigg)}^{-\alpha}\frac{h^{'}(s)}{\Gamma(1-\alpha)}ds.\end{gathered}$$\[kd2\]
\[l5\]\[Fractional integration by parts\] If $\alpha>0,p\geq1,0\leq a<b\leq\infty$ and $\rho,c\in\mathbb{R}$ be such that $\rho\geq{c},$ then, for $g,h\in{X_{c}^{p}(a,b)},$ the following relation hold: $$\begin{gathered}
\label{fip}
\int_{a}^{b}t^{\rho-1}g(t){(^{\rho}I_{a+}^{\alpha}h)}(t)dt=\int_{a}^{b}t^{\rho-1}h(t){(^{\rho}I_{b-}^{\alpha}g)}(t)dt.\end{gathered}$$
The proof is straightforward. Using Dirichlet formula, we obtain $$\begin{aligned}
\int_{a}^{b}t^{\rho-1}g(t){(^{\rho}I_{a+}^{\alpha}h)(t)}dt&=\int_{a}^{b}t^{\rho-1}g(t)\int_{a}^{t}s^{\rho-1}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{\alpha-1}\frac{h(s)}{\Gamma(\alpha)}{ds}~{dt}\\
&=\int_{a}^{b}s^{\rho-1}h(s)\int_{s}^{b}t^{\rho-1}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{\alpha-1}\frac{g(t)}{\Gamma(\alpha)}{dt}~{ds}\\
&=\int_{a}^{b}t^{\rho-1}h(t){(^{\rho}I_{b-}^{\alpha}g)}(t)dt.\end{aligned}$$
\[d8\][@oo] The generalized Katugampola fractional derivatives, $^{\rho}D_{a+}^{\alpha,\beta}$ (left-sided) and $^{\rho}D_{b-}^{\alpha,\beta}$ (right-sided), of order $0<\alpha<1$ and type $0\leq\beta\leq1$ are respectively defined by $$\begin{gathered}
\label{gkl}
{(^{\rho}D_{a+}^{\alpha,\beta}h)}(t)={({^{\rho}I_{a+}^{\beta(1-\alpha)}}\delta_\rho{{^{\rho}I_{a+}^{(1-\beta)(1-\alpha)}}}h)}(t),\end{gathered}$$ $$\begin{gathered}
\label{gkr}
{(^{\rho}D_{b-}^{\alpha,\beta}h)}(t)={(-{^{\rho}I_{b-}^{\beta(1-\alpha)}}\delta_\rho{{^{\rho}I_{b-}^{(1-\beta)(1-\alpha)}}}h)}(t)\end{gathered}$$ for the function to which right-hand side expressions exist and $\rho>0.$
\[r1\] For $0<\alpha<1,0\leq\beta\leq1,$ the generalized Katugampola fractional derivative ${^{\rho}D_{a+}^{\alpha,\beta}}$ can be written in terms of Katugampola fractional derivative as $$\begin{gathered}
{^{\rho}D_{a+}^{\alpha,\beta}}={^{\rho}I_{a+}^{\beta(1-\alpha)}}\delta_\rho{^{\rho}I_{a+}^{1-\gamma}}={^{\rho}I_{a+}^{\beta(1-\alpha)}}~~{^{\rho}D_{a+}^{\gamma}},\qquad \gamma=\alpha+\beta(1-\alpha).\end{gathered}$$
[@ki]\[l6\] Let $\alpha>0,0<a<b<\infty,0\leq\gamma<1$ and $g\in{C_{\gamma,\rho}[a,b]}.$ If $\alpha>\gamma,$ then $$\begin{gathered}
{(^{\rho}I_{a+}^{\alpha}g)}(a)=\lim_{x\to{a+}}{(^{\rho}I_{a+}^{\alpha}g)}(t)=0, \\
{(^{\rho}I_{b-}^{\alpha}g)}(b)=\lim_{x\to{b-}}{(^{\rho}I_{b-}^{\alpha}g)}(t)=0.\end{gathered}$$
\[yi\]\[Young’s inequality\] If $\varphi$ and $\eta$ are non-negative real numbers and $m$ and $m'$ are positive real numbers such that $\frac{1}{m}+\frac{1}{m'}=1,$ then we have $$\begin{gathered}
\varphi\eta\leq\frac{\varphi^m}{m}+\frac{\eta^{m'}}{m'}.\end{gathered}$$
Non-existence result
====================
The proof of following theorem is based on the test function method developed by Mitidieri and Pokhazhaev in [@mp] and recently used in [@neh; @nehh; @newnec; @newne; @oldnex].
Assume that $\mu\in\mathbb{R}$ and $1<m<\frac{1+\mu}{1-\alpha},~\mu>-\alpha.$ Then, Problem does not admit global non-trivial solutions in $C_{1-\gamma,\rho}^{\gamma}[a,b],$ when $x_a>0$.
On the contrary, assume that a non-trivial solution exists for all time $t>a.$ Let $\phi\in{C^1([a,\infty))}$ be a test function satisfying $\phi\geq0$ and non-increasing such that $$\begin{gathered}
\label{1}
\phi(t)=\begin{cases}
1, & a\leq t\leq\theta{T}, \\
0, & t\geq{T},
\end{cases}\end{gathered}$$ for some $T>a$ and some $\theta\leq\frac{1}{2}$ such that $a<\theta{T}<T$. Multiplying the inequality in by $\phi(t)$ and integrating over $[a,T],$ we obtain $$\begin{gathered}
\label{2}
\int_{a}^{T}\phi(t){(^{\rho}D_{a+}^{\alpha,\beta}x)}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ Observe that the integral in left-hand side exists and the one in the right-hand side exists for $m<\frac{1+\mu}{1-\alpha}$ when $x\in{C_{1-\gamma,\rho}^{\gamma}[a,b]}.$ Moreover, from Definition \[d8\], we can write as $$\begin{gathered}
\int_{a}^{T}\phi(t){({^{\rho}I_{a+}^{\beta(1-\alpha)}}\delta_\rho{^{\rho}I_{a+}^{1-\gamma}}x)}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt\end{gathered}$$ $$\begin{gathered}
\label{3}
\int_{a}^{T}t^{\rho-1}\phi(t){({^{\rho}I_{a+}^{\beta(1-\alpha)}}\delta_\rho{^{\rho}I_{a+}^{1-\gamma}}x)}(t)t^{1-\rho}dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ In accordance with Lemma \[l5\] (after extending by zero outside $[a,T]$), from we may deduce that $$\begin{gathered}
\label{4}
\int_{a}^{T}t^{\rho-1}{(\delta_\rho{^{\rho}I_{a+}^{1-\gamma}}x)}(t){({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(t)t^{1-\rho}dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ An integration by parts in yields $$\begin{aligned}
\label{5}
[{(^{\rho}I_{a+}^{1-\gamma}x)}(t){({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(t)]{\big|}_{t=a}^{T}-\int_{a}^{T}&{({^{\rho}I_{a+}^{1-\gamma}}x)}(t)\frac{d}{dt}{({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(t)dt\nonumber\\
&\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{aligned}$$ Using ${({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(T)=0$ (Lemma \[l6\]) and initial condition ${({^{\rho}I_{a+}^{1-\gamma}}x)}(a+)=x_a,$ we obtain $$\begin{aligned}
\label{6}
-x_a{({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(a)&-\int_{a}^{T}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t)\frac{d}{dt}{({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(t)dt\nonumber\\
&~~~\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{aligned}$$ Multiplying by $\frac{t^{\rho-1}}{t^{\rho-1}}$ inside the integral in the left-hand side of expression , we see that $$\begin{gathered}
\label{7}
{\hspace{-0.2cm}}L=\int_{a}^{T}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t)(-\delta_\rho){({^{\rho}I_{T-}^{\beta(1-\alpha)}}\phi)}(t)\frac{dt}{t^{\rho-1}}\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ From Definition \[d7\] (for $n=1$), we have $$\begin{gathered}
\label{8}
L=\int_{a}^{T}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t){({^{\rho}D_{T-}^{1-\beta(1-\alpha)}}\phi)}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ and from Lemma \[l4\], we see that $$\begin{aligned}
L=\int_{a}^{T}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t)&\bigg[\frac{\phi(T)}{\Gamma(\beta(1-\alpha))}{\bigg(\frac{T^\rho-a^\rho}{\rho}\bigg)}^{\beta(1-\alpha)}\\
&-\int_{t}^{T}s^{\rho-1}{\bigg(\frac{s^\rho-t^\rho}{\rho}\bigg)}^{\beta(1-\alpha)-1}\frac{\phi^{'}(s)}{\Gamma(\beta(1-\alpha))}ds\bigg]dt.\end{aligned}$$ Since $\phi(T)=0$ we get $$\begin{gathered}
\label{9}
L=-\int_{a}^{T}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t){({^{\rho}I_{T-}^{\beta(1-\alpha)}}\delta_\rho\phi)}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ Therefore $$\begin{gathered}
L=-\int_{a}^{T}t^{\rho-1}{({^{\rho}I_{a+}^{1-\gamma}}x)}(t){({^{\rho}I_{T-}^{\beta(1-\alpha)}}\delta_\rho\phi)}(t)\frac{dt}{t^{\rho-1}}\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ Lemma \[l5\] again allows us to write $$\begin{gathered}
L=-\int_{a}^{T}\delta_\rho\phi(t){({^{\rho}I_{a+}^{\beta(1-\alpha)}}{({^{\rho}I_{a+}^{1-\gamma}}x)})}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt,\end{gathered}$$ and by Lemma \[l1\] we obtain $$\begin{gathered}
\label{10}
L=-\int_{a}^{T}\delta_\rho\phi(t){{({^{\rho}I_{a+}^{1-\alpha}}x)}}(t)dt\geq\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{|x(t)|}^m\phi(t)dt.\end{gathered}$$ On the other hand $$\begin{aligned}
\label{11}
\int_{a}^{T}\delta_\rho\phi(t){{({^{\rho}I_{a+}^{1-\alpha}}x)}}(t)dt&=\int_{a}^{T}\delta_\rho\phi(t)\int_{a}^{t}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{-\alpha}\frac{s^{\rho-1}x(s)}{\Gamma(1-\alpha)}ds~dt\nonumber\\
&\leq\int_{a}^{T}|\delta_\rho\phi(t)|\int_{a}^{t}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{-\alpha}\frac{s^{\rho-1}|x(s)|}{\Gamma(1-\alpha)}ds~dt.\end{aligned}$$ As $\phi$ is non-increasing, we have $\phi(s)\geq\phi(t)$ for all $t\geq{s}$ and $\frac{1}{\phi^{{1}/{m}}(s)}\leq\frac{1}{\phi^{{1}/{m}}(t)}$ for $m>1.$ Also it is clear that $\phi^{'}(t)=0,\,\,t\in[a,\theta{T}].$ Therefore $$\begin{aligned}
\label{12}
L&\leq{\int_{a}^{T}|\delta_\rho\phi(t)|\int_{a}^{t}\frac{s^{\rho-1}}{\Gamma(1-\alpha)}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{-\alpha}\frac{|x(s)|{\phi^{{1}/{m}}(s)}}{{\phi^{{1}/{m}}(s)}}ds~dt}\nonumber\\
&\leq{\int_{\theta{T}}^{T}\frac{|\delta_\rho\phi(t)|}{\phi^{{1}/{m}}(t)}{\int_{a}^{t}\frac{s^{\rho-1}}{\Gamma(1-\alpha)}{\bigg(\frac{t^\rho-s^\rho}{\rho}\bigg)}^{-\alpha}{|x(s)|{\phi^{{1}/{m}}(s)}}}ds~dt}.\end{aligned}$$ Definition \[d4\] allows us to write $$\begin{gathered}
\label{13}
L\leq\int_{\theta{T}}^{T}t^{\rho-1}\frac{|\delta_\rho\phi(t)|}{\phi^{{1}/{m}}(t)}{\big({^{\rho}I_{a+}^{1-\alpha}{|x|{\phi^{{1}/{m}}}}}\big)}(t)\frac{dt}{t^{\rho-1}}.\end{gathered}$$ Moreover, it is easy to see that $\frac{\delta_\rho\phi(t)}{\phi^{{1}/{m}}(t)}\in{L_p},$ for otherwise, we consider $\phi^{\lambda}(t)$ with some sufficiently large $\lambda.$ Thus, we can apply Lemma \[l5\] to obtain $$\begin{gathered}
\label{14}
L\leq\int_{\theta{T}}^{T}{|x(t)|{\phi^{{1}/{m}}}}(t){\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}(t)dt.\end{gathered}$$ Next, we multiply by ${\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\frac{\mu}{m}}{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\frac{-\mu}{m}}$ inside the integral in the right-hand side of : $$\begin{gathered}
\label{15}
L\leq\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}(t){|x(t)|{\phi^{{1}/{m}}}}(t)\frac{{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\frac{\mu}{m}}}{{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\frac{\mu}{m}}}dt.\end{gathered}$$ For $-\alpha<\mu<0,$ we have ${\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}}<{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}}$ because $t<T$ and for $\mu>0$ we obtain ${\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}}<{\theta}^{\frac{\mu}{m}}{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}}$ because $t>\theta T.$ It follows that $$\begin{gathered}
{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{-\frac{\mu}{m}}<\max{\{1,{\theta}^{{\mu}/{m}}\}}{\bigg(\frac{T^\rho-a^\rho}{\rho}\bigg)}^{-\frac{\mu}{m}}.\end{gathered}$$ Therefore, $$\begin{gathered}
\label{16}
L\leq\max{\{1,{\theta}^{{\mu}/{m}}\}}{\bigg(\frac{T^\rho-a^\rho}{\rho}\bigg)}^{-\frac{\mu}{m}}\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}(t){{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\mu}{m}}}{|x(t)|{\phi^{{1}/{m}}}}(t)dt.\end{gathered}$$ A simple application of Young’s inequality (Theorem \[yi\]) with $m$ and $m'$ such that $\frac{1}{m}+\frac{1}{m'}=1$ in the right-hand side of yields $$\begin{aligned}
\label{17}
L&\leq\frac{1}{m}\int_{\theta{T}}^{T}{\bigg({|x(t)|{\phi^{{1}/{m}}}}(t){{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\mu}{m}}}\bigg)}^{m}(t)dt\nonumber\\
&\hspace{0.3cm}+\frac{{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}}{m'}
\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}^{m'}(t)dt\nonumber\\
&\leq\frac{1}{m}\int_{a}^{T}{|x(t)|}^{m}{\phi}(t){{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{{\mu}}}dt\nonumber\\
&\hspace{0.3cm}+\frac{{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}}{m'}
\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}^{m'}(t)dt.\end{aligned}$$ Clearly, from and , we see that $$\begin{aligned}
\label{18}
\frac{{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}}{m'}
&\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}^{m'}(t)dt\nonumber\\
\geq{\bigg(1-\frac{1}{m}\bigg)}&\int_{a}^{T}{|x(t)|}^{m}{\phi}(t){{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{{\mu}}}dt\end{aligned}$$ or $$\begin{aligned}
\label{19}
{\bigg(\frac{1}{m'}\bigg)}\int_{a}^{T}{|x(t)|}^{m}{\phi}(t){{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{{\mu}}}dt&\leq\frac{{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}{(\max{\{1,{\theta}^{\frac{\mu}{m}}\}})}^{m'}}{m'}\nonumber\\
&~~~~\times\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}\frac{|\delta_\rho\phi|}{\phi^{{1}/{m}}}\bigg)}^{m'}(t)dt.\end{aligned}$$ Therefore by Definition \[d5\], we have $$\begin{aligned}
\label{20}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}&{|x(t)|}^{m}dt
\leq\frac{{\big(\frac{{(\theta{T})}^{\rho}-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}}{{(\Gamma(1-\alpha))}^{m'}}{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}\nonumber\\
&\times\int_{{\theta}T}^{T}{\bigg(\int_{t}^{T}{\bigg(\frac{s^\rho-t^\rho}{\rho}\bigg)}^{-\alpha}s^{\rho-1}{\frac{|\delta_\rho\phi(s)|}{\phi^{1/m}(s)}}ds\bigg)}^{m'}(t)dt\end{aligned}$$ The change of variable $t=\sigma{T}$ in the right-hand side integral yields $$\begin{aligned}
\label{21}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}&{\phi(t)}{|x(t)|}^{m}dt
\leq\frac{{\big(\frac{{(\theta{T})}^{\rho}-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}}{{(\Gamma(1-\alpha))}^{m'}}{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}\nonumber\\
\times\int_{{\theta}}^{1}&{\bigg(\int_{\sigma{T}}^{T}{\bigg(\frac{s^\rho-{(\sigma{T})}^\rho}{\rho}\bigg)}^{-\alpha}s^{\rho-1}{\frac{|\delta_\rho\phi(s)|}{\phi^{1/m}(s)}}ds\bigg)}^{m'}(\sigma)Td\sigma.\end{aligned}$$ Another change of variable $s=rT$ therein yields $$\begin{aligned}
\label{22}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}&{\phi(t)}{|x(t)|}^{m}dt
\leq\frac{{\big(\frac{{(\theta{T})}^{\rho}-a^\rho}{\rho}\big)}^{\frac{-\mu{m'}}{m}}}{{(\Gamma(1-\alpha))}^{m'}}{(\max{\{1,{\theta}^{\frac{\mu}{m}}\}})}^{m'}\nonumber\\
\times\int_{{\theta}}^{1}&{\bigg(\int_{\sigma}^{T}{\bigg(\frac{{(rT)}^\rho-{(\sigma{T})}^\rho}{\rho}\bigg)}^{-\alpha}{(rT)}^{\rho-1}{\frac{|\delta_\rho\phi(r)|}{\phi^{1/m}(r)}}dr\bigg)}^{m'}(\sigma)Td\sigma.\end{aligned}$$ We may assume that the integral in the right-hand side of is bounded, i.e. $$\begin{gathered}
\label{23}
\frac{{(\max{\{1,{\theta}^{{\mu}/{m}}\}})}^{m'}}{{(\Gamma(1-\alpha))}^{m'}}\int_{{\theta}}^{1}{\bigg(\int_{\sigma}^{T}{\bigg(\frac{{r}^\rho-{\sigma}^\rho}{\rho}\bigg)}^{-\alpha}{r}^{\rho-1}{\frac{|\delta_\rho\phi(r)|}{\phi^{1/m}(r)}}dr\bigg)}^{m'}(\sigma)d\sigma\leq{C},\end{gathered}$$ for some positive constant $C$, for otherwise we consider $\phi^{\lambda}(r)$ with some sufficiently large $\lambda$. Therefore $$\begin{gathered}
\label{24}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}{|x(t)|}^{m}dt
\leq{C}{\bigg(\frac{{(\theta{T})}^{\rho}-a^\rho}{\rho}\bigg)}^{\frac{-\mu{m'}}{m}}.\end{gathered}$$ If $\mu>0,$ then $$\begin{gathered}
{\bigg(\frac{{(\theta{T})}^{\rho}-a^\rho}{\rho}\bigg)}^{\frac{-\mu{m'}}{m}}\to0\end{gathered}$$ as $T\to\infty.$ Finally, from , we obtain $$\begin{gathered}
\label{25}
\lim_{T\to\infty}\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}{|x(t)|}^{m}dt=0.\end{gathered}$$ This is a contradiction since the solution is assumed to be nontrivial.In case $\mu=0$ we have $-\frac{\mu{m'}}{m}=0$ as the relation ensures that $$\begin{gathered}
\label{26}
\lim_{T\to\infty}\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}{|x(t)|}^{m}dt\leq{C}.\end{gathered}$$ Moreover, it is clear that $$\begin{aligned}
\label{27}
{\bigg(\frac{{(\theta{T})}^\rho-a^\rho}{\rho}\bigg)}^{-\frac{\mu}{m}}
\int_{\theta{T}}^{T}&{\bigg({^{\rho}I_{T-}^{1-\alpha}}{\frac{|\delta_\rho\phi|}{\phi^{1/m}}}\bigg)}(t){\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\mu}{m}}{|x(t)|\phi^{1/m}(t)}dt\nonumber\\
&\leq{\bigg(\frac{{(\theta{T})}^\rho-a^\rho}{\rho}\bigg)}^{-\frac{\mu}{m}}{\bigg[\int_{\theta{T}}^{T}{\bigg({^{\rho}I_{T-}^{1-\alpha}}{\frac{|\delta_\rho\phi|}{\phi^{1/m}}}\bigg)}(t)dt\bigg]}^{\frac{1}{m'}}\nonumber\\
&\hspace{1cm}\times{\bigg[\int_{\theta{T}}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\mu}{m}}{|x(t)|\phi^{1/m}(t)}dt\bigg]}^{\frac{1}{m}}.\end{aligned}$$ This relation together with implies that $$\begin{gathered}
\label{28}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^\mu\phi(t){|x(t)|}^mdt\leq{K}{\bigg[\int_{\theta{T}}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\mu}{m}}{|x(t)|\phi^{1/m}(t)}dt\bigg]}^{\frac{1}{m}}\end{gathered}$$ for some positive constant $K$, with $$\begin{gathered}
\label{29}
\lim_{T\to\infty}\int_{\theta{T}}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}{|x(t)|}^{m}dt=0\end{gathered}$$ due to the convergence of the integral in . This is again contradiction.
If $\mu<0,$ we have ${\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}}\leq{\big(\frac{T^\rho-a^\rho}{\rho}\big)}^{-\frac{\mu}{m}},$ because $-\frac{\mu}{m}>0$ and $t<T$. Similar to previous one, the expression $\frac{|\phi^{'}(r)|}{\phi^{1/m}(r)}$ may be assumed bounded and hence it can be shown that $$\begin{gathered}
\label{30}
\int_{a}^{T}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}{\phi(t)}{|x(t)|}^{m}dt\leq{C}{\bigg(\frac{T^\rho-a^\rho}{\rho}\bigg)}^{-m'-\mu{m'/m}}\end{gathered}$$ for some positive constant $C.$ This completes the proof.
Illustrations
=============
In this section, we prove the sharpness of exponent $\frac{\mu+1}{1-\alpha}$ i.e. the solutions exist for exponents strictly bigger than $\frac{\mu+1}{1-\alpha}.$ We need the following Lemma.
\[l7\] Suppose that $0<\alpha<1,0\leq\beta\leq1$ and $\rho>0,\xi>0.$ Then, the following identity holds for generalized Katugampola fractional derivative ${(^{\rho}D_{a+}^{\alpha,\beta})}:$ $$\begin{gathered}
\bigg({^{\rho}D_{a+}^{\alpha,\beta}}{\bigg(\frac{s^\rho-a^\rho}{\rho}\bigg)}^{\xi-1}\bigg)(t)=\frac{\Gamma(\xi)}{\Gamma(\xi-\alpha)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\xi-\alpha-1},\quad t>a>0.\end{gathered}$$
From Lemma \[l2\], for $\gamma=\alpha+\beta-\alpha\beta$ we see that $$\begin{gathered}
\label{31}
\bigg({^{\rho}I_{a+}^{1-\gamma}}{\bigg(\frac{s^\rho-a^\rho}{\rho}\bigg)}^{\xi-1}\bigg)(t)=\frac{\Gamma(\xi)}{\Gamma(\xi-\gamma+1)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\xi-\gamma}.\end{gathered}$$ Therefore $$\begin{gathered}
\label{32}
\delta_\rho\bigg({^{\rho}I_{a+}^{1-\gamma}}{\bigg(\frac{s^\rho-a^\rho}{\rho}\bigg)}^{\xi-1}\bigg)(t)=
\frac{\Gamma(\xi)}{\Gamma(\xi-\gamma)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\xi-\gamma-1}\end{gathered}$$ which yields $$\begin{gathered}
\label{33}
\bigg({^{\rho}D_{a+}^{\gamma}}{\bigg(\frac{s^\rho-a^\rho}{\rho}\bigg)}^{\xi-1}\bigg)(t)
=\frac{\Gamma(\xi)}{\Gamma(\xi-\gamma)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\xi-\gamma-1}\end{gathered}$$ Applying ${^{\rho}I_{a+}^{\beta(1-\alpha)}}$ on both sides of and using Lemma \[l2\], again in the light of Definition \[d6\] we conclude the Lemma.
Consider the following differential equation involving generalized Katugampola fractional derivative of order $0<\alpha<1$ and type $0\leq\beta\leq1:$ $$\begin{gathered}
\label{34}
{(^{\rho}D_{a+}^{\alpha,\beta}g)}(t)=\lambda{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}[g(t)]^{m},\quad t>a>0,m>1,\end{gathered}$$ with $\lambda,\mu\in\mathbb{R^+},\lambda\neq0,\rho>0.$
Look for the solution of the form $g(t)=c{\big(\frac{t^\rho-a^\rho}{\rho}\big)}^{\nu}$ for some $\nu\in\mathbb{R}.$Our aim is to find the values of $c$ and $\nu.$ By using Lemma \[l7\], we have $$\begin{gathered}
\label{35}
{\bigg({^{\rho}D_{a+}^{\alpha,\beta}\bigg[c{\bigg(\frac{s^\rho-a^\rho}{\rho}\bigg)}^{\nu}\bigg]}\bigg)}(t)=\frac{c\Gamma(\nu+1)}{\Gamma(\nu-\alpha+1)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\nu-\alpha}.\end{gathered}$$ Therefore, plugging this expression in , we get $$\begin{gathered}
\label{36}
\frac{c\Gamma(\nu+1)}{\Gamma(\nu-\alpha+1)}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\nu-\alpha}=\lambda{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\mu}
{\bigg[c{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\nu}\bigg]}^{m}.\end{gathered}$$ We obtain $\nu=\frac{\alpha+\mu}{1-m}$ and $c=\frac{\Gamma(\frac{\alpha+\mu}{1-m}+1)}{\lambda{\Gamma(\frac{m\alpha+\mu}{1-m}+1)}}.$ If $\frac{m\alpha+\mu}{1-m}>-1$ means $m>(\frac{1+\mu}{1-\alpha}),$ then has the exact solution: $$\begin{gathered}
\label{37}
y(t)={\bigg[\frac{\Gamma(\frac{\alpha+\mu}{1-m}+1)}{\lambda{\Gamma(\frac{m\alpha+\mu}{1-m}+1)}}\bigg]}^{\frac{1}{m-1}}{\bigg(\frac{t^\rho-a^\rho}{\rho}\bigg)}^{\frac{\alpha+\mu}{1-m}}.\end{gathered}$$ This solution satisfies the initial condition with $$\begin{gathered}
x_a={\bigg[\frac{\Gamma(\frac{\alpha+\mu}{1-m}+1)}{\lambda{\Gamma(\frac{m\alpha+\mu}{1-m}+1)}}\bigg]}^{\frac{1}{m-1}},\,\,\text{when}\,\,\frac{\alpha+\mu}{1-m}\geq\gamma-1>-1.\end{gathered}$$
From the above non-existence theorem, we observe that no solutions can exists for certain values of $m$ and the exponent $\mu$ for all time $t>a.$ Clearly, we determine the range of values of $m$ for which solutions do not exists globally. Also note that, sufficient conditions for non-existence of solutions provide the necessary conditions for existence of solutions.
Conclusions
===========
We successfully employed the fractional integration by parts and the method of test functions to obtain the necessary conditions for non-existence of global solutions for a wide class of fractional differential equations. The obtained results are well illustrated with suitable examples. Result discussed in this paper necessarily generalized / improved the existing results in literature.
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abstract: 'A precessing source frame, constructed using the Newtonian orbital angular momentum ${\boldsymbol{L}}_{\rm N}$, can be invoked to model inspiral gravitational waves from generic spinning compact binaries. An attractive feature of such a precessing convention is its ability to remove all spin precession induced modulations from the orbital phase evolution. However, this convention usually employs a post-Newtonian (PN) accurate precessional equation, appropriate for the PN accurate orbital angular momentum ${\boldsymbol{L}}$, to evolve the ${\boldsymbol{L}}_{\rm N}$-based precessing source frame. This influenced us to develop inspiral waveforms for spinning compact binaries in a precessing convention that explicitly employ ${\boldsymbol{L}}$ to describe the binary orbits. Our approach introduces certain additional 3PN order terms in the evolution equations for the orbital phase and frequency with respect to the usual ${\boldsymbol{L}}_{\rm N}$-based implementation of the precessing convention. We examine the practical implications of these additional terms by computing the match between inspiral waveforms that employ ${\boldsymbol{L}}$ and ${\boldsymbol{L}}_{\rm N}$-based precessing conventions. The match estimates are found to be smaller than the optimal value, namely $0.97 $, for a non-negligible fraction of unequal mass spinning compact binaries.'
address: |
$^1$Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India\
$^2$Inter University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune 411007, India
author:
- 'A Gupta$^{1,2}$ and A Gopakumar$^{1}$'
title: 'Post-Newtonian analysis of precessing convention for spinning compact binaries'
---
Introduction
============
Inspiralling compact binaries containing spinning neutron stars and (or) black holes (BHs) are key sources for the network of second generation interferometric gravitational wave (GW) detectors [@SS_lr]. These instruments include the two advanced LIGO (aLIGO) observatories [@Harry10], the advanced Virgo [@virgo2], the KAGRA [@KS11], the GEO-HF [@GEO2] and the planned LIGO-India [@LIGO_I_Unni]. In contrast, massive spinning BH binaries are one of the most exciting sources for the space-based GW observatory like the planned eLISA [@eLISA]. GWs from such inspiralling compact binaries, whose components are specified by their masses and spins, can be accurately modeled using perturbative approaches to tackle the underlying Einstein field equations [@LR_LB]. Therefore, the optimal detection technique of [*matched filtering*]{} can be employed to detect and characterize inspiral GWs from such binaries [@JK_LRR; @Vitale14]. This involves cross correlating the interferometric output data with a bank of templates that theoretically model inspiral GWs from spinning compact binaries. A successful detection demands that at least one template should remain in phase as much as possible with the buried weak GW signals in the frequency windows of various GW observatories [@DIS98].
During the GW emission induced inspiral, dynamics and associated GWs from compact binaries can be accurately described using the post-Newtonian (PN) approximation to general relativity [@LR_LB; @Will]. The PN description provides interesting quantities, required for various template constructions, as an asymptotic series in terms of certain dimensionless parameter. For binaries in quasi-circular orbits, it is usual to use $x=(G\, m\, \omega/c^3)^{2/3}$ as the PN expansion parameter while constructing inspiral templates, where $m$ and $\omega$ stand for the total mass and orbital (angular) frequency of the binary [@BDI; @Boyle_07]. Currently, GW frequency and associated phase evolution, crucial inputs to construct various inspiral template families, are known to 3.5PN order for non-spinning compact binaries [@BFIJ; @BDFI]. In other words, PN corrections to the above two quantities are computed to the $x^{7/2}$ order beyond the leading quadrupolar (Newtonian) order for such binaries. Moreover, the amplitudes of the two GW polarization states, $h_{\times}$ and $ h_+$, for non-spinning binaries are available to 3PN order [@BF3PN]. Very recently, detailed computations led to the determination of the dynamics of such binaries to the 4PN order [@4PN]. Binaries that contain compact objects with intrinsic rotations, the spin effects enter the dynamics and GW emission via spin-orbit and spin-spin interactions [@BO_75; @Tulczyjew]. In binaries containing maximally spinning BHs, the spin-orbit coupling (linear in the spins) first appears at the 1.5PN order, while the spin-spin interaction (which is quadratic in spins) first occurs at the 2PN order [@LK_95]. Additionally, ${\boldsymbol{S}}_1$, ${\boldsymbol{S}}_2$ and ${\boldsymbol{L}}$, the two spin and orbital angular momenta, for generic spinning compact binaries precess around the total angular momentum ${\boldsymbol{J}}={\boldsymbol{L}}+{\boldsymbol{S}}_1+{\boldsymbol{S}}_2$ due to spin-orbit and spin-spin interactions. This forces substantial modulations of the emitted GWs from inspiralling generic spinning compact binaries compared to their non-spinning counterparts [@LK_95; @ACST]. Therefore, it is important to incorporate various effects due to the intrinsic rotations of compact objects while constructing inspiral GW templates for spinning compact binaries. At present, GW frequency evolution and amplitudes of $h_{\times}$ and $ h_+$ for BH binaries having maximally spinning components are fully determined to 2.5PN and 2PN orders, respectively, while incorporating all the relevant spin induced effects [@LK_95; @ABFO; @BBF; @Alvi; @FBB; @BFH]. Moreover, the on-going detailed computations are providing various higher PN order spin-orbit and spin-spin contributions to the dynamics of spinning compact binaries in general orbits and to the orbital frequency evolution for quasi-circular inspiral. At present, the spin-orbit contributions to binary dynamics and GW frequency evolution are available up to the next-to-next-to-leading order (2PN order) beyond the leading order [@Bohe2013; @2PN_SO] while adapting the MPM (Multipolar post-Minkowskian) approach [@LB_2009]. In contrast, the higher order spin-spin contributions to the binary dynamics are usually tackled in the Arnowitt-Deser-Misner canonical formalism [@GS_2009] and in the Effective Field Theory formalism [@GR_04; @Porto2006] (Note that the spin-orbit effects in the Effective Field Theory formalism are computed, for example, in [@EFT_SO]). These approaches provide various spin(1)-spin(2) and spin-squared contributions to the orbital dynamics [@GS_group]. Moreover, various source multipole moments needed to obtain the spin contributions to GW luminosity at the 3PN order and GW polarization states to the 2.5PN order were computed in [@porto].
The construction of time-domain $h_{\times}$ and $h_+$ associated with inspiralling generic spinning compact binaries requires us to numerically solve a set of PN accurate differential equations for ${\boldsymbol{S}}_1, {\boldsymbol{S}}_2,{\boldsymbol{L}}_{\rm N}, x $ and the orbital phase [@LK_95], where ${\boldsymbol{L}}_{\rm N}$ is the Newtonian orbital angular momentum. The numerical integration provides temporal evolutions for the orbital phase, the associated angular frequency and the two angles that specify the orientation of the orbital plane in an inertial frame associated with the direction of ${\boldsymbol{J}}$ at the initial epoch. These variations are incorporated into the PN accurate expressions for $h_{\times}$ and $ h_+$ to obtain PN accurate time-domain inspiral waveforms from such binaries [@LK_95]. In this approach, the differential equation for the orbital phase explicitly depends on the precessional motion of the orbital plane [@LK_95; @ACST]. Therefore, it is not possible to express the orbital phase as an integral of the orbital frequency as usually done in the case of non-spinning compact binaries [@DIS98]. A decade ago, Buonanno, Chen and Vallisneri proposed an approach, referred to as the precessing convention, that factorizes the generic spinning binary waveform into a [*carrier signal*]{} and a [*modulated amplitude*]{} term [@BCV]. In this approach, the phase of the carrier signal ($\Phi_p$) essentially coincides with the accumulated orbital phase such that $\dot {\Phi}_{p} \equiv \omega$. Moreover, the precessional dynamics of the orbital plane only influences the modulated amplitude part of inspiral waveform even for generic compact binaries. Therefore, the precessing convention disentangles the precessional effects from its non-precessional counterparts while modeling both the amplitude and the phase evolutions of inspiral GWs from such astrophysical systems. This convention was employed to model inspiral GW signals from compact binaries containing misaligned single-spin and to probe its data analysis benefits [@PBCV04; @BCPTV05]. Very recently, inspiral-merger-ringdown waveforms for generic spinning BH binaries, invoking the effective-one-body approach [@TD], also adapted the precessing convention to model GWs from the inspiral part [@Pan_etal_14]. We note that this convention requires a [*precessing source frame*]{} which is usually based on the Newtonian orbital angular momentum ${\boldsymbol{L}}_{\rm N}$. However, [@Pan_etal_14] employed both ${\boldsymbol{L}} $ and ${\boldsymbol{L}}_{\rm N}$ to model GWs during the late part of the binary inspiral just prior to the plunge.
In this paper, we develop a prescription to compute PN accurate inspiral waveforms for generic spinning compact binaries while using the PN accurate orbital angular momentum ${\boldsymbol{L}} $ to construct the precessing source frame. This is influenced by the usual practice of employing precessional equation appropriate for ${\boldsymbol{L}}$ to evolve ${\boldsymbol{L}}_{\rm N}$ and the associated precessing source frame while constructing inspiral waveforms via the precessing convention of [@BCV]. We show that the use of such an adiabatic approximation, namely employing an orbital averaged differential equation for ${\boldsymbol{L}}_{\rm N}$, can lead to PN corrections to $\dot {\Phi}_{p} = \omega$. These observations motivated us to provide a set of PN accurate equations to obtain temporally evolving quadrupolar order inspiral GW polarization states for generic spinning compact binaries in the ${\boldsymbol{L}} $-based precessing convention. In our approach, the spin precession induced modulations enter the differential equation for the orbital phase only at the 3PN order. Moreover, the ${\boldsymbol{L}} $-based convention requires us to include additional 3PN order terms in the differential equation for $x$, compared to the usual ${\boldsymbol{L}}_{\rm N}$-based approach. We explore the practical implications of these additional terms with the help of [*match*]{} computations, detailed in [@DIS98; @BO96]. The match computations involve two inspiral families where one is constructed via our ${\boldsymbol{L}}$-based precessing convention and therefore incorporate the above mentioned 3PN order terms. The other family is based on the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code, developed by the LIGO Scientific Collaboration (LSC) [@LAL], that implemented the ${\boldsymbol{L}}_{\rm N}$-based precessing convention of [@BCV]. In this code, the precessional dynamics is fully 2PN accurate while the differential equation for $x$ incorporates spin-orbit contributions to 3.5PN order. For match computations, we employ PN accurate relation connecting ${\boldsymbol{L}}_{\rm N}$ and ${\boldsymbol{L}}$ to construct two waveform families with physically identical initial orbital and spin orientations. We terminate the two inspiral waveform families when their respective $x$ parameter reaches $0.1$ which roughly corresponds to orbital separations $\sim10\,G\,m/c^2$. These considerations allow us to attribute the reported match estimates to the above mentioned additional 3PN order terms present in our differential equations for the orbital phase and frequency. We find that the match estimates are less than the optimal 0.97 value for a non-negligible fraction of unequal mass spinning compact binaries. It may be recalled that such an optimal match value roughly corresponds to a $10\%$ loss in the ideal event rate. In what follows, we briefly summarize the usual implementation of the ${\boldsymbol{L}}_{\rm N}$-based precessing convention and explore the consequence of employing an orbital averaged precessional equation for ${\boldsymbol{L}}_{\rm N}$. In section \[Sec\_L\], we detail the construction of quadrupolar order GW polarization states in our ${\boldsymbol{L}}$-based precessing convention. The match estimates involving these two inspiral families with physically equivalent initial configurations and associated discussions are listed in section \[result\] while section \[Sec\_dis\] provides a brief summary.
Inspiral waveforms in ${\boldsymbol{L}}_{\rm N}$ and ${\boldsymbol{L}}$-based precessing conventions
=====================================================================================================
We begin by summarizing the usual implementation of the ${\boldsymbol{L}}_{\rm N}$-based precessing convention and explore the consequence of employing an orbital averaged differential equation for ${\boldsymbol{L}}_{\rm N}$.
The ${\boldsymbol{L}}_{\rm N}$-based precessing convention {#SecII_A}
-----------------------------------------------------------
The precessing convention, introduced in [@BCV], aims to remove all the spin precession induced modulations from the orbital phase evolution. In this approach, the orbital phase $\Phi_p(t)$ is written as an integral of the orbital frequency $\omega(t)$, namely $\Phi_p(t)=\int \omega(t')\, dt'$, even for generic spinning compact binaries. This feature is crucial to ensure that the inspiral waveform for a precessing spinning compact binary can be written as the product of a non-precessing carrier waveform and a modulation term that contains all the precessional effects. This is how the approach disentangles the precessional effects from their non-precessional counterparts both in the amplitude and phase of inspiral waveforms. It should be noted that in the absence of precessing convention, the orbital phase of a generic spinning binary is given by $\int [ \omega(t') -\dot{\alpha}'(t')\,\cos \iota'(t') ] \, dt'$, where $\iota'$ and $\alpha'$ specify the orientation of ${\boldsymbol{L}}_{\rm N}$ in an inertial frame associated with the initial direction of ${\boldsymbol{J}}$ [@LK_95].
To obtain inspiral waveforms for spinning compact binaries in their precessing convention, [@BCV] employed certain [*precessing source frame*]{} $( {\boldsymbol{e}}_1^l, {\boldsymbol{e}}_2^l, {\boldsymbol{e}}_3^l \equiv {\boldsymbol{l}})$, where ${\boldsymbol{l}}$ is the unit vector along ${\boldsymbol{L}}_{\rm N}$. The basis vectors of this triad satisfy the evolution equations $\dot{{\boldsymbol{e}}}^l_{1,2}={\boldsymbol{\Omega}}^l_e \times {\boldsymbol{e}}^l_{1, 2}$ and $\dot{{\boldsymbol{e}}}^l_{3}\equiv \dot{{\boldsymbol{l}}}={\boldsymbol{\Omega}}_{k} \times {\boldsymbol{l}}$. The angular frequency ${\boldsymbol{\Omega}}^l_e$ is constructed in such a manner that these three basis vectors always form an orthonormal triad. This is possible with the following expression for ${\boldsymbol{\Omega}}^l_e$, namely ${\boldsymbol{\Omega}}^l_e= {\boldsymbol{\Omega}}_k - ({\boldsymbol{\Omega}}_k \cdot {\boldsymbol{l}}){\boldsymbol{l}}$, where $\Omega_k$ is the usually employed precessional frequency for ${\boldsymbol{l}}$. The expression for ${\boldsymbol{\Omega}}_{k}$ that includes the dominant order spin-orbit and spin-spin contributions can be obtained by collecting the terms that multiply $\hat {{\boldsymbol{L}}}_{\rm N}$ $(\equiv {\boldsymbol{l}})$ in equation (9) of [@BCV]. With the help of ${\boldsymbol{e}}_1^l $ and ${\boldsymbol{e}}_2^l$, an orbital phase $\Phi_p(t)$ may be defined such that ${\boldsymbol{n}} = \cos \Phi_p \, {\boldsymbol{e}}^l_1 +\sin \Phi_p \,{\boldsymbol{e}}^l_2$, where ${\boldsymbol{n}}$ is the unit vector along the binary separation vector ${\boldsymbol{r}}$. Additionally, one may define a co-moving frame (${\boldsymbol{n}}, {\boldsymbol{\lambda}} = {\boldsymbol{l}} \times {\boldsymbol{n}}, {\boldsymbol{l}}$) such that the time derivative of ${\boldsymbol{n}}$ is given by $\dot{{\boldsymbol{n}} } = \dot{\Phi}_p {\boldsymbol{\lambda}} + {\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$. It was argued in [@BCV] that ${\boldsymbol{\Omega}}^l_e$ should only be proportional to ${\boldsymbol{n}}$ which ensures that $\dot{{\boldsymbol{n}} } \cdot {\boldsymbol{\lambda}} = \dot{\Phi}_p$. This leads to the desirable expression for $\dot{\Phi}_p$, namely $\dot{\Phi}_p=\omega$, while employing the adiabatic condition for the sequence of circular orbits: $\dot{{\boldsymbol{n}} } \cdot {\boldsymbol{\lambda}} = \omega$. This adiabatic condition is equivalent to $\dot{{\boldsymbol{n}}} \cdot \dot{{\boldsymbol{n}}} \equiv \omega^2$ that provides another version of the PN independent relation connecting the linear and orbital angular velocities, $ v^2 \equiv r^2\,\omega^2$.
A close inspection reveals that this approach usually employs an orbital averaged differential equation for ${\boldsymbol{L}}_{\rm N}$ to evolve the precessing source frame while constructing PN accurate inspiral waveforms. It turns out that the differential equation for ${\boldsymbol{L}}_{\rm N}$ in such an adiabatic approximation is identical to the evolution equation for the PN accurate orbital angular momentum ${\boldsymbol{L}}$ [@GG1; @GS11]. In what follows, we explore the effect of such an adiabatic approximation on the equation for $\dot {{\boldsymbol{n}}}$ in the (${\boldsymbol{n}}, {\boldsymbol{\lambda}}, {\boldsymbol{l}}$) co-moving frame and on the derivation of $\dot{\Phi}_p$ equation. The usually employed expression for ${\boldsymbol{\Omega}}_k$ while considering only the leading order spin-orbit interactions may be written as $$\begin{aligned}
\label{Eq_l_omega_k}
{\boldsymbol{\Omega}}_k &= \frac{c^3}{Gm}\, x^3\,\Bigg\{ \delta_1\,q\, \chi_1\, {\boldsymbol{s}}_1
+\frac{\delta_2}{q}\, \chi_2\, {\boldsymbol{s}}_2 \Bigg\} \,,\end{aligned}$$ where $q=m_1/m_2$ ($m_1 \geq m_2$) is the mass ratio and $\delta_{1,2} = \eta/2 + 3\,(1\mp \sqrt{1-4\eta})/4$ while $\eta=m_1\,m_2/m^2$ is the symmetric mass ratio. The Kerr parameters $\chi_1$ and $\chi_2$ of the two compact objects of mass $m_1$ and $m_2$ specify their spin angular momenta by ${\boldsymbol{S}}_{1,2}=G\, m_{1,2}^2\, \chi_{1,2}\,{\boldsymbol{s}}_{1,2}/c$, where ${\boldsymbol{s}}_1$ and ${\boldsymbol{s}}_2$ are the unit vectors along ${\boldsymbol{S}}_1$ and ${\boldsymbol{S}}_2$. It is straightforward to show that the above equation is identical to $\omega^2$ terms that multiply $\hat { \bf L}_{\rm N}$ on the right hand side of equation (9) in [@BCV]. To explore the implication of using the above expression for ${\boldsymbol{\Omega}}_k$ to construct ${\boldsymbol{\Omega}}^l_e $, we revisit the arguments detailed in the appendix B of [@BCV]. These arguments, crucial to obtain $\dot{\Phi}_p=\omega$, require that ${\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$, appearing in the equation $\dot{{\boldsymbol{n}} } = \dot{\Phi}_p {\boldsymbol{\lambda}} + {\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$, should be zero. A closer look reveals that [@BCV] did not use the explicit expression for ${\boldsymbol{\Omega}}^l_e$ to show that ${\boldsymbol{\Omega}}^l_e$ lies along ${\boldsymbol{n}}$. Instead, the authors arrived at such a conclusion with the help of the following two steps. First, it was noted that $ \dot {{\boldsymbol{l}}}$ should be $\propto {\boldsymbol{\lambda}}$ (see lines around equations (B3), (B4) and (B5) in [@BCV]). This was inferred by employing the definition for ${\boldsymbol{l}}$ (${\boldsymbol{l}} = {\boldsymbol{n}} \times {\boldsymbol{\lambda}}$), the resulting time derivative for ${\boldsymbol{l}}$ ($ \dot {{\boldsymbol{l}}} = \dot {{\boldsymbol{n}}} \times {\boldsymbol{\lambda}} + {\boldsymbol{n}} \times \dot { {\boldsymbol{\lambda}}}$), the adiabatic condition for circular orbits ($\dot {{\boldsymbol{n}}} = \omega \,{\boldsymbol{\lambda}})$ and the time derivative for ${\boldsymbol{\lambda}}$ in the co-moving frame ($\dot {{\boldsymbol{\lambda}}} = - \dot{\Phi}_p \, {\boldsymbol{n}} +
{\boldsymbol{\Omega}}^l_e \times {\boldsymbol{\lambda}} $). In the second step, [@BCV] invoked the requirement that $ \dot {{\boldsymbol{l}}}$ should also be given by ${\boldsymbol{\Omega}}^l_e \times {\boldsymbol{l}}$ in the precessing source frame. With the help of the above two arguments, namely $ \dot {{\boldsymbol{l}}} \propto {\boldsymbol{\lambda}} $ and $ \dot {{\boldsymbol{l}}}= {\boldsymbol{\Omega}}^l_e \times {\boldsymbol{l}}$, [@BCV] concluded that ${\boldsymbol{\Omega}}^l_e$ should lie along ${\boldsymbol{n}}$ as ${\boldsymbol{\lambda}} = {\boldsymbol{l}} \times {\boldsymbol{n}}$. We would like to emphasize that [@BCV] never invoked their explicit expression for ${\boldsymbol{\Omega}}^l_e$ to show that ${\boldsymbol{\Omega}}^l_e$ can have components only along ${\boldsymbol{n}}$ which is essential to obtain the relation $\dot{\Phi}_p=\omega$. In fact, it is straightforward to show with the help of equation (\[Eq\_l\_omega\_k\]) that ${\boldsymbol{\Omega}}^l_e$ can have components along ${\boldsymbol{\lambda}}$ since ${\boldsymbol{\Omega}}^l_e \cdot {\boldsymbol{\lambda}} = (c^3/Gm)\, x^3\,( \delta_1\,q\, \chi_1\, {\boldsymbol{s}}_1 \cdot {\boldsymbol{\lambda}}
+\delta_2 \, \chi_2/q \,{\boldsymbol{s}}_2 \cdot {\boldsymbol{\lambda}}) \neq 0$, in general. This results in the following 1.5PN accurate expression for $ \dot {{\boldsymbol{n}}}$ $$\begin{aligned}
\label{Eq_ndot}
\dot {{\boldsymbol{n}} } &= \dot{\Phi}_p {\boldsymbol{\lambda}} + \frac{c^3}{Gm}\, x^3\,\Bigg\{ \delta_1\,q\, \chi_1\, \bigl [{\boldsymbol{s}}_1 \times {\boldsymbol{n}} - ({\boldsymbol{s}}_1 \cdot {\boldsymbol{l}})\, {\boldsymbol{\lambda}} \bigr]
+\frac{\delta_2}{q}\, \chi_2\, \bigl [{\boldsymbol{s}}_2 \times {\boldsymbol{n}} - ({\boldsymbol{s}}_2 \cdot {\boldsymbol{l}})\, {\boldsymbol{\lambda}} \bigr] \Bigg\} \,.\end{aligned}$$ Clearly, the 1.5PN order terms that arise from ${\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$ in the above equation will not be zero for generic spinning compact binaries.
Interestingly, it is still possible to equate $\dot{\Phi}_p$ to $ \omega$ by employing the adiabatic condition $\dot{{\boldsymbol{n}} } \cdot {\boldsymbol{\lambda}} = \omega$ as $( {\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}} ) \cdot {\boldsymbol{\lambda}} =0$ even in the presence of non-vanishing 1.5PN order ${\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$ term. However, the equivalent version of the adiabatic condition, namely $\dot{{\boldsymbol{n}}} \cdot \dot{{\boldsymbol{n}}} \equiv \omega^2$, forces the differential equation for $\Phi_p$ to contain 3PN order corrections in addition to the standard $\omega$ term. In other words, two equivalent versions of the same adiabatic condition for circular orbits, namely $\dot{{\boldsymbol{n}} } \cdot {\boldsymbol{\lambda}} = \omega$ and $\dot{{\boldsymbol{n}}} \cdot \dot{{\boldsymbol{n}}} \equiv \omega^2$, provide different evolution equations for $\Phi_p$. This is the unexpected consequence of employing precessional equation appropriate for ${\boldsymbol{L}}$ to evolve the ${\boldsymbol{L}}_{\rm N}$-based precessing source frame. It is not difficult to deduce that this essentially arises from the non-vanishing 1.5PN order ${\boldsymbol{\Omega}}^l_e \times {\boldsymbol{n}}$ contributions to $ \dot {{\boldsymbol{n}}}$ listed in the above equation. The following arguments can also be used to clarify why the above two versions of the same adiabatic condition for the circular orbits result in two different expressions for $ \dot{\Phi}_p$. In our opinion, this arises by identifying ${\boldsymbol{k}} \times {\boldsymbol{n}}$ to be ${\boldsymbol{\lambda}} ={\boldsymbol{l}} \times {\boldsymbol{n}} $, where ${\boldsymbol{k}}$ is the unit vector along ${\boldsymbol{L}}$. Strictly speaking, the precessing source frame of [@BCV] is based on ${\boldsymbol{L}}$ rather than ${\boldsymbol{L}}_{\rm N}$ due to the use of ${\boldsymbol{\Omega}}_k$ that provides the precessional equation for ${\boldsymbol{L}}$ [@GG1; @GS11]. This implies that their co-moving triad is rather ${\boldsymbol{L}}$-based and the expression for $ \dot {{\boldsymbol{n}}} $ got components along ${\boldsymbol{k}} \times {\boldsymbol{n}}$ instead of ${\boldsymbol{\lambda}} = {\boldsymbol{l}} \times {\boldsymbol{n}}$. Therefore, $\dot{{\boldsymbol{n}} } \cdot {\boldsymbol{\lambda}} $ results in PN corrections to $ \dot{\Phi}_p$ as $ ({\boldsymbol{k}} \times {\boldsymbol{n}}) \cdot {\boldsymbol{\lambda}}$ is unity only at the leading order (this may be deduced from our equation (\[Eq\_l\_k\]) listed below). This leads to a differential equation for $ \dot{\Phi}_p$ that involves PN corrections to $\omega$. The use of precessional equation appropriate for ${\boldsymbol{L}}$ while implementing the precessing convention of [@BCV] motivated us to develop a ${\boldsymbol{k}}$-based precessing convention for constructing inspiral waveforms for spinning compact binaries. This should also allow to explore the practical implications of using the adiabatic approximation to evolve ${\boldsymbol{L}}_{\rm N}$ in the usual implementation of the precessing convention.
Inspiral waveforms via an ${\boldsymbol{L}} $-based precessing convention {#Sec_L}
--------------------------------------------------------------------------
Influenced by the above arguments and [@BCV], we first introduce a ${\boldsymbol{k}}$-based precessing source frame: (${\boldsymbol{e}}_1$,${\boldsymbol{e}}_2$, ${\boldsymbol{e}}_3 \equiv {\boldsymbol{k}}$). The precessional dynamics of ${\boldsymbol{e}}_1, {\boldsymbol{e}}_2$ and ${\boldsymbol{e}}_3$ are provided by $\dot {{\boldsymbol{e}}}_{1,2,3}={\boldsymbol{\Omega}}_e \times {\boldsymbol{e}}_{1,2,3}$, where $ {\boldsymbol{\Omega}}_e \equiv {\boldsymbol{\Omega}}_k - ( {\boldsymbol{\Omega}}_k \cdot {\boldsymbol{k}})\, {\boldsymbol{k}}$ and $ \Omega_k$ is the precessional frequency of ${\boldsymbol{k}}$. It should be obvious that $\dot {{\boldsymbol{e}}}_{3} = {\boldsymbol{\Omega}}_e \times {\boldsymbol{e}}_{3} $ is identical to $\dot {{\boldsymbol{e}}}_{3} = {\boldsymbol{\Omega}}_k \times {\boldsymbol{e}}_{3}$ as $ {\boldsymbol{e}}_{3} \equiv {\boldsymbol{k}}$. It is possible to construct a ${\boldsymbol{k}}$-based co-moving triad (${\boldsymbol{n}}, {\boldsymbol{\xi}}={\boldsymbol{k}} \times {\boldsymbol{n}}, {\boldsymbol{k}}$) and introduce an orbital phase $\Phi$ such that
\[Eq\_n\_xi\] $$\begin{aligned}
{\boldsymbol{n}} &= \cos \Phi \, {\boldsymbol{e}}_1 + \sin \Phi \, {\boldsymbol{e}}_2\,, \\
{\boldsymbol{\xi}} &= - \sin \Phi \, {\boldsymbol{e}}_1 + \cos \Phi \, {\boldsymbol{e}}_2 \,.\end{aligned}$$
It is fairly straightforward to obtain following expressions for the time derivatives of ${\boldsymbol{n}}$ and ${\boldsymbol{\xi}}$
$$\begin{aligned}
\label{Eq_ndot_xidot}
\dot { {\boldsymbol{n}}} &= \dot {\Phi}\, {\boldsymbol{\xi}} + {\boldsymbol{\Omega}}_e \times {\boldsymbol{n}} \,,\\
\dot {{\boldsymbol{\xi}}} &= - \dot {\Phi}\, {\boldsymbol{n}} + {\boldsymbol{\Omega}}_e \times {\boldsymbol{\xi}} \,.\end{aligned}$$
We are now in a position to obtain the differential equation for $\Phi$. This is derived with the help of the frame independent adiabatic condition for circular orbits, namely $ \dot { {\boldsymbol{n}}} \cdot \dot { {\boldsymbol{n}}} \equiv \omega^2$. Employing the above expression for $ \dot { {\boldsymbol{n}}}$ in such an adiabatic condition leads to $$\label{Eq_omega2}
\omega^2 = \dot{\Phi}^2 + \Omega_{e\xi}^2 \,,$$ where $\Omega_{e\xi} ={\boldsymbol{\Omega}}_e \cdot {\boldsymbol{\xi}} $ is given by $$\label{Eq_Omega_e_xi}
\Omega_{e\xi} =\frac{c^3}{Gm}\, x^3\,\Bigg\{ \delta_1\,q\, \chi_1\,
\left( {\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}} \right) +\frac{\delta_2}{q}\, \chi_2\, \left( {\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}} \right) \Bigg\} \,.$$ This results in the following 3PN accurate differential equation for $\Phi$ $$\label{Eq_phidot}
\dot {\Phi} = \frac{c^3}{G\, m}\, x^{3/2} \, \biggl\{ 1- \frac{x^3}{2}\, \Bigl[\delta_1 \, q\, \chi_1 \, ({\boldsymbol{s}}_1\cdot {\boldsymbol{\xi}})
+ \frac{\delta_2}{q}\, \chi_2\, ({\boldsymbol{s}}_2\cdot {\boldsymbol{\xi}})\Bigr]^2\biggr\} \,.$$ These additional terms appear at the 3PN order as we employ equation (\[Eq\_l\_omega\_k\]) for ${\boldsymbol{\Omega}}_k$ that only incorporates the leading order spin-orbit interactions appearing at 1.5PN order. It is not difficult to deduce that the inclusion of 2PN order spin-spin interaction terms in ${\boldsymbol{\Omega}}_k$ can lead to certain 4PN order contributions to the $\dot {\Phi}$ equation. Additionally, the presence of the above 3PN contributions to $\dot {\Phi}$ equation may be attributed to the fact that the orbital velocity ${\boldsymbol{v}}$ can have non-vanishing PN order components along ${\boldsymbol{L}}$ while considering generic spinning compact binaries [@GG1].
The use of ${\boldsymbol{L}}$ to describe binary orbits also modifies the evolution equation for $\omega$ (or $x$). This is because the spin-orbit interactions are usually incorporated in terms of ${\boldsymbol{s}}_1 \cdot {\boldsymbol{l}}$ and ${\boldsymbol{s}}_2 \cdot {\boldsymbol{l}}$, in the literature [@LK_95; @BCV]. These terms require modifications due to the following 1.5PN order relation connecting ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$ $$\begin{aligned}
\label{Eq_l_k}
{\boldsymbol{l}} &=& {\boldsymbol{k}} + x^{3/2}\, \biggl \{-\frac{1}{2}\, \eta \, \bigl [\chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}})+ \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}}) \bigr]\, {\boldsymbol{n}} \nonumber \\
&&+ \bigl [2\, X_1^2 \, \chi_1 \, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})+ \eta\, \chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}}) \nonumber \\
&&+ 2\, X_2^2 \, \chi_2 \, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}})+ \eta\, \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}}) \bigr]\, {\boldsymbol{\xi}} \biggr\} \,,\end{aligned}$$ where $X_1=m_1/m$ and $X_2=m_2/m$. The above relation can easily be extracted, for example, from equations (6.10) and (7.10) in [@BBF]. This relation leads to certain additional 3PN order contributions to $\dot{x}$ while describing the binary orbits with ${\boldsymbol{k}}$. These additional terms are, for example, with respect to equation (3.16) of [@Bohe2013] that provides PN accurate expression for $dx/dt$ while invoking ${\boldsymbol{l}}$ to describe binary orbits. Our additional contributions to $\dot x$ appear at 3PN order as the dominant spin-orbit interactions, in terms of ${\boldsymbol{s}}_1 \cdot {\boldsymbol{l}}$ and ${\boldsymbol{s}}_2 \cdot {\boldsymbol{l}}$, contribute to the $x$ evolution equation at 1.5PN order. The differential equation for $x$ in our approach may be written as $$\begin{aligned}
\label{Eq_dxdt}
\frac{ d x}{dt} &= \frac{dx}{dt}(Equation~(3.16)\, in \, \cite{Bohe2013}; \, {\boldsymbol{l}} \rightarrow {\boldsymbol{k}}) \nonumber \\
&\quad+\frac{64}{5}\frac{c^3}{Gm}\eta\, {x}^5
\biggl \{
x^3\, \biggl[
-\frac{47}{3}\, X_1^2\, \chi_1\, \Bigl[-\frac{1}{2}\, \eta\, \Bigl(\chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}})+\chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}})\Bigr)({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}}) \nonumber \\
&\quad+\Bigl(2\, X_1^2\, \chi_1\, ({\boldsymbol{s}}_1\cdot {\boldsymbol{\xi}})+\eta\, \chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})
+2\, X_2^2\, \chi_2\, ({\boldsymbol{s}}_2\cdot {\boldsymbol{\xi}})
+\eta\, \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}}) \Bigr) ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})\Bigr] \nonumber \\
&\quad-\frac{47}{3}\, X_2^2\, \chi_2\, \Bigl[-\frac{1}{2}\, \eta\, \Bigl(\chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}})
+\chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}})\Bigr)({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}}) \nonumber \\
&\quad+\Bigl(2\, X_1^2\, \chi_1\, ({\boldsymbol{s}}_1\cdot {\boldsymbol{\xi}})
+\eta\, \chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})
+2\, X_2^2\, \chi_2\, ({\boldsymbol{s}}_2\cdot {\boldsymbol{\xi}})
+\eta\, \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}}) \Bigr) ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}})\Bigr] \nonumber \\
&\quad-\frac{25}{4}\, (X_1-X_2)\, X_2\, \chi_2\, \Bigl[-\frac{1}{2}\, \eta\, \Bigl(\chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}})
+\chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}})\Bigr)({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}}) \nonumber \\
&\quad+\Bigl(2\, X_1^2\, \chi_1\, ({\boldsymbol{s}}_1\cdot {\boldsymbol{\xi}})+\eta\, \chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})
+2\, X_2^2\, \chi_2\, ({\boldsymbol{s}}_2\cdot {\boldsymbol{\xi}})
+\eta\, \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}}) \Bigr) ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}})\Bigr] \nonumber \\
&\quad+\frac{25}{4}\, (X_1-X_2)\, X_1\, \chi_1\, \Bigl[-\frac{1}{2}\, \eta\, \Bigl(\chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}})+\chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{n}})\Bigr)({\boldsymbol{s}}_1 \cdot {\boldsymbol{n}}) \nonumber \\
&\quad+\Bigl(2\, X_1^2\, \chi_1\, ({\boldsymbol{s}}_1\cdot {\boldsymbol{\xi}})+\eta\, \chi_1\, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})
+2\, X_2^2\, \chi_2\, ({\boldsymbol{s}}_2\cdot {\boldsymbol{\xi}})
+\eta\, \chi_2\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{\xi}}) \Bigr) ({\boldsymbol{s}}_1 \cdot {\boldsymbol{\xi}})\Bigr] \biggr]
\biggr \} \,,\end{aligned}$$ where the first term is adapted from equation (3.16) in [@Bohe2013] by replacing its ${\boldsymbol{l}}$ vectors by our ${\boldsymbol{k}}$ vectors. Clearly, these additional [*spin*]{}-squared terms contribute to $dx/dt$ at 3PN order. Note that the use of ${\boldsymbol{k}}$ for ${\boldsymbol{l}}$ in the leading order spin-spin contributions to $dx/dt$, as listed in equation (1) of [@BCV], results in additional 3.5PN order [*spin*]{}-cubed terms. Such contributions to our $dx/dt$ equation are neglected as the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code that implement the ${\boldsymbol{L}}_{\rm N}$-based precessing convention does not include any [*spin*]{}-cubed terms in its differential equation for $x$.
In what follows, we model inspiral GWs from spinning compact binaries in our ${\boldsymbol{k}}$-based precessing convention. We begin by displaying the following quadrupolar order expressions for the two GW polarization states:
\[Eq\_hp\_hx\] $$\begin{aligned}
h_{\times}|_{\rm Q}(t) &=& 2\, \frac{ G\, m\, \eta \, x}{c^2\, R'}\, (2\, \xi_x\, \xi_y -2\, n_x\, n_y)\,, \\
h_{+}|_{\rm Q}(t) &=& 2\, \frac{ G\, m\, \eta \, x}{c^2\, R'}\, (\xi_x^2 - \xi_y^2- n_x^2 + n_y^2) \,,\end{aligned}$$
where $\xi_{x,y}$ and $n_{x, y}$ are the $x$ and $y$ components of ${\boldsymbol{\xi}}$ and ${\boldsymbol{n}}$ in an inertial frame associated with ${\boldsymbol{N}}$, the unit vector that points from the source to the detector, while $R'$ is the distance to the binary. These $x$ and $y$ components can be expressed in terms of the Cartesian components of ${\boldsymbol{e}}_1$ and ${\boldsymbol{e}}_2$ via equations (\[Eq\_n\_xi\]). We note that the above expressions for the quadrupolar order $h_{\times}$ and $h_+$ are written in the so-called frame-less convention [@LK_95]. These expressions emerge from the following standard definitions for the quadrupolar order GW polarization states:
$$\begin{aligned}
h_{\times}|_{\rm Q}(t) &=& \frac{1}{2}\,(p^i\, q^j + q^i\, p^j)\, h^{\rm TT}_{ij}|_{\rm Q}\,, \\
h_{+}|_{\rm Q}(t) &=& \frac{1}{2}\,(p^i\, p^j - q^i\, q^j)\, h^{\rm TT}_{ij}|_{\rm Q}\,,\end{aligned}$$
where ${\boldsymbol{p}}$ and ${\boldsymbol{q}}$ are the two polarization vectors forming, along with ${\boldsymbol{N}}$, an orthonormal right-handed triad. To obtain equation (\[Eq\_hp\_hx\]), we used the following expression for the quadrupolar order transverse-traceless part of the far-zone field $h^{\rm TT}_{ij}|_{\rm Q}$ $$\begin{aligned}
h^{\rm TT}_{ij}|_{\rm Q}= \frac{4\, G\, m\, \eta\, x}{c^2\, R'}\, (\xi^i\, \xi^j- n^i\, n^j)\,.
$$ In the frame-less convention, we let the components of ${\boldsymbol{p}}$ and ${\boldsymbol{q}}$ in the ${\boldsymbol{N}}$-based inertial frame to be ${\boldsymbol{p}}=(1,0,0)$ and ${\boldsymbol{q}}=(0,1,0)$. Clearly, we need to specify how the Cartesian components of ${\boldsymbol{\xi}}$ and ${\boldsymbol{n}}$ vary in time to obtain temporally evolving GW polarization states for inspiralling generic spinning compact binaries. Therefore, we require to solve numerically the differential equations for $\Phi, x, {\boldsymbol{e}}_1$ and ${\boldsymbol{e}}_2$ to obtain $h_{\times}|_{\rm Q}(t)$ and $h_{+}|_{\rm Q}(t)$. We use equation (\[Eq\_phidot\]) for $\Phi$ while the differential equation for $x$ is given by equation (\[Eq\_dxdt\]) that contains all the non-spinning contributions accurate up to 3.5PN order and the usual spin contributions that are fully 2PN accurate. These contributions are provided, for example, by equation (3.16) in [@Bohe2013] and are listed in the \[appendix\], where we have replaced ${\boldsymbol{l}}$ by ${\boldsymbol{k}}$. These specific PN order choices are influenced by the fact that the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code also employs an evolution equation for $\omega$ that incorporates such PN contributions. We note that this routine implements the ${\boldsymbol{l}}$-based [precessing convention]{} of [@BCV] to construct inspiral templates to search for GWs from generic spinning binaries. The differential equations for ${\boldsymbol{e}}_1$ and ${\boldsymbol{e}}_2$ in our approach are given by
\[Eq\_e1\_e2\_dot\] $$\begin{aligned}
\label{Eq_e1dot}
\dot{{\boldsymbol{e}}_1} &=& {\boldsymbol{\Omega}}_e \times {\boldsymbol{e}}_1 = ({\boldsymbol{\Omega}}_k - ({\boldsymbol{\Omega}}_k \cdot {\boldsymbol{k}}))\times {\boldsymbol{e}}_1\,, \\
\dot{{\boldsymbol{e}}_2} &=& {\boldsymbol{\Omega}}_e \times {\boldsymbol{e}}_2 = ({\boldsymbol{\Omega}}_k - ({\boldsymbol{\Omega}}_k \cdot {\boldsymbol{k}}))\times {\boldsymbol{e}}_2\,,\end{aligned}$$
where we use the following expression for ${\boldsymbol{\Omega}}_k$ that incorporates the leading order spin-orbit and spin-spin interactions $$\begin{aligned}
\label{Eq_Omega_k}
{\boldsymbol{\Omega}}_k &=\frac{c^3}{Gm}\, x^{3}\,\Bigg\{ \delta_1\,q\, \chi_1\, {\boldsymbol{s}}_1
+\frac{\delta_2}{q}\, \chi_2\, {\boldsymbol{s}}_2
-\frac{3}{2}\,x^{1/2}\,\eta\, \chi_1\, \chi_2\, \bigl[ ({\boldsymbol{k}} \cdot {\boldsymbol{s}}_1)\, {\boldsymbol{s}}_2
+ ({\boldsymbol{k}} \cdot {\boldsymbol{s}}_2 )\, {\boldsymbol{s}}_1 \bigr] \Bigg\} \,.\end{aligned}$$ It is fairly straightforward to verify that this expression is identical to the coefficient of $\hat { \bf L}_{\rm N}$ appearing on the right hand side of equation (9) in [@BCV]. The above equations imply that we also need to invoke the precessional equations for ${\boldsymbol{s}}_1$, ${\boldsymbol{s}}_2$ and ${\boldsymbol{k}}$ (or ${\boldsymbol{e}}_3$) to tackle numerically the dynamics of such binaries. The three coupled equations for ${\boldsymbol{s}}_1$, ${\boldsymbol{s}}_2$ and ${\boldsymbol{k}}$ that include the leading order spin-orbit and spin-spin interactions read
\[eq:s1\_s2\_dot\] $$\begin{aligned}
\label{eq:kdot}
{\dot {{\boldsymbol{s}}}_{1}} &= \frac{c^3}{Gm}\, x^{5/2}\,
\Bigg\{\delta_1 \left({\boldsymbol{k}}\times {\boldsymbol{s}}_1\right)
+ \frac{1}{2}\,x^{1/2}\, \bigg[ X_2^2\,\chi_2\,({\boldsymbol{s}}_2 \times {\boldsymbol{s}}_1)
-3\, X_2^2\,\chi_2\,({\boldsymbol{k}} \cdot {\boldsymbol{s}}_2) \, ({\boldsymbol{k}} \times {\boldsymbol{s}}_1) \bigg] \Bigg\} \,, \\
{\dot {{\boldsymbol{s}}}_{2}} &= \frac{c^3}{Gm}\,x^{5/2}\,
\Bigg\{\delta_2 \left({\boldsymbol{k}}\times {\boldsymbol{s}}_2\right)
+ \frac{1}{2}\,x^{1/2}\,\bigg[ X_1^2\,\chi_1\,({\boldsymbol{s}}_1 \times {\boldsymbol{s}}_2)
-3\, X_1^2\,\chi_1\,({\boldsymbol{k}} \cdot {\boldsymbol{s}}_1) \, ({\boldsymbol{k}} \times {\boldsymbol{s}}_2) \bigg] \Bigg\} \,, \\
{\dot {{\boldsymbol{k}}}} &= \frac{c^3}{Gm}\, x^{3}\,\Bigg\{ \delta_1\,q\, \chi_1\,
\left( {\boldsymbol{s}}_1\times{\boldsymbol{k}}\right)
+\frac{\delta_2}{q}\, \chi_2\, \left( {\boldsymbol{s}}_2\times{\boldsymbol{k}}\right) \nonumber \\
&\quad-\frac{3}{2}\,x^{1/2}\,\eta\, \chi_1\, \chi_2\, \biggl[ ({\boldsymbol{k}} \cdot {\boldsymbol{s}}_1)\, ({\boldsymbol{s}}_2 \times {\boldsymbol{k}})
+ ({\boldsymbol{k}} \cdot {\boldsymbol{s}}_2 )\, ({\boldsymbol{s}}_1 \times {\boldsymbol{k}}) \biggr] \Bigg\} \,.\end{aligned}$$
It is not very difficult to verify that the above equations for ${\dot {{\boldsymbol{s}}}_{1}}$ and ${\dot {{\boldsymbol{s}}}_{2}}$ are identical to equations (2) and (3) in [@BCV] while the differential equation for ${\boldsymbol{k}}$, as expected, arises from the usual conservation of total angular momentum ${\boldsymbol{J}}$. This conservation implies that $L\,\dot{{\boldsymbol{k}}}=-S_1\, \dot{{\boldsymbol{s}}_1}-S_2\, \dot{{\boldsymbol{s}}_2}$. We would like to state again that the equations (\[Eq\_phidot\]), (\[Eq\_dxdt\_l\]), (\[eq\_sk\]) and (\[Eq\_dxdt\]) provide the differential equations for $\Phi$ and $x$ in the present implementation of ${\boldsymbol{k}}$-based precessing convention. Strictly speaking, the use of equation (\[Eq\_Omega\_k\]) for ${\boldsymbol{\Omega}}_k$ requires us to include the additional 4PN and 3.5PN contribution to $d\Phi/dt$ and $dx/dt$, respectively. However, we do not incorporate such spin-quartic and spin-cubic terms in our present work.
In practice, we numerically solve simultaneously the differential equations for ${\boldsymbol{e}}_1, {\boldsymbol{k}}$, ${\boldsymbol{s}}_1$, ${\boldsymbol{s}}_2$, $\Phi$ and $x$ to obtain temporally evolving Cartesian components of ${\boldsymbol{n}}$ and ${\boldsymbol{\xi}}$. The resulting variations in these Cartesian components are imposed on the expressions for $h_{\times,+}|_{\rm Q}(t)$, given by equations (\[Eq\_hp\_hx\]). This leads to inspiral waveforms for generic spinning compact binaries in our ${\boldsymbol{k}}$-based precessing convention. Note that we do not solve the differential equation for ${\boldsymbol{e}}_2$. This is because the temporal evolution of ${\boldsymbol{e}}_2$ can be estimated using the relation ${\boldsymbol{e}}_2(t)={\boldsymbol{k}}(t)\times {\boldsymbol{e}}_1(t)$. This implies that we solve 12 differential equations for the Cartesian components of ${\boldsymbol{e}}_1$, ${\boldsymbol{k}}$, ${\boldsymbol{s}}_1$ and ${\boldsymbol{s}}_2$ along with the differential equations for $\Phi$ and $x$ to track the time evolution for the Cartesian components of ${\boldsymbol{n}}$ and ${\boldsymbol{\xi}}$. The required initial values for the Cartesian components of ${\boldsymbol{e}}_1$, ${\boldsymbol{k}}$, ${\boldsymbol{s}}_1$ and ${\boldsymbol{s}}_2$ are given by freely choosing the following five angles: $\theta_{10}$, $\phi_{10}$, $\theta_{20}$, $\phi_{20}$ and $\iota_0$. These angles specify the above four unit vectors in the ${\boldsymbol{N}}$-based inertial frame at the initial epoch such that
\[eq:16\] $$\begin{aligned}
\label{eq:s1_s2_1}
{\boldsymbol{s}}_1 &= \left ( \sin \theta_{10}\,\cos \phi_{10}, \sin \theta_{10}\,\sin \phi_{10}, \cos \theta_{10} \right )\,,\\
\label{eq:s1_s2_2}
{\boldsymbol{s}}_2 &= \left ( \sin \theta_{20}\,\cos \phi_{20}, \sin \theta_{20}\,\sin \phi_{20}, \cos \theta_{20} \right )\,,
\\
{\boldsymbol{k}} &= (\sin \iota_0 ,0 ,\cos \iota_0)\,,\\
{\boldsymbol{e}}_1 &= (\cos \iota_0 ,0 ,-\sin \iota_0)\,.\end{aligned}$$
This choice is also influenced by the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code of LSC. Additionally, we let the initial $x$ value to be $x_0=(G\, m\, \omega_0/c^3)^{2/3}$ where $\omega_0=10\pi$ Hz and the initial phase $\Phi_0$ to be zero.
We move on to explain how to specify initial conditions, physically identical to equations (\[eq:16\]), while constructing inspiral waveforms based on the ${\boldsymbol{l}}$-based precessing convention. Clearly, the initial orientations of two spin vectors in the inertial ${\boldsymbol{N}}$ frame should be identical in the two approaches. However, the orientation of ${\boldsymbol{l}}$ in such an inertial frame is different from that of ${\boldsymbol{k}}$. We compute ${\boldsymbol{N}} \cdot {\boldsymbol{l}}$ using equation (\[Eq\_l\_k\]) that provide 1.5PN accurate relation connecting ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$. This PN relation makes the value of ${\boldsymbol{N}} \cdot {\boldsymbol{l}}$ at $x_0$ to depend on $\iota_0, m, \eta, \chi_1, \chi_2, \Phi_0$ and the four angles that specify ${\boldsymbol{s}}_1$ and ${\boldsymbol{s}}_2$ in the inertial ${\boldsymbol{N}}$ frame. We observe that \[${\boldsymbol{N}} \cdot {\boldsymbol{l}} (x_0) - {\boldsymbol{N}} \cdot {\boldsymbol{k}} (x_0)$\] is maximum for equal mass maximally spinning compact binaries and the difference is usually less than $0.1 \%$. The value of ${\boldsymbol{N}} \cdot {\boldsymbol{l}} (x_0) $ specifies the initial orientation of ${\boldsymbol{l}}$ in the inertial ${\boldsymbol{N}}$ frame as we usually let its azimuthal angle to be zero along with $\Phi_p(x_0)$ (we have verified that the changes in these angles play no role in our match computations). The difference in ${\boldsymbol{N}} \cdot {\boldsymbol{l}} (x_0)$ and ${\boldsymbol{N}} \cdot {\boldsymbol{k}} (x_0)$ values leads to slightly different values for ${\boldsymbol{k}} \cdot {\boldsymbol{s}}_1$ and ${\boldsymbol{l}} \cdot {\boldsymbol{s}}_1$ at the initial epoch. Therefore, the orbital frequency and phase evolutions are slightly different in the two approaches even in the absence of our 3PN order additional terms. We observe that the differences in these two dot products, namely ${\boldsymbol{k}} \cdot {\boldsymbol{s}}_1$ and ${\boldsymbol{l}} \cdot {\boldsymbol{s}}_1$, have their maximum values for equal mass maximally spinning compact binaries. In the next section, we purse the [*match*]{} computations to probe the implications of additional 3PN order terms present in the frequency and phase evolution equations while constructing our ${\boldsymbol{k}}$-based inspiral waveforms.
Match computations involving the above two families of inspiral waveforms {#result}
==========================================================================
We employ the [*match*]{}, detailed in [@DIS98; @BO96], to compare inspiral waveforms constructed via the above described ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based precessing conventions. Our comparison is influenced (and justified) by the fact that the precessing source frames of the above two conventions are functionally identical. This should be evident from the use of [*the same*]{} precessional frequency, appropriate for ${\boldsymbol{k}}$, to obtain PN accurate expressions for both the ${\boldsymbol{l}}$-based ${\boldsymbol{\Omega}}_e^l $ and ${\boldsymbol{k}}$-based ${\boldsymbol{\Omega}}_e $. Therefore, the [*match*]{} estimates probe influences of the additional 3PN order terms present in the differential equations for $\Phi$ and $x$ in our approach. Additionally, we have verified that these 3PN order terms are not present in the usual implementation of the precessing convention as provided by the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code.
Our match ${\cal M}(h_l, h_k)$ computations involve $h_l $ and $ h_k$, the two families of inspiral waveforms arising from the ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based precessing conventions. The $ h_l$ inspiral waveform families are adapted from the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code of LSC while $ h_k$ families, as expected, arise from our approach (equations (\[Eq\_hp\_hx\])). It should be noted that we employ the quadrupolar (Newtonian) order expressions for $h_{\times, +}$ while computing $h_l$ and $h_k$ in the present analysis. However, the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code can provide waveforms that include 1.5PN order corrections to their amplitudes. We would like to stress that the two families involved in our ${\cal M}(h_l, h_k)$ computations are characterized by identical values of $m, \eta, \chi_1 $ and $ \chi_2$. Additionally, the initial orientations of the two spins in the ${\boldsymbol{N}}$-based inertial frame were also chosen to be identical. The computation of ${\boldsymbol{N}} \cdot {\boldsymbol{l}}$ from ${\boldsymbol{N}} \cdot {\boldsymbol{k}}$ with the help of equation (\[Eq\_l\_k\]) ensures that ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$ orientations at the initial epoch are physically equivalent. Therefore, our ${\cal M}(h_l, h_k)$ computations indeed compare two waveform families with physically equivalent orbital and spin configurations at the initial epoch. To obtain a specific ${\cal M}(h_l, h_k)$ estimate, we first compute an overlap function between the relevant $h_l $ and $ h_k$ inspiral waveforms: $$\mathcal{O}(h_l, h_k) = < \hat h_l, \hat h_k> =
\frac{\langle h_l|h_k \rangle}{\sqrt{\langle h_l|h_l \rangle \, \langle h_k|h_k\rangle}} \,,$$ where $\hat h_l$ and $\hat h_k$ stand for the normalized $h_l(t)$ and $h_k(t)$ waveforms, respectively. The angular bracket between $h_l$ and $h_k$ defines certain noise weighted inner product, namely $$\langle h_l | h_k \rangle= 4\, {\rm Re}\, \int_{f_{\rm low}}^{f_{\rm cut}} \,
\frac{\tilde h_l^*(f)\, \tilde h_k(f)}{S_{\rm h}(f)} df \,.$$ In the above equation, $\tilde h_l(f)$ and $\tilde h_k(f)$ stand for the Fourier transforms of $h_l(t)$ and $h_k(t)$, while $S_{\rm h}(f)$ denotes the one-sided power spectral density. We have invoked the zero-detuned, high power sensitivity curve of aLIGO [@LIGO_2010] in our ${\cal M}(h_l, h_k)$ computations. The upper cut-off frequency $f_{\rm cut}$ is chosen to be $c^3/(G\, m\, \pi\, 10^{3/2})$ while the lower cut-off frequency $f_{\rm low}$, associated with the GW detector, equals $10$ Hz. The match ${\cal M}(h_l,h_k)$ is computed by maximizing the $\mathcal{ O}(h_l, h_k)$ over two extrinsic variables, namely the time of arrival $t_0$ and the overall phase $\phi_0$ of GW at time $t_0$ [@Veitch14]. This leads to $$\label{Eq_match}
{\cal M }= \max_{t_0, \phi_0}\, \mathcal{O}(h_l, h_k)\,.$$ The maximizations of over $t_0$ and $\phi_0$ are performed by following [@DIS98]. We perform the maximization over $t_0$ via the FFT algorithm while the maximization over $\phi_0$ requires us to deploy two orthogonal templates. Let us emphasize that we terminate $h_l $ and $ h_k$ inspiral waveform families when their respective $x$ parameters reach $0.1$ which roughly corresponds to orbital separations $\sim 10 \, G\, m/c^2 $. This choice arguably ensures the validity of PN approximation to describe the temporal evolutions of the above two families in our $[f_{\rm low}$-$f_{\rm cut}]$ frequency window. Therefore, it is reasonable to associate the departure of ${\cal M}$ estimates from unity to the additional 3PN order contributions to the differential equations for the orbital phase and the associated angular frequency. We move on to list results of some of our ${\cal M}$ computations. It is not very difficult to realize that extensive ${\cal M}$ computations that deal with all aLIGO relevant spin and binary configurations will be rather difficult to achieve. Therefore, we restrict our attention to a selected number of binaries to compare the $h_l(t)$ and $h_k(t)$ inspiral waveform families. In our match computations, we mainly consider binaries having total mass $m\geq30M_{\odot}$ due to the following two reasons. First, it is comparatively expensive (computationally) to generate lengthy inspiral waveforms for low mass binaries in the aLIGO frequency window. Additionally, we are interested in to explore the dependence of our ${\cal M}$ estimates on the mass ratio $q$ in the $[1-10]$ range. Clearly, low $m$ binaries can lead to secondary BHs having masses lower than the usual neutron star masses for high $q$ cases. In figure \[figure:q\_M\_Phi\], we consider maximally spinning BH binaries, characterized by the total mass $m=30M_{\odot}$, while varying the mass ratio $q$ from unity to $10$. The left and right panel plots are for configurations having initial dominant spin-orbit misalignments $\tilde\theta_1(x_0)$ $(\cos^{-1}({\boldsymbol{k}} \cdot {\boldsymbol{s}}_1))$ given by $30^{\circ}$ and $ 60^{\circ}$, respectively. Additionally, we let the initial orbital plane orientation in the ${\boldsymbol{N}}$-based inertial frame to take two values leading to edge-on ($\iota_0= 90^{\circ}$) and face-on ($\iota_0= 0^{\circ}$) binary orientations. These binary configurations, characterized by two different spin-orbit misalignments and orbital plane orientations in the ${\boldsymbol{N}}$-based inertial frame, are obtained by choosing appropriately different values for the more massive BH initial spin orientation, namely $\theta_{1}(x_0)$ in equation (\[eq:s1\_s2\_1\]). For example, we choose $\theta_{10}$ to be $30^{\circ}$ and $60^{\circ}$ to get $\tilde\theta_1(x_0)$ equals $30^{\circ}$ and $60^{\circ}$, respectively, for face-on binaries. However, in the case of edge-on binaries, we require to choose $\theta_{10}$ to be $60^{\circ}$ and $30^{\circ}$, respectively. All other initial angular variables, appearing in equations (\[eq:s1\_s2\_1\]) and (\[eq:s1\_s2\_2\]), were chosen to be $\phi_{10}=0^{\circ}$, $\theta_{20}=20^{\circ}$, $\phi_{20}=90^{\circ}$ (we have verified that the ${\cal M}$ estimates are rather insensitive to the choice of these initial angular variables). Let us note that ${\boldsymbol{l}}$ orientations (from ${\boldsymbol{N}}$) for these configurations will be slightly different from $0^{\circ}$ or $90^{\circ}$ due to the 1.5PN accurate relation between ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$.
$\begin{array}{ccc}
\includegraphics[width=62mm,height=60mm]{Plots_M_Phi_q_ks1_30_10m_T4_14Feb15.eps}&
\includegraphics[width=62mm,height=60mm]{Plots_M_Phi_q_ks1_60_10m_T4_14Feb15.eps}
\end{array}$
The upper and lower row plots in figure \[figure:q\_M\_Phi\] are for $\Delta \Phi$, the accumulated orbital phase differences in the $[f_{\rm low}$-$f_{\rm cut}]$ frequency interval while dealing with the $h_l $ and $ h_k$ inspiral waveforms and the associated ${\cal M}(h_l,h_k)$ estimates, respectively. We find that the variations in ${\cal M}$ estimates are quite independent of the initial orbital plane orientations. However, variation in match estimates do depend on the initial dominant spin-orbit misalignment and the mass ratio $q$. The left panel plots show gradual decrease in ${\cal M}$ values as we increase the $q$ value and this variation is reflected in the gradual increase of $\Delta \Phi$. Incidentally, this pattern is also observed for configurations having somewhat smaller initial dominant spin-orbit misalignments. However, the ${\cal M}$ estimates are close to unity for tiny $\tilde \theta_1(x_0)$ values and this is expected as precessional effects are minimal for such binaries. Therefore, the effect of the above discussed additional 3PN order terms are more pronounced for high $q$ compact binaries having [*moderate*]{} dominant spin-orbit misalignments. This picture is modified for binaries having substantial dominant spin-orbit misalignments as evident from the ${\cal M}$ plots in the right panel of figure \[figure:q\_M\_Phi\]. For such binaries, the ${\cal M}$ estimates dip to a minimum and recover as we vary $q$ from $1$ to $10$ for both edge-on and face-on orbital plane orientations. In contrast, we observe a gradual increase in $\Delta \Phi (q)$. The monotonic increment in $\Delta \Phi (q) $ plots is essentially due to the presence of $X_1-X_2$ terms in the additional 3PN order contributions to $dx/dt$, given by equation (\[Eq\_dxdt\]). This is because $X_1-X_2 $ terms are absent for equal mass binaries which leads to smaller $\Delta \Phi $ estimates for smaller $q$ value binaries. However, it is not possible to explain the observed ${\cal M}(q)$ variations purely in terms the displayed $\Delta \Phi (q)$ values, especially when precessional effects are substantial as in the $\tilde\theta_1(x_0)=60^{\circ}$ cases. We observe that the initial values of angles like $\cos^{-1}({\boldsymbol{k}} \cdot {\boldsymbol{j}})$ and $\cos^{-1}({\boldsymbol{N}} \cdot {\boldsymbol{j}})$ also influence the precessional modulations in the waveforms, where ${\boldsymbol{j}}$ is the unit vector along ${\boldsymbol{J}}$. Therefore, we examined how the initial values of these angles vary as functions of $q$. We find monotonic increase (decrease) in the initial values of $\cos^{-1}({\boldsymbol{k}} \cdot {\boldsymbol{j}})$ $( \cos^{-1}({\boldsymbol{N}} \cdot {\boldsymbol{j}}) )$ when $q$ value is varied from 1 to 10. Therefore, we speculate that the combined effect of such variations and the non-negligible $\Delta \Phi (q) $ values may provide a possible explanation for the dip in match estimates around q=5. This is because initial values of the above mentioned angles do define the way various dot products, involving ${\boldsymbol{n}}$, ${\boldsymbol{\xi}}$, ${\boldsymbol{s}}_1$ and ${\boldsymbol{s}}_2$, vary in time. It should be noted that these dot products are present in the additional 3PN order terms in the $dx/dt$ expression given by equation (\[Eq\_dxdt\]).
This plausible explanation is tested in figure \[figure:th1\_M\_Phi\], where we plot $\Delta \Phi (q) $ and ${\cal M} (q)$ estimates for binary configurations having three different $m$ values. It is not difficult to infer that the above listed arguments should also hold for such compact binaries. The initial dominant spin-orbit misalignment is again chosen to be $60^{\circ}$ while considering only edge-on configurations. All other spin and orbital orientations at the initial epoch are identical to the cases displayed in the right panel plots of figure \[figure:q\_M\_Phi\]. Clearly, the plots in figure \[figure:th1\_M\_Phi\] are qualitatively similar to those in the right panel plots of figure \[figure:q\_M\_Phi\]. However, the dip in ${\cal M}$ estimates shifts to higher $q$ values for higher $m$ compact binaries. We conclude from the above two figures that GW data analysis relevant differences between the above two inspiral families are more pronounced for unequal mass BH binaries. Incidentally, we also find similar behavior while plotting ${\cal M}(q)$ and $\Phi(q)$ for maximally spinning $m = 20\,M_{\odot}$ BH binaries. In this case, the minimum ${\cal M}$ value occurs around $q=3$ and this is consistent with the trend observed in figure \[figure:th1\_M\_Phi\]. It is reasonable to suspect that the spin-squared additional terms are influential only for maximally spinning BH binaries. However, we observe qualitatively similar ${\cal M} (q)$ estimates for binary configurations having moderately spinning BHs ($\chi_1=\chi_2=0.75$) as well as having mildly spinning BHs ($\chi_1=\chi_2=0.5$). Moreover, the ${\cal M} (q)$ computations indicate that the effect of additional 3PN order terms in $d\Phi/dt$ and $dx/dt$ equations are non-negligible even for single-spin compact binaries. In figure \[figure:th1\_M\_Phi\_LN\], we plot $\Delta \Phi (q) $ and ${\cal M} (q)$ while considering ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based single-spin waveforms for $m=30M_{\odot},\tilde\theta_1(x_0)=60^{\circ} $ binaries in edge-on orientations. We find that these plots fairly resemble their double spin counterparts shown in the right panel plots of figure \[figure:q\_M\_Phi\]. Additionally, the variations in the initial values of $\cos^{-1}({\boldsymbol{k}} \cdot {\boldsymbol{j}})$ and $ \cos^{-1}({\boldsymbol{N}} \cdot {\boldsymbol{j}}) $ are similar to the double spin binaries, while varying $q$ value from $1$ to $10$. Therefore, we conclude that it may be beneficial to keep the additional 3PN order terms in the differential equations for $\Phi$ and $x$ while modeling inspiral GWs even from single-spin binaries via the precessing convention.
![ Plots that display variations in $\Delta \Phi$ and $\mathcal{M}$ as functions of $q$ for edge-on maximally spinning BH binaries having $m=30\,M_{\odot},40\,M_{\odot}$ and $50\,M_{\odot} $ while keeping $\tilde\theta_1(x_0)$ to be $60^{\circ}$. All other initial angular parameters are identical to those used in the right panel plots of figure \[figure:q\_M\_Phi\]. The position of the dip in $\mathcal{M}(q)$ plots is shifted towards higher $q$ values for higher $m$ binaries. A possible explanation relies on the more influential contributions from the various dot products that appear in equation (\[Eq\_dxdt\]) for such binaries having $q$ roughly in the $4$ to $ 9$ range. []{data-label="figure:th1_M_Phi"}](Plots_M_Phi_q_Diff_M_10m_T4_14Feb15.eps){width="82mm" height="74mm"}
![ Plots that compare *single-spin* SpinTaylorT4 inspiral waveform families in the ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based approaches. These $\Delta \Phi(q)$ and $\mathcal{M}(q)$ plots are for edge-on $m=30M_{\odot} $ binaries containing maximally spinning dominant BH having $\tilde \theta_1(x_0)=60^{\circ}$. The qualitative and quantitative nature of these plots are essentially identical to the double spin binaries of figure \[figure:q\_M\_Phi\]. []{data-label="figure:th1_M_Phi_LN"}](Plots_M_Phi_q_30_60_OneSpin_14Sep15.eps){width="82mm" height="74mm"}
The LSC also developed [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT2 code that employs a slightly different version of $dx/dt$ equation while implementing the ${\boldsymbol{l}}$-based precessing convention. Note that it is possible to construct various PN approximants to model inspiral GWs by using the same PN accurate expressions for the conserved energy, far-zone energy flux and the [*energy balance argument*]{} [@Boyle_07]. The differential equation for $x$ in the SpinTaylorT2 approximant arises from the following considerations. We begin by displaying (symbolically) the 3.5PN accurate expression for $dx/dt$, employed in the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code, as $$\begin{aligned}
\frac{dx}{dt}\Big |_{\rm T4}= \frac{64}{5}\frac{c^3}{Gm}\eta\, x^5\, \Big\{ 1+ A_1\, x + A_{1.5}\, x^{3/2} + A_{2}\, x^2 + A_{2.5}\, x^{5/2} + A_{3}\, x^3 + A_{3.5}\, x^{7/2} \Big\}\,,\end{aligned}$$ where the coefficients $A_i$ are functions of $\eta, \pi, \log (6\,x) $ and $\gamma_{\rm E}$ (Euler’s constant) when including only the non-spinning contributions. However, the coefficients from $A_{1.5}$ to $A_{3.5}$ additionally depend on $X_1, X_2, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{l}}),
({\boldsymbol{s}}_2 \cdot {\boldsymbol{l}}),
({\boldsymbol{s}}_1 \cdot {\boldsymbol{s}}_2), \chi_1$ and $\chi_2$ to incorporate various spin contributions. The explicit expressions for these $A_i$ coefficients may be obtained either from equation (3.16) in [@Bohe2013] or from the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code itself. With these inputs, one defines the differential equation for $x$ in the SpinTaylorT2 approximant to be [$$\begin{aligned}
\label{Eq_T2}
\frac{dx}{dt}\Big |_{\rm T2}&= \frac{64}{5}\frac{c^3}{Gm}\eta\, \frac{x^5}{ \Big\{ 1+ A_1\, x + A_{1.5}\, x^{3/2} + A_{2}\, x^2 + A_{2.5}\, x^{5/2} + A_{3}\, x^3 + A_{3.5}\, x^{7/2} \Big\}^{-1}}\,, \\
&=\frac{64}{5}\frac{c^3}{Gm}\eta\, \frac{x^5}{ \Big\{ 1+ A'_{1}\, x + A'_{1.5}\, x^{3/2} + A'_{2}\, x^2 + A'_{2.5}\, x^{5/2} + A'_{3}\, x^3 + A'_{3.5}\, x^{7/2} \Big\}}\,.\end{aligned}$$ ]{} This construction ensures that the coefficients $A'_i$ are going to depend on various $A_i$ coefficients due to the binomial expansion of the denominator of equation (\[Eq\_T2\]) that includes all the $x^{7/2}$ contributions. For example, the $A'_{3.5}$ coefficient is going to depend explicitly on all the $A_i$ coefficients whereas the $A'_{2}$ coefficient depends only on the $A_1$ and $A_2$ coefficients. We observe that the resulting spin-squared terms are usually not incorporated into the $A'_3$ and $A'_{3.5}$ coefficients in the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT2 code of LSC.
![ Plots for $\Delta \Phi$ and ${\cal M}$ as functions of $q$ that invoke ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based SpinTaylorT2 approximants. We are dealing with edge-on maximally spinning BH binaries having $m=30\, M_{\odot} $ and $\tilde \theta_1 (x_0) =60^{\circ}$. The $\Delta \Phi$ and ${\cal M}$ estimates are similar to those obtained using SpinTaylorT4 approximant as shown in the right panel plots of figure \[figure:q\_M\_Phi\]. []{data-label="figure:M_Phi_q_T2"}](Plots_M_Phi_q_30_60_90_T2_20Feb15.eps){width="82mm" height="74mm"}
We construct the differential equation for $x$ in our SpinTaylorT2 approximant in a similar manner. Therefore, the various $A'_i$ coefficients are given in terms of $A_i$ coefficients which can be extracted from our equations (\[Eq\_dxdt\]), (\[Eq\_dxdt\_l\]) and (\[eq\_sk\]) for $dx/dt$. For example, the expressions for $A_2$ and $A'_2$ in our approach are given by [$$\begin{aligned}
A_2 &= \Bigl ( \frac{34103}{18144}+\frac{13661}{2016}\eta +\frac{59}{18}\eta^2 \Bigr )
- \frac{1}{48} \eta \chi_1 \chi_2 \,\Bigl ( 247 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{s}}_2)
-721 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}}) ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}}) \Bigr ) \nonumber \\
&\quad+X_1^2\, \chi_1^2\, \Bigl( \frac{5}{2}\, (3\, ( {\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2-1)+\frac{1}{96}\, (7-({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2)\Bigr) \nonumber \\
&\quad+X_2^2\, \chi_2^2\, \Bigl( \frac{5}{2}\, (3\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2-1)
+\frac{1}{96}\, (7-({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2)\Bigr) \,, \\
A'_2&= -A_2 + A_1^2 \,, \nonumber \\
&= -\Big [ \Bigl ( \frac{34103}{18144}+\frac{13661}{2016}\eta +\frac{59}{18}\eta^2 \Bigr )
- \frac{1}{48} \eta \chi_1 \chi_2 \,\Bigl ( 247 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{s}}_2)
-721 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}}) ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}}) \Bigr ) \nonumber \\
&\quad+X_1^2\, \chi_1^2\, \Bigl( \frac{5}{2}\, (3\, ( {\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2-1)+\frac{1}{96}\, (7-({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2)\Bigr) \nonumber \\
&\quad+X_2^2\, \chi_2^2\, \Bigl( \frac{5}{2}\, (3\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2-1)
+\frac{1}{96}\, (7-({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2)\Bigr) \Big ]
+ \left [ -\frac{743}{336}-\frac{11\eta}{4}\right ]^2 \,. \end{aligned}$$ ]{} In the ${\boldsymbol{k}}$-based SpinTaylorT2 approximant, we incorporate our additional 3PN accurate spin-spin terms which contribute [*only*]{} to the $A_3'$ coefficient. In other words, $A_3' = -A_3 + A_{1.5}^2 + 2A_1A_2 - A_1^3$ contains our additional 3PN order spin-squared terms via the SpinTaylorT4 based $A_3$ coefficient. We would like to emphasize that we are not incorporating possible 3PN and 3.5PN order spin-spin terms arising due the binomial expansion. This is to make sure that the ${\boldsymbol{l}}$ and ${\boldsymbol{k}}$-based SpinTaylorT2 approximants differ only by our additional 3PN order spin-spin terms. We explore the implications of these additional 3PN order terms again with the help of ${\cal M}(h_l, h_k)$ computations, where $h_l$ and $h_k$ now stand for the ${\boldsymbol{l}}$-based and ${\boldsymbol{k}}$-based waveform families that employ the above described PN accurate expressions for $dx/dt$. In figure \[figure:M\_Phi\_q\_T2\], we plot ${\cal M}(q)$ and $\Delta \Phi(q)$ for edge-on $m=30\,M_{\odot}$ maximally spinning BH binaries having dominant spin-orbit misalignment equal to $60^{\circ}$ at $x_0$. The ${\cal M}$ and $\Delta \Phi$ estimates are similar to those obtained using SpinTaylorT4 approximant as shown in the right panel plots of figure \[figure:q\_M\_Phi\].
It may be argued that the inclusion of our 3PN order spin-squared terms to the differential equation for $x$ is not desirable. This is because we do not incorporate 3PN order contributions to $dx/dt$ that arise from the next-to-leading order spin-spin interactions. These terms are not included as they are yet to be computed in the literature. Clearly, the present match computations should be repeated when these 3PN order contributions are explicitly available. However, this should not prevent one from probing the data analysis implications of the adiabatic approximation employed in the ${\boldsymbol{L}}_{\rm N}$-based precessing convention of [@BCV]. This could be especially useful while invoking PN accurate differential equation for $x$ that incorporate higher order spin contributions as pursued in [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 and [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT2 codes, developed by the LSC to implement the ${\boldsymbol{L}}_{\rm N}$-based precessing convention. Hopefully, the data analysis implications of our 3PN order terms should provide additional motivation to explicitly obtain the next-to-leading order spin-spin contributions to $dx/dt$. In contrast, it should be noted that the additional PN contribution to $d \Phi/dt$, given by equation (\[Eq\_phidot\]), are fully accurate to 3PN order. The fact that such additional precessional effects enter the expression for $d \Phi/dt$ only at the 3PN order should justify naming our approach as ${\boldsymbol{k}}$-based precessing convention. It is worthwhile to remember that precession induced modulations enter the differential equation for the orbital phase at 1.5PN order in the absence of precessing convention.
Conclusions {#Sec_dis}
===========
We developed a precessing convention to model inspiral GWs from generic spinning compact binaries that employs PN accurate orbital angular momentum ${\boldsymbol{L}}$ to describe binary orbits and to construct the required precessing source frame. The main motivation for our approach is the usual practice of using PN accurate precessional equation, appropriate for ${\boldsymbol{L}}$, to evolve the Newtonian orbital angular momentum ${\boldsymbol{L}}_{\rm N}$ while constructing inspiral waveforms. We showed that this practice leads to higher order PN corrections to $\dot {\Phi}_{p} = \omega$ equation. A set of differential equations and the quadrupolar order GW polarization states in certain frame-less convention are developed to model inspiral GWs in our ${\boldsymbol{L}}$-based approach. We explained why the differential equations for the orbital phase and frequency will have additional 3PN order terms in our approach compared to the usual ${\boldsymbol{L}}_{\rm N}$-based implementation of precessing convention. The influence of these additional 3PN order terms were explored with the help of [*match*]{} computations involving ${\boldsymbol{L}}$ and ${\boldsymbol{L}}_{\rm N}$-based inspiral waveforms for spinning compact binaries with physically equivalent orbital and spin configurations at the initial epoch. We adapted both [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 and [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT2 codes, developed by the LSC, while implementing ${\boldsymbol{L}}_{\rm N}$-based precessing convention. The resulting match estimates indicate that our additional 3PN order terms should not be neglected for a substantial fraction of unequal mass BH binaries. It will be useful to pursue our match computations for an extended range of the relevant parameter space. The present computations should also be extended by invoking 1.5PN order amplitude corrections to both families of inspiral waveforms in the frame-less convention (the [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 and [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT2 codes do incorporate such amplitude corrections). This requires us to add certain additional 1.5PN order amplitude corrections to the usual expressions for $h_{\times}$ and $ h_+$ as noted in [@GG1]. Investigating the influence of these additional 3PN (1.5PN) order terms in phase (amplitude) while estimating the GW measurement accuracies of compact binary parameters will be interesting. It should also be worthwhile to probe the influence of these additional terms during the construction of inspiral-merger-ringdown waveforms from generic spinning BH binaries with the help of the effective-one-body approach.
3.5PN accurate expression for $dx/dt$ {#appendix}
======================================
We list below the regular contributions to $dx/dt$ that we employ while implementing our version of the SpinTaylorT4 approximant. These contributions, adapted from [@Bohe2013], read [$$\begin{aligned}
\label{Eq_dxdt_l}
\frac{ d x}{dt} &= \frac{64}{5}\frac{c^3}{Gm}\eta\, {x}^5
\biggl \{
1+x \left [ -\frac{743}{336}-\frac{11\eta}{4}\right ] + x^{3/2}\,\biggl [4\, \pi -\frac{47}{3}\, s_{k} -\frac{25}{4}\, (X_1-X_2)\,\sigma_{k} \biggr ] \nonumber \\
&\quad+x^2\,\biggl[\Bigl ( \frac{34103}{18144}+\frac{13661}{2016}\eta +\frac{59}{18}\eta^2 \Bigr )
- \frac{1}{48} \eta \chi_1 \chi_2 \,\Bigl ( 247 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{s}}_2)
-721 ({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}}) ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}}) \Bigr ) \nonumber \\
&\quad+X_1^2\, \chi_1^2\, \Bigl( \frac{5}{2}\, (3\, ( {\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2-1)+\frac{1}{96}\, (7-({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})^2)\Bigr) \nonumber \\
&\quad+X_2^2\, \chi_2^2\, \Bigl( \frac{5}{2}\, (3\, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2-1)
+\frac{1}{96}\, (7-({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})^2)\Bigr)\biggr] \nonumber \\
&\quad+ x^{5/2}\,\biggl[-\frac{4159}{672}\, \pi-\frac{5861}{144}\, s_{k}-\frac{809}{84}\, (X_1-X_2)\, \sigma_{k} \nonumber \\
&\quad+\eta\, \Bigl( -\frac{189}{8}\, \pi +\frac{1001}{12}\, s_{k} + \frac{281}{8}\, (X_1-X_2)\, \sigma_{k}\Bigr) \biggr] \nonumber \\
&\quad + x^3\, \biggl[ \frac{16447322263}{139708800} +\frac{16}{3}\, \pi^2 -\frac{1712}{105}\, \gamma_{\rm E}-\frac{856}{105}\, \ln[16x]
-\frac{188}{3}\, \pi\, s_{k} \nonumber \\
&\quad-\frac{151}{6}\, \pi\, (X_1-X_2)\, \sigma_{k}+\eta\, \Bigl( -\frac{56198689}{217728}+\frac{451}{48}\, \pi^2\Bigr)
+\frac{541}{896}\, \eta^2-\frac{5605}{2592}\, \eta^3 \biggr] \nonumber \\
&\quad+x^{7/2}\, \biggl[ \Bigl(-\frac{4323559}{18144}+\frac{436705}{672}\, \eta - \frac{5575}{27}\, \eta^2\Bigr)\, s_{k} \nonumber \\
&\quad+(X_1-X_2)\, \Bigl( -\frac{1195759}{18144}+\frac{257023}{1008}\, \eta-\frac{2903}{32}\, \eta^2 \Bigr)\, \sigma_{k} \nonumber \\
&\quad+\pi\, \Bigl( -\frac{4415}{4032}+\frac{358675}{6048}\, \eta + \frac{91495}{1512}\, \eta^2\Bigr)\biggr]
\biggr \} \,,\end{aligned}$$ ]{} where $s_k$ and $\sigma_k$ are given by
\[eq\_sk\] $$\begin{aligned}
s_{k}&= X_1^2\, \chi_1 \, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}}) +X_2^2\, \chi_2 \, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}})\,, \\
\sigma_{k}&= X_2\, \chi_2 \, ({\boldsymbol{s}}_2 \cdot {\boldsymbol{k}}) -X_1\, \chi_1 \, ({\boldsymbol{s}}_1 \cdot {\boldsymbol{k}})\,.\end{aligned}$$
We would like to stress that in the above equations we have merely replaced ${\boldsymbol{l}}$ appearing in equation (3.16) of [@Bohe2013] by ${\boldsymbol{k}}$. This differential equation for $x$ is also employed in the usual implementation of the SpinTaylorT4 approximant, as provided by [<span style="font-variant:small-caps;">lalsuite</span>]{} SpinTaylorT4 code, while using ${\boldsymbol{l}}$ to describe the binary orbits.
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|
---
abstract: 'Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourier domain. While this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive with classic Nyquist-based schemes. This is especially due to the observation that soliton pulses generally exhibit a large time-bandwidth product. We consider the phase modulation of spectral amplitudes of higher order solitons, taking into account their varying spectral and temporal behavior when propagating along the fiber. For second and third order solitons, we numerically optimize the pulse shapes to minimize the time-bandwidth product. We study the behavior of multi-soliton pulse duration and bandwidth and generally observe two corner cases where we approximate them analytically. We use these results to give an estimate on the minimal achievable time-bandwidth product per eigenvalue.'
author:
- 'Alexander Span, Vahid Aref, Henning Bülow, and Stephan ten Brink [^1][^2]'
bibliography:
- 'references\_nft.bib'
title: 'Time-Bandwidth Product Perspective for Multi-Soliton Phase Modulation'
---
Multi-Soliton, NFT, Nonlinear Fiber Transmission, Time-Bandwidth Product.
Introduction
============
optical technology has increased the transmission capacities of fiber channels to a point where the Kerr-nonlinearity becomes a limiting factor. Increasing transmit power leads to strong nonlinear distortions which need to be compensated. This is, however, quite complex and achieves only small gains, since the signal interacts nonlinearly with the noise along the channel.
The Nonlinear Fourier Transform (NFT), that has recently regained attention [@yousefi2014nft], shows a potential path to dealing with nonlinear fiber channels, as described by the Nonlinear Schroedinger Equation (NLSE). The NFT maps pulses to a nonlinear spectrum where the nonlinear crosstalk is basically absent, and the signal evolution along the fiber can be described by simple equations. This motivates to modulate data in this nonlinear spectral domain. The nonlinear spectrum consists of two parts: The *continuous spectrum*, which is analogous to the conventional Fourier transform, and the *discrete spectrum* which describes the solitonic component of a pulse. Multi-solitons of order $N$ ($N-$solitons), as considered in this paper, are special solutions to the NLSE with discrete nonlinear spectrum only, and can be described in the nonlinear spectrum by $N$ pairs of eigenvalue and a corresponding spectral amplitude.
Modulation of the continuous spectrum has been studied theoretically and experimentally in [@le2014nonlinear; @le2016; @Le2017b; @prilepsky2014nonlinear; @gemechu2017] and, so far, shows a higher spectral efficiency when compared to discrete spectrum modulation. Up to $2.3$ $\frac{\mathrm{bit}}{\mathrm{s \, Hz}}$ has been demonstrated in [@Le2018]. Combined modulation of both the discrete and the continuous spectrum was demonstrated in [@Tavakkolnia2015; @aref2016condis; @Aref2018]. Discrete spectrum modulation has been intensively studied already two decades ago for on-off keying of first order solitons [@mollenauer2006solitons]. Multi-solitons have been proposed to increase the spectral efficiency [@yousefi2014nft]. Since then, eigenvalue and spectral amplitude modulation have been introduced and experimentally demonstrated.
Characterization of the spectral efficiency for soliton transmission is, however, still an open problem. First, the noise statistics in the nonlinear spectrum are not fully understood. Second, there is generally no closed-form solution for multi-solitons and their pulse duration and bandwidth. Furthermore, the temporal and (linear) spectral properties may change due to modulation and during propagation. It is, therefore, not clear how to modulate the discrete spectrum in order to achieve a small time-bandwidth product. This is, however, important as the spectral efficiency will be proportional to the inverse of the time-bandwidth product. We have already investigated the behavior of pulse duration and bandwidth for multi-solitons of order $N=2,3$ with imaginary eigenvalues in [@Span2017]. We also have *numerically optimized* their pulse shapes to minimize the time-bandwidth product in a spectral phase modulation scenario. In this paper, we extend that work and give an *analytical estimation* for general soliton orders and also consider eigenvalues with nonzero real part. We consider a scenario where the spectral amplitudes of $N$ predefined eigenvalues are independently phase modulated. We give a definition of the time-bandwidth product that takes into account the variations due to modulation and propagation along an ideal fiber. We give some general properties for multi-solitons that preserve the time-bandwidth product and reduce the number of optimization parameters. We numerically optimize the eigenvalue constellation and the absolute value of the corresponding spectral amplitudes to minimize this time-bandwidth product of the resulting multi-soliton pulses for $N \leq 3$. We distinguish the cases of purely imaginary eigenvalues or eigenvalues parallel to the real axis. Studying the behavior of multi-soliton pulse duration and bandwidth, we generally observe two limit cases, where either the pulse duration or the bandwidth becomes minimal, respectively. We derive analytical approximations for them in these limit cases. We use those results to estimate the minimal achievable time-bandwidth product for general soliton orders $N$. Numerical observations show that this may serve as a conjectured lower-bound.
The paper is outlined as follows: In Sec. \[sec:pre\], we review the basics of the NFT and some numerical algorithms. In Sec. \[sec:multi\_soliton\_transmission\], recent works on soliton modulation are summarized. We then motivate the time-bandwidth product optimization for the soliton phase modulation scenario regarding spectral efficiency and give a reasonable definition for the time-bandwidth product. The soliton pulse optimization problem is statet in Sec. \[sec:num\_pulse\_opt\] and numerical optimization resulting in the ideal soliton pulses are presented. In Sec. \[sec:TBP\_higherN\], analytical approximations for the minimal achievable time-bandwidth product are derived. Conclusions are given in Sec.\[sec:conclusion\].
Preliminaries - NFT and INFT {#sec:pre}
============================
In this section, we briefly introduce the basics of NFT and its notation and show how multi-solitons can be constructed via the Inverse NFT (INFT) using the Darboux transform [@matveev1991darboux].
The pulse propagation in single polarization along an ideal lossless and noiseless fiber is characterized by the normalized standard Nonlinear Schr[ö]{}dinger Equation (NLSE) $$\label{NLSE}
\frac{\partial}{\partial z}q(t,z)+j\frac{\partial^2}{\partial t^2}q(t,z)+2j|q(t,z)|^2q(t,z)=0.$$ The physical pulse $Q(\tau,\ell)$ at location $\ell$ along the fiber is then described by $$\label{eq:Q_physical}
Q\left(\tau,\ell\right)=\sqrt{P_0}\,\,\, q\left(\frac{\tau}{T_0},\ell \frac{\left|\beta_2\right|}{2T_0^2}\right) \text{ with } P_0\cdot T_0^2=\frac{\left|\beta_2\right|}{\gamma},$$ where $\beta_2<0$ is the chromatic dispersion, $\gamma$ is the Kerr nonlinearity of the fiber, and $T_0$ determines the symbol rate. The closed-form solutions of the NLSE can be described in a nonlinear spectrum defined by the following so-called Zakharov-Shabat system [@shabat1972exact] $$\label{eq:ZS}
\frac{\partial }{\partial t}\left(\begin{matrix}u_1(t;z)\\ u_2(t;z)\end{matrix}\right)=
\left(\begin{matrix}
0 & q\left(t,z\right) e^{2j\lambda t} \\-q^*\left(t,z\right) e^{-2j\lambda t} & 0
\end{matrix}\right)
\left(\begin{matrix}u_1(t;z)\\ u_2(t;z)\end{matrix}\right),
\qquad
\lim_{t \to -\infty} \left(\begin{matrix}u_1(t;z)\\ u_2(t;z)\end{matrix}\right) \to \left(\begin{matrix}1 \\ 0 \end{matrix}\right)
%\mathrm{for} \quad t \to -\infty$$ under the assumption that $q(t,z)\to 0$ decays sufficiently fast as $|t|\to \infty$ (faster than any polynomial). The nonlinear Fourier coefficients (Jost coefficients) are defined as $$\begin{aligned}
a\left(\lambda;z\right) =\lim_{t\to \infty}u_1(t;z)
\qquad \qquad b\left(\lambda;z\right) =\lim_{t\to \infty}u_2(t;z).\end{aligned}$$ The set of simple roots of $a(\lambda;z)$ with positive imaginary part is denotes as $\Omega$. Theses roots are called *eigenvalues* as they do not change in terms of $z$, i.e., $\lambda_k(z)=\lambda_k$. The nonlinear spectrum is usually described by the following two parts:
- Continuous part: spectral amplitude ${Q_c(\lambda;z)=b(\lambda;z)/a(\lambda;z)}$ for real frequencies $\lambda\in\mathbb{R}$.
- Discrete part: $\{\lambda_k,Q_d(\lambda_k;z)\}$ where $\lambda_k\in\Omega$, and $Q_d(\lambda_k;z)=b(\lambda_k;z)/\frac{\partial a(\lambda;z)}{\partial \lambda}|_{\lambda=\lambda_k}$.
Some algorithms to compute the nonlinear spectrum by numerically solving the Zakharov-Shabat system are reviewed in [@yousefi2014nft; @wahls2015fast; @SergeiK.Turitsyn2017].
An $N-$soliton pulse is described by the discrete part only and the continuous part is equal to zero (for any $z$). The discrete part contains $N$ pairs of eigenvalue and the corresponding spectral amplitude, i.e., $\{\lambda_k,Q_d(\lambda_k;z)\},1\leq k\leq N $. We rewrite the eigenvalues as $\lambda_k=j\sigma_k+\omega_k$ with $\sigma_k\in \mathbb{R}^+,\omega_k \in \mathbb{R}$ and denote the spectral phases as $\varphi_k=\arg\left\{Q_d(\lambda_k)\right\}$. An important property of the nonlinear spectrum is its simple linear evolution when the pulse propagates along the nonlinear fiber [@yousefi2014nft]. We define $Q_d(\lambda_k)=Q_d(\lambda_k;z=0)$. $$\label{eq:Qd_evol}
Q_d(\lambda_k;z)=Q_d(\lambda_k)\exp(-4j\lambda_k^2z),$$
This transformation is linear and depends only on its own eigenvalue $\lambda_k$. This property motivates the modulation of data over independently evolving spectral amplitudes.
The INFT is used to map the modulated nonlinear spectrum at the transmitter to the corresponding time domain pulse that is launched into the fiber. Some fast INFT algorithms can be found in [@wahls2015],[@wahls2016]. For the special case of the spectrum without the continuous part, the Darboux transform can generate the corresponding multi-soliton pulse [@matveev1991darboux]. Alg. \[alg:DT2\] shows the pseudo-code of a variant of this transform[@aref2016control]. It generates an $N-$soliton signal $q\left(t\right)$ by recursively adding a pairs of $\{\lambda_k,Q_d(\lambda_k)\}$. The advantage of this algorithm is that it is exact with a low computational complexity and it can be used to express some basic properties of multi-soliton pulses.
Multi-Soliton Transmission {#sec:multi_soliton_transmission}
==========================
Motivation and Scenario {#sec:multi_soliton_motivation}
-----------------------
Soliton pulses are special solutions of the NLSE and are therefore matched to the nonlinear fiber channel. first order solitons have the special property of keeping their pulse shape during propagation along the (ideal) nonlinear fiber. Thus, they have a simple analytical description, and can easily be detected.
The basic idea of eigenvalue modulation for $1-$solitons dates back to [@Hasegawa1993] where 1-out-of-$N$ imaginary eigenvalues were selected for transmission. A corresponding lower bound on the capacity per soliton with eigenvalue modulation was derived in [@Shevchenko2015] and a capacity achieving probabilistic eigenvalue shaping scheme was shown in [@Buchberger2018]. A noise model for the discrete spectrum was first introduced in [@zhang2015spectral]. By using higher order solitons and exploiting the spectral amplitudes, many follow-up studies aimed at designing modulation formats that allow to increase the number of bits being conveyed by a soliton. Classical time domain modulation and detection by means of the corresponding eigenvalues was demonstrated in [@matsuda2014]. Other works directly modulated the discrete spectrum. On-off keying of up to ten predefined eigenvalues was demonstrated in [@dong2015nonlinear; @aref2016onoff]. Multi-eigenvalue position encoding in [@hari2016multieigenvalue] achieved spectral efficiencies above $3$ $\frac{\mathrm{bit}}{\mathrm{s \, Hz}}$. However, only for short fiber lengths (compared to the dispersion length) or for dominating nonlinearity with small dispersion. Exploiting the modulation of the spectral amplitudes has been demonstrated experimentally [@aref2015experimental; @geisler2016] with up to $14$ bits per soliton [@buelow20167eigenvalues]. The noise statistics for the perturbation of the discrete spectrum have been investigated in [@Zhang2017; @Buelow2018]. Correlations between different degrees of freedom in the nonlinear spectrum have been investigated also in [@Gui2017], potentially allowing to further increase the amount of bits per soliton.
Under general conditions, all of these discrete spectrum-based transmission schemes imply the drawback of low spectral efficiency, owing to the, generally, large time-bandwidth product of soliton pulses. For $1-$solitons, their pulse shape is known to be “sech” in time and (linear) frequency domain. Although $N$-th order solitons have $N$ times more degrees of freedom for modulation and can, therefore, ideally carry $N$ times more information bits, they are not necessarily spectrally more efficient. In most of the previous works, authors have either not focused on spectral efficiency or have investigated a “capacity per soliton”, neglecting the time-bandwidth product of the soliton pulses.
Therefore our goal is to investigate the behavior of pulse duration and bandwidth for multi-soliton pulses in an ideal and noiseless transmission scenario. We consider the transmission of multi-solitons with $N$ predefined and fixed eigenvalues $\lambda_k$, $1\leq k \leq N$, along an ideal optical fiber (described by the NLSE), where each spectral amplitude $Q_d(\lambda_k)$ is independently phase modulated. The absolute values of the spectal amplitudes $|Q_d(\lambda_k)|$ are not modulated. Note that each spectral amplitude could carry an arbitrary QAM-constellation. Variation of $|Q_d(\lambda_k)|$, however, leads to pulse position modulation, may decrease spectrally efficiency. For this defined scenario, we optimize multi-soliton pulses to have a small time-bandwidth product.
The pulse shape of a multi-soliton not only depends on $\lambda_k$ and $|Q_d(\lambda_k)|$, but also on the phase combinations of $\varphi_k$. Modulating these phases, therefore, leads to different transmit pulses that can have different pulse durations $T$ and bandwidths $B$. Besides, the $Q_d(\lambda_k)$ are transformed during propagation according to . Since all eigenvalues are necessarily distinct, the spectral amplitudes $Q_d(\lambda_k)$, and thus, pulse shape, pulse duration and bandwidth vary along the fiber. Such pulse changing effects do not occur for Nyquist-based signalling over linear channels. Fig. \[fig:soliton\_propagation\_example\] illustrates two examples of multi-solitons and their propagation along the link. Fig. \[fig:soliton\_propagation\_example\] (a) shows a $3-$soliton with imaginary eigenvalues and Fig. \[fig:soliton\_propagation\_example\] (b) shows a $2-$ soliton with nonzero eigenvalue real part, each for two different initial phase combinations of the $\varphi_k$. The pulse shape is shown at the transmitter, once along the fiber and at the receiver. Fig. \[fig:soliton\_propagation\_example\] (c) and (d) show the evolution of the corresponding bandwidth $B$ and pulse duration $T$ during propagation.
table [3Sol\_example-1\_Tx.dat]{};
table [3Sol\_example-1\_mid.dat]{};
table [3Sol\_example-1\_Rx.dat]{};
table [3Sol\_example-2\_Tx.dat]{};
table [3Sol\_example-2\_mid.dat]{};
table [3Sol\_example-2\_Rx.dat]{};
at (2,0.7) [(a)]{};
table [3Sol\_example-3\_Tx.dat]{};
table [3Sol\_example-3\_mid.dat]{};
table [3Sol\_example-3\_Rx.dat]{};
table [3Sol\_example-4\_Tx.dat]{};
table [3Sol\_example-4\_mid.dat]{};
table [3Sol\_example-4\_Rx.dat]{};
at (2,0.7) [(b)]{}; (-4.3,-0.5) to (2.1,-0.5); at (-3.2,-0.75) [$z_0$]{}; at (-1.2,-0.75) [$z_1$]{}; at (0.9,-0.75) [$z_2$]{};
table [3Sol\_example-1\_Bw.dat]{}; table [3Sol\_example-2\_Bw.dat]{}; table [3Sol\_example-3\_Bw.dat]{}; table [3Sol\_example-4\_Bw.dat]{};
at (7,1) [(c)]{};
table [3Sol\_example-1\_Tw.dat]{}; table [3Sol\_example-2\_Tw.dat]{}; table [3Sol\_example-3\_Tw.dat]{}; table [3Sol\_example-4\_Tw.dat]{};
at (7,0.4) [(d)]{};
To characterize the spectral efficiency, we need to find a suitable definition for the time-bandwidth product of the soliton pulses. In an ideal tranmission case for a single channel scenario, this definition only needs to consider the bandwidth at the transmitter and the receiver $z \in \{0,L\}$ where the unknown variable $L$ is the link length. The bandwidth restriction is only given by the ability of transmitter and receiver to sample the signals. Assuming an ideal fiber transmission and amplification, the bandwidth (and its variation) along the link has no influence. For pulse duration, however, the evolution along the whole link $z \in [0,L]$ needs to be taken into account in order to avoid intersymbol interference in a train of soliton pulses.
If the PSK constellation size for $Q_d(\lambda_k)$ is large enough, almost all phase combinations of $\varphi_k$ (and the different resulting bandwidths $B$ and pulse durations $T$) occur at the transmitter due to modulation. For simplification and to be independent of the phase modulation format, we thus consider the maximization of $T$ and $B$ over all existing combinations of $\varphi_k$. The absolute value of the spectral phases $|Q_d(\lambda_k)|$ is not modulated, however it may change as well during propagation according to . Note that $|Q_d(\lambda_k)|$ remains fixed for purely imaginary eigenvalues. Based on these considerations, we define the time-bandwidth product as $\widehat{T} \cdot \widehat{B}$ with $$\begin{aligned}
\label{eq:TBP_def}
& \widehat{T}=\max_{\substack{\varphi_k, 1\leq k\leq N \\ |Q_d(\lambda_k,z)|, z \in [0,L]}} T
\qquad \qquad \qquad
\widehat{B}=\max_{\substack{\varphi_k, 1\leq k\leq N \\ |Q_d(\lambda_k,z)|, z \in \{0,L\}}} B\end{aligned}$$ The pulse duration $T$ is maximized over all existing spectral phase combinations $\varphi_k$ and all spectral amplitudes $|Q_d(\lambda_k)|$ occuring *at and between* transmitter and receiver. The bandwidth is maximized as well over all existing spectral phase combinations $\varphi_k$ but only over spectral amplitudes $|Q_d(\lambda_k)|$ occuring *at* the positions of transmitter and receiver. Based on this definition, we aim to find multi-soliton pulses with minimum time-bandwidth product $\widehat{T}\cdot \widehat{B}$. Eigenvalues $\lambda_k$, the magnitude of the spectral amplitudes $\left|Q_d(\lambda_k)\right|$ at the transmitter, the transmission distance $L$ and the soliton order $N$ are left as parameters for this optimization. We observe that $\widehat{T} \cdot \widehat{B}$ is in general increasing with the soliton order $N$. However, a higher order soliton has also $N$ dimensions for encoding data. To have a fair comparison, we introduce the notion of “time-bandwidth product per eigenvalue” defined as $$\label{eq:TBP_per_eigenvalue}
\overline{T \cdot B}_N=\frac{\widehat{T} \cdot \widehat{B}}{N}.$$
In the subsequent sections, we address the following questions:
- What is the minimum $\overline{T \cdot B}_N$ for a given $N$?
- What are optimal choices for $\{\lambda_k\}_{k=1}^N$, $\{|Q_d(\lambda_k)|\}_{k=1}^N$ and how do optimal pulses look like?
- Does the optimum “time-bandwidth product per eigenvalue” decrease with increasing $N$?
Definition of Pulse Duration and Bandwidth {#sec:def}
------------------------------------------
Before we minimize , the pulse duration $T$ and the bandwith $B$ need to be defined. One observes arbitrary multi-solitons having unbounded support and exponentially decaying tails in time and (linear) frequency domain. Their pulse shape depends on their nonlinear spectrum $\{\lambda_k,Q_d(\lambda_k;z)\}$ and changes by modulation and during propagation. Despite pulse shape and peak power variation, the energy of the multi-soliton *remains constant* and is equal to $\sum_{k=1}^N 4\sigma_k$. This motivates the definition of pulse duration and bandwidth in terms of energy:
\[def:TB\_def\_energy\]
The pulse duration $T(\varepsilon)$ (and bandwidth $B(\varepsilon)$, respectively) is defined as the smallest interval (frequency band) containing $E_\mathrm{trunc} = (1 - \varepsilon)E_\mathrm{total}$ of the total soliton energy
Note that truncation causes small perturbations of the eigenvalues. In practical transmission systems, the perturbations become even larger due to intersymbol interference of adjacent truncated soliton pulses. Thus, there is a trade-off: $\varepsilon$ must be kept small such that the truncation causes only small perturbations, but large enough to have a small time-bandwidth product. In the following we use a practically reasonable $\varepsilon=10^{-4}$. Then, the time-bandwidth product of first order solitons becomes $\overline{T \cdot B}_1 \approx 9.9$. Another possible definition for $T$ and $B$ is:
\[def:TB\_def\_threshold\]
The pulse duration ${T(\alpha)=T_+ - T_-}$ and the bandwidth ${B(\alpha)=B_+ - B_-}$ are defined as the smallest interval such that
[2]{}
- ${t\notin [T_-,T_+], \quad |q(t)|\leq \alpha \cdot \mathrm{max}\{|q(t)|\}}$
- ${f\notin [B_-,B_+], \quad |Q(f)|\leq \alpha \cdot \mathrm{max}\{|Q(f)|\}}$
$\alpha$ is chosen such that the resulting $T$ and $B$ are identical to the energy based definition for a first order soliton. Note that the peak $\mathrm{max}\{q(t)\}$ changes for different multi-soliton shapes.
Multi-Soliton Pulse optimization {#sec:num_pulse_opt}
================================
A first approach to minimize $\overline{T \cdot B}_N$ for a given $N$ is exhaustive search. For all combinations of $\{\lambda_k\}_{k=1}^N$, $\{|Q_d(\lambda_k)|\}_{k=1}^N$ and $L$, we evaluate $\widehat{T}\cdot \widehat{B}$ to finally select the optimum among them: $$\label{eq:TBP_minimization}
(\widehat{T}\cdot \widehat{B})^\star=\min_{\substack{\lambda_k,1\leq k\leq N \\ |Q_d(\lambda_k)|,1\leq k\leq N \\ L}} \widehat{T}\cdot \widehat{B}$$ We find the optimum set of eigenvlaues $\{\lambda_k^\star\}$ and set of absolute value of spectral amplitudes at the transmitter $\{|Q_d^\star(\lambda_k)|\}$ such that the time-bandwidth product $\widehat{T}\cdot \widehat{B}$ defined by becomes minimal. For each given nonlinear spectrum, we construct the time domain signal $q(t)$ via Alg. \[alg:DT2\] and determine $T$ and $B$ numerically from $q(t)$ and its Fourier transform according to the definitions in Sec. \[sec:def\]. We rewrite $Q_d(\lambda_k)$ as where the amplitude scaling is $\eta_k \in \mathbb R^+$. $$\begin{aligned}
\label{eq:Qd_eta}
Q_d(\lambda_k) & =\eta_k \cdot |Q_\mathrm{d,init}(\lambda_k)| \cdot \exp(j\varphi_k)
\\
\mathrm{with} \qquad Q_\mathrm{d,init}(\lambda_k) & =\left(\lambda_k-\lambda_k^*\right) \prod_{m=1,m\ne k}^N \frac{\lambda_k-\lambda_m^*}{\lambda_k-\lambda_m}.\label{eq:Qd_0}\end{aligned}$$ Thus, it is equivalent to optimize $\eta_k$ instead of $|Q_d(\lambda_k)|$. Note that the scaling factor $\eta_k$ is identical to the magnitude of the Fourier coefficient $|b(\lambda_k)|$. The benefit of this representation will become clear in the following section. We consider two special cases for the eigenvalues, either located on the imaginary axis with $\lambda_k=j\sigma_k$ or parallel to the real axis $\lambda_k=j\sigma+\omega_k$. W.l.o.g., we can assume ${\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_N}$. To indicate a “net”-gain when $N$ grows, we normalize $\overline{T \cdot B}_N$ by $\overline{T \cdot B}_1$, i.e. the time-bandwidth product of a first order soliton which can be derived analytically.
Transformations Preserving the Time-Bandwidth Product {#sec:sol_prop}
-----------------------------------------------------
To reduce the number of parameters to optimize, we show that the product $T\cdot B$ is preserved under the transformations following in Thm. \[th:SolProperties\]. The proofs can be found in Appx. \[sec:sol\_prop\_proof\].
\[th:SolProperties\] Assume that an $N$-soliton signal $q(t)$ corresponds to the nonlinear spectrum given by $\left\{\lambda_k=j\sigma_k+\omega_k\right\}_{k=1}^N$, $\left\{\eta_k\right\}_{k=1}^N$, $\left\{\varphi_k\right\}_{k=1}^N$. Then the following properties hold:
- $\left\{\varphi_k-\varphi_0\right\}$ corresponds to $\exp(j\varphi_0) q(t)$.
- $\left\{\exp(2\sigma_k t_0)\eta_k\right\}$, $\left\{\varphi_k-2\omega_k t_0\right\}$ corresponds to $q(t-t_0)$.
- $\left\{\lambda_k / \sigma_0 \right\}$ corresponds to $\frac{1}{\sigma_0} q(t / \sigma_0)$ for $\sigma_0>0$.
- $\left\{\omega_k-\omega_0\right\}$ leads to $\exp(2j\omega_0 t) q(t)$.
- $\left\{1/\eta_k\right\}$, $\left\{-\omega_k\right\}$ corresponds to $q(-t)$.
- Setting $\left\{\eta_k=1\right\}$, $\left\{\omega_k=0\right\}$ leads to symmetric pulses ${q(t)=q(-t)}$.
- $\{-\varphi_k\}$, $\{-\omega_k\}$ corresponds to $q^*(t)$
From these properties, simple restrictions on the optimization parameters that do not influence the time-bandwidth product of a multi-soliton can be assumed: (i) One can assume $\varphi_N=0$, (ii) one of the $\eta_k$ can be chosen arbitrarily, e.g. it suffices to assume $\eta_N = 1$, (iii) one of the $\sigma_k$ can be chosen arbitrarily, e.g. $\sigma_N=0.5$, (iv) one of the $\omega_k$ can be chosen arbitrarily, e.g. ${\omega_1=0}$ or ${\omega_1=-\omega_2}$, (v) it suffices to assume $\eta_2\in (0,1]$ or $\eta_2\in [1,\infty)$ when $\omega_k=0$ and (vi) the sign of one of the $\omega_k$ can be chosen arbitrarily, e.g. $\omega_1>0$.
Eigenvalues on the Imaginary Axis
---------------------------------
First, we find optimum solitons with eigenvalues limited to the imaginary axis, $\lambda_k=j\sigma_k$. In this case, the transformation simplifies to a phase rotation only; the $|Q_d(\lambda_k;z)|$ do not change during propagation. This means we can skip the maximization in terms of $|Q_d(\lambda_k;z)|$ along $z$ in .Thus the result will be independent of the link length $L$. Following properties i)-iii) and v) in Sec. \[sec:sol\_prop\], we choose $\varphi_N=0$, $\eta_N=1$, $\eta_{N-1}>1$ and $\sigma_N=0.5$. We define $$\label{eq:DeltaT_eta}
\Delta t_k= \ln \left( \eta_k \right) / 2\sigma_k,$$ which, conceptually, is the temporal shift of the individual first order solitonic components by scaling $\eta_k$. We check all combinations of the free parameters $\sigma_k$ and $\Delta t_k$ to find the minimum $(\widehat{T} \cdot \widehat{B})^\star$ according to . Note that the solution is independent of the propagation distance $L$ in this case, since $|Q_d(\lambda_k;z)|$ does not change along the fiber. We uniformly quantize $\varphi_k$ to $m\cdot 2 \pi/128$, $m=1,\dots,128$. For each chosen combination of $\{\Delta t_k\}$ and $\{\sigma_k\}$ we calculate $\widehat{T}$ and $\widehat{B}$ in as the maximum of $T$ and $B$ for the resulting $128^{N-1}$ phase differences for $\{\varphi_k\}$. For a first coarse estimate of the minimal $\widehat{T} \cdot \widehat{B}$, the eigenvalues $\sigma_k$ are quantized in steps of $0.1$ in the range $[0.5,1.5]$, whereas we search $\Delta t_k$ in steps of $0.25$ in $[0,5]$ for $k=2$ and $[-5,5]$ for $k=3$. We increase the resolution around the observed minimum to $0.02$ for eigenvalues $\sigma_k$ and $0.05$ for $\Delta t_k$. Choosing the optimum values $\{\Delta t_k^\star\}$ and $\{\sigma_k^\star\}$ that achieve the minimal $(\widehat{T}\cdot \widehat{B})^\star$ results in the optimum soliton pulses that are shown in Fig. \[fig:optimum\_pulse\_2Sol\_imag\] for four different phase combinations of $\{\varphi_k\}$. Optimization is done for $N=2$, Fig. \[fig:optimum\_pulse\_2Sol\_imag\] (a) and $N=3$, Fig. \[fig:optimum\_pulse\_2Sol\_imag\] (b). The optimum values $\{\Delta t_k^\star\}$ and $\{\sigma_k^\star\}$ are given in Tab. \[tab:imag\_const\_result\] for the two definitions used for $T$ and $B$ (energy based and threshold based). The optimum second and third order sets of pulses show an improvement in the time-bandwidth product per eigenvalue of $\overline{T \cdot B}_2 / \overline{T \cdot B}_1\approx 0.89$ and $\overline{T \cdot B}_3 / \overline{T \cdot B}_1\approx 0.84$ using the energy based definition for pulse duration and bandwidth described in Sec. \[sec:def\]. It turns out that the optimum pulses look similar to a train of first order solitons. The intuitive reason is that the nonlinear superposition of (overlapping) first order pulses causes large bandwidth expansions for some extreme choices of $\varphi_k$ due to the nonlinear interaction. Therefore, the first order components are slightly separated to reduce this bandwidth expansion with the cost of small increase in the pulse duration.
Eigenvalue Real Part Constellation {#sec:real_const}
----------------------------------
Next, we consider the pulse optimization for solitons with eigenvalues parallel to the real axis $\lambda_k=j\sigma+\omega_k$. We check all combinations of the free parameters $\omega_k$, $\Delta t_k$ to find the minimum $(\widehat{T} \cdot \widehat{B})^\star$ according to . Following property i) and iii), we choose ${\varphi_1=0}$ and $\sigma=0.5$. Following property iv), one can set $\omega_1=-\omega_2$ for the optimization of the time-bandwidth product $\hat{T}\cdot \hat{B}$. Considering property ii) for the given specific case $\sigma_k=\sigma$ one can further choose $\eta_2=1/\eta_1$ which is identical to $\Delta t_2=-\Delta t_1$ (see ). Note that property iv) and ii) allow also other choices; all $\omega_k$ could be shifted by some $\omega_0$ or all $\eta_k$ could be scaled by some $\exp(2\sigma t_0)$ without affecting $T \cdot B$ *at a specific* $z$. For the real part eigenvalue constellation, however, $|Q_d(\lambda_k)|$, and thus $\Delta t_k$ are changing during propagation (see ). The $\omega_k \neq 0$ lead to a frequency shift of the first order solitonic component associated to this eigenvalue. This leads to solitonic components with different propagation velocities proportional to $\omega_k$. Consequently, multi-solitons with distinct $\omega_k$ will eventually be separated into their first order components during propagation as $z\to \infty$. This is associated to an increased required time frame for each pulse. According to property vi), we can set the signs of $\omega_1$ and $\omega_2$ depending on the signs of $\Delta t_1$ and $\Delta t_2$ such that the solitonic components will move slowly towards each other, collide at the soliton pulse center and then separate again during propagation [@Aref2018]. This means $\omega_1=-\omega_2>0$ if $\Delta t_2=-\Delta t_1>0$. This minimizes the overall pulse duration for all $z\in[0,L]$.
We uniformly qunatize $\varphi_k$ to $m\cdot 2 \pi/128$, $m=1,\dots,128$. For each chosen combination of $\{\Delta t_k \coloneqq \Delta t_k(z=0)\}$ and $\{\omega_k\}$ we calculate $\widehat{T}$ and $\widehat{B}$ in as the maximum of $T$ and $B$ for the resulting $128^{N-1}$ phase combinations of $\{\varphi_k\}$. For $N=2$, we find the minimum of $\widehat{T} \cdot \widehat{B}$ by exhaustive search for all combinations for $\omega_1$ and $\Delta t_1$, where we quantize $\omega_1$ in steps of $0.05$ in the range $[0,1]$ and $\Delta t_1$ in steps of $0.2$ in $[-4,0]$. For $N=3$ we search additionally the best combination of $\omega_3$ and $\Delta t_3$ quantized in $[-1,1]$ in steps of $0.1$ and $[-3,0]$ in steps of $0.2$ respectively. We increase the resolution around the observed optima to $0.01$ for $\omega_k$ and $0.05$ for $\Delta t_k$. Due to the precompensation, the optimization result is only valid for the specific (normalized) transmission distance that follows from the optimum parameters ${L^\star=\left|\frac{\Delta t_1^\star}{2\omega_1^\star}\right|}$. The respective optimization results $\{\Delta t_1^\star\}$ and $\{\omega_1^\star\}$ that minimize $\widehat{T} \cdot \widehat{B}$ are given in Tab. \[tab:real\_const\_result\] for $N=2,3$ and the two definitions (energy based and threshold based) for $T$ and $B$.
table [T\_2Sol\_real\_energy.dat]{};
table [T\_3Sol\_real\_energy-1.dat]{};
table [B\_2Sol\_real\_energy.dat]{};
table [B\_3Sol\_real\_energy-1.dat]{};
Note that multiple parameter sets achieve similar optimization results for $\widehat{T} \cdot \widehat{B}$. However the achieved $\widehat{T} \cdot \widehat{B}$ are very close. Thus the given parameter set $\{\omega_k^\star\}$ and $\{\Delta t_k^\star\}$ result in soliton pulses very close to the optimum. Using the resulting optimized third order pulses, an improvement in the time-bandwidth product per eingenvalue of $\overline{T \cdot B}_3 / \overline{T \cdot B}_1\approx 0.73$ can be achieved in this scenario. For $N=2$ we achieve $\overline{T \cdot B}_2 / \overline{T \cdot B}_1\approx 0.75$. The resulting optimum soliton pulse for $N=3$ is shown in Fig. \[fig:optimum\_pulse\_3Sol\_real\] for different phase combinations of $\{\varphi_k\}$ at transmitter $z=0$, along the fiber $z=L/2$ and at the receiver $z=L$. Note that, althoug changing during propagation, the $\Delta t_k$ will keep their respective relation. From , it follows that after propagation along a distance ${L=\left|\frac{\Delta t_k(z=0)}{2\omega_k }\right|}$, the $\Delta t_k(z)$ are transformed to $\Delta t_k(z=L)=-\Delta t_k(z=0)$. Thus the resulting multi-solitons achieve their $\widehat{T}$ at $z=0$ as well as at $z=L$ and have the same $\widehat{B}$ at $z=0$ and at $z=L$. This scenario with eigenvalues parallel to the real axis outperforms the scenario with purely imaginary eigenvalues, however, only for the fixed normalized propagation distance that matches to the optimum soliton parameters as ${L^\star=\left|\frac{\Delta t_1^\star}{2\omega_1^\star}\right|}$. For the optimum parameters from Tab. \[tab:real\_const\_result\], we get $L \approx 2$. The intuitive reason is that due to changing $\Delta t_k$ along the propagation, we can avoid large bandwidths at transmitter and receiver by “shifting” the bandwidth expansion to the center of the transmission link. For propagation distances larger than $L^\star$, the solitons further separate and the overall pulse duration gets larger which reduces the achievable time-bandwidth product gain. We illustrate this influence of the changing $\Delta t_k$ on the pulse duration and bandwidth during propagation. We define where $T_\mathrm{max}$ and $B_\mathrm{max}$ are the maximum pulse duration and bandwidth that could be observed at a *specific* $z$ along the fiber. $$\label{eq:TmaxBmax}
T_{\max}=\underset{\varphi_k, 1\leq k\leq N}{\max} T \qquad B_{\max}=\max_{\varphi_k, 1\leq k\leq N} B,$$ Comparing with , the time-bandwidth product is given by the maximization of $T_{\max}$ and $B_{\max}$ along $z=[0,L]$ or $z=\{0,L\}$, respectively. We show the evolution of $T_{\max}$ and $B_{\max}$ during propagation along $z\in [0,L]$ in Fig. \[fig:TBP\_vs\_z\_2Sol\_real\_opt\] for the optimal $2-$ and $3-$solitons. We observe these solitons indeed achieving their maximum $T_{\max}$ at $z=0$ as well as at $z=L$ leading to $\widehat{T}=\left.T_{\max}\right|_{z=0}$. Furthermore they have the same $B_{\max}$ at $z=0$ and at $z=L$ giving $\widehat{B}=\left.B_{\max}\right|_{z=0}$.
Estimation of Time-Bandwidth Product {#sec:TBP_higherN}
====================================
From the optimization results in the previous section, we observe the improvement in the time-bandwidth product per eigenvalue being small and slowly decreasing in $N$. The complexity of the numerical brute-force optimization, however, grows exponentially in the number of eigenvalues. Instead of finding the optimal soliton pulses, we estimate the smallest possible $(\widehat{T}\cdot \widehat{B})^\star$, to see how the time-bandwidth product per eigenvalue could potentially be decreased for higher $N$.
Finding the joint minimization $\min \widehat{T}\cdot \widehat{B}$ in is hard. Therefore we look at$$\label{eq:minTminB}
(\widehat{T}\cdot \widehat{B})^\star \geq \min_{\{\lambda_k\}_{k=1}^N} \left( \min_{\substack{\{\Delta t_k\}_{k=1}^N \\ L}} \widehat{T} \min_{\substack{\{\Delta t_k\}_{k=1}^N \\ L}} \widehat{B} \right)$$ where we minimize pulse duration and bandwidth individually with respect to $\Delta t_k$. For each minimization, the minima are achieved for zero propagation distance, $L=0$ (pulse duration and bandwidth change during propagation). Then using and the definition , we get $$\label{eq:minTminB2}
(\widehat{T}\cdot \widehat{B})^\star \geq \min_{\{\lambda_k\}_{k=1}^N} \left( \min_{\{\Delta t_k\}_{k=1}^N } T_{\max} \min_{\{\Delta t_k\}_{k=1}^N } B_{\max} \right)$$
We first ask how pulse duration $T_{\max}$ and bandwidth $B_{\max}$ change in terms of $\{\Delta t_k\}$. Fig. \[fig:T\_B\_vs\_ln(eta)\] illustrates $T_\mathrm{max}$ and $B_\mathrm{max}$ in terms of $\Delta t_2$ for two different exemplary $2-$solitons. We observe that the smallest $T_\mathrm{max}$ is attained at $\Delta t_2=0$, whereas $B_\mathrm{max}$ reaches its maximum. On the other hand, $B_\mathrm{max}$ reaches its minimum for $\Delta t_2 \to \infty$, for which $T_\mathrm{max}$ grows unboundedly. We observe this behavior for different eigenvalues and soliton orders $N$. The intuitive reason is, that each multi-soliton splits into its first order components when $|\Delta t_k - \Delta t_l| \to \infty \quad \forall k>l$; meaning $N$ seperate 1-solitons without any interaction. As $|\Delta t_k - \Delta t_l|$ decreases, the distance between these 1-solitons decreases. This reduces the overall pulse duration $T_\mathrm{max}$, however the bandwidth $B_\mathrm{max}$ increases due to higher nonlinear interaction. We use these observations to estimate $$\begin{aligned}
\label{eq:TB_limitCases}
\min_{\{\Delta t_k\}_{k=1}^N } T_{\max} \qquad & \mathrm{when} \qquad \Delta t_k = \Delta t \quad \forall k=1,\dots,N \nonumber
\\
\min_{\{\Delta t_k\}_{k=1}^N } B_{\max} \qquad & \mathrm{when} \qquad \Delta t_k - \Delta t_l \to \infty \quad \forall k>l\end{aligned}$$
In the following, we give analytical approximations for $T_\mathrm{max}$ and $B_\mathrm{max}$ in these respective limit cases. Based on the Darboux transform Alg. \[alg:DT2\], we derive the following approximations on pulse duration given in Thm. \[th:T\_imag\] and Thm. \[th:T\_real\]. For detailed derivations see App. \[sec:proof\_Th12\].
\[th:T\_imag\] Consider an $N-$soliton with eigenvalues $\lambda_k=j\sigma_k$, ${\sigma_k \in \mathbb{R}^+}$, and spectral amplitude scaling $\eta_k=\exp(2\sigma_k \Delta t)$ (${\Delta t_k = \Delta t}$), $1\leq k \leq N$. We assume (w.l.o.g) $\sigma_N=\underset{k}{\min}\,{\sigma_k}$. This soliton’s pulse duration $T_\mathrm{max}(\varepsilon)$, according to Def. \[def:TB\_def\_energy\], can be well approximated by $$\begin{aligned}
\label{eq:Tmin_imag_const}
T_{\max}(\varepsilon) \approx T_\mathrm{lim,im} =
\frac{1}{2\sigma_N}\left( \ln\left(\frac{2}{\varepsilon}\frac{\sigma_N}{\sum_{k=1}^{N}\sigma_k}\right) + 2\sum_{k=1}^{N-1} \ln \left(\left|\frac{\sigma_N +\sigma_k}{\sigma_N-\sigma_k}\right|\right)\right)\end{aligned}$$ The approximation becomes tight for $\varepsilon \to 0$.
\[th:T\_real\] Consider an $N-$soliton with eigenvalues $\lambda_k=j\sigma+\omega_k$ parallel to the real axis ($\sigma \in \mathbb{R}^+, \omega_k \in \mathbb{R}$) and spectral amplitude scaling $\eta_k$. We denote $A_r=\left|\sum_{k=r}^{N} a_{r,k}\right|$ where $a_{r,k}$ are calculated from in Alg. \[alg:DT2\] such that $\rho_k^{(k-1)}(t)=\sum_{r=1}^k a_{r,k} \rho_r^{(0)}(t)$. This soliton’s pulse duration $T_\mathrm{max}(\varepsilon)$, according to Def. \[def:TB\_def\_energy\], can be well approximated by . The minimization of this approximation is attained when $\eta_k=\eta, \quad \forall 1\leq k \leq N$ and we denote $T_\mathrm{lim,re}=\underset{\eta_k}{\min}\, T_\mathrm{max}(\varepsilon)$. $$\label{eq:T_real}
T_\mathrm{max}(\varepsilon) \approx \frac{1}{2\sigma}\ln\left( \frac{2}{N \varepsilon} \left(\sum_{r=1}^{N} \eta_r A_r \right) \left(\sum_{r=1}^{N} \frac{1}{\eta_r} A_r \right)\right)$$
As *numerically* observed, the minimum of $B_\mathrm{max}$ is attained for multi-solitons being separated into their first order components where $\eta_k/\eta_l \to \infty \quad \forall k > l$ (which is equivalent to $\Delta t_k - \Delta t_l \to \infty \quad \forall k>l$). In this case, the pulse is a linear superposition of fundamental solitons. From their available analytical description, we derive the following approximations on bandwidth given in Thm. \[th:B\_imag\] and Thm. \[th:B\_real\]. For details see Appx. \[sec:proof\_Th34\].
\[th:B\_imag\] Consider an $N-$soliton with imaginary eigenvalues $\lambda_k=j\sigma_k$, $\sigma \in \mathbb{R}^+$, separated into its first order components where $\eta_k/\eta_l \to \infty \quad \forall k > l$ and $\sigma_1=\underset{k}{\max} \, \sigma_k$. Its bandwidth $B_{\max}$, according to Def. \[def:TB\_def\_energy\], can be well approximated by $$\label{eq:Bmin_real_const_energy}
B_{\max}(\varepsilon) \approx B_\mathrm{lim,im} = \frac{2 \sigma_1}{\pi^2} \ln\left(\frac{2}{\varepsilon}\frac{\sigma_1}{\sum_{k=1}^{N}\sigma_k}\right)$$
\[th:B\_real\] Consider an $N-$soliton with eigenvalues ${\lambda_k=j\sigma+\omega_k}$ parallel to the real axis ($\sigma \in \mathbb{R}^+, \omega_k \in \mathbb{R}$) separated into its first order components where $\eta_k/\eta_l \to \infty, \quad \forall k > l$. Its bandwidth $B_{\max}$, according to Def. \[def:TB\_def\_energy\], can be well approximated by $$\begin{aligned}
\label{eq:Bmin_real_const_energy}
B_{\max}(\varepsilon) \approx B_\mathrm{lim,re} =
\frac{2\sigma}{\pi^2} \left( \ln\left(\frac{2}{\varepsilon N}\right) + \ln \left( \sum_{k=1}^N \exp(\frac{\pi \omega_k}{2 \sigma})\cdot \sum_{k=1}^N \exp(-\frac{\pi \omega_k}{2 \sigma})\right) \right)\end{aligned}$$
Note that the two limiting cases of minimal $T_\mathrm{max}$ and minimal $B_\mathrm{max}$ can not be attained simultaneously (compare Fig. \[fig:T\_B\_vs\_ln(eta)\]). However, ${\underset{\Delta t_k}{\min}\, T_\mathrm{max}\cdot \underset{\Delta t_k}{\min}\, B_\mathrm{max}}$ in is a lower bound on the achievable $T_\mathrm{max}B_\mathrm{max}$ for which we can given an analytical expression using the approximations in Thm. \[th:T\_imag\],\[th:T\_real\],\[th:B\_imag\],\[th:B\_real\]. We approximate ${\underset{\Delta t_k}{\min}\, T_\mathrm{max}\cdot \underset{\Delta t_k}{\min}\, B_\mathrm{max}}$ as $T_\mathrm{lim,im/re} \cdot B_\mathrm{lim,im/re}$ and thus as $$\left(\widehat{T}\cdot \widehat{B}\right)^\star \gtrapprox \min_{\{\lambda_k\}_{k=1}^N} \left(T_\mathrm{lim,im/re} \cdot B_\mathrm{lim,im/re} \right).$$
table [B\_vs\_DeltaT\_imagConst.dat]{};
table [B\_vs\_DeltaT\_realConst.dat]{};
coordinates [(0,1.92) (7,1.92)]{};
coordinates [(0,1.38) (7,1.38)]{};
table [T\_vs\_DeltaT\_imagConst.dat]{};
table [T\_vs\_DeltaT\_realConst.dat]{};
coordinates [(0,11) (7,11)]{}; coordinates [(0,11) (7,10)]{};
For the time-bandwidth product per eigenvalue, following definition , we have $$\label{eq:TBPmin_approx}
\overline{T \cdot B}_N \gtrapprox \frac{1}{N} \min_{\{\lambda_k\}_{k=1}^N} \left( T_\mathrm{lim,im/re} \cdot B_\mathrm{lim,im/re} \right).$$ The above lower bound can be minimized numerically. We normalize again by $\overline{T \cdot B}_1$ to indicate the gain compared to a first order pulse. In Fig. \[fig:TBP\_LB\], the blue “$\square$” line shows the minimization result for imaginary eigenvalues while the red “$\circ$” line corresponds to the case with eigenvalues parallel to the real axis. The brute-force optimized $\overline{T \cdot B}_N / \overline{T \cdot B}_1$ in Sec. \[sec:num\_pulse\_opt\] are marked correspondingly in Fig. \[fig:TBP\_LB\]. Non-filled marks indicate the optimization result for defining pulse duration and bandwidth according to Def. \[def:TB\_def\_energy\], whereas filled marks correspond to Def.\[def:TB\_def\_threshold\]. The analytical result indicates that the time-bandwidth product per eigenvalue can be decreased for higher soliton orders, however the improvement is small and only slowly growing in $N$. We observe the same behavior for the optimum soliton pulses obtained from brute-force optimizaiton. However, the gap between analytical approximation and practically achieved values seems to become larger for higher $N$. One can observe the eigenvalue constellation with non-zero real part potentially achieving a higher gain. However, to be competitive to classical Nyquist-based transmission schemes, multi-solitons should achieve a time-bandwidth product in the range of $1$ (time-bandwidth product of a classical Nyquist pulse). But this target value, indicated by the black dashed line in Fig. \[fig:TBP\_LB\], appears to have a large gap to the values achievable by the soliton pulses.
table [TBP\_per\_EV\_LB\_real\_const\_energy.dat]{};
table [TBP\_per\_eigenvalue\_LB\_imag\_energy.dat]{};
coordinates [(0,0.1) (20,0.1)]{};
coordinates [(2,0.89)]{}; coordinates [(2,0.74)]{}; coordinates [(3,0.84)]{}; coordinates [(3,0.71)]{};
coordinates [(2,0.86)]{}; coordinates [(3,0.855)]{}; coordinates [(2,0.75)]{}; coordinates [(3,0.72)]{};
Conclusion {#sec:conclusion}
==========
We studied the pulse duration and bandwidth of multi-solitons for a transmission scenario where the phases of spectral amplitudes are modulated. We define the time-bandwidth product such that it takes into account the pulse variations due to modulation and propagation. For second and third order solitons, we numerically optimized the location of eigenvalues and magnitudes of spectral amplitudes in order to minimize the time-bandwidth product. The optimized multi-soliton pulses turn out to look similar to a train of first order pulses. This shape reduces the nonlinear interaction and thus the bandwidth expansion. These optimum pulses decrease the time-bandwidth product per degree of freedom (eigenvalue). However the improvement is slow in the soliton order $N$ and quite small. For arbitrary solitons, we generally identify two limit cases where either pulse duration or bandwidth becomes minimal for given eigenvalues. We give analytical approximations on pulse duration and bandwidth in these limit cases and numerically minimize them in terms of eigenvalues. The result is an approximation on the smallest possible time-bandwidth product per degree of freedom as a function of the soliton order $N$. One should note that the maximization of pulse duration and bandwidth over all phase combinations of spectral amplitudes is a worst case assumption that simplifies the optimization and clearly increases the time-bandwidth product result. Better results may be achievable by, e.g., considering only a finite number of phases and tuning magnitudes of spectral amplitudes accordingly.
Proof of Theorem \[th:SolProperties\] {#sec:sol_prop_proof}
=====================================
Properties (i) to (iv) are already shown in [@yousefi2014nft] using the Zakharov-Shabat system . Property (v\*) was partially shown in [@Haus1985]. In more details, it is shown in [@Span2017] that (v) is a necessary and sufficient condition for the pulse symmetry. Here, we prove all properties using the recursive Darboux transform in Alg. \[alg:DT2\]. Using , , the initialization in is with the resulting time domain signal from given as . $$\begin{aligned}
\rho_k^{(0)}(t) & =\eta_k \exp(j\varphi_k) \exp(2j\omega_k t) \exp(-2\sigma_k t).\label{eq:rho_init_etak}
\\
q^{(N)}(t) & = -4 \sum_{k=1}^{N} \sigma_k \frac{\rho_k^{*(k-1)}(t)}{1+\left|\rho_k^{(k-1)}(t)\right|^2}.\label{eq:sig_rho}\end{aligned}$$
Let $\tilde{\rho}_k^{(i)}(t)$ and $\tilde{q}^{(N)}(t)$ denote the transformed $\rho_k^{(i)}(t)$ and, respectively, the transformed $q^{(N)}(t)$, if we apply one of the properties (i) to (vi). First we prove properties (i) to (iv) and (vi):
Applying one of the properties (i) to (iv), the transformed Darboux initializations are
\_k\^[(0)]{}(t) & = \_k (2j\_k t) (j\_k-j\_0) (-2\_k t) = (-j\_0)\_k\^[(0)]{}(t) \[eq:prop1\_init\]\
\_k\^[(0)]{}(t) & = \_k (2\_k t\_0) (j\_k-j2\_k t\_0+2j\_k t) (-2\_k t) = \_k\^[(0)]{}(t-t\_0) \[eq:prop2\_init\]\
\_k\^[(0)]{}(t) & =\_k (j\_k) (2j t) (-2 t) =\_k\^[(0)]{}() \[eq:prop3\_init\]\
\_k\^[(0)]{}(t) & =\_k (j\_k) (2j(\_k-\_0) t) (-2 \_k t) =(-2j\_0 t)\_k\^[(0)]{}(t) \[eq:prop4\_init\]\
\_k\^[(0)]{}(t) & = \_k (-2j\_k t) (-j\_k) (-2\_k t) = \_k\^[\* (0)]{}(t) \[eq:prop6\_init\]
For the update equation , it is easy to check that if one of the above relations holds between $\tilde{\rho}_k^{(j)}(t)$ and $\rho_k^{(j)}$ for all ${k=j+1,\dots,N}$, the same relation will hold between $\tilde{\rho}_k^{(j+1)}$ and $\rho_k^{(j+1)}$ for all $k>j+1$. Since the respective relation holds at $j=0$, we conclude that the relation is preserved for any $j=1,\dots,N-1$. Thus, all the above properties hold in particular between $\tilde{\rho}_k^{(k-1)}$ and $\rho_k^{(k-1)}$ for all $k=1,\dots,N$. Following , the transformed signal $\tilde{q}^{(N)}(t)$ is $$\label{eq:sig_rho_tilde}
\tilde{q}^{(N)}(t) = -4 \sum_{k=1}^{N} \sigma_k \frac{\tilde{\rho}_k^{*(k-1)}(t)}{1+\left|\tilde{\rho}_k^{(k-1)}(t)\right|^2}.$$ Replacing $\tilde{\rho}_k^{(k-1)}$ in by either of the above (i), (ii), (iii), (iv), (vi), one concludes
[2]{}
- Given , $\quad \tilde{q}^{(N)}(t) = \exp(j\varphi_0) q^{(N)}(t)$
- Given , $\tilde{q}^{(N)}(t) = q^{(N)}(t-t_0)$
- Given , $\tilde{q}^{(N)}(t) = \frac{1}{\sigma_0} q^{(N)}(\frac{t}{\sigma_0})$
- Given , $\tilde{q}^{(N)}(t) = \exp(2j\omega_0 t) q^{(N)}(t)$
- Given , $\tilde{q}^{(N)}(t) = q^{*\, (N)}(t)$
Now we prove property (v): By replacing $\{\eta_k\}\rightarrow \{1/\eta_k\}$ and $\{\omega_k\}\rightarrow \{-\omega_k\}$ , the transformed Darboux initialization writes as
\[eq:rho\_init\_etak\_inv\] \_k\^[(0)]{}(t) = & (j\_k) (-2j\_k t) (-2\_k t) =. &
In addition, the update rule taking into account the change of eigenvalues ${\omega_k \to -\omega_k}$, or equivalently ${\lambda_k \to -\lambda_k^*}$, rewrites as $$\begin{aligned}
\label{eq:rho_update_etak}
\tilde{\rho}_k^{(j+1)}(t) =
\frac{(\lambda_{j+1}^* - \lambda_k^*)\tilde{\rho}_{k}^{(j)}(t) +\frac{\lambda_{j+1} - \lambda_{j+1}^*}{1+|\tilde{\rho}_{j+1}^{(j)}(t)|^2}(\tilde{\rho}_{k}^{(j)}(t)-\tilde{\rho}_{j+1}^{(j)}(t))}
{\lambda_{j+1} - \lambda_k^*-\frac{\lambda_{j+1} - \lambda_{j+1}^*}{1+|\tilde{\rho}_{j+1}^{(j)}(t)|^2}\left(1 + \tilde{\rho}_{j+1}^{*(j)}(t)\tilde{\rho}_k^{(j)}(t) \right)}.\end{aligned}$$
If the relation $\tilde{\rho}_k^{(j)}(t) \rho_k^{*(j)}(-t)=1$ holds for all ${k=j+1,\dots,N}$, it will hold for $j+1$ as well. That means $\tilde{\rho}_k^{(j+1)}(t) \rho_k^{*(j+1)}(-t)=1$ for all $k>j+1$. Let us verify, that the update rule indeed preserves this property. Given the property for all ${k=j+1,\dots,N}$ at an arbitrary $j=0,\dots,N-1$, i.e. $\tilde{\rho}_k^{(j)}(t)=\frac{1}{\rho_k^{*(j)}(-t)}$, the update rule can be rewritten as $$\begin{aligned}
\label{eq:prop5_induction1}
\tilde{\rho}_k^{(j+1)}(t) =
\frac{(\lambda_{j+1}^* - \lambda_k^*)\frac{1}{\rho_{k}^{*(j)}(-t)} +\frac{\lambda_{j+1} - \lambda_{j+1}^*}{1+\frac{1}{|\rho_{j+1}^{(j)}(-t)|^2}}\left(\frac{1}{\rho_{k}^{*(j)}(-t)}-\frac{1}{\rho_{j+1}^{*(j)}(-t)}\right)}
{\lambda_{j+1} - \lambda_k^*-\frac{\lambda_{j+1} - \lambda_{j+1}^*}{1+\frac{1}{|\rho_{j+1}^{(j)}(-t)|^2}}\left(1 + \frac{1}{\rho_{j+1}^{(j)}(-t)\rho_k^{*(j)}(-t)} \right)}.\end{aligned}$$
To prove the induction step, let us assume the induction statement is correct. Then the update rule should preserve property . That means $$\begin{aligned}
\tilde{\rho}_k^{(j+1)}(t) & = \frac{1}{\rho_k^{*(j+1)}(-t)} \label{eq:prop5}
\\
& = \frac{(\lambda_{k}^* - \lambda_{j+1})-\frac{\lambda_{j+1}^* - \lambda_{j+1}}{1+|\rho_{j+1}^{(j)}(-t)|^2} (1+\rho_{j+1}^{(j)}(-t) \rho_k^{*(j)}(-t))}{(\lambda_{k}^* - \lambda_{j+1}^*)\rho_k^{*(j)}(-t) + \frac{\lambda_{j+1}^* - \lambda_{j+1}}{1+|\rho_{j+1}^{(j)}(-t)|^2} \left(\rho_{k}^{*(j)}(-t) - \rho_{j+1}^{*(j)}(-t) \right)}
\label{eq:prop5_induction2}\end{aligned}$$
One can now verify, that and are identical. The time domain signal according to for the transformation is given by $$\label{eq:signal_prop5}
\tilde{q}^{(N)}(t) = -4 \sum_{k=1}^{N} \sigma_k \frac{\tilde{\rho}_k^{*(k-1)}(t)}{1+\left|\tilde{\rho}_k^{(k-1)}(t)\right|^2}.$$
We have shown that property is valid for all $k>j+1$ and therefore as well for $\tilde{\rho}_k^{(k-1)}(t)$. Using this property in , the transformed signal rewrites as $$\label{eq:rho_rho_tilde_symmetry}
\tilde{q}^{(N)}(t) = -4 \sum_{k=1}^{N} \sigma_k \frac{\rho_k^{*(k-1)}(-t)}{1+\left|\rho_k^{(k-1)}(-t)\right|^2}.$$
Comparing with the original signal , one finds $$\label{eq:q_symmetry}
\tilde{q}^{(N)}(t)= q^{(N)}(-t)$$
Now we prove the special case (v\*) when $\eta_k=1$ and $\omega_k=0$. From , we have $\tilde{\rho}_k^{(0)}(t)=\rho_k^{(0)}(t)$ from which we can immediately conclude $\tilde{q}^{(N)}(t)=q^{(N)}(t)$. But from , we know $\tilde{q}^{(N)}(t)= q^{(N)}(-t)$. Thus, one can conclude $q^{(N)}(t)=q^{(N)}(-t)$.
Proof of approximations Thm. \[th:T\_imag\] and Thm. \[th:T\_real\] {#sec:proof_Th12}
===================================================================
To approximate the soliton pulse duration, we derive analytical expressions for the pulse in its tails $t \to \pm \infty$. W.l.o.g. we assume the eigenvalues $\lambda_k=\omega_k+j\sigma_k$ ($\sigma_k \in \mathbb{R}^+$, $\omega_k \in \mathbb{R}$) being order as follows: $\sigma_1\geq \sigma_2 \geq ... \geq \sigma_N$
\[lem:Darboux\_iteration\_limit\] Consider the limits $t \to \pm \infty$.\
a) For arbitrary $\lambda_k=\omega_k+j\sigma_k$, the update rule in the Darboux algorithm \[alg:DT2\] simplifies to $$\begin{aligned}
\rho_k^{(j+1)}(t) & \to \rho_k^{(j)}(t)\frac{\lambda_k-\lambda_{j+1}^*}{\lambda_k-\lambda_{j+1}}-\frac{2j\sigma_{j+1}}{\lambda_k-\lambda_{j+1}}\rho_{j+1}^{(j)}(t), \qquad t \to \infty \label{eq:rho_update_t_infty_general}
\\
\frac{1}{\rho_k^{(j+1)}(t)} & \to \frac{1}{\rho_k^{(j)}(t)}\frac{\lambda_k-\lambda_{j+1}^*}{\lambda_k-\lambda_{j+1}}-\frac{2j\sigma_{j+1}}{\lambda_k-\lambda_{j+1}}\frac{1}{\rho_{j+1}^{(j)}(t)}, \qquad t \to - \infty.
\label{eq:rho_update_t_-infty_general}\end{aligned}$$ b) Thus, each $\rho_k^{(j)}(t) \propto \exp(-2\sigma_k t)$ preserves its exponential order for all iterations
Assume for some $j=0,\dots,N-1$, we have $$\label{eq:rho_prop}
\rho_k^{(j)}(t) \propto \exp(-2\sigma_k t) \qquad \forall k=j+1, \dots, N$$ which is true for $j=0$, as the Darboux initialization is $\rho_k^{(0)}= \eta_k \exp(j\varphi_k)\cdot \exp(2j\omega_k t)\cdot \exp(-2\sigma_k t)$. Then one can easily verify by neglecting all terms\
${\rho_k^{(j)}(t)\cdot \rho_l^{(j)}(t) \propto \exp(-2(\sigma_k+\sigma_l) t) \to 0}$ ${\forall k,l=j+1, \dots, N}$ in for the limit $t\to + \infty$. Similarly, for $t \to - \infty$, considering only the dominant terms $\rho_k^{(j)}(t) \propto \exp(-2 \sigma_k t) \to \infty$ and ${\rho_k^{(j)}(t)\cdot \rho_l^{(j)}(t) \propto \exp(-2(\sigma_k+\sigma_l) t) \to \infty}$ $\forall k,j=j+1, \dots, N$ in leads to . Recalling the ordering $\sigma_1\geq \sigma_2 ... \geq \sigma_N$ and $k>j+1$, Lem. \[lem:Darboux\_iteration\_limit\] b) follows from and .
The update equation is a linear combination of $\rho_k^{(j)}(t)$ and $\rho_{j+1}^{(j)}(t)$. Similarly, from update equation , $\frac{1}{\rho_k^{(j+1)}(t)}$ is a linear combination of $\frac{1}{\rho_k^{(j)}(t)}$ and $\frac{1}{\rho_{j+1}^{(j)}(t)}$. Therefore, $$\label{eq:a_rk_superpos}
\rho_k^{(k-1)}(t)=\sum_{r=1}^k a_{r,k} \rho_r^{(0)}(t)
\qquad \qquad
\frac{1}{\rho_k^{(k-1)}(t)}=\sum_{r=1}^k \tilde{a}_{r,k} \frac{1}{\rho_r^{(0)}(t)}.$$ The values of $a_{r,k}$ and $\tilde{a}_{r,k}$ are finite and depend on the eigenvalues. $a_{r,k}$ and $\tilde{a}_{r,k}$ can recursively be computed using the update equations and . Since the update coefficients in and are the same, one can expect that $\tilde{a}_{r,k}=a_{r,k}$. Now, we express the tails of an $N$-soliton in terms of $a_{r,k}$. We denote the envelope of $|x(t)|$ or its spectrum $|X(f)|$ as $\widehat{|x(t)|}$ and $\widehat{|X(f)|}$.
\[lem:sig\_tails\_gen\] Consider a multi-soliton pulse with $N$ eigenvalues $\lambda_k=\omega_k+j\sigma_k$ ($\sigma_k \in \mathbb{R}^+$, $\omega_k \in \mathbb{R}$) and spectral amplitude scaling $\eta_k$. $a_{r,k}$ are the scalars according to .
a\) As $t \to \pm \infty$, the tails of the multi soliton pulse tend to $$\begin{aligned}
\label{eq:q_abs_gen_limit}
|q^{(N)}(t)| \underset{t \to \pm \infty}{\to}
\left|4 \sum_{r=1}^{N} \eta_r^{\pm 1} \exp(\mp 2\sigma_r t) \exp(\pm j\varphi_r \pm 2j\omega_r t) \sum_{k=r}^{N} \sigma_k a_{r,k}\right|.\end{aligned}$$
b\) The signal envelope in the tails $t \to \pm \infty$ is given as $$\begin{aligned}
\label{eq:q_abs_limit_superpos}
\widehat{\left|q^{(N)}(t)\right|} \underset{t \to \pm \infty}{\to} 4 \sum_{r=1}^{N} \eta_r^{\pm1} \exp(\mp 2\sigma_r t) \left|\sum_{k=r}^{N} \sigma_k a_{r,k}\right| \end{aligned}$$
Consider the signal update rule of the Darboux transformation in the limits $t \to \pm \infty$. Using Lem. \[lem:Darboux\_iteration\_limit\] b), the signal update rule simplifies to $$\begin{aligned}
q^{(N)}(t) & \to -4 \sum_{k=1}^{N} \sigma_k \rho_k^{*(k-1)}(t), \quad t \to \infty \qquad \mathrm{where} \quad \rho_k^{(k-1)}(t) \to 0 \label{eq:q_limit+}
\\
q^{(N)}(t) & \to -4 \sum_{k=1}^{N} \sigma_k \frac{1}{\rho_k^{(k-1)}(t)}, \quad t \to -\infty, \qquad \mathrm{where} \quad \rho_k^{(k-1)}(t) \to \infty.\label{eq:q_limit-}\end{aligned}$$ Using in the above equations, we get $$\begin{aligned}
\label{eq:q_limit_superpos+}
q^{(N)}(t) &\underset{t \to \infty}{\to} -4 \sum_{k=1}^{N} \sigma_k \rho_k^{*(k-1)}(t)
= -4 \sum_{k=1}^{N} \sigma_k \sum_{r=1}^k a^*_{r,k} \rho_r^{*\, (0)}(t)
= -4 \sum_{r=1}^{N} \rho_r^{*\, (0)}(t) \sum_{k=r}^N \sigma_k a^*_{r,k}
\\
q^{(N)}(t) &\underset{t \to -\infty}{\to} -4 \sum_{k=1}^{N} \sigma_k \frac{1}{\rho_k^{(k-1)}(t)}
= -4 \sum_{k=1}^{N} \sigma_k \sum_{r=1}^{k} a_{r,k} \frac{1}{\rho_r^{(0)}(t)}
= -4 \sum_{r=1}^{N} \frac{1}{\rho_r^{(0)}(t)} \sum_{k=r}^{N} \sigma_k a_{r,k}
\label{eq:q_limit_superpos-}\end{aligned}$$ Note further, that the index $k$ ranges only from $r$ to $N$, since all $a_{r,k}$ with $k<r$ are zero (see , ). Then, replacing $\rho_r^{(0)}(t)$ with the initialization leads to $$\begin{aligned}
& q^{(N)}(t) \underset{t \to \infty}{\to} -4 \sum_{r=1}^{N} \eta_r \exp(- 2\sigma_r t) \exp(-j\varphi_r-2j\omega_r t) \sum_{k=r}^{N} \sigma_k a_{r,k}^* \label{eq:q_+limit_superpos2}
\\
& q^{(N)}(t) \underset{t \to -\infty}{\to} -4 \sum_{r=1}^{N} \frac{1}{\eta_r} \exp(2\sigma_r t) \exp(-j\varphi_r-2j\omega_r t) \sum_{k=r}^{N} \sigma_k a_{r,k} \label{eq:q_-limit_superpos2}\end{aligned}$$
It follows from and that $$\begin{aligned}
|q^{(N)}(t)| \underset{t \to \pm \infty}{\to} &\left|4 \sum_{r=1}^{N} \eta_r^{\pm 1} \exp(\mp 2\sigma_r t) \exp(\pm j\varphi_r \pm 2j\omega_r t) \sum_{k=r}^{N} \sigma_k a_{r,k}\right| \label{eq:q_abs_lim}
\\
\leq &4 \sum_{r=1}^{N} \eta_r^{\pm 1} \exp(\mp 2\sigma_r t) \left| \sum_{k=r}^{N} \sigma_k a_{r,k}\right| \label{eq:q_env_lim}\end{aligned}$$
We consider the envelope of the signal which is the maximum absolute value of the signal for all possible phase combinations $\{\varphi_k\}_{k=1}^N$. For each given time instance $t_0$, there exists one combination of $\{\varphi_k\}$ such that all terms $\exp(-j\varphi_r-2j\omega_r t) \sum_{k=r}^{N} a_{r,k}$ in have identical phase. Therefore we consider the case $\arg\left\{e^{j\varphi_r+2j\omega_r t_0}\sum_{k=r}^{N} a_{r,k}\right\}
=\arg\left\{e^{j\varphi_s+2j\omega_s t_0}\sum_{k=s}^{N} a_{s,k}\right\}$ for all $r,s=1,...,N$. The signal envelope among all phase combinations $\{\varphi_k\}$ for $t \to \pm \infty$ is .
\[lem:sig\_tails\_imag\] Consider an $N-$soliton with imaginary eigenvalues $\lambda_k=j\sigma_k$, $\sigma_k \in \mathbb{R}^+$ and spectral amplitude scaling $\eta_k$. The multi soliton pulse in the tails $t \to \pm \infty$ is given as $$\begin{aligned}
\left|q^{(N)}(t)\right| \to 4 \eta_N^{\pm 1} \exp(\mp 2\sigma_N t) \sigma_N \prod_{k=1}^{N-1}\left|\frac{\sigma_N+\sigma_k}{\sigma_N-\sigma_k}\right| %\nonumber
%\\
%\left|q^{(N)}(t \to -\infty) \right| \to 4 \frac{1}{\eta_N} \exp(2\sigma_N t) \sigma_N \prod_{k=1}^{N-1}\left|\frac{\sigma_N+\sigma_k}{\sigma_N-\sigma_k}\right| \nonumber\end{aligned}$$
Consider the general signal limits in Lem. \[lem:sig\_tails\_gen\] and limit the eigenvalues to be imaginary $\lambda_k=j\sigma_k$. Under the (practically relevant) assumption of well separated eigenvalues, all terms but the leading exponential term given by the smallest eigenvalue $\sigma_N=\underset{k}{\min}\, \sigma_k$ can be neglected in the limit $t \to \pm \infty$. Consequently, the absolute value of the pulse in in its tails simplifies to where $a_{N,N}$ can be derived from . If $\Delta t_k=\Delta t$ (symmetric pulses), $|q^{(N)}(t)|$ converges to its limit fast; convergence takes longer when $|\Delta t_N-\Delta t_l|$ ($l\neq N$) is large. $$\begin{aligned}
\label{eq:q_abs_limit_superpos_imag}
\left|q^{(N)}(t)\right| \underset{t \to \pm \infty}{\to} 4 \eta_N^{\pm 1} \exp(\mp 2\sigma_N t) \sigma_N \left|a_{N,N}\right| \qquad \mathrm{with} \quad a_{N,N} = \prod_{k=1}^{N-1}\frac{\sigma_N+\sigma_k}{\sigma_N-\sigma_k}\end{aligned}$$
\[lem:sig\_tails\_real\] Consider a multi-soliton pulse with $N$ eigenvalues parallel to the real axis ${\lambda_k=\omega_k+j\sigma}$ ($\sigma \in \mathbb{R}^+$, $\omega_k \in \mathbb{R}$) and spectral amplitude scaling $\eta_k$. We denote $A_r=\left|\sum_{k=r}^{N} a_{r,k}\right|$. The soliton pulse in the tails $t \to \pm \infty$ is bounded by and the signal envelope converges to . $$\begin{aligned}
\left|q^{(N)}(t)\right| & \leq 4 \sigma \exp\left(\mp 2\sigma \left(t \mp \frac{1}{2 \sigma}\ln\left(\sum_{r=1}^{N} \eta_r^{\pm 1} A_r \right) \right) \right) \label{eq:signal_bound_real}
\\
\widehat{\left|q^{(N)}(t)\right|} & \to 4 \sigma \exp\left(\mp 2\sigma \left(t \mp \frac{1}{2 \sigma}\ln\left(\sum_{r=1}^{N} \eta_r^{\pm 1} A_r \right) \right) \right) \label{eq:signal_envelope_real}\end{aligned}$$
Consider the general soliton signal and its envelope for $t \to \pm \infty$ in Lem. \[lem:sig\_tails\_gen\]. By limiting all eigenvalues to have identical imaginary part $\lambda_k=\omega_k+j\sigma$, we can conclude $$\begin{aligned}
\left|q^{(N)}(t)\right| \leq 4 \sigma \exp(\mp 2\sigma t) \sum_{r=1}^{N} \eta_r^{\pm 1} \left|\sum_{k=r}^{N} a_{r,k}\right|
\qquad \widehat{\left|q^{(N)}(t)\right|} \to 4 \sigma \exp(\mp 2\sigma t) \sum_{r=1}^{N} \eta_r^{\pm 1} \left|\sum_{k=r}^{N} a_{r,k}\right|.\end{aligned}$$
Consider a multi-soliton pulse with imaginary eigenvalues $\lambda_k=j\sigma_k$ in the limits $t \to \pm \infty$ according to Lem. \[lem:sig\_tails\_imag\]. Since the signal tails have the same slope for $t \to -\infty$ and $t \to \infty$, the smallest pulse duration $T=T_+ -T_-$ according to Def. \[def:TB\_def\_energy\] is attained when the pulse is truncated such that the same fraction $\frac{\varepsilon}{2} E_\mathrm{tot}$ of the total soliton energy $E_\mathrm{tot}=4\sum_{k=1}^N \sigma_k$ is lost in both of the respective tails. Then, the pulse duration can be calculated from $\frac{\varepsilon}{2} E_\mathrm{tot} =\int_{-\infty}^{T_-} \left|q^{(N)}(t)\right|^2 \partial t = \int_{T_+}^{\infty} \left|q^{(N)}(t)\right|^2 \partial t$. Using the signal tails in Lem. \[lem:sig\_tails\_imag\] we get ${\frac{\varepsilon}{2} E_\mathrm{tot} \approxeq \int_{-\infty}^{T_-}16 \frac{\sigma_N^2}{\eta_N^2} \exp\left(4\sigma_N t \right) |a_{N,N}|^2 \partial t
\approxeq \int_{T_+}^{\infty} 16 \sigma_N^2 \eta_N^2 \exp\left(-4\sigma t \right) |a_{N,N}|^2 \partial t}$. Solving the equation for $T_+$ and $T_-$ leads to the pulse duration in Th. \[th:T\_imag\]. Note that the accuracy of this approximation decreases if $\Delta t_k$ are very different and solitonic components become separated (unless for very small $\varepsilon \to 0$), since in Lem. \[lem:sig\_tails\_imag\] becomes less precise for smaller $|t|$.
Consider a multi-soliton with eigenvalues parallel to the real axis $\lambda_k=\omega_k+j\sigma$ in its tails $t \to \pm \infty$ according to Lem. \[lem:sig\_tails\_real\]. Since the signal tails have the same slope for $t \to -\infty$ and $t \to \infty$, the smallest pulse duration $T=T_+ -T_-$ according to Def. \[def:TB\_def\_energy\] is attained when the pulse is truncated such that the same fraction $\frac{\varepsilon}{2} E_\mathrm{tot}$ of the total soliton energy $E_\mathrm{tot}=4\sum_{k=1}^N \sigma_k=4N\sigma$ is lost in both of the respective tails. Then, the pulse duration can be calculated from where we use the signal envelope in Lem. \[lem:sig\_tails\_real\] to approximate the integral: $$\begin{aligned}
\label{eq:outage_energy_EVreal}
\frac{\varepsilon}{2} E_\mathrm{tot} =\int_{-\infty}^{T_-} \left|q^{(N)}(t)\right|^2 \partial t = \int_{T_+}^{\infty} \left|q^{(N)}(t)\right|^2 \partial t
\approxeq 4 \sigma \exp\left(\pm 4\sigma \left(T_\mp \pm \frac{1}{2 \sigma} \ln\left(\sum_{r=1}^N \eta_r^{\mp 1} A_r \right)\right)\right).\end{aligned}$$ Solving for $T_+$ and $T_-$ leads to the pulse duration in Th. \[th:T\_real\]. Showing the approximation of $T_\mathrm{max}(\varepsilon)$ in being minimized for $\eta_r=\eta$, needs the proof of $\left. T_\mathrm{max}(\varepsilon)\right|_{\eta_r} \geq \left. T_\mathrm{max}(\varepsilon)\right|_{\eta_r=\eta}$. Using this rewrites as ${\sum_{r=1}^{N} \eta_r A_r \cdot \sum_{r=1}^{N} \frac{1}{\eta_r} A_r \geq \left(\sum_{r=1}^{N} A_r \right)^2}$. Since $\eta_r, A_r \in \mathbb{R}$, we have ${\sum_{r=1}^{N} \left|\sqrt{\eta_r A_r}\right|^2 \cdot \sum_{r=1}^{N} \left|\sqrt{\frac{1}{\eta_r} A_r}\right|^2
\geq \left|\sum_{r=1}^{N} A_r \right|^2}$ which is the Cauchy-Schwarz inequality.
Proof of approximations Thm. \[th:B\_imag\] and Thm. \[th:B\_real\] {#sec:proof_Th34}
===================================================================
\[lem:sep\_sol\_spectrum\] Consider an $N-$soliton with $\lambda_k=\omega_k+\sigma_k$, temporally separated into its first order soliton components. Its (linear) spectrum becomes with the corresponding envelope . $$\begin{aligned}
Q(f) & =-\pi \sum_{k=1}^N \exp\left(j (-\varphi_k-2\omega_k \Delta t_k-2\pi f \Delta t_k) \right) \mathrm{sech}\left(\frac{\pi^2}{2\sigma_k}\left(f+\frac{\omega_k}{\pi}\right)\right) \label{eq:soliton_spectrum}
\\
\widehat{|Q(f)|} & =\pi \sum_{k=1}^N \mathrm{sech}\left(\frac{\pi^2}{2\sigma_k}\left(f+\frac{\omega_k}{\pi}\right)\right)\label{eq:soliton_spectrum_envelope}\end{aligned}$$
The spectrum of fundamental soliton solution with eigenvalues $\lambda_k=j\sigma_k+\omega_k$, spectral amplitude scaling $\eta_k=\exp(2\sigma_k \Delta t_k)$ and spectral phase $\varphi_k$ is well known to be . $$\label{eq:one_soliton_spectrum}
Q(f) =-\pi \exp\left(j (-\varphi_k-2\omega_k \Delta t_k-2\pi f \Delta t_k)\right) \mathrm{sech}\left(\frac{\pi^2}{2\sigma_k}\left(f+\frac{\omega_k}{\pi}\right)\right).$$ Assuming a multi-soliton being separated into its first order components, it can be represented as a linear superposition of first order solitons leading to . We can conclude from that $|Q(f)| \leq \pi \sum_{k=1}^N \mathrm{sech}\left(\frac{\pi^2}{2\sigma_k}\left(f+\frac{\omega_k}{\pi}\right)\right)$. Furthermore, for each frequency point $f_0$, there exists one combination of $\{\varphi_k\}$ such that all terms ${\exp(-j\varphi_k-2j\omega_k \Delta t_k-2j\pi f \Delta t_k)}$ in have identical phase. Thus, the envelope of the spectrum among all phase combinations $\{\varphi_k\}$ is .
Consider the signal spectrum in Lem. \[lem:sep\_sol\_spectrum\] for imaginary eigenvalues $\lambda_k=j\sigma_k$. In the limits $f\to \pm \infty$, $\mathrm{sech}(x)$ decays exponentially. Thus all but the leading exponential term, determined by $\sigma_1=\underset{k}{\max} \, \sigma_k$, can be neglected in . Thus we get $$\label{eq:Q_imag_limit}
|Q(f)| \to 2\pi \exp \left(\mp \frac{\pi^2 f}{2 \sigma_1} \right), \quad f \to \pm \infty$$ Using and noting its symmetry, the bandwidth $B$ according to Def. \[def:TB\_def\_energy\] can be estimated from\
${\frac{\varepsilon}{2}E_\mathrm{tot} = \int_{B/2}^\infty |Q(f)|^2 \partial f
\approxeq
%\int_{B/2}^\infty 4 \pi^2 \exp \left(- \frac{\pi^2 f}{\sigma_1} \right) \partial f =
4 \sigma_1 \exp(-\frac{\pi^2 B}{2 \sigma_1})}$. Solving for $B$ leads to the approximation in Th. \[th:B\_imag\].
Consider the envelope spectrum in Lem. \[lem:sep\_sol\_spectrum\] for eigenvalues parallel to the real axis $\lambda_k=\omega_k + j\sigma$. In the limits $f \to \pm \infty$, the envelope spectrum writes as $$\begin{aligned}
\widehat{|Q(f)|} & \to 2\pi \exp\left(\mp \frac{\pi^2 f}{2 \sigma}\right)\sum_{k=1}^N \exp\left(\mp \frac{\pi \omega_k}{2\sigma}\right), \quad f \to \pm \infty.\end{aligned}$$ We can derive an approximation of the bandwidth $B=B_+ - B_-$ according to Def. \[def:TB\_def\_energy\] from $\frac{\varepsilon}{2 }E_\mathrm{tot} \approxeq \int_{-\infty}^{B_-} \widehat{|Q(f)|}^2 \partial f =\int_{B_+}^{\infty} \widehat{|Q(f)|}^2 \partial f
\approxeq 4 \sigma \exp(\mp \frac{\pi^2 B_\pm}{\sigma}) \left(\sum_{k=1}^N \exp\left(\mp\frac{\pi \omega_k}{2\sigma}\right)\right)^2$. Solving for $B_+$ and $B_-$ leads to the bandwidth approximation in Thm. \[th:B\_real\].
[^1]: A. Span and S. ten Brink are with the Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany
[^2]: V. Aref and H. Bülow are with Nokia Bell Labs, Stuttgart, Germany.
|
---
abstract: |
We have carried out a multi-fractal analysis of the distribution of galaxies in the three Northern slices of the Las Campanas Redshift Survey. Our method takes into account the selection effects and the complicated geometry of the survey. In this analysis we have studied the scaling properties of the distribution of galaxies on length scales from $20 h^{-1} {\rm Mpc}$ to $200 h^{-1} {\rm Mpc}$. Our main results are: (1) The distribution of galaxies exhibits a multi-fractal scaling behaviour over the scales $20 h^{-1} {\rm
Mpc}$ to $80 h^{-1} {\rm Mpc}$, and, (2) the distribution is homogeneous on the scales $80h^{-1} {\rm Mpc}$ to $200 h^{-1} {\rm Mpc}$. We conclude that the universe is homogeneous at large scales and the transition to homogeneity occurs somewhere in the range $80 h^{-1} {\rm Mpc}$ to $100 h^{-1} {\rm Mpc}$.
author:
- Somnath Bharadwaj
- 'A. K. Gupta'
- 'T. R. Seshadri'
title: Nature of Clustering in the Las Campanas Redshift Survey
---
Introduction {#intro}
============
Many surveys have been carried out to chart the positions of galaxies in large regions of the universe around us, and many more surveys which go deeper into the universe are currently underway or are planned for the future. These surveys give us detailed information about the distribution of matter in the universe, and identifying the salient features that characterize this distribution has been a very important problem in cosmology. The statistical properties, the geometry and the topology are some of the features that have been used to characterize the distribution of galaxies, and a large variety of tools have been developed and used for this purpose.
The correlation functions which characterize the statistical properties of the distributions have been widely applied to quantify galaxy clustering. Of the various correlation functions (2-point, 3- point, etc...) the galaxy-galaxy two point correlation function $\xi(r)$ is very well determined on small scales (Peebles 1993 and references therein) and it has been found to have the form $$\xi(r)=\left( \frac{r}{r_0} \right)^{-\gamma} \hspace{0.5cm}
{\rm with} \hspace{0.5cm} \gamma=1.77 \pm 0.04 \hspace{0.5cm} {\rm and}
\hspace{0.5cm} r_0= 5.4 \pm 1 h^{-1} {\rm Mpc}$$
This power-law form of the two point correlation function suggests that the universe exhibits a scale invariant behaviour on small scales $r<r_0$. The two point correlation function becomes steeper at larger scales $r > r_0$. It is, however, not very well determined on very large scales where the observations are consistent with the correlation function being equal to zero. The standard cosmological model and the correlation function analysis are both based on the underlying assumption that the universe is homogeneous on very large scales and the indication that the correlation function vanishes at very large scales is consistent with this.
Fractal characterization is another way of quantifying the gross features of the galaxy distribution. Fractals have been invoked to describe many physical phenomena which exhibit a scale invariant behaviour and it is very natural to use fractals to describe the clustering of galaxies on small scales where the correlation function analysis clearly demonstrates a scale invariant behaviour.
Coleman and Pietronero ([@fractal]) applied the fractal analysis to galaxy distributions and concluded that it exhibits a self-similar behaviour up to arbitrarily large scales. Their claim that the fractal behaviour extends out to arbitrarily large scales implies that the universe is not homogeneous on any scale and hence it is meaningless to talk about the mean density of the universe. These conclusions are in contradiction with the Cosmological Principle and the entire framework of cosmology, as we understand today, will have to be revised if these conclusions are true.
On the other hand, several others ([@martjones; @borgani]) have applied the fractal analysis to arrive at conclusions that are more in keeping with the standard cosmological model. They conclude that while the distribution of galaxies does exhibit self similarity and scaling behaviour, the scaling behaviour is valid only over a range of length scales and the galaxy distribution is homogeneous on very large scales. Various other observations including the angular distribution of radio sources and the X-ray background testify to the universe being homogeneous on large scales ([@smooth]; [@scos]).
Recent analysis of the ESO slice project ([@guzzo]) also indicates that the universe is homogeneous over large scales. The fractal analysis of volume limited subsamples of the SSRS2 ([@yx]) studies the spatial behaviour of the conditional density at scales up to $40
h^{-1} \rm Mpc$. Their analysis is unable to conclusively determine whether the distribution of galaxies is fractal or homogeneous and it is consistent with both the scenarios. A similar analysis carried out for the APM-Stromlo survey ([@xz]) seems to indicate that the distribution of galaxies exhibits a fractal behaviour with a dimension of $D=2.1 \pm 0.1$ on scales up to $40 h^{-1} \rm Mpc$. In a more recent paper ([@X]) the fractal analysis has been applied to volume limited subsamples of the Las Campanas Redshift Survey. This uses the conditional density to probe scales up to $200 h^{-1} \rm Mpc$. They find evidence for a fractal behaviour with dimension $D \simeq 2$ on scales up to $20 \hbox{--} 40 h^{-1} \rm Mpc$. They also conclude that there is a tendency to homogenization on larger scales ($50 \hbox{--} 100 h^{-1} \rm Mpc$) where the fractal dimension has a value $D \simeq 3$, but the scatter in the results is too large to conclusively establish homogeneity and rule out a fractal universe on large scales.
In this paper we study the scaling properties of the galaxy distribution in the Las Campanas Redshift Survey (LCRS) ([@lcrs]). This is the deepest redshift survey available at present. Here we apply the multi-fractal analysis ([@martjones; @borgani]) which is based on a generalization of the concept of a mono-fractal. In a mono-fractal the scaling behaviour of the point distribution is the same around each point and the whole distribution is characterized by a single scaling index which corresponds to the fractal dimension. A multi-fractal allows for a sequence of scaling indices known as the multi-fractal spectrum of generalized dimensions. This allows for the possibility that the scaling behaviour is not the same around each point. The spectrum of generalized dimensions tells how the scaling properties of the galaxy distribution changes from the very dense regions (clusters) to the sparsely populated regions (voids) in the survey.
In this paper we compute the spectrum of generalized dimensions ($D_q$ [*vs*]{} $q$) by calculating the Minkowski-Bouligand dimension ([@borgani]) for both volume limited and magnitude limited subsamples of the LCRS. We also investigate how the spectrum of generalized dimensions depends on the length scales over which it is measured and whether the distribution of galaxies in the LCRS exhibits homogeneity on very large scales or if the fractal nature extends to arbitrarily large scales. .
We next present a brief outline of the organization of the paper. Section \[gendim\] describes the method we adopt to compute the spectrum of generalized dimensions. In section \[survey\] we describe the basic features of the LCRS and discuss the issues related to the processing of the data so as to bring it into a form usable for our purpose. Section \[analysis\] gives the details of the method of analysis specifically in the context of LCRS. The discussion of the results are presented in section \[results\] and the conclusions in section \[conc\]..
In several parts of the analysis it is required to use definite values for the Hubble parameter $H_0 (= 100 h {\rm km/s/Mpc})$ and the decceleration parameter $q_0$, and we have used $h=1$ and $q_0=.5$.
Generalized Dimension {#gendim}
=====================
A fractal point distribution is usually characterized by its dimension and there exists a large variety of ways in which the dimension can be defined and measured. Of these possibilities two which are particularly simple and can be easily applied to a finite distribution of points are the box-counting dimension and the correlation dimension. In this section we discuss the “working definitions” of these two quantities that we have adopted for analyzing a distribution of a finite number of points. For more formal definitions of these dimensions the reader is referred to Borgani (1995) and references therein. The formal definitions usually involve the limit where the number of particles tends to infinity and they cannot be directly applied to galaxy distributions.
We first consider the box-counting dimension. In calculating the box-counting dimension for a distribution of points, the space is divided into identical boxes and we count the number of boxes which contain at least one point inside them. We then progressively reduce the size of the boxes while counting the number of boxes with at least one point inside them at every stage of this process. This gives the number of non-empty boxes $N(r)$ as a function of the size of one edge of the box $r$ at every stage of the procedure. If the number of non-empty boxes exhibits a power-law scaling as a function of the size of the box i.e. $$N(r) \propto r^{D} \label{eq:gd1}$$ we then define $D$ to be the box-counting dimension. In practice the nature of the scaling may be different on different length scales and we look for a sufficiently large range of $r$ over which $ N(r)$ exhibits a particular scaling behaviour and we then use equation (\[eq:gd1\]) to obtain the box-counting dimension valid over those scales. So finally we may get more than one value of box-counting dimension for the distribution, each value of the box counting dimension being valid over a limited range of length scales.
To compute the correlation dimension for a point distribution with N points we proceed by first labeling the points using an index j which runs from $1$ to $N$. We then randomly select $M$ of the $N$ points and the index $i$ is used to refer to these $M$ randomly chosen points.
For every point $i$, we count the total number of points which are within a distance $r$ from the $i^{th}$ point and this quantity $n_i(r)$ can be written as $$n_i(r) =\sum_{j=1}^{N} {\Theta}(r-\mid {\vec {x_i}}-{\vec {x_j } }\mid)
\label{eq:gd2}$$ where $\vec {x_i }$ is the position vector of the $i^{th}$ point and $\Theta$ is the Heavy-side function. $\Theta=0$ for $x<0$ and $\Theta=1$ for $x{\ge}0$. We next divide $n_i(r)$ by the total number of points $N$ to calculate $p_i(r)$, the probability of finding a point within a distance $r$ from the $i {\rm th}$ point. We then average the quantity, $p_i(r)$, over all the $M$ randomly selected centers to determine the probability of finding a point within a distance $r$ of another point and we denote this by $C_2(r)$ which is given by, $$\begin{aligned}
C_2(r) = \frac{1}{M N} \sum_{i=1}^{M} n_i (r) \,. \label{eq:gd3}\end{aligned}$$
If the probability $C_2$ exhibits a scaling relation $$C_{2}(r) \propto r^{D_2} \label{eq:gd4}$$ we then define $D_2$ to be the correlation dimension.
As with the box-counting dimension, the nature of the scaling behaviour may be different on different length scales and we may then get more than one value for the correlation dimension, each different value being valid over a range of scales.
It is very clear that $C_2(r)$ - which is the probability of finding a point within a sphere of radius $r$ centered on another point, is closely related to the volume integral of the two point correlation function. In a situation where the two point correlation function exhibits a power-law behaviour $\xi(r)=(r/r_o)^{-\gamma}$ on scales $r<r_0$, we expect the correlation dimension to have a value $D_2=3-\gamma$ over these scales.
For a mono-fractal the box-counting dimension and the correlation dimension will be the same, and for a homogeneous, space filling point distribution they are both equal to the dimension of the ambient space in which the points are embedded.
The box-counting dimension and the correlation dimension quantify different aspects of the scaling behaviour of a point distribution and they will have different values in a generic situation. The concept of a generalized dimension connects these two definitions and provides a continuous spectrum of dimensions $D_q$ for a range of the parameter $q$. The definition of the Minkowski-Bouligand dimension $D_q$ ([@falconer; @feder]) closely follows the definition of the correlation dimension the only difference being that we use the $(q-1) {\rm th}$ moment of the galaxy distribution $n_i(r) $ (eq. \[eq:gd2\]) around any point. Equation (\[eq:gd3\]) can then be generalized to define $$\begin{aligned}
C_{q}(r)= {\frac{1}{N M}}\sum_{i=1}^{M}[n_i(<r)]^{q-1} \label{eq:gd5} \,.\end{aligned}$$ which is used to define the generalized dimension $$D_q=\frac{1}{q-1} \frac {d{\ln}C_{q}(r)}{d{\ln}r} \,. \label{eq:gd6}$$
The quantity $C_{q}(r)$ may exhibit different scaling behaviour over different ranges of length scales and we will then get more than one spectrum of generalized dimensions each being valid over a different range of length scales.
From equations (\[eq:gd5\]) and (\[eq:gd6\]) it is clear that the the generalized dimension $D_q$ corresponds to the correlation dimension at $q=2$. In addition $D_q$ corresponds to the box-counting dimension at $q=1$.
For a mono-fractal the generalized dimension is a constant i.e. $D_q=D$ which reflects the fact that for a mono-fractal the point distribution is characterized by a unique scaling behaviour. For a generic multi-fractal the values of $D_q$ will be different for different values of $q$. The positive values of $q$ give more weight-age to the over-dense regions. Thus, for $q > 0$, $D_q$ probes the scaling behaviour of the distribution of points in the over-dense regions like inside clusters etc. The negative values of $q$, on the other hand, give more weight-age to the under-dense regions and, hence, for negative $q$, $D_q$ probes the scaling behaviour of the distribution of points in the under-dense regions like voids.
Finally it should be pointed out that the Minkowski-Bouligand generalized dimension $D_q$ is one of the many possible definitions of a generalized dimension. The minimal spanning tree used by van der Weygaert and Jones ([@Weygaert]) is another possible method which can be used. The Minkowski-Bouligand generalized dimension has the advantage of being easy to compute. In addition the various selection effects which have to be taken into account when analyzing redshift surveys can be easily accounted for when determining the Minkowski-Bouligand generalized dimension and hence we have chosen this particular method for the multi-fractal characterization of the galaxy distribution in LCRS,
A Brief Description of the Survey and the Data. {#survey}
===============================================
The LCRS consists of 6 alternating slices each subtending $80^{\circ}$ in right-ascension and $1.5^{\circ}$ in declination, 3 each in the Northern and Southern Galactic Caps centered at $\delta = -3^{\circ}, -6^{\circ}, -12^{\circ}$ and $\delta = -39^{\circ}, -42^{\circ}, -45^{\circ}$ respectively. The survey extends to a redshift of $\sim .2$ corresponding to $600
h^{-1}{\rm Mpc}$ in the radial direction. The survey contains about 24000 galaxies distributed with a mean redshift of $z=.1$ corresponding to $300 h^{-1} {\rm Mpc}$.
We next elaborate a little on the shape of the individual slices. Consider two cones both with the same axis and with their vertices at the same point. Let the angle between the first cone and the axis be $90^{\circ}-(\delta - .75^{\circ})$ and the second cone and the axis be $90^{\circ}-(\delta + .75^{\circ})$ so that the angle between the two cones is $1.5^{\circ}$. Next truncate both the cones at a radial distance of $600 h^{-1} {\rm Mpc}$ from the vertex. Finally, a slice centered at a declination $\delta$ corresponds to a $80^{\circ}$ wedge of the region between these two cones. The effect of the extrinsic curvature of the cones is small for the three northern slices and we have restricted our analysis to only these three slices for which we have neglected the effect of the curvature.
Each slice in the LCRS is made up of $1.5^\circ$ x $1.5^\circ$ fields some of which were observed with a $50$ object fibre system and others with a $112$ object fibre system. Of the three northern slices the one at $\delta=-12^\circ$ is exclusively made up of 112 fibre fields while the slice at $\delta=-6^\circ$ is mostly 50 fibre, and the slice at $\delta=-3^\circ$ has got both 50 and 112 fibre fields.
For each field, redshifts were determined for those galaxies which satisfy the magnitude limits and the central brightness limits of the survey. These limits are different for the 50 fibre and the 112 fibre fields. In addition, for those fields where the number of galaxies satisfying the criteria for inclusion in the survey exceeded the number of fibres, the redshifts were determined for only a fraction of the galaxies in the field. This effect is quantified by the “galaxy sampling function” $f$ which varies from field to field and is around $80 \%$ for the 112 fibre fields and around half this number for the 50 fibre fields. In addition to the field to field variation of the galaxy sampling function there are two other effects which have to be accounted for when analyzing the galaxy distribution. They are, (1). Apparent Magnitude and Surface Brightness Incompleteness, and, (2). Central Surface Brightness Selection. These are quantified by two factors $F$ and $G$, respectively, which are discussed in detail in Lin [*et al.*]{} (1996). The survey data files provide the product of these three factors $sf=f \cdot F \cdot
G$ for each galaxy and the contribution from the $i \, th$ galaxy has to be weighted with the factor $$W_i=\frac{1}{f_i \cdot F_i \cdot G_i} \label{eq:s1}$$ when analyzing the survey.
The factor $W_i$ discussed above takes into account the effects of the field-to-field sampling fraction and the incompleteness as a function of the apparent magnitude and central surface brightness. In addition, the selection function $s(r)$ has also to be taken into account, and this depends on both the differential luminosity function $\phi(M)$ and the magnitude limits of the survey. The luminosity function of LCRS has been studied by Lin [*et al.*]{} (1996) who have determined the luminosity function for different sub-samples of LCRS.
They find that the Schechter form with the parameters $M^{\star}=-20.29 + 5 \log(h)$, $\alpha=-0.70$ and $\phi^{\star}=
0.019 h^{3} {\rm Mpc}^{-3}$ provides a good fit for the luminosity function in the absolute magnitude range $-23.0 \ge M \ge -17.5$. They have obtained these parameters from the analysis of the combined Northern and Southern 112 fibre fields and we shall refer to the Schechter luminosity function with these set of parameters as the NS112 luminosity function. The analysis of Lin [*et al.*]{} (1996) shows that this luminosity function can be used for the Northern 50 fibre fields in addition to the Northern and Southern 112 fibre fields, and we have used the NS112 luminosity function for most of our analysis.
Lin [*et al.*]{} (1996) have also separately provided the luminosity function determined using just the Northern 112 fibre fields. This has the Schechter form with the parameters $M^{\star}=-20.28 + 5 \log(h)$, $\alpha=-0.75$ and $\phi^{\star}= 0.018 h^{3} {\rm Mpc}^{-3}$ and we refer to this as the N112 luminosity function. We have used this in some of our analysis of the $\delta=-12^{\circ}$ slice which contains only 112 fibre fields.
The selection function $s(z)$ quantifies the fact that the fraction of the galaxies which are expected to be included in the survey varies with the distance from the observer. For a magnitude limited survey the apparent magnitude limits $m_1$ and $m_2$ can be converted to absolute magnitude limits $M_1(z)$ and $M_2(z)$ at some redshift $z$. In addition if we impose further absolute magnitude criteria $M_1 \ge M
\ge M_2$, then the selection function can be expressed as $$s(z)=\int^{min[M_2(z),M_2]}_{max[M_1(z),M_1]} \phi(M) d M
{\Big /} \int^{M_2}_{M_1} \phi(M) d M \,. \label{eq:s2}$$
The apparent magnitude limits are different for the 50 and 112 fibre fields and we have used the appropriate magnitude limits and the N112/ NS112 luminosity functions to calculate the selection function at the redshift of each of the galaxies. This is then used to calculate a weight factor for each of the galaxies, and the contribution of the $i \, th$ galaxy in the survey has to be weighed by $$w_i=\frac{W_i}{ s(z_i)} \,. \label{eq:s3}$$
Another effect that we have to correct for arises because of the fact that we would like to treat the distribution of galaxies in each slice as a two dimensional distribution. Each slice consists of galaxies that are contained within a thin conical shell of thickness $1.5^o$ and we construct a two dimensional distribution by collapsing the thickness of the slice. The thickness of each slice increases with the distance from the observer and in order to compensate for this effect we weigh each galaxy by the inverse of the thickness of the slice at its red-shift. Taking this effect into account the weight factor gets modified to $$w_i=\frac{W_i}{z_i s(z_i)} \,. \label{eq:s4}$$ which we use to weigh the contribution from the $i \, th $ galaxy in the LCRS.
We should also point out that through the process of flattening the conical slices and collapsing its thickness, the three dimensional galaxy distribution has been converted to a 2-dimensional distribution and the whole of our multi-fractal analysis is for a planar 2-dimensional point distribution.
In our analysis we have considered various subsamples of LCRS all chosen from the 3 Northern slices. In addition to the apparent magnitude limits of the survey we have imposed further absolute magnitude and redshift cutoffs to construct both volume and apparent magnitude limited subsamples whose details are presented in Table I.
Method of Analysis {#analysis}
==================
We first extract various subsamples of LCRS using the criteria given in Table I for each of the subsample. For each subsample we next calculate the weight function $w_i$ (equation \[eq:s4\]) for all the galaxies in the subsample. In addition the 3-dimensional distribution of galaxies in the sub-sample is converted into a corresponding 2-dimensional distribution using the steps outlined in the previous section and we finally have a collection of $N$ galaxies distributed over a region of a plane.
We next choose $M$ of these galaxies at random and count the number of galaxies inside a circle of radius $r$ drawn around each of these $M$ randomly chosen galaxies. In determining this we use a modified version of equation (\[eq:gd2\]) where each galaxy in the circle has an extra weight factor $w_j$ as calculated in the previous section, i.e. $$n_i(r) =\sum_{j=1}^{N} w_j {\Theta}(r-\mid {\vec {x_i}}-{\vec {x_j } }\mid)
\label{eq:az1} \,.$$
The different moments of this quantity are averaged over the $M$ galaxies to obtain $C_q(r)$ defined in equation (\[eq:gd5\]) for a range of $q$. The exercise is repeated with circles of different radii (different values of $r$) to finally obtain $C_q(r)$ for a large range of $r$.
It should be noted that the region from which the $M$ points can be chosen at random depends on the size of the circle which we are considering. For very large values of $r$ a large region around the boundaries of the survey has to be excluded because a circle of radius $r$ drawn around a galaxy in that region will extend beyond the boundaries of the survey. As a consequence for large values of $r$ we do not have many galaxies which can serve as centers, while for small values of $r$ there are many galaxies which can serve as centers for circles of radius $r$. For $r$ between $80 h^{-1} {\rm
Mpc}$ to $200 h^{-1} {\rm Mpc}$ we use $M=60$ which is of the same order as the the total number of galaxies available for use as centers. To estimate the statistical significance of our results at this range of length-scales we have randomly divided the 60 centers into independent groups of 20 centers and repeated the analysis for each of these. We have used the variation in the results from the different subsamples to estimate the statistical errors for our results on large scales. In the range $r<80 h^{-1} {\rm Mpc}$ we have used $M=100$ which is only a small fraction of the total number of galaxies which can possibly serve as centers which is around 1500. At this range of length-scales it is possible to choose many independent sets of 100 centers. We have performed the analysis for a large number of such sets of 100 centers and these have been used to estimate the mean generalized dimension $D_q$ and the statistical errors in the estimated $D_q$ at small scales. For both the range of length-scales considered we have tried the analysis making changes in the number of centers and we find that the results do not vary drastically as we vary the number of centers used in the analysis.
The value of the generalized dimension $D_q$ is determined for a fixed value of $q$ by looking at the scaling behaviour of $C_q(r)$ as a function of $r$ ([*e.g.*]{} Figures \[scaleq0\] and \[scaleq2\]) We have considered $q$ in the range $-10 \le q \le +10$. In principle we could have considered arbitrarily large (or small) values of $q$ also, but the fact that there are only a finite number of galaxies in the survey implies that only a finite number of the moments can have independent information. This point has been discussed in more detail by Bouchet [*et.al.*]{} ([@bouchet]).
In addition to the subsamples of galaxies listed in table 1, we have also carried out our analysis for mock versions of these subsamples of galaxies. The mock versions of each subsample contains the same number of galaxies as the actual subsample. The galaxies in the mock versions are selected from a homogeneous random distribution using the same selection function and geometry as the actual subsample. We have carried out the whole analysis for many different random realizations of each of the subsamples listed in Table I. The main aim of this exercise was to test the reliability of the method of analysis adopted here.
Results and Discussion {#results}
======================
We first discuss our analysis of the mock subsamples. Since the effect of the selection function and the geometry of the slices have both been included in generating these subsamples, our analysis of these subsamples allows us to check how well these effects are being corrected for. In the ideal situation for all the mock subsamples we should recover a flat spectrum of generalized dimensions with $D_q=2$ corresponding to a homogeneous point distribution. The actual results of the multi-fractal analysis of the mock subsamples are presented below where we separately discuss the behaviour of $D_q$ at small scales $(r<80 h^{-1} {\rm Mpc})$ and at large scales $(r>80 h^{-1}
{\rm Mpc})$.
The results for mock versions of the subsample d-12.1 are shown in figure (\[mock12\]). This is a magnitude limited subsample from a slice that has only 112 fibre fields and it contains the largest number of galaxies. We get a nearly flat spectrum with $D_q=2$ corresponding to a homogeneous point distribution at both small and large scales. Similar results are also obtained for mock versions of the other subsamples of the $\delta=-12^\circ$ slice.
The analysis of mock versions of the subsample d-03.1 which contains both 112 and 50 fibre fields gives a spectrum with a weak $q$ dependence (figure \[mock3\]). This effect is more noticeable at small scales than at large scales. The analysis of mock versions of the d-06.1 subsample (figure \[mock6\]) gives similar results at small scales. At large scales we get a nearly flat curve with $D_q \simeq 1.8$ This subsample d-06.1 has mostly 50 fibre fields and it has around half the number of galaxies as the d-12.1 subsample.
We thus find that the analysis is most effective for the subsample from the $\delta=-12^\circ$ slice where $D_q$ shows very little $q$ dependence and $2.1 \le D_q \le 1.9$. For the other two slices we find a weak $q$ dependence with $2.2 \le D_q \le 1.8$. This clearly demonstrates that our method of multi-fractal analysis correctly takes into account the different selection effects and the complicated sampling and geometry for all the subsamples that we have considered.
We next discuss our analysis of the actual data. The analysis of the curves corresponding to $C_q(r)$ versus $r$ for the different subsamples shows the existence of two very different scaling behaviour - one at small scales and another at large scales, with the transition occurring around $80 h^{-1} {\rm
Mpc}$ to $100 h^{-1} {\rm Mpc}$. The scaling behaviour of $C_q(r)$ is shown in figure \[scaleq0\] and figure \[scaleq2\] for $q=0$ and $q=2$, respectively for the subsample d-12.1 The other subsamples all exhibit a similar behaviour. Based on this we have treated the scales $20 h^{-1} {\rm Mpc} \le r \le80 h^{-1} {\rm Mpc}$ (small scales) and $80 h^{-1} {\rm Mpc} \le r \le 200 h^{-1} {\rm Mpc}$ (large scales) separately and the multi-fractal analysis has been performed separately for the small and large scales. Figures \[dq12\], \[dq3\] and \[dq6\] show the spectrum of generalized dimensions $D_q \, vs \, q$ at both small and large scales for three of the subsamples.
We find that at small scales the plots of $D_q$ versus $q$ for the actual data (figures \[dq12\], \[dq3\], \[dq6\] ) are quite different from the corresponding plots for the mock versions of the data (figures \[mock12\], \[mock3\], \[mock6\]). This clearly shows that the distribution of galaxies is not homogeneous over the scales $20h^{-1} \rm{Mpc} \le r
\le 80 h^{-1} \rm{Mpc}$. In addition we find that all the subsamples exhibit a multi-fractal behaviour over this range of length-scales. The interpretation of the different values of the multi-fractal dimension $D_q$ is complicated by the geometry of the survey and we do not attempt this here.
At large scales the behaviour of the generalized dimension $D_q$ is quite different. For the subsample d-12.1 the spectrum shows a weak $q$ dependence (figure \[dq12\]) and $D_q$ shows a gradual change from $D_q \simeq 2$ to $D_q \simeq 1.8$ as $q$ varies from $-10$ to $10$. This is quite different from the behaviour at small scales where the change in $D_q$ is larger and more abrupt. The behaviour of the other subsamples of the $\delta=-12^{\circ}$ slice are similar. For the subsample d-03.1 we find that the spectrum is nearly flat (figure \[dq3\]) with $D_q \simeq 2$ and for d-06.1 (figure \[dq6\]) the spectrum is nearly flat with $D_q \simeq 1.8$. These values are within the range we recover from our analysis of the mock subsamples which are constructed from an underlying random homogeneous distribution of galaxies. This agreement between the actual data and the random realizations with $2.2 \le D_q \le 1.8$ in all the subsamples shows that the distribution of galaxies in LCRS is homogeneous at the large scales.
The work presented here contains significant improvements on the earlier work of Amendola & Palladino (1999) on two counts and these are explained below:\
(1). Unlike the earlier work which has analyzed volume limited subsamples of one of the slices ($\delta=-12^{\circ} $) of the LCRS we have analyzed both volume and magnitude limited subsamples of all the three northern slices of the LCRS. The magnitude limited samples contain more than four times the number of galaxies in the volume limited samples and they extend to higher redshifts. This allows us to make better use of the data in the LCRS to improve the statistical significance of the results and to probe scales larger than those studied in the previous analysis.\
(2). We have calculated the full spectrum of generalized dimensions which has information about the nature of clustering in different environments. The integrated conditional density used by the earlier workers is equivalent to a particular point $(q=2)$ on the spectrum and it does not fully characterize the scaling properties of the distribution of galaxies.
Conclusion. {#conc}
===========
Here we present a method for carrying out the multi-fractal analysis of both magnitude and volume limited subsamples of the LCRS. Our method takes into account the various selection effects and the complicated geometry of the survey.
We first apply our method to random realizations of the LCRS subsamples for which we ideally expect a flat spectrum of generalized dimensions with $D_q =2$. Our analysis gives a nearly flat spectrum with $1.8 \le D_q
\le 2.2$ on large scales. The deviation from the expected value includes statistical errors arising from the finite number of galaxies and systematic errors arising from our treatment of the selection effects and the complicated geometry. The fact that the errors are small clearly shows that our method correctly accounts for these effects.
Our analysis of the actual data shows the existence of two different regimes and the distribution of galaxies on scales $20 h^{-1} {\rm
Mpc} \le r \le 80 h^{-1} {\rm Mpc}$ shows clear indication of a multi-fractal scaling behaviour. On large scales $80 h^{-1} {\rm
Mpc} \le r \le 200 h^{-1} {\rm Mpc}$ we find a nearly flat spectrum with $1.8 \le D_q \le 2.2$. This is consistent with our analysis of the random realizations which have been constructed from a homogeneous underlying distribution of galaxies.
Based on the above analysis we conclude that the distribution of galaxies in the Las Campanas Redshift Survey is homogeneous at large scales with the transition to homogeneity occurring somewhere around $80 h^{-1} {\rm Mpc}$ to $100 h^{-1} {\rm Mpc}$.
TRS would like to thank thank T. Padmanabhan, K. Subramanian, J. S. Bagla, F. S. Labini and L. Pietronero for several useful discussions. AKG and TRS gratefully acknowledge the project grant (SP/S2/009/94) from the Department of Science and Technology, India. All the authors are extremely grateful to the LCRS team for making the catalogue publicly available.
[ddd 9999]{} Amendola L. and Palladino E., 1999, Ap.J., In press, astro-ph/9901420 Borgani S., 1995, [*Phys. Rep.*]{} [**251**]{}, 1 Bouchet F. R., Schaeffer R. and Davis M., 1991, [*Ap J*]{}, [**383**]{}, 19 Cappi A., Benoist C., Da Costa L.N., Maurogordato S., 1998, astro-ph/9804085 Coleman P H. and Pietronero L., 1992, [*Phys. Rep.*]{} [**213**]{}, 311. Falconer K, 1990 in [*Fractal Geometry: Mathematical Foundations and Applications, John Wiley.*]{} Feder J, 1989, in [*Fractals, Plenum Press.*]{} Guzzo , 1998, [*New Astronomy*]{} [**2**]{}, 517. Labini F. Sylos & M. Montuori, 1997, astro-ph/9711134 Labini F S, Montuori M. and Pietronero L., 1998, [*Phys. Rep.*]{} [**293**]{}, 61. Lin, H., Kirshner, R. P., Shectman, S.A., Landy, S. D., Oemler, A., Tucker, D. L. and Schechter, P. L. 1996, [*ApJ*]{}, [**471**]{}, 617. Martinez V J., 1991, in [*Applying Fractals in Astronomy, eds. A. Heck and J.M.Perdang, Springer-Verlag, Page 135.*]{} Martinez V J. and Jones B J T., 1990, [*MNRAS*]{} [**242**]{}, 517. Padmanabhan T., 1992 in [*Structure Formation in the Universe, Cambridge University Press.*]{} Peebles P. J. E., 1998, preprint, astro-ph 9806201 Peebles P. J. E., 1993, in [*Principles of Physical Cosmology, Princeton University Press*]{} Shectman, S. A., Landy, S. D., Oemler, A., Tucker, D. L., Lin, H., Kirshner, R. P. and Schechter, P. L. 1996, [*ApJ*]{}, [**470**]{}, 172. van der Weygaert R, and Jones B J T., 1992, [*Phys. Lett.*]{} [**A169**]{}, 145. Wu Kelvin K. S., Ofer Lahav, Martin J. Rees,, 1998, preprint , astro-ph/9804062
[**Table 1.**]{}
[lllllll]{} & & &Absolute&Luminosity&Number of&Vol./Mag.\
subsample&$\,\,\,\delta$&z range& Magnitude range& Function&Galaxies &Limited\
\
d-12.1&$-$12.0&0.017-0.2&$-$23.0 - $-$17.5&NS112&4458&M\
d-12.2&$-$12.0&0.017-0.2&$-$23.0 - $-$17.5&N112&4458&M\
d-12.3&$-$12.0&0.05-0.1&$-$21.0 - $-$20.0&N112&869 &V\
d-12.4&$-$12.0&0.065-0.125&$-$21.5 - $-$20.5&N112&923&V\
d-06.1&$-$6.0&0.017-0.2&$-$23.0 - $-$17.5&NS112&2316&M\
d-03.1&$-$3.0&0.017-0.2&$-$23.0 - $-$17.5&NS112&4055&M\
|
---
author:
- Maša Lakićević
- Stefan Kimeswenger
- Stefan Noll
- Wolfgang Kausch
- Stefanie Unterguggenberger
- Florian Kerber
bibliography:
- 'BESOMolecfit\_arxiv.bib'
date: Received 10 November 2015
title: 'Study of the atmospheric conditions at Cerro Armazones using astronomical data[^1]'
---
[We studied the precipitable water vapour (PWV) content near Cerro Armazones and discuss the potential use of our technique of modelling the telluric absorbtion lines for the investigation of other molecular layers. The site is designated for the European Extremely Large Telescope (E-ELT) and the nearby planned site for the [Č]{}erenkov Telescope Array (CTA).]{} [Spectroscopic data from the Bochum Echelle Spectroscopic Observer (BESO) instrument were investigated by using [line-by-line radiative transfer model (LBLRTM)]{} radiative transfer models for the Earth’s atmosphere with the telluric absorption correction tool [molecfit]{}. All observations from the archive [in the period from]{} December 2008 to the end of 2014 were investigated. [The dataset completely covers the El Ni[ñ]{}o event registered in the period 2009-2010.]{} Models of the 3D Global Data Assimilation System (GDAS) were used for further comparison. Moreover, for those days with coincidence of data from a similar study with VLT/X-shooter and microwave radiometer LHATPRO data at Cerro Paranal, a direct comparison is presented.]{} [[This analysis shows that the site has systematically lower PWV values, even after accounting for the decrease in PWV expected from the higher altitude of the site with respect to Cerro Paranal, using the average atmosphere found with radiosondes.]{} We found that GDAS data are not a suitable method for predicting of local atmospheric conditions – they usually systematically overestimate the PWV values. Due to the large sample, we were furthermore able to characterize the site with respect to symmetry across the sky and variation with the years and within the seasons. This kind of technique of studying the atmospheric conditions is shown to be a promising step into a possible monitoring equipment for CTA.]{}
Introduction
============
Site selection and characteristics are often based on long–term campaigns with mainly ground–based facilities [@GTC98; @TMT1; @ELT1; @ELT2], calibrated by a few in situ verifications. These campaigns measure meteorological parameters like temperature, humidity, wind speed and direction, ambient pressure near the surface and seeing conditions. All these facilities are calibrated on other sites with already existing enhanced astronomical facilities, assuming similar ratios of ground layer to general properties of the atmosphere [@ELT3ground]. In the context of site testing for Giant Magellan Telescope (GMT), Thirty Meter Telescope (TMT) and European Extremely Large Telescope (E-ELT), the determination of atmospheric precipitable water vapour (PWV) from optical absorption spectroscopy has been established as a standard approach [@Querel11].
Another innovation was the use of microwave radiometers deployed at several potential sites [@otarola10; @Kerber10pwv; @Kerber12pwv]. [Multi-wavelength]{} radiometers, [as e.g. the Low Humidity And Temperature PROfiling microwave radiometer (LHATPRO[^2]) used at ESO,]{} are capable of providing profiles of the distribution of water vapour in the atmosphere. The commonly accepted verification for such [measurements with balloon-borne radiosondes was shown by @Kerber12pwv. The results]{} agree well with [those of ]{} the radiosondes. Hence, further measurements can be verified with radiometer data directly (at least in desert zones like Cerro Paranal). The LHATPRO is a radiometer measuring at a frequency of 183 GHz and is hence especially suited for operations in extremely dry conditions such as the Atacama desert [@Rose]. It has been deployed on Paranal as a tool to support VLT science operations and has been operational since 2012 [@Kerber12pwv]. Its measurement have been instrumental in documenting episodes of extremely low PWV [well below 0.5 mm, @Kerber14mnras].
For the making of local atmospheric models, based on the astronomical data, all the above mentioned facilities are combined. [The global models,]{} from worldwide weather and climate data calculations (e.g. 3D Global Data Assimilation System (GDAS)[^3] models) [are used as starting point for the fitting procedures on the astronomical data, which scale the modelled molecular abundance profiles [@Smette15]]{}. This allows the determination of individual absorption of molecules like ozone, required e.g. for the data reduction of experiments like the [Č]{}erenkov Telescope Array (CTA). Furthermore, this allows one to characterize a site for its homogeneity and asymmetries in the water vapour content caused e.g. by dominating local ground layer air streams, [which is important for observations with the E-ELT in the infrared spectral bands]{}. An even more sophisticated solution was presented recently, using local refinement calculations around the site [@Lascaux15; @Valparaiso11; @Valparaiso15]. [The numerical weather prediction models, in particular those running at high horizontal and vertical resolution, require extensive computing time, and they are not always readily available to support the operations and astronomical observations program.]{}
@Noll12 designed a detailed sky model of Cerro Paranal, for ESO. In a follow–up project, a software [molecfit]{} was designed for the Earth’s lower atmosphere in local thermodynamic equilibrium [@Smette15]. [[Molecfit]{}]{} is applied to the astronomical spectroscopic data to determine the atmospheric conditions at the time of the observation. It uses the radiative transfer code LNFL/LBLRTM[^4] [@CLO05]. The latter incorporates the spectral line parameter database [aer\_v\_3.2]{}, based on HITRAN[^5] [@Rothman_Hitran]. [Furthermore, [molecfit]{} uses a model of the altitude-dependent chemical composition of the atmosphere obtained from a combination of]{} the ESO MeteoMonitor[^6], a standard atmosphere[^7] and the GDAS model by the National Oceanic and Atmospheric Administration (NOAA; time-dependent profiles of the temperature, pressure, and humidity). The code calculates the amount of atmospheric molecules by scaling the atmospheric profiles iteratively with the radiative transfer code LNFL/LBLRTM [@CLO05]. By fitting the telluric absorption features of the theoretical spectrum to the observed one, a final best-fit profile is achieved, which is assumed to be representative to the state of the atmosphere at the time of observation. This finally allows one to determine the column densities of the incorporated molecular species.
In [@Kausch14] we showed the verification of the results against the same , which was used for a comparison with the balloon experiments in [@Kerber12pwv]. [We found that the median difference between the PWV from [molecfit]{} and from was 0.12mm with $\sigma\,=\,$0.35mm]{}. The program package is designed for a robustness against inaccurate initial values. Thus, it is suited for an automatic batch mode capability, as included for pipelines. Moreover, it is able to run ’on the fly’ in monitoring during operations; e.g. as proposed in @TMT2015 for TMT on Mauna Kea. [@Kimeswenger15] thus proposed a small telescope with a fiber optic spectrograph for a site of the [Č]{}erenkov Telescope Array (CTA) operations.
We present, to our knowledge, for the first time a spatially resolved study for the designated site for the E-ELT which is also nearby the recently proposed location for the [Č]{}erenkov Telescope Array (CTA). Moreover our study covers six years and includes [El Ni[$\tilde {\rm n}$]{}o]{} [a quasi periodic change of the south pacific climate; @elnino]. Both were not included in the previous site study for the TMT [@otarola10], covering only about 200 days of radiometer data and containing no information on the spatial distribution. We were able to characterize the water vapor distribution and discuss the potential of this technique used here for further extensions towards CTA.
The data
========
The astronomical observations were made with BESO, a fibre-fed high-resolution spectrograph at Observatorio Cerro Armazones (OCA), located 22 km from Cerro Paranal in the Atacama desert in Chile [@Steiner06; @Steiner08; @Hodapp10; @Fuhrmann11]. The location is 1.45 km SW (direction 221$\degr$) from the main summit of Cerro Armazones and often called Cerro Murphy, at an altitude of 2817 m, about 200 m below the site. The proposed CTA site is 15.4 km SW (direction 227$\degr$) from our observatory. The instrument covers wavelengths from 370 to 840 nm, with a mean instrumental [spectral resolution of about R$\sim$50000]{}. The products of a dedicated data reduction pipeline [@Fuhrmann11] were used for this study. These fully reduced data sets were provided by the Bochum team. The spectra from 2008 to 2010 were taken at the 1.5 m Hexapod telescope. Since 2011, spectra were taken at the 0.83 m IRIS telescope. The observations cover typically 16 days on average monthly. The observation days were selected not only by means of best weather conditions, but mainly by the science cases of the observers such as deriving orbits and radial velocities of binary stars. These data are without spectrophotometric calibration. The data selection was obtained by an automated statistical quality check during the analysis itself, which is described in the following section in detail. There are some differences and inhomogeneities of the data obtained with the two telescopes, such as existing header keywords and the length of the obtained spectrum. For a small fraction of less than 100 files a unique reconstruction of observing time, airmass, etc. was not possible, and thus they were rejected.
In total, data from 996 nights out of the period from 2008 to 2014 exist. The observed stars are distributed over the whole sky, with a slight concentration towards the Galactic plane, covering the sky well until a zenith angle of $ z \approx 60\degr$. Only a handful of stars exceed this limit. The sample distribution is shown in Fig. \[FigVibStab\].
The precipitable water vapour (PWV) data for the Cerro Paranal sample were taken from @Kausch15. This data set contains several months of LHATPRO microwave radiometry data campaign covering various seasons in 2012 and about 3 years of data taken in the infrared by the VLT instrument X-shooter.
![An example of whole spectrum taken by BESO; the blue and red fitting region are shown with blue and red colour, respectively.[]{data-label="spectrum_example"}](fig_2.png){width="87mm"}
The methods and the analysis
============================
Analysis of the BESO sample with [molecfit]{}
---------------------------------------------
An example of a BESO spectrum is shown in Fig. \[spectrum\_example\]. The spectra contain telluric absorption bands of the O$_{3}$, O$_{2}$ and H$_{2}$O molecules [@Smette15]. For each spectra we separately fitted H$_{2}$O in the two narrow ranges, marked in the figure. [[Molecfit]{}]{} calculates the PWV values normalized to zenith. The fits for the two ranges were obtained independently in order to check for the quality of the pipeline reduced data. As these ranges originate from the same populations of the roto-vibrational H$_{2}$O (211$\leftarrow$000) band and thus from the same volume along the line of sight, they should give the same results. The so called [*blue range*]{} covered the wavelength from 816.01 to 820.60 nm, while the [*red range*]{} was defined from 824.43 to 828.67 and from 829.02 to 830.84 nm. In the red range we had to skip the region from 828.67 to 829.02nm, due to a strong instrumental artifact which most likely originates from an overlap of orders in the echelle spectrograph. The fitting of the sample spectrum for the two ranges is given in Fig. \[spectrum\].
For a detailed description of the methods and how to use [molecfit]{} see [@Smette15].
During the manual investigation of a small subset of data, to optimize the initial conditions for the automated fitting routine, some spectra failed to be fit automatically. This was due to bad background calibration, bad matching of orders of the echelle spectrograph, or some instrument and data reduction pipeline artifacts. Since for the large data sample a manual selection of the individual fits was not possible, we used a test data set of three months to develop the selection criteria for the automatic rejection of spectra where the fits failed. In a tradeoff between completeness and reliability we decided towards the conservative approach of higher reliability and thus probably missing some good data points. After the rejection, the PWV from the blue range was adopted for the final result (see below), while the red range was only used as a control set. We applied three criteria to select the results:
1. We selected five narrow windows in both ranges (having in total 108 and 106 spectral pixels, respectively) on top of the strongest spectral features. They were selected to compare the obtained model and the data[, without having to include a large amount of pixels where no absorption is applicable]{}. The latter pixels were only used for [molecfit]{} to derive the stellar continuum. The model from fitting the whole range, was compared with the data only in those windows, building a $\chi^{2}$ value. We only used those PWV results where the degree of freedom of the fit (5 in our case) corresponds to a statistical confidence level of $\alpha = 0.05$.
2. We used the [two-sample]{} Kolmogorov-Smirnov test to check if the observed and fitted spectrum are identical. We adopted only those results with a confidence level of $\alpha = 0.05$. This test is known to be very stable for undefined non-normal distribution functions [@KS].
3. Finally, the differences were calculated as $$\Delta_{\rm PWV} = \left|{\left({PWV_{\rm red}-PWV_{\rm blue}}\right)/ \left({{\genfrac{}{}{}{2}{1}{2}} \left({PWV_{\rm red}+PWV_{\rm blue}}\right)}\right)\,}\right|$$ and are shown in Fig. \[error\_hist\]. The main distribution until the 2$\,\sigma$ level is representing a normal distribution with an [*rms*]{} of 7.95%. Thus, this value is adopted as typical individual error of the measurements. At larger $\Delta_{\rm PWV}$ systematic errors distort the distribution function. Only spectra for which the independently derived PWV values from the two ranges do not differ more than 30% ($\equiv 3.8\,\sigma$) were selected for the sample.
After applying these three criteria, we obtained PWV estimates based on 5879 spectra, from 918 nights for the final sample. That corresponds to about 63% of all archive files. As we still cover 92% out of all the 996 nights, no bias towards better PWV by the selection process has been introduced.
While at normal conditions the values from them are expected to coincide very well, in extreme conditions (extremely dry and thus weak absorption or extremely wet and thus saturation effects), the values could have larger differences. In the red fitting range molecular absorptions are weaker. Therefore, that part of [the]{} spectrum is more difficult to fit in case of dry conditions. Also, instrumental defects, most likely remnants of fringing, causing periodic wave structures in the continuum of the star, are stronger in the [red spectral]{} range. Manual control samples also showed that these can be a source for some of the larger differences. After the selection mentioned above based on the control set and the statistical reliability, we adopted for the further analysis only the results from the blue range, since the H$_2$O absorption is stronger there. Furthermore, the range covers a shorter wavelength region in total and thus is less affected from inter–order errors in the data reduction of the echelle spectrograph. It is assumed that the fits are more accurate there. Finally, for the spectra that were not rejected, only an extremely weak systematic effect between the results of the two ranges was found (Fig. \[compare3\]).
Homogeneity of the Sky
----------------------
An important issue for a site is the homogeneity on the sky. A large number of observations are required for such an analysis in order to obtain statistically strong results. In our case we could combine data from the full six year period [for separating seasonal and short-term variations from large timescale trends.]{} We tested the spatial homogeneity of the PWV above the site by dividing the sky in 30$^{\circ}$ azimuth and 10$^{\circ}$ zenith distance bins, resulting in a polar coordinate grid of 9$\times$12 fields. As [molecfit]{} already recalculates the column density with respect to the zenith solution of the profile, the results are cleaned already from effects due to the airmass. For each bin we calculated a 1.5$\,\sigma$ clipped mean. [The bins where the zenith distance is less than]{} 60$^{\circ}$ have an average PWV of 3-4mm, and the distribution is fairly homogeneous and independent of the direction (Fig. \[mean\]). Especially no East-West asymmetry can be identified due to the $\sim$35 km distant Pacific coast and/or the air upstreaming the mountains. Bins with higher zenith distances ($>$60$^{\circ}$) are statistically not reliable, as they contain only 1 to 10 measurements over the whole period of 6 years and are not used for analysis. The same is true for the two fields where zenith distance is between $50^{\circ}$ and $60^{\circ}$ at an azimuth is between 0$^{\circ}$ and 60$^{\circ}$, having less than 25 measurements.
These findings are consistent with the results obtained by [@Querel14] for Paranal. Using all-sky scans[, obtained every 6 hours over a period of 21 months]{} (down to $27.5^{\circ}$ elevation), they report that the PWV over Paranal is remarkably uniform with a median variation of 0.32 mm (peak to valley) or 0.07 mm (rms). The homogeneity is a function of the absolute PWV but the relative variation is fairly constant at 10% (peak to valley) and 3% (rms). Such variations will not have a significant impact on the analysis of astronomical data and they conclude that observations are representative for the whole sky under most conditions.
Comparison and Verification with Cerro Paranal
----------------------------------------------
We compare BESO PWV results with the ones from Cerro Paranal [@Kausch15] to verify our results (see Fig. \[sample\_compare\]). They used X-shooter data in the same way, by fitting the water vapour bands with [molecfit]{}. While the typical seasonal variations follow nicely the same trends at both sites, there is no clear signature of the [moderate-to-strong]{} [El Ni[$\tilde {\rm n}$]{}o]{} [@elnino] during the Chilean summer 2009/2010. Interestingly enough, @EWASS15 reported an unusually large number of extremely dry ($\le 0.50$gr/m$^3$ H$_2$O) and very dry ($\le 0.65$gr/m$^3$ H$_2$O) events for the years 2008, 2009 and 2010 by measuring the absolute humidity at 30-m height above Paranal. These events were 2–3 times more frequent than in other years. Thus, it seems that the 2009/2010 [El Ni[$\tilde {\rm n}$]{}o]{} event did not cause more humid conditions at Cerro Paranal.
To get widely bias–free and comparable data sets, only those spectra of the X-shooter and BESO samples were selected which were taken nearly simultaneously (within 1.5 hours). This results in 3154 pairs of data. The resulting median distribution as function of the months for both sites is shown in Fig. \[evol\][, where the uncertainty varies between 4% and 16%]{}. The difference of the two sites is especially obvious during the more humid summer. For nights with coincident observations the median PWV for Cerro Murphy (BESO) is 2.43 mm, while it is 2.96 mm for Cerro Paranal (X-shooter). This [difference of 22%]{} is significantly larger than expected by the difference of the altitude (2817 and 2635 m for Cerro Murphy and Cerro Paranal, respectively), [using the scale-height of 1800 m found by balloon-born radiosondes from Antofagasta for the altitudes above 2km by @Otarola2011. As they showed, the scale height in the Atacama desert can be lower under dry conditions. However, a random sampled spot check of the [molecfit]{} results from our sample with 20 probes around the median PWV revealed that it was well in agreement at these average conditions]{}. Using [molecfit]{}, including ground–based meteorological data [@Smette15], with data from Cerro Paranal leads to a theoretical difference of only about 8% due to the altitude. [The PWV as function of site altitude given by @otarola10 leads to 8.2%. This was obtained by comparing the 5 tested TMT candidate sites, covering an altitude range from 2290m to 4480m.]{} To supplement, we compare the BESO PWV with the values derived from GDAS as well as with those observed within 1.5 hours with X-shooter and with the verification data set of the LHATPRO radiometer [@Kausch14; @Kausch15]. In Fig. \[compall\] we analyzed the linear correlation ($y=Ax$) of the BESO data with those from Cerro Paranal. The resulting coefficients and R$^{2}$ values are given in Table \[table:1\]. The BESO and X-shooter data show nearly identical slope (0.85) as BESO and the verification data set from LHATPRO radiometer (0.84); with $R^2$ of 0.89 and 0.93, respectively. Thus, one can conclude that the LHATPRO radiometer verification as well can be used for BESO.
The $R^2$ of BESO with GDAS data is 0.78 (Fig. \[compall\]). As we can see, using GDAS data, which works fine for the prediction far from the coast, on the other side of the Andes mountains, at the Auger observatory [@GDAS_AUGER], does not work in this zone of the Atacama desert. GDAS predictions systematically overestimate the PWV by 52%. Similar results were found for Cerro Paranal, but using already locally improved GDAS profiles merged with ground station meteorological data [@Kausch15].
Data $A$ R$^2$
------------ --------------------- -- -------
X-shooter 0.851$\,\,\pm$0.003 0.89
Radiometer 0.838$\,\,\pm$0.009 0.93
GDAS 0.561$\,\,\pm$0.003 0.78
: \[prva\] Comparison of BESO PWV with the ones from Cerro Paranal and GDAS data for Cerro Murphy. Linear fitting $y\,=\,A\,x$ was performed between pairs of PWV data.[]{data-label="table:1"}
Conclusions and Outlook
=======================
PWV over Cerro Murphy
---------------------
As already shown for Cerro Paranal in @Kausch15, spectral feature fitting by [molecfit]{} can successfully be used to investigate the PWV content in the Earth’s atmosphere. The resulting PWV from Cerro Murphy, southwest of Cerro Armazones, are systematically lower than those from Cerro Paranal. The effect is stronger than expected from the altitude difference of 182 m[, assuming the water vapor vertical distribution at the median conditions]{}. The PWV is homogeneous and does not show any major sky direction asymmetries or trends within the six years of the study, although it covers the El Ni[ñ]{}o event 2009/10. Thus, it is possible to parameterize the sky only by means of zenith distance, without including azimuth dependencies. We show that a very small telescope facility is able to provide an atmospheric monitoring capability with results fully comparable to those from the large telescopes. Moreover, it is clear that the interpolated data from GDAS are too inaccurate. Recently, two independent local refinement calculations of these coarse grid weather models were presented by @Lascaux15 and @Valparaiso15. Certainly, a comparison study of these methods with our results would be a further step. For this or other studies we could provide our resulting atmosphere profiles electronically.
Outlook to atmosphere studies for CTA
-------------------------------------
Up to now, data reduction uses only simple integrated profiles through the whole atmosphere as function of the airmass, or if very high accuracy is required only uses nearby calibrators (e.g. differential photometry or iso-airmass telluric standards). The latter is not always possible or requires a large calibration plan. The first one suffers from missing information on the azimuth dependence, and is not sufficient for describing light propagation of sources located in the atmosphere as it is the case for $\gamma$-ray astronomy if using [Č]{}erenkov radiation.
If we use individual species behaving differently due to distributions other than the classical exponential scale height, real height profiles are especially important. The distribution of water vapour is strongly concentrated to the lower 10 km and thus do not follow these simple dependencies as normally used for wide band filter extinction curves.
Molecules in some distinct layers have to be treated specifically. An important example for this is ozone. Nearly all ozone absorption occurs in the stratosphere. It has to be removed from the extinction curve derived by photometry [@photometer; @FRAM] for the CTA data reduction. For measuring the variation of the tropospheric layers, our methods of ground-based spectroscopy presented here will be not sensitive enough. With the standard profiles of our sky model [@Noll12] we calculated the average fraction of the tropospheric ozone from the ground at 2km to 10km. As it only contributes about 6% to the total ozone absorption throughout the whole atmosphere, the error introduced by assuming a constant low fraction of tropospheric ozone on the total calculation is small. The strong high atmosphere contribution has to be derived separately from other sources of extinction and subtracted from the overall absorption. The Chappuis bands at $500 < \lambda < 700$ nm are absorbing and scattering significantly up to 4% at zenith. That is already essential, as the flux calibration specification for CTA is 5-7%. However, they only show very shallow and highly blended absorption structures. Our spectra do not cover the blue to the ultraviolet light sufficiently to also model the ozone variation in the Huggins bands ($\lambda\,<\,400$ nm), which are stronger and show more characteristic features. Thus, they would be more appropriate for measuring the total ozone column. Therefore, a proposed CTA installation should extend to the ultraviolet until 320 nm to fit the profile from the Huggins bands. The contribution of the Chappuis bands can be calculated by our radiative transfer codes thereafter [@Kimeswenger15 Fig. 2].
We would like to thank the referee for his comments and the very detailed suggestions. Furthermore we thank Angel Ot[á]{}rola for providing us his full presentation of the EWASS15 conference. We thank Christian Westhues and Thomas Dembsky from Ruhr Universit[ä]{}t Bochum for their help in accessing the BESO archive. M.L. was funded by the Comit[é]{} Mixto ESO-Gobierno de Chile in the project [*Investigaci[ó]{}n Atmosf[é]{}rica con Instrumentaci[ó]{}n Astron[ó]{}mica*]{}. [Molecfit]{} was designed in the framework of the Austrian ESO In-Kind project funded by BM:wf under contracts BMWF-10.490/0009-II/10/2009 and BMWF-10.490/0008-II/3/2011. S.N and S.U. are funded by the Austrian Science Fund (FWF): P26130 and W.K. is supported by the project IS538003 (Hochschulraumstrukturmittel) provided by the Austrian Ministry for Research, Investigation and Economy (BM:wfw).
[^1]: Based on archival observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile and of the Cerro Armazones Observatory facilities of the Ruhr Universit[ä]{}t Bochum.
[^2]: Radiometer Physics GmbH, <http://www.radiometer-physics.de/>
[^3]: <http://www.ready.noaa.gov/gdas1.php>
[^4]: <http://rtweb.aer.com/lblrtm_frame.html>
[^5]: <http://www.cfa.harvard.edu/hitran/>
[^6]: <http://archive.eso.org/asm/ambient-server>
[^7]: <http://www.atm.ox.ac.uk/RFM/atm/>
|
---
abstract: 'A water Cherenkov detector array, LHAASO-WCDA, is proposed to be built at Shangri-la, Yunnan Province, China. As one of the major components of the LHAASO project, the main purpose of it is to survey the northern sky for gamma ray sources in the energy range of 100 GeV¨C 30 TeV. In order to design the water Cherenkov array efficiently to economize the budget, a Monte Carlo simulation is proceeded. With the help of the simulation, cost performance of different configurations of the array is obtained and compared with each other, serving as a guide for the more detailed design of the experiment in the next step.'
address: |
$^1$ School of Physical Science and Technology, Southwest Jiaotong University, Chengdu, 610031, China\
$^2$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China\
$^3$ Normal University of Hebei, Shijiazhuang, 050016, China\
author:
- |
LI Hui-Cai $^{1;2}$CHEN Ming-Jun $^{2}$ JIA Huan-Yu $^{1}$ GAO Bo $^{2}$\
WU Han-Rong $^{2}$ YAO Zhi-Guo $^{2}$ YUO Xiao-Hao $^{2;3}$ ZHOU Bin $^{2}$ ZHU Feng-Rong $^{1}$
title: Study on optimization of water Cherenkov detector array of LHAASO project for surveying VHE gamma rays sources
---
LHAASO-WCDA, gamma rays source, cost performance, optimization
96.50.sd, 07.85.-m
[2]{}
Introduction
============
The detection of Very-High-Energy (VHE, $>100$ GeV) cosmic gamma rays has been campaigned vigorously since the first detection of TeV gamma radiations from the Crab Nebula by the Whipple experiment in 1989 [@lab1]. Two major detection approaches exist in this research field: Imaging Air Cherenkov Telescopes (IACTs) [@lab2] and ground particle detector arrays [@lab3]. The formers win in angular resolution ($<0.1^\circ$) and background rejection power so that they possess a better sensitivity in morphology observation and spectrum measurement. However, because of low duty-circle ($\sim$10%) and narrow field of view ($<5^\circ$ typically), their capability in full sky survey and long-term monitoring of sources is very limited. This aspect is fortunately complemented by ground particle arrays, which can work all the time (duty cycle $>95\%$) and monitor a large piece of the sky ($>$2$\pi$/3) simultaneously, as shown by precedent experiments such as Tibet AS$\gamma$, Milagro, and ARGO-YBJ. One of the techniques used in this kind of approach, the water Cherenkov, has the unique advantage of much better background rejection power than other options like plastic scintillator and RPC, which is well demonstrated by simulations and has verified by Milagro experiment.
Targeting gamma astronomy in energy band from 100 GeV to 30 TeV, the water Cherenkov detector array (WCDA) [@lab4] of the Large High Altitude Air Shower Observatory (LHAASO) [@lab5], covering an area $90,000\rm\;m^2$, has been proposed to be built at Shangri-la( 4300 m a.s.l.), Yunnan, China. Much upon the experience of Milagro experiment, the current official configuration of WCDA is 5 m $\times$ 5 m cell-sized, and incorporated a single photomultiplier at a water depth of 4 m for each cell [@lab4]. The simulation of the array in this configuration shows a very good performance in sensitivity to gamma ray sources, particularly at the energy band around 5 TeV. But it is not necessarily the best configuration, for instance for gamma rays at low energies, and especially when the cost factor is taken into account. This paper is just to carry out this study, to see what the best configuration of the array is in the sense of cost performance, via tuning the cell size, the water depth and the number of PMTs.
In section 2, the scheme of the optimization is introduced, and next the optimization procedures and results based on Monte Carlo simulations are presented, seen in section 3.
Optimization scheme
===================
To simplify the optimization procedure, some discontinuous values of the detector configuration parameters regarding the cell size, the water depth and the number of PMTs in each cell are proposed to be taken as the starting point of the simulation. Figure 1 shows the sketch drawing of 4 neighboring cell detectors in two view angles, where parameters $L$, $H$, and $N$ are subjected to be tuned. Three cell sizes such as 4 m $\times$ 4 m, 5 m $\times$ 5 m and 6 m $\times$ 6 m are selected; three effective water depths such as 3 m, 4 m and 5 m are chosen; five groups of PMT quantities in each cell such as 1, 2, 3, 4 centered and 4 scattered are adopted - as shown in figure 2, where the verbal adjectives of the last two groups with 4 PMTs represent where the four PMTs are located, near the center or much separated in the cell. Each iteration of these 3 kinds of elements can be combined to form a particular configuration. Altogether there are $3 \times 3 \times 5 = 45$ configurations to be proceeded, whose cost performance are due evaluated and compared.
{width="0.60\linewidth"} {width="0.30\linewidth"}
{width="0.95\linewidth"}
Simplified simulations
----------------------
A full simulation should be the best approach for calculating the performance of each configuration. But in practice it would be exhausted due to the fact that the simulation procedure is quite time-consuming, especially the process of production and tracing of the Cherenkov lights in the water. As to the array of current design, which is just one example of above 45 configurations, 3 months were taken with a PC farm of 100 CPU cores. In order to accomplish the optimization for all above configurations, years of computing time are expected. It is of course not a practical solution. To get over this obstacle, a simple and efficient optimization procedure is adopted and proceeded, as follows.
First of all, in order to lessen the burden of iterations, in the simulation, for each cell, overall 9 PMTs are put into the detector configuration at the same time, see figure 3. These 9 PMTs can be easily classified afterwards in the off line into the 5 groups required for the optimization. This kind of overall configuration would not change the simulation results much, as the area of 9 PMTs is very limited comparing the whole cell bottom, and as they react more-or-less like the surrounding curtains or plastics in black of each cell for the purpose of avoiding reflections of Cherenkov lights. With help of this treatment, the total number of iterations can be reduced to be 3$\times$3 = 9.
{width="0.80\linewidth"}
Special scenarios, which mean simplified detector setups and or simplified sets of primary shower parameters like energy, direction and projection area, are adopted in the study for sake of speeding up the simulation. What we concern is the performance comparison between different detector configurations which can be told by simulations in some special scenarios; the absolute performance values for real cases are important too but they are out of the scope of this study. After that, two simulation approaches dealing with two different special scenarios are carried out.
- Single cell approach\
As the simplest scenario, just a single cell detector is used for the simulation. Secondary particles of hundreds of primary gamma showers are thrown one by one onto the cell detector, and then their interaction and transport processes in the water and in the PMT are simulated. Counting the number of particles that fired at least 1 PMT, compared with the number that thrown onto the cell, the efficiency $\eta$ of the cell detector is obtained. The efficiency for water Cherenkov is usually less than 30$\%$, but ample photons in the shower secondary particles compensate the loss of efficiency when comparing with other type of arrays such as plastic scintillator or RPCs that detect only charged particles. For the full coverage detector, if major part of the shower particles fall into the array, the efficiency value would not differentiate much from above $\eta$ obtained with the simplest case. The performance parameter in this case is then defined as $\sqrt\eta$, as the sensitivity is usually inverse proportional to the square root of the number of hits, especially for showers with scarcely distributed hits at the low energies. Just because of this reason, this approach is principally for low energy gamma ray sources.
- Array approach\
As a more complex scenario, the configuration of an array of 150 m $\times$ 150 m is adopted, but simulated with simplified primary particle sources ¨C gammas and protons with fixed energies, vertically incident, and bombarding at the center of the array. Analyses on angular resolution and proton-gamma discrimination are performed, so that somehow realistic performance results for each configuration are obtained. In the circumstance we are interested, usually gamma showers generate 2-3 times more particles at the ground than that of proton with the same energy, so in the proton-gamma discrimination, the proton energy is deliberately chosen to double the energy of its gamma partner, for instance 0.5 TeV gamma versus 1 TeV proton, and 1 TeV gamma versus 2 TeV proton, and so on, see figure 4 for the details.
{width="0.90\linewidth"}
In this study, air shower cascade is simulated with Corsika v6.990 [@lab6], and QGSJETII model [@lab7] is used for high energy hadronic interactions. For avoiding losing the shower information, the kinetic energy cut for secondary particles in Corsika is set to much lower values than that of the Cherenkov production threshold in the water, i.e., 50 MeV for hadrons and muons, 0.3 MeV for pions, photons and electrons. The detector is supposed to be at the altitude of 4300 m a.s.l., and the geometrical setup and the particle propagation are simulated with a code developed on the framework of Geant4 v 9.1.p01 [@lab8].
Cost performance
----------------
The cost performance rather than the sole performance of the detector is adopted for the comparison of different configurations. Different configurations with a same array dimension may bring different cost. For example, thicker water depth needs higher pool bank in the construction, larger water volume for filling up, and stronger water purification and recirculation ability in the operating, all of which requires more money; more PMTs in a cell means more PMTs to be ordered and tested, and more electronic channels and cables to be manufactured ¨C that cost more money too. Assuming the detector array is fixed in size of 150 m $\times$ 150 m, the cost of every configuration are carefully evaluated, based on the engineering design reports. The performance of the detector $P$, here defined as the inverse of the flux sensitivity, divided by the cost $C$, the cost performance $P_{\rm C}$ is then calculated, that is $$\label{four} P_{\rm C}=\frac{\alpha \cdot P}{C}=\frac{\alpha \cdot Q/\theta}{C_{\rm base}+C_{\rm depth}+C_{\rm PMT}}.$$ Where $C_{\rm base}$ is the cost of fundamental pool construction including the pond basement and the roof, which actually is a constant in this study, $C_{\rm depth}$ is the cost dependent on the water depth, $C_{\rm PMT}$ is the cost dependent on the total number of PMTs, and $\alpha$ is a constant normalization parameter which sets $P_{\rm C} = 1 $ for the case of the pool in the current official configuration, that is 5 m $\times$ 5 m in cell size, 4 m in water depth, and 1 PMT in each cell. Regardless of the number of PMTs, which is taken account already in $C_{\rm PMT}$, the cell size alone doesn¡¯t affect much on the cost so its contribution is trivial and ignored. Moreover, comparing with $C_{\rm base}$ and $C_{\rm PMT}$, the influence of the water depth to the cost $C_{\rm depth}$ is small. For approach $i$ in the above, the performance is set to $$\label{two} P=\sqrt\eta.$$ Where $\eta$ is the efficiency above-mentioned. For approach $ii$, the performance is set to $$\label{three} P=Q/\theta.$$ Where $Q$ is the quality factor for proton-gamma discrimination, and $\theta$ is the angular resolution to primary gammas. Those 2 factors are all dependent on the PMT threshold applied in the off line analysis, the $P$ here is the maximum value while ranging the PMT threshold.
Simulation results
===================
Single cell approach
--------------------
A detector cell is constructed in the framework of GEANT4. 8-inch PMTs of type Hamamatsu R5912 are used, whose geometrical description and boundary effect code is taken from GenericLAND package [@lab9]. The water absorption length to lights, whose dependence on wavelengths is parameterized according to the curve shape of the measurement of pure water, is scaled to 27 m at 400 nm.
Around 10,000 showers of primary gamma at energy 1 TeV are generated in Corsika. The total number of their secondary particles arriving at the ground amounts to $10^6$, most of which are photons. As to different primary energies, the energy distributions of secondary particles are eventually very close, though actually the energy of primary particle is not important at all in this study. The secondary particles are thrown and tracked in the cell detector one by one, with the particle that fires at least 1 PMT being counted, and finally the efficiency for each configuration is obtained, shown in figure 5. From the plot it is seen that the efficiency is dropping while increasing the cell size, raising the water depth, or reducing the number of PMTs. For the first two trends, it can be explained as the opening angle of PMTs to tracks turning smaller, so that the chance of Cherenkov lights produced along the track hitting the PMT becomes less. The energies of air shower secondary particles are rather low, so usually the tracks of their own or of small showers they initiate appear only at the top part of the water and have scrambled directions. The configuration with water depth lower than 3 m is not simulated, but other studies show that there is a turning point in the efficiency curve around 3 m.
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The cost of the 150 m $\times$ 150 m array composed of single cells in each of 45 configurations is estimated in the way mentioned in previous section. With equation 1 and 2, the cost performance for each configuration is then obtained, shown in figure 6. It is obvious that the group with single PMT in a cell has better cost performance than that of other groups. As to the water depth, due to the cost changes in the same way as the efficiency, the detector at lower water depth has a better cost performance. It means a shallow water depth is desired for detecting low energy gamma rays. The cell size gains some 20% too when the cell size is 6 m $\times$ 6 m, because of the reduction of PMT cost.
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Array approach
--------------
In this approach, primary gammas with 3 energies, 0.5, 1 and 2 TeV, are simulated, and that of primary protons, as the background, are set to 1, 2 and 4 TeV correspondingly. For brevity, only the case of 1 TeV gamma and 2 TeV proton is introduced here. As mentioned above, the showers are vertically incident, with core projected to the center of the array. Around 20,000 gamma showers and more proton showers are simulated.
To further lessen the $P_{\rm C}$ express, the configuration with 4 PMTs scattered is abandoned. Both simulations on E/M particles and muons show no obvious differences.
### Gamma proton discrimination
Water Cherenkov array has an extraordinary property in Gamma proton discrimination. Muons or sub-cores in hadronic showers can produce unevenly lateral distribution of hit intensity on PMTs. When the brightest PMT outside a certain range (e.g., 45 m in radius) of the reconstructed core is chosen, using whose inverse signal amplitude $(1/cxPE)$ to weight the number of fired PMTs $(nPMT)$, the resulting distribution of compactness (= $nPMT/cxPE$) between gamma and proton turns quite different, especially at high energies. Tuning the threshold of compactness, the best quality factor ($Q_{\rm max}$), defined as $$\label{three}
Q_{\rm max}=\frac{\xi_{\rm s}}{\sqrt{\xi_{\rm b}}},$$ is found, where $\xi_{\rm s}$ and $\xi_{\rm b}$ are retained fractions of gamma and proton respectively. Figure 7 shows the $Q_{\rm max}$ for these 36 configurations. The prominent phenomenon is that $Q_{\rm max}$ turns bigger when water depth is higher. The reason behind is as follows: When water depths raises, the cxPE of proton formed most by muon signals suffers little from the water depth hence keeps constant, but $nPMT$ turns smaller due to the dropping of efficiency as shown in previous single cell approach; Gamma shower has very few penetrating muon particles, whose $cxPE$ produced mainly by soft components changes in the way very similar as $nPMT$, so the ratio $nPMT/cxPE$ varies very little.
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### Angular resolution
Only gamma showers are meaningful for this analysis. With the hit information, using the official program developed for water Cherenkov arrays, the shower direction is reconstructed. Selecting events with the compactness threshold used by the $Q$ optimization, from the distribution of the opening angle between the reconstructed and the real directions, the angular resolution is obtained. The Rayleigh distribution is assumed so that the Gaussian-like standard deviation [@lab10] is assigned. Plots in figure 8 show values $1/\theta$ for different configurations. Similar trend as efficiency plots (figure 5) is observed, because the angular resolution is heavily dependent on the number of hits, which is actually determined by the efficiency.
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### Cost performance
Due to very weak dependence of angular resolution on the compactness threshold around the optimized value, we have not bothered to optimize the performance with two factors in equation 3 in company. Tests to two cases of 36 configurations show also no difference at all. The performance and cost performance is finally calculated for the 36 configurations with equation 1 and 3, whose results is shown in figure 9, 10. Controlled by the $Q_{\rm max}$, both the performance and cost performance turn better at deeper water depth, totally contrasting to the trend in approach $i$. If taking the statistical errors into account, configurations with less PMTs in a cell, e.g., 1 or 2, have better cost performance. The cell size affects the cost performance too, but the influence for cases with 1 or 2 PMTs in a cell is marginal.
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Discussions and conclusion
==========================
Besides only simple scenarios are investigated, there are still some other realistic situations not being considered yet. For example, for the water quality, the absorption length 27 m at 400 nm might be too good to be maintained. If an absorption length 15 m is used, the middle plot in figure 10 would look like the one in figure 11 - the rising trend going with the water depth is depressed. Another issue is the accidentally coincident muon, which is not taken into account in the simulation. For configurations with several PMTs in a cell, the muon may fire all these PMTs altogether, troubling the reconstruction and even the trigger.
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Bigger cell size is more preferred by the simulation, no matter which approach, $i$ or $ii$. But the difference between 5 m $\times$ 5 m and 6 m $\times$ 6 m is trivial, with an effect $<10\%$. In practice, a cell size in between can be chosen out of engineering considerations.
Contrasting results on water depth optimization are found between the two approaches. As what has mentioned in section 2, these two approaches actually focus on gamma rays at different energy ranges. At low energies, gamma proton discrimination with compactness analysis does not work very well, so the angular resolution, i.e., efficiency, dominates in the performance. This point is proved by the simulation of lower primary energies in approach $ii$, where gamma is 0.5 TeV and proton is 1 TeV, and the result shows that the 4 m rather than 5 m water depth is the best. Considering that this kind of water Cherenkov detector array is not very sensitive at low energies such as below 1 TeV [@lab4], it is more proper to optimize the cost performance towards higher energies - in another word, deeper water depth is more preferred. But at the same time, the water quality issue should not be ignored. A water depth in between 4 m and 5 m shall be appropriate.
No critical difference on PMT quantity selection is found in two approaches. One PMT in a cell shall be the best option.
In summary, based on this simulation work, a configuration of cell size in between 5 m $\times$ 5 m and 6 m $\times$ 6 m, water depth in between 4 m and 5 m, and 1 PMT in a cell is the best in cost performance for the water Cherenkov detector array of LHAASO.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to express their gratitude to the Milagro Collaboration for beneficial discussions on their experiences. This work is partly supported by NSFC (11175147) and the Knowledge Innovation Fund of IHEP, Beijing.\
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---
abstract: 'High-dispersion spectra of 333 stars in the open cluster NGC 6819, obtained using the HYDRA spectrograph on the WIYN 3.5m telescope, have been analyzed to determine the abundances of iron and other metals from lines in the 400 Å region surrounding the Li 6708 Å line. Our spectra, with signal-to-noise per pixel (SNR) ranging from 60 to 300, span the luminosity range from the tip of the red giant branch to a point two magnitudes below the top of the cluster turnoff. We derive radial and rotational velocities for all stars, as well as \[Fe/H\] based on 17 iron lines, \[Ca/H\], \[Si/H\], and \[Ni/H\] in the 247 most probable, single members of the cluster. Input $T_{eff}$ estimates for model atmosphere analysis are provided by $(B-V)$ colors merged from several sources, with individual reddening corrections applied to each star relative to a cluster mean of $E(B-V)$ = 0.16. Extensive use is made of ROBOSPECT, an automatic equivalent width measurement program; its effectiveness on large spectroscopic samples is discussed. From the sample of likely single members, \[Fe/H\] = $-0.03 \pm 0.06$, where the error describes the median absolute deviation about the sample median value, leading to internal precision for the cluster below 0.01 dex. The final uncertainty in the cluster abundance is therefore dominated by external systematics due to the temperature scale, surface gravity, and microturbulent velocity, leading to \[Fe/H\] = $-0.02 \pm 0.02$ for a sub-sample restricted to main sequence and turnoff stars. This result is consistent with our recent intermediate-band photometric determination of a slightly subsolar abundance for this cluster. \[Ca/Fe\], \[Si/Fe\], and \[Ni/Fe\] are determined to be solar within the uncertainties. NGC 6819 has an abundance distribution typical of solar metallicity thin disk stars in the solar neighborhood.'
author:
- 'Donald B. Lee-Brown and Barbara J. Anthony-Twarog'
- 'Constantine P. Deliyannis'
- Evan Rich
- 'Bruce A. Twarog'
title: |
Spectroscopic Abundances in\
the Open Cluster, NGC 6819 [^1]
---
Introduction
============
NGC 6819 is an old (2.3 Gyr) open cluster whose fundamental properties have garnered increasing attention in recent years. As detailed in @AT14, (hereinafter Paper I), its age places it in a sparsely populated range of open cluster ages, making it an invaluable testbed for stellar models of stars just above the Sun’s mass. It was chosen as a key cluster in our program to map the evolution of Li among stars of varying mass as they evolve from the main sequence to the tip of the giant branch and beyond. Delineation of such a map requires reliable estimates of stellar temperature, luminosity, and metallicity, especially iron, and, indirectly, the cluster reddening, distance modulus and age.
Data on individual cluster stars and the surrounding field have expanded due to the cluster’s location in the [*Kepler*]{} field, bringing the added potential for asteroseismic insight into the structure of individual stars in a variety of evolutionary states [@GI10]. Reliable radial velocities are available for a large sample of stars extending well below the cluster turnoff [@H09] (hereinafter H09) and improved proper-motion memberships by @PL13 (hereinafter PL) have supplanted the older data of @SA72. Comprehensive broad-band [@RV98; @KA01; @YA13] and intermediate-band (Paper I) photometric surveys allow precise determination of each star’s position in the color-magnitude diagram (CMD), while definitively demonstrating (PL, Paper I) that the cluster suffers from variable foreground reddening with a range of $\Delta$$E(B-V)$ = 0.06 mag, not surprising given the cluster’s galactic coordinates $(l,b = 74\arcdeg, +8\arcdeg$) and a distance of 2.4 kpc from the Sun. Until recently, the weakest link in the discussion of the properties of the cluster and its stars has been the cluster metallicity. As detailed in Paper I, metallicity estimation from cluster members has been plagued by small number statistics often coupled with incorrect assumptions of uniform and/or anomalously high reddening, leading to a consensus view that NGC 6819 was metal-rich, with \[Fe/H\] $\sim$ +0.1. Precise intermediate-band photometry on the extended Stromgren system (Paper I) strongly contradicts this claim, producing \[Fe/H\]$ = -0.06 \pm 0.04$ from 278 single, unevolved F stars.
Although a fuller discussion of the Lithium abundances for our large sample in NGC 6819 is in preparation [@d15], one preliminary result - the identification of a Li-rich giant star - was reported in @AT13. We employ a combination of spectral synthesis and curve-of-growth analysis to convert measured equivalent widths for the Li line at 6707.8 Å to Lithium abundances, A(Li). Our curve-of-growth analysis method depends on subtracting an estimated contribution by an iron line at 6707.4 Å from the equivalent width, for which accurate temperature and iron abundance information is required. Given the criticality of a precise estimate of the stellar Fe abundance for derivation of Li from spectroscopy, especially among the giants, it was decided that all measurable lines within the wavelength range of our HYDRA high dispersion spectra of the Li 6708 Å region would be used to estimate the cluster metallicity for Fe and other elements. In addition to testing the photometric abundance, following the pattern laid out in our earlier investigations of the open clusters NGC 3680 [@AT09] and NGC 6253 [@AT10], such data could reveal the effects of stellar evolution on surface abundances through comparison of the red giants to the turnoff stars, while pinpointing the place of the cluster within the global context of galactic chemical evolution.
The layout of the paper is as follows. Sec. 2 details the spectroscopic observations and reduction of images to uniformly wavelength-calibrated and continuum-fitted spectra. Sec. 3 explains the collation of radial-velocity, photometric, proper-motion, and reddening data used to define the crucial temperatures, surface gravities, and microturbulent velocities for single-star cluster members that, together with appropriate model atmospheres, translate the spectroscopic measures to abundances. Sec. 4 supplies a detailed examination of ROBOSPECT, an automated line-measurement program used for this study which makes internally consistent and rapid abundance estimation for hundreds of stars feasible. Sec. 5 presents the abundance results for individual stars and the cluster as a whole, delineating the impact of the parametric uncertainties on our final results, and places the data within the context of both stellar and galactic evolution. Sec. 6 summarizes our conclusions.
Spectroscopic Data
==================
Spectroscopic data were obtained for target stars in NGC 6819 using the WIYN 3.5-meter telescope [^2] and HYDRA multi-object spectrograph over 13 nights from September and October 2010, June 2011 and February 2013. The positions of the target stars within the cluster $(V, B-V)$ CMD are shown in Figure 1. Six configurations were designed to position fibers on a total of 333 stars. The brightest configurations include stars near $V \sim 11$, while the fainter configurations reach to $V \sim 16.5$. Individual exposures ranged from 10 to 90 minutes, with accumulated totals of 2.5 to 4.5 hours for stars in the brightest configurations, 10 to 13 hours for the configuration with stars of intermediate brightness, and over 14 hours total for the faintest configurations. Our spectra cover a wavelength range $\sim$ 400 Å wide centered on 6650 Å, with dispersion of 0.2 Å per pixel. Examination of Thorium-Argon lamp spectra indicates that the line resolution comprises 2.5 pixels, yielding a spectral resolution over 13,000.
The data were processed using standard reduction routines in IRAF[^3]. These routines included, in order of application, bias subtraction, division by the averaged flat field, dispersion correction through interpolation of the comparison spectra, throughput correction for individual fibers using daytime sky exposures in the same configuration, and continuum normalization. After flat field division and before the dispersion correction, the long-exposure program images were cleaned of cosmic rays using “L. A. Cosmic”[^4] [@VD01]. Real-time sky subtraction was accomplished by using the dozens of fibers not assigned to stars and exposed to the sky for each integration. Composite spectra for each of the six configurations were constructed by additive combination.
The signal-to-noise ratio per pixel (SNR) may be estimated two ways: first, by direct inspection of the spectra within IRAF’s SPLOT utility, using mean values and r.m.s. scatter from a relatively line-free region or, second, by construction from output files of the ROBOSPECT software suite. We found the ROBOSPECT values to be entirely consistent with hand-measured SNR values and quote the ROBOSPECT-derived values in Table 1, computed from the relatively line-free 6680-6694 Å region. SNR estimates reflect the statistics characterizing the summed composite spectra.
Stellar Properties
==================
Radial Velocities and Proper Motions
------------------------------------
An initial spectroscopic sample of probable cluster members was taken from the valuable radial-velocity survey of NGC 6819 by H09. All stars brighter than $V \sim 16.75$ with radial-velocity membership probabilities greater than 50% were identified as candidates for the present study. Stars classed as double-lined spectroscopic binaries were eliminated; single-lined systems were retained since the existence of the companion would have minimal impact on line measurement. Stars were not eliminated based upon their position in the CMD to avoid biasing the sample against stars undergoing anomalous evolution.
Individual stellar radial velocities were derived from each summed composite spectrum utilizing the Fourier-transform, cross-correlation facility FXCOR in IRAF. In this utility, program stars are compared to stellar templates of similar effective temperature ($T_{eff}$) over the wavelength range from 6575 Å to 6790 Å, as well as a narrower region in the vicinity of H$\alpha$ alone. Typical uncertainties in the individual radial velocities were estimated at 1.15 km/sec. The FXCOR utility also provides measures of the line widths within each spectrum, from which rotational velocities may be inferred. Observations of a pair of radial-velocity standards were obtained during each run and processed using the same procedure applicable to the cluster. Comparison of the velocity zero-points using standard values from the General Catalog of Radial Velocities [@WI53] allowed transformation of the cluster data to the standard system, within the uncertainties of the measurements.
As a first check on our spectroscopic data, we can compare our measured radial velocities with those of H09 to identify discrepant stars or long-term variables. Eliminating 29 stars classed as spectroscopic binary members of NGC 6819, the remaining 304 single stars have a mean radial velocity of 2.65 $\pm$ 1.36 (s.d.) km/sec. The same sample from H09 has a mean velocity of 2.38 $\pm$ 0.99 (s.d.) km/sec. The dispersion in the residuals between these samples is 1.06 km/sec; the predicted dispersion from the quoted errors for each star is 1.19 km/sec. The residuals for only three stars (WOCS 1007, 2016 and 56018) fall more than three sigma from the mean; one of these, WOCS 2016, has unusually broad lines (presumably due to high rotational velocity) which leads to a larger than average uncertainty in the final radial-velocity estimate. By contrast, the sample of 29 spectroscopic binaries has a mean cluster velocity of 3.45 km/sec and a dispersion of 9.8 km/sec; the same stars from H09 exhibit a dispersion of 9.78 km/sec. We conclude that all single star members as classified by H09 are confirmed as such by our data. Table 1 contains the derived mean radial and estimated rotational velocities for each star in our survey.
At the start of our program, the only NGC 6819 proper-motion study available was that of @SA72, which proved inadequate for reliable identification of probable members, particularly at the fainter limit of interest. Fortunately, the comprehensive survey by PL covers the appropriate range in both area and depth, generating probabilities for all stars in our sample except one. Of 332 stars selected via radial velocity, 59 have proper-motion membership probabilities below 50%, 43 of which are in single digits. The PL values for individual stars may be found in Table 1. The different stellar categories are identified in the CMD, Fig. 1.
Effective Temperature, Surface Gravity, and Microturbulent Velocity
-------------------------------------------------------------------
In keeping with our approach to spectroscopic abundance determination for previous clusters in this program, our default scheme for determining model atmosphere input temperatures is based upon photometric color, specifically $B-V$. Four sources of broad-band $BV$ data exist. As a first step, we merged the CCD photometry presented by H09 as PHOT98 and PHOT03. From 923 stars brighter than $V$ = 16.7, excluding 5 stars with absolute residuals greater than 0.10 mag, the mean offsets, in the sense (PHOT98 - PHOT03), in $V$ and $B-V$ are $-0.005 \pm 0.022$ and $+0.003 \pm 0.022$, respectively. While the dispersion in both sets of residuals was satisfactory, further analysis revealed that a large fraction of the scatter in $V$ is the result of a nonlinear radial gradient among the residuals, reaching $\Delta V = -0.05$ mag for stars near the core of the cluster but $+0.01$ for stars in the outer regions of the frame. @MI14 have independently discovered the same effect in a comparison between the $VI$ photometry of @YA13 and PHOT03, implying that the primary source of the trend in the current comparison must lie with the PHOT03 database. Additional evidence in support of this conclusion comes from a direct comparison between the $V$ magnitudes from @YA13 and the intermediate-band data from Paper I where the residuals do not exhibit a radial dependence. More important from our perspective is the absence of a radial trend among the residuals in $B-V$ between the two samples, PHOT98 and PHOT03. Application of a small color term, $(B-V)_{03} = 0.986(B-V)_{98}+ 0.006$, reduces the scatter in the residuals to $\pm 0.021$ mag. The PHOT98 $B-V$ data were converted to the PHOT03 system and averaged for stars common to both samples.
Next, the composite H09 data were transformed to our adopted standard of @RV98. From 470 stars with $V$ brighter than 16.5 common to the two datasets, removing 10 stars with residuals in $B-V$ greater than 0.05 mag, the mean residual, in the sense (RV-H09), is +0.006 $\pm$ 0.014 mag. A weak color dependence among the residuals was found, $(B-V)_{RV} = 1.007(B-V)_{H09}$, and applied prior to the merger of the two databases, eliminating the small zero-point offset.
Finally, a transformation was derived between the $B-V$ indices of @KA01 and the previously merged $B-V$ data on the @RV98 system; comparisons of the $V$ mags of @RV98 and @KA01 have already been discussed in Paper I.
Eliminating 11 stars with absolute residuals greater than 0.1 mag and applying a small color term, $(B-V)_{RV} = 1.025(B-V)_{KA} - 0.018$, 846 stars brighter than $V = 16.5$ exhibit a mean residual in $B-V$ of $0.000 \pm 0.018$ mag.
All stars with absolute residuals greater than 0.05 mag were individually checked. If $B-V$ estimates were available from more than two sources and one source was clearly the origin of the discrepancy, that value was dropped. If only two sources of $B-V$ existed, an independent check on the predicted $B-V$ was attempted using the published $V-I$ data of @YA13. In cases where no resolution of the discrepancy was possible, the averaged value from all sources was retained. Averaged $B-V$ indices from the four transformed primary data sets are given in Table 1 along with the number of sources and each star’s estimated individual reddening correction.
As noted earlier, PL identified and mapped variable reddening across the field of NGC 6819, a result confirmed in Paper I. We used the map of individual reddening values derived by PL to estimate by spatial interpolation the degree of reddening affecting each of the stars in our spectroscopic sample. Individual reddening estimates, with a range of $\pm 0.033$ about our adopted mean value of $E(B-V)=0.16$, were applied to each star’s $(B-V)$ color. A temperature for each star based on its dereddened $(B-V)$ color was then derived using two primary color-temperature calibrations.
For dwarfs, we continue to use a calibration [@DE02] consistent with previous spectroscopic studies by this group, namely:
$T_{eff} = 8575 - 5222.7 (B-V)_0 + 1380.92 (B-V)_0^2 + 701.7 (B-V)_0[{\rm [Fe/H]} - 0.15]$ K.
For giants, the $T_{eff}$-color-\[Fe/H\] calibration of @RA05 was used. We note that for 13 stars, it was possible to obtain temperatures from both color-temperature calibrations. Temperatures derived from the @DE02 calibration are $41 \pm 6$ K higher for these mostly subgiant stars.
Surface gravity estimates (log $g$) were obtained by direct comparison of $V$ magnitudes and $B-V$ colors for our sample of 333 stars to isochrones from the $Y^2$ compilation [@DE04], constructed for a scaled solar composition with \[Fe/H\] = -0.06 and an age of 2.3 Gyr, essentially the same as the comparison presented in Paper I. The isochrone’s predicted magnitudes and colors were adjusted to match the cluster’s reddening, $E(B-V)=0.16$ and apparent distance modulus, $12.40$. For stars that appear to be blue stragglers, surface gravities were estimated by comparing photometric information to a grid of younger isochrones of similar composition.
We are fortunate to have independent confirmation of the log $g$ values for the giant branch stars, thanks to their status as [*Kepler*]{} objects of interest. We compared log $g$ values from our isochrone match to log $g$ values inferred using asteroseismology from [*Kepler*]{} data, as presented by @BA11 for 21 giants. The seismological gravities are in remarkable agreement with the isochrone-inferred gravities, differing by an insignificant amount, $-0.02 \pm 0.04$ where the listed error is the standard deviation.
Input estimates for the microturbulent velocity parameter were constructed using various prescriptions. For dwarfs within appropriate limits of $T_{eff}$ and log $g$, the formula of @ED93 was used; a similar formulation by @RA13 extends to slightly lower temperatures and gravities and was used for some stars. For giants, a gravity-dependent formula, $V_t= 2.0 - 0.2$ log $g$, was used. For some subgiants and candidate blue stragglers, no suitable formula for $V_t$ was found other than the purely gravity-dependent expression employed by the SDSS collaboration in their DR10 data release and discussion of the abundance analysis pipeline for APOGEE spectra [@AH14].
Spectroscopic Processing With ROBOSPECT
=======================================
With the expansion in spectroscopic samples from a few dozen stars in previous cluster work [@AT09; @AT10] to a few hundred in this and future analyses, manual measurement of equivalent widths (EW) for individual lines is now prohibitive. To overcome this obstacle, the automated line-measuring program, ROBOSPECT [@WH13] (hereinafter WH), has been utilized. For full details, the reader is referred to WH; a brief outline of the program operation and our procedure is provided here. ROBOSPECT measures EW by first determining the continuum level and constructing a noise profile across the wavelength range of interest using an iterative process. With the initial continuum and noise levels established, spectral lines are tagged at locations specified by a user-supplied line list, and other potential lines are automatically identified based on significant deviations of the local spectrum from the continuum. For the automatic line identification, the significance threshold is user-specified and based on the current noise solution. After lines are identified though these two processes, they are subtracted from the spectrum and the continuum process is repeated and the noise level refined. Through multiple iterations, the best-fit continuum and individual line solutions are reached. For this study, each spectrum was individually corrected in ROBOSPECT for radial velocity and run through 25 iterations of continuum fitting and line estimation using a gaussian line profile with three-sigma automatic line identification and no least-squares line deblending. All other parameters for the program were set to default values.
An issue of obvious concern with any automated procedure is the potential for unreliable EW due to low SNR spectra, line blending, or inaccurate radial-velocity correction. With interactive measurement, such issues can be flagged and corrected or eliminated but, with ROBOSPECT, flawed EW measures propagate to followup steps in the abundance determination process, potentially distorting the final results. It is expected that the uncertainty in EW will increase with decreasing SNR, potentially reaching a level where the subsequent uncertainty in a star’s calculated abundance renders it useless. Similarly, we can expect the EW uncertainty to increase as individual lines suffer increased blending from neighboring lines. Finally, a significant error in radial velocity will cause ROBOSPECT to misidentify the wavelength of line centers. To ensure that our abundances are minimally affected by these issues, a variety of tests were conducted to determine if our line list and ROBOSPECT input parameters lead to robust results over the range in SNR and radial-velocity uncertainties spanned by our spectra. Since ROBOSPECT will be used in future analyses with similar input parameters, the level of detail supplied is somewhat greater than usual.
The line list used during this study was constructed by visually identifying relatively isolated, medium-strength (EW $\sim$ 10 to 200 mÅ) lines in the solar spectrum, using the solar atlas of @WH98 as a guide. Atomic information for each line was retrieved from the VALD database [@KU99]. Log $gf$ values given by VALD were then modified to force-fit abundances from ROBOSPECT analysis of solar spectrum EW to abundances in the 2010 version of the abundance analysis software, MOOG [@SN73]. We used day-time sky spectra, taken as part of the calibration data sets, as solar spectra for this normalization. MOOG analysis was conducted using the [*abfind*]{} driver and a solar model [@KU95] with the following atmospheric parameters: ($T_{eff}, {\rm log\ } g, v_{t}, [Fe/H]$) = (5770 K, 4.40, 1.14 km/s, 0.00).
Our final line list contains 22 lines of interest (17 Fe, 3 Ni, 1 Ca, 1 Si), presented along with the relevant atomic parameters in Table 2. To minimize the impact of blending, all selected lines are at least 0.5 Å from any line with EW greater than 5 mÅ in the solar spectrum. To further minimize measurement distortions in spectra with lower SNR, the line list input for ROBOSPECT included any significant feature within 2 Å of a line of interest. EW for these features were disregarded during the abundance analysis. The extraneous entries were added after observing unusual amounts of scatter among the EW for lines with close neighbors, particularly at lower SNR.
To evaluate the performance of our line list and to quantify the effects of a spectrum’s SNR on EW values, we tested ROBOSPECT on 100 solar spectra, 25 each with mean SNR of 160, 130, 95 and 70. These represent typical SNR for our program stars (see Fig. 2). We note that WH include an evaluation of ROBOSPECT’s performance over a wide range of line strengths and SNR; our test results are consistent with theirs. The mean standard deviations in EW for our four test samples were 4.2, 4.7, 7.5, and 11.1 mÅ, in order of decreasing mean SNR. The uncertainty this scatter introduces into our abundance results is small relative to the uncertainty due to external systematics. At all SNRs tested, we observed no significant correlation between the mean EW of a line and its associated uncertainty. Taken together, these results indicate that our line list is robust over the range in mean EW and SNR of interest.
The effect of potential errors in the determined radial velocity on EW measurement was evaluated by artificially increasing and decreasing our best estimate of radial velocity by a flat amount in the set of 100 solar spectra. For a shift of $\pm$2.5 km/s, corresponding to a radial-velocity uncertainty greater than that of 99% of our program stars, the mean deviation in EW between the true and adjusted spectra was $-0.7$ mÅ. When the shift was increased to 5.0 km/s, corresponding to a radial-velocity uncertainty greater than that of all program stars (including the rapid rotator WOCS 2016), the mean deviation in EW increased to $-3.5$ mÅ. In both cases, ROBOSPECT correctly identified essentially the same number of entries from the line list as it had with the correct radial velocity. We conclude that the effect of radial-velocity error on our abundance results is negligible.
Finally, we compared ROBOSPECT results with manual EW measurements for 18 random program stars using a 10-line subset of our line list. Manually obtained EWs were measured using IRAF’s SPLOT tool. ROBOSPECT’s results largely agree with the manual values, with ROBOSPECT’s EW on average being 7.7 $\pm$4.2 mÅ (s.d.) lower than the manual results. We note that this consistent underestimate of EW by ROBOSPECT (or, over-estimate using SPLOT and interactive measuring) is similar in solar spectra and in the spectra of program stars, and is not a function of SNR to any detectable extent, so we anticipate no effect on our abundance estimates based on differential analysis with respect to the sun.
Using our measured radial velocities and constructed line list, EWs were measured for all 333 stars in our observing program. We discarded negative EW (ROBOSPECT’s designation of emission lines) due to non-convergent fitting solutions, artifacts of cosmic ray removal, large noise spikes, or nonexistent lines in the measured spectrum. Approximately 4% of the measured lines of interest returned negative values.
Abundance Determinations
========================
Model atmospheres were constructed for each program star using the grid of @KU95 and the input $T_{eff}$, log $g$ and microturbulent velocity values listed in Table 1. As a reminder, each star’s $T_{eff}$ estimate is based on its $B-V$ color, individually dereddened based on its spatial position within the field of the cluster. For any star that is not actually a cluster member, this spatially interpolated reddening correction may be spurious; the gravity estimate based on an isochrone matched to the cluster CMD almost certainly would be as well. For suspected binaries, the color might reflect the combined light of the two stars and affect the assigned temperature. While this might not be a large issue for SB1, a few candidate SB2 were highlighted by examination of spectra: WOCS 7009, 26007, 35025, 9004 and 60021. As a conservative approach, we restricted our analysis to stars with membership probablity $\geq 50$% (PL) and no evidence of binarity. We also excluded from our final analysis several stars with $(B-V)_0$ greater than 1.35, for which severe line-blending makes reliable measurement of equivalent widths problematic. This left us with a sample of 247 single, member stars.
Each star’s measured equivalent widths and model serve as input to the [*abfind*]{} routine of MOOG to produce individual \[A/H\] estimates for each measured line for each star. A large volume of spectroscopic information is available for interpretation from over 7000 measured lines, with up to 17 \[Fe/H\] values for each star. Before constructing median \[Fe/H\] values for each star and for the sample as a whole, the following filters were applied to the individual abundance estimates: abundances based on equivalent width measurements smaller than three times the expected error in equivalent width for each star based on the star’s SNR, were not considered, nor were equivalent widths for very strong lines (EW $\geq 200$ mÅ). We note that the consequences of using 200 or 250 mÅ as the upper limit for EW are negligible. Estimated abundances from individual lines deviant by more than 1 dex from solar were not included in the determination of \[Fe/H\] for each star.
Figures 2 and 3 show the spread of stellar \[Fe/H\] estimates for each star as a function of the individual spectrum SNR and unreddened $B-V$ color for each star. A few outliers with SNR below 50 are obvious, as are a few of the brightest (and coolest) stars with higher than typical abundance. Figure 3 suggests that abundances for the cooler stars appear to be slightly below the range for the stars nearer the turnoff. We further divided our sample of 247 likely members into stars bluer than and redder than $(B-V)_0 = 0.60$ to separate out the cooler and more evolved stars and re-examined the effect of filters that precede the construction of each star’s median \[Fe/H\] abundance. Do some of our 17 iron lines produce abundances in dwarfs that are unacceptably noisy?
By examining the effect of imposing different criteria for excluding small EW measures, we were able to highlight several iron lines, which for dwarfs, lead to biased \[Fe/H\] values in the sense that the abundance values for these lines in dwarf stars are higher than the ensemble median value by several tenths. The reason for this is clear; if the median EW for a particular absorption line is small, imposing a 3-sigma cut will preferentially preserve stars in the sample with larger EW, and consequently larger \[Fe/H\] values. We excluded lines in dwarfs for which imposing 3-sigma cuts changed the median EW for the line by more than 25% in comparison with a more generous 1-sigma cut. As a reminder, these lines are measured in the solar spectrum at considerably higher SNR so we may still place considerable confidence in the log $gf$ values determined from those lines.
As this still leaves information from several thousand separate iron line measurements, we were able to further explore the consequences of these filters. A diagnostic diagram is presented in Figure 4, demonstrating the lack of a trend in the median \[Fe/H\] for each of our 17 iron lines with wavelength, where the median value is constructed from the full sample of 247 probable-member, single stars for lines denoted by black symbols; the six lines denoted by red symbols represent measures for giant stars only. The upper panel of this figure shows the MAD ([*median absolute deviation*]{}) statistic for each line, a robust estimator of variance, constructed by comparing the \[Fe/H\] value for each star to the median value for that particular absorption line.
For the subset of 247 single member stars bluer than $(B-V)_0 < 1.35$, the median \[Fe/H\] value is $-0.03 \pm 0.06$, with abundances for the other elements that are entirely consistent with solar: $-0.01 \pm 0.10, -0.01 \pm 0.06 $ and $0.00 \pm 0.06$ for \[Ca/H\], \[Si/H\] and \[Ni/H\], respectively. Of these 247 stars, 200 are main sequence and turnoff stars, with a median iron abundance of $-0.02 \pm 0.05$, while the remaining 47 cool and evolved stars show a lower \[Fe/H\], $-0.09 \pm 0.05$. We note that this apparent discrepancy between giant and dwarf \[Fe/H\] values is nearly identical to that found in NGC 3680 [@AT09].
For all of these quantities, the indicated error is the MAD statistic; for a large sample size drawn from a normally distributed population, the normalized MAD statistic, MADN = 1.48 MAD, approximates well the sample standard deviation, permitting an estimate of the traditional standard error of the mean through division by $(N-1)^{1/2}$. For the sample of 247 single, probable-member stars, the MAD statistic implies a value for MADN = 0.09; as this statistic is a good proxy for a standard deviation, a standard error of the mean of 0.01 is implied. These values can be confirmed by computing averages, standard deviations and s.e.m. values in linear space; for these stars, the average \[Fe/H\], computed in linear space and then converted to logarithmic values, is $-0.02^{+0.07}_{-0.13}$ (standard deviations).
Table 3 includes the median \[Fe/H\], along with the number of iron lines included in the estimate, as well as estimates of \[Ca/H\], \[Si/H\] and \[Ni/H\] for 333 stars. Stars not part of the set of 247 single, probable members stars are flagged with a note indicating non-member of binary status. The statistic that accompanies each \[Fe/H\] estimate is the MAD. We note that the 29 stars not included in these estimates of cluster \[Fe/H\] due to evidence of binarity, yield a similar median \[Fe/H\] value if the suspected double-lined spectroscopic binaries are excluded. For member stars which are mostly SB1, \[Fe/H\]$ = -0.05 \pm 0.06$, consistent with the more conservative sample that excludes binaries.
Conversions of photometric parameters to $T_{eff}$ and $V_t$ are fundamentally discontinuous for dwarf and giant stars; to assess the effects of parameter choices on the derived abundances, each star was reanalyzed with atmospheric parameters incremented and decremented by the following amounts: 100 K for $T_{eff}$, 0.25 for log $g$ and 0.25 km/sec for $V_t$. A summary of the extensive results is shown in Figure 5 where once again median values and variances indicated by median absolute deviations are compiled and shown for stars in different color bins. This figure illustrates some well-known dependences of abundance determinations on stellar parameters, namely that main sequence star abundances are very sensitive to $T_{eff}$ estimates, while evolved stars demonstrate a higher sensitivity to surface gravity and consequently to microturbulent velocity estimation schemes.
Although the discrepancy between the giant and near-turnoff samples is small, it is of value to explore modest parameter changes that could eliminate it. Reference to Figure 5 suggests that errors in log $g$ have relatively little impact; we remind the reader that the log $g$ values employed are consistent with values suggested by asteroseismic analyses of NGC 6819 giants. The relation used for microturbulent velocities for the evolved stars is more tenuous; a fairly modest decrement of 0.2 km/sec would raise the giant abundances to match the \[Fe/H\] representative of the near-turnoff stars. Finally, increased abundances for both regions of the CMD would accompany higher temperatures, though a simple increment to the mean cluster reddening will have differing effects for the two classes of stars, both because of the slight color-dependence of the effect on the abundance as shown in Figure 5 and the larger increment implied for warmer stars resulting from the same higher reddening.
To define the final estimate of \[Fe/H\] for the cluster, we default to the dwarfs, where the sensitivity to the changes in the three key input parameters ($T_{eff}$, $V_t$, log $g$) is collectively minimized, the color range is small, and the parametric differences between the program stars and the solar reference spectra are significantly smaller than for the evolved stars. For log $g$ and $V_t$ we adopt 0.1 dex and 0.2 km/sec as generous estimates for the uncertainties. From Fig. 5, these translate into 0.003 and 0.008 dex, respectively. As usual, the more sensitive parameter is the temperature, tied to the $B-V$ color transformation, which has two primary components, the uncertainty in the mean cluster reddening, $E(B-V)$, and the uncertainty in the photometric zero point. From Paper I, the photometric internal and external errors combined lead to an uncertainty of 0.007 in $E(B-V)$, while the photometric merger of the four broad-band surveys above implies an uncertainty in the $B-V$ zero-point of 0.004 mag. Combined, the $(B-V)_0$ scale is reliable to 0.008 mag. On our temperature scale, this translates to approximately 32 K or, from Fig. 5, 0.012 dex in \[Fe/H\]. Combining all three parameters, one arrives at a final external uncertainty of 0.015 dex, much larger than the internal precision of the spectroscopic average. From 200 turnoff stars, the mean \[Fe/H\] for NGC 6819 is determined to be $-0.02 \pm 0.02$.
Among the prior determinations of NGC 6819’s heavy element abundance, the result published by @BR01 has anchored a string of moderately super-solar values. Their study of three clump giants utilized the high dispersion ($R \sim 40,000$) spectrograph on the Galileo Italian National Telescope (TNG). Though the sample size was small, their analysis incorporated hundreds of lines, including nearly 100 iron lines, six of which were Fe II lines to permit an accurate verification of surface gravity. The reddening value of E$(B-V) = 0.14$ they adopted, however, was somewhat smaller than the mean value derived in Paper I and utilized here, E$(B-V) = 0.16$. We repeated parameter determinations for the three giants studied by @BR01, including independent determinations of spatially-dependent reddening and $T_{eff}$ values, log $g$ values from isochrone comparisons and microturbulent velocity values based on the surface gravities. Using the estimates of $\Delta$\[Fe/H\] for each of the principal atmospheric parameters cited by @BR01, their equivalent width analysis would yield an average abundance 0.07 dex lower with our atmospheric parameters, lowering their \[Fe/H\] for the cluster from $+0.09$ to $+0.02 \pm 0.03$, essentially solar.
Discussion and Conclusions
==========================
High dispersion spectra of 333 potential members of the old open cluster, NGC 6819, have been processed and analyzed. From the wavelength region near Li 6708 Å, abundances have been obtained for Fe, Ni, Ca, and Si for 247 highly probable, single-star members ranging from the tip of the giant branch to well below the top of the CMD turnoff. Using up to 17 Fe lines per star, the cluster exhibits a slightly subsolar \[Fe/H\], a conclusion which remains unchanged if the data are sorted into evolved stars (47) or turnoff/unevolved stars (200). Due to the cumulative impact of a few thousand Fe lines, the standard error of the mean for \[Fe/H\] tied to internal errors is below 0.01 dex, leaving the systematics of the temperature, log $g$, and microturbulent velocity scales as the overwhelming source of uncertainty in the final cluster metallicity. Once again, thanks to the size of the sample, we can minimize the impact of these parameters by looking only at the 200 unevolved stars since the red giant abundances have a higher degree of sensitivity to microturbulent velocity. From these stars alone, \[Fe/H\] = $-0.02 \pm 0.02$, where the uncertainty includes both internal and external errors. The lower metallicity compared to the commonly adopted value of \[Fe/H\] = +0.09 [@BR01] continues a trend first indicated by the intermediate-band photometry of Paper I, where precision photometry was used to derive a reliable cluster reddening and confirm the existence of variable reddening across the face of the cluster (PL), leading to subsolar \[Fe/H\] from either $m_1$ or $hk$ indices.
Within the context of Galactic evolution, NGC 6819 exhibits no features which distinguish it from a typical thin-disk population formed within the last 5 Gyrs within 1 kpc of the Sun’s galactocentric distance. Its \[Fe/H\] places it at the mean for open clusters at the solar galactocentric radius, a distribution which exhibits a scatter at only the $\pm$0.1 dex level [@TA97]. Its abundance ratios relative to Fe are all consistent with solar, a pattern which holds true within the scatter for Ni, Ca, and Si for stars with \[Fe/H\] $\sim$0.0, whether they are classed as members of the thin or thick disk population (see @HI14 and references therein). NGC 6819 forms a rich and ideal link for tests of stellar evolution theory midway in age between NGC 5822 [@CA11] and M67 [@SA04], a characteristic which will prove valuable in the analysis of the evolution of Li for intermediate mass stars evolving off the main sequence and up the giant branch.
The authors express their thanks to Imants Platais for supplying additional data to compute spatially-dependent reddening estimates in the cluster. Drs. Ryan Maderak and Jeff Cummings assisted as visiting astronomers at KPNO for runs in 2010, supplying support and advice for the spectroscopic data collection. NSF support for this project was provided to ER while he was an undergraduate at the University of Kansas through NSF grant AST-0850564 via the CSUURE REU program at San Diego State University, supervised by Eric Sandquist. NSF support for this project to BJAT, BAT and DLB through NSF grant AST-1211621, and to CPD through NSF grant AST-1211699 is gratefully acknowledged. Extensive use was made of the WEBDA [^5] database operated at the Department of Theoretical Physics and Astrophysics of the Masaryk University, the TOPCAT suite [^6] the Vienna VALD, of the MOOG suite of spectroscopic analysis software, and of ROBOSPECT software.
Ahn, C. P., Alexandroff, R., Allende Prieto, C., et al. 2014, , 211, 17 Anthony-Twarog, B. J., Deliyannis, C. P., Twarog, B. A., Croxall, K. V., & Cummings, J. 2009, , 138, 1171 Anthony-Twarog, B. J., Deliyannis, C. P., Twarog, B. A., Cummings, J. D., & Maderak, R. M. 2010, , 139, 2034 Anthony-Twarog, B. J., Deliyannis, C. P. & Twarog, B. A. 2014, , 148, 151 (Paper I) Anthony-Twarog, B. J., Deliyannis, C. P., Rich, E. & Twarog, B. A. 2013, , 767, 19 Basu, S., Grundahl, F., Stello, D., et. al. 2011, , 729, L10 Bragaglia, A., Carretta, E., Gratton, R. G., et al. 2001, , 121, 327 Carraro, G., Anthony-Twarog, B.J., Costa, E., Jones, B.J. & Twarog, B.A. 2011, , 142, 127 Deliyannis, C.P., Twarog, B.A., Lee-Brown, D.B. & Anthony-Twarog, B.J. 2015, in preparation. Deliyannis, C. P., Steinhauer, A., & Jeffries, R. D. 2002, , 577, L39 Demarque, P., Woo, J. -H., Kim, Y. -C., & Yi, S. K. 2004, , 155, 667 ($Y^2$) Edvardsson, B., Andersen, J., Gustafsson, B. et al. 1993, , 275, 101 Gilliland, R. L., Brown, T. M., Christensen-Dalsgaard, J., et al. 2010, , 122, 131 Hinkel, N. R., Timmes, F. X., Young, P. A., Pagano, M. D., & Turnbull, M. C. 2014, , 148, 54 Hole, K. T., Geller, A. M., Mathieu, R. D., et al. 2009, , 138, 159 (H09) Kalirai, J. S., Richer, H. B., Fahlman, G. G., et al. 2001, , 122, 266 Kupka, F., Piskunov, N., Ryabchikova, T. A., Stempels, H. C., & Weiss, W. W. 1999, , 138, 119 Kurucz, R. L. 1995, in IAU Symp. 149, The Stellar Populations of Galaxies, ed. B. Barbuy & A. Renzini (Dordrecht: Kluwer), 225 Milliman, K. E., Mathieu, R. D., Geller, A. M., et al. 2014, , 148, 38 Platais, I., Gosnell, N. M., Meibom, S., et al. 2013, , 146, 43 (PL) Ramírez, I. & Meléndez, J. 2005, , 626, 465 Ramírez, I., Allende Prieto, C. & Lambert, D. L. 2013, , 764, 78 Rosvick, J., & VandenBerg, D. A. 1998, , 115, 1516 Sanders, W. L. 1972, , 19, 155 Sandquist, E. L. 2004, , 347 101 Sneden, C. 1973, , 184, 839 Twarog, B. A., Ashman, K., & Anthony-Twarog, B. J. 1997, , 114, 2556 van Dokkum, P. G. 2001, , 113, 1420 Wallace, L, Hinkle, K. & Livingston, W. 1998, [*An atlas of the spectrum of the solar photosphere from 13,500 to 28,000 $cm^{-1}$ (3570 to 7405 Å)*]{}, NSO Technical Report \#98-001. Waters, C. Z., & Hollek, J. K. 2013, , 125, 1164 (WH) Wilson, R. E., 1953, Carnegie Inst. Washington D.C. Publ. 601 Yang, S.-C., Sarajedini, A., Deliyannis, C. P., et al. 2013, , 762, 3
[^1]: WIYN Open Cluster study LXV
[^2]: The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatory.
[^3]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^4]: http://www.astro.yale.edu/dokkum/lacosmic/, an IRAF script developed by P. van Dokkum (van Dokkum 2001); spectroscopic version.
[^5]: http://www.univie.ac.at/webda
[^6]: http://www.star.bristol.ac.uk/ mbt/topcat/
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abstract: 'The NIS Directive introduces obligations for the security of the network and information systems of operators of essential services and of digital service providers and require from the national competent authorities to assess their compliance to these obligations. This paper describes a novel cybersecurity maturity assessment framework (CMAF) that is tailored to the NIS Directive requirements and can be used either as a self assessment tool from critical national infrastructures either as an audit tool from the National Competent Authorities for cybersecurity.'
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title: A NIS Directive compliant Cybersecurity Maturity Assessment Framework
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Introduction
============
Cyber attacks could contribute towards the collapse of a state if they initiate or prolong the failure of Critical National Infrastructures (CNI). Nations are becoming reliant on the cyber domain to provide services that keep a nation running: power grids, water supplies, communications, transportation and finance are all increasingly becoming cyber dependant. The NIS Directive (Directive (EU) 2016/1148) (European Union, 2016) has certain obligations that each member state should follow, with a major goal of enhancing cybersecurity posture across the EU. Having to cope with the obligations of the NIS directive and to meet strict deadlines, Greece has taken some steps forward [@maglaras2018nis]. The directive and a relevant national Law (Greek National Law 4577/2018)(Greek Government, 2018) introduce obligations for the Security Of The Network And Information Systems Of Operators Of Essential Services (OES) and the Security Of The Network And Information Systems Of Digital Service Providers (DSP) and require from the National Competent Authorities (NCA) to assess the compliance of operators of essential services and digital service providers with these obligations.
More specifically, the Greek National Law 4577/2018 (Greek Government, 2018) states that “the National Cybersecurity Authority (NCSA), acting as the NCA for cybersecurity, in collaboration with the relevant CSIRTs and other organizations and entities as appropriate assesses the technical and organizational measures implemented by OES, in order to manage risks related to the security of network and information systems used in their activities, regarding their suitability and their proportionality“. Additionally, NCSA ”assesses the suitability of the measures implemented by DSP for the avoidance and the minimization of the impact of incidents affecting the security of network and information systems used for the provision of the basic services, aiming to assure their business continuity".
Moreover, the objectives of the NCA across the EU reach further than just the collection of evidence send by the assessed entities and include the vision to reach a common level of cybersecurity posture. In order to effectively meet this vision, an initial step for NCSA was to conduct an assessment of the current cybersecurity posture of public sectors’ main ICT services, using a structured questionnaire. This assessment revealed inconsistencies and major misalignment among different entities [@drivas2019cybersecurity]. As a result, the below additional targets are proposed:
- Standardization of the collected feedback
- Assignment of a specific level of security, based in the implemented controls per category
- Analysis of the outputs and extraction of relevant statistical information regarding the level achieved per industry, category and service
- Implementation of comparisons between subsequent assessments, in order to monitor progress
- Extraction of possible correlations or contrasts between the information security posture among stakeholders
- Conduction of further analysis and definition of best practices
The minimum security requirements that OES and DSP have to comply with, have been defined in Decision 1027 published in 3739, B, 08.10.2019 Official Gazette of the Greek Government (Greek Government, 2019). This set of requirements, covering areas like Risk Management, Access Control, Physical and Environmental Controls etc, is generic in phrasing and does not provide specifications regarding the requirements implementation. For example, for the area of physical and environmental security, the requirement is that the installations of data centers and information processing facilities shall be protected against physical or environmental risk through suitable and relevant policies and measures based on a risk management strategy (Greek Government, 2019). This phrasing, although mandated by the fact that the entities required to comply with these requirements have a great variety in terms of business operation, size, security posture and technical and organizational capability, is difficult to be monitored effectively by the NCA in order to achieve the objectives mentioned above.
What was needed in order to facilitate the fulfillment of the NCSA’s objectives, especially the measurable ones, was a tool to standardize the possible maturity levels of the organizations. For this purpose a specific assessment framework, the Cybersecurity Maturity Assessment Framework (CMAF), was designed and tested for implementation. The CMAF could help identify the strengths and weaknesses of an organization’s processes and examine how closely these processes comply to related identified best practices or guidelines.
The assessment framework consists of the security controls against which the organization’s processes are appraised and the scale, based on which the rating of compliance of the organization’s processes is evaluated. Based on the proposed targets mentioned above the assessment model should incorporate the following characteristics:
- Cover the full extent of the security requirements complying to the NIS directive obligations
- Be able to be used as a self-assessment tool
- Be able to be used as a basis for an independent assessment
- Provide clear results regarding the security posture of the organizations
- Be able to be used as a benchmarking tool per industry, type of organization and area of operation
- Be able to be used as a guide for security requirements implementation by the organizations
- Be measurable
- Be easily extractable
Design of the framework
-----------------------
In order to design the CMAF, a combination of literature review regarding security requirements and a review of existing frameworks (related to cybersecurity or other well established areas) was conducted. At the time of the conduction of this review, there was only a limited number of established frameworks in the field, although during the past months, some more have been introduced. The literature review regarding security requirements included 16 basic documents. These documents were published by organizations like ENISA, ISO, CIS, European Union, NIST, ISACA and others. The review regarding existing frameworks included frameworks or models from organizations like: CMMI, CIS, ENISA, Department of Homeland Security – USA, Citigroup, U.S.Department of Energy and others.
The maturity scale
------------------
After the review of the existing frameworks, it was decided that the CMAF would be based in a 6-levels maturity scale (Figure \[figure1\]). For each level of the maturity scale, a different seal was selected. The seals represent a series of concentric cycles. The lowest possible score being represented by one cycle and the highest by six. The color of each cycle has been selected from the PH scale – red being the lowest and blue the highest.
{width="\textwidth"}
- [**Maturity Level 5: Efficient - Optimized**]{}
The organization has implemented methods for the continuous improvement of the implemented controls and the security posture of the organization. A full risk-based approach is followed and a cost-benefit balance is applied. The necessary controls are implemented, measured and controlled at the described level.
- [**Maturity Level 4: Effective - Quantitatively Managed**]{}
The organization has set relevant objectives. The objectives (were possible S.M.A.R.T.) are being monitored, measured, analyzed and evaluated. The necessary controls are implemented, measured and controlled at the described level.
- [**Maturity Level 3: Advanced - Defined**]{}
There is a standardized method regarding the fulfillment of the requirement. The necessary controls are implemented, measured and controlled at the described level.
- [**Maturity Level 2: Basic - Managed**]{}
There is a concrete plan regarding the fulfillment of the requirement. The necessary controls are implemented, but are partially measured and partially controlled.
- [**Maturity Level 1: Initial - Reactive**]{}
The organization has started implemented the requirement, but the extent of the implementation is partial or reactive.
- [**Maturity Level 0: Incomplete - Not existing**]{}
The requirement under examination is not implemented or it is implemented only partially or ad hoc.
The requirements
----------------
The basis of the CMAF and thus the requirements of the framework, are those that have been published under the sections “Common Security Policy” and “Baseline Security Requirements”, in Ministerial Decree 1027 - 3739, B, 08.10.2019 Official Gazette of the Greek Government (Greek Government, 2019). These requirements are grouped under the following categories:
- Information Security Policy
- Business Environment
- Asset Management
- Risk Assessment
- Risk Management Strategy
- Supply Chain Risk Management
- Self-Assessment and Improvement
- Policies, Processes and Procedures for the protection of essential services
- Identity Management and Access Control
- Physical and Environmental Security
- Systems and Applications Security
- Data Security
- Backups
- Security Technologies
- Systems Testing
- Change Management
- Awareness and Training
- Threat Detection
- Incident Management
- Business Continuity
- Disaster Recovery
For each of these categories the applicable controls have been recognized, analyzed and assigned to the appropriate level, as described in the maturity scale above.
The framework structure
-----------------------
The structure of the resulting framework is the following:
- A. IDENTIFICATION
Requirement A1. Business Environment
Requirement A2. Asset Management
Requirement A3. Risk Assessment
Requirement A4. Risk Management Strategy
Requirement A5. Supply Chain Risk Management
Requirement A6. Self-Assessment – Improvement
- B. PROTECTION
Requirement B7. Policies, Processes and Procedures for the protection of essential services.
Sub - Requirement B7.1 : Information Security Policy, Processes and Procedures
Sub - Requirement B7.2 : Communication and acceptance
Requirement B8. Identity Management and Access Control
Sub - Requirement B8.1 Asset Management
Sub - Requirement B8.2. Access control for privileged accounts
Sub - Requirement B8.3. Management of equipment for administrative purposes
Sub - Requirement B8.4. Access Control
Sub - Requirement B8.5. Authentication mechanisms
Requirement B.9. Physical and Environmental security
Requirement B.10. Systems and Applications security
Sub - Requirement B.10.1.: Systems Security
Sub - Requirement B.10.2.: Application Security
Sub - Requirement B.10.3.: Security in Application Development
Requirement B.11. Data Security
Sub - Requirement B.11.1.: Encryption
Sub - Requirement B.11.2.: Data Classification
Requirement B.12. Backups
Requirement B.13. Security Technologies
Sub - Requirement B.13.1. : Traffic filtering
Sub - Requirement B.13.2.: Segregation of systems
Sub - Requirement B.13.3. : Malware protection
Requirement B.14. Systems Testing
Sub - Requirement B.14.1.: Security Assessments
Sub - Requirement B.14.2.: Compliance Checking
Requirement B.15. Change Management
Requirement B.16. Awareness and Training
- C. DEFENSE
Requirement C.17. Threat Detection
Requirement C.18. Incident Management
Requirement C.19. Business Continuity
Requirement C.20 Disaster Recovery
Framework validation
--------------------
Since the framework was primarily based on literature review, it was decided that before it’s release, it should be validated through an implementation pilot project. The pilot project should include organizations from different sectors, of different sizes and with different security postures. To achieve this, the following pilots were selected for validation purposes:
- A mid-sized OES from healthcare sector, that is expected to have a low level of maturity
- A mid-sized OES from Digital Infrastructures sector, that is expected to have a medium level of maturity
- A large-sized OES from air-transport sector, that is expected to have a high level of maturity
The pilot assessments were carried out by an experts team consisting of: One expert from the project team, involved in the development of the CMAF with a deep knowledge of the framework itself and with high security control auditing skills, and two experts from the NCSA, tasked with the monitoring of the framework implementation process and the review of the related outcomes.
The assessments were carried out via the following methods: Table-top assessment, Interview and On site visit. The results of the assessments and the improvements proposed, were gathered, analyzed and incorporated in the final version of the model. In Figure \[figure2\] the graphical representation of the assessment result of a fictitious organization, that is produced from the model implementation, is presented.
![Graphical representation of the security assessment[]{data-label="figure2"}](graph1.png){width="50.00000%"}
Related Work
============
Authors in [@rea2017maturity] describe a cybersecurity capability maturity model as a means by which an organisation can assess its current level of maturity of its practices. They provide a comparative study of cybersecurity capability maturity models that builds on a previous review [@rea2017comparative]. The research presents an assessment of the differences, advantages and disadvantages of a systematic review of published studies from 2012 to 2017.The method was based on a modified taxonomy of software improvement environments across five categories proposed by Halvorson and Conradi [@halvorsen2001taxonomy]:
1. General: The broad attributes of the improvement environment.
2. Process: Describing how the environment is used.
3. Organisation: The features that articulate the relationship between the organisation and the environment in which it is used.
4. Quality: The indicators used to determine quality in the environment.
5. Result: A statement of the required outcomes, associated costs, and the methods used for evaluation.
Rea-Guaman et al [@rea2017comparative] reduce the categories to three, arguing that quality and result do not support the comparison of cybersecurity capability maturity models.
The research considers the Systems Security Engineering Capability Maturity Model (SSE-CMM), Cybersecurity Capability Maturity Model (C2M2) [@christopher2014cybersecurity], Community Cyber Security Maturity Model (CCSMM) [@white2011community], National Initiative for Cybersecurity Education – Capability Maturity Model (NICE) [@curtis2015cybersecurity].
The analysis concludes that all four models require a degree of customisation, and that SSE-CMM is the most established of those reviewed. The paper assessed C2M2 as the only model focused on cybersecurity. It went on to state that C2M2 and CCSMM are designed to be implemented in conjunction with the NIST framework and that recent updates to the models had not been identified. It further concluded that only SSE-CMM and C2M2 provided detailed considerations of risk. The research did not present a model to address the issues identified within those reviewed.
Sabillon [@sabillon2017comprehensive] reviews the best practices and methodologies in cybersecurity assurance and audit and presents the Cyber Security Audit Model (CSAM) for use by organisations and nation states to validate audit, preventative, forensic and detective controls. CSAM comprises 18 domains, with one limited to nation states and the remaining 17 applicable to organisations in general. All domains have at least one sub-domain, controls, checklists, sub-controls, and scorecard. The authors compare CSAM to the NIST Cyber Security Framework (CSF) version 1.1 and the Audit First Methodology [@donaldson2015enterprise], highlighting the differences. The work describes the CSAM model and states it was tested, implemented and validated in a Canadian higher education institution, although no results are presented, and its efficacy is not evidenced.
The research does not indicate whether CSAM requires specific expertise for its use, or whether self-assessment is feasible. Neither does it present whether the model provides any actionable outcomes following its use.
Akinsanya [@akinsanya2019current] performed a literature review of cyber security models for healthcare organisations adopting cloud computing. The analysis considers the following:
- Information Security Focus Area Maturity Model (ISFAM)[@van2010design]
- Cloud Security Capability Maturity Model (CSCMM)
- UK National Health Service (NHS) National Infrastructure Maturity Model (NIMM)[@savidas2009your]
- Health Information Network (HIN) Capability Maturity Model
The analysis found that ISFAM was targeted on small to medium enterprises with a focus of use within software development environments and could not demonstrate a capacity to integrate emerging technologies such as cloud computing. CSCMM was identified as able to support a range of organisational sizes but deemed too technically complex to implement in healthcare. The NHS NIMM was assessed as relevant to the cyber security assessment of a healthcare organisation. It was considered to provide an assessment of an organisation’s cyber security maturity and provided a basis to define which steps were required to achieve a greater level of maturity. It demonstrated a capacity to support platform independence but did not show an ability to accommodate the characteristics of cloud computing or its resulting threats. The researchers judged that the HIN Capability Maturity Model has similar characteristics and constraints as the NHS NIMM.
The researchers concluded that maturity models provide a compliance model that could not support the complexity of the emerging cyber environment, particularly for healthcare organisations adopting cloud technologies. The conclusions highlight three specific areas of concern and requirements for further work:
- Cyber security maturity models should focus on more than standards compliance.
- Any new measures of maturity introduced should be provided with adequate metrics to make them meaningful.
- The model should be extensible to accommodate the dynamic nature of the cyber security threat landscape.
The research focuses specifically on healthcare and the adoption of cloud computing and does not consider adoption beyond this industry or technology.
Miron and Muita [@miron2014cybersecurity] examine cybersecurity maturity models to identify the standards and controls available to providers of critical national infrastructures (CNI). The research considers the cyber threats to CNI and the impact of a loss of such services. The authors document nine cybersecurity capability maturity models assessed as applicable to CNI. These are presented based on their applicability to either specific CNI sub-sectors or their general cyber security focus. The review concludes that, of the models considered, none are designed to address the scenario of a CNI operator with a multiplicity of interdependent systems. Instead, the authors propose, the models are described at a high level and focus on CNI or industry sub-sectors and present the need for a model to support municipal governments.
Adler [@adler2013dynamic] states that capability maturity models are inherently static and diagnostic, in that they identify maturity gaps but are not directly actionable. The research proposes a methodology that follows three stages:
1. Model: Captures the organisation’s current-state cybersecurity maturity levels, formulates a maturity improvement plan, and identifies influences factors that will shape the organisation’s cybersecurity posture.
2. Simulate: Produces dynamic simulations of scenarios to determine how particular cybersecurity situations could evolve within the organisation, potential interventions, and likely outcomes and impacts.
3. Analyse: Assesses the projected mitigation costs and risk reduction benefits, comparing outcomes across alternate plans and scenarios.
The stages are illustrated through an analysis based upon extension to the Electric Sub-sector Cybersecurity Capability Maturity Model (ES-C2M2) [@stevens2014electricity]. ES-C2M2 is a lightweight adaptation of the Resilience Maturity Model [@caralli2010cert]. This extension describes that the requirement for any process improvement elements within the maturity model should provide explicit guidance for improving performance levels towards a desired end-state.
The research describes a set of decision support tools implemented in a proprietary software package but does not provide data or results from experimental use of the proposed approach.
Le and Hoang in [@le2016can] investigate existing cybersecurity maturity models to examine their strengths and weaknesses. They provide a comparison of 12 models mapped against five levels of maturity proposed by Humphrey (1989), concluding that models require relevant quantitative metrics for measurable and actionable assessment. No examples of such metrics are provided.
Almuhammadi and Alsaleh in [@almuhammadi2017information] review the NIST Cyber Security Framework (CSF) for critical infrastructures in order to assess how it can be applied to cybersecurity maturity models and to determine if any gaps exist. The authors propose that one of the key benefits of a cybersecurity capability maturity model is that it provides a structure to allow stakeholders to reach a consensus and set agreed priorities. The authors state that the NIST CSF does not provide organisations with a mechanism to measure the progress of a NIST implementation, or the maturity levels and information security processes’ capabilities. They assess that the framework focuses on high-level information security requirements and is applicable for the development of information security programmes and policies. They contrast this to other frameworks such as COBIT, ISO 27001, and the ISF Standard of Good Practice (SoGP) for Information Security. They argue that these focus on information security technical and functional controls, and that they are applicable for developing checklists and conducting compliance/audit assessments. The research proposes an information security maturity model that is able to measure the implementation progress of a security programme over time and assesses the reliability of the IT services that underpin a business. No examples of such measures are provided, and no experimental data is presented.
Discussion
==========
The National Competent Authorities for cybersecurity, especially those who need to comply with the NIS directive, could use the proposed maturity assessment framework in order to request from the applicable organizations the implementation of self-assessments based on the framework and collect the results. The authorities can review the collected responses, analyze the data and produce valuable conclusions. Also, comments regarding possible improvements can be collected by the authorities from the applicable organizations as well as from the dedicated staff dealing with the assessment of the results. Adapted version of the CMAF or other similar models incorporating specific requirements for cloud services and OT/ IoT environments could also be used. Based on the findings of the cybersecurity assessment of the Critical National Infrastructures, additional cyber security controls may be applied. Those security controls can be classified according to legal, technical, organizational, capacity building, and cooperation aspects
Conclusions
===========
As Critical National Infrastructures are becoming more vulnerable to cyber attacks, their protection becomes a significant issue for any organization, as well as a nation. Similar to other information technology (IT) processes, cybersecurity often follows a lifecycle model of prediction, protection, detection, and reaction. Moreover, an assessment is an activity that helps identify the strengths and weaknesses of an organization’s processes and examine how closely these processes relate to identified best practices and guidelines. In order to help in the evaluation of the cybersecurity posture of CNI, a novel cybersecurity maturity assessment framework, the CMAF, is presented in this paper. The proposed framework consists of 20 security categories, 6 maturity levels and and can be used both as a self assessment and as an external audit tool, facilitating organisations to perform a gap analysis and receive graphical representation of their security posture. Information that will be collected from the framework can be used, after proper aggregation and anonymisation processes, from National Competent Authorities in order to identify common security gaps and prioritise future security programmes and funding actions on a national or European level.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thankfully acknowledge the support of the CONCORDIA H2020 (GA no. 830927) EU project.
Acknowledgment {#acknowledgment .unnumbered}
==============
|
[**M. Baillargeon$^a$, F. Boudjema$^b$, C. Hamzaoui$^a$ and J. Lindig$^c$** ]{}
$^a$ Département de physique, Université du Québec à Montréal\
C.P. 8888, Succ. Centre Ville, Montréal, Québec, Canada, H3C 3P8\
$^b$ Laboratoire de Physique Théorique LAPTH\
Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, Cedex, France.\
$^c$ Institute for Theoretical Physics, Leipzig University\
Augustusplatz 10, 04109 Leipzig, Germany
[**Abstract**]{}
Introduction
============
Within the three-family standard model, the Yukawa interaction provides ten physical measurable parameters, the six quark masses and the four parameters of the CKM mixing matrix[@CKM]. Although ten is certainly a large number if the model is to be viewed as a fundamental theory, this number of parameters in fact emerges from two $3\times 3$ Yukawa matrices which in total amount to as many as 36 parameters! Clearly there is a large number of redundant parameters. In order to better corner the mechanism of symmetry breaking it could be advantageous to work with mass matrices that exhibit only the least number of parameters, [*i.e.*]{} ten. Having to deal with a minimal set not only eases the computational task, like going from the mass matrices to the mixing matrix, but when confronting the set with the data this may even help in better exhibiting patterns of mass matrices that hint towards further relations between some of the elements of the set. In this case one can entertain the existence of symmetries and models beyond the standard model that can explain the approximate relations. At the same time this general minimum approach could also reveal whether some relations, like those that relate some mixing angles to ratios of masses, are in fact not specific to a particular constrained model but are rather generic in a much wider class of models.\
Most of the [*constrained*]{} matrices that aim at relating the mixing angles to the quark masses, and hence reduce the number of parameters to much less than the needed 10, are based on so called texture zeroes mass matrices [@RRR]. These are based either on some specific “beyond the standard model” scenario or by postulating some ad-hoc ansatz. In ansätze were these zeroes are not related to any symmetry only the non-zero elements are counted as parameters, although it is clear that there are numerous ways of keeping to ten independent parameters. For instance instead of zeroes one can take some elements to be equal or have any other definite relations between them [@democratic; @usy]. Democratic [@democratic] mass matrices with all elements equal are a case in mind, these are a one-parameter model which when written in an appropriate basis can be turned into matrices with all elements but one being zero. In our approach we keep within the popular textures zeroes paradigm and look for those bases were only the minimum number of (non-zero) independent parameters appears explicitly.
Obviously, dealing with less than ten parameters invariably leads to relations between masses and mixings. In many cases and for a certain range of masses the less-than-ten parameter descriptions may turn out not to be supported by data if the textures are over-predictive. On the other hand if one works with, at least, ten independent parameters then one should always reproduce the data since there should be possible to make a one-to-one mapping. A general classification of symmetric/hermitian textures zeroes with a number of parameters less than ten has been given in [@RRR], while Branco Lavoura and Mota [@BrancoMota] (BLM) have been the first to point out that for non hermitian matrices some textures zero à la Fritzsch [@Fritzsch1] were just a rewriting of the mass matrices in a special basis and thus the zeroes of the much celebrated Fritzsch ansatz were “ void" of any physical content. In fact the BLM approach for non hermitian matrices still involves twelve parameters, the extra two being related to the phase conventions taken for the CKM[^1].
Recently an approach based on BLM has been pursued by some authors [@Falcone; @Koide; @Harayama], taking a specific pattern of non hermitian matrices and in some cases re-expressing one of the mass matrices with the help of the phenomenological parametrization of the CKM matrix. In this talk we will concentrate on hermitian texture zero matrices having zeroes in the non-diagonal entries. The case with zeroes on the diagonal will be presented in a longer communication [@ourpapermass]. Note that we differ from [@RRR] in that we still have ten parameters. It is known that within the standard model one can always express the mass matrices in a basis were they are hermitian [@Frampton; @BrancoMota; @Mahermitian]. Also, in the case of non hermitian matrices our results should be understood as applying to the hermitian square matrices, $H=MM^\dagger$ . We supply a systematic list of all possible texture zeroes that contain the minimal set of ten parameters and show how these textures can be reached from a general set of two hermitian matrices for up and down quarks, through specific weak basis transformation which we construct explicitly. Among all the patterns that we list, one shows a particularly very simple and appealing structure which has a direct connection to the Wolfenstein parametrization [@Wolfenstein]. For this we have been able to analytically construct a compact exact formulae for the mixing matrix.
Simple Texture Zeroes Quark Mass Matrices and the Choice of Basis
=================================================================
The key observation as concerns the search of a suitable basis, ideally one with the maximum number of zeroes, is that starting from any set of matrices for the up and down quarks, the physics is invariant if one performs a weak basis transformation on the fields. In the case of the standard model, one can choose any right-handed basis for both the up quark fields ($u_R$) and the down quark fields ($d_R$), as well as any basis for the doublets of left-handed fields ($Q_L$). All these bases are related to each other through unitary transformations, $u_R \rightarrow V_u u_R \;
;\; d_R \rightarrow V_d d_R \;;\; Q_L \rightarrow U_L Q_L$. Therefore all sets of mass matrices related to each other through
$$M^{\prime}_u=U^\dagger_L M_u V_u \;\;\;\;\;\; M^{\prime}_d=U^\dagger_L M_d V_d$$
give rise to the same physics (same masses and mixing angles in the charged current).
For hermitian matrices this means that weak basis transformations involve only a single unitary transformation, [*i.e.*]{}, $U_L=V_u=V_d=U$ and therefore one can use either the set $M_u, M_d$ or the set $M'_u, M'_d$ with $M'_f=U^\dagger M_f U$. In the case of hermitian matrices, one is starting with a set of 18 parameters and the task is to find a unitary matrix $U$ which can absorb 8 redundant parameters. This should always be possible since a $3 \times 3$ unitary matrix has nine real parameters, but since an overall phase transformation $U=e^{i\phi} {\bf 1}$ does not affect weak bases transformations, a unitary matrix provides the required number of variables to absorb the redundant parameters.\
Phase transformations
---------------------
One special case of this type of unitary transformations which always proves useful, even in the case of non-hermitian matrices, is the one provided by unitary phase transformations $U_{ij}=e^{i\phi_i}\delta_{ij}$. Because a global phase does not affect the transformation, we set $\phi_1=0$ without loss of generality. This type of matrix is therefore a two-parameter matrix . Applying this type of transformation on both $M_u, M_d$ one has the freedom to choose $\phi_{2,3}$ such that two phases out of the six contained in the hermitian $M_u, M_d$ can be set to zero. The only restriction is that one can not, in general, simultaneously remove the phases of both $M_u(ij)$ and $M_d(ij)$ ([*i.e.*]{} for the same $(ij)$). In any case, two parameters, or rather phases, out of the 18 can always be removed this way.
The simple case of a basis where one matrix is diagonal
-------------------------------------------------------
It is always possible to take $U=U_d^D$ ($U_u^D$), that is the unitary matrix that diagonalises the down (up) matrix. In these specific bases where one matrix is diagonal, the other, non-diagonal matrix, will then have no zero in general but 9 real parameters (of which 3 can be taken as phases in the non-diagonal entries). Applying an extra phase transformation removes two phases and therefore one does indeed end up with 3 parameters in $M_d$ (the masses) and 7 in $M_u$ making up a total of ten which is the minimal number.\
It is worth mentioning that similar bases (where one of the matrices is diagonal) have been studied in the literature but for the case of non-hermitian matrices [@Falcone; @Koide; @Harayama]. It is easy to see that one can easily recover these bases. Indeed, one can apply on our hermitian matrices, the following transformations: assuming one is starting with a diagonal $M_u$ take $U=V_u$ as a phase transformation or simply just the unit matrix, then it is always possible to choose $V_d$ such that $M_d$ turns into a non-hermitian matrix but with extra zeroes. We leave the proof and a discussion of these kind of (diagonal) bases to our longer communication [@ourpapermass].
Non trivial cases: Non diagonal matrices with no diagonal zero
--------------------------------------------------------------
In the above simple case one had three zeroes[^2]. In fact requiring that one maintains 10 parameters, and in the case of hermitian matrices where the zeroes are set on the off-diagonal elements, three is the maximum number of zeroes. The non trivial cases are when these three zeroes are shared between the up-quark and down-quark matrices, that is one off-diagonal zero in one matrix and two off-diagonal zeroes in the other. Indeed, having more than three off-diagonal zeroes, four say, one is left with the six real parameters on the diagonals plus two complex numbers which reduce to two real numbers after a phase transformation has been applied and thus leading to only 8 real parameters. Therefore, by requiring off-diagonal zeroes the problem is rather simple: one only has to combine a matrix with one off-diagonal zero with a matrix with two off-diagonal zeroes. For each of these matrices there are three possibilities of where to put the zero. All in all, one counts 18 such possibilities or patterns. These are displayed in Table 1.
$M_u$ $M_d$ $M_u$ $M_d$
--- ------- ----------------- ---- ----------------- -------
1 (1,2) (1,3) and (2,3) 10 (1,3) and (2,3) (1,2)
2 (1,2) (1,2) and(2,3) 11 (1,2) and (2,3) (1,2)
3 (1,2) (1,2) and(1,3) 12 (1,2) and (1,3) (1,2)
4 (1,3) (1,3) and (2,3) 13 (1,3) and (2,3) (1,3)
5 (1,3) (1,2) and (2,3) 14 (1,2) and (2,3) (1,3)
6 (1,3) (1,2) and (1,3) 15 (1,2) and (1,3) (1,3)
7 (2,3) (1,3) and (2,3) 16 (1,3) and (2,3) (2,3)
8 (2,3) (1,2) and (2,3) 17 (1,2) and (2,3) (2,3)
9 (2,3) (1,2) and (1,3) 18 (1,2) and (1,3) (2,3)
: \[allforms\]Location of the zeroes for the 18 different forms.
All of these combinations can in fact be classified in only two distinct cases which can not be obtained from each other by a simple relabeling of the axes. Denoting the two arbitrary hermitian mass matrices in those bases by $M_u=A^{\prime}$ and $M_d=B^{\prime}$, these cases are explicitly: $$\label{case1} A^{\prime}=\left(\begin{array}{ccc} A^{\prime}_{11} & 0 & 0 \\ 0 &
A^{\prime}_{22} & A^{\prime}_{23} \\ 0 & A^{\prime*}_{23} & A^{\prime}_{33} \\
\end{array}\right),\; B^{\prime}=\left(\begin{array}{ccc} B^{\prime}_{11} & 0 &
B^{\prime}_{13} \\ 0 & B^{\prime}_{22} & B^{\prime}_{23} \\ B^{\prime*}_{13} &
B^{\prime*}_{23} & B^{\prime}_{33} \end{array}\right)\;\mbox{\rm \underline{case I}}$$ and $$\label{case2} A^{\prime}=\left(\begin{array}{ccc} A^{\prime}_{11} & 0 & 0 \\ 0 &
A^{\prime}_{22} & A^{\prime}_{23} \\ 0 & A^{\prime*}_{23} & A^{\prime}_{33}
\end{array}\right),\; B^{\prime}=\left(\begin{array}{ccc} B^{\prime}_{11} &
B^{\prime}_{12} & B^{\prime}_{13} \\ B^{\prime*}_{12} & B^{\prime}_{22} & 0 \\
B^{\prime*}_{13} & 0 & B^{\prime}_{33} \end{array}\right)\;\mbox{\rm \underline{case
II}}.$$
where $A^{\prime}_{11}$ is an eigenvalue. Of course, one can exchange the role of $A^{\prime}$ and $B^{\prime}$ so that $A^{\prime}=M_d$ and $B^{\prime}=M_u$.
To prove the existence of these bases and show how they are reached, it is easiest to first move to the basis where $A$ is diagonal.
Denoting the eigenvalues of $A$ by $\lambda_i$, $(i=1,2,3)$ and $A^{\prime}_{11}=\lambda_1$, we have in the eigenbasis of A $$\label{diagonal}
A=\left(\begin{array}{ccc}
\lambda_1 & 0 & 0 \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3\end{array}\right),\;
B=\left(\begin{array}{ccc}
B_{11} & B_{12} & B_{13} \\
B^*_{12} & B_{22} & B_{23} \\
B^*_{13} & B^*_{23} & B_{33} \end{array}\right).$$
The unitary matrix which leads to the form for $A^{\prime}$ in both Eq. \[case1\] (Case I) and Eq. \[case2\] (case II) is simply $$U = \left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & x_2 & x_3 \\
0 & y_2 & y_3\end{array}\right)$$ with the complex numbers $x_2,x_3,y_2,y_3$ subject to the orthonormality conditions. It is then trivial to find the appropriate combinations of $x_2,x_3,y_2,y_3$ that lead to either $B^{\prime}$ in the above two cases [@ourpapermass]. For instance in the first case, requiring $B^{\prime}_{12}=x_2
B_{12} + x_3 B_{13}=0$ gives the appropriate $U$. All other cases with two off-diagonal zeroes in one matrix and one in the other are treated in an analogous way. The proofs are obtained from case I and case II just by relabeling the indices. Of course, the case where the two quark mass matrices make up between them only two-zeroes, being much less constrained, is always easier to construct.
CKM matrices from off-diagonal texture zeroes hermitian matrices
================================================================
The advantage of texture zeroes matrices yet accommodating all the ten physical parameters is that they allow to easily express the mixing matrix solely in term of the elements describing the mass matrices. One could then work backward and use the hierarchy observed in a particular parametrization of the CKM mixing matrix, together with the hierarchy in the masses, to exhibit further correlations in the elements of the mass matrices expressed in a simple basis that already exhibits zeroes.\
Recently, Rašin [@Rasin] has devised a procedure to express the CKM matrix as a function of the mass matrices in the general case where no zero element is found in neither $M_u$ nor $M_d$. He uses a product of rotation matrices and phase matrices to diagonalise a general $3\times 3$ matrix. However, even when we require $M_u$ to be diagonal, which is a special case of [@Rasin], we are still left with large formulae which include sines and cosines of angles for which only the tangent is explicitly known. These results do not give compact expressions for the CKM matrix elements. Only when more zeroes are imposed do the results simplify. Even with the simple textures that are displayed in Table \[allforms\], the recipe given in [@Rasin] leads to tedious and complicated formulae [@ourpapermass] which moreover come with an ambiguity in determining the signs of the sines and cosines. We will show that, with the textures that are displayed in Table \[allforms\], there exists a more compact way of expressing the CKM that does not make use of any sines or cosines but exhibits the masses and the elements of the mass matrices explicitly.\
Each combination in Table \[allforms\] will lead to a particular parametrization of the mixing matrix. We concentrate on parametrization 14 not only to illustrate how the diagonalisation of the matrices is carried out exactly, and hence how one expresses the CKM, but also because it leads to a parametrization of the Kobayashi-Maskawa matrix which is directly related to the Wolfenstein parametrization [@Wolfenstein].
To achieve this, we first apply a weak basis phase transformation to the form 14, such that the only remaining phase is located in the up quark matrix. Thus one is dealing with
$$\label{huhd14} M_u=\left(\begin{array}{ccc} u & 0 & y e^{i\phi} \\ 0 & \lambda_c & 0 \\ y
e^{-i\phi} & 0 & t
\end{array}\right),\;\;
M_d=\left(\begin{array}{ccc} d & x & 0 \\ x & s & z \\ 0 & z & b
\end{array}\right)$$
Note that this parametrization allows to have as input, at the level of the mass matrices, the physical mass of the charm quark, $\lambda_c$. In what follows all physical masses will be denoted by $\lambda_i$, the index $i$ being a flavour index.
These mass matrices are diagonalised through the following unitary matrices $$U_u=\left(\begin{array}{ccc} \sqrt{\frac{u_t}{\lambda_{ut}}}& 0 &
\sqrt{\frac{u_u}{\lambda_{ut}}}e^{i\phi}\\ 0 & 1 & 0 \\
-\sqrt{\frac{u_u}{\lambda_{ut}}}e^{-i\phi}&0&\sqrt{\frac{u_t}{\lambda_{ut}}}
\end{array}\right),
U_d = \left(\begin{array}{ccc} \sqrt{\frac{b_d d_s d_b}{\Delta\lambda_{ds}\lambda_{db}}}
& \sqrt{\frac{d_d b_s d_b}{\Delta\lambda_{ds}\lambda_{sb}}} \sigma & \sqrt{\frac{d_d d_s
b_b}{\Delta\lambda_{db}\lambda_{sb}}}
\\
-\sqrt{\frac{d_d b_d}{\lambda_{ds}\lambda_{db}}} & \sqrt{\frac{d_s
b_s}{\lambda_{ds}\lambda_{sb}}} & \sqrt{\frac{d_b b_b}{\lambda_{db}\lambda_{sb}}}
\\
\sqrt{\frac{d_d b_s b_b}{\Delta\lambda_{ds}\lambda_{db}}} & -\sqrt{\frac{b_d d_s
b_b}{\Delta\lambda_{ds}\lambda_{sb}}} \sigma & \sqrt{\frac{b_d b_s
d_b}{\Delta\lambda_{db}\lambda_{sb}}}
\end{array}\right)$$ Such that
$$U^\dagger_{u,d} \;M_{u,d}\; U_{u,d}=\left(\begin{array}{ccc} \lambda_{u,d}&0&0\\
0&\lambda_{c,s}&0\\ 0&0&\lambda_{t,b}
\end{array}\right)$$
and $$\begin{aligned}
x_i &=& |x - \lambda_i| \;\;\; ({\it e.g.} \;\;u_t=|u-\lambda_t|)\\ \lambda_{ij} &=&
|\lambda_i - \lambda_j| \\ \Delta &=& |b - d| \\ \sigma &=& \mbox{sign of}\; (b - d)\end{aligned}$$
Expressing the diagonalising matrices, $U_u,U_d$, with the help of the physical masses keeps the expressions of these matrices very compact. As $V_{\rm CKM}= U_u^\dagger U_d$, we can now write the CKM matrix exactly: $$\begin{aligned}
V_{us} & = & \frac{\sigma\left(\sqrt{u_t d_d b_s d_b} +
\sqrt{u_u b_d d_s b_b} e^{i\phi}\right)}
{\sqrt{\Delta \lambda_{ut} \lambda_{ds} \lambda_{sb}}} \\
V_{ub} & = & \frac{\sqrt{u_t d_d d_s b_b} -
\sqrt{u_u b_d b_s d_b} e^{i\phi}}
{\sqrt{\Delta \lambda_{ut} \lambda_{db} \lambda_{sb}}}
\label{vub} \\ V_{cd} & = & - \sqrt{\frac{d_d b_d}{\lambda_{ds}\lambda_{db}}} \label{vcd}
\\ V_{cb} & = & \sqrt{\frac{d_b b_b}{\lambda_{db}\lambda_{sb}}} \label{vcb} \\ V_{td} & =
& \frac{\left(\sqrt{u_u b_d d_s d_b} e^{-i\phi} +
\sqrt{u_t d_d b_s b_b}\right)}
{\sqrt{\Delta \lambda_{ut} \lambda_{ds} \lambda_{db}}} \\
V_{ts} & = & \frac{\sigma\left(\sqrt{u_u d_d b_s d_b} e^{-i\phi} -
\sqrt{u_t b_d d_s b_b}\right)}
{\sqrt{\Delta \lambda_{ut} \lambda_{ds} \lambda_{sb}}} \\
J & \equiv & \frac{\det\left[M_u,M_d\right]}{2i \lambda_{uc} \lambda_{ut} \lambda_{ct}
\lambda_{ds} \lambda_{db} \lambda_{sb}}
= \frac{\sqrt{u_u u_t d_d d_s d_b b_d b_s b_b}}{\lambda_{ut} \lambda_{ds} \lambda_{db}
\lambda_{sb}} \sin\phi\end{aligned}$$
We see that these expressions are surprisingly simple given that they come from the mass matrices. Moreover, contrary to some ansätze, this type of CKM matrix can always be made to fit the data.
Nonetheless, we are now in a position to exploit the mass hierarchies. One can take $x_x$ as a small perturbation, which means that in fact $\lambda_f \simeq f$ where $f$ refers to a diagonal element. In other words this assumption amounts to requiring that the diagonal elements of the mass matrices deviate very little from their corresponding eigenvalues. We then have from eq. \[vcd\] and \[vcb\], $$\begin{aligned}
d_d & \simeq & |V_{cd}|^2 \lambda_s, \\ b_b & \simeq & |V_{cb}|^2 \lambda_b.\end{aligned}$$
We also have from eq.\[vub\]
$$\begin{aligned}
V_{ub} \simeq \frac{1}{\sqrt{\lambda_t}}\biggl( \sqrt{ \frac{d_d b_b}{\lambda_b} }-
\sqrt{u_u} \;e^{i\phi}\biggr) \simeq - \sqrt{ \frac{u_u}{\lambda_t} }\; e^{i\phi}\end{aligned}$$
where we have made the additional assumption that the terms involving the down-quarks are quadratic in the “perturbation" $d_d \times b_b$ compared to the term originating from the up quark matrix: $u_u$. This additional assumption is stronger than the previous ones since it also compares the strengths of the off-diagonal elements of the up [*and* ]{} down quark matrices. In any case with these mild assumptions one can now trade $d_d,b_b,
u_u$, [*i.e.*]{} $d,b,u$ for the moduli of $V_{cd}, V_{cb}$ and $V_{ub}$ and physical masses (up to some signs).
Taking into account the size of $d_d$, $b_b$ and $u_u$, we can now write $$\begin{aligned}
\label{ourpara} V_{\rm CKM}& \simeq & \left(\begin{array}{ccc} V_{ud} &
\sqrt{\frac{d_d}{\lambda_s}} & - \sqrt{\frac{u_u}{\lambda_t}}e^{i\phi} \\
-\sqrt{\frac{d_d}{\lambda_s}} & V_{cs} & \sqrt{\frac{b_b}{\lambda_b}} \\ \sqrt{\frac{d_d
b_b}{\lambda_s \lambda_b}} + \sqrt{\frac{u_u}{\lambda_t}} e^{-i\phi} &
-\sqrt{\frac{b_b}{\lambda_b}} & V_{tb}
\end{array}\right), \\
J & = & \sqrt{\frac{u_u d_d b_b}{\lambda_s \lambda_b \lambda_t}} \sin\phi.\end{aligned}$$
It is interesting to see that in this parametrization the $V_{CKM}$ can be split into elements which originate either solely from the down-quark sector[^3] or the up-quark sector. To recover a phenomenologically viable mixing matrix, one could thus concentrate on each sector separately. Moreover this parametrization is equivalent to the standard Wolfenstein parametrization $$V_W = \left(\begin{array}{ccc} 1 - \frac{1}{2}\lambda^2 & \lambda & \lambda^3 A (\rho - i
\eta) \\ -\lambda & 1 - \frac{1}{2}\lambda^2 & \lambda^2 A \\ \lambda^3 A (1 - \rho - i
\eta) & -\lambda^2 A & 1 - {\cal O}(\lambda^4)
\end{array}\right).$$ with $$\begin{aligned}
\lambda &=& \sqrt{\frac{d_d}{\lambda_s}} \nonumber\\ A &=& \frac{\lambda_s}{d_d}
\sqrt{\frac{b_b}{\lambda_b}} \nonumber\\ \rho &=& - \sqrt{\frac{u_u \lambda_s
\lambda_b}{\lambda_t d_d b_b}} \cos\phi \nonumber\\ \eta &=& \sqrt{\frac{u_u \lambda_s
\lambda_b}{\lambda_t d_d b_b}} \sin\phi\end{aligned}$$
Asking for maximal CP violation [@maxcp] sets $\phi = \pi/2$ and leads to $\rho = 0$.
From the form of the $V_{CKM}$ matrix it is now an easy matter to find phenomenologically viable quark mass matrices. Most direct from our study is the [*general*]{} feature that if $d=(M_d)_{11}=0$, then $d_d=\lambda_d=m_d$ and therefore one has the rather successful prediction [@GattoSartori; @Weinbergmass]: $V_{us}\simeq \lambda \simeq
\sqrt{\lambda_d/\lambda_s}=\sqrt{m_d/m_s}$. Moreover, introducing the perturbative parameter $\epsilon \ll 1$ and with all other parameters of order 1, we may write the hierarchical matrices:
$$\label{huhde} M_u=\lambda_t \left(\begin{array}{ccc} 0 & 0 & c \; \epsilon^3 \; e^{i\phi}
\\ 0 & \lambda_c/\lambda_t & 0 \\
c \; \epsilon^3 \; e^{-i\phi} & 0 & 1
\end{array}\right),\;\;
M_d=\lambda_b \left(\begin{array}{ccc} 0 & a \; \epsilon^3 & 0 \\ a \; \epsilon^3 &
\epsilon^2 & b \; \epsilon^2 \\ 0 & b \; \epsilon^2 & 1
\end{array}\right)$$
This leads to $V_{us}=\sqrt{m_d/m_s}=a \; \epsilon$ whereas $|V_{ts}|=b\; \epsilon^2=(b/a^2)\; |V_{us}|^2$ with ($b,a \sim 1$). Therefore, if one identifies $\lambda=a \epsilon$ then $A=b/a^2$. We could “adjust" $a,b$ (and $c$) to better fit the data. The forms in Eq. \[huhde\] bear some resemblance to those presented in [@HouWong], but note that we arrive at these forms from a rather different approach.
Also in the down sector, the ansatz reproduces the correct ratio of masses. Note also that with the ansatz for the up-quark, copied somehow on that of the down quark, we get: $|V_{ub}|\simeq c \; \epsilon^3$.
Conclusions
===========
We have shown that without any assumption on the mass matrices apart from hermiticity, it is always possible to find a quark basis such that 3 off-diagonal elements are vanishing, allowing to diagonalise unambiguously the mass matrices and obtain the mixing matrix. The case where either $M_u$ or $M_d$ is diagonal (and therefore all the 6 vanishing elements are contained in one single matrix) is of special interest but leads to lengthy formulae for the CKM matrix entries. In all other cases, we arrived at compact formulae for the mixing matrix. These compact formulae that express without any approximation the $V_{CKM}$ matrix in terms of the masses and other elements of the mass matrices can be compared to popular parametrizations of the CKM matrix. The exact forms that we find make it transparent which further assumptions one can make ([*i.e*]{} more zeroes) to simplify the structure of the mass matrices and yet be compatible with the data. We have given one such example, and in passing we have shown how starting from the general 10 parameter bases, the mere assumption of one extra zero in $(M_d)_{11}$ gives the famous relation[@GattoSartori; @Weinbergmass] $V_{us}=\sqrt{m_d/m_s}$ which is seen then to be rather generic to a large class of models and ansätze. From there one can add more constraints, for example we have presented a new ansatz which can be made to fit the data quite well.
Acknowledgements
================
The work of M. B. is supported by La Fondation de l’Université du Québec à Montréal and the work of C. H. is supported in part by N.S.E.R.C. of Canada.
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[^1]: One should be fair and say that, sometimes, keeping one or two of the redundant parameters may prove useful. However we will stick with the minimalist description.
[^2]: Since one is dealing with hermitian matrices, the number of zeroes is that contained on one side of the diagonal.
[^3]: A similar observation has also been made in [@HouWong].
|
---
abstract: 'Synchrotron X-ray diffraction (XRD) experiments were performed on the network compounds Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ at temperatures between 15 and 800 K. The ferroelectric phase of the parent BaAl$_2$O$_4$ is largely suppressed by the Sr-substitution and disappears for $x\geq0.1$. Structural refinements reveal that the isotropic atomic displacement parameter ($B_{\rm iso}$) in the bridging oxygen atom for $x\geq0.05$ is largely independent of temperature and retains an anomalously large value in the adjacent paraelectric phase even at the lowest temperature. The $B_{\rm iso}$ systematically increases as $x$ increases, exhibiting an especially large value for $x\geq0.5$. According to previous electron diffraction experiments for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ with $x\geq0.1$, strong thermal diffuse scattering occurs at two reciprocal points relating to two distinct soft modes at the M- and K-points over a wide range of temperatures below 800 K \[Y. Ishii $et$ $al.$, Sci. Rep. **[6]{}, 19154 (2016)\]. Although the latter mode disappears at approximately 200 K, the former does not condense, at least down to 100 K. The anomalously large $B_{\rm iso}$ observed in this study is ascribed to these soft modes existing in a wide temperature range.**'
author:
- 'S. Kawaguchi'
- 'Y. Ishii'
- 'E. Tanaka'
- 'H. Tsukasaki'
- 'Y. Kubota'
- 'S. Mori'
title: 'Giant Thermal Vibrations in the Framework Compounds Ba$_{1-x}$Sr$_x$Al$_2$O$_4$'
---
Introduction
============
Since the discovery of the incipient perovskite-type quantum paraelectric oxides[@QP-STO; @QP-KTO], the quantum criticality in ferroelectrics has attracted substantial interest in condensed-matter physics. On the border of ferroelectricity, the dielectric constant is enhanced in the vicinity of absolute zero. This non-classical behavior was recently quantitatively explained[@QCFerro-NatPhys], and the relevance of the ferroelectric quantum criticality to other condensed states, such as superconductivity in doped SrTiO$_3$[@STO-super1; @STO-super2], has been reported. The recently observed quantum critical behavior in other new ferroelectric compounds, $e.g.$, the organic ferroelectric TSCaCl$_{2(1-x)}$Br$_{2x}$[@OrgFerro] and TTF-QBr$_{4-n}$I$_n$ complexes[@QP-TTF] and the multiferroic Ba$_2$CoGe$_2$O$_7$[@Ba2CoGe2O7], has prompted fascinating studies on the quantum criticality in ferroelectrics.
![\[crystal\_str\] (Color online) Crystal structures of BaAl$_2$O$_4$. (a) The high-temperature phase (space group $P6_322$) and (b) the low-temperature phase ($P6_3$). (c) The split atom model for the $P6_322$ high-temperature phase. In this structure model, the atomic positions of Ba/Sr and O2 are split into two sites along the $c$-axis, and the O1 site is split into three sites around the threefold axis.](Fig1_Kawaguchi){width="88mm"}
BaAl$_2$O$_4$ is a chiral improper ferroelectric[@Stokes] without an inversion center that crystallizes in a stuffed tridymite-type structure comprising a corner-sharing AlO$_4$ tetrahedral network with six-member cavities occupied by Ba ions. This compound undergoes a structural phase transition from a high-temperature phase with a space group of $P6_322$ to a low-temperature phase with $P6_3$ at approximately 450 K[@Ishii_BaAl2O4; @Huang1; @Huang2]. This transition is accompanied by the significant tilting of the Al-O-Al bond angle along the $c$-axis, giving rise to the enlargement of the cell volume to $2a\times2b\times c$. The crystal structures of the high-temperature and low-temperature phases are shown in Figs. \[crystal\_str\](a) and (b), respectively.
This compound possesses low-energy phonon modes, which are associated with a tilting of the AlO$_4$ tetrahedra around the shared vertices without a large distortion in each AlO$_4$ block[@Perez-Mato]. Such low-energy phonon modes have also been reported in SiO$_2$ modifications[@RUMs1; @RUMs2], nepheline[@nepheline], and ZrW$_2$O$_8$[@ZrW2O8-1] and are often called rigid unit modes (RUMs). RUMs can sometimes act as soft modes and cause structural phase transitions, as observed in quartz[@quartz2], tridymite[@Pryde], and nepheline[@nepheline]. RUMs have also been suggested as the origin of a negative thermal expansion, as in ZrW$_2$O$_8$[@ZrW2O8-1; @ZrW2O8-2].
Low-energy phonon modes, such as RUMs, can be observed as thermal diffuse scattering in electron and X-ray diffractions (XRD). In electron diffraction experiments of BaAl$_2$O$_4$, a characteristic honeycomb-type diffuse scattering pattern has been reported over a wide range of temperatures below 800 K[@Abakumov; @Ishii_BSAO]. A similar pattern was also observed in Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ with $x=0.4$[@Fukuda]. In the structural refinements of the $x=0.4$ sample, significant disorder was noted in the two oxygen sites, O1($2d$) and O2($6g$), in the initial model of $P6_322$ symmetry, as shown in Fig \[crystal\_str\](a). In those analyses, a split atom model, as shown in Fig. \[crystal\_str\](c), was employed, in which the positions of Ba/Sr($2b$), O1($2d$), and O2($6g$) are off-center from their ideal positions and split into the less symmetric $4e$, $6h$ and $12i$ sites, respectively. This type of disorder has also been reported in Sr$_{0.864}$Eu$_{0.136}$Al$_2$O$_4$[@HYamada] and BaGa$_2$O$_4$[@BaGa2O4].
We recently investigated the low-energy phonon modes in BaAl$_2$O$_4$ in detail via synchrotron XRD using single crystals and first principles calculations[@Ishii_BaAl2O4]. According to the calculations, this compound possesses two unstable phonon modes at the M- and K-points with nearly the same energies. Both of these unstable phonon modes gave rise to strong diffuse scattering intensities in a wide range of temperatures below 800 K. Interestingly, their intensities sharply increase towards $T_{\rm C}$, indicating that the two modes soften simultaneously.
Furthermore, the ordered phase with the $P6_3$ superstructure has been reported to be substantially suppressed by a small amount of Sr-substitution for Ba[@Ishii_BSAO; @Rodehorst]. According to our electron diffraction experiments on Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ with precisely controlled Sr concentrations, no superstructure is observed, at least down to 100 K, for $x\geq0.1$. In the temperature and compositional window of $T<200$ K and $x\geq0.1$, although the soft mode at the K-point disappears, the soft mode at the M-point survives and shows further fluctuation as the temperature decreases[@Ishii_BSAO]. These findings imply the presence of a new quantum critical state induced by the soft modes in Ba$_{1-x}$Sr$_x$Al$_2$O$_4$.
In the present study, we performed synchrotron powder XRD on Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ at 15–800 K, revealing an unusually large and temperature-independent thermal vibration at the bridging oxygens.
Experimental
============
Polycrystalline samples of Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ ($x$ = 0, 0.02, 0.05, 0.06, 0.1, 0.3, and 0.5) were synthesized using a conventional solid-state reaction. The sample preparation procedure is described elsewhere[@Ishii_BSAO; @Tanaka]. The obtained samples were stored in a vacuum. The synchrotron powder XRD patterns were obtained in the temperature range of 15–800 K at the BL02B2 beamline of SPring-8[@02B2]. The samples to be measured were crushed into fine powder and filled into a fused quartz capillary with a diameter of 0.2 mm. The diffraction intensities were recorded using an imaging plate and multiple microstrip solid-state detectors. The incident X-ray beam was monochromatized to 25 keV using a Si (111) double-crystal. The temperature was controlled with flowing helium and nitrogen gases.
The structure refinements were performed via the Rietveld method using the JANA2006 software package[@JANA]. The split atom model was employed for the structural refinements of the high-temperature phase. For the fitting of several profiles obtained at high temperatures, the Ba/Sr atom was placed on the $2b$ site of the $P6_322$ average structure rather than on the $4e$ site of the split atom model because the atomic displacement from the $2b$ site is so small that the $4e$ site cannot be distinguished from the $2b$ site at these temperatures. The $P6_3$ structure model was used for the low-temperature phase below $T_{\rm C}$. Several profiles just below $T_{\rm C}$ were analyzed using the $P6_322$ split atom model because of the poor fitting results obtained using the $P6_3$ structure model. The space groups and the atomic positions used for the refinements are summarized in Table S1.
Results and Discussion
======================
The obtained diffraction profiles for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ with $x=0.02$ and 0.1 are shown in Figs. \[SXRD\_Profiles\](a) and (b), respectively, and the profiles for the other compositions are displayed in Fig. S1. The superlattice reflections of the low-temperature $P6_3$ phase can be clearly seen below 400 K for $x=0.02$, as indicated by arrows in Fig. \[SXRD\_Profiles\](a). These superlattice reflections were also observed for $x=0.05$ and 0.06, but they could not be observed for $x\geq0.1$. Thus, the structural phase transition does not occur down to 15 K for $x\geq0.1$.
![\[SXRD\_Profiles\] (Color online) Synchrotron powder XRD profiles obtained at 15–600 K for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ with (a) $x = 0.02$ and (b) 0.1. All of the indices throughout the paper are based on the $P6_322$ parent phase. The superlattice reflection grows below 400 K for $x=0.02$, as marked by the arrows, indicating the structural phase transition from $P6_322$ to $P6_3$. No superlattice reflections can be seen for the $x=0.1$ sample down to 15 K.](Fig2_Kawaguchi){width="88mm"}
![\[PhaseDiagram\] (Color online) Phase diagram for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$. $T_{\rm C}$ abruptly decreases as $x$ increases. Anomalously large and temperature-independent $B_{\rm iso}$ values were observed in the region of $x\geq0.1$, which is labeled as a giant thermal vibration. $T_{\rm f}$ denotes the temperature at which the K-point mode disappears, as described in the text.](Fig3_Kawaguchi){width="64mm"}
$T_{\rm C}$ was defined as the temperature at which the superlattice reflections appear. A phase diagram for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ is shown in Fig. \[PhaseDiagram\]. The $T_{\rm C}$ values determined in this study were plotted together with the previously reported data[@Ishii_BSAO; @Ishii_BaAl2O4]. As shown in this figure, $T_{\rm C}$ becomes largely suppressed as $x$ increases, in agreement with previous reports[@Rodehorst; @Ishii_BSAO]. Synchrotron XRD experiments using single crystals[@Ishii_BaAl2O4] revealed a continuous variation of the superlattice intensities at $T_{\rm C}$, indicating a second-order phase transition. These findings indicate that Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ may plausibly show a new quantum critical state.
We performed structural refinements on the Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ sample using the Rietveld method. Fig. \[Rietveld\] displays the powder diffraction pattern for $x=0.2$ at 15 K and the fitting results obtained using the split atom model. The refinement using the initial model yielded a poor fitting result; the obtained $R$-factors based on the weighted profile ($R_{\rm WP}$) and the Bragg-integrated intensities ($R_{\rm I}$) were 5.14 and 9.25 %, respectively. In contrast, Fig. \[Rietveld\] shows that the split atom model accurately reproduces the experimental profile with high reliability ($R_{\rm WP}$ = 4.11 % and $R_{\rm I}$ = 3.20 %). The refined structural parameters are listed in Table \[Models\]. One important finding is that the isotropic atomic displacement parameter ($B_{\rm iso}$) of the O1 site is extraordinarily large, which is rare in oxide insulators. Because the $R$-factors are sufficiently small, this large $B_{\rm iso}$ should have an important physical meaning. Such a large $B_{\rm iso}$ in the O1 site has also been reported in previous works on Ba$_{0.6}$Sr$_{0.4}$Al$_2$O$_4$[@Fukuda], for which it was ascribed to the thermal diffuse scattering observed in the diffraction experiments, and Sr$_{0.864}$Eu$_{0.136}$Al$_2$O$_4$[@HYamada] at room temperature.
![\[Rietveld\] (Color online) Synchrotron powder XRD patterns (triangles) of Ba$_{0.8}$Sr$_{0.2}$Al$_2$O$_4$ at 15 K. The solid red curve represents the result of the structure refinement using the split atom model. The difference curve is shown in the lower part of the figure. Vertical lines indicate the positions of possible Bragg peaks. The fitting result for the high-$Q$ region is depicted in the inset. The Ba/Sr atom is placed at the $4e$ site.](Fig4_Kawaguchi){width="88mm"}
Atom site $g$ $x$ $y$ $z$ $B_{\rm iso}$ (Å$^{2}$)
------- ------- ----- ----------- ----------- ----------- -------------------------
Ba/Sr 4$e$ 1/2 0 0 0.2414(1) 0.392(7)
Al 4$f$ 1 1/3 2/3 0.9451(1) 0.93(2)
O1 6$h$ 1/3 0.3863(3) 2$x$ 3/4 2.1(2)
O2 12$i$ 1/2 0.3593(8) -0.002(1) 0.0289(3) 0.77(5)
: \[Models\] Results of the structure refinement for Ba$_{0.8}$Sr$_{0.2}$Al$_2$O$_4$ at 15 K using the split atom model. The cation ratio of Ba:Sr is fixed to 0.8:0.2.
![\[Biso\] (Color online) Temperature dependence of $B_{\rm iso}$ of Ba/Sr, Al, the bridging oxygen (O$_{\rm br}$) and the basal oxygen (O$_{\rm bas}$) atoms for (a) $x = 0$, (b) 0.05, and (c) 0.5. $T_{\rm C}$ is indicated by an arrow. (d) $B_{\rm iso}$ at 15 K plotted as a function of $x$. (e) The displacement of the Ba/Sr atom ($-\Delta z$) from the ideal $2b$ site of $z=0.25$. The $-\Delta z$ values obtained using the $P6_322$ split atom model are indicated by the open symbols. Below 420 K, the Ba site for $x=0$ is split into two sites of $P6_3$, $2a$ and $6c$, as indicated by the closed symbols. ](Fig5_Kawaguchi){width="87mm"}
The temperature dependences of the obtained $B_{\rm iso}$ for the $x$ = 0, 0.05, and 0.5 samples are shown in Figs. \[Biso\](a), (b), and (c), respectively. In these figures, the bridging oxygen, O$_{\rm br}$, and the basal oxygen, O$_{\rm bas}$, represent the O1 and O2 atoms in the $P6_322$ structure, respectively. In the structural refinements for the $P6_3$ structure, the atoms were categorized into four groups: the Ba group of Ba1–2, the Al group of Al1–4, the basal oxygen group of O1–4, and the bridging oxygen group of O5–6. The calculations were performed under the constraint that an equal $B_{\rm iso}$ is used within each group. As shown in Fig. \[Biso\](a), the $B_{\rm iso}$ of each atom for $x=0$ gradually decreases as the temperature decreases, and rapidly decreases at $T_{\rm C}$, except for the Ba atom, as marked by an arrow in Fig. \[Biso\](a). This is probably because of the structural phase transition accompanying the condensation of one of the soft modes and the resulting suppression of the thermal vibrations. The $B_{\rm iso}$ values of all atoms are small at 15 K. The sudden drops in the $B_{\rm iso}$ at $T_{\rm C}$ were also observed for the $x=0.02$ sample. The temperature dependences of the $B_{\rm iso}$ for all compositions are shown in Fig. S2.
The full width at half maximum (FWHM) of the superlattice reflection has been reported to increase as $x$ increases[@Ishii_BSAO], as observed in this study (Fig. S3). Thus, the long-range ordering of the low-temperature $P6_3$ structure is strongly suppressed by the Sr-substitution. In Fig. \[Biso\](b), no sudden drop in $B_{\rm iso}$ can be observed for $x=0.05$, although the superlattice reflections develop below 320 K in the $x = 0.05$ sample, as shown in Fig. S1(c). This is probably because of the suppression of the long-range ordering of the low-temperature structure in the $x=0.05$ sample.
In contrast, the observed $B_{\rm iso}$ for $x=0.5$ is surprisingly large, as seen in Fig. \[Biso\](c). In addition, it is largely independent of the temperature, resulting in anomalously large values even at the lowest temperature. Such a temperature-independent large $B_{\rm iso}$ is also observed for $x=0.1$ and 0.3, as shown in Figs. S2(e) and (f). Other atoms also exhibit larger $B_{\rm iso}$ over the whole temperature range than that of $x=0$. Because the $B_{\rm iso}$ values for Ba/Sr, Al, and O$_{\rm bas}$ are sufficiently small at 15 K, and the refinements yield the satisfactorily small $R$-factors, as shown in Figs. S4 and S5, the extraordinarily large $B_{\rm iso}$ observed in O$_{\rm br}$ cannot be attributed to the fitting error in the refinements. Fig. \[Biso\](d) represents the $B_{\rm iso}$ for each atom at 15 K as a function of $x$. These values systematically increase as $x$ increses, particularly for the O$_{\rm br}$ atom. The $B_{\rm iso}$ for the O$_{\rm br}$ atoms are especially large for $x\geq0.06$, indicating that the O$_{\rm br}$ atom for $x\geq0.06$ exhibits an unusually large thermal vibration down to absolute zero. Notably, this enhancement in $B_{\rm iso}$ is observed on the border of the ferroelectric phase. The Ba/Sr atom displaces only along the $c$-axis in the split atom model. The displacements of the Ba/Sr atoms ($-\Delta z$) from the ideal $2b$ site of $z=0.25$ are plotted in Fig. \[Biso\](e) as a function of temperature. These gradually increase as the temperature decreases. For $x=0$, the Ba site splits into the $2a$ and $6c$ sites of the $P6_3$ low-temperature structure at $T_{\rm C}$, as indicated by the closed symbols in Fig. \[Biso\](e), because of the structural phase transition. Below $T_{\rm C}$, the $-\Delta z$ of Ba at the $6c$ site increases as the temperature decreases. For $x=0.1$, 0.3, and 0.5, the $-\Delta z$ increases as $x$ increases. Notably, the $-\Delta z$ for $x=0.5$ exceeds that for $x=0$ in the ferroelectric phase, although the $x=0.5$ sample does not exhibit a structural phase transition.
The temperature and compositional window of the anomalously large $B_{\rm iso}$ is illustrated in Fig. \[PhaseDiagram\]. In general, the thermal vibration is large at high temperatures and becomes suppressed as temperature decreases, and thus $B_{\rm iso}$ is expected to decrease as the temperature drops. However, the $B_{\rm iso}$ of the O$_{\rm br}$ atom in this system with $x\geq0.1$ does not follow this general trend; instead, it exhibits fairly large values even at 15 K. According to our previous studies, this system possesses structural instabilities at the M- and K-points, leading to two energetically competing soft modes[@Ishii_BaAl2O4]. Mode analyses have revealed that the O$_{\rm br}$ atom shows a remarkable in-plane vibration around the ideal $2d$ site. In addition, strong thermal diffuse scattering because of these soft modes has been observed in the electron diffraction patterns for $x\geq0.1$ over a wide range of temperatures between 100 and 800 K[@Ishii_BSAO]. Clearly, these soft modes are responsible for the unusually large $B_{\rm iso}$ observed in this study. The large enhancement in $B_{\rm iso}$ by the Sr substitution indicates “a giant thermal vibration,” which might be a phonon-related quantum critical phenomenon. For the $x\geq0.1$ sample, the M-point soft mode does not condense but survives down to at least 100 K, whereas the K-point mode weakens and disappears below 200 K[@Ishii_BSAO]. The temperature at which the K-point mode disappears, $T_{\rm f}$, is indicated by a broken line in Fig. \[PhaseDiagram\]. Below this line, the M-point mode fluctuates with short-range correlations in nanoscale regions[@Ishii_BaAl2O4]. The anomalously large and temperature-independent $B_{\rm iso}$ even at low temperatures can be attributed to this fluctuating M-point mode.
Large thermal displacements have also been reported in the metallic phase of VO$_2$[@VO2-1; @VO2-2], which exhibits a drastic metal-insulator transition (MIT). Several approaches, including thermal diffuse scattering just above the MIT[@VO2-TDS], the symmetry analyses[@VO2-Group], and the phonon dispersion calculations[@VO2-cal], indicate the presence of a soft mode. ZrW$_2$O$_8$ is another example that shows a large $B_{\rm iso}$ in its oxygen atoms, comprises a corner-sharing polyhedral network, and possesses RUMs. However, the $B_{\rm iso}$ values of these oxygen atoms have been reported to decrease as the temperature decreases[@ZrW2O8_Biso]. No compound showing a temperature-independent and anomalously large thermal vibration even at low temperature has been reported to date. To clarify the nature of the fluctuating state in Ba$_{1-x}$Sr$_x$Al$_2$O$_4$, measurements of the physical properties, such as the dielectric constant and specific heat, are now in progress.
Conclusions
===========
We performed synchrotron XRD experiments for Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ at 15–800 K and analyzed their crystal structures via the Rietveld method using the split atom model. $T_{\rm C}$ becomes substantially suppressed as $x$ increases, and the structural phase transition disappears for $x\geq0.1$. We observed an anomalously large and temperature-independent $B_{\rm iso}$ in the bridging oxygen for the $x\geq0.1$ samples, with the value systematically increasing as $x$ increases. These anomalously large $B_{\rm iso}$ values can be attributed to the giant thermal vibration arising from the existence of soft modes over a wide temperature range.
This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) from the Japan Society for the Promotion of Science (JSPS). The synchrotron radiation experiments were performed at BL02B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposals No. 2015A2058, No. 2015A1510, and No. 2015B1488).
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abstract: 'In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such an area in searched parameters’ space that generates simulated curve crossing as many acquired experimental points as possible, but at least half of them. Such a task is pretty easy to attack with interval calculations. The problem is, however, that interval calculations operate on guaranteed intervals, that is on pairs of numbers determining minimal and maximal values of measured quantity while in vast majority of cases our measured quantities are expressed rather as a pair of two other numbers: the average value and its standard deviation. Here we propose the combination of interval calculus with basic notions from probability and statistics. This approach makes possible to obtain the results in familiar form as reliable values of searched parameters, their standard deviations, and their correlations as well. There are no assumptions concerning the probability density distributions of experimental values besides the obvious one that their variances are finite. Neither the symmetry of uncertainties of experimental distributions is required (assumed) nor those uncertainties have to be ‘small.’As a side effect, outliers are quietly and safely ignored, even if numerous.'
address: 'Institute of Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland'
author:
- 'Marek W. Gutowski'
bibliography:
- '<your-bib-database>.bib'
title: Reliable uncertainties in indirect measurements
---
data analysis,reliable computations,guaranteed results, safety critical applications ,scientific computations
Introduction {#intro}
============
Practically all known data fitting procedures are based on minimization of distance between measured and simulated values. Yet there exist various, equally good, distances (metrics) in $n$-dimensional space $\mbox{\mm R}^n$. Besides the best known Euclidean distance there exist many other ones, like Manhattan (taxi driver) or Chebyshev distance. Those three produce exactly the same values of $d(x,y)$ when $n=1$, but still another metrics, defined as$d(x,y)=c\log\,(1+\left|x-y\right|/c)$ (with any fixed $c>0$) will differ.
The choice of this or other metrics is therefore not unique. Exactly for this reason we have various families of fitting procedures[@atmos] at our disposal, with (weighted) least squares regression (LSQ) being most popular.
However, the real question is: why at all we are resorting to distance–based routines? The results of individual measurements are conventionally reported in form $y_{0}\pm\sigma_{y}$, with $y_{0}$ being the most likely numerical estimate of true value $\hat{y}$, and $\sigma_{y}$ its standard deviation. Such result is usually drawn as a section of a straight line $\left[\,y_{0}-\sigma_{y}\,,\,y_{0}+\sigma_{y}\,\right]$, when presented in graphs. This doesn’t mean that the true value $\hat{y}=y_{0}$, nor that $\hat{y}$ is even contained within the interval shown! Indeed, when $\hat{y}$ is normally distributed (a common but often unfounded[@notnormal] assumption), the chance for its true value to be located outside this interval is roughly equal to $1/3$ — rather far from being negligible.This simple observation makes all distance–based procedures questionable at least. Besides, we often fit our data after non-linear transformation, performed just to visualize them as forming a straight line in the new coordinate system. After such a transformation, the original center of uncertainty interval no longer corresponds to the center of its image, thus making the notion of distance even more dubious.
So, perhaps we should deal with intervals of type $\left[\,y_0-\delta,\,y_0+\delta\,\right]$ instead, where $\delta$ is maximal uncertainty of a measurement? This way the true value $\hat{y}$ is guaranteed to reside inside the interval shown. But is it indeed? No, at least not in experimental practice, when unpredictable interferences do happen (or just data storage/transmission errors). Additionally, under this approach, our task would be to find a set of curves passing through *all* the measured values expressed as guaranteed intervals. Informally speaking, we should find a ‘thick’ simulated curve and then evaluate somehow the uncertainties of all its fitted parameters. However, it is easy to see that a single outlier may make this task impossible. On the other hand, when the said outlier is ‘small’ enough, then the fitted curve will be unrealistically narrow, thus suggesting incredibly good precision of its parameters. No wonder that this approach didn’t gain much popularity in scientific laboratory practice (notable exceptions are proposals given by Zhilin[@EmpDataFit; @Zhilin] or Kieffer[@ch7]), nevertheless it is developed in the field of various engineering applications under the keywords like tolerance problem[@Remote; @Tolerable], confidence regions[@FIR; @B09], or guaranteed state estimation[@Jaul; @KieffWalt].
In many cases the LSQ approach leads to solving a set of linear equations.This is also true for interval versions of those procedures. It appears, however, that interval linear systems have definitely more specific features than their classical predecessors. Specifically, they may have solutions of many different kinds. This fact was most likely first recognized by Shary, who also proposed a classification scheme for solutions of interval systems of linear equations (applicable to non-linear systems as well). The literature concerning linear interval equations is now quite impressive[@EmpDataFit; @Remote; @Tolerable; @Jaul; @veryOldShary; @MCS; @middleShary; @Irene; @Skalna; @Kolev-lin; @Kolev1; @SergeyIrene; @inner; @Kolev] while non-linear systems are investigated less frequently[@ch7; @FIR; @KieffWalt; @Kolev-non; @Kolev-nonlin] and often presented at conferences only[@Tucker].Needless to say that majority of mentioned papers is of purely mathematical nature, without explicit relations to physical problems.
The combination of interval calculus with probability is tempting but certainly uneasy to satisfy. The papers in this area are appearing only recently[@FIR; @Sandia; @Krein2009; @KreinShary; @me] and still remain scarce. Our approach, as presented below, tries to follow this trend but seems more general and at the same time generally easier to follow.
Bridging the gap between intervals, probability and statistics
==============================================================
Interval calculus
-----------------
For some reasons, interval calculus remains still largely unknown to the general computing public. It is much younger than the idea of complex numbers but it is with us already since $\sim\,$1960[@Ray] and proves to be very useful, especially in computer calculations.In fact, it was developed first of all to put under strict control the uncertainties of results produced by various computers operating with finite precision arithmetic, often with different machine word lengths and/or representations of real numbers, thus necessarily introducing rounding errors.Informally, interval arithmetic makes possible to compute guaranteed ranges of mathematical expressions when exact ranges of all their components are known. By *guaranteed* we mean that the so obtained results contain true ranges with certainty. It doesn’t mean, however, that those ranges are equal. It happens quite often that interval result overestimates the true result – but never underestimates it.
For those completely unfamiliar with interval computations, we provide a brief introduction in \[A\] and \[B\]. Readers interested in more details are referred to classical books[@Ray; @book-intvals; @Neum] or to the nice Wikipedia page[@web-intvals]. Here we only mention the notation used in the rest of this article: symbols like $x$, $p$, or $f$ are for real variables, parameters or functions, while their interval counterparts will be written in different font as $\mathsf{x}$, $\mathsf{p}$, or $\mathsf{f}$, respectively.Greek letters always represent real quantities.
Connection with probability and statistics
------------------------------------------
At the first sight, there cannot be any direct connection between interval calculus and probability or statistics. While intervals are always guaranteed to contain the true values then probability and statistics operate rather with imprecise quantities, describing them in terms of most likely (expected) values and estimating their standard deviations. On the other hand, the very existence of well known term*confidence interval* strongly suggests that such connection *is*possible.
One might also ask why not to simply rewrite well known procedures, like Least Squares Method, into their interval formulations? This is indeed possible when experimental uncertainties are known as guaranteed intervals (i.e. containing true values with probability equal exactly to unity). Doing so we are lead to system of interval equations. But even in the simplest case, when all equations are linear, we encounter few serious problems. First, we have to decide which kind of solutions we are looking for – as there are many possible classes to choose from.In what follows, we will consider only the so called *united solutions* set.United solution set is an interval generalization of ordinary set of solution of a system of equations. In usual arithmetics, the vector $\vec{p}$ of unknowns belongs to the solution set iff for all considered equations the following equality holds: $$L_i(\vec{p}) = R_i(\vec{p}),$$ where $L_i(\cdot)$ and $R_i(\cdot)$ are left- and right-hand sides of equation $i$, respectively.
However, when operating with intervals, both $L_i(\cdot)$ and $R_i(\cdot)$ are not just real numbers, but intervals, each containing infinitely many numbers. In this case a subset ${\cal S}$ of searched parameters space certainly *does not belong* to the united solutions set when $$L_i({\cal S})\,\cap\,R_i({\cal S}) = \varnothing\quad {\rm for\ at\ least\ one\ } i$$ ($\varnothing$ is an empty set). It may be somewhat unexpected, but the opposite, i.e. $L_i({\cal S})\,\cap\,R_i({\cal S}) \ne \varnothing$ for all $i$does *not* guarantee that ${\cal S}$ contains at least one true solution of our system. In other words, ${\cal S}$ is only a set of *possible* solutions.
One more comment is in order here. It is rather unlikely that the solution set ${\cal S}$ is a single multidimensional interval. More often it is a rich composition of many ‘small’ intervals (boxes), sometimes counted in thousands. It is not a comfortable situation when computer memory (or disk space) required merely to store such a set greatly exceeds the storage needed for original data. Additionally, any simple operation on ${\cal S}$ becomes time-consuming task as it has to be performed on each member of set ${\cal S}$. This is probably the main reason why interval computations are still rare, even in cases when the observed data can be considered to be *guaranteed* intervals. Of course, one might use less precise description of solution set ${\cal S}$, say in form of intervals describing minimal and maximal values of each parameter in turn.The drawback is that set ${\cal S}$ will usually occupy only a very small part of so defined single big box.
In further consideration we will need only one fact from probability and statistics, namely the famous Chebyshev inequality (1874): $$\label{Cheb}
\Pr\,\left(\,\left|\,x^{\phantom{|}}\!-\mathrm{E}\left(x\right)\,\right|\,>\,
\xi\sigma\,\right)\,\leqslant\,\frac{1}{\xi^2}\quad\mathrm{valid\ for\ }\quad \xi> 1.$$ It quantifies the probability of large deviations of measured value $x$ from its expected value $\mathrm{E}(x)$. It is valid for any probability density function, if only $\mathrm{E}(x)$ and variance $\sigma^2$ both exist and are finite.
The algorithm
=============
Preliminaries
-------------
As usually, we start with a set of $N$ uncertain measurements $y_1\pm\sigma_1,\,y_2\pm\sigma_2,\,\ldots,\,y_N\pm\sigma_N$, obtained at the corresponding values of control variables $\mathsf{x}_1,\,\mathsf{x}_2,\,\ldots,\,\mathsf{x}_N$.Control variables, $\mathsf{x}$’s, are often just real numbers but may be multidimensional entities and/or uncertain as well. We also have a model $f$, containing $k$ unknown parameters $p_1,\,p_2,\,\ldots,\,p_k$ and relating uniquely every $y_i$ with $x_i$.The relation $f$ most often takes the form of algebraic equation $$\label{explicit}
y_i=f\left(x_i, p_1, p_2,\,\ldots,\,p_k\right),\quad\,i=1,\,\ldots,\,N$$ (one equation for each individual measurement $y_i$, taken at always the same, fixed set of unknown parameters $p_1, p_2, \ldots, p_k$).
Sometimes our problem is more complicated and cannot be written in explicit form, as in (\[explicit\]), but rather as an implicit formula $$\label{implicit}
f\left(x_i, y_i, p_1, p_2,\,\ldots,\,p_k\right)=0,\quad\,i=1,\,\ldots,\,N$$ For purely numerical reasons (see \[A\]) it may be desirable — if possible — to write relation (\[implicit\]) in still another, but equivalent form $$\begin{aligned}
\label{twosided}
f_{\mathrm L}\left(x_i, y_i, p_1, p_2,\,\ldots,\,p_k\right)\!\!\!&\!=\!&\!\!\!%
f_{\mathrm R}\left(x_i, y_i, p_1, p_2,\,\ldots,\,p_k\right)\\
&&\ \qquad\qquad i=1,\,\ldots,\,N\nonumber,\end{aligned}$$ where $f_{\mathrm L}$ and $f_{\mathrm R}$, treated separately, have all their (interval) arguments appearing at most once, i.e. without repetitions.
From now on, our measurements will be represented as intervals: $y_i\pm\sigma_i\,\rightarrow\,\mathsf{y}_i = \left[\,y_i-\xi\sigma_i,\
y_i+\xi\sigma_i\,\right]$, with $\xi$, called *extension factor*, equal to one unless noted otherwise. Note that in interval calculus the above range should be guaranteed to contain the true value with probability equal to exactly one. This requirement is satisfied only when $\sigma_i$ is equal to maximum absolute deviation, as specified by measuring instrument maker, and $\xi=1$.But even then, we may face the problem of outliers; either because our instrument is malfunctioning or due to undetected data transmission errors. In all other situations, when $\sigma_i$ is a standard deviation of the measurement $y_i$, even for arbitrarily large $\xi$ we have $$\mathrm{Pr}\,\bigl(\,(\mathrm{true\ value\ of}\ y(x_i))\in\,%
\left[\,y_i-\xi\sigma_i,\,y_i+\xi\sigma_i\,\right]\,\bigr) < 1$$ and the inequality is sharp.This observation may suggest that interval calculus is completely unsuitable for the kind of calculations we would like to perform.It will be shown below that such a view is unjustified.
Main idea
---------
Our idea is illustrated in Fig. \[straight\]. When presented a set of uncertain measurements supposed to lay on a straight line, we can quickly estimate its slope and offset by simply using a ruler. It is rather difficult to say which so obtained line is the ‘best,’ but after few trials we are able to estimate the sensible ranges of both relevant parameters.Our algorithm only formalizes those simple actions. Its main steps are:
1. Start with the box of all fitted parameters, large enough to contain the solution, and make it the first and only element of list $ \mathcal{L}$. Establish *unit lengths* for all searched parameters (for explanation see the end of section \[details\]).
2. Pick the largest box $\mathsf{V}$ from list $\mathcal{L}$ and remove it from list.
3. Bisect the box $\mathsf{V}$, by halving its longest edge, to obtain two offspring boxes, $\mathsf{V}_{\mathrm L}$ and $\mathsf{V}_{\mathrm R}$.
4. Perform admissibility tests on boxes $\mathsf{V}_{\mathrm L}$ and $\mathsf{V}_{\mathrm R}$. Discard box, if it appears certainly unsuitable, or append it to the list $\mathcal{L}$ otherwise.
5. Stop when the list $\mathcal{L}$ is empty or contains only elements being either small or certified boxes. Otherwise go to step $1$.
Details of operations {#details}
---------------------
- Bisection means halving the longest edge of the box $\mathsf{V}$.More precisely: if $V_{c}^{m}$ is the center of the longest edge $m$ of the original box $\mathsf{V}$ then $\mathsf{V}_{\mathrm L}^{m}=[\underline{\mathsf{V}}^{m}, V_{c}^{m}]$ and $\mathsf{V}_{\mathrm R}^{m}=[V_{c}^{m}, \overline{\mathsf{V}}^{m}]$, while all the remaining components ($\ne{m}$) are exact copies of those of parent box $\mathsf{V}$.This way $\mathsf{V}=\mathsf{V}_{\mathrm L}\,\cup\,\mathsf{V}_{\mathrm R}$, what means that no point within the original search area will ever be missed by the algorithm. On the other hand we also have $\mathsf{V}_{\mathrm L}\,\cap\,\mathsf{V}_{\mathrm R}\ne\varnothing$, since offspring boxes always share a common face. We will need this feature at later stages.
- Testing means counting ‘hits.’ By ‘hit’ we understand the event$\mathsf{f}\left(\mathsf{x}_i, \mathsf{V}\right)\,\cap\,\mathsf{y}_{i}\,\ne\,\varnothing$ (compare with formula (\[explicit\])), or non-empty intersection of $\mathsf{f}_{\mathrm L}\left(\mathsf{x}_i, \mathsf{y}_i, \mathsf{V}\right)$ and $\mathsf{f}_{\mathrm R}\left(\mathsf{x}_i, \mathsf{y}_i, \mathsf{V}\right)$ — when formula (\[twosided\]) is at work. Box $\mathsf{V}$ should pass the test, when number of hits exceeds number of misses (empty intersections). But we shouldn’t ignore constraints, if there are any. Violating of at least one constraint immediately invalidates the box, if only this violation is certain.
For example, if we require two unknown parameters $p_m$ and $p_n$ to be equal, then the investigated box $\mathsf{V}$ should be discarded only when the intersection of its corresponding components is empty: $\mathsf{V}^{m}\cap\mathsf{V}^{n}=\varnothing$.Non-empty intersection means that our constraint has a chance to be satisfied in current box and therefore $\mathsf {V}$ should be retained for further investigations (if there is no other certainly violated constraint within this box, of course).
It may happen, when the task is to satisfy formula (\[explicit\]), that in given box $\mathsf{V}$ the following inclusion occurs: $\mathsf{f}\left(\mathsf{x}_i, \mathsf{V}\right) \subseteq \mathsf{y}_i$, for $i=1, 2, \ldots, N$, whenever $\mathsf{f}\left(\mathsf{x}_i, \mathsf{V}\right)\,\cap\,\mathsf{y}_{i}\,\ne\,\varnothing$. Such a box may be safely called *certified* as it needs not to be bisected further. This is because any subset of $\mathsf{V}$ also satisfies this inclusion. It is therefore a good idea to put such box aside and never test it again.
- *Small boxes* are those with diameter not exceeding unity. But how can we compare searched parameters of different nature, expressed in various units, like meters, degrees or seconds? For this we need to arbitrarily establish *unit lengths*, individually for each searched parameter. This way the lengths of all edges of our boxes will become dimensionless numbers. Adopted unit lengths should not exceed accuracies we expect to get, but making them too small will result in significant increase of computation time.
What next?
----------
### $\mathcal{L}$ is empty
We are done, but certainly not satisfied, when $\mathcal{L}$ is empty.What could be the reason not to obtain any result at all? Apart from obvious mistake of processing data obtained from different experiment, or mistakenly searching parameters outside their true ranges, we can think about the validity of our formula $f$. Maybe our model $f$ is simply too rough and is therefore unable to replicate observed features? Maybe it is only applicable within some range of control parameters and not outside it?
Less obvious reason for emptiness of the list $\mathcal{L}$ is perfectly adequate model evaluated on *too precise* data.By *too precise* data we mean those with grossly underestimated uncertainties, including the case when they are being presented as equal to zero to the algorithm.It is evident that cheating doesn’t pay.
Yet, the case of *too precise* data need not to be completely at lost. It is possible to get $\mathcal{L}\ne\varnothing$ in another run, with significantly enlarged unit lengths. Of course, the standard deviations of so obtained results may be very disappointing. This is the price for poor quality/inconsistent measurements.
When none of the above mentioned cases applies and the list $\mathcal{L}$ in nevertheless empty, then we can conclude that our model $f$ is *certainly* inadequate to the problem under investigations.
### $\mathcal{L}$ is non-empty
In this case we should check whether the convex hull of all boxes has no common parts with any face of the initial search domain.The presence of some boxes at the original boundary usually means that either the initial search domain was too small (not covering all solutions) or the corresponding unit length was selected too large.The second possibility will certainly occur in ‘pathological’ cases, when we don’t want to evaluate some parameter(s) and therefore deliberately and forcibly fix their values by setting widths of their search intervals to zero.There exist still other possibilities, to be discussed later, but in any case the algorithm should issue a detailed warning after encountering such a situation.
So, there is at least one box present on the list $\mathcal{L}$.Yet even a single box contains infinite number of solutions, what is in sharp contrast with results delivered by other point-type routines, Monte Carlo investigations, or even population-based approaches, like genetic algorithms. By the way, the list $\mathcal{L}$ with exactly one member will be an exception rather than the rule.More often $\mathcal{L}$ will consist of much more boxes, perhaps counted in thousands. How should we report our results?
Reporting results
-----------------
Well, first thing is to check how many solutions were found.The obtained boxes need not to make a simply connected set, they may form few disjoint clusters. Is it possible? Yes, think about fitting two non-overlapping spectral peaks (their positions, amplitudes and half-widths) located on noisy background. Without constraints we will get *two* solutions, showing exactly the same two peaks but in different order.
For this reason, the next step should be to recover the individual simply connected components, that is clusters of neighboring boxes.It is the place where the property $\mathsf{V}_{\mathrm L}\,\cap\,\mathsf{V}_{\mathrm R}\ne\varnothing$ will be exploited extensively. Indeed, this part of algorithm often appears the most time-consuming one. Only after this step is completed, it is possible to process/report each solution, one after another.
### Original uncertainties are guaranteed limits
In this case the final processing is rather simple. All we have to do is to compute the convex hull of each cluster. This way the guaranteed lower and upper limit of each searched parameter are determined, in full concordance with usual rules of interval computations.Final report will typically include extremal values for each parameter as well as centers of those intervals.
One can think about the hulls of certified boxes making each cluster.Looks like that by doing so, we may get ‘more accurate’ (tight) estimates of searched parameters, see Fig. \[wrapping\].Unfortunately, this is a bad idea. First, the cluster might contain no certified boxes at all! This will almost certainly happen whenever adopted unit lengths are too large. Secondly, we will loose the rigor of interval computations.
In some applications it is essential to know guaranteed tolerances of searched parameters. If so, then we need as a solution the ‘biggest’ box covering certified boxes, and only those boxes. However, the solution having this property is not unique — it depends on what the word ‘biggest’ means in every particular case. In practice, some parameters may be easy to control, while others only with excessive cost, and so on. Thus it may be a matter of user preferences which solution is preferable. The case of linear equations was extensively studied by Shary[@Shary], and similar problems – mostly related to robotics – were presented at numerous conferences. Nevertheless, this topic is is out of scope of the current paper.
### Original uncertainties and standard deviations
At first sight, the rigor of interval computations becomes doubtful, when we operate on data expressed in familiar form, i.e. as a pair: measurement result, $y$, and its standard deviation, $\sigma_y$. This is because no interval of type $\left[y-\xi\sigma_y, y+\xi\sigma_y\right]$ guarantees that the true value, $\tilde{y}$, is located within these limits, no matter how large (but finite) is the positive extension factor $\xi$.In some cases we can find the exact value of probability of such event, most notably when the measured quantity, $y$, follows normal (Gaussian) distribution, or — generally — when the probability distribution is known (preferably in analytical form).In all other cases we can use the already mentioned Chebyshev inequality (\[Cheb\]) to rigorously estimate probabilities of interest.
1ex Before we proceed further, let us explain our point of view on data fitting process, to our best knowledge never presented before.In short: data fitting process is like a final step of any ordinary (direct) measurement.And here is why. During direct measurements, the measuring instrument, say a ruler, seems to deliver immediate answer how long is the investigated object. In reality things are slightly more complicated, even in simple cases like that one. Here we have a light, which is reflected, both from our instrument and from the object under study, and which finally reaches our eye.Still later on, our impression of reality is transferred to our brain, which decides the final outcome of the measurement. This complicated process may be formally described as a superposition of several transformations. So, between the input signal(s) and the final numerical outcome(s) there are intermediate steps, some of them performed inside measuring instrument. Generally, all what this machinery is doing is selection of a single number from the real line and presenting it as the final result. Such an action is usually repeated several times, what makes possible to estimate the most probable value of measured quantity and its standard deviation.
This is exactly what our algorithm is doing, with the small exception that its ‘measurements’ are repeated infinitely many times. Thanks to this observation we can think about every point within the obtained cluster of boxes as being the result of a single measurement – why not? This way our algorithm becomes a last part in chain of transformations normally performed by measuring instrument. What remains, usually the experimenter’s task (and her computer, perhaps), is to derive simplest statistical properties of the bunch of measurements. But now finding expected values of searched parameters, their standard deviations, and correlation coefficients as well, is a next to trivial task. Best of all, it can be done without any tricks, or simplifications, just by following appropriate definitions.
In conclusion: for most popular types of measurements we are able to find and present not only the extremal values of fitted parameters but also highly desired, reliable estimates of their standard deviations and correlation coefficients, for every separate cluster in turn.
1ex *Final note.* It is tempting to treat on unequal basis the boxes differing by number of hits. Assigning higher weights to points located in boxes with higher number of hits will certainly result in smaller values of standard deviations of fitted parameters – but is it well justified? At this moment this remains an open question.
The meaning of extension factor
===============================
Before the algorithm starts, it needs to know its input data in interval form. This is easy when input data are known within guaranteed limits.No extension factor is needed then, correct intervals are already known.
Let’s discuss all other cases now. As already mentioned, no value of extension factor $\xi$ makes certain that true value is covered by so created interval. Assume that we know nothing about the distribution(s) ruling our measurements, except that its average value and variance do exist and are both finite. Our goal is to find such values of unknown parameters that resulting curve (or a hyper-surface in multidimensional case) hits more than half of our uncertain measurements. Suppose such set of parameters indeed makes sense (exists). If so, then hits are binomially distributed, with probability $p$ of success in a single trial equal to the probability of true value being located within the inspected interval. One might think, that all what is needed to hit more than half of measurements is to set $(\xi^2=2)\,\equiv\,(\xi=\sqrt{2})$ in Chebyshev inequality.Unfortunately, this guess is correct only for a single measurement or for infinitely many measurements. In all other cases we need to find smallest $\,0 < p < 1$ satisfying inequality: $$\label{Poiss}
\sum_{k=0}^{\lfloor\,(N+1)/2\,\rfloor -1} {{N}\choose{k}}\,p^k\,q^{N-k}\, \leqslant
\sum_{k=\lfloor\,(N+1)/2\,\rfloor}^{N} {{N}\choose{k}}\,p^k\,q^{N-k}$$ where $N$ is the number of uncertain data points, $q=1-p$, and $\lfloor\,\cdot\,\rfloor$ means integer part of the argument. From Chebyshev inequality (\[Cheb\]) we immediately have $1-p \leqslant 1/\xi^2$ and thus the minimal value of extension factor $\xi =1/\sqrt{1-p}$.
In order to satisfy the inequality (\[Poiss\]), one has to resort to numerical calculations. The results, for $N\leqslant{30}$, are presented in Table \[tabelka\], together with minimal required probability of success (hit) in single attempt ($p=\,$min. Pr). Somewhat unexpectedly, the sequence $\left\{\xi_N\right\}$ appears to consist of two distinct subsequences: one for even and the other for odd $N$, as illustrated in Fig. \[sequence\]. Both subsequences are decreasing and converge to the same limit: $\lim_{N\rightarrow\infty}\,\xi_N=\sqrt{2}$.For practical purposes extension factors may be approximated by $$\begin{aligned}
\xi_N&\approx&\sqrt{2}\left(1+\frac{3}{2N}\right)\qquad\mathrm{for~}\ N\ \mathrm{even}\\
\xi_N&\approx&\sqrt{2}\left(1+\frac{1}{N}\right)~\ \qquad\mathrm{for~}\ N\ \mathrm{odd}\end{aligned}$$ Both approximations are from below, with the ratio of true extension factor and its approximated value dropping below $1.0010$ for $N\geqslant{35}$ ($N$ odd) or $N\geqslant{54}$ ($N$ even).
![Extension factor, $\xi$ (decreased by $\sqrt{2}$), *vs.* the number of fitted data entities, $3\leqslant{N}\leqslant{1000}$.Two distinct subsequences are clearly visible, both well approximated by straight lines. Note the logarithmic scale in both coordinates. []{data-label="sequence"}](extfact.eps){width="0.7\columnwidth"}
It is worth noticing that for $N\in[2,6]$ (see Table \[tabelka\]) we should use , i.e. we should extend original intervals, even for normally distributed uncertainties. One might think that for normally distributed data it should make a sense to shrink their intervals (apply $\xi<1$) whenever $N\geqslant{7}$. Unfortunately, this is not recommended, as in such case we are loosing the solid ground of Chebyshev inequality (\[Cheb\]). In effect, to avoid this unreasonable temptation, we leave the last column of Table \[tabelka\] mostly unfilled.
Note also the unexpected relation $$\Pr\,(N-1) > \Pr\,(N) < \Pr\,(N+1)$$ valid for odd $N$. Analogous inequality is satisfied by $\xi$, what suggests potentially more tight estimates for data sets containing odd number of measurements — at least when using the proposed approach. In fact, we are dealing here with the old truth: odd number of voters will never generate a tie what might happen when the number of voters is even.Another surprising observation is that the general idea *more is better* indeed works, but only for $N\,\geqslant\,4$.
1ex *Final remark:* Some measurements deliver not just a single number but rather few components at once, say two or three components of a vector. The extension factor, $\xi$, should be modified in such situations accordingly. Namely, it should be replaced with $\xi\leftarrow\,\xi^\frac{1}{D}$, where $D$ is the number of individual components making single measurement. Not doing so will result in needlessly overestimated uncertainties of fitted parameters.
$N$ $N_{\mathrm{min}}$ $\xi$ unknown min. Pr $\xi$ Gauss
----- -------------------- --------------- ---------- -------------
1 1 $ \sqrt{2}$ 0.500000
2 2 1.847759 0.707107 1.051801
3 2 2.201664 0.793701 1.263802
4 3 2.507033 0.840896 1.408092
5 3 1.785116 0.686190 1.007259
6 4 1.944591 0.735550 1.115935
7 4 1.657220 0.635884
8 5 1.766335 0.679481
9 5 1.594986 0.606915
10 6 1.678127 0.644900
11 6 1.558150 0.588110
12 7 1.625364 0.621471
13 7 1.533792 0.574923
14 8 1.590223 0.604557
15 8 1.516488 0.565167
16 9 1.565126 0.591773
17 9 1.503560 0.557658
18 10 1.546302 0.581774
19 10 1.493535 0.551699
20 11 1.531658 0.573738
21 11 1.485532 0.546856
22 12 1.519939 0.567140
23 12 1.478997 0.542843
24 13 1.510348 0.561625
25 13 1.473559 0.539463
26 14 1.502354 0.556947
27 14 1.468964 0.536577
28 15 1.495588 0.552929
29 15 1.465029 0.534084
30 16 1.489787 0.549441
: Extension factor $\xi$ *vs.* the number of collected measurements $N$, valid for unknown distribution of their uncertainties.$N_{\mathrm{min}}=1+\lfloor\,N/2\,\rfloor$ is the required minimal number of hits. The column marked as ‘min. Pr’ shows minimal probability of a single measurement to guarantee that $\mathrm{Pr}%
\left(N_\mathrm{hit}\geqslant{N}_\mathrm{min}\right)>1/2$.The last column shows extension factors for normally distributed measurements. Values below unity are not shown.
\[ext\] \[tabelka\]
An example
==========
As an example we present how our algorithm deals with experimental data on Newton gravitational constant $G$. This fundamental physical constant still remains the least precisely known[@troublesomeG]. The 11 original measurements, presented in CODATA report[@CODATA] and not repeated here, are not quite consistent, as can be seen from shaded area in Fig. \[G-covering\]. No more than four of them overlap anywhere within the full range, with small interval $[6.67520,\,6.67532]\,\times{10}^{-11}\,$m$^3$kg$^{-1}$s$^{-2}$ being not covered at all, and another one, $[6.67315,\,6.67324]\,\times{10}^{-11}\,$m$^3$kg$^{-1}$s$^{-2}$, near the center of ‘solid body,’ containing just one measurement.The CODATA committee found no good reason to discard any of those $11$ measurements, so they decided to extend all uncertainties by the factor 14 before proceeding. The resulting bell-shaped ‘coverage curve,’ which may be considered to be proportional to the probability density function, is also shown in Fig. \[G-covering\]. It remains unclear what were the other steps of procedure, but the final result, the recommended value of Newton gravitational constant, is: $G = 6.67384(80)\times{10}^{-11}\,$m$^3\,$kg$^{-1}\,$s$^{-2}$.
We find only the narrow range $[6.673449,\,6.673636]\,\times{10}^{-11}\,$m$^3\,$kg$^{-1}$s$^{-2}$ being covered by minimally required majority of (expanded) measurements ($6$ out of $11$). The result produced by our algorithm is: $G = 6.673542(54)\times{10}^{-11}\,$m$^3\,$kg$^{-1}\,$s$^{-2}$.Not surprisingly our result is roughly 14 times more precise.
![Coverage of the search domain with original measurements (shaded part), with data modified by extension factor $\xi=1.558150$ (shifted up by $2$ hits for clarity) – as appropriate for $N=11$ measurements of unknown origin, and following methodology of CODATA 2010 report, i.e. with $\xi=14$. []{data-label="G-covering"}](G-covers.eps){width="0.7\columnwidth"}
Advantages and disadvantages
============================
Advantages:
- recovering unknown parameters from implicit dependencies is equally easy as from explicit formulas
- robustness against outliers, up to $50\,$%
- obtained uncertainties (variances) are never underestimated
- no need to apply any ‘error propagation law,’ often questionable
- high flexibility. The same algorithm may be used for detecting outliers, hypotheses testing, solving systems of nonlinear equations, possibly containing uncertain parameters, or just for simulations of complex, implicit models with uncertain parameters.The presented approach may be also useful in metrology, for inter-laboratory comparisons.
- uncertainties in both coordinates do not pose any problem and are handled naturally, including non-linear and implicit cases. In fact, we have found in literature only two articles[@xy; @Krystek] describing straight line fitting with uncertainties in both variables; general non-linear cases seem not to be discussed at all.
Disadvantages:
- worst case complexity is exponential in the number of fitted parameters, thus
- impractical when the number of unknowns is large
- the computed standard deviations may be overestimated by unknown factor.
Conclusions
===========
It is hoped that the presented algorithm will soon replace a great deal of existing optimization procedures. In author’s opinion, interval computations deserve to become soon as familiar to experimentalists as are complex numbers to electrical engineers.
Brief introduction to interval computations {#A}
===========================================
Some definitions and important properties
-----------------------------------------
An *interval* ${\mathsf x}$ is a compact and finite subset of a real axis: $${\mathsf x} = \left\{\,\mbox{\mm R}^{\phantom{!}}\!\ni x:\,
\underline{\mathsf x} \leqslant x \leqslant \overline{\mathsf x}\,
\right\}\, \mathop{=}^{\mathrm{def}}\,
\left[\,\underline{\mathsf x},\,\overline{\mathsf x}\,\right]\,\subset\,\mbox{\mm R},$$ where both $\underline{\mathsf x}$ and $\overline{\mathsf x}$ are finite.It may be thought to be a representation of a real number, certainly located somewhere between $\underline{\mathsf x}$ and $\overline{\mathsf x}$, inclusive, but unknown otherwise. A special case is $\mathsf{x} = \left[\,a,\,a\,\right]$ (a.k.a. *thin interval* or *singleton*), identified with the real number $a$. The set of all intervals is usually denoted as $\mbox{\mm IR}$.
It is easy to define interval counterparts of ordinary arithmetic operations: $$\begin{aligned}
\mathsf{x}+\mathsf{y}&=& \left[\,\underline{\mathsf{x}}+\underline{\mathsf{y}},\,
\overline{\mathsf{x}}+\overline{\mathsf{y}}\,\right]\\
\mathsf{x}-\mathsf{y}&=& \left[\,\underline{\mathsf{x}}-\overline{\mathsf{y}},\,
\overline{\mathsf{x}}-\underline{\mathsf{y}}\,\right]%
\qquad\left(\,\mathsf{a}\!\ne\!0\,\Rightarrow\,\mathsf{a}\!-\!\mathsf{a}\!\ne\!0\ %
\right) \\
\mathsf{x}\,\cdot\,\mathsf{y}&=&\left[\,\min\,{\mathcal Z},\, \max\,{\mathcal Z}\,\right]\end{aligned}$$ where ${\mathcal Z}$ is a four–element set: $\mathcal{Z}= \left\{\,
\underline{\mathsf{x}}\,\underline{\mathsf{y}},\,
\underline{\mathsf{x}}\,\overline{\mathsf{y}},\,
\overline{\mathsf{x}}\,\underline{\mathsf{y}},\,
\overline{\mathsf{x}}\,\overline{\mathsf{y}}\,\right\}$. Division is defined, for $\mathsf{y}\,\notni\,0$ as: $\mathsf{x}\,/\,\mathsf{y} = \mathsf{x}\cdot{1/\mathsf{y}},$ where $1/\mathsf{y}=
\left[\,1/\overline{\mathsf y},\,1/\underline{\mathsf y}\,\right]$, and remains undefined otherwise (as usually). It may be checked that so defined arithmetic operations produce all possible results of $x\,\boxdot\,y$ for any pair $(x,y)$ satisfying $x\,\in\,\mathsf{x}$ and $y\,\in\,\mathsf{y}$, and *only* those results (here $\boxdot$ stands for any of $+$, $-$, $\cdot$ or $/$). However, more complicated arithmetic expressions may happen to overestimate the true range. Specifically, we generally have: $\mathsf{x}\,\left(\mathsf{y}+\mathsf{z}\right)
\subseteq\, \mathsf{x}\,\mathsf{y}+ \mathsf{x}\,\mathsf{z}$ for $\mathsf{x, y, z}\,\in\,\mbox{\mm{IR}}$, not the equality. Nevertheless, the following theorem holds[@Ray]: 1 ex Theorem (*Fundamental Theorem of Interval Arithmetic*)\
Let $f(x_1, x_2 , \ldots, x_n)$ be an explicitly defined real function. Then evaluating $f$ ‘in interval mode’ over any interval inputs $\left(%
\mathsf{x}_1, \mathsf{x}_2, \ldots, \mathsf{x}_n\right)$ is guaranteed to give a set $\mathsf{f}$ that contains the range of $f$ over those inputs. 1 ex The above theorem is true, but in practice we often obtain overestimated results, i.e. intervals wider than necessary. To avoid such undesired situations, we should — whenever possible — write complex interval expressions in form with each interval variable appearing exactly once.For example, to compute the resistance $R$ of two resistors $R_1$ and $R_2$, connected in parallel, we normally use the formula $R=R_1\cdot{R}_2/(R_1+R_2)$.When $R_1$ and $R_2$ are uncertain, it is better to compute their equivalent resistance as $\mathsf{R}=\left(1/\mathsf{R}_1 +
1/\mathsf{R}_2\right)^{-1}$.
There is another subtlety, not mentioned until now. Our algorithm extensively exploits the ‘obvious’ property: $\mathsf{x} \subset \mathsf{y} \Longrightarrow
\mathsf{f}(\mathsf{x}) \subseteq \mathsf{f}(\mathsf{y})$, which need not to be true. Functions $\mathsf{f}$ satisfying this relations are called *monotonously inclusive*. At this place it is enough to say that all ordinary (‘calculator’) functions have this property. Nevertheless exceptions sometimes happen and among the suspected functions are those containing min and/or max.
When dealing with interval computations on a computer, that is with finite precision, it is also very important to properly round all the intermediate results, as well as the final one. Proper rounding means outwards rounding, i.e. lower (left) endpoint has to be rounded towards minus infinity, while the other one — towards plus infinity. Fortunately, the existing interval software packages have this feature built in. Sometimes, however, it is highly recommended to perform such an action explicitly.
Interval $n$–dimensional vectors, i.e. objects belonging to Cartesian product $\mbox{\mm IR}^n=\mbox{\mm IR}\times\mbox{\mm IR}%
\times\ldots\times\mbox{\mm IR}$, are often called *boxes*, for obvious reasons. We will need to know how large are our boxes. The box’s *diameter* is a real number, defined as the length of its longest edge: $$\label{diam}
\mathrm{diam}\,\left(\,\mathsf{x}_1,\,\ldots\,\mathsf{x}_{\mathrm{N}}\,\right)%
= \max\,\left(\,\overline{x}_1-\underline{x}_1,\,\ldots\,,\,%
\overline{x}_{\mathrm{N}}-\underline{x}_{\mathrm{N}}\,\right)$$
Set theory operations on intervals {#B}
==================================
Intervals are sets and therefore also the set–theory operations may be performed on them. Here we sketch only two:
- intersection $$\mathsf{a}\,\cap\,\mathsf{b} = \left[\max\,\left(\underline{\mathsf{a}},
\underline{\mathsf{b}}\right),\,
\min\,\left(\overline{\mathsf{a}},\overline{\mathsf{b}}\right)\right]$$ When $\mathsf{a}$ and $\mathsf{b}$ happen to be disjoint, then the above formula will necessarily produce illegal result, not an element of $\mbox{\mm IR}$, i.e. the one with left endpoint value higher than right endpoint. If this is the case, then we should replace the so obtained result with an *empty interval*, see below.
- convex hull $$\mathsf{hull} \left(\mathsf{a},\mathsf{b}\right) =
\left[\,\min\,\left(\underline{\mathsf{a}},\,\underline{\mathsf{b}}\right),\,
\max\,\left(\overline{\mathsf{a}},\overline{\mathsf{b}}\right) \right]$$ This operation is an interval counterpart of union of two sets, with result being again an element of $\mbox{\mm IR}$.We always have $\mathsf{a}\cup\mathsf{b}\subseteq\,
\mathsf{hull} \left(\mathsf{a},\mathsf{b}\right)$, with equality occurring only for arguments having non–empty intersection. Therefore we can say that interval $\mathsf{hull}$ possibly overestimates ordinary union of $\mathsf{a}$ and $\mathsf{b}$.
- empty interval\
It is easy to see that arbitrary illegal interval $\mathsf{a}=%
\left[\underline{\mathsf{a}}, \overline{\mathsf{a}}\right]$with $\underline{\mathsf{a}} > \overline{\mathsf{a}}$, doesn’t guarantee the satisfaction of the otherwise obvious property $\mathsf{a}\,\cup\,\mathsf{b} = \varnothing\,\cup\,\mathsf{b} =%
\mathsf{b}$, as one might expect.Therefore we need to define an empty interval in a special form, the one making possible to always obtain correct results during computer calculations. The suitable choice is $$\varnothing = [\,HUGE,\,-HUGE\,],$$ where $HUGE>0$ is the largest machine number.
Acknowledgments {#acknowledgments .unnumbered}
===============
Current author’s interest in interval computations started after Ramon E. Moore published the paper[@seminal] (1977) showing the power of interval Newton method applied to nonlinear equations. Author is deeply indebted to him as well as to R. Baker Kearfott, Vladik Kreinovich, Sergey P. Shary and Sergei I. Zhilin for words of encouragement and occasional e-mail discussions on interval-related problems.
For long time the interval computations were developed with no visible link to physical problems, until the largely overlooked conference presentation[@SIAM2002] given in 2002.This article presents a working algorithm following the ideas sketched there.
This work has been done as a part of author’s regular duties in the Institute of Physics, Polsh Academy of Sciences, and was not funded otherwise.
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---
abstract: 'Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants $\tau$, $\varepsilon$, and $\Upsilon$. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.'
address: 'Samantha Allen: Department of Mathematics, Indiana University, Bloomington, IN 47405 '
author:
- Samantha Allen
bibliography:
- 'mybiblio.bib'
title: Using Secondary Upsilon Invariants to Rule out stable equivalence of knot complexes
---
Introduction
============
In general, the study of torus knots and their concordance invariants has been a frequent topic of investigation. One early highlight was Litherland’s proof of the independence of torus knots using Tristram-Levine signature functions in [@Litherland]. Because of their role in studying algebraic curves, research on invariants of torus knots continues. In particular, the Ozsváth–Stipsicz–Szabó Upsilon function has been used in [@borodzik-hedden] and [@feller]. Recently, Feller and Krcatovich (in [@feller-krcatovich]) determined relationships among the Upsilon functions of torus knots. Our goal here is to use the secondary Upsilon invariants, defined by Kim and Livingston in [@kim-livingston], to show that these relationships do not extend to stabilized knot complexes of torus knots.
Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. In [@hom2], Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. The concordance invariants $\tau$, $\varepsilon$, $\Upsilon$, $\U^2$ are all invariants of the stable equivalence class (see [@hom1; @hom2; @oss; @os2; @kim-livingston]). We give an example of a pair of torus knots which have identical Upsilon invariants (by Feller and Krcatovich [@feller-krcatovich]) but differing secondary Upsilon invariants, and thus have knot complexes which are not stably equivalent.
\[main thm\] The knot complex $\cfk ^{\infty}(T(5,7))$ is not stably equivalent to the knot complex $\cfk ^{\infty}(T(2,5)\cs T(5,6))$.
Similar procedures show that $\cfk ^{\infty}(T(7,9))$ is not stably equivalent to $\cfk ^{\infty}(T(2,7)\cs T(7,8))$, and, in fact, the following general theorem holds.
\[general thm\] For all $p\geq 5$ odd, the knot complex $\cfk ^{\infty}(T(p, \, p+2))$ is not stably equivalent to $\cfk ^{\infty}(T(2,p)\cs T(p, \, p+1))$.
Furthermore, in their original paper introducing the secondary [*Upsilon*]{} invariants, Kim and Livingston [@kim-livingston] showed that $\U ^2$ is stronger than $\U$ for a single pair of knots, as well as a family of complexes which are not known to be knot complexes. We give the first example of a family of knots for which $\U ^2$ is stronger.
Knot complexes, $\cfki $ {#basics}
========================
To each knot $K \subset S^3$, we can associate a chain complex $\cfki$ (see [@os2]). It is equipped with a grading (called the [*Maslov*]{} grading) having the property that the boundary map decreases gradings by 1. The complex $\cfki$ is also bifiltered — each element $x$ has an [*algebraic*]{} and an [*Alexander*]{} filtration, denoted by $\alg (x)$ and $\alex (x)$ respectively. We consider these complexes up to bifiltered chain homotopy equivalence, denoted here by $\simeq$.
We can represent $\cfki$ as a diagram in the $(\alg,\alex)$–plane, as in Figure \[fig34-labeled\]. Let $\calb$ be a bifiltered basis for $\cfki$. Then each element $x\in \calb$ is represented by the point $(\alg (x), \alex (x))$ and the boundary map is indicated by arrows; for example, $\partial (b) = a+c$. (We discuss the case when two basis elements have the same filtration levels below.) Throughout this paper, when it will cause no confusion, we will use the $(i,j)$ coordinates interchangeably with the basis element at those filtration levels. Here white dots represent elements at grading 0 and black dots represent elements at grading 1.
The chain complex $\cfki$ also has a compatible $\F[U,U^{-1}]$ structure, where $\F$ is the field of two elements. The action of $U$ decreases both filtration levels by 1 and the Maslov grading by 2. To form the diagram of the full complex, we take all integer diagonal translates of the diagram shown. Unless we need to use the $U$-action, we will hide this structure in any diagrams.
In general, for a given knot $K$, $\cfki$ may have multiple elements at the same filtration levels. In this case, we use a grid to show the complex and each bifiltered basis element at filtration level $(i,j)$ will be shown in the unit square above and to the right of the point $(i,j)$. For $L$–space knots, however, the complex $\cfki$ is always a [*staircase*]{} complex, as in Figure \[fig34-labeled\]. In this case, the height and width of each step is determined by the gaps in the exponents of the Alexander polynomial of $K$; the Alexander polynomial can be written as $$\Delta_K(t)=\sum_{i=0}^d (-1)^i t^{a_i}$$ for some $\{a_i\}$ and $\cfki$ is a staircase of the form $$[a_1-a_0, a_2-a_1,...,a_d-a_{d-1}]$$ where the indices alternate between horizontal and vertical steps. For more details, see [@os3] and [@borodzik-livingston]. For example, for $K=T(3,4)$, $$\Delta_K(t)=1-t+t^3-t^5+t^6,$$ so $\cfki$ is of the form $$[1-0, 3-1, 5-3, 6-5] = [1,2,2,1]$$ as shown.
For use in later sections, we record some properties of the complex $\cfki$.
For knots $K,J\subset S^3$,
1. $\cfk ^{\infty}(K)\otimes \cfk ^{\infty}(J)\simeq\cfk ^{\infty}(K\cs J)$.
2. $\cfk ^{\infty}(-K)\simeq \cfk ^{\infty}(K)^*$. In terms of the diagram, $\cfk ^{\infty}(K)^*$ is obtained from $\cfk ^{\infty}(K)$ via a $180\degree$ rotation (each bifiltered basis element $(i,j)$ in $\cfk ^{\infty}(K)$ is represented by $(-i,-j)$ in $\cfk ^{\infty}(-K)$) and reversing all arrows.
3. if $K$ and $J$ are concordant knots, then there are acyclic complexes $A_1$ and $A_2$ such that $\cfk ^{\infty}(K)\oplus A_1 \simeq \cfk ^{\infty}(J)\oplus A_2$.
The third property leads to the following definition.
([@hom2]). Two knot complexes $\cfk ^{\infty}(K_1)$ and $\cfk ^{\infty}(K_2)$ are called [*stably equivalent*]{} if there are acyclic complexes $A_1$ and $A_2$ such that $\cfk ^{\infty}(K_1)\oplus A_1 \simeq \cfk ^{\infty}(K_2)\oplus A_2$.
See, for instance, [@os1] for a more detailed description of the $\cfki$ complex and [@hom2] for more discussion on stable equivalence.
The Upsilon Invariant
=====================
For a knot $K$ and $t\in [0,2]$, a filtration can be defined on $\cfki$ by the function $$\frac{t}{2}\,\alex (x)+\left(1-\frac{t}{2}\right)\alg (x).$$ Based on this filtration, we define a family of subcomplexes of $\calf_{t,s} \subset \cfki$ by $$\calf_{t,s}=\left\{x\in \calb \left| \left(\frac{t}{2}\alex (x)+\left(1-\frac{t}{2}\right)\alg (x)\right)\leq s\right\}\right.$$ for $t\in [0,2]$ and $s\in\R$ where $\calb$ is a bifiltered basis for $\cfki$. The subcomplex is independent of the choice of basis. Diagrammatically, the subcomplex $\calf_{t,s}$ is represented as a half-space with boundary line $$\frac{t}{2}j+\left(1-\frac{t}{2}\right)i = s$$ which has slope $m=1-\frac{2}{t}$ and $j$–intercept $b=\frac{2s}{t}$. We call this boundary line the [*support line*]{} and denote it by $\call_{t,s}$.
Let $$\gamma_K(t) = \text{min} \left\{s\, |\, H_0(F_{t,s}) \longrightarrow H_0(\cfki) \text{ is surjective} \right\}.$$ In [@oss], Ozsváth, Stipsicz, and Szabó define the knot invariant Upsilon $\U_K(t)$ for $t\in[0,2]$. In [@livingston1], it is shown that $\U_K(t)=-2\cdot \gamma_K(t).$
For knots $K,J \subset S^3$,
1. $\U_K(t)$ is piecewise linear.
2. $\U_{K\cs J}(t) = \U_{K}(t)+\U_{J}(t)$.
3. $\U_{-K}(t)=-\U_{K}(t)$.
4. If $K$ is slice, $\U_K(t)=0.$
Based on these properties, we get the following corollary:
If $K,J \subset S^3$ are concordant knots, then $\U_K(t)=\U_J(t)$. \[ups conc\]
The [*Upsilon*]{} invariant is also a stable equivalence invariant. Feller and Krcatovich gave the following relation.
\[feller-krcatovich\] Let $p<q$ be coprime integers. Then $$\U_{T(p, \, q)}(t)=\U_{T(p,\, q-p)}(t)+\U_{T(p, \, p+1)}(t).$$
Thus $\U$ cannot differentiate between the stable equivalence classes of $T(p,q)$ and $T(p,q-p)\cs T(p, \, p+1)$.
Secondary Upsilon Invariants
============================
In [@kim-livingston], Kim and Livingston defined the family of secondary [*Upsilon*]{} invariants $\U_{K,t}^{2}(s)$. For our purposes, we may restrict to a situation where the definition is simple. We will consider only knots $K$ such that $\Delta\U'_K(t)>0$ and we will define $\U_{K,t}^2(t)$ (removing the dependence on $s$ in the original definition) at $t$ which are singularities of $\U_K'(t)$.
Let $K\subset S^3$ and $t\in[0,2]$ and denote $$\calf_t:=\calf_{t, \gamma_K(t)}.$$ Let $t_0\in [0,2]$ be a singularity of $\U_K'(t)$. If $\U_K'(t)>0$, then, for $\delta$ small enough, the set of cycles which are not boundaries $\mathcal{C}_{t_0}$ in $\calf_{t_0}$ is split into two disjoint sets $\mathcal{C}_{t_0+\delta}$ and $\mathcal{C}_{t_0-\delta}$; the sets of cycles which are not boundaries in $\calf_{t_0+\delta}$ and $\calf_{t_0-\delta}$ respectively.
For each $t_0\in [0,2]$ which is a singularity of $\U_K'(t)$, let $$\gamma_{K,t_0}^2(t_0)=\text{min} \left\{ r \left|
\begin{array}{c}
\exists \,x_{\pm}\in \calc _{t_{0} \pm \delta} \text{ such that } x_- \text{ and } x_+ \text{ represent }\\ \text{the same class in } H_0(\calf_{t_0}+\calf_{t_0,r})
\end{array} \right\}\right..$$ Then the secondary [*Upsilon*]{} invariants defined by Kim and Livingston [@kim-livingston] are given by $$\Upsilon_{K,t_0}^2(t_0) = -2\cdot (\gamma_{K,t_0}^2(t_0)-\gamma_{K}(t_0)).$$
As an example, in Figure \[fig34-ups2\], we see that $\U'_{T(3,4)}(t)$ has a singularity at $t_0 = \frac{2}{3}$. Then $\calc_{t_0+\delta}=\{x_+\}$ and $\calc_{t_0-\delta}=\{x_-\}$ where $x_+$ and $x_-$ are represented by the points $(1,1)$ and $(0,3)$ respectively. Let $z$ be the point represented by $(1,3)$. We see that $\partial z= x_++x_-$, which implies $\U^2_{T(2,3),\frac{2}{3}}(\frac{2}{3})=-2\cdot \left(\frac{5}{3}-1\right)=-\frac{2}{3}$.
\[ups2 conc\] $\U^2_{K,t}(s)$ is a stable equivalence invariant.
Results
=======
We begin with a proof of Theorem \[main thm\], then use the same procedure to prove the general theorem.
The case of $p=5$
-----------------
$ $ To prove Theorem \[main thm\], we compute that when $t_0=\frac{4}{5}$$$\U^2_{T(5,7),t_0}(t_0)\neq \U^2_{T(2,5)\cs T(5,6),t_0}(t_0)$$ and then apply Theorem \[ups2 conc\].
$\U^2_{T(5,7),\frac{4}{5}}(\frac{4}{5})=-\frac{8}{5}.$ \[prop57\]
As in Section \[basics\], we can compute $\cfk ^{\infty}(T(5,7))$ from the gaps in the exponents of the Alexander polynomial: $$\begin{array}{cc}
\Delta_{T(5,7)}(t) = &
1-t+t^5-t^6+t^7-t^8+t^{10}-t^{11}+t^{12}-t^{13}\\&+t^{14}-t^{16}+t^{17}-t^{18}+t^{19}-t^{23}+t^{24}.
\end{array}$$ So $\cfk ^{\infty}(T(5,7))$ is a staircase complex of the form $$[1,4,1,1,1,2,1,1,1,1,2,1,1,1,4,1].$$ This yields a bifiltered graded basis $\calb$ for $\cfk ^{\infty}(T(5,7))$. See Figure \[fig57\] for the diagram for $\cfk ^{\infty}(T(5,7))$.
To prove the proposition, we first compute $\gamma\,_{T(5,7)}(\frac{4}{5})$. Recall that $$\gamma\,_{T(5,7)}\left(\frac{4}{5}\right)=\text{min} \left\{s\, |\, H_0(F_{\frac{4}{5},s}) \longrightarrow H_0(\cfki) \text{ is surjective} \right\}.$$ So we need to find the minimal $s$ such that $\call_{\frac{4}{5},s}$ contains a bifiltered basis element (or multiple elements) in $\cfk ^{\infty}(T(5,7))$. We compute that $\call_{\frac{4}{5},s}$ has slope $m=-\frac{3}{2}$ and $j$–intercept $b=\frac{5s}{2}$. In Figure \[fig57-ups2\], one can see that $\call_{\frac{4}{5},s}$ with minimal $s$ passes through the points $(1,8)$ and $(3,5)$. The $j$-intercept of this line is $\frac{19}{2}$ corresponding to an $s$ value of $\frac{19}{5}$. Thus $\gamma\,_{T(5,7)}(\frac{4}{5})=\frac{19}{5}$. Note that near $t=\frac{4}{5}$, the line $\call_{t,s}$ pivots around the two points $(1,8)$ and $(3,5)$. This causes a change in slope in $\U_K$ and so $t=\frac{4}{5}$ is a singulariy of $\U_K'$.
Now, we turn our attention to secondary Upsilon. We have that $$\calc_{\frac{4}{5}}=\{(1,8),(3,5)\},$$ and for small enough $\delta$ $$\calc_{\frac{4}{5}-\delta}=\{(1,8)\} \text{ and } \calc_{\frac{4}{5}+\delta}=\{(3,5)\}.$$ To determine $\U^2_{T(5,7),\frac{4}{5}}(\frac{4}{5})$, we compute how far the line of slope $-\frac{3}{2}$ needs to be moved so that the elements represented by $(1,8)$ and $(3,5)$ are homologous in $\calf_{\frac{4}{5},r}$. In the diagram we see that we need $\calf_{\frac{4}{5},r}$ to contain the elements represented by $(3,7)$ and $(2,8)$. The minimal $r$ which accomplishes this is $r=\frac{23}{5}$, as shown in Figure \[fig57-ups2\]. So we have that $\gamma^2_{T(5,7),\frac{4}{5}}(\frac{4}{5}) = \frac{23}{5}$. Thus $$\U^2_{T(5,7),\frac{4}{5}}(\frac{4}{5}) = -2\cdot \left(\frac{23}{5}-\frac{19}{5}\right) = -\frac{8}{5}.$$
$\U^2_{T(2,5)\cs T(5,6),\frac{4}{5}}(\frac{4}{5})<-\frac{8}{5}.$
We follow the same procedure as in the proof of Proposition \[prop57\]. The chain complexes $\cfk ^{\infty}(T(2,5))$ and $\cfk ^{\infty}(T(5,6))$ are both computed from their Alexander polynomials and then the tensor product is taken to produce $\cfk ^{\infty}(T(2,5)\cs T(5,6))$ as shown in Figures \[fig25+56-sep\] and \[fig25+56-tensor\].
[m[2.1in]{} m[2.1in]{}]{} &\
$\cfk ^{\infty}(T(2,5))$ &$\cfk ^{\infty}(T(5,6))$
\
Considering lines of slope $-\frac{3}{2}$ (corresponding to $t=\frac{4}{5}$), an analysis of the bifiltered basis elements in $\cfk ^{\infty} (T(2,5)\cs T(5,6))$ reveals that at $j$–intercept $b=\frac{19}{2}$ corresponding to $s=\frac{19}{5}$, and for no smaller $b$ or $s$, the line $\call_{\frac{4}{5},s}$ contains basis elements. In fact, the line contains exactly two bifiltered basis elements – those represented by $(1,8)$ and $(3,5)$ in Figure \[fig25+56-tensor\] and arising from the tensor product elements $(0,2)\otimes (1,6)$ and $(0,2)\otimes (3,3)$ respectively. Denote by $A$ the element represented by $(1,8)$ and $B$ the element represented by $(3,5)$.
Note that Theorem \[feller-krcatovich\] implies that the singularities of $\U'_{T(5,7)}(t)$ and $\U'_{T(2,5)\cs T(5, 6)}(t)$ occur at the same $t$–values. So $\frac{4}{5}$ is a singularity of $\U'$ and we compute the secondary Upsilon invariant at $t_0=\frac{4}{5}$. Now, suppose that $A$ and $B$ are homologous in $\calf_{\frac{4}{5},\frac{23}{5}}$. Then, since both $A$ and $B$ are at Maslov grading 0, there would be basis elements $x_1, x_2, ... \,, x_k$ in $\calf_{\frac{4}{5},\frac{23}{5}}$ at Maslov grading 1 such that $$\partial(b_1x_1+b_2x_2+\cdots +b_kx_k)=A+B
\label{boundary}$$ for some $b_i\in\Z_2$. We compute for all basis elements in $\cfk ^{\infty}(T(2,5)\cs T(5,6))$ which are at Maslov grading 1 (note that these must be tensor products of one element at grading 0 and one at grading 1), the value of $s$ for which the element is on the line $\call_{\frac{4}{5},s}$. See Figure \[tensor table\] for the full list of computations.
[| c || c | c || c |]{}
------------------------------------------------------------------------
------------------------------------------------------------------------
$\cfk ^{\infty}(T(2,5))_0\otimes \cfk ^{\infty}(T(5,6))_1$& $i$&$j$ &$s = \frac{2}{5}j+\frac{3}{5}i$\
------------------------------------------------------------------------
$(0,2)\otimes(1,10)$ & 1&12 & $27/5$\
$(0,2)\otimes(3,6)$& 3 & 8 & $25/5$\
$(0,2)\otimes(6,3)$& 6&5 & $28/5$\
$(0,2)\otimes(10,1)$& 10&3 & $36/5$\
$(1,1)\otimes(1,10)$& 2&11& $28/5$\
$(1,1)\otimes(3,6)$& 4&7& $26/5$\
$(1,1)\otimes(6,3)$& 7&4 & $29/5$\
$(1,1)\otimes(10,1)$& 11&2 & $37/5$\
$(2,0)\otimes(1,10)$& 3&10 & $29/5$\
$(2,0)\otimes(3,6)$& 5&6& $27/5$\
$(2,0)\otimes(6,3)$& 8&3& $30/5$\
$(2,0)\otimes(10,1)$& 12&1 & $38/5$\
------------------------------------------------------------------------
------------------------------------------------------------------------
$\cfk ^{\infty}(T(2,5))_1\otimes \cfk ^{\infty}(T(5,6))_0$& $i$ & $j$ &$s = \frac{2}{5}j+\frac{3}{5}i$\
------------------------------------------------------------------------
$(1,2)\otimes (0,10)$ & 1&12 & $27/5$\
$(1,2)\otimes (1,6)$ & 2&8 & $22/5$\
$(1,2)\otimes (3,3)$ & 4&5 & $22/5$\
$(1,2)\otimes (6,1)$ & 7&3 & $27/5$\
$(1,2)\otimes (10,0)$ & 11&2 & $37/5$\
$(2,1)\otimes (0,10)$ & 2&11 & $28/5$\
$(2,1)\otimes (1,6)$ & 3&7 & $23/5$\
$(2,1)\otimes (3,3)$ & 5&4 & $23/5$\
$(2,1)\otimes (6,1)$ & 8&2 & $28/5$\
$(2,1)\otimes (10,0)$ & 12&1 & $38/5$\
Our search results in exactly four elements within the desired range: $$x_1=(1,2)\otimes (1,6), x_2=(1,2)\otimes (3,3), x_3=(2,1)\otimes (1,6), \text{ and } x_4=(2,1)\otimes (3,3).$$ Taking the boundaries, we get: $$\begin{gathered}
\partial (x_1)=((0,2)+(1,1))\otimes (1,6) = (0,2)\otimes (1,6)+(1,1)\otimes (1,6)=A+(1,1)\otimes (1,6),\\
\partial (x_2)=((0,2)+(1,1))\otimes (3,3)=(0,2)\otimes (3,3)+(1,1)\otimes (3,3)=B+(1,1)\otimes (3,3),\\
\partial (x_3)=((2,0)+(1,1))\otimes (1,6)=(2,0)\otimes (1,6)+(1,1)\otimes (1,6),\\
\partial (x_4)=((2,0)+(1,1))\otimes (3,3)=(2,0)\otimes (3,3)+(1,1)\otimes (3,3).\end{gathered}$$ Notice that if Equation \[boundary\] is to hold, it must be that $b_1=b_2=1$. Since $$\partial(x_1+x_2)=A+B+(1,1)\otimes (1,6)+(1,1)\otimes (3,3),$$ we need $b_3=b_4=1$ in order to counteract the extra contributions of $x_1$ and $x_2$. However, $$\partial(x_1+x_2+x_3+x_4)=A+B+(2,0)\otimes (1,6)+(2,0)\otimes (3,3)$$ and we are left without options. So $A$ and $B$ are not homologous in $\calf_{\frac{4}{5},\frac{23}{5}}$. Thus $$\gamma^2_{T(2,5)\cs T(5,6), \frac{4}{5}}\left(\frac{4}{5}\right) > \frac{23}{5}$$ and so $$\U^2_{T(2,5)\cs T(5,6),\frac{4}{5}}(\frac{4}{5})=-2\cdot\left(\gamma^2_{T(2,5)\cs T(5,6), \frac{4}{5}}\left(\frac{4}{5}\right)-\frac{19}{5}\right)<-2\cdot \left(\frac{23}{5}-\frac{19}{5}\right)=-\frac{8}{5}.$$
Thus, since $\U^2$ is a stable equivalence invariant, it follows that $\cfk ^{\infty}(T(5,7))$ is not stably equivalent to $\cfk ^{\infty}(T(2,5)\cs T(5,6))$. This completes the proof of Theorem \[main thm\].
Proof of the general theorem
----------------------------
$ $ The proof of Theorem \[general thm\] follows similar steps to those of Theorem \[main thm\]. We will:
1. Construct the knot complex for $T(p,\,p+2)$
2. Compute $\U^2_{T(p,\,p+2),\frac{4}{p}}(\frac{4}{p})$
3. Construct the knot complex for $T(2,\,p)\cs T(p, \,p+1)$
4. Show that $\U^2_{T(2,\,p)\cs T(p, \,p+1),\frac{4}{p}}(\frac{4}{p}) <\U^2_{T(p,\,p+2),\frac{4}{p}}(\frac{4}{p})$
In steps $(1)$ and $(3)$ we will use the relationship between the semigroup generated by $p,q$ and the Alexander polynomial $\Delta_{T(p, \,q)}(t)$ given in [@borodzik-livingston]: $$\frac{\Delta_{T(p,q)}(t)}{1-t} = \sum_{s\in S_{p,q}}t^s.
\label{semigroup alex}$$ This relationship combined with the method given in Section \[basics\] describes the staircase.
In step $(2)$ we show that $t_0=\frac{4}{p}$ is a singularity of $\U'_{T(p,\,p+2)}(t)$ and identify the two pivot points in the complex at this $t$–value. Then we compute $\U^2_{T(p,\,p+2),\frac{4}{p}}(\frac{4}{p})$ from the staircase complex by showing that the two pivot points become homologous in $\calf_{\frac{4}{p}, \frac{p^2+p-7}{p}}$.
Finally, in step $(4)$, we see that, as in step (2), $t_0=\frac{4}{p}$ is a singularity of $\U'_{T(2,\,p)\cs T(p,\,p+1)}(t)$ and there are precisely two bifiltered basis elements acting as pivot points for $\U_{T(2,\,p)\cs T(p, \,p+1)}(t)$ at this $t$–value. In order to show that these two elements do not become homologous in $\calf_{\frac{4}{p}, \frac{p^2+p-7}{p}}$, we compute that, as in the proof of Theorem \[main thm\], there are precisely four bifiltered basis elements at Maslov grading 1 in this subcomplex and no combination of the four has boundary equal to the sum of the pivot points.
$ $**Step (1):** Let $S_{p,\,q}$ be the semigroup generated by $p$ and $q$, i.e., $S_{p,\,q}=\{np+mq\;|\;n,m\in \Z_{\geq 0}\}.$ We have that $$\arraycolsep=1.4pt\def\arraystretch{1}
\begin{array}{rll}
S_{p,\,p+2} =\{&0,&\\
&p, p+2&\\
&2p, 2p+2, 2p+4,&\\
&3p, 3p+2, 3p+4, 3p+6,&\\
&\vdots&\\
&np, np+2, np+4, ... , np+2n,&\\
&\vdots&\\
&(p-1)p, (p-1)p+2, ... , (p-1)p+2(p-1)\;\}\cup \Z_{\geq (p-1)(p+1)}.&
\end{array}
\label{S_p,p+2}$$ The following is a relationship between the Alexander polynomial of $T(p,q)$ and its semigroup, given in [@borodzik-livingston], $$\frac{\Delta_{T(p,q)}(t)}{1-t} = \sum_{s\in S_{p,\,q}}t^s,$$ in other words, $$\Delta_{T(p,q)}(t)=\sum_{s\in S_{p,\,q}}t^s-t^{s+1}.$$ Since $T(p, \, p+2)$ is an $L$–space knot, $\cfk^{\infty}(T(p, \, p+2))$ is then a staircase of the form $$[a_1-a_0, a_2-a_1,...,a_d-a_{d-1}]$$ where $d=p^2-1$ and $$\Delta_{T(p, \, p+2)}(t) = \sum_{i=0}^d (-1)^it^{a_i}.$$
Order the elements in the semigroup $S_{p,\,p+2}=\{s_0, s_1, s_2, ...\}$ such that $s_i<s_{i+1}$. Note that $S_{p,\,p+2}$ as shown in (\[S\_p,p+2\]) is in increasing order through the element $s_{i^*}=\left( \frac{p+1}{2} \right) p+2\left(\frac{p-1}{2}\right)$, and for $s_i\leq s_{i^*}$, $s_i-s_{i-1}>1$. So we have that $\cfk^{\infty}(T(p, \, p+2))$ is a staircase with initial portion: $$[(s_0+1)-s_0, s_1-(s_0+1), (s_1+1)-s_1, s_2-(s_1+1),..., (s_{{i^*}-1}+1)-s_{{i^*}-1}, s_{i^*}-(s_{{i^*}-1}+1)]$$ $$=[1, s_1-(s_0+1), 1, s_2-(s_1+1),..., 1, s_{i^*}-(s_{{i^*}-1}+1)].
\label{staircase}$$ On the one hand, adding the first steps through $s_i^*$, we have $$\sum_{s_i\leq s_{i^*}} 1+s_i-(s_{i-1}+1) = s_{i^*},$$ and on the other hand, adding the first $d/2$ steps $$d/2 = \sum_{i=1}^{d/2} a_i-a_{i-1}$$ by symmetry of the $\cfk^{\infty}(T(p, \, p+2))$ staircase. Since $s_i^*>d/2$, this implies that the full staircase is (\[staircase\]), where the pattern is truncated and reflected after the $(d/2)$th term. More precisely, $\cfk^{\infty}(T(p, \, p+2))$ is a staircase of the form $$[1, p-1,1,1,1,p-3, 1,1,1,1,1,p-5, ... ,\underbrace{1,1, ... ,1}_{2j+1} , p-(2j+1), ... ]$$ where the pattern is truncated and reflected after the $(p^2-1)/2$th term. This gives us a bifiltered basis $\mathcal{B}$ for $\cfk^{\infty}(T(p, \, p+2))$.
**Step (2):** Note that the points $$A=\left(1, \frac{(p-1)(p+1)}{2}-(p-1)\right) \text{ and } B=\left(3, \frac{(p-1)(p+1)}{2}-(p-1)-1-(p-3)\right)$$ both lie on the line $\mathcal{L}_{\frac{4}{p},\frac{p^2-p-1}{p}}$ given by $$\, j=-\frac{p-2}{2}i+\frac{p^2-p-1}{2}.$$ A computation shows that all other points in the diagram of $\cfk^{\infty}(T(p, \, p+2))$ are above this line. Thus $\gamma\,_{T(p, \, p+2)}(\frac{4}{p})=\frac{p^2-p-1}{p}$. So near $t=\frac{4}{p}$, the line $\call_{t,s}$ pivots around the two points $A$ and $B$. This causes a change in slope in $\U_{T(p, \, p+2)}$ and so $t=\frac{4}{p}$ is a singulariy of $\U_{T(p, \, p+2)}'$ and $\calc_{\frac{4}{p}}=\{A,B\}$.
We now compute $\U^2_{T(p,\,p+2),\frac{4}{p}}(\frac{4}{p})$. For small enough $\delta$, $$\calc_{\frac{4}{p}-\delta}=\{A\} \text{ and } \calc_{\frac{4}{p}+\delta}=\{B\}.$$ To determine $\U^2_{T(p, \, p+2),\frac{4}{p}}(\frac{4}{p})$, we compute how far the line of slope $-\frac{(p-2)}{2}$ needs to be moved so that the elements represented by $A$ and $B$ are homologous in $\calf_{\frac{4}{p},r}$.
Based on the staircase, we see that we need $\calf_{\frac{4}{p},r}$ to contain the basis elements represented by $A+(1,0)$ and $A+(2,-1)$. The minimal $r$ which accomplishes this is $r=\frac{p^2+p-7}{p}$. So we have that $\gamma^2_{T(p, \, p+2),\frac{4}{p}}(\frac{4}{p}) = \frac{p^2+p-7}{p}$. Thus $$\U^2_{T(p, \, p+2),\frac{4}{p}}(\frac{4}{p}) = -2\cdot \left(\frac{p^2+p-7}{p}-\frac{p^2-p-1}{p}\right) = -4\cdot \frac{p-3}{p}.$$
**Step (3):** The diagrams for the chain complexes $\cfk ^{\infty}(T(2,p))$ and $\cfk ^{\infty}(T(p, \, p+1))$ are computed from their semigroups. We have that $$S_{2,p}=\{0,2,4, ... , p-1\}\cup\Z_{\geq p},$$ so $$\Delta_{T(2,p)} = 1-t+t^2-t^3+t^4-t^5+\cdots +t^{p-1}$$ thus the staircase for $\cfk ^{\infty}(T(2,p))$ is $$[\underbrace{1,1,1,1, ... , 1}_{p-1}].$$ Similarly, $$\arraycolsep=1.4pt\def\arraystretch{1}
\begin{array}{rll}
S_{p,\,p+1} =\{&0,p, p+1,2p, 2p+1, 2p+2, ... ,&\\
&(p-2)p, (p-2)p+1, ... , (p-2)p+(p-2)\;\}\cup \Z_{\geq (p-1)p}&
\end{array}$$ so $$\Delta_{T(p, \, p+1)} = 1-t+t^p-t^{p+2}+t^{2p}-t^{2p+3}+\cdots + t^{(p-2)p}-t^{(p-2)p+(p-1)}+t^{(p-1)p},$$ and thus the staircase for $\cfk ^{\infty}(T(p, \, p+1))$ is $$[1, p-1, 2, p-2, ... , j, p-j, ..., p-1, 1].$$
From these staircase descriptions, a bifiltered basis $\calb_{2,\,p} = \{\alpha_i\}_{i=0}^p$, $\calb_{p,\,p+1} = \{\beta_i\}_{i=0}^{2p-1}$ for each complex can be determined: $$\alpha_{2i} \text{ is represented by } \left(i, \frac{p-1}{2}-i\right),$$ $$\alpha_{2i+1} \text{ is represented by } \left(i+1, \frac{p-1}{2}-i\right),$$ $$\beta_{2i} \text{ is represented by } \left(\sum_{n=1}^i n, \frac{(p-1)p}{2}-\sum_{n=1}^i (p-n)\right),$$ $$\beta_{2i+1} \text{ is represented by } \left(\sum_{n=1}^{i+1} n, \frac{(p-1)p}{2}-\sum_{n=1}^i (p-n)\right).$$ Here even-indexed elements are at Maslov grading 0, while odd-indexed elements are at Maslov grading 1. A bifiltered basis for the tensor product is the tensor product of the bases $\calb_{2,\,p}\otimes\calb_{p,\,p+1}=\{\alpha_i\otimes\beta_j\}$.
**Step (4):** In the tensor product, the points $\alpha_0\otimes \beta_2$ and $\alpha_0\otimes \beta_4$ are at the same filtration levels as $A$ and $B$ respectively. Thus they lie on a line of slope $-\frac{p-2}{2}$ (corresponding to $t=\frac{4}{p}$) with $j$–intercept $\frac{p^2-p-1}{2}$ (corresponding to $s=\frac{p^2-p-1}{p}$). We need to confirm that all other bifiltered basis elements in the tensor prodcut lie above this line.
First, note that $\alpha_{2i+1}\otimes \beta_{2j}$, $\alpha_{2i+1}\otimes \beta_{2j+1}$, and $\alpha_{2i}\otimes \beta_{2j+1}$ are at higher filtration levels than $\alpha_{2i}\otimes \beta_{2j}$. So we will show that for all $(i,j)\neq (0,1)\text{ or } (0,2)$, $\alpha_{2i}\otimes \beta_{2j}$ is above line $\call_{\frac{4}{p},\frac{p^2-p-1}{p}}$ given by $$y=-\frac{p-2}{2}\,x+\frac{p^2-p-1}{2}.$$ The element $\alpha_{2i}\otimes \beta_{2j}$ is represented by $$\left(i+\sum_{n=1}^j n, \frac{p-1}{2}-i+\frac{(p-1)p}{2}-\sum_{n=1}^j (p-n) \right)=\left(i+\frac{j(j+1)}{2}, \frac{p^2-1}{2}-i-jp+\frac{j(j+1)}{2}\right).$$ We test the inequality $$y\leq-\frac{p-2}{2}x+\frac{p^2-p-1}{2},$$ at the $x$– and $y$–values above and find that $$\frac{p^2-1}{2}-i-jp+\frac{j(j+1)}{2}\leq-\frac{p-2}{2}\left(i+\frac{j(j+1)}{2}\right)+\frac{p^2-p-1}{2},$$ $$-i-jp+\frac{j^2+j}{2}\leq -\frac{p-2}{2}i-\frac{p-2}{2}\cdot \frac{j^2+j}{2}-\frac{p}{2},$$ $$\frac{p-4}{2} i \leq -\frac{p}{2}\cdot\frac{j^2+j}{2}+jp-\frac{p}{2,}$$ $$\frac{p-4}{2}i\leq -\frac{p}{4}j^2+\frac{3p}{4}j-\frac{p}{2},$$ $$i\leq \frac{2}{p-4} \left(-\frac{p}{4}j^2+\frac{3p}{4}j-\frac{p}{2}\right),$$ $$i\leq -\frac{p}{2(p-4)} (j^2 -3j+2),$$ $$i\leq - \frac{p}{2(p-4)} (j-2)(j-1). \label{ineq1}$$ Inequality \[ineq1\] holds only for $i=0$ and $j=1$ or $2$. So for all other values of $i$ and $j$, $\alpha_{2i}\otimes\beta_{2j}$ is above the line $\call_{\frac{4}{p},\frac{p^2-p-1}{p}}$.
Theorem \[feller-krcatovich\] implies that the singularities of $\U'_{T(p, \, p+2)}(t)$ and $\U'_{T(2,p)\cs T(p, \, p+1)}(t)$ occur at the same $t$–values. Thus $\frac{4}{p}$ is a singularity of $\U'_{T(2,p)\cs T(p, \, p+1)}$ and so we can consider the secondary Upsilon invariant of $T(2,p)\cs T(p, \, p+1)$ at $t_0=\frac{4}{p}$. Now, suppose that $\alpha_0\otimes\beta_2$ and $\alpha_0\otimes\beta_4$ are homologous in $\calf_{\frac{4}{p},\frac{p^2+p-7}{p}}$. Then, since both $\alpha_0\otimes\beta_2$ and $\alpha_0\otimes\beta_4$ are at Maslov grading 0, there would be basis elements $x_1, x_2, ... \,, x_k$ in $\calf_{\frac{4}{p},\frac{p^2+p-7}{p}}$ at Maslov grading 1 such that $$\partial(b_1x_1+b_2x_2+\cdots +b_kx_k)= \alpha_0\otimes\beta_2+\alpha_0\otimes\beta_4
\label{boundary2}$$ for some $b_i\in\Z_2$. Bifiltered basis elements at Maslov grading 1 have the form
$$\begin{aligned}
\alpha_{2i}\otimes\beta_{2j+1} &= \left(i+\sum_{n=1}^{j+1}n\, , \,\frac{p-1}{2}-i+\frac{(p-1)p}{2}-\sum_{n=1}^j(p-n) \right)\\
&=\left(i+\frac{(j+1)(j+2)}{2} \, , \, \frac{p^2-1}{2} -i-jp+\frac{j(j+1)}{2}\right)\end{aligned}$$
or $$\begin{aligned}
\alpha_{2i+1}\otimes\beta_{2j} &= \left(i+1+\sum_{n=1}^{j}n\, , \,\frac{p-1}{2}-i+\frac{(p-1)p}{2}-\sum_{n=1}^j(p-n) \right)\\
&=\left(i+1+\frac{j(j+1)}{2} \, , \, \frac{p^2-1}{2} -i-jp+\frac{j(j+1)}{2}\right).\end{aligned}$$ To determine which elements of Maslov grading 1 are in $\calf_{\frac{4}{p},\frac{p^2+p-7}{p}}$, we determine which of the above satisfy the inequality $$y\leq -\frac{p-2}{2}\,x+\frac{p^2+p-7}{2}.$$ For $\alpha_{2i}\otimes\beta_{2j+1}$ we have $$\begin{aligned}
\frac{p^2-1}{2} -i-jp+\frac{j(j+1)}{2} &\leq -\frac{p-2}{2}\left(i+\frac{(j+1)(j+2)}{2}\right) + \frac{p^2+p-7}{2} \\
\frac{p-4}{2}i &\leq -\frac{p-2}{2}\cdot \frac{j^2+3j+2}{2} - \frac{j^2+j}{2} +jp +\frac{p-6}{2}\\
\frac{p-4}{2}i&\leq -\frac{p}{4}j^2+\frac{p+4}{4}j-2 \\
i&\leq \frac{2}{p-4} \left(-\frac{p}{4}j^2+\frac{p+4}{4}j-2 \right) \\
i&\leq -\frac{p}{2(p-4)} \left(j^2-\left(1+\frac{4}{p}\right)j+\frac{8}{p} \right).\end{aligned}$$ Since the right-hand side of the final inequality is negative for $j\geq 0$, the element $\alpha_{2i}\otimes\beta_{2j+1}$ is not in $\calf_{\frac{4}{p},\frac{p^2+p-7}{p}}$ for any $i, j$. For $\alpha_{2i+1}\otimes\beta_{2j}$ we have $$\begin{aligned}
\frac{p^2-1}{2} -i-jp+\frac{j(j+1)}{2} &\leq -\frac{p-2}{2}\left(i+1+\frac{j(j+1)}{2}\right) + \frac{p^2+p-7}{2} \\
\frac{p-4}{2}i &\leq -\frac{p-2}{2}\cdot \frac{j^2+j+2}{2} - \frac{j^2+j}{2}+jp +\frac{p-6}{2}\\
\frac{p-4}{2}i&\leq -\frac{p}{4}j^2+\frac{3p}{4}j-2 \\
i&\leq \frac{2}{p-4} \left(-\frac{p}{4}j^2+\frac{3p}{4}j-2 \right) \\
i&\leq -\frac{p}{2(p-4)} \left(j^2-3j+\frac{8}{p} \right) \\
i&\leq -\frac{p}{2(p-4)}\left(j-\frac{3+\sqrt{9-\frac{32}{p}}}{2} \right) \left(j-\frac{3-\sqrt{9-\frac{32}{p}}}{2} \right).\end{aligned}$$ This inequality only holds when $$\frac{3-\sqrt{9-\frac{32}{p}}}{2}\leq j\leq \frac{3+\sqrt{9-\frac{32}{p}}}{2} \text{ and } 0\leq i\leq -\frac{p}{2(p-4)}\left(\left(\frac{3}{2}\right)^2-3\cdot \frac{3}{2}+\frac{8}{p}\right),$$ which is when $$1\leq j \leq 2 \text{ and } 0\leq i\leq 1.$$Thus there are four elements within the desired range: $$\alpha_1\otimes\beta_2,\, \alpha_1\otimes\beta_4,\, \alpha_3\otimes\beta_2, \text{ and } \alpha_3\otimes\beta_4.$$ Taking the boundaries, we get: $$\begin{aligned}
\partial (\alpha_1\otimes\beta_2) &= \partial \alpha_1 \otimes \beta_2 +\alpha_1 \otimes \partial \beta_2\\
&= (\alpha_0 +\alpha_2) \otimes \beta_2 + \alpha_1 \otimes 0 \\
& = \alpha_0 \otimes \beta_2 + \alpha_2 \otimes \beta_2, \\
\partial (\alpha_1\otimes\beta_4) &= \partial \alpha_1 \otimes \beta_4 +\alpha_1 \otimes \partial \beta_4\\
&= (\alpha_0 +\alpha_2) \otimes \beta_4 + \alpha_1 \otimes 0 \\
& = \alpha_0 \otimes \beta_4 + \alpha_2 \otimes \beta_4, \\
\partial (\alpha_3\otimes\beta_2) &= \partial \alpha_3 \otimes \beta_2 +\alpha_3 \otimes \partial \beta_2\\
&= (\alpha_2 +\alpha_4) \otimes \beta_2 + \alpha_3 \otimes 0 \\
& = \alpha_2 \otimes \beta_2 + \alpha_4 \otimes \beta_2, \\
\partial (\alpha_3\otimes\beta_4) &= \partial \alpha_3 \otimes \beta_4 +\alpha_3 \otimes \partial \beta_4\\
&= (\alpha_2 +\alpha_4) \otimes \beta_4 + \alpha_3 \otimes 0 \\
& = \alpha_2 \otimes \beta_4 + \alpha_4 \otimes \beta_4. \\\end{aligned}$$ Notice that if Equation \[boundary2\] is to hold, it must be that $\alpha_1\otimes\beta_2$ and $\alpha_1\otimes\beta_4$ have coefficients of 1. Since $$\partial(\alpha_1\otimes\beta_2 + \alpha_1\otimes\beta_4)=\alpha_0\otimes\beta_2+\alpha_0\otimes\beta_4+\alpha_2\otimes\beta_2 + \alpha_2 \otimes\beta_4,$$ it must be that $\alpha_3\otimes\beta_2$ and $\alpha_3\otimes\beta_4$ also have coefficients of 1. However, $$\partial(\alpha_1\otimes\beta_2+\alpha_1\otimes\beta_4+\alpha_3\otimes\beta_2 + \alpha_3 \otimes\beta_4)=\alpha_0\otimes\beta_2+\alpha_0\otimes\beta_4+\alpha_4\otimes\beta_2 + \alpha_4 \otimes\beta_4$$ and we are left without options. So $\alpha_0\otimes\beta_2$ and $\alpha_0\otimes\beta_4$ are not homologous in $\calf_{\frac{4}{5},\frac{23}{5}}$. Thus $$\gamma^2_{T(2,p)\cs T(p, \, p+1), \frac{4}{p}}\left(\frac{4}{p}\right) > \frac{p^2+p-7}{p}$$ and so $$\begin{aligned}
\U^2_{T(2, p)\cs T(p, \, p+1),\frac{4}{p}}\left(\frac{4}{p}\right)&=-2\cdot\left(\gamma^2_{T(2, p)\cs T(p, \, p+1), \frac{4}{p}}\left(\frac{4}{p}\right)-\frac{p^2-p-1}{p}\right)\\
&<-2\cdot \left(\frac{p^2+p-7}{p}-\frac{p^2-p-1}{p}\right)\\
&=-4\cdot \frac{p-3}{p}\end{aligned}$$ as desired.
Since $\U^2$ is a stable equivalence invariant, it follows that $\cfk ^{\infty}(T(p, \, p+2))$ is not stably equivalent to $\cfk ^{\infty}(T(2, p)\cs T(p, \, p+1))$.
It may be that steps similar to those of the proof of Theorem \[general thm\] can be used to generalize it. The following is a conjecture of the author.
For all $p\geq 5$ and $2\leq k \leq p-2$ such that gcd($p, k)=1$, the knot complex $\cfk ^{\infty}(T(p, \, p+k))$ is not stably equivalent to $\cfk ^{\infty}(T(k, p)\cs T(p, \, p+1))$.
Note that the Feller-Krcatovich relationships among the Upsilon functions of torus knots do extend to stable equivalence in some cases. For example, with a change of basis one can see that the knot complexes $\cfk ^{\infty}(T(2,3) \cs T(2,3))$ and $\cfk ^{\infty}(T(2,5))$ are stably equivalent. In a recent paper, Kim, Krcatovich, and Park [@kim-krcatovich-park] gave a condition for the knot complex of the connected sum of two $L$–space knots to be stably equivalent to a staircase complex. Using this result, we can see that $\cfk ^{\infty}(T(p-1, \, p) \cs T(p, \, p+1))$ and $\cfk ^{\infty}(T(p, \,2p-1))$ are stably equivalent. As a result, we limit our conjecture to $k\leq p-2$ and $p\geq 5$.
|
---
abstract: 'We derive relations between polarized transverse momentum dependent distribution functions (TMDs) and the usual parton distribution functions (PDFs) in the 3D covariant parton model, which follow from Lorentz invariance and the assumption of a rotationally symmetric distribution of parton momenta in the nucleon rest frame. Using the known PDF $g_{1}^{q}(x)$ as input we predict the $x$- and $\mathbf{p}_{T}$-dependence of all polarized twist-2 naively time-reversal even (T-even) TMDs.'
address:
- '$^{1}$ [Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia]{}'
- '$^{2}$ [Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.]{}'
- '$^{3}$ [Institute of Physics AS CR, Na Slovance 2, CZ-182 21 Prague 8, Czech Rep.]{}'
author:
- 'A.V. Efremov$^{1}$, P. Schweitzer$^{2}$, O. V. Teryaev$^{1}$, $^{3}$'
title: |
The polarized TMDs in the covariant\
parton model approach[^1]
---
\[sec1\]
TMDs [@tmds; @Mulders:1995dh] open a new way to a more complete understanding of the quark-gluon structure of the nucleon. Indeed, some experimental observations can hardly be explained without a more accurate and realistic 3D picture of the nucleon, which naturally includes transverse motion. The azimuthal asymmetry in the distribution of hadrons produced in deep-inelastic lepton-nucleon scattering (DIS), known as the Cahn effect [@Cahn:1978se], is a classical example. The intrinsic (transversal) parton motion is also crucial for the explanation of some spin effects [@Airapetian:1999tv]–[@Adams:2003fx].
In previous studies we discussed the covariant parton model, which is based on the 3D picture of parton momenta with rotational symmetry in the nucleon rest frame [@Zavada:1996kp]–[@Efremov:2010cy].
In this model we studied all T-even TMDs and derived a set of relations among them [@Efremov:2009ze]. It should be remarked that some of the relations among different TMDs were found (sometimes before) also in other models [@Jakob:1997wg]–[@Pasquini:2010pa].
In the recent paper [@Efremov:2010mt] we further develop and broadly extend our studies [@Zavada:2009sk]–[@Efremov:2009vb] of the relations between TMDs and PDFs. The formulation of the model in terms of the light-cone formalism [@Efremov:2009ze] allows us to compute the leading-twist TMDs by means of the light-front correlators $\phi(x,\mathbf{p}_{T})_{ij}$ [@Mulders:1995dh] as:$$\begin{aligned}
\frac{1}{2}\,\mathrm{tr}\left[ \gamma^{+}\;\phi(x,\mathbf{p}_{T})\right] &
=f_{1}^{q}(x,\mathbf{p}_{T})-\frac{\varepsilon^{jk}p_{T}^{j}S_{T}^{k}}{M}\,f_{1T}^{\perp q}(x,\mathbf{p}_{T}),\label{e1}\\
\frac{1}{2}\,\mathrm{tr}\left[ \gamma^{+}\gamma_{5}\phi(x,\mathbf{p}_{T})\right] & =S_{L}g_{1}^{q}(x,\mathbf{p}_{T})+\frac{\mathbf{p}_{T}\mathbf{S}_{T}}{M}g_{1T}^{\bot q}(x,\mathbf{p}_{T}),\label{e2}\\
\hspace{-8mm}\frac{1}{2}\,\mathrm{tr}\left[ i\sigma^{j+}\gamma_{5}\phi(x,\mathbf{p}_{T})\right] & =S_{T}^{j}\,h_{1}^{q}(x,\mathbf{p}_{T})+S_{L}\,\frac{p_{T}^{j}}{M}\,h_{1L}^{\perp q}(x,\mathbf{p}_{T})\label{ee2}\\
& +\frac{(p_{T}^{j}p_{T}^{k}-\frac{1}{2}\,\mathbf{p}_{T}^{2}\delta^{jk})S_{T}^{k}}{M^{2}}\,h_{1T}^{\perp q}(x,\mathbf{p}_{T})+\frac{\varepsilon
^{jk}p_{T}^{k}}{M}\,h_{1}^{\perp q}(x,\mathbf{p}_{T}).\;\;\nonumber\end{aligned}$$ In the present contribution we report about new results related to the polarized distributions [@Efremov:2010mt].
In our approach all polarized leading-twist T-even TMDs are described in terms of the *same* polarized covariant 3D distribution $H(p^{0})$. This follows from the compliance of the approach with relations following from QCD equations of motion [@Efremov:2009ze]. As a consequence all polarized TMDs can be expressed in terms a single generating function $K^{q}(x,\mathbf{p}_{T})$ as follows $$\renewcommand{\arraystretch}{2.2}\begin{array}
[c]{rcrcl}g_{1}^{q}(x,\mathbf{p}_{T}) & = & \displaystyle\frac{1}{2x}\left( \left(
x+\frac{m}{M}\right) ^{2}-\frac{\mathbf{p}_{T}^{2}}{M^{2}}\right) & \times &
K^{q}(x,\mathbf{p}_{T})\;,\\
h_{1}^{q}(x,\mathbf{p}_{T}) & = & \displaystyle\frac{1}{2x}\left( x+\frac
{m}{M}\right) ^{2} & \times & K^{q}(x,\mathbf{p}_{T})\;,\\
g_{1T}^{\perp q}(x,\mathbf{p}_{T}) & = & \displaystyle\frac{1}{x}\left(
x+\frac{m}{M}\right) \; & \times & K^{q}(x,\mathbf{p}_{T})\;,\\
h_{1L}^{\perp q}(x,\mathbf{p}_{T}) & = & \displaystyle-\,\frac{1}{x}\left(
x+\frac{m}{M}\right) \; & \times & K^{q}(x,\mathbf{p}_{T})\;,\\
h_{1T}^{\perp q}(x,\mathbf{p}_{T}) & = & \displaystyle-\frac{1}{x}\, & \times
& K^{q}(x,\mathbf{p}_{T})\;.
\end{array}
\label{Eq:all-TMDs}$$ with the generating function $K^{q}(x,\mathbf{p}_{T})$ defined (in the compact notation of [@Efremov:2009ze]) by $$K^{q}(x,\mathbf{p}_{T})=M^{2}x\int\mathrm{d}\{p^{1}\}\;\;,\;\;\;\;\mathrm{d}\{p^{1}\}\equiv\frac{\mathrm{d}p^{1}}{p^{0}}\;\frac{H^{q}(p^{0})}{p^{0}+m}\;\delta\left( \frac{p^{0}-p^{1}}{M}-x\right) \,.
\label{Eq:generating-function}$$ We have shown that due to rotational symmetry the following relations hold: $$K^{q}(x,\mathbf{p}_{T})=M^{2}\frac{H^{q}(\bar{p}^{0})}{\bar{p}^{0}+m},\qquad\bar{p}^{0}=\frac{1}{2}\,xM\,\left( 1+\frac{\mathbf{p}_{T}^{2}+m^{2}}{x^{2}M^{2}}\right) , \label{Eq:generating-function-2}$$ $$\pi x^{2}M^{3}H^{q}\!\left( \frac{M}{2}x\right) =2\int_{x}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(x)-x\;\frac{\mathrm{d}g_{1}^{q}(x)}{\mathrm{d}x}, \label{Eq:relation-g1-Hp}$$ where we took the limit $m\rightarrow0$ in (\[Eq:relation-g1-Hp\]). In that limit we obtain for the generating function (\[Eq:generating-function-2\]) the result $$K^{q}(x,\mathbf{p}_{T})=\frac{H^{q}(\frac{M}{2}\xi)}{\,\frac{M}{2}\xi}=\frac{2}{\pi\xi^{3}M^{4}}\left( 2\int_{\xi}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(\xi)-x\;\frac{\mathrm{d}g_{1}^{q}(\xi)}{\mathrm{d}\xi}\right) ,\quad\xi=\,x\,\left( 1+\frac{\mathbf{p}_{T}^{2}}{x^{2}M^{2}}\right) . \label{e3}$$ and from (\[Eq:all-TMDs\]) we obtain $$g_{1}^{q}(x,\mathbf{p}_{T})=\frac{2x-\xi}{\pi\xi^{3}M^{3}}\left( 2\int_{\xi
}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(\xi)-\xi\;\frac
{\mathrm{d}g_{1}^{q}(\xi)}{\mathrm{d}\xi}\right) . \label{e4}$$ This relation yields for $g_{1}^{q}(x,\mathbf{p}_{T})$, with the LO parameterization of [@lss] for $g_{1}^{q}(x)$ at $4\,\mathrm{GeV}^{2}$, the results shown in Fig. \[ff3\].
![The TMD $g_{1}^{q}(x,\mathbf{p_{T}})$ for $u$- (*upper panel*) and $d$-quarks (*lower panel*). **Left panel**: $g_{1}^{q}(x,\mathbf{p_{T}})$ as function of $x$ for $p_{T}/M=0.10$ (dashed), 0.13 (dotted), 0.20 (dash-dotted line). The solid line corresponds to the input distribution $g_{1}^{q}(x)$. **Right panel**: $g_{1}^{q}(x,\mathbf{p_{T}})$ as function of $p_{T}/M$ for $x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted line).[]{data-label="ff3"}](ggg1.pdf){width="12cm"}
![The TMDs $h_{1}^{q}(x,\mathbf{p_{T}})$, $g_{1T}^{\bot q}(x,\mathbf{p_{T}})$, $h_{1T}^{\perp q}(x,\mathbf{p_{T}})$ for $u$- and $d$-quarks. ** Left panel**: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13 (dotted), 0.20(dash-dotted lines). **Right panel**: The TMDs as functions of $p_{T}/M$ for $x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted lines).[]{data-label="ff4"}](ggg2.pdf "fig:"){width="10.7cm"}\
![The TMDs $h_{1}^{q}(x,\mathbf{p_{T}})$, $g_{1T}^{\bot q}(x,\mathbf{p_{T}})$, $h_{1T}^{\perp q}(x,\mathbf{p_{T}})$ for $u$- and $d$-quarks. ** Left panel**: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13 (dotted), 0.20(dash-dotted lines). **Right panel**: The TMDs as functions of $p_{T}/M$ for $x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted lines).[]{data-label="ff4"}](ggg3.pdf "fig:"){width="10.7cm"}\
![The TMDs $h_{1}^{q}(x,\mathbf{p_{T}})$, $g_{1T}^{\bot q}(x,\mathbf{p_{T}})$, $h_{1T}^{\perp q}(x,\mathbf{p_{T}})$ for $u$- and $d$-quarks. ** Left panel**: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13 (dotted), 0.20(dash-dotted lines). **Right panel**: The TMDs as functions of $p_{T}/M$ for $x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted lines).[]{data-label="ff4"}](ggg4.pdf "fig:"){width="10.7cm"}
The remarkable observation is that $g_{1}^{q}(x,\mathbf{p}_{T})$ changes sign at the point $p_{T}=Mx$, which is due to the prefactor (this is the definition of the variable $\bar{p}^{1}$ in the limit $m\rightarrow0$) $$2x-\xi=x\left( 1-\left( \frac{p_{T}}{Mx}\right) ^{2}\right) =-2\bar{p}^{1}/M \label{m23}$$ in (\[e4\]). The expression in (\[m23\]) is proportional to the quark longitudinal momentum $\bar{p}^{1}$ in the proton rest frame, which is determined by $x$ and $p_{T}$ [@Efremov:2010mt]. This means, that the sign of $g_{1}^{q}(x,p_{T})$ is controlled by sign of $\bar{p}^{1}$. To observe these dramatic sign changes one may look for multi-hadron jet-like final states in SIDIS. Performing the cutoff for transverse momenta from below and from above, respectively, should affect the sign of asymmetry.
There is some similarity to $g_{2}^{q}(x)$ which also changes sign, and is given in the model by [@Zavada:2007ww] $$g_{2}^{q}(x)=\frac{1}{2}\int H^{q}(p^{0})\left( p^{1}-\frac{\left(
p^{1}\right) ^{2}-p_{T}^{2}/2}{p^{0}+m}\right) \:\delta\left( \frac
{p^{0}-p^{1}}{M}-x\right) \frac{d^{3}p}{p^{0}}. \label{sp11}$$ The $\delta-$function implies that, for our choice of the light-cone direction, large $x$ are correlated with large and negative $p^{1}$, while low $x$ are correlated with large and positive $p^{1}$. Thus, $g_{2}(x)$ changes sign, because the integrand in (\[sp11\]) changes sign between the extreme values of $p^{1}$. Let us remark, that the calculation of $g_{2}(x)$ based on the relation (\[sp11\]) well agrees [@Zavada:2002uz] with the experimental data.
The other TMDs (\[Eq:all-TMDs\]) can be calculated similarly and differ, in the limit $m\to0$, by simple $x$-dependent prefactors $$h_{1}^{q}(x,\mathbf{p}_{T}) =\frac{x}{2}K^{q}(x,\mathbf{p}_{T}), \;\;
g_{1T}^{\perp q}(x,\mathbf{p}_{T}) = K^{q}(x,\mathbf{p}_{T}),\;\;
h_{1T}^{\perp q}(x,\mathbf{p}_{T}) =-\frac{1}{x}K^{q}(x,\mathbf{p}_{T}).\label{e5}$$ The resulting plots are shown in Fig. \[ff4\]. We do not plot $h_{1L}^{\perp
q}$ since this TMD is equal to $-g_{1T}^{\perp q}$ in our approach [@Efremov:2009ze]. Let us remark, that $g_{1}^{q}(x,\mathbf{p}_{T})$ is the only TMD which can change sign. The other TMDs have all definite signs, which follows from (\[Eq:all-TMDs\], \[e5\]). Note also that pretzelosity $h_{1T}^{\perp q}(x,\mathbf{p}_{T})$, due to the prefactor $1/x$, has the largest absolute value among all TMDs. Noteworthy, pretzelosity is related to quark orbital angular momentum in some quark models [@She:2009jq; @Avakian:2010br], including the present approach [@Efremov:2010cy].
To conclude, let us remark that an experimental check of the predicted TMDs requires care. In fact, TMDs are not directly measurable quantities unlike structure functions. What one can measure for instance in semi-inclusive DIS is a convolution with a quark fragmentation function. This naturally dilutes the effects of TMDs, and makes it difficult to observe for instance the prominent sign change in the helicity distribution, see Fig. \[ff3\]. A dedicated study of the phenomenological implications of our results is in progress.
**Acknowledgements.** A. E. and O. T. are supported by the Grants RFBR 09-02-01149 and 09-02-00732, and (also P.Z.) Votruba-Blokhitsev Programs of JINR. P. Z. is supported by the project AV0Z10100502 of the Academy of Sciences of the Czech Republic. The work was supported in part by DOE contract DE-AC05-06OR23177. We would like to thank also Jacques Soffer and Claude Bourrely for helpful comments on an earlier stage of this study.
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[^1]: Contribution to the Proceedings of the 19th International Spin Physics Symposium (SPIN2010), Jülich, Germany, September 27 - October 2, 2010
|
---
author:
- M Wang
- W J Sun
- 'B H Sun[^1]'
- 'J Li[^2]'
- 'L H Zhu[^3]'
- Y Zheng
- G L Zhang
- L C He
- W W Qu
- |
F Wang\
T F Wang
- C Xiong
- C Y He
- G S Li
- J L Wang
- X G Wu
- S H Yao
- C B Li
- H W Li
- S P Hu
- J J Liu
bibliography:
- 'mybib.bib'
date: 'Received: date / Revised version: date'
title: 'The $\Delta I$=2 bands in $^{109}$In: possible antimagnetic rotation'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
The rotational bands consisting of electric quadrupole transitions are usually related to the rotation of the deformed nucleus around an axis perpendicular to the symmetry axis of the deformed density distribution. With the development of theoretical and experimental research [@Frauendorfcon; @CLARK1992247; @197Pb; @BALDSIEFEN1992252], a novel rotation has been found in weakly deformed or near-spherical nuclei, and is interpreted as a result of shears mechanism, i.e., the gradual closing of the angular momentum vector of relatively few high-$j$ proton particles ($j_\pi$) and high-$j$ neutron holes ($j_\nu$). Such kind of rotation is introduced as magnetic rotation. Up to now, numerous magnetic rotational bands have been observed in $A \sim 110$ mass region using the HI-13 tandem accelerator at the China Institute of Atomic Energy (CIAE), such as $^{106}$Ag [@M1_106Ag; @M1_106Ag_cpc; @Shapecoexistence106Ag], $^{107}$Ag [@107Ag; @M1_107Ag], $^{112}$In [@MR_112In_LXQ; @MR112In; @MR_112In], $^{113}$In [@113In; @113In_cpl] and $^{115}$In [@MR_115In].
Antimagnetic rotation (AMR) is another exotic rotation observed in near-spherical nuclei [@Frauendorf2001; @meng2013progress; @6]. The angular momentum is increased by simultaneous closing of the two blades of protons and neutrons toward the total angular momentum vector, which is so called “two-shears-like mechanism". Because the transverse magnetic moments of the valence nucleons are anti-aligned, there are no $M1$ transitions in antimagnetic rotational bands. AMR is characterized by weak $E2$ transitions and decreasing $B(E2)$ values with increasing spin, which reflects the nearly spherical core. The large $\mathfrak {J}^{(2)}/B(E2)$ ratio of the order of 100 $\hbar ^{2}$MeV$^{-1}$$e$$^{-2}$b$^{-2}$, compared with around 10 $\hbar ^{2}$MeV$^{-1}$$e$$^{-2}$b$^{-2}$ for well-deformed nucleus, is also a typical feature [@Frauendorf2001; @meng2013progress; @6].
The antimagnetic rotation is expected to be realized in the same mass region with magnetic rotation. Experimentally, they have been observed simultaneously in the $A \sim 110$ mass region. Especially for Cd isotopes, the positive parity yrast bands after the alignment of neutrons at sufficiently high frequencies are perfect candidates for the two-shears-like mechanism. Up to now, the antimagnetic rotational bands have been identified in $^{105}$Cd [@anti105Cd], $^{106}$Cd [@anti106Cd], $^{107}$Cd [@anti107Cd], $^{108}$Cd [@anti106108Cd; @antimr_108Cd] and $^{110}$Cd [@anti110Cd]. For In isotopes, when an additional proton occupies the $g_{7/2}$ or $d_{5/2}$ orbital, the “two-shears-like mechanism" can also be expected. In fact, the rotational bands in $^{108,110}$In [@anti108110In; @Sun2016], $^{112}$In [@antiMR_112In] and $^{113}$In [@AMR113In] have been taken as candidates for antimagnetic rotation.
In our previous work [@meng2018], the triaxial deformation, shape evolution and possible chirality for the dipole bands in $^{109}$In were discussed in detail. However, it was unclear for the underlying nuclear structure of the $\Delta I$=2 bands. In this paper, the level scheme of those bands has been extended by eleven $\gamma$ rays. The $\Delta I$=2 rotational bands in $^{109}$In are investigated based on the systematic discussion, and the configurations have been suggested. The experimental results are compared with the tilted axis cranking relativistic mean-field (TAC-RMF) approach [@meng2013progress; @6]. Candidates for antimagnetic rotational bands in $^{109}$In will be discussed.
EXPERIMENT AND RESULTS {#exp}
======================
The experiment was carried out using the HI-13 tandem accelerator at the China Institute of Atomic Energy (CIAE) in Beijing. Excited states in $^{109}$In were populated using the $^{100}$Mo($^{14}$N, 5$n$)$^{109}$In fusion-evaporation reaction, and the beam energy was 78 MeV. The target consisted of a 0.5 mg/cm$^{2}$ foil of $^{100}$Mo with a backing of 10 mg/cm$^{2}$-thick $^{197}$Au. The $\gamma$ rays were detected by an array composed of nine BGO-Compton-suppressed HPGe detectors, two low-energy photon (LEP) HPGe detectors, and one clover detector. A total of 84$\times$10$^{6}$ $\gamma$-$\gamma$ coincidence events were sorted into a fully symmetrized $E_{\gamma}$-$E_{\gamma}$ matrix, and analyzed using the software package RADWARE [@Radford] for the $\gamma$-ray coincidence relationship.
The data from the detectors at around 40$^{\circ}$ on one axis and at around 140$^{\circ}$ on the other axis were sorted into an asymmetric DCO matrix. By analyzed this asymmetric DCO matrix, the ratios of directional correlation of oriented states (DCO) can be obtained. The DCO ratios of the known $\gamma$-rays of nuclei produced in the present experiment were taken as the expected value. When the gate is set on a quadrupole transition, the expected value of stretched quadrupole transitions and pure dipole transitions are around 1.0 and 0.5 in the present array geometry. Analogously, when the gate is set on dipole transitions, the DCO ratios distribute from 1.5 to 2 for quadrupole transitions and from 0.5 to 1.3 for dipole transitions. When the gate is set on pure dipole transitions, the ratios are around 1 for pure dipole transitions.
The partial level scheme focused on the $\Delta I$=2 bands in $^{109}$In is shown in Fig. \[band78\]. The placements of $\gamma$ rays in the level scheme were determined through the observed coincidence relationships, intensity balances, and energy summations. Compared with the results reported in Ref. [@meng2018], the level scheme of $^{109}$In has been revised by adding eleven new $\gamma$ rays.
{width="17cm"}
Band 7 is an $\Delta I$=2 band and extended to (41/2$^{+}$) state at energy of 8460.8 keV. In Fig. \[AMR\](a), the 1000.4 keV transition has no coincidence with the newly identified $\gamma$-rays with energies of 1174.4 and 1024.8 keV, while it has coincidene with the 1026.3 keV transition which decays from 11/2$^{+}$ state at 1026.3 keV to 9/2$^{+}$ ground state, and the transitions with energies of 829.6, 888.2, 864.8, 643.3, 1109.6 keV, etc. In the spectrum gated on 443.6 keV transition, shown in Fig. \[AMR\](b), the transitions decay out from higher levels of band 7 and 9 can be seen, along with the linking transitions with energies of 1109.6, 888.2 keV. The peak with a centroid energy of 1025 keV can been seen in Fig. \[AMR\](b), which is composed of 1024.8 and 1026.3 keV transitions. This indicates the existence of 1024.8 keV transition. Moreover, the 1174.4 and 1024.8 keV transitions have mutual coincidence with the transitions of band 7, but has no coincidence with the 888.2, 1109.6 keV transitions and the $\gamma$ rays in band 9. Therefore, the 1174.4 and 1024.8 keV transitions are placed on the top of band 7, and the sequence of those transitions is determined by the intensity. The 864.8 keV transition has no coincidence with the 1109.6 and 643.3 keV transitions. Such coincident relationship along with the energy summation restricts the position of 643.4 and 1109.6 keV $\gamma$ rays to (31/2$^{+}$) $\rightarrow$ 29/2$^{(+)}$. The $R_{DCO}$ of 643.4 keV transition is consistent with the $\Delta I$=2 transition, which is similar to the 829.6 and 1000.4 keV transitions of band 9. Therefore, the 1109.6 keV transition is taken as a linking transition of band 9 and 7, and the 643.4 keV $\gamma$ ray is placed between the state at 7149.8 keV and the state at 6506.5 keV. The alignment analysis in Sec. \[config\] also supports such placement. In summary, the $\gamma$-rays with the energies of 1024.8, 1174.4 keV belong to band 7. Band 9 is built on the level at 6506.5 keV and decays to band 7 through the linking transitions with energies of 1109.6, 888.2 keV.
$E_{\gamma}$(keV)$^{a}$ $E_{i}$$\rightarrow$$E_{f}$ $I_{\gamma}$($\%$)$^{b}$ $R_{DCO}$(D)$^{c}$ $R_{DCO}$(Q)$^{d}$ $I_{i}^{\pi}$$\rightarrow$$I_{f}$$^{\pi}$ Band
------------------------- ----------------------------- -------------------------- -------------------- -------------------- ------------------------------------------- -----------------
355.7 2447.3$\rightarrow$2091.6 $<$0.1 (9/2$^{+}$)$\rightarrow$(5/2$^{+}$) 7
402.0 1428.3$\rightarrow$1026.3 23(1) 0.61(17) 13/2$^{+}$$\rightarrow$11/2$^{+}$
443.6 4742.8$\rightarrow$4299.2 4.0(3) 0.82(17) 25/2$^{(+)}$$\rightarrow$21/2$^{(+)}$ 7
463.0 5218.7$\rightarrow$4755.7 2.6(2) 1.07(16) 27/2$^{+}$$\rightarrow$23/2$^{+}$ 8
469.7 5396.8$\rightarrow$4927.1 3(1) 0.64(8) 29/2$^{(+)}$$\rightarrow$27/2$^{-}$ 7$\rightarrow$5
475.9 5218.7$\rightarrow$4742.8 0.7(4) 0.60(10) 27/2$^{+}$$\rightarrow$25/2$^{(+)}$ 8$\rightarrow$7
521.2 2968.5$\rightarrow$2447.3 $<$0.1 (13/2$^{+}$)$\rightarrow$(9/2$^{+}$) 7
596.2 2318.5$\rightarrow$1722.2 0.3(2) 0.9(2) 11/2$^{+}$$\rightarrow$(7/2)$^{+}$ 8
605.6 2318.5$\rightarrow$1712.5 0.7(2) 0.7(2) 11/2$^{+}$$\rightarrow$(9/2)$^{+}$
614.2 1712.5$\rightarrow$1099.4 0.2(1) (9/2)$^{+}$$\rightarrow$5/2$^{+}$
623.6 1722.2$\rightarrow$1099.4 0.3(1) (7/2)$^{+}$$\rightarrow$5/2$^{+}$
631.1 5849.8$\rightarrow$5218.7 2.9(3) 1.04(11) 31/2$^{+}$$\rightarrow$27/2$^{+}$ 8
643.3 7149.8$\rightarrow$6506.5 0.5(2) 1.0(3) (35/2$^{+}$)$\rightarrow$(31/2$^{+}$) 9
645.7 4299.2$\rightarrow$3653.5 0.8(3) 1.1(3) 21/2$^{(+)}$$\rightarrow$17/2$^{(+)}$ 7
654.0 5396.8$\rightarrow$4742.8 3.2(2) 0.83(10) 29/2$^{(+)}$$\rightarrow$25/2$^{(+)}$ 7
658.6 4755.7$\rightarrow$4097.1 1.2(1) 0.93(9) 23/2$^{+}$$\rightarrow$19/2$^{+}$ 8
673.5 2101.8$\rightarrow$1428.3 69(3) 19/2$^{+}$$\rightarrow$13/2$^{+}$
673.7 2102.0$\rightarrow$1428.3 14.7(7) 1.62(23) 17/2$^{+}$$\rightarrow$13/2$^{+}$
685.0 3653.5$\rightarrow$2968.5 0.3(3) 1.1(6) 17/2$^{(+)}$$\rightarrow$(13/2$^{+}$) 7
816.3 6666.1$\rightarrow$5849.8 1.5(1) 1.19(16) 35/2$^{+}$$\rightarrow$31/2$^{+}$ 8
829.6 7979.4$\rightarrow$7149.8 1.2(2) 0.96(29) (39/2$^{+}$)$\rightarrow$(35/2$^{+}$) 9
837.0 3155.5$\rightarrow$2318.5 1.0(3) 0.93(7) 15/2$^{+}$$\rightarrow$11/2$^{+}$ 8
864.8 6261.6$\rightarrow$5396.8 4.2(3) 0.80(7) 33/2$^{(+)}$$\rightarrow$29/2$^{(+)}$ 7
888.2 7149.8$\rightarrow$6261.6 1.6(2) 0.78(17) (35/2$^{+}$)$\rightarrow$33/2$^{(+)}$ 9$\rightarrow$7
893.0 2995.0$\rightarrow$2102.0 7.3(7) 0.46(8) 19/2$\rightarrow$17/2$^{+}$
941.6 4097.1$\rightarrow$3155.5 1.0(1) 1.10(13) 19/2$^{+}$$\rightarrow$15/2$^{+}$ 8
973.0 7639.1$\rightarrow$6666.1 1.6(2) 0.98(13) 39/2$^{+}$$\rightarrow$35/2$^{+}$ 8
1000.4 8979.8$\rightarrow$7979.4 0.6(1) 1.4(5) (43/2$^{+}$)$\rightarrow$(39/2$^{+}$) 9
1024.8 7286.4$\rightarrow$6261.6 2.6(7) (37/2$^{+}$)$\rightarrow$33/2$^{(+)}$ 7
1026.3 1026.3$\rightarrow$0 31(1) 0.59(7) 11/2$^{+}$$\rightarrow$9/2$^{+}$
1099.4 1099.4$\rightarrow$0 0.5(1) 5/2$^{+}$$\rightarrow$9/2$^{+}$
1109.6 6506.5$\rightarrow$5396.8 $<$0.1 (31/2$^{+}$)$\rightarrow$29/2$^{(+)}$ 9$\rightarrow$7
1143.4 8782.5$\rightarrow$7639.1 0.4(2) 1.00(26) 43/2$^{+}$$\rightarrow$39/2$^{+}$ 8
1174.4 8460.8$\rightarrow$7286.4 0.3(2) (41/2$^{+}$)$\rightarrow$(37/2$^{+})$ 7
1304.2 4299.2$\rightarrow$2995.0 0.8(1) 0.54(11) 21/2$^{(+)}$$\rightarrow$19/2
1428.3 1428.3$\rightarrow$0 100 13/2$^{+}$$\rightarrow$9/2$^{+}$
1551.7 3653.5$\rightarrow$2101.8 0.5(2) 17/2$^{(+)}$$\rightarrow$(19/2$^{+})$
Uncertainties are between 0.2 and 0.5 keV depending upon their intensity.
Intensities are normalized to the 1428.3 keV transition with $I_{\gamma}=100$.
DCO ratios gated by dipole transitions.
DCO ratios gated by quadrupole transitions.
The DCO of the 888.2 keV transition needs special explanation. In our early work [@meng2018], the 888.2 keV transition is thought to be an $E$2 transition and belong to band 7. Nevertheless, the DCO of the 888.2-keV transition is 0.78(17), which is not a strict proof for an $E$2 transition. Now the 888.2-keV transition is suggested as a linking transition between band 9 and 7, considering the newly found transitions and coincidence relationships. The DCO information of the newly found $\gamma$ rays with energy of 1109.6 keV can not been extracted for the weak intensity. However, if we suppose that the linking transitions with energies of 1109.6 and 888.2 keV are $E$2 transitions, there will be a 37/2$^{+}$ state at energy of 7149.8 keV, which is lower than that of the (37/2$^{+}$) state at 7286.4 keV of band 7. It will be inconsistent with the intensity of the 643.3 and 1024.8 keV transitions. Therefore, we suggest those linking transitions are $\Delta I$=1 transitions, and the bandhead of band 9 at energy of 6506.5 keV is (31/2$^{+}$).
The newly identified 645.7 and 685.0 keV transitions can been seen in the spectrum gated on 443.6 keV transition, as shown in Fig. \[AMR\](b). Though the 521.2 and 355.7 keV transitions can not been identified in Fig. \[AMR\](b) for their weak intensities, each of the 645.7, 685.0, 521.2, 355.7 keV transitions has mutual coincidence with their cascade $\gamma$ rays. Therefore, they are taken as the member of band 7, and the sequence of those four $\gamma$ rays are determined by the intensities.
The 443.6 keV transition has been identified as a $\Delta I$=2 transition in the early work [@meng2018]. The $R_{DCO}$ of 645.7 and 685.0 keV transition extracted from the spectrum gated on the 443.6 keV transition are around 1, which correspond to $\Delta I$=2 transitions. While it is difficult to extract the DCO information of the 521.2 and 355.7 keV transitions, we suggest them as $\Delta I$=2 transition considering that they are the intraband transitions of band 7. The parity of band 7 is suggested to be positive according to the alignment analysis in Sec. \[config\].
Band 8 consists of nine $\Delta I$=2 transitions and is extended to the 43/2$^{+}$ state at 8782.5 keV. From the 941.6 keV gated spectrum, as shown in Fig. \[AMR\](c), all the members of band 8 can be identified, along with the 605.6, 614.2, 623.6 and 1099.4 keV transitions. In the early work\
[@meng2018], the decay path of band 8 is not clear. With the observed 605.6 and 596.2 keV transitions, the band 8 are connected to the known (9/2)$^{+}$ state at 1712.5 keV and the (7/2)$^{+}$ state at 1722.2 keV [@TOI]. The 605.6 keV transition is a linking transition, which links the 11/2$^{+}$ state at 2318.5 keV to the (9/2)$^{+}$ state at 1712.5 keV. Because the band 8 is composed of $\Delta I$=2 transitions, the $\gamma$ ray with energy of 596.2 keV is taken as a member of band 8, and the (7/2)$^{+}$ state at 1722.2 keV is assigned as the bandhead of band 8. The positive bandhead also supports the parity assignment of band 8 in Ref. [@meng2018].
DISCUSSION
==========
Systemic discussion and configuration assignment {#config}
------------------------------------------------
The experimental alignment as a function of rotational frequency for bands 7, 8 and 9 is shown in Fig. \[band7\], and that of the yrast band and band 10 of the neighboring even-even nucleus $^{108}$Cd is shown as a comparison. The configuration of band 8 is assigned as $\pi g_{7/2}g^{-2}_{9/2}$ before the backbend and $\pi g_{7/2}g^{-2}_{9/2}\otimes\nu h_{11/2}^{2}$ after backbend in Ref. [@meng2018]. In this work, the bandhead of band 8 has been observed. The whole behaviour of band 8 before the backbend also supports the configuration assignment.
The band 10 of $^{108}$Cd is built on a non-aligned excitation into the $\nu$$h_{11/2}$ subshell [@108Cd]. Before the backbend, the initial aligned spin of band 7 is nearly 2.5$\hbar$ greater than that of the band 10 of $^{108}$Cd, which can be caused by the occupation of the odd proton in the $d_{5/2}$ orbital. A sharp backbend in both bands occurs at around 0.28 MeV and with similar gains in aligned spin, which is consistent with $h_{11/2}$ neutron pair alignment. Therefore, the band 7 before the backbend should build on a non-aligned neutron excitation into the $\nu$$h_{11/2}$ subshell associating with the $\pi$$d_{5/2}$ orbital, and the backbend can be attributed to the alignment of the neutrons in the $h_{11/2}$ orbitals. Band 9, which decays to band 7 through two linking transitions, can be related to different neutron excitations in comparison with band 7. The alignments of band 9 is about 3 $\hbar$ greater than that of band 7, which could be attributed to the midshell $g_{7/2}(d_{5/2})$ neutron alignment.
According to the configuration assignment, bands 7 and 8 are believed to be related to $1p1h$ proton excitations from the $g_{9/2}$ orbital to one of the $g_{7/2}$ and $d_{5/2}$ orbitals above the shell gap. Similar bands related to the $g_{7/2}$ orbital in $^{107, 111, 113}$In [@SMB_107In; @MR_111In; @AMR113In] are summarized in Fig \[energy\]. Even though several corresponding levels in $^{107}$In have not been observed, the isotopic regularity of the level energies is significant. It is worth noting that the excitation energies of $1p1h$ proton excitation from $g_{9/2}$ to $g_{7/2}$ orbital in $^{109, 111, 113}$In are within 1$\sim$2 MeV relative to the ground state, and decreases with the increasing neutron number. The proton-neutron residual interaction may play an important role in $1p1h$ excitation from $\pi$$g_{9/2}$ to $\pi$$g_{7/2}$ orbital at such low energy. It reduces the energy spacing between the $\pi$$g_{9/2}$ and $\pi$$g_{7/2}$ orbitals, and its impact is enhanced when more neutrons are occupying the midshell.
Moreover, when the additional proton of indium nuclei is occupying the $g_{7/2}$ or $d_{5/2}$ orbital, the rotational bands after the $h_{11/2}$ neutrons alignment at high frequencies are perfect candidates for the two-shears-like mechanism, such as the rotational bands in $^{108, 110, 112, 113}$In [@anti108110In; @Sun2016; @antiMR_112In; @AMR113In]. The dynamic moment of inertia $\mathfrak{J}^{(2)}$ is a sensitive probe of the nuclear collectivity. The $\mathfrak{J}^{(2)}$ and rotational frequency can be extracted experimentally by the following formulae,
$$\hbar\omega_{\rm exp}=\frac{1}{2}E_\gamma(I \rightarrow I-2)$$
$$\mathfrak{J}^{(2)}\approx\frac{dI}{d\omega}=\frac{4}{E_\gamma(I+2 \rightarrow I)-E_\gamma(I \rightarrow I-2)}$$
$\mathfrak{J}^{(2)}$ of bands 7 and 8 after the backbend in $^{109}$In are shown in Fig. \[J2\]. The typical antimagnetic rotational bands in $^{106}$Cd, $^{108}$Cd [@anti106108Cd; @antimr_108Cd] are also shown for comparison. As shown in Fig. \[J2\], $\mathfrak{J}^{(2)}$ stays around 23 MeV$^ {-1}$$\hbar$$^{2}$ as rotation frequency increases for bands 7 and 8 after backbend, and have the similar pattern with that of AMR bands in $^{106,108}$Cd. Such small and stable value of $\mathfrak{J}^{(2)}$ indicates that bands 7 and 8 after backbend in $^{109}$In are much less collective, and can be candidates for antimagnetic rotation.
{width="12cm"}
{width="12cm"}
Theoretical interpretation
--------------------------
In the following, the rotational structure of two positive parity bands in $^{109}$In are investigated by tilted axis cranking relativistic mean-field (TAC-RMF) approach. In contrast to its non-relativistic counterparts [@1], the relativistic mean field (RMF) approach including point-coupling or mesonic exchange interaction [@lj16; @lj17; @lj18], takes the fundamental Lorentz symmetry into account from the very beginning so that naturally takes care of the important spin degree of freedom and time-odd fields, resulting in great successes on many nuclear phenomena [@1; @2; @3; @4; @5; @6]. Moreover, without any additional parameters, the rotation excitations can be described self-consistently with the tilted axis cranking relativistic mean-field (TAC-RMF) approach [@meng2013progress; @6]. In particular, the TAC-RMF model has been successfully used in describing magnetic rotation (MR) and AMR microscopically and self consistently in different mass regions [@meng2013progress; @6], and especially the 110 region, such as the AMR bands in $^{105, 109, 110}$Cd [@zhao2012prc; @zhao2011prl; @zhao2012covariant; @peng2015magnetic; @zhang2014competition] and $^{108, 110, 112, 113}$In\
[@Sun2016; @antiMR_112In; @AMR113In], and also the MR bands in $^{113,114}$In [@113In; @MR_114In]. In the present TAC-RMF calculations, the point-coupling interaction PC-PK1 [@zhao2010new] is used for the Lagrangian without any additional parameters. A basis of 10 major oscillator shells is adopted for the solving of the Dirac equation and pairing correlations are neglected. In order to describe bands 7 and 8 in $^{109}$In, the configurations $\pi d_{5/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ and $\pi g_{7/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ are adopted in the TAC-RMF calculations, respectively.
The calculated results for the $\pi d_{5/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ and $\pi g_{7/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ configuration are shown in Fig. $\ref{EI}$ and Fig. $\ref{spin}$ in comparison with the experimental data for bands 7 and 8 after the backbend. It could be seen that the TAC-RMF calculations including both energy spectrum and rotational frequency based on the assigned configurations are in a good agreement with the experimental data, supporting the configuration assignment.
{width="12cm"}
After the backbend in bands 7 and 8, the calculated $\mathfrak{J}^{(2)}$ well reproduce the data and the $\mathfrak{J}^{(2)}$ values are around 20-25 MeV$^ {-1}$$\hbar$$^{2}$, which are much smaller than the typical values ($\sim 35$ MeV$^ {-1}$$\hbar$$^{2}$) for the $A=110$ rigid spherical rotor. This indicates that bands 7 and 8 are not based on a collective behavior, but most likely an antimagnetic rotation, as discussed in Section. \[config\].
Weak $E2$ transition is one of the typical characteristics of AMR and reflects the small deformation of the core which causes large ratios of $\mathfrak{J}^{(2)}$ to the reduced transition probability $B(E2)$ values. Furthermore, the $B(E2)$ values decrease with increasing angular momentum rather rapidly. The $B(E2)$ values and $\mathfrak{J}^{(2)}/B(E2)$ ratios as functions of the rotational frequency in the TAC-RMF calculations for the assigned configurations of bands 7 and 8 are given respectively in Fig. \[BJ\]. The $B(E2)$ values are shown to be decrease smoothly with increasing rotational frequency, while the $\mathfrak{J}^{(2)}/B(E2)$ ratios show rising tendencies for both bands 7 and 8. It should be noted that the calculated $\mathfrak{J}^{(2)}/B(E2)$ ratios for those two bands are around $100-120\;\hbar^2$MeV$^ {-1}e^{-2}b^{-2}$, which are much higher than that for a typical deformed rotational band ($\sim 10$\
$\;\hbar^2$MeV$^ {-1}e^{-2}b ^{-2}$ [@Frauendorf2001]) and also in agreement with the expectations from AMR bands [@zhao2012prc; @antiMR_112In; @Sun2016].
The decrease of the $B(E2)$ values can be attributed to the evolution of the nuclear deformation. As shown in the inset of Fig. \[BJ\](a), with increasing rotational frequency, the nucleus undergoes a smooth decrease in $\beta$ deformation with a rather small and steady triaxiality ($\gamma\leq 10^\circ$) for both bands 7 and 8, which is responsible for the falling tendency of $B(E2)$ values with rotational frequency.
In order to examine the two-shears-like mechanism for bands 7 and 8, $J_{\pi+\nu}$ (the angular momentum vectors of neutrons and the low-$\Omega$ proton) and $j_\pi$ (the two high-$\Omega$ $g_{9/2}$ proton holes) in the TAC-RMF calculations have been extracted and shown in Fig. \[twoS\]. Taking band 8 with the configuration $\pi g_{7/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ as an example. The angular momentum $J_{\pi+\nu}$ is related to all the neutron levels and the occupied low-$\Omega$ $g_{7/2}$ proton in the intrinsic system. At the bandhead ($\hbar\omega=\;$0.2 MeV), the two $j_\pi$ are nearly perpendicular to $J_{\pi+\nu}$ and pointing opposite to each other, which form the blades of the two shears. As the rotational frequency increases, the gradual alignment of the $g_{9/2}$ proton hole vectors $j_\pi$ toward $J_{\pi+\nu}$ generates angular momentum, while the direction of the total angular momentum stays unchanged. This leads to the closing of the two shears simultaneously by moving one blade toward the other, demonstrating the two-shears-like mechanism in band 8. A similar mechanism can also been seen in TAC-RMF calculations with assigned configuration of $\pi d_{5/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ for band 7 as shown in Fig. \[twoS\].
SUMMARY
=======
In summary, the $\Delta I$=2 rotational bands populated in the $^{100}$Mo($^{14}$N, 5$n$)$^{109}$In reaction have been modified and extended by eleven new $\gamma$ rays. The systematic discussion has been made and the configurations for the $\Delta I$=2 rotational bands have been assigned. The dynamic moment of inertia shows that bands 7 and 8 after backbend are much less collective.
The experimental data of bands 7 and 8 in $^{109}$In have been compared with the TAC-RMF calculations, and good agreements have been obtained. The predicted $B(E2)$, deformation $\beta$ and $\gamma$, as well as $\mathfrak{J}^{(2)}/B(E2)$ ratios in TAC-RMF calculations based on the $\pi d_{5/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ and $\pi g_{7/2}g_{9/2}^{-2}\otimes\nu h_{11/2}^2$ configurations have been discussed and the characteristic features of AMR for the bands 7 and 8 after the backbend have been shown. The two-shears-like mechanism for bands 7 and 8 show that they can be candidate antimagnetic rotational bands. Further experimental investigation such as life-time measurements are expected for a conclusive interpretation.
We thank the crew of the HI-13 tandem accelerator at the China Institute of Atomic Energy for their help in steady operation of the accelerator and for preparing the target. This work is partially supported by the National Natural Science Foundation of China under Contracts No. 11375023, No. 11575018, No. U1867210, No. 11675063, U1832211, and 11922501.
[^1]: *e-mail:* [email protected]
[^2]: *e-mail:* [email protected]
[^3]: *e-mail:* [email protected]
|
---
author:
- |
Ch. Brouder\
Laboratoire de minéralogie cristallographie\
Universités Paris 6, Paris 7, IPGP, Case 115, 4 place Jussieu\
75252 Paris [cedex]{} 05, France
title: 'Runge-Kutta methods and renormalization'
---
[essai1]{}
Introduction
============
The purpose of this paper is to point out a link between two apparently remote concepts: renormalization and Runge-Kutta methods.
Renormalization enables us to remove infinities from quantum field theory. Recently, Kreimer discovered a Hopf algebra of rooted trees that brings order and beauty in the intricate combinatorics of renormalization [@Kreimer98]. He established formulas that automate the subtraction of infinities to all orders of the perturbation expansion, and proved the effectiveness of his method for the practical computation of renormalized quantities in joint works with Broadhurst [@Broadhurst] and Delbourgo [@Delbourgo]. Moreover, his approach shines new light on the problem of overlapping divergences [@KreimerOD; @Krajewski] and on the mechanics of the renormalization group [@Kreimer]. Furthermore, Connes and Kreimer revealed a deep connection between the algebra of rooted trees (ART) and a Hopf algebra of diffeomorphisms [@Connes].
On the other hand, Runge published in 1895 [@Runge] an efficient algorithm to compute the solution of ordinary differential equations. For an equation of the type $dy/ds=f(y(s))$, he defines recursively $k_1=f(y_n)$, $k_2=f(y_n+h k_1/2)$, $y_{n+1}=y_n+hk_2$. His algorithm was improved in 1901 by Kutta, and became known as the Runge-Kutta method. It is now one of the most widely used numerical methods.
In 1972, Butcher published an extraordinary article where he analyzed general Runge-Kutta methods on the basis of the ART. He showed that the Runge-Kutta methods form a group[^1] and found explicit expressions for the inverse of a method or the product of two methods. He also defined sums over trees that are now called B-series in honour of Butcher.
Altough the Hopf algebra structure of ART is implicit all along his paper, Butcher did not mention it[^2]. Important developments were made in 1974 by Hairer and Wanner [@Hairer74]. Since then, B-series are used routinely in the analysis of Runge-Kutta methods.
Our main purpose is to show that the results and concepts established by Kreimer fit nicely into the Runge-Kutta language, and that the tools developed by Butcher have a range of application much wider than the numerical analysis of ordinary differential equations.
The present expository paper will be reasonably self-contained. After an introduction to rooted trees, the genetic relation between ART and differentials is presented. Then Butcher’s approach to Runge-Kutta methods is sketched. Several B-series are calculated and the connection with the Hopf structure of ART is exhibited. The application of Runge-Kutta methods to renormalization is exposed using a toy model which is solved non perturbatively. Finally, the solution of non-linear partial differential equations is written as a formal B-series.
The rooted trees
================
A rooted tree is a graph with a designated vertex called a root such that there is a unique path from the root to any other vertex in the tree [@Tucker]. Several examples of rooted trees are given in the appendix, where the root is the black point and the other vertices are white points (the root is at the top of the tree). The length of the unique path from a vertex $v$ to the root is called the level number of vertex $v$. The root has level number 0. For any vertex $v$ (except the root), the father of $v$ is the unique vertex $v'$ with an edge common with $v$ and a smaller level number. Conversely, $v$ is a son of $v'$. A vertex with no sons is a leaf. Rooted trees are sometimes called pointed trees or arborescences.
The tree with one vertex is ${\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}$, the “tree” with zero vertex is designated by $1$.
Operations and functions on trees
---------------------------------
An important operation is the merging of trees. If $t_1$,…,$t_k$ are trees, $t=B^+(t_1,t_2,\dots,t_k)$ is defined as the tree obtained by creating a new vertex $r$ and by joining the roots of $t_1$,…,$t_k$ to $r$, which becomes the root of $t$. This operation is also denoted by $t=[t_1,t_2,\dots,t_k]$, but we avoid this notation because of the possible confusion with commutators.
In [@Connes], Connes and Kreimer defined a natural growth operator $N$ on trees: $N(t)$ is the set of $|t|$ trees $t_i$, where each $t_i$ is a tree with $|t|+1$ vertices obtained by attaching an additional leaf to a vertex of $t$. For example $N(1)={\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}$, $$\begin{aligned}
N({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}},\quad N({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})={
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\quad
N({
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}})={
\parbox{8mm}{
\begin{fmfchar*}(8,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.5w,0.1h)}{o2}
\fmfforce{(0.9w,0.1h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}+2\,{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.5h)}{o1}
\fmfforce{(0.1w,0.1h)}{o2}
\fmfforce{(0.9w,0.5h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{o1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}.\end{aligned}$$ Some trees may appear with multiplicity.
A number of functions on rooted trees have been defined independently by several authors:
$|t|$ designates the number of vertices of a tree $t$ (alternative notation is $r(t)$, $\rho(t)$ and $\# t$). Clearly, $|B^+(t_1,t_2,\dots,t_k)|=|t_1|+|t_2|+\cdots+|t_k|+1$.
The tree factorial $t!$ is defined recursively as $$\begin{aligned}
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}!&=&1\\
B^+(t_1,t_2,\dots,t_k)!&=&|B^+(t_1,t_2,\dots,t_k)|\, t_1! t_2! \cdots t_k!.\end{aligned}$$ (an alternative notation is $\gamma(t)$). The notation $t!$ is taken from Kreimer [@Kreimer] because $t!$ generalizes the factorial of a number. Besides $t!$ has also similarities with the product of hooklengthes of a Young diagram in the representation theory of the symmetric group [@Fomin]. A few examples may be useful $$\begin{aligned}
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}!&=&2,\quad {
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}!=6,\quad {
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}!= 12,\quad {
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}!= 24,
\quad{
\parbox{8mm}{
\begin{fmfchar*}(8,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.5w,0.1h)}{o2}
\fmfforce{(0.9w,0.1h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}!=4.\end{aligned}$$
On $CM(t)$
----------
$CM(t)$ was defined in [@Kreimer] as the number of times tree $t$ appears in $N^n(1)$ where $n=|t|$ is the number of vertices of $t$. In the literature ([@Butcher63], [@Butcher], p.92, [@Hairer], p.147), $CM(t)$ is written as $\alpha(t)$ and considered as the number of “heap-ordered trees” with shape $t$, where a heap-ordered tree with shape $t$ is a labelling of each vertex of $t$ (i.e. a bijection between the vertices and the set of numbers $0,1,\dots |t|-1$) such that the labels decrease along the path going from any vertex to the root. This is called a monotonic labelling in [@Hairer], p.147.
$$\begin{aligned}
\parbox{10mm}{
\begin{fmfchar*}(10,10)
\fmftop{i1}
\fmfforce{(0,0)}{o2}
\fmfforce{(0,0.6h)}{o1}
\fmfright{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{o1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=0, label=\mbox{\small$0$},label.angle=0,label.dist=0,decor.size=90}{i1}
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\quad\quad
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\fmfv{decor.shape=circle,decor.filled=0, label=\mbox{\small$2$},label.angle=0,label.dist=0,decor.size=90}{o3}
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}\end{aligned}$$
There are $(n-1)!$ heap-ordered trees with $n$ vertices. This can be seen by a recursive argument. Take a heap-ordered tree $t$ with $n$ vertices, and make $n$ labeled trees by adding a new vertex with label $n$ to each vertex of $t$. Then all the created trees are heap-ordered (because the added vertex is a leaf and all other labels are smaller than $n$). Furthermore, all the heap-ordered trees created by this process from the set of heap-ordered trees with $n$ vertices are different. Therefore, there are at least $n!$ heap-ordered trees with $n+1$ vertices. On the other hand, in each heap-ordered tree $t$ with $n+1$ vertices, the vertex labeled $n$ is a leaf, therefore $t$ can be created from a heap-ordered tree with $n$ vertices by adding this leaf with label $n$. So there are exactly $n!$ heap-ordered trees with $n+1$ vertices. We shall give a non combinatiorial proof of this fact in the sequel.
Since $N(t)$ is defined by the addition of a leaf to all the vertices of $t$, $\alpha(t)$ is the number of heap-ordered trees with shape $t$. This number has been calculated in [@Butcher63] (see also. [@Butcher], p.92): $$\begin{aligned}
\alpha(t)=\frac{|t|!}{t!S_t},\end{aligned}$$ where $S_t$ is the symmetry factor of $t$, defined in [@Broadhurst; @Kreimer] and in [@Butcher] where it is denoted by $\sigma(t)$.
Note that there is a simple correspondence between the permutations of $n-1$ numbers and the heap-ordered trees. Let $(p_1,\dots,p_{n-1})$ be a permutation of $(1,\dots,n-1)$, then
- $p_1$ is a subroot, labeled $p_1$
- for $i$=2 to $n-1$
- if all $p_j$ for $1\le j\le i$ are such that $p_j > p_i$, then $p_i$ is a subroot, labeled $p_i$
- otherwise, let $p_j$ be first number such that $p_j < p_i$, in the series $p_{i-1},p_{i-2},\dots,p_1$, then the $i$-th vertex, labeled $p_i$, is linked to $p_j$ by a line
- when all $(p_1,\dots,p_{n-1})$ have been processed, all subroots are linked to a common root, labeled $0$
On the other hand, starting from a heap-ordered tree $t$, $t$ is arranged so that the set of all vertices with a given level number are ordered with labels increasing from right to left. Then the permutation is built by gathering the labels through a depth-first search (backtracking) of the tree from left to right. For instance, the permutation corresponding to the three labeled trees of the above example are (312), (231) and (213).
Finally, we use the term algebra of rooted trees and not Hopf algebra of rooted trees because, thanks to the work of Butcher, the Hopf structure is only one aspect of the ART.
Differentials and rooted trees
==============================
Assume that we want to solve the equation $(d/ds)x(s)=F[x(s)]$, $x(s_0)=x_0$, where $s$ is a real, $x$ is in $\mathbb{R}^N$ and $F$ is a smooth function from $\mathbb{R}^n$ to $\mathbb{R}^N$, with components $f^i(x)$. This is the equation of the flow of a vector field.
Calculation of the $n$-th derivative
------------------------------------
Let us write the derivatives of the $i$-th component of $x(s)$ with respect to $s$:
$$\begin{aligned}
\frac{d^2x^i(s)}{ds^2}&=&\frac{d}{ds} f^i[x(s)]
= \sum_j\frac{\partial f^i}{\partial x_j}[x(s)] \frac{dx^j}{ds}
= \sum_j\frac{\partial f^i}{\partial x_j}[x(s)] f^j[x(s)]\end{aligned}$$
$$\begin{aligned}
\frac{d^3x^i(s)}{ds^3}&=&\frac{d}{ds}
\left(\sum_j\frac{\partial f^i}{\partial x_j}[x(s)] f^j[x(s)]\right)\\
&=& \sum_{jk}\frac{\partial^2 f^i}{\partial x_j\partial x_k}[x(s)]
f^j[x(s)] f^k[x(s)] +
\sum_{jk}\frac{\partial f^i}{\partial x_j}[x(s)]
\frac{\partial f^j}{\partial x_k}[x(s)] f^k[x(s)].\end{aligned}$$
A simplified notation is now required. Let $$\begin{aligned}
f^i &=& f^i[x(s)] \\
f^i_{j_1j_2\cdots j_k} &=&
\frac{\partial^k f^i}{\partial x_{j_1}\cdots\partial x_{j_k}}[x(s)],\end{aligned}$$ so that $$\begin{aligned}
\frac{dx^i(s)}{ds}&=& f^i\quad
\frac{d^2x^i(s)}{ds^2}= f^i_j f^j\quad
\frac{d^3x^i(s)}{ds^3}= f^i_{jk} f^j f^k + f^i_j f^j_k f^k,\end{aligned}$$ where summation over indices appearing in lower and upper positions is implicitly assumed.
With this notation, we can write the next term as $$\begin{aligned}
\frac{d^4x^i(s)}{ds^4}&=& f^i_j f^j_k f^k_l f^l
+ f^i_j f^j_{kl} f^k f^l + 3 f^i_{jk} f^j_l f^k f^l
+ f^i_{jkl} f^j f^k f^l \\
&=&
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\quad
\quad
\quad
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\quad
\quad
\quad
\quad
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\quad
\quad
\quad
\quad
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}\end{aligned}$$
This relation between differentials and rooted tree was established by Arthur Cayley in 1857 [@Cayley]. With this notation, there is a one-to-one relation between a rooted tree with $n$ vertices and a term of $d^n x(s)/ds^n$
Elementary differentials
------------------------
A little bit more formally, we can follow Butcher ([@Butcher], p.154.) and call “elementary differentials” the $\delta_t$ defined recursively for each rooted tree $t$ by: $$\begin{aligned}
\delta^i_\bullet &=& f^i \nonumber\\
\delta^i_t &=&
f^i_{j_1j_2\cdots j_k} \delta^{j_1}_{t_1}
\delta^{j_2}_{t_2} \cdots \delta^{j_k}_{t_k}
\quad\mathrm{when}\quad t=B^+(t_1,t_2,\cdots,t_k). \label{defdeltat}\end{aligned}$$
Using this correspondence between rooted trees and differential expressions, we establish the identity: $$\begin{aligned}
N\delta_{t}\equiv\frac{d\delta_t}{ds}=\delta_{N(t)},\end{aligned}$$ where $N(t)$ is the natural growth operator of rooted trees defined in Ref.[@Connes].
So that the solution of the flow equation is $$\begin{aligned}
x(s)&=&x_0+\int_{s_0}^s ds' \exp[s'N]\delta_\bullet\nonumber\\
&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) \delta_t(s_0), \label{Butcherflow}\end{aligned}$$ where $|t|$ is the number of vertices of $t$, and $\alpha(t)$ is called $CM(t)$ in [@Kreimer].
Runge-Kutta methods
===================
We shall see that sum over trees appear quite naturally with differential equations. So, if one is given a function $\phi$ that assigns a value (e.g. a real, a complex, a vector) to each tree $t$, is there a function $f$ such that $\phi(t)=\delta_t$. Generally, the answer is no. Consider a function $\phi$ such that all components are equal (and denoted also by $\phi$): $$\begin{aligned}
\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1,\quad \phi({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})=a,\quad \phi({
\parbox{3mm}{
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}})=b, \end{aligned}$$ so that for any $i$, $f^i=1$, $f^i_j f^j=a$ and $f^i_j f^j_kf^k=b$. The first two equations give $\sum_jf^i_j=a$, so that the third gives $f^i_j f^j_kf^k=\sum_j f^i_j a=a^2$, and $\phi$ cannot be represented as elementary differentials (i.e. it cannot be the $\delta_t$) of a function $f$ if $b\not=a^2$. In fact, the number of functions reachable as elementary differentials is rather narrow.
Given such a function $\phi$ over rooted trees, we extend it to a homomorphism of the algebra of rooted trees by linearity and $\phi(tt')=\phi(t)\phi(t')$ where the componentwise product was used on the right-hand side. If vector flows are not enough to span all possible $\phi$, what more general equation can do that? As we shall see now, the answer is the Runge-Kutta methods.
Butcher’s approach to the Runge-Kutta methods
---------------------------------------------
To solve a flow equation $dx(s)/ds=F[x(s)]$, some efficient numerical algorithms are known as Runge-Kutta methods. They are determined by a $m\times m$ matrix $a$ and an $m$-dimensional vector $b$, and at each step a vector $x_n$ is defined as a function of the previous value $x_{n-1}$ by: $$\begin{aligned}
X_i &=& x_{n-1}+h\sum_{j=1}^m a_{ij} F(X_j)\\
x_n &=& x_{n-1}+h\sum_{j=1}^m b_j F(X_j),\end{aligned}$$ where $i$ range from $1$ to $m$. If the matrix $a$ is such that $a_{ij}=0$ if $j\ge i$ then the method is called explicit (because each $X_i$ can be calculated explicitly), otherwise the method is implicit.
In 1963, Butcher showed that the solution of the corresponding equations: $$\begin{aligned}
X_i(s) &=& x_0+(s-s_0)\sum_{j=1}^m a_{ij} F(X_j(s))\\
x(s) &=& x_0+(s-s_0)\sum_{j=1}^m b_j F(X_j(s)),\end{aligned}$$ is given by $$\begin{aligned}
X_i(s)&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) t! \sum_{j=1}^m a_{ij} \phi_j(t) \delta_t(s_0) \nonumber\\
x(s)&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) t! \phi(t) \delta_t(s_0). \label{Bseries}\end{aligned}$$
These series over trees are called B-series in the numerical analysis literature, in honour of John Butcher ([@Hairer], p.264). The homomorphism $\phi$ is defined recursively as a function of $a$ and $b$, for $i=1,\dots,m$: $$\begin{aligned}
\phi_i({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi_i(B^+(t_1\cdots t_k))&=&\sum_{j_1,\dots,j_k}
a_{ij_1}\dots a_{ij_k}\phi_{j_1}(t_1)\dots\phi_{j_k}(t_k)\\
\phi(t)&=&\sum_{i=1}^m b_i \phi_i(t).\end{aligned}$$ Comparing Eqs.(\[Butcherflow\]) and (\[Bseries\]) it is clear that the Runge-Kutta approximates the solution of the original flow equation up to order $n$ if $\phi(t)=1/t!$ for all trees with up to $n$ vertices.
In 1972 [@Butcher72], Butcher made further progress. Firstly he showed that Runge-Kutta methods are “dense” in the space of rooted tree homomorphisms. More precisely, he showed that given any finite set of trees $T_0$ and any function $\theta$ from $T_0$ to $\mathbb{R}$, then there is a Runge-Kutta method (i.e. a matrix $a$ and a vector $b$) such that the corresponding $\phi$ agrees with $\theta$ on $T_0$ (see also [@Butcher] p.167).
Further developments
--------------------
Furthermore, Butcher proved that the combinatorics he used to study Runge-Kutta methods in 1963 [@Butcher63] was hiding an algebra. If ($a$,$b$) and ($a'$,$b'$) are two Runge-Kutta methods, with the corresponding homomorphisms $\phi$ and $\phi'$, then the product homomorphism is defined (in Hopf algebra terms) by $$\begin{aligned}
(\phi\star\phi')(t)&=&m[(\phi\otimes\phi')\Delta(t)].\end{aligned}$$
Butcher proved that the $\phi$ derived from Runge-Kutta methods form a group. Again, this is nicely interpreted within the Hopf structure of the ART. For instance, the inverse of the element $\phi$ is simply defined by $\phi^{-1}(t)=\phi[S(t)]$, where $S$ is the antipode. This concept of inverse is quite important in practice since it is involved in the concept of self-adjoint Runge-Kutta methods, which have long-term stability in time-reversal symmetric problems ([@Hairer], p.219). The adjoint is defined within our approach by $\phi^*(t)=(-1)^{|t|}\phi[S(t)]$.
On the other hand, Butcher found an explicit expression for all the Hopf operations of the ART. Given the method ($a$,$b$) for $\phi$, he expressed the method ($a'$,$b'$) for $\phi\circ S$ ($S$ is the antipode) in (simple) terms of ($a$,$b$). Moreover, ([@Butcher], p.312 et sq.), if ($a$,$b$) and ($a'$,$b'$) are two Runge-Kutta methods (with dimensions $m$ and $m'$, respectively), corresponding to $\phi$ and $\phi'$, the method ($a"$,$b"$) corresponding to the convolution product $(\phi\star\phi')$ is $$\begin{aligned}
a''_{ij}&=&a_{ij}\mathrm{\quad if\quad} 1\le i\le m
\mathrm{\quad and\quad} 1\le j\le m,\\
a''_{ij}&=&a'_{ij}\mathrm{\quad if\quad} m+1\le i\le m+m'
\mathrm{\quad and\quad} m+1\le j\le m+m',\\
a''_{ij}&=&b_{j}\mathrm{\quad if\quad} m+1\le i\le m+m'
\mathrm{\quad and\quad} 1\le j\le m,\\
a''_{ij}&=&0 \mathrm{\quad if\quad} 1\le i\le m
\mathrm{\quad and\quad} m+1\le j\le m+m',\\
b''_i &=& b_i \mathrm{\quad if\quad} 1\le i\le m,\\
b''_i &=& b'_i \mathrm{\quad if\quad} m+1\le i\le m+m'.\end{aligned}$$
In 1974, Hairer and Wanner ([@Hairer], p.267) built upon the work of Butcher and proved the following important result: if we denote $$\begin{aligned}
B(\phi,F)&=&1+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) t! \phi(t) \delta_t(s_0) \label{Hairer1}\end{aligned}$$ then $$\begin{aligned}
B(\phi',B(\phi,F))&=&B(\phi\star\phi',F), \label{Hairer2}\end{aligned}$$ where $B(\phi',B(\phi,F))$ is the same as Eq.(\[Hairer1\]), with $\phi(t)$ replaced by $\phi'(t)$ and $\delta_t$ replaced by $\delta'(t)$ (i.e. $\delta'(t)$ is calculate as $\delta_t$, but with the function $B(\phi,F)(s)$ instead of the function $F(x(s))$).
In other words, the group of homomorphisms acts on the right on the functions $F$.
The continuous limit
====================
In his seminal article [@Butcher72], Butcher did not restrict his treatment to finite sets of indices. It is possible to consider the continuous limit of Runge-Kutta methods. A possible form of it is an integral equation, which we write artitrarily between 0 and 1: $$\begin{aligned}
X_u(s) &=& x_0+(s-s_0)\int_0^1 dv a(u,v) F(X_v(s))\\
x(s) &=& x_0+(s-s_0)\int_0^1 b(u) du F(X_u(s)),\end{aligned}$$ the solution of which are $$\begin{aligned}
X_u(s)&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) t! \int_0^1 dv a(u,v) \phi_v(t) \delta_t(s_0)\\
x(s)&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) t! \phi(t) \delta_t(s_0).\end{aligned}$$ The homomorphism $\phi$ is defined recursively as a function of $a$ and $b$: $$\begin{aligned}
\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi_u(B^+(t_1\cdots t_k))&=&\int_0^1 du_1 a(u,u_1) \phi_{u_1}(t_1)
\dots\int_0^1 du_k a(u,u_k) \phi_{u_k}(t_k)\\
\phi(t)&=&\int_0^1 du b(u) \phi_u(t).\end{aligned}$$ Continuous RK-methods do not seem to have been much used, except for an example in Butcher’s book ([@Butcher] p.325).
Butcher’s example \[Butcherexample\]
------------------------------------
It will be useful in the following to have the results of a modified version of Butcher’s example. So, we consider: $$\begin{aligned}
X_u(s)&=&x_0+(s-s_0)\int_0^u F[X_v(s)] dv \label{Picard}\\
x(s)&=&x_0+(s-s_0)\int_0^1 F[X_u(s)] du\nonumber,\end{aligned}$$ which corresponds to $a(u,v)=1_{[0,u]}(v)$, $b(u)=1$. This Runge-Kutta method will be used again in the sequel, and will be referred to as the “simple integral method”.
If we take the derivative of Eq.(\[Picard\]) with respect to $u$ we obtain $$\begin{aligned}
\frac{d}{du} X_u(s)=(s-s_0) F[X_u(s)],\end{aligned}$$ so $X_u(s)=y(s_0+(s-s_0)u)$, where $y(s)$ is the solution $$\begin{aligned}
y(s)=x_0+\int_{s_0}^s F[y(s')] ds'.\end{aligned}$$ Moreover $$\begin{aligned}
x(s)&=&x_0+(s-s_0)\int_0^1 F[X_u(s)] du\\
&=&x_0+(s-s_0)\int_0^1 F[y(s_0+(s-s_0)u)] du\\
&=&x_0+\int_{s_0}^s F[y(s')] ds' = y(s).\end{aligned}$$
The corresponding homomorphism $\phi(t)$ is defined by $$\begin{aligned}
\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi_u(B^+(t_1\cdots t_k))&=&\int_0^u du_1 \phi_{u_1}(t_1)
\dots\int_0^u du_k \phi_{u_k}(t_k)\\
\phi(t)&=&\int_0^1 du \phi_u(t).\end{aligned}$$ Using the facts that $|B^+(t_1\cdots t_k)|=(|t_1|+\cdots+|t_k|+1)$ and $B^+(t_1\cdots t_k)!=(|t_1|+\cdots+|t_k|+1)t_1!\dots t_k!$ it is proved that the solutions of these equations are $$\begin{aligned}
\phi_u(t)&=&\frac{|t|u^{|t|-1}}{t!}\\
\phi(t)&=&\frac{1}{t!}.\end{aligned}$$
If we introduce $\phi(t)=1/t!$ into Eq.(\[Bseries\]) we obtain Eq.(\[Butcherflow\]). So we confirm that the solution of the equation $$\begin{aligned}
x(s)=x_0+\int_{s_0}^s F[x(s')] ds'\end{aligned}$$ is $$\begin{aligned}
x(s)&=&x_0+
\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\alpha(t) \delta_t(s_0). \end{aligned}$$
First applications
------------------
The above example can already bring some interesting applications. But we must start by giving a way to calculate $\delta_t(s_0)$ in a simple case.
### Calculation of $\delta_t(s_0)$ \[sectiondeltat\]
To obtain specific results, we must choose a particular function $F$. The simplest choice is to take a vector function $F$, with components $f^i(x)=f(\sum_j x_j/N)$, where $N$ is the dimension of the vector space and $f$ has the series expansion $$\begin{aligned}
f(s)=\sum_{n=0}^\infty \frac{f^{(n)}(0) s^n}{n!}.\end{aligned}$$
From the definition of $\delta_t$ in Eq.(\[defdeltat\]), one can show recursively that, for $i=1,\dots,N$, $\delta^i_t(0)$ is independent of $i$ (and will be denoted $\delta_t$) and $$\begin{aligned}
\delta_\bullet &=& f(0) \nonumber\\
\delta_t &=& f^{(k)}(0) \delta_{t_1} \delta_{t_2} \cdots \delta_{t_k}
\quad\mathrm{when}\quad t=B^+(t_1,t_2,\cdots,t_k). \label{defdeltatsimple}\end{aligned}$$ In ref.[@Kreimer], Kreimer defined a similar quantity, that he called $B_t$. Here $\delta_t$ and $B_t$ will be used as synonymous.
The simplest case is $f(s)=\exp s$ and $s_0=0$, where $f^{(n)}(0)=1$ and $\delta_t=1$ for all trees $t$.
### Weighted sum of rooted trees
If we take $f=\exp$, $s_0=0$ and $x_0=0$ in Butcher’s example (see section \[Butcherexample\]), we have to solve the equation $$\begin{aligned}
x(s)=\int_{0}^s \exp[x(s')] ds'\end{aligned}$$ which can be differentiated to give $x'(s)=\exp(x(s))$ with $x(0)=0$. This has the solution $$\begin{aligned}
x(s)=-\log(1-s)=\sum_{n=1}^\infty \frac{s^n}{n}.\end{aligned}$$ On the other hand, the corresponding homomorphism is $\phi(t)=1/t!$ and the B-series for this problem is $$\begin{aligned}
x(s)&=&
\sum_t \frac{s^{|t|}}{|t|!}
\alpha(t).\end{aligned}$$ Comparing the last two results, we find $$\begin{aligned}
\sum_{|t|=n} \alpha(t)=(n-1)!\end{aligned}$$ in other words, the number of heap-ordered trees with $n$ vertices is $(n-1)!$.
### Derivative of inverse functions
We can try to extend the last example to an arbitrary function $f(x)$. The equation to solve becomes $$\begin{aligned}
x(s)=\int_{0}^s f[x(s')] ds',\end{aligned}$$ or $x'(s)=f(x(s))$ with $x(0)=0$. Let $$\begin{aligned}
S(x)=\int_{0}^x \frac{dy}{f(y)},\end{aligned}$$ which gives us $s=S(x)$, or $x(s)=S^{-1}(s)$, where $S^{-1}$ is the inverse function of $S$. If $f=\exp$, $S(x)=1-\exp(-x)$ and we confirm that $x(s)=-\log(1-s)$.
We can use this result to calculate the derivatives of a function $x(s)$, given as the inverse of a function $S(x)$. To do this, we define $f(x)=1/S'(x)$ and, using Eq.(\[Butcherflow\]), we obtain $$\begin{aligned}
x^{(n)}(0) &=&\sum_{|t|=n} \alpha(t) \delta_t,\end{aligned}$$ where $\delta_t$ is calculated from $f(s)$ using Eq.(\[defdeltatsimple\]) in section \[sectiondeltat\].
This method can also be calculated to find the function $f$ satisfying given values for $$\begin{aligned}
a_n=\sum_{|t|=n} \alpha(t) \delta_t,\end{aligned}$$ where $\delta_t$ is calculated from $f$. For instance, if we want $$\begin{aligned}
\sum_{|t|=n} \alpha(t) \delta_t=n!,\end{aligned}$$ we must take $f(s)=1+s^2$.
### Other sums over trees \[sectionothersum\]
We give now further examples of sums over trees, that will be used in the sequel. For instance, assume that we need to compute $$\begin{aligned}
S=\sum_{|t|=n} \frac{\alpha(t)}{t!}.\end{aligned}$$ This term comes in the Butcher series with $\phi(t)=1/(t!)^2$. Since this $\phi(t)$ is the square of the previous one, the corresponding Runge-Kutta method can be realized as the tensor product of two “simple integral methods” (see section \[Butcherexample\]). In other words $$\begin{aligned}
a(u,u',v,v')&=&a(u,v) a(u',v')=1_{[0,u]}(v)1_{[0,u']}(v')
\quad b(u,u')=b(u)b(u')=1.\end{aligned}$$ and the Runge-Kutta method is now $$\begin{aligned}
X_{uu'}(s)&=&x_0+(s-s_0)\int_0^u dv \int_0^{u'} dv'f[X_{vv'}(s)] \\
x(s)&=&x_0+(s-s_0)\int_0^1du \int_0^1du' f[X_{uu'}(s)].\end{aligned}$$ The corresponding homomorphism $\phi(t)$ is given by $$\begin{aligned}
\phi_{uu'}({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi_{uu'}(B^+(t_1\cdots t_k))&=&\int_0^u du_1 \int_0^{u'} du'_1
\phi_{u_1u'_1}(t_1)
\dots\int_0^u du_k \int_0^{u'} du'_k \phi_{u_ku'_k}(t_k)\\
\phi(t)&=&\int_0^1 du \int_0^1 du'\phi_{uu'}(t).\end{aligned}$$ The solutions of these equations are $$\begin{aligned}
\phi_{uu'}(t)&=&\frac{|t|^2(uu')^{|t|-1}}{(t!)^2}\\
\phi(t)&=&\frac{1}{(t!)^2},\end{aligned}$$ so that, from Eq.(\[Bseries\]) $$\begin{aligned}
X_{uu'}(s)&=&x_0+\sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\frac{\alpha(t)(uu')^{|t|}}{t!} \delta_t(s_0). \end{aligned}$$ The conclusion is that $X_{uu'}(s)$ is in fact a function of $uu'$ and not of $u$ and $u'$. More precisely, we know from the general formula Eq.(\[Bseries\]) that the B-series for the solution of $x(s)=x_0+(s-s_0)\int_0^1du \int_0^1du' f[X_{uu'}(s)]$ is $$\begin{aligned}
x(s)&=&x_0+ \sum_t \frac{(s-s_0)^{|t|}}{|t|!}
\frac{\alpha(t)}{t!} \delta_t(s_0),\end{aligned}$$ so that $X_{uu'}(s)=x(s_0+(s-s_0)uu')$. If we use the successive changes of variables $w=uu'$, $v'=s_0+(s-s_0)w$ and $v=s_0+(s-s_0)u$ we find $$\begin{aligned}
x(s)&=&x_0+(s-s_0)\int_0^1du \int_0^1du' f[x(s_0+(s-s_0)uu')]\\
&=&x_0+(s-s_0)\int_0^1\frac{du}{u} \int_0^u dw f[x(s_0+(s-s_0)w)]\\
&=&x_0+\int_0^1\frac{du}{u} \int_{s_0}^{s_0+(s-s_0)u}
dv' f[x(v')]\\
&=&x_0+\int_{s_0}^{s}\frac{dv}{v-s_0} \int_{s_0}^{v}
dv' f[x(v')].\end{aligned}$$ With the initial values $x_0=s_0=0$ this gives us $$\begin{aligned}
x(s) &=&\int_{0}^{s}\frac{dv}{v} \int_{0}^{v} dv' f[x(v')],
\label{doublefact}\end{aligned}$$ or $sx''+x'=f(x)$ with $x(0)=0$ and $x'(0)=f(0)$. If we take again $f(x)=\exp(x)$ we find $sx''+x'=\exp(x)$ with $x(0)=0$ and $x'(0)=1$, so that $$\begin{aligned}
x(s)=-2\log(1-s/2)=\sum_{n=1}^\infty \frac{s^n}{n2^{n-1}}.\end{aligned}$$ Comparing this with the B-series $$\begin{aligned}
x(s)&=&
\sum_t \frac{s^{|t|}}{|t|!}
\frac{\alpha(t)}{t!}\end{aligned}$$ we obtain $$\begin{aligned}
S=\sum_{|t|=n} \frac{\alpha(t)}{t!}=\frac{(n-1)!}{2^{n-1}},\end{aligned}$$ which is the result found by Kreimer in [@Kreimer] using combinatorial arguments.
As a final example, we can consider the Runge-Kutta method $a(u,v)=1$, $b(u)=1$ which gives $\phi(t)=1$ for all trees $t$. The equation for $x(s)$ is now a fixed point problem $x(s)=s\exp(x(s))$, whose well-known solution is $$\begin{aligned}
x(s)&=&
\sum_n \frac{s^n}{n!} n^{n-1},\end{aligned}$$ so that $$\begin{aligned}
\sum_{|t|=n} \alpha(t) t!=n^{n-1}.\end{aligned}$$
These examples show that B-series can be used as generating series for sums over trees.
The antipode \[sectionantipode\]
--------------------------------
The Hopf algebra structure of the ART entails an antipode $S$. If $\phi(t)$ is an homomorphism, the action of the antipode on $\phi$ can be written as $S(\phi)(t)=\phi(S(t))$. If the Runge-Kutta method for $\phi$ is $A_u$, $B$, then the Runge-Kutta method for $\phi^S=S(\phi)$ is $A^S_u=A_u-B$, $B^S=-B$. It is useful to see it working on simple cases: $$\begin{aligned}
\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1=\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\\
\phi^S({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&B^S(\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))=-B(\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))=-\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\\
\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&A^S_u(\phi^S_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))=A_u(\phi_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))-B(\phi_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))=
\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})-\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\\
\phi^S({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&-B(\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}))=-\phi({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})+B(1)\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})=
-\phi({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})+\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\end{aligned}$$
The convolution \[sectionconvolution\]
--------------------------------------
The convolution of $\phi$ and $\phi'$ is defined as $\phi''(t)=(\phi\star\phi')(t)=m[(\phi\otimes\phi')\Delta(t)]$.
Let $A_u$, $B$ and $A'_u$, $B'$ be the Runge-Kutta methods of, respectively, $\phi(t)$ and $\phi'(t)$. To be specific, we consider that $u$ varies from 0 to 1. Then the Runge-Kutta method for $\phi''$ is $A''_u$, $B''$, where $u$ varies from 0 to 2 and $$\begin{aligned}
A''_u(X_v)&=&A_u(X_v)\quad\mathrm{if}\quad 0\le u\le 1
\quad\mathrm{and}\quad 0\le v\le 1\\
A''_u(X_v)&=&0\quad\mathrm{if}\quad 0\le u\le 1
\quad\mathrm{and}\quad 1\le v\le 2\\
A''_u(X_v)&=&B(X_v) \quad\mathrm{if}\quad 1\le u\le 2
\quad\mathrm{and}\quad 0\le v\le 1\\
A''_u(X_v)&=&A'_{u-1}(X_{v-1}) \quad\mathrm{if}\quad 1\le u\le 2
\quad\mathrm{and}\quad 1\le v\le 2\\
B''(X_v)&=&B(X_v)\quad\mathrm{if}\quad 0\le v\le 1\\
B''(X_v)&=&B'(X_{v-1})\quad\mathrm{if}\quad 1\le v\le 2.\end{aligned}$$
Again, we show the formula in action: $$\begin{aligned}
\phi''_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi''({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&B(1)+B'(1)=\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})+\phi'({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\\
\phi''_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&A''_u(\phi''_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))=A_u(1) 1_{[0,1](u)}
+(B(1)+A'_{u-1}(1))1_{[1,2](u)}\\
\phi''({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&B(A_u(1))+B'((B(1)+A'_{u-1}(1)))=
B(\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))+B(1)B'(1)+B'(\phi'u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))\\
&=& \phi({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})+\phi({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})\phi'({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})+\phi'({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}).\end{aligned}$$
Runge-Kutta methods for renormalization
=======================================
In this section, we shall follow closely Kreimer’s paper [@Kreimer] and define, for each operation on homomorphisms, a corresponding transformation of the Runge-Kutta methods. Instead of attempting a general theory, we consider a specific example in detail.
Runge-Kutta method for bare quantities
--------------------------------------
We consider that a given bare physical quantity can be calculated as a sum over trees, and that the corresponding Runge-Kutta method has been found as a pair of linear operators $A_u$ and $B$. The usual combinatorial proof show that the solution of the equations (we take $s_0=0$) $$\begin{aligned}
X_u(s) &=& x_0+s A_u[f(X_v(s))]\\
x(s) &=& x_0+s B[f(X_u(s))],\end{aligned}$$ is $$\begin{aligned}
X_u(s)&=&x_0+\sum_t \frac{s^{|t|}}{|t|!}
\alpha(t) t! A_u[\phi_v(t)] \delta_t\\
x(s)&=&x_0+\sum_t \frac{s^{|t|}}{|t|!}
\alpha(t) t! \phi(t) \delta_t,\end{aligned}$$ where, as usually, $$\begin{aligned}
\phi_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi_u(B^+(t_1\cdots t_k))&=&A_u[\phi_{u_1}(t_1)]
\dots A_u[\phi_{u_k}(t_k)]\\
\phi(t)&=&B[\phi_u(t)].\end{aligned}$$ Here $x(s)$ is the sum giving the bare quantity of interest. In the examples developed by Broadhurst and Kreimer [@Broadhurst], the quantity of interest is $$\begin{aligned}
x(s)&=&\sum_t \frac{s^{|t|}}{|t|!} B_t,\end{aligned}$$ where $B_t$ is obtained recursively from given $B_n$ by $$\begin{aligned}
B_\bullet &=& B_1 \nonumber\\
B_t &=& B_{|t|} \delta_{t_1} \delta_{t_2} \cdots \delta_{t_k}
\quad\mathrm{when}\quad t=B^+(t_1,t_2,\cdots,t_k). \end{aligned}$$
In the renormalization problems considered by Broadhurst and Kreimer, the $B_n$ are defined from a function $L(\delta)$ regular (and equal to 1) at the origin, by $$\begin{aligned}
B_n &=&\frac{L(n\epsilon)}{n\epsilon}.\end{aligned}$$
A pair of operators giving $\phi(t)=B_t$ can be defined as $$\begin{aligned}
A_u(X_v) &=& \frac{1}{\epsilon} \int_0^u L(\epsilon \frac{d}{dv}v)X_v,
\quad B(X_v)=A_1(X_v).\end{aligned}$$
The quantity of interest $x(s)$ is then obtained by tensoring $A_u$ with the “simple integral method” to obtain $\phi(t)=B_t/t!$.
The only thing that we need in the following is the action of $A_u$ on a monomial $v^{n-1}$ $$\begin{aligned}
A_u(v^{n-1}) &=& \frac{1}{\epsilon} \int_0^u L(\epsilon \frac{d}{dv}v)v^{n-1}=
\frac{1}{\epsilon} \int_0^u v^{n-1}dv L(n \epsilon )=B_n u^n. \label{actiondeAu}\end{aligned}$$
$S_R$, the “renormalized antipode”
----------------------------------
In ref.[@Kreimer], Kreimer defines recursively a renormalized antipode[^3] depending on a renormalization scheme $R$. We take as an example the toy model used by Kreimer, where $R[\phi]=\langle\phi\rangle$ is the projection of $\phi$ on the pole part of the Laurent series in $\epsilon$ inside the bracket.
Following the results of section \[sectionantipode\], the Runge-Kutta method for $S_R(\phi)$ can be obtained from the Runge-Kutta method of $\phi$ by $A^S_u(X)=A_u(X)-\langle A_1(X)\rangle $, $B^S(X)=-\langle A_1(X)\rangle $. Working out the first examples using Eq.(\[actiondeAu\]), we find, $$\begin{aligned}
\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&1\\
\phi^S({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&\langle A_1(1) \rangle=\langle B_1 \rangle\\
\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&A_u(\phi_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))-\langle A_1(\phi_v({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}))\rangle=
A_u(1)-\langle A_1(1)\rangle=B_1 u- \langle B_1 \rangle\\
\phi^S({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&-\langle A_1(\phi^S_u({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}))\rangle=
-\langle B_2 B_1 \rangle + \langle\langle B_1 \rangle B_1 \rangle.\end{aligned}$$
Renormalized quantities
-----------------------
Finally, the renormalized quantities $x^R(s)$ are obtained from the convolution of $S_R(\phi)$ with $\phi$. To obtain the corresponding Runge-Kutta method, we use the results of section \[sectionconvolution\]. However, the domain where $1\le u\le 2$ is not used, and the Runge-Kutta method for the renormalized quantity is $A^R_u(X)=A_u(X)-\langle A_1(X)\rangle $, $B^R(X)=A_1(X)-\langle A_1(X)\rangle $. It may seem surprising that such a simple equation encodes the full combinatorial complexity of renormalization. It is not even necessary to work examples out, because $A^R_u(X)=A^S_u(X)$ so that $\phi^R_u(t)=\phi^S_u(t)$, and the only difference comes from the action of $B^R$.
For a real calculation of $x^R(s)$, we do not need $A^R_u$ and $B^R$ which give us $\phi(t)=\Gamma(t)$, but the tensor product of this method with the “simple integral method” to obtain $\phi(t)=t! \Gamma(t)$. In detail, the equation for the renormalized quantity $x^R(s)$ is $$\begin{aligned}
X^R_{uu'}(s)&=&\frac{s}{\epsilon}\int_0^u dv \int_0^{u'} dv'
L(\epsilon \partial_v v) e^{X_{vv'}(s)}-
\langle \frac{s}{\epsilon}\int_0^1 dv \int_0^{u'} dv'
L(\epsilon \partial_v v) e^{X_{vv'}(s)}\rangle \\
x^R(s)&=&\frac{s}{\epsilon}\int_0^1 dv \int_0^{1} dv'
L(\epsilon \partial_v v) e^{X_{vv'}(s)}-
\langle \frac{s}{\epsilon}\int_0^1 dv \int_0^{1} dv'
L(\epsilon \partial_v v) e^{X_{vv'}(s)}\rangle.
\label{Req1}\end{aligned}$$
For a general renormalization scheme $R$, one replaces $\langle A_u(X) \rangle$ by $R[A_u(X)]$. Finally, Chen’s lemma for renormalization schemes [@Kreimer] is obtained from Hairer and Wanner’s theorem Eq.(\[Hairer2\]).
Renormalization of Kreimer’s toy model
======================================
In this section, we use Runge-Kutta methods to renormalized explicitly Kreimer’s toy model for even functions $L(\epsilon)$. In [@Broadhurst], remarkable properties of the renormalized sum of diagrams with “Connes-Moscovici weights” were noticed.
Equation for the renormalized quantity
--------------------------------------
The role of the sum over $u'$ in Eq.(\[Req1\]) is to add a factor $1/t!$, as in section[\[sectionothersum\]]{}. Therefore, the same reasoning can be used to show that $X^R_{uu'}(s)$ is in fact a function of $su'$ and we write $X^R_{uu'}(s)=X^R_{u}(su')$, which defines the function $X^R_{u}(s)$. The equation for $X^R_{u}(s)$ can be found from Eq.(\[Req1\]) and the relation $X^R_{u}(s)=X^R_{us}(1)$ as
$$\begin{aligned}
X^R_{u}(s)&=&\frac{1}{\epsilon}\int_0^u dv \int_0^{s} ds'
L(\epsilon \partial_v v) e^{X_{v}(s')}-
\langle \frac{1}{\epsilon}\int_0^1 dv \int_0^{s} ds'
L(\epsilon \partial_v v) e^{X_{v}(s')}\rangle \\
x^R(s)&=&\frac{1}{\epsilon}\int_0^1 dv \int_0^{s} ds'
L(\epsilon \partial_v v) e^{X_{v}(s')}-
\langle \frac{1}{\epsilon}\int_0^1 dv \int_0^{s} ds'
L(\epsilon \partial_v v) e^{X_{v}(s')}\rangle.
\label{Req2}\end{aligned}$$
To solve this equation, we expand $X^R_{u}(s)$ in a power series over $u$: $$\begin{aligned}
X^R_{u}(s)&=&\sum_{n=0}^{\infty} a_n(s)u^n.\end{aligned}$$ A standard identity gives us $$\begin{aligned}
\exp(X^R_{u}(s))&=&\sum_{n=0}^{\infty} \lambda_n(a)u^n,\quad\mathrm{where}\\
\lambda_n(a)&=&\sum_{|\alpha|=n}\frac{a_1^{\alpha_1}\cdots a_n^{\alpha_n}}
{{\alpha_1}!\cdots{\alpha_n}!}\quad\mathrm{with}\quad
|\alpha|=a_1+2\alpha_2+\dots+n\alpha_n.\end{aligned}$$ $\lambda_n(a)$ depends on $s$ through its arguments $a_i(s)$. The sets of $\alpha_i$ for a given $n$ can be obtained from the partitions of $n$: $(\mu_1,\dots,\mu_n)$, where $\mu_1 \ge \cdots \ge \mu_n$ by $\alpha_n=\mu_n$, $\alpha_i=\mu_i-\mu_{i+1}$ for $i<n$.
The first $\lambda_n(a)$ are $$\begin{aligned}
\lambda_0(a)&=&1\quad
\lambda_1(a)=a_1\quad
\lambda_2(a)=a_2+\frac{a_1^2}{2}\quad
\lambda_3(a)=a_3+a_1a_2+\frac{a_1^3}{6}.\end{aligned}$$
Solution of the equation
------------------------
Introducing the series expansions for $X^R_{u}(s)$ and $\exp(X^R_{u}(s))$ into Eq.(\[Req2\]) we obtain $$\begin{aligned}
\sum_{n=0}^{\infty} a_n(s)u^n&=&\sum_{n=0}^{\infty} B_{n+1}
\int_0^s e^{a_0(s')} \lambda_n(a) ds' u^{n+1}-
\langle\sum_{n=0}^{\infty} B_{n+1}
\int_0^s e^{a_0(s')} \lambda_n(a) ds' \rangle\end{aligned}$$ or $$\begin{aligned}
a_0(s)&=&-\langle\sum_{n=0}^{\infty} B_{n+1}
\int_0^s e^{a_0(s')} \lambda_n(a) ds' \rangle\nonumber\\
a_n(s)&=&B_n \int_0^s e^{a_0(s')} \lambda_{n-1}(a) ds'\quad\mathrm{for}\quad n>0.
\label{eqa0an}\end{aligned}$$
To solve this equation, we need to go back to the equation for the bare quantity $$\begin{aligned}
X^0_{u}(s)&=&\frac{1}{\epsilon}\int_0^u dv \int_0^{s} ds'
L(\epsilon \partial_v v) e^{X^0_{v}(s')}.
\label{eqbare}\end{aligned}$$ Again $X^0_{u}(s)$ is a function of $su$, we define $X^0(s)=X^0_{s}(1)$ which satisfies $$\begin{aligned}
X^0(s)&=&\frac{1}{\epsilon}\int_0^s \frac{du}{u} \int_0^{u} dv
L(\epsilon \partial_v v) e^{X^0(v)}.\end{aligned}$$
The solution of this equation is given by the B-series $$\begin{aligned}
X^0(s)&=&\sum_n {\bar{\alpha}}_n s^n\quad\mathrm{with}\quad
{\bar{\alpha}}_n = \sum_{|t|=n} \frac{\alpha(t)B_t}{|t|!}.
\label{bareeq}\end{aligned}$$ On the other hand, we can also expand $e^{X^0(v)}$ using the functions $\lambda_n(\bar{a})$. Identifying both sides of Eq.(\[bareeq\]), we obtain the relation $$\begin{aligned}
{\bar{a}}_n&=&\frac{B_n}{n}\lambda_{n-1}(\bar{a}).\label{abar}\end{aligned}$$
With this identity, we can now prove that, for the renormalized quantities, $$\begin{aligned}
a_n(s)&=&(g(s))^n{\bar{a}}_n,\quad\mathrm{where}\quad g(s)=\int_0^s\exp(a_0(s'))ds'.
\label{aid}\end{aligned}$$ Since $\lambda_0(a)=1$ and ${\bar{a}}_n=B_1$, this equation is true for $n=1$, from Eq.(\[eqa0an\]). If Eq.(\[aid\]) is true up to $n-1$, then $\lambda_{n-1}(a)=(g(s))^{n-1}\lambda_{n-1}(\bar{a})$ and the derivative of Eq.(\[eqa0an\]) gives us $$\begin{aligned}
a'_n(s)&=&B_n e^{a_0(s)} \lambda_{n-1}(a)=
B_n g'(s) (g(s))^{n-1} \lambda_{n-1}(\bar{a})=n(g(s))^{n-1} \bar{a}_n,\end{aligned}$$ by Eq.(\[abar\]). Integrating this equation with the condition $a_n(s)=0$ gives Eq.(\[aid\]) at level $n$.
By this we have proved that the flow for the renormalized quantity is a reparametrization of the flow for the bare quantity: $X^R_u(s)=a_0(s)+X^0(u g(s))$ and $X^R(s)=a_0(s)+X^0(g(s))$.
To determine $a_0(s)$ we proceed step by step. In Eq.(\[bareeq\]) we expand $L(\epsilon \partial_v v)$ over $\epsilon$. The first term is just 1, and we obtain Eq.(\[doublefact\]) with the solution $x(s)=-2\log(1-s/(2\epsilon))$. For the renormalized quantity, the most singular term becomes $X^0(g(s))=-2\log(1-g(s)/(2\epsilon))$. Since $X^R(s)$ is regular, this singular term must be compensated by a corresponding term in $a_0(s)$. By equating the most singular terms we obtain $a_0(s)=-2\log(1-g(s)/(2\epsilon))$. We know from Eq.(\[aid\]) that $a_0(s)=\log(g'(s))$, and we obtain the most singular terms as the solution of $g'(s)=1/(1-g(s)/(2\epsilon))^2$, which is: $$\begin{aligned}
g(s)&=&\frac{s}{1+\frac{s}{2\epsilon}}\\
a_0(s)&=& -2\log(1+\frac{s}{2\epsilon}).\end{aligned}$$ By expanding $a_0(s)$ as a series in $s$, we obtain the most singular term observed in [@Broadhurst] and proved in [@Kreimer]. One notices that the singularity of the non-pertubative term $a_0(s)$ is logarithmic, and much smoother than the singularities coming from the expansion over $s$ (i.e. the perturbative expression).
Differential equation for the finite part
-----------------------------------------
In general, one should proceed now with the next singular term. To obtain it we denote $Y(s)=X^0(g(s))$, this change of variable gives the equation for $Y(s)$: $$\begin{aligned}
Y(s)&=&\frac{1}{\epsilon}\int_0^s \frac{g'(u)du}{g(u)} \int_0^{u} dv
g'(v) L(\epsilon+\epsilon \frac{g(v)}{g'(v)}
\partial_v ) e^{Y(v)}.\end{aligned}$$
Now we can write $Y(s)=X^R(s)-a_0(s)$, and notice that the term $-a_0(s)$ on the left-hand side is compensated by a term on the right-hand side where $L=1$ and $\exp(X^R(s))=1$. We obtain the equation for $X^R(s)$: $$\begin{aligned}
X^R(s)&=&\frac{1}{\epsilon}\int_0^s \frac{du}{u(1+\frac{u}{2\epsilon})}
\int_0^{u} dv\left[
\frac{1}{(1+\frac{v}{2\epsilon})^2}
L(\epsilon
\partial_v v+\frac{v^2}{2}
\partial_v){(1+\frac{v}{2\epsilon})}^2 e^{X^R(v)}
-1\right].\end{aligned}$$ The nice aspect of the previous equation is that it seems to have a limit as $\epsilon$ goes to zero. In fact, it has a limit when $L$ is even, as we shall show now.
Writing $\bar{X}(s)=\lim_{\epsilon\rightarrow 0}X^R(s)$, and taking the limit $\epsilon\rightarrow 0$ in the previous equation, we obtain $$\begin{aligned}
\bar{X}(s)&=&2\int_0^s \frac{du}{u^2}
\int_0^{u} dv\left[
\frac{1}{v^2}
L(\frac{v^2}{2} \partial_v )v^2 e^{\bar{X}(v)}
-1\right],\end{aligned}$$ or, in differential form: $$\begin{aligned}
\frac{1}{2}(s^2\bar{X}'(s))'&=&\frac{1}{s^2}
L(\frac{s^2}{2}\frac{d}{ds})s^2 e^{\bar{X}(s)}
-1.\end{aligned}$$
If $\bar{X}(s)$ and $ L(\delta)$ are expanded as $$\begin{aligned}
\bar{X}(s)&=& \sum_{n=1}^\infty b_n s^n\quad\mathrm{and}\quad
L(\delta)= 1+\sum_{n=1}^\infty L_n \delta^n,\quad\mathrm{so that} \nonumber\\
L(\frac{s^2}{2}\frac{d}{ds})&=& 1+\sum_{n=1}^\infty L_n (\frac{s^2}{2}\frac{d}{ds})^n,
\label{renoreq}\end{aligned}$$ we obtain the following relation for the term in $s$: $b_1s=(b_1+L_1/2)s$. If $L_1$ is not zero, we obtain a contradition and must proceed with the withdrawal of divergences. For simplicity, we shall assume that $L_1=0$. Then $b_1$ becomes a free parameter of $\bar{X}(s)$. All terms $b_n$ with $n>1$ can now be determined from $b_1$ and $L_n$ ($n>1$). All terms are regular.
In [@Broadhurst], the function $L(\delta)$ was taken even. Then $L_1=0$, and their results correspond to $b_1=0$. Broadhurst and Kreimer have also used a function $L(\epsilon,\delta)$. The present treatment can be applied to this more general situation, with the only change that $$\begin{aligned}
L_n=n!\lim_{\epsilon\rightarrow 0}\lim_{\delta\rightarrow 0}
\frac{d^n}{d\delta^n} L(\epsilon,\delta).\end{aligned}$$
Clearly, Eq.(\[renoreq\]) is much faster to solve than computing the sum over trees. For instance, the expansion could be calculated up to 20 loops (i.e. $b_{20}$) within a few seconds with a computer.
Alternative point of view
-------------------------
There is an alternative way to solve Eq.(\[eqbare\]) for the bare quantity. We define a function $f(s)$ from $L(\delta)$ by $$\begin{aligned}
f(s)&=& \sum_{n=0}^\infty \frac{L(n\epsilon+\epsilon)}{n!} s^n=
L(\epsilon \frac{d}{ds}s) e^s.\end{aligned}$$ A relation between $f(s)$ and $L(\delta)$ can also be established through the Mellin transforms of $f$ and $L$ as $M(f)(z)=M(L)(\epsilon-\epsilon z)\Gamma(z)$.
With $f(s)$ we can write the equation for the bare quantity as $$\begin{aligned}
X^0(s)&=&\frac{1}{\epsilon}\int_0^s \frac{du}{u} \int_0^{u} dv
f(X^0(v)). \label{eqavecf}\end{aligned}$$
Alternatively, one can go from $f$ to $L$ and consider the results of the toy model as a method to renormalize equations of the type (\[eqavecf\]).
n-dimensional problems
======================
For applications to classical field theory, we need to develop Runge-Kutta methods for the n-dimensional analogue of the flow equation: non-linear partial differential equations. The purpose of the present section is to indicate how B-series can be used for this case[^4]. The method apply to equations of the form $L\psi(\mathbf{r})= F[\psi(\mathbf{r})]$, where $L$ is a differential operator (e.g. the nonlinear Schrödinger equation $\Delta\psi=\psi^3$).
Formulation
-----------
We need two starting elements: a function $\psi_0(\mathbf{r})$ which is the solution of $L\psi_0(\mathbf{r})=0$, and a Green function $G(\mathbf{r},\mathbf{r}')$, that is a solution of the equation $L_rG(\mathbf{r},\mathbf{r}')=\delta(\mathbf{r}-\mathbf{r}')$, with given boundary conditions. The function $\psi_0(\mathbf{r})$ will play the role of an initial value, and the Green function will decide in which “direction” you move from the initial value. It will also state, in some sense, the boundary conditions of the solution $\psi(\mathbf{r})$.
Using these two functions, the differential equation $L\psi(\mathbf{r})=F[\psi(\mathbf{r})]$ is transformed into $\psi(\mathbf{r})=\psi_0(\mathbf{r})+\int d\mathbf{r}'
G(\mathbf{r},\mathbf{r}')F[\psi(\mathbf{r}')]$. The action of $L$ enables us to go from the second to the first equation.
The combinatorics is the same as for the standard Runge-Kutta method, and the result is $$\begin{aligned}
\psi(\mathbf{r})&=&\psi_0(\mathbf{r})+\sum_t \frac{\alpha(t) t!}{|t|!}
\int d\mathbf{r}' G(\mathbf{r},\mathbf{r}')
\phi_{r'}(t),\label{Bseries_n}\end{aligned}$$ where $\phi_{r}(t)$ is defined recursively by $$\begin{aligned}
\phi_{r}({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&F[\psi_0(\mathbf{r})]\nonumber\\
\phi_{r}(B^+(t_1\cdots t_k))&=&F^{(k)}[\psi_0(\mathbf{r})]
\int dr_1 G(\mathbf{r},\mathbf{r}_1)\phi_{r_1}(t_1)
\dots \int dr_k G(\mathbf{r},\mathbf{r}_k)\phi_{r_k}(t_k). \label{Bphi_n}\end{aligned}$$
If $\psi$ is a vector field, the solution is the same, and equations (\[Bphi\_n\]) get indices: $$\begin{aligned}
\phi^i_{r}({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})&=&f^i[\psi_0(\mathbf{r})]\\
\phi^i_{r}(B^+(t_1\cdots t_k))&=&F^i_{j_1\dots j_k}[\psi_0(\mathbf{r})]
\int dr_1 G^{j_1}_{j'_1}(\mathbf{r},\mathbf{r}_1)\phi^{j'_1}_{r_1}(t_1)
\dots \int dr_k G^{j_k}_{j'_k}(\mathbf{r},\mathbf{r}_k)\phi^{j'_k}_{r_k}(t_k),\end{aligned}$$ where $G^i_j(\mathbf{r},\mathbf{r}')$ is a component of the matrix Green function.
In the previous sections, the series (\[Bseries\]) was written as a function of $\phi(t)$ (describing the effect of the Runge-Kutta method ($a$,$b$)) and $\delta_t$ (describing the effect of the function $F[x]$). In the present case, this separation is no longer possible, and $\phi(t)$ combines both pieces of information.
Examples
--------
In this section, equation (\[Bseries\_n\]) is applied to the one-dimensional problem and to the Schrödinger equation.
### The one-dimensional case
It is instructive to observe how the one-dimensional case is obtained from Eq.(\[Bseries\_n\]). The differential operator is $L=d/ds$, so the initial function $\psi_0(s)$ must satisfy $d/ds\psi_0(s)=0$: $\psi_0(s)$ is a constant that we write $x_0$. For the Green function $G(s,s')$, we have the equation $LG(s,s')=\delta(s-s')$, so $G(s,s')=\theta(s-s') +C(s')$, where $\theta(s)$ is the step function and $C(s')$ a function of $s'$. To determine $C(s')$, we note that, in the “simple integral method”, there is an integral from $s_0$ to $s$. From the Green function $G(s,s')=\theta(s-s') -\theta(s_0-s')$, we obtain $$\begin{aligned}
\int_{-\infty}^{\infty} G(s,s') f(s') ds'&=& \int_{s_0}^{s} f(s') ds'\end{aligned}$$ which is the required expression.
Now, the role of $\psi_0$ and the Green function is clear for the one-dimensional case: $\psi_0$ gives the initial value $x_0$ and $G$ specifies (among other things) the starting point $s_0$. To complete the derivation of the one-dimensional case, we note that $\psi_0(s)=x_0$ does not depend on $s$, so the terms $F^{(k)}[\psi_0(s)]=F^{(k)}[x_0]$ are independent of $s$ and can be grouped together to build $\delta_t$ as in (\[defdeltat\]). On the other hand, the integration over Green functions build up $(s-s_0)^{|t|}/t!$ and we obtain Eq.(\[Butcherflow\]).
### The Schrödinger equation I
If we write the Schrödinger equation as $(E+\Delta)\psi(\mathbf{r})=V(\mathbf{r})\psi(\mathbf{r})$, we can apply Eq.(\[Bseries\_n\]) with $F[\psi]=V(\mathbf{r})\psi$. We take for $\phi_0(\mathbf{r})$ a solution of $(E+\Delta)\psi_0(\mathbf{r})=0$ and for $G(\mathbf{r},\mathbf{r}')$ the scattering Green function (e.g. $G(\mathbf{r}-\mathbf{r}')=-e^{i\sqrt{E}|\mathbf{r}-\mathbf{r}'|}/(4\pi
|\mathbf{r},\mathbf{r}'|)$ in three dimensions).
The calculation of $\phi(t)$ is straightforward because, in a such a linear problem, $F^{(k)}=0$ for $k>1$. Hence, the only rooted trees that survive are those with one branch. For these trees $\alpha(t)=1$ and $t!=|t|!$ and we obtain $$\begin{aligned}
\psi(\mathbf{r})=\psi_0(\mathbf{r})+\int d\mathbf{r}_1
G(\mathbf{r},\mathbf{r}_1) V(\mathbf{r}_1)\psi_0(\mathbf{r}_1)+
\int d\mathbf{r}_1 d\mathbf{r}_2
G(\mathbf{r},\mathbf{r}_1) V(\mathbf{r}_1)
G(\mathbf{r}_1,\mathbf{r}_2) V(\mathbf{r}_2)\psi_0(\mathbf{r}_2)+\cdots\end{aligned}$$ where we recognize the Born expansion of the Lippmann-Schwinger equation.
### The Schrödinger equation II
We can also treat the Schrödinger equation in an alternative way as the system of equations: $$\begin{aligned}
(E+\Delta)\psi(\mathbf{r})&=& V(\rho)\psi(\mathbf{r}) \\
\frac{\partial \rho_i}{\partial r_j} &=& \delta_{ij}.\end{aligned}$$ This is a matrix differential equation. We give index 0 to the first line, and index $i$ (running from 1 to the dimension of space) to the other lines, called the space lines. The purpose of the space lines is just to ensure that $\rho=\mathbf{r}$. This is a standard trick to take the $\mathbf{r}$ dependence of $V$ into account in the expansion (see e.g. [@Hairer] p.143). As initial value we take $\psi_0(\mathbf{r})$ and $\rho_0=0$, the matrix Green function is diagonal and it is equal to the scattering wave function for line 0 and to $\theta(r_i-r'_i)-\theta(-r'_i)$ for line $i$.
For $\phi_r({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})$, the zero-th component is $V(0)\psi_0(\mathbf{r})$ and the space components are 1, for all the other trees, the space components are 0 and the zero-th component of the simplest tree is $$\begin{aligned}
\phi_r({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}})&=&V(0)^2 \int d\mathbf{r}'
G(\mathbf{r},\mathbf{r}') \psi_0(\mathbf{r}')+
\sum_i r_i\partial_i V(0) \psi_0(\mathbf{r})\\
\phi_r({
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}})&=&V(0)^3 \int d\mathbf{r}_1 G(\mathbf{r},\mathbf{r}_1) +
V(0) \int d\mathbf{r}' G(\mathbf{r},\mathbf{r}')
\sum_i r'_i\partial_i V(0) \psi_0(\mathbf{r}')\\
\phi_r({
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}})&=&2V(0)\sum_i r_i\partial_i V(0)
\int d\mathbf{r}'
G(\mathbf{r},\mathbf{r}') \psi_0(\mathbf{r}')+
\sum_{ij} r_i r_j\partial_i\partial_j V(0).\end{aligned}$$ The expressions become more and more complex, but their derivation is made systematic by the recurrence relation.
Conclusion
==========
Butcher’s approach to Runge-Kutta methods was applied to some simple renormalization problems. Since Cayley, it is clear that the ART is ideally suited to treat differentials. This was confirmed here by presenting a B-series solution of a class of non-linear partial differential equations.
The recursive nature of B-series make them computationally efficient: $\phi_u(t)$ can be obtained by a simple operation from the $\phi_u(t')$ of smaller order $t'$. This is why B-series can be automated and implemented in a computer.
Butcher’s approach has still much to offer. In the numerical analysis literature, B-series have been generalized to treat flow equations on Lie groups. The main change [@Munthe98] is to replace the algebra of rooted trees by the algebra of planar trees (also called ordered trees [@Owren]). The elementary differentials get then a “quantized calculus” flavor, especially in the definition given Munthe-Kaas [@Munthe95] in terms of commutators with the vector field $F=f^i\partial_i$ (see also Ginocchio). Using this generalized ART, extended work has been carried out recently for the numerical solution of differential equations on Lie groups (see Ref.[@Munthe98; @Owren] and the web site [http://www.math.ntnu.no/num/synode]{}).
B-series have been generalized in other directions, e.g. stochastic differential equations [@Komori] and differential equations of the type $dy/ds=f(y,z)$, $g(y,z)=0$, which are called differential algebraic equations [@HairerII].
It is our hope that Butcher’s approach can be applied to quantum field theory.
Acknowledgements
================
It is my great pleasure to thank Dirk Kreimer and Alain Connes for interest, encouragement and discussions.
Appendix
========
For further reference, the action of the coproduct and the antipode on the first few trees are given here.
Coproduct
---------
$$\begin{aligned}
\Delta 1 &=& 1 \otimes 1 \\
\Delta {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}&=& {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes 1 + 1 \otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
\Delta {\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}&=& {\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\otimes 1 + 1 \otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}&=& {
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}&=& {
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}+2
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}&=& {
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}&=& {
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+ 2
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.5h)}{o1}
\fmfforce{(0.1w,0.1h)}{o2}
\fmfforce{(0.9w,0.5h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{o1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}&=& {
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.5h)}{o1}
\fmfforce{(0.1w,0.1h)}{o2}
\fmfforce{(0.9w,0.5h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{o1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.5h)}{o1}
\fmfforce{(0.1w,0.1h)}{o2}
\fmfforce{(0.9w,0.5h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{o1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\otimes{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
\Delta
{
\parbox{8mm}{
\begin{fmfchar*}(8,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.5w,0.1h)}{o2}
\fmfforce{(0.9w,0.1h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}&=& {
\parbox{8mm}{
\begin{fmfchar*}(8,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.5w,0.1h)}{o2}
\fmfforce{(0.9w,0.1h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}\otimes 1 + 1 \otimes
{
\parbox{8mm}{
\begin{fmfchar*}(8,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.5w,0.1h)}{o2}
\fmfforce{(0.9w,0.1h)}{o3}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmf{vanilla}{i1,o3}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o3}
\end{fmfchar*}
}}+
3{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+
3{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\otimes
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\end{aligned}$$
Antipode
--------
$$\begin{aligned}
S(1) &=& 1 \\
S({\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}})
&=& -{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left({\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}\right)
&=& -{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+ {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left( {
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}\right)
&=& -{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+2 {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}-{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left( {
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\right)
&=& -{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+2 {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}-{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left( {
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}\right)
&=& -{
\parbox{3mm}{
\begin{fmfchar*}(3,12)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,v2}
\fmf{vanilla}{v2,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v2}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+2 {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}-3{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left( {
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}\right)
&=& -{
\parbox{6mm}{
\begin{fmfchar*}(6,10)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmfforce{(0.5w,0.45h)}{v1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmf{vanilla}{v1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}+2 {\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{
\parbox{3mm}{
\begin{fmfchar*}(3,8)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,v1}
\fmf{vanilla}{v1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{v1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}
}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{
\parbox{5mm}{
\begin{fmfchar*}(5,6)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.1w,0.1h)}{o1}
\fmfforce{(0.9w,0.1h)}{o2}
\fmf{vanilla}{i1,o1}
\fmf{vanilla}{i1,o2}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o2}
\end{fmfchar*}
}}-3{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,5)
\fmfforce{(0.5w,0.9h)}{i1}
\fmfforce{(0.5w,0.1h)}{o1}
\fmf{vanilla}{i1,o1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\fmfv{decor.shape=circle,decor.filled=0,decor.size=30}{o1}
\end{fmfchar*}}}+
{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}{\parbox{3mm}{
\begin{fmfchar*}(3,3)
\fmfforce{(0.5w,0.5h)}{i1}
\fmfv{decor.shape=circle,decor.filled=1,decor.size=30}{i1}
\end{fmfchar*}}}\\
S\left( {
\parbox{6mm}{
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[10]{}
D. Kreimer. On the [H]{}opf algebra structure of perturbative quantum field theory. , 2:303–???, 1998. q-alg/9707029.
D.J. Broadhurst and D. Kreimer. Renormalization automated by [H]{}opf algebra. , 1999. To be published, hep-th/9810087.
D. Kreimer and R. Delbourgo. Using the [H]{}opf algebra structure of [QFT]{} in calculations. 1999. hep-th/9903249.
D. Kreimer. On overlapping divergences. 1998. hep-th/9810022.
T. Krajewski and R Wulkenhaar. On [K]{}reimer’s [H]{}opf algebra structure of [F]{}eynman graphs. 1998. hep-th/9810022.
D. Kreimer. Chen’s iterated intregral represents the operator product expansion. 1999. hep-th/9901099.
A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. , 199:203–42, 1998.
C. Runge. Ueber die numerische [A]{}uflösung von [D]{}ifferentialgleichungen. , 46:167–78, 1895.
E. Hairer and G. Wanner. On the [B]{}utcher group and general multi-value methods. , 13:1–15, 1974.
A. Dür. . Springer, Berlin, 1986.
A. Tucker. . Wiley, New York, 1980.
S.V. Fomin and N. Lulov. On the number of rim hook tableaux. , 87:4118–23, 1997.
J.C. Butcher. Coefficients for the study of [R]{}unge-[K]{}utta integration processes. , 3:185–201, 1963.
J.C. Butcher. . Wiley, Chichester, 1987.
E. Hairer, S.P. N[ø]{}rsett, and G. Wanner. . Springer, Berlin, second edition, 1993.
A. Cayley. On the theory of the analytical forms called trees. , 13:172–6, 1857.
J.C. Butcher. An algebraic theory of integration methods. , 26:79–106, 1972.
H. Munthe-Kaas. unge-[K]{}utta methods on [L]{}ie groups. , 38:92–111, 1998.
B. Owren and A. Marthinsen. Runge-[K]{}utta methods adapted to manifolds and based on rigid frames. , 39:116–42, 1999.
H. Munthe-Kaas. Lie-[B]{}utcher theory for [R]{}unge-[K]{}utta methods. , 35:572–87, 1995.
Y. Komori, T. Mitsui, and H. Sugiura. Rooted tree analysis of the order condition conditions of [ROW]{}-type scheme for stochastic differential equations. , 37:43–66, 1997.
E. Hairer and G. Wanner. . Springer, Berlin, 1991.
[^1]: Hairer and Wanner called it the Butcher group [@Hairer74].
[^2]: The opposite of the present Hopf structure of ART was discussed in 1986 by Dür ([@Dur], p.88-90), together with the corresponding Lie algebra (identical, up to a sign, with the one defined in [@Connes]).
[^3]: In Hopf algebra terms $S_R(\phi)(t)=-R[\phi(t)+m[(S_R\otimes Id)(\phi\otimes \phi)P_2\Delta(t)]$.
[^4]: Kreimer was independently aware of the possibility to use B-series for non-linear partial differential equations.
|
---
author:
- 'D.'
title: 'Are galaxies shy ?'
---
Introduction
============
Until 1996, there was little evidence that most galaxies were “shy”, i.e. that they would hide their stars behind a veil of dust and turn red when forming stars, radiating the bulk of their luminosity in the infrared (IR) at a given epoch of their history. Ten years before, IRAS had unveiled a population of luminous IR galaxies exhibiting such a “shy” behavior, the so-called LIGs and ULIGs (with 12$\geq
log_{10}\left(L_{\rm IR}/L_{\odot}\right)\geq$ 11 and $log_{10}\left(L_{\rm IR}/L_{\odot}\right)\geq$ 12 respectively), which are responsible for the shape of the bolometric luminosity function of local galaxies above $\sim$ 10$^{11}~L_{\odot}$ (Sanders & Mirabel 1996). But integrated over the whole local luminosity function, LIGs and ULIGs only produce $\sim$ 2$\%$ of the total integrated luminosity and overall only $\sim$ 30$\%$ of the bolometric luminosity of local galaxies is radiated in the IR above $\lambda \sim$ 5$\mu$m. The discovery of an extragalactic background in the IR at least as large as the UV-optical-near IR one, the so-called cosmic infrared background (CIRB), with the COBE satellite (Puget et al. 1996, see references in Elbaz et al. 2002b) implied that shyness must have been more common among galaxies in the past than it is today. This was confirmed with the detection of an excess of faint mid IR (MIR) galaxies by ISOCAM onboard ISO (Elbaz et al. 1999), as well as in the far IR (FIR) with ISOPHOT onboard ISO (Dole et al. 2001) and in the sub-millimeter with SCUBA at the JCMT (see Smail et al. 2001). This excess is relative to expectations based on galaxies in the local universe. It implies that galaxies were more luminous in the IR regime and/or more numerous in the past (Chary & Elbaz 2001, Franceschini et al. 2001).
Mid infrared as a star formation indicator
==========================================
Chary & Elbaz (2001) and Elbaz et al. (2002b) demonstrated that the MIR luminosity of local galaxies is correlated with their integrated IR luminosity (8-1000$\mu$m). Hence MIR flux densities can be converted into $L_{\rm IR}$ and used to compute star formation rates (SFR). The sensitivities of the deepest surveys performed in the MIR (0.1 mJy at 15$\mu$m), FIR (120 mJy at 170$\mu$m) and sub-millimeter (2 mJy at 850$\mu$m) with ISOCAM, ISOPHOT and SCUBA and in the radio (40$\mu$Jy at 1.4 GHz, i.e. 21 cm, with the VLA and WSRT) to IR galaxies are compared in Fig. \[FIG:sfr\]a as a function of redshift (see also Elbaz et al. 2002b). Fig. \[FIG:sfr\]a shows that ISOCAM was the most sensitive instrument among the four selected and that it was able to detect nearly all luminous IR galaxies below $z\sim$ 1. A similar result is obtained using either the proto-typical spectral energy distribution (SED) of M 82 or the library of 100 template SEDs from Chary & Elbaz (2001) constructed to reproduce the correlations between MIR-FIR and sub-millimeter luminosities of local galaxies.
An indication that the SEDs in the IR of distant galaxies ressemble local ones comes from the distant “clone” of Arp 220 serendipitously discovered in the field of a QSO (PC 1643+4631). This galaxy, HR10 ($z=$ 1.44) known as an extremely red object (ERO) was detected in the radio, MIR and sub-millimeter with a SFR around 1000 M$_{\odot}$ yr$^{-1}$ (see Elbaz et al. 2002a and references therein).
The spatial resolution (4 arcsec PSF FWHM) of ISOCAM provided the possibility to identify rather easily optical counterparts to these galaxies and to determine their redshift. Due to limited telescope time allocation, their redshift distribution was inferred from a sub-sample of galaxies in the Hubble Deep Field North (HDFN, Aussel et al. 1999) and their luminosities and star formation rates are presented in the Fig. \[FIG:sfr\]b. About 75$\%$ of the galaxies brighter than about 0.1 mJy at 15$\mu$m, and responsible for the steep slope of the number counts, belong to the class of LIGs ($\sim$55$\%$) and ULIGs ($\sim$20$\%$). Their redshifts spread over the $z=$ 0.5-1.3 range with a median around ${\bar z}=$ 0.7-0.8.
The fraction of IR light produced by active nuclei was computed from the cross-correlation of ISOCAM with the deepest X-ray surveys from the Chandra and XMM-Newton observatories in the HDFN (41 MIR galaxies) and Lockman Hole (103 MIR galaxies) respectively. Less than 20$\%$ of the ISOCAM galaxies appear to be dominated by an AGN at 15$\mu$m ((12$\pm$5)$\%$ in the HDFN and (13$\pm$4)$\%$ in the Lockman Hole, see Fadda et al. 2001).
On the shyness of galaxies
==========================
The MIR-FIR correlations observed in the local universe (Elbaz et al. 2002b) can be used to compute the contribution of the luminous IR galaxies unveiled by ISOCAM below $z\sim$ 1.5 to the CIRB. Elbaz et al. (2002b) computed a contribution of (16$\pm$5) nW m$^{-2}$ Hz$^{-1}$ as compared to the peak value of the CIRB of (25$\pm$7) nW m$^{-2}$ Hz$^{-1}$ measured with COBE at $\lambda \sim$ 140$\mu$m. Hence luminous IR galaxies below $z\sim$ 1.5 are responsible for the bulk of the CIRB. Since the CIRB contains most photons radiated by galaxies over the history of the universe, this means that luminous IR galaxies represent a common phase for galaxies. Chary & Elbaz (2001) have studied the range of possible parameters for the evolution of galaxies in luminosity and density over the history of the universe, that would fit number counts from ISOCAM, ISOPHOT and SCUBA as well as the CIRB and the redshift distribution of ISOCAM galaxies. The major result of this study is that although a level of degeneracy remains in the choice of the parameters ruling the evolution of galaxies, existing observations set a strong constraint on the relatively recent ($z<1.5$) evolution of the number and luminosity density of luminous IR galaxies. The best fit is obtained for the cosmic history of star formation shown in the Fig. \[FIG:mstar\]a, where the relative roles of ULIGs, LIGs and “normal” galaxies are differentiated. Fig. \[FIG:mstar\]a implies that we are living at an epoch when “normal” galaxies ($L_{\rm
bol}<$ 10$^{11}$ $L_{\odot}$) contribute dominantly to the global star formation activity in the local universe, whereas above $z\sim$ 0.3 the reverse was true: the bulk of the cosmic density of star formation was due to luminous IR galaxies. Hence galaxies in general must have experienced a period of shyness, such as local LIGs and ULIGs, when they formed the bulk of their present-day stars. Fig. \[FIG:mstar\]b represents the fraction of present-day stars plus remnants formed as a function of lookback time or redshift for a given IMF (here from Gould et al. 1996). The total mass is comparable to the local density of baryons in the local universe (5$\pm$3$\times 10^{8}~M_{\odot}$ Mpc$^{-3}$, Fukugita et al. 1998). The error bar on the computed stellar mass is as large $\sim$ 50$\%$ (including the conversion from MIR to FIR and FIR to SFR), but this result suggests that the bulk of present-day stars formed at a time when their host galaxies experienced such a phase of shyness 5 to 10 Gyr ago, i.e. between $z=$ 0.5 and 2 for an age of the universe of 12.6 Gyr in our cosmology (H$_o$= 75 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm matter}$= 0.3, $\Omega_{\Lambda}= 0.7$). The shyness of galaxies seems to be the result of galaxy encounters since all ISOCAM galaxies in the HDFN are either merging or members of small groups of galaxies (see Aussel, in this conference). The fact that the CIRB peaks around $\lambda\sim$ 140$\mu$m was already an indication that it must originate from this redshift range since galaxies SEDs peak above $\lambda \sim$ 60$\mu$m.
Aussel, H., Cesarsky, C.J., Elbaz, D., Starck, J.L. 1999, A&A 342, 313 Bressan, A., et al. 1993, A&AS, 100, 647 Chary, R.R., Elbaz, D. 2001, ApJ 556, 562 Elbaz, D., Cesarsky, C.J., Fadda, D., et al. 1999, A&A 351, L37 Elbaz, D., Flores, H., Chanial, P. et al. 2002a, A&A 381, L1 Elbaz, D., Cesarsky, C.J., Chanial, P. et al. 2002b, A&A, in press (astro-ph/0201328) Fadda, D., Flores, H., Hasinger, G., et al. 2001, A&A, in press (astro-ph/0111412) Franceschini, A., Aussel, H., Cesarsky, C.J., Elbaz, D., Fadda, D. 2001, A&A 378, 1 Fukugita, M., Hogan, C. J. & Peebles, P. J. E. 1998, ApJ, 503, 518 Gispert, R., Lagache, G., Puget, J.L. 2000, A&A 360, 1 Gould, A., Bahcall, J. N., Flynn, C. 1996, ApJ 465, 759 Prantzos, N., & Silk, J. 1998, ApJ 507, 229 Puget, J.L., Abergel, A., Bernard, J.P., et al. 1996, A&A 308, L5 Sanders, D.B., Mirabel, I.F. 1996, ARA&A 34, 749 Smail, I., et al. 2001, MNRAS in press (astro-ph/0112100)
|
---
abstract: 'The presence of weak intergalactic magnetic fields can be studied by their effect on electro-magnetic cascades induced by multi-TeV $\gamma$-rays in the cosmic radiation background. Small deflections of secondary electrons and positrons as the cascade develops extend the apparent size of the emission region of distant TeV $\gamma$-ray sources. These $\gamma$-ray halos can be resolvable in imaging atmospheric Cherenkov telescopes and serve as a measure of the intergalactic magnetic field strength and coherence length. We present a method of calculating the $\gamma$-ray halo for isotropically emitting sources by treating magnetic deflections in the cascade as a diffusion process. With this ansatz the moments of the halo follow from a set of simple diffusion-cascade equations. The reconstruction of the angular distribution is then equivalent to a classical moment problem. We present a simple solution using Padé approximations of the moment’s generating function.'
author:
- Markus Ahlers
title: |
Gamma-ray halos as a measure of intergalactic magnetic fields:\
a classical moment problem
---
Introduction {#sec:introduction}
============
The presence of large-scale magnetic fields in cosmic environments can be probed by various astronomical techniques. Synchrotron radiation of relativistic electrons can be detected by its characteristic linear polarization and spectrum. Faraday rotation of linearly polarized emission tests the birefringent properties of a dilute magnetized plasma filling intergalactic space. Zeeman splitting of an atom’s energy levels can be observed by the corresponding shift of spectral lines from astrophysical masers. With these standard methods it has been possible to identify micro-Gauss magnetic fields coherent over galactic scales in many galaxies and galaxy clusters [@Kronberg:1993vk; @Beck:2008ty].
The origin of these large-scale magnetic fields is unclear. It is assumed that galactic magnetic fields can be maintained and amplified via a dynamo mechanism, where the kinetic energy of a turbulent interstellar plasma is converted into magnetic energy [@Kulsrud:2007an]. However, this requires initial seed fields of unknown origin, possibly pre-galactic or primordial [@Grasso:2000wj; @Widrow:2002ud]. The strength and correlation length of primordial intergalactic magnetic fields (IGMFs) can be limited by their effect on various stages in cosmic history. The strongest bounds on the strength of primordial IGMFs arise from the study of temperature anisotropies in the cosmic microwave background (CMB) [@Barrow:1997mj]. The limits are of the order of nano-Gauss for a correlation lengths larger than a few Mpc. Simulations of large-scale structure formation favor long-range IGMF with a strength of the order of pico-Gauss [@Dolag:2004kp].
It has been suggested that weak IGMFs of the order of fempto-Gauss can be probed by their effect on electro-magnetic cascades initiated by the emission of distant multi-TeV $\gamma$-ray sources [@Aharonian:1993vz; @Plaga:1995]. High-energy $\gamma$-rays produce pairs of electrons/positrons in the cosmic infrared/optical background (CIB) with an interaction length of the order of 100 Mpc. The secondary leptons lose their energy via inverse-Compton scattering off the background photons and produce secondary $\gamma$-rays at somewhat lower energies. If these photons are still above the pair-production threshold the cycle repeats. In this way the electro-magnetic energy of the cascade is continuously shifted into the GeV-TeV energy region. In the presence of magnetic fields secondary leptons are deflected off the line-of-sight and secondary $\gamma$-rays inherit this deflection. This will attenuate the flux originally emitted towards the observer. However, $\gamma$-rays initially emitted away from the observer can be scattered back into the line-of-sight and partially compensate for this loss.
There are various ways to infer the strength $B_0$ and correlation length $\lambda_B$ of the IGMF from this effect. For small deflections and isotropically emitting sources (or sufficiently large jet opening-angles) the net effect will be an extended emission region of secondary $\gamma$-rays [@Aharonian:1993vz; @Neronov:2007zz; @Eungwanichayapant:2009bi; @Elyiv:2009bx; @Dolag:2009iv]. For burst-like $\gamma$-ray sources this can also cause an observable time-delay between the primary burst and secondary $\gamma$-rays [@Plaga:1995; @Murase:2008pe]. In the case of a hard TeV $\gamma$-ray emission the secondary component can dominate over the attenuated primary $\gamma$-rays. Non-observation of the point-source in the GeV-TeV band can then imply a lower limit on the magnetic field depending on the instrument’s resolution [@Aharonian:1993vz]. These methods have been applied to various TeV $\gamma$-ray sources [@Neronov:1900zz; @d'Avezac:2007sg; @Tavecchio:2010mk; @Dolag:2010ni; @Dermer:2010mm; @Taylor:2011bn; @Neronov:2011ni] and indicate the presence of an IGMF. The inferred lower limits on its strength range from $10^{-18}$ G to $10^{-15}$ G, depending on many systematic uncertainties like the primary emission spectrum, the CIB and the coherence length of magnetic fields.
Besides the systematic uncertainties of these methods, there are also some technical challenges in calculating the energy and angular spectrum of the $\gamma$-ray halos. A straightforward Monte-Carlo calculation of the electro-magnetic cascade can become numerically expensive; since energy is conserved in the cascade the number of $\gamma$-rays, electrons and positrons in the cascade increases by one order of magnitude for every decade of the energy shift. At every step of the cascade each particle will have accumulated a deflection angle with respect to the line-of-sight which depends on its history in the cascade. In order to accumulate a satisfactory resolution in energy and angular extend of the halos it is necessary to sample over many cascades.
In the absence of deflections by magnetic fields the electro-magnetic cascade can be calculated efficiently by analytical methods using cascade equations and the method of matrix doubling [@Protheroe:1992dx]. We will show in this paper that there is a straightforward extension of this method to the case of isotropic emitters and small deflections in magnetic fields. The key observation is that the deflection $\theta$ of electrons and positrons in the cascade in combination with inelastic losses to photons can be treated as a diffusion process in $\theta$-space where the diffusion coefficient depend on the particle’s Larmor radius and the inverse-Compton energy loss length. We derive diffusion-cascade equations that describe the evolution of the moments of the $\theta$-distribution and give a simple method how these moments can be used to reconstruct the distribution.
We will begin in section \[sec:cascades\] by a discussion of electro-magnetic cascades from $\gamma$-ray point sources in the presence of weak IGMFs. In section \[sec:diffusion\] we will motivate the extension of the Boltzmann equations by a diffusion term in $\theta$-space and give an extended set of cascade equations for the moments of the $\theta$-distribution. We discuss in section \[sec:reconstruction\] how the full $\theta$-distribution can be reconstructed efficiently from a finite number of moments via explicit inverse Laplace transformations of Padé approximations of the moment’s generating function. We will test our method in section \[sec:example\] by two examples and compare our results to previous studies. Finally, we conclude in section \[sec:conclusion\].
We work throughout in natural Heaviside-Lorentz units with $\hbar=c=\epsilon_0=\mu_0=1$, $\alpha=e^2/(4\pi)\simeq1/137$ and $1~{\rm G} \simeq 1.95\times10^{-2}{\rm eV}^2$.
Electro-magnetic Cascades {#sec:cascades}
=========================
The driving processes of the electro-magnetic cascade in the cosmic radiation background (CRB) are inverse Compton scattering (ICS) with CMB photons, $e^\pm+\gamma_{\rm bgr}\to e^\pm+\gamma$, and pair production (PP) with CMB and CIB radiation, $\gamma+\gamma_{\rm bgr}\to e^++e^-$ [@Blumenthal:1970nn; @Blumenthal:1970gc]. In particular, the interaction length of multi-TeV $\gamma$-rays depend on the CIB background at low redshift and is of the order of a few 100 Mpc. We show the relevant interaction lengths and energy loss lengths in the left panel of Fig. \[fig:fig1\]. High energetic electrons and positrons may also lose energy via synchrotron radiation in the intergalactic magnetic field, but this contribution is in general negligible for the small magnetic field strength considered here. Further processes contributing to the electro-magnetic cascade are double pair production, $\gamma+\gamma_{\rm bgr}\to e^++e^-+e^++e^-$, and triple pair production, $e^\pm+\gamma_{\rm bgr}\to e^\pm+e^++e^-$ [@Protheroe:1992dx; @Lee:1996fp]. These contributions can be neglected for cascades initiated by multi-TeV $\gamma$-rays considered here. Also, interactions on the cosmic radio background are negligible in this case.
For the calculation of the flux from a $\gamma$-ray point-source it is convenient to start from the evolution of a comoving number density $Y_\alpha = n_\alpha/(1+z)^3$ (GeV${}^{-1}$ cm${}^{-1}$) in a spatially homogeneous and isotropic universe. The Boltzmann equations of electrons/positrons ($Y_e$) and $\gamma$-rays ($Y_\gamma$) is given by $$\label{eq:boltzmann}
\dot {Y}_\alpha(E) = \partial_E(HE{Y}_\alpha)- \Gamma_\alpha{Y}_{\alpha}(E)+\sum_{\beta=e,\gamma}\int_E{\rm d}E'\gamma_{\beta\alpha}(E',E){Y}_{\beta}(E')+\mathcal{L}_\alpha(E)\,,$$ together with the Friedman-Lemaître equations describing the cosmic expansion rate $H(z)$ as a function of the redshift $z$, $H^2 (z) = H^2_0\,[\Omega_{\rm m}(1 + z)^3 + \Omega_{\Lambda}]$, normalized to its present value of $H_0 \sim70$ kms$^{-1}$Mpc$^{-1}$. We consider the usual “concordance model” dominated by a cosmological constant with $\Omega_{\Lambda} \sim 0.7$ and a (cold) matter component, $\Omega_{\rm m} \sim 0.3$ [@Nakamura:2010zzi]. The time-dependence of the redshift is given by ${\rm d}z = -{\rm d} t\,(1+z)H$. The first term in the r.h.s. of Eq. (\[eq:boltzmann\]) accounts for the continuous energy loss due to the adiabatic expansion of the Universe. The second and third terms describe the interactions with background photon fields involving particle losses ($\alpha \to$ anything) and particle generation $\alpha\to\beta$. The angular-averaged (differential) interaction rate, $\Gamma_\alpha$ ($\gamma_{\alpha\beta}$) is defined as $$\begin{gathered}
\label{eq:diffgamma1}
\Gamma_{\alpha}(z,E_\alpha) =
\frac{1}{2}\int_{-1}^1{\rm d}\cos\theta\int{\rm d}
\epsilon\,(1-\beta
\cos\theta) n_\gamma(z,\epsilon)\sigma^{\rm tot}_{\alpha\gamma}\,,\\
\gamma_{\alpha\beta}(z,E_\alpha,E_\beta) = \Gamma_\alpha(z,E_\alpha)\,\frac{{\rm d} N_{\alpha\beta}}{{\rm d} E_\beta}(E_\alpha,E_\beta)\,,\label{eq:diffgamma2}\end{gathered}$$ where $n_\gamma(z,\epsilon)$ is the energy distribution of background photons at redshift $z$ and ${\rm d} N_{\alpha\beta}/{\rm d} E_\beta$ is the angular-averaged distribution of particles $\beta$ after interaction of a particle $\alpha$. Besides the contribution of the CMB we use the CIB from Ref. [@Franceschini:2008tp]. Due to the cosmic evolution of the radiation background density the interaction rates (\[eq:diffgamma1\]) and (\[eq:diffgamma2\]) scale with redshift. The CMB evolution follows an adiabatic expansion, $n_\gamma(z,\epsilon) = (1+z)^2\,n_\gamma(0,\epsilon/(1+z))$, and we assume the same evolution of the CIB for simplicity. We refer to Ref. [@Ahlers:2009rf] for a list of the redshift scaling relations of the interaction rates in Eqs. (\[eq:boltzmann\]). The last term in Eq. (\[eq:boltzmann\]), $\mathcal{L}_\alpha$, accounts for the emission rate of particles $\alpha$ per comoving volume.
In the limit of small deflections of particles via magnetic fields, the flux from a $\gamma$-ray point source at redshift distance $z^\star$ with emission rate $Q_\gamma$ is equivalent to an angular-averaged flux from a sphere at redshift $z^\star$. Hence, the solution of $Y$ at $t=0$ is equivalent to the point source flux $J$ (GeV${}^{-1}$ cm${}^{-2}$ s${}^{-1}$) by replacing the emission rate density $\mathcal{L}$ in (\[eq:boltzmann\]) by $$\label{eq:PS}
\mathcal{L}_\gamma^\star(z,E) = \frac{Q_\gamma(E)}{4\pi d_C^2(z^\star)}H(z^\star)\delta(z-z^\star)\,,$$ where the comoving distance of the source (in a flat universe) is given by $d_C(z) \equiv \int_0^z{\rm d}z'/H(z')$. Note, that we can also use the ansatz (\[eq:PS\]) for a cosmic ray (CR) point source located at redshift $z^\star$, where the electro-magnetic emission is in the form of cosmogenic $\gamma$-rays, electrons and positrons produced during CR propagation [@AhlersSalvado].
Angular Diffusion in Intergalactic Magnetic Fields {#sec:diffusion}
==================================================
The $\gamma$-ray cascade can only contribute to a GeV-TeV point-source flux if the deflections of secondaries off the line-of-sight are sufficiently low. The scattering angle of secondaries is only of the order of $\epsilon/m_e$ and can be neglected for the optical/infra-red background photon energies $\epsilon$. However, electrons and positrons can be deflected in the IGMF. We can estimate the extend of the cascaded $\gamma$-ray emission by simple geometric arguments following [@Neronov:2007zz]. Deflection of electrons and positrons will be small if the energy loss length $\lambda_e$ of electrons/positrons via inverse Compton scattering (ICS) is much smaller than the Larmor radius given as $R_L = E/eB \simeq{1.1} (E_{\rm TeV}/B_{\rm fG}){\rm Mpc}$. Here and in the following we use the abbreviations $E = E_{\rm TeV}{\rm TeV}$, etc. For center of mass energies much lower than the electron mass, corresponding to energies below PeV in the CMB frame, electrons and positrons interact quickly on kpc scales but with low inelasticity proportional to their energy, $\lambda_e\simeq 0.4~{\rm Mpc}/E_{\rm TeV}$. The typical size of the point-spread function (PSF) of imaging atmospheric Cherenkov telescopes (IACTs) is of the order of $\theta_{\rm PSF}\simeq0.1^\circ$. Hence, magnetic deflections become important if $\theta_{\rm PSF}\lesssim \lambda_e/R_L$ or $E\lesssim 14~{\rm TeV} \sqrt{B_{\rm fG}/\theta_{\rm PSF, 0.1^\circ}}$.
In the following we are going to study these magnetic deflections more quantitatively. For simplicity, we will start with a regular IGMF that fills the space between the source and the observer and has the component $B_\perp$ perpendicular to the line-of-sight. We also assume that the source is emitting $\gamma$-rays isotropically.[^1] Due to charge conservation in the cascade electrons and positrons will be produced in equal rates and will be deflected in opposite directions. For small scattering and isotropic emission we can assume that leptons that are lost by deflections out of the line-of-sight are replenished by the corresponding leptons deflected into the line-of-sight. Effectively, we can hence assume that the total number of electrons/positrons within the line-of-sight remains constant by these deflections while the scattering angle $\theta$ is broadened by the magnetic field. The width of this $\theta$-distribution, ${\mathcal Y}_e(E,\theta)$, is determined by the energy loss length via ICS. Secondary $\gamma$-rays will inherit the $\theta$-distribution of the parent leptons and will appear as extended halos.
The energy loss length via ICS with CMB photons is much smaller than the typical distance of TeV $\gamma$-ray sources or the interaction length of PP in the CIB. This indicates that we can treat magnetic deflections in the cascade as a diffusive process of the angle $\theta$. The mean free path of the electrons/positrons corresponds to the energy loss rate in ICS and the diffusion velocity is the inverse Larmor radius. Hence, the diffusion coefficient $D$ is of the order of $\lambda^2_{\rm ICS}/R^2_L$. A more rigorous derivation (see Appendix \[sec:appI\]) shows that the evolution of the $\theta$-distributions of leptons and $\gamma$-rays, ${\mathcal Y}_e(E,\theta)$ and ${\mathcal Y}_\gamma(E,\theta)$ respectively, can be described by the coupled set of differential equations, $$\begin{aligned}
\label{eq:diffregular}
\dot {\mathcal Y}_\gamma&\simeq \partial_E(HE{\mathcal Y}_\gamma)-\Gamma_\gamma {\mathcal Y}_\gamma + \sum_{\alpha=e,\gamma}\int_E{\rm d}E' \gamma_{\alpha \gamma}(E',E) {\mathcal Y}_\alpha(E') + \mathcal{L}^\star_\gamma\delta(\theta)\,,\\
\dot {\mathcal Y}_e &\simeq \partial_E(HE{\mathcal Y}_e)-\Gamma_e {\mathcal Y}_e + \sum_{\alpha=e,\gamma}\int_E{\rm d}E' \gamma_{\alpha e}(E',E) {\mathcal Y}_\alpha(E') + \mathcal{L}^\star_e\delta(\theta)+ \int\limits_E{\rm d}E'{\mathcal D}_{\rm reg}(E',E)\partial^2_{\theta} {\mathcal Y}_e(E')\,.\end{aligned}$$ The diffusion matrix of electrons/positrons in a regular magnetic field is given by $$\label{eq:Dregular}
{\mathcal D}_{\rm reg}(E',E) = \frac{1}{E\,\Gamma_{\rm ICS}(E)}\frac{e^2B_\perp^2}{E'^2\,\langle x\rangle(E')}\,,$$ where $\langle x\rangle(E)$ is the inelasticity of ICS with interaction rate $\Gamma_{\rm ICS}(E)$. For cosmological sources the redshift scaling of the diffusion matrix (\[eq:Dregular\]) can also become important. For primordial magnetic fields scaling as $B_\perp(z) = (1+z)^2B_\perp(0)$ and ICS with CMB photons the redshift dependence is given by the simple relation ${\mathcal D}_{\rm reg}(z,E',E) = (1+z)^4{\mathcal D}_{\rm reg}(0,(1+z)E',(1+z)E)$.
This formalism has the advantage that we can calculate the moments of the $\theta$-distribution by an extended set of cascade equations as we will see in the following. Firstly, we introduce the quantities $$\label{eq:Thetan}
{Y}_{e/\gamma}^{(n)} \equiv \frac{1}{(2n)!}\int\limits_{-\infty}^\infty{\rm d}\theta\,\theta^{2n}\,{\mathcal Y}_{e/\gamma}\quad \text{(regular)}\,.$$ At leading order we have ${Y}^{(0)} = Y$ as the solution of Eq. (\[eq:boltzmann\]) and for $n\geq1$ the quantities ${Y}^{(n)}$ correspond to the scaled moments of the $\theta$-distribution.[^2] It is easy to see that the quantities ${Y}^{(n)}$ ($n>0$) follow the coupled set of differential equations, $$\label{eq:Thetanevol}
\dot {Y}^{(n)}_\alpha(E) = \partial_E(HE{Y}^{(n)}_\alpha)- \Gamma_\alpha{Y}_{\alpha}^{(n)}(E)+\sum_{\beta=e,\gamma}\int_E{\rm d}E'\gamma_{\beta\alpha}(E',E){Y}_{\beta}^{(n)}(E')
+ \delta_{e\alpha}\int_E{\rm d}E'{\mathcal D}(E',E){Y}_\alpha^{(n-1)}(E')\,,$$ in addition to Eqs. (\[eq:boltzmann\]). Note, that electro-magnetic interactions of photons and leptons that drive the cascade happen on time-scales much shorter than the rate of adiabatic losses in the Universe. We can hence treat the interaction rates as constant over small time-intervals and neglect the energy loss terms $\partial_E(HE{Y}_\alpha^{(n)})$ in Eqs. (\[eq:boltzmann\]) and (\[eq:Thetanevol\]). We show in Appendix \[sec:appII\] that this system of equations can then be solved efficiently by a generalization of the conventional cascade equations.
We next consider a randomly oriented IGMF field with a coherence length $\lambda_B$ much smaller then the distance $d$ of the source. In this case we have to replace Eq. (\[eq:diffregular\]) by the evolution of radial diffusion on a sphere of the form[^3] $$\label{eq:diffrandom}
\dot {\mathcal Y}_e\simeq \partial_E(HE{\mathcal Y}_e)-\Gamma_e {\mathcal Y}_e + \sum_{\alpha=e,\gamma}\int_E{\rm d}E' \gamma_{\alpha e}(E',E) {\mathcal Y}_\alpha(E') + \mathcal{L}^\star_\gamma\delta(\theta) + \int\limits_E{\rm d}E'{\mathcal D}_{\rm rnd}(E',E)\theta^{-1}\partial_{\theta}\left[\theta\partial_\theta {\mathcal Y}_e(E')\right] \,,$$ with diffusion coefficient (see Appendix \[sec:appI\]) $$\label{eq:Drandom}
{\mathcal D}_{\rm rnd}(E',E) \simeq \frac{1}{3}\frac{\min(1,\lambda_B\Gamma_{\rm ICS}(E))}{E\,\Gamma_{\rm ICS}(E)}\frac{e^2B_0^2}{E'^2\langle x\rangle(E')}\,.$$ Here, a factor $1/3$ accounts for the random orientation of the magnetic field w.r.t. the line-of-sight. Analogously to the diffusion in a regular magnetic field we can define moments of the diffusion in random IGMFs by $$\label{eq:twodimThetan}
{Y}_{e/\gamma}^{(n)} \equiv \frac{2\pi}{(2^nn!)^2}\int\limits_{0}^\infty{\rm d}\theta\,\theta\,\theta^{2n}\,{\mathcal Y}_{e/\gamma}\quad \text{(random)}\,,$$ which follow the same differential equations (\[eq:Thetanevol\]) with diffusion matrix ${\mathcal D}_{\rm rnd}$.
So far we have only considered the diffuse scattering of the photons along their initial trajectory. How does this translate into the observed morphology of the $\gamma$-ray signal? Deflections of electrons close to the source at distance $d$, [*e.g.*]{} by the magnetic field of the source itself, will have a weaker impact on the observed angular distribution than deflections close to the observer. If the cascade experiences a deflection $\Delta \theta$ at a distance $r$ from the observer we can approximate the corresponding angular displacement $\Delta \theta'$ in the observer’s frame via $\Delta \theta'/\Delta\theta \simeq (d-r)/d$. We can account for this scaling in the cascade equation by introducing the corresponding scaling in the convection velocity $R_L^{-1}$ or, equivalently, by a scaling of the diffusion matrix of the form ${\mathcal D}' \simeq ((d-r)/d)^2 {\mathcal D}$. In practice, this requires that we repeat the calculation of transfer matrices after sufficiently small propagation distances, for which we then also account for the variation of (differential) interaction rates $\Gamma$ ($\gamma$) with redshift and adiabatic energy loss. With this simple modification the moments ${Y}^{(n)}$ reflect the angular distribution of $\gamma$-ray halos, as long as scattering in the magnetic field is small and the source is emitting isotropically.
![[**Left panel:**]{} The interaction length (solid lines) and energy loss length (dashed lines) from various contributions of the CRB. We show the rates separately for the CMB and CIB. Also shown is the inverse Hubble scale (dotted line). [**Right panel:**]{} The spectra of $\gamma$-rays from a source at $120$ Mpc with injection spectrum $Q_\gamma\sim E^{-2}\exp(-E/300\,{\rm TeV})$ (gray line) following Ref. [@Elyiv:2009bx]. We show the contribution of surviving primary $\gamma$-rays (dashed line) and secondary cascaded $\gamma$-rays (solid line) separately.[]{data-label="fig:fig1"}](intlen.pdf "fig:"){height="2.7in"}![[**Left panel:**]{} The interaction length (solid lines) and energy loss length (dashed lines) from various contributions of the CRB. We show the rates separately for the CMB and CIB. Also shown is the inverse Hubble scale (dotted line). [**Right panel:**]{} The spectra of $\gamma$-rays from a source at $120$ Mpc with injection spectrum $Q_\gamma\sim E^{-2}\exp(-E/300\,{\rm TeV})$ (gray line) following Ref. [@Elyiv:2009bx]. We show the contribution of surviving primary $\gamma$-rays (dashed line) and secondary cascaded $\gamma$-rays (solid line) separately.[]{data-label="fig:fig1"}](gamma_spectrum.pdf "fig:"){height="2.7in"}
As an example, we study in the following an isotropic $\gamma$-ray point-source at a distance of about 120 Mpc – as Mrk 421 – with a $\gamma$-ray injection spectrum of the form $Q_\gamma \sim E^{-2}\exp(-E/300\,{\rm TeV})$. This particular example has been studied in Ref. [@Elyiv:2009bx] and hence our results are directly comparable. In the left panel of Fig. \[fig:fig1\] we show the source spectrum, [*i.e.*]{} the spectrum that would be visible without the CRB (thin gray line) together with the electron/positron and $\gamma$-ray spectrum after propagation. The total $\gamma$-ray spectrum (dotted green line) can be decomposed into a “primary” component (dashed green line) of surviving $\gamma$-rays and a “cascaded” component (solid green line) from $\gamma$-rays of the cascade. The $\gamma$-ray flux is strongly suppressed beyond 10 TeV due to the PP with the CIB and secondary $\gamma$-rays from ICS with the CMB peak between 0.1-1 TeV.
We will assume in the following that the cascade develops in a weak IGMF with strength $B_0=10^{-15}$ G and a coherence length $\lambda_B=1$ Mpc extends. For the reconstruction of the $\gamma$-ray halo it is convenient to first subtract the moments of the surviving primary $\gamma$-rays that don’t take part in the cascade, $$Y^{(n)}_{\gamma,\,{\rm halo}} = Y^{(n)}_{\gamma,\,{\rm total}} - Y^{(n)}_{\gamma\,{\rm primary}}\,.$$ In our example we assume a point-source with sufficiently small angular extend, corresponding to the case $Y^{(0)}_{\gamma,\,{\rm primary}}=Y_{\gamma,\,{\rm primary}}$ and vanishing higher moments. In general, the higher moments of the primary source with an angular extend $2\theta_s$ can be approximated by $$Y^{(n)}_{\rm primary} \simeq Y^{(0)}_{\rm primary}\frac{\theta_s^{2n}}{n!(n+1)!4^n}\,.$$
The size of the first non-trivial moment ${Y}_{\gamma,\,{\rm halo}}^{(1)}/{Y}^{(0)}_{\gamma,\,{\rm halo}}$ already serves as a first indicator for the size of the $\gamma$-ray halo. If this is much larger than the PSF of an IACT the flux of secondary $\gamma$-rays will be strongly isotropized and can only be constrained by the diffuse $\gamma$-ray background (see [*e.g.*]{} [@Ahlers:2010fw]). We will show in the following that we can use the spectrum of moments to reconstruct the $\gamma$-ray halo for small deflection angles. This will also give an indication at which energies the contribution of secondary $\gamma$-rays contribute to the point-source spectrum.
Reconstruction of the Angular Distribution {#sec:reconstruction}
==========================================
The moments of the $\gamma$-ray halo serve as a measure for its angular distribution. How we can reconstruct the angular distribution from a limited number of moments? As a first step it is convenient to define a distribution $f(E,x)$ by the transformation $$\label{eq:Ndis}
{\mathcal{Y}}_{\gamma, {\rm halo}}(E,\theta) \equiv Y_{\gamma, {\rm halo}}(E)\int\limits_0^\infty {\rm d}x\left[\frac{1}{(2\pi x)^{\frac{d}{2}}}e^{-\theta^2/2x}\right]f(E,x)\,,$$ for regular ($d=1$) or random ($d=2$) magnetic fields. This transformation is motivated by the observation that the kernel $G_d(x,\theta) = [\ldots]$ corresponds to a Green’s function of the d-dimensional diffusion equation, $(\partial_x - \sum_i\partial^2_{\theta_i}) G_d(x,\vec\theta) = 0$ and $G_d(0,\vec\theta)=\prod_i\delta(\theta_i)$ with $\sum_i\theta_i^2 =\theta^2$. We can then identify the quantities ${Y}^{(n)}/{Y}^{(0)}$ as (scaled) moments of the distribution $f(E,x)$ for both, regular and random fields: $$\label{eq:mun}
\mu_n(E) \equiv \int\limits_0^\infty {\rm d}x\,x^n\,f(E,x) = {2^nn!}\frac{{Y}_{\gamma, {\rm halo}}^{(n)}(E)}{{Y}_{\gamma, {\rm halo}}^{(0)}(E)}\,.$$
We hence arrive at a classical (Stieltjes) moment problem [@Akhierzer:1965] of finding the distribution $f$ from its moments $\mu_n$. From the differential equations (\[eq:Thetanevol\]) and the definition (\[eq:mun\]) it is easy to see that we can find a constant $\mathcal{C}$ such that $\mu_n(E) < {\mathcal C}n![2d\max_{E'\geq E}(\mathcal{D}(E',E))]^n$, where $d$ the distance to the source. This is a sufficient condition for a determinate moment problem [@Akhierzer:1965], [*i.e.*]{} there exists a unique solution $f$ satisfying Eq. (\[eq:mun\]). Note, that the reconstruction of $f$ from the [*complete*]{} set of moments $\mu_n$ is trivial. For instance, we can express $f$ by an infinite sum of Laguerre polynomials which are orthogonal on $[0,\infty)$ under the measure $\exp(-x)$. However, this method does not prove convenient if there are only a finite number of $\mu_n$ at our disposal. The truncation of the expansion after the first $N+1$ basis function leads typically to rapidly oscillating solutions. Alternatively, we can reconstruct the distribution by a sequence of approximations $f$, which are maxima of an entropy functional [@Mead:1983qg], where the condition (\[eq:mun\]) are introduced via Lagrange multipliers. This problem can then be reduced to a minimization problem of an $N$-dimensional effective potential.
In our case we choose a different approach, which is suitable for the particular form of the distribution. First, we introduce the Laplace transform of the potential $f$ as $$\label{eq:genfunction}
\hat f(E,s) = \mathcal{L}\lbrace f(E,x)\rbrace\equiv \int\limits_0^\infty {\rm d}x e^{-sx}f(E,x) = \sum\limits_{k=0}^\infty\frac{(-s)^k}{k!}\mu_k\,.$$ The Laplace transform $\hat f$ corresponds to a generating function of the moments, $(-1)^n\partial^n_sf(E,s)|_{s=0} = \mu_n(E)$, and the solution to the moment problem corresponds to the inverse Laplace transform $f(E,x) = \mathcal{L}^{-1}\lbrace\hat f(E,s)\rbrace$. However, in practice we have only a finite number of moments $N+1$ and the truncation of the alternating series (\[eq:genfunction\]) does not converge for large $s$.
![[**Left panel:**]{} The first 15 non-trivial moments $\mu_n$ at three different $\gamma$-ray energies for the example shown in the right panel of Fig. \[fig:fig1\] and assuming a IGMF with strength $B_0=10^{-15}$ G and $\lambda_B=1$ Mpc. The dashed lines show the momenta reconstructed by the Padé approximations $\hat f_{[4,5]}$ that are fixed by the first 6 calculated moments. The approximation reproduces the higher moments well. [**Right panel:**]{} The $\gamma$-ray halos reconstructed from the moments shown in left panel. We also indicate the typical size of the PSF for IACTs.[]{data-label="fig:fig2"}](moments.pdf "fig:"){height="2.7in"}![[**Left panel:**]{} The first 15 non-trivial moments $\mu_n$ at three different $\gamma$-ray energies for the example shown in the right panel of Fig. \[fig:fig1\] and assuming a IGMF with strength $B_0=10^{-15}$ G and $\lambda_B=1$ Mpc. The dashed lines show the momenta reconstructed by the Padé approximations $\hat f_{[4,5]}$ that are fixed by the first 6 calculated moments. The approximation reproduces the higher moments well. [**Right panel:**]{} The $\gamma$-ray halos reconstructed from the moments shown in left panel. We also indicate the typical size of the PSF for IACTs.[]{data-label="fig:fig2"}](2Dspectrum.pdf "fig:"){height="2.7in"}
We can find an approximate solution by replacing the truncated series by a Padé approximation – a method which is well-known to chemistry, engineering or nuclear physics [@Baker:1981]. We are approximating $\hat f$ by a rational function $\hat f_{[M,M+1]}(s) = P(s)/Q(s)$ where $P$ and $Q$ are polynomials of degree $M$ and $M+1$, respectively. The coefficents of $P$ and $Q$ are determined by matching the first $2M+1$ terms of the Taylor expansion of $f_{[M,M+1]}$ to the truncated series. Clearly, for $N+1$ calculated moments we can only consider $M\leq N/2$ for the approximation. Since $\deg(Q)>\deg(P)$ the Padé approximation is finite as $s\to \infty$, in contrast to the truncated series it approximates. If we write the denominator via its roots $s_i$ with multiplicity $m_i$, $Q(s) = \prod_{i=1}^n(s-s_i)^{m_i}$, the inverse Laplace transform of the rational function $f_{[M,M+1]}$ has the simple form $$\label{eq:fapprox}
f(E,x) \simeq \mathcal{L}^{-1}\lbrace\hat f_{[M,M+1]}\rbrace = \sum_{i=1}^{n}\sum_{j=1}^{m_i}\frac{c_{ij}(E)}{(j-1)!}x^{j-1}e^{xs_i(E)}\,,$$ where the coefficients $c_{ij}$ follow from an expansion into partial fraction. Note, however, that for a general Padé approximation it is not guaranteed that all $\Re(s_i)<0$ and hence the approximation (\[eq:fapprox\]) can be unstable even if the exact solution (\[eq:genfunction\]) is stable itself. However, by lowering the degree of approximation $M$ it is in general possible to obtain a stable Padé approximation that fulfills the necessary criteria. This can be done by trial and error – as we do here for simplicity – or by an algorithmic procedure [@Hutton]. We will show in the following section, that this procedure is stable and reproduces the moments of the distribution well. Finally, the distribution $N(\theta)$ can be obtained from Eqs. (\[eq:Ndis\]) and (\[eq:fapprox\]).
We illustrate this procedure for the cascade spectrum shown in the right panel Fig. \[fig:fig1\]. In the left panel of Fig. \[fig:fig2\] we show the first 15 non-trivial moments $\mu_n$ of the distribution $f$ for $\gamma$-ray halos at $10^{2.5}$, $10^3$ and $10^{3.5}$ GeV. The dashed line shows the moments calculated via the Padé approximation $\hat{f}_{[4,5]}$. Note, that this approximation is determined by the first eight non-trivial moments, but also reproduces all the higher moments of our calculation satisfactorily. Using Eqs. (\[eq:Ndis\]) and (\[eq:fapprox\]) we can derive the angular distribution of the halos which are shown in the right panel of Fig. \[fig:fig2\]. For illustration we normalize the distribution as $N(\theta,E)={\mathcal Y}_{\gamma,{\rm halo}}(\theta,E)/Y_{\gamma,{\rm total}}(E)$. Note, that not all of this $\gamma$-ray halo will be resolvable in IACTs. We are indicating in the plot the typical size of the PSF of $0.1^\circ$. We will discuss in the following section the size of these $\gamma$-ray halos in more detail.
![[**Left panel:**]{} The size of the extended $\gamma$-ray halo defined by Eqs. (\[eq:Nasym\]) and (\[eq:numeric\]) for a source at $z=0.031$ with spectrum $Q_\gamma(E) \sim E^{-2}\exp(-E/300{\rm TeV})$. [**Right panel:**]{} A model for the $\gamma$-ray spectrum of the blazar source 1ES0229+200 located at $z=0.14$. The blue data points show the H.E.S.S. observation [@Aharonian:2007wc] and the red lines correspond to the upper flux limits from Fermi-LAT (taken from Ref. [@Tavecchio:2010mk]). We assume a source spectrum of the form $Q_\gamma \propto E^{-2/3}\Theta(20{\rm TeV}-E)$. The solid green line shows the spectrum of secondary $\gamma$-rays without deflections in the IGMF. The dotted green lines indicate the part of the cascaded $\gamma$-ray spectrum within $0.1^\circ$ around the source for an IGMF with coherence length $\lambda_B=1$ Mpc and strength $B_0=10^{-16}$ G, $10^{-15}$ G and $10^{-14}$ G, respectively. []{data-label="fig:fig3"}](thetacut.pdf "fig:"){height="2.7in"}![[**Left panel:**]{} The size of the extended $\gamma$-ray halo defined by Eqs. (\[eq:Nasym\]) and (\[eq:numeric\]) for a source at $z=0.031$ with spectrum $Q_\gamma(E) \sim E^{-2}\exp(-E/300{\rm TeV})$. [**Right panel:**]{} A model for the $\gamma$-ray spectrum of the blazar source 1ES0229+200 located at $z=0.14$. The blue data points show the H.E.S.S. observation [@Aharonian:2007wc] and the red lines correspond to the upper flux limits from Fermi-LAT (taken from Ref. [@Tavecchio:2010mk]). We assume a source spectrum of the form $Q_\gamma \propto E^{-2/3}\Theta(20{\rm TeV}-E)$. The solid green line shows the spectrum of secondary $\gamma$-rays without deflections in the IGMF. The dotted green lines indicate the part of the cascaded $\gamma$-ray spectrum within $0.1^\circ$ around the source for an IGMF with coherence length $\lambda_B=1$ Mpc and strength $B_0=10^{-16}$ G, $10^{-15}$ G and $10^{-14}$ G, respectively. []{data-label="fig:fig3"}](PSF.pdf "fig:"){height="2.7in"}
Size of the Extended Halos {#sec:example}
==========================
The size of the extended halo serves as a measure of the IGMF. Typically, the low-$\theta$ form of the halo derived from the approximation (\[eq:fapprox\]) depend on a few roots $s_i$ with large real component $|\Re(s_i)|$. In this case, the $\theta$-distribution is in the form of a modified Bessel function for a random IGMF with $\lambda_B\ll d$. The sub-halos have the form $$\label{eq:Nasym}
N(\theta) \sim \frac{|\Re(s_i)|}{\pi}K_0(\sqrt{2|\Re(s_i)|}\theta) \sim \frac{1}{\sqrt{8\pi\theta}}\frac{1}{\theta_i^{3/2}}e^{-\theta/\theta_i}\,,$$ where in the last step we took the asymptotic form of $K_0$ at large $\theta$ and introduced the characteristic size of the sub-halo, $\theta_{i}=(2|\Re(s_i)|)^{-1/2}$. Hence, there is a simple relation between the measurable size of the halo and the simple poles of the Padé approximation.
In the cases shown in the right panel of Fig. \[fig:fig2\] the leading order halo is below the typical instrument’s resolution of $\theta_{\rm PSF}=0.1^\circ$. Instead, the next-to-leading-order root will determine the size of the halo. In general, we hence define the size of the leading (observable) halo as $$\label{eq:numeric}
\theta_{\rm cut} ={\rm min}(\lbrace\theta_i\rbrace | \theta_i>\theta_{\rm PSF})\,.$$ For our test spectrum we show the parameter $\theta_{\rm cut}$ in the left panel of Fig. \[fig:fig3\] for various magnetic field strengths and $\gamma$-ray energies between $100$ GeV to a few TeV. As before we consider a coherence length of $\lambda_B=1$ Mpc. The size of the halo in this energy range follows approximately $\theta_{\rm cut} \propto E^{-1}$ as the fit shows. This agrees with the findings of Ref. [@Elyiv:2009bx] (Fig. 7) derived from a Monte-Carlo study.
Another interesting situation occurs if the cascaded $\gamma$-ray spectrum dominates over the primary $\gamma$-ray emission. This can happen for injection spectra that are considerably harder than $E^{-2}$. In this case the detection sensitivity of the cascaded GeV-TeV spectrum depends on the size of the halo and the resolution of the telescope. As an example we consider the emission of the blazar source 1ES0229+200 located at redshift $z=0.14$, which has been detected by its TeV $\gamma$-ray emission by H.E.S.S. [@Aharonian:2007wc]. The spectrum is shown in the right panel of Fig. \[fig:fig3\] as the blue data. Following Ref. [@Tavecchio:2010mk] we model the $\gamma$-ray emission spectrum as $Q_\gamma \propto E^{-2/3}\Theta(20{\rm TeV}-E)$ (thin gray line). The surviving primary $\gamma$-rays are shown as a dashed green line and secondary cascaded $\gamma$-rays by a solid line.
It is easy to understand the shape of the various spectra. Primary $\gamma$-rays close to $E_{\rm max}$ interact with the CIB to produce electron/positron pairs. This is a slow process happening on typical scales of the order of a few 100 Mpc (see left panel of Fig. \[fig:fig1\]). The leptons quickly lose energy via ICS with CMB photons at a rate $b_{\rm ICS} = E/\lambda_e$; their spectrum in quasi-equilibrium ($\partial_tY_e\simeq 0$) follows the differential equation $\partial_E(b_{\rm ICS}Y_e) \simeq \Gamma_{\rm PP}Y_\gamma$. Thus, the Comptonized electron spectrum for $E\ll E_{\rm max}$ has the form $Y_e\sim E_e^{-2}$. The typical photon energy from ICS of a background photon with energy $\epsilon$ is given by $E_\gamma \simeq \epsilon(E_e/m_e)^2$. The resulting photon spectrum at $E\ll E_{\rm max}$ follows from energy conservation in ICS, $\partial_t Y_\gamma \simeq ({\rm d}E_e/{\rm d}E_\gamma)(b_{\rm ICS}/E_\gamma)Y_e\sim (E_e/E_\gamma)^2/(2\lambda_e)Y_e \sim E_\gamma^{-3/2}$. The plateau of the full cascaded spectrum shown in the right panel of Fig. \[fig:fig3\] is slightly softer than this since a part of the inverse-Compton spectrum is still above pair-production threshold and enters a second cascade cycle.
We also show the expected contribution of secondary $\gamma$-rays confined within the PSF of a typical IACT with $\theta_{\rm PSF}=0.1^\circ$ assuming an IGMF with coherence length $\lambda_B=1$ Mpc and strength $B_0=10^{-16}$, $B_0=10^{-15}$ and $B_0=10^{-14}$ G, respectively. The deflection of an electron of the Comptonized spectrum is approximately $\theta_e \sim \lambda_{e}/R_L/4$ following from $\partial_t(\theta_e Y_e) \simeq Y_e/R_L$ and $Y_e\sim E_e^{-2}$. This is consistent with the results of our diffusion ansatz since the first moment of the electron/positron distribution follows $Y^{(1)}_e/Y^{(0)}_e \simeq (\lambda_e/R_L/4)^2/2$. If the typical deflection $\theta_e$ exceeds $\theta_{\rm PSF}$ we expect to see a reduction in the point-source flux by a geometric factor $(\theta_{\rm PSF}/\theta_e)^{2} \propto E_e^{-4} \sim E_\gamma^{-2}$. For ICS in the CMB the transition is expected to occur close to the energy $$E_{\rm cr} \simeq 0.2\,\sqrt{\frac{B_{\rm fG}}{\theta_{{\rm PSF}, 0.1^\circ}}}\,{\rm TeV}\,.$$ This agrees well with the reduced cascade flux (“$\theta<0.1^\circ$”; dotted lines) shown in the plot.
Before we conclude we would like to emphasize a subtlety concerning the contribution of the CIB in ICS. As can be seen from the summary of interaction/loss lengths in the left panel of Fig. \[fig:fig1\], the contribution of the CIB to the total energy loss of ICS is negligible. The $\gamma$-ray spectrum $Y_\gamma$ is hence almost independent of this contribution, but this is not the case for the higher moments $Y^{(n)}_\gamma$. To see this, let us consider a fully Comptonized electron/positron spectrum $Y_e\sim E_e^{-2}$. Following our previous arguments we have $\partial_t (\theta_\gamma Y_\gamma) \sim ({\rm d}E_e/{\rm d}E_\gamma)(b_{\rm ICS}/E_\gamma)\theta_e Y_e \simeq (E_e/E_\gamma)^2/(8R_L)Y_e$ for the Comptonized electron spectrum. The growth of the deflection is hence proportional to $\sqrt{\epsilon}$ and optical photons are expected to contribute much stronger than CMB photons. However, the fraction of photons that contribute with this large deflection is negligible. Inverse-Compton scattering by the CIB will form a shallow plateau of $\gamma$-rays that are negligible for the calculation of the moments of the central halo from the CMB contribution. We can hence neglect this contribution in the calculation of moments which improves the quality of the halo reconstruction at low $\theta$.
Conclusion {#sec:conclusion}
==========
We have discussed a novel technique of calculating extended halos of TeV $\gamma$-ray sources in the presence of intergalactic magnetic fields. The method builds on standard cascade equations that account for all particle interactions with the background radiation and treats the effect of secondary electron/positron deflections in intervening magnetic fields by a diffusion ansatz. The moments of the angular distribution can be calculated efficiently by an extended set of cascade equations. The first moments of the distribution already serve as a good estimator of the halo size. We have shown how the full distribution can be reconstructed from further moments via an inverse Laplace transformation of the moment’s generating function using Padé approximations.
Our method applies to situations where the emission of $\gamma$-rays is isotropic or within a sufficiently large jet opening-angle. The $\gamma$-ray halo is expected to show further structure in the more general case. For instance, $\gamma$-ray emission into narrow jets are expected to produce additional breaks in the halo profile [@Neronov:2007zz] and non-spherical geometries in the case of an off-axes emission [@Neronov:2010bd]. For the illustration of the method we have considered a steady $\gamma$-ray emission. In the case of pulsed or short-lived $\gamma$-ray sources there will be a time-delay between primary and secondary $\gamma$-rays due to the increased path length of the leptons. This can also serve as a measure for the intergalactic magnetic field.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by US National Science Foundation Grant No PHY-0969739 and by the Research Foundation of SUNY at Stony Brook.
Derivation of Eqs. (\[eq:diffregular\]) and (\[eq:diffrandom\]) {#sec:appI}
===============================================================
We assume in the following that the magnetic field is perpendicular to the line-of-sight of the source. In this setup electrons and positrons will be deflected by an angle $\theta$ in a plane normal to the magnetic field. For isotropic emission and small deflections the leptons deflected off the line-of-sight are replenished by leptons initially streaming away from the observer. The net effect is a broadening of the $\theta$-distribution (${\mathcal Y}_\pm$) due to a convection term with opposite sign for electrons ($-$) and positrons ($+$). The transport equations take the form $$\begin{aligned}
\nonumber
\dot {\mathcal Y}_{\pm}(E,t,\theta) =&\pm\frac{1}{R_L(E)}\partial_\theta {\mathcal Y}_{\pm}(E,t,\theta) - \Gamma_e(E) {\mathcal Y}_{\pm}(E,t,\theta)\\ &+ \int_E{\rm d}E' \bigg(\frac{1}{2}\gamma_{\gamma e}(E',E) {\mathcal Y}_\gamma(E',t,\theta) + \gamma_{ee}(E',E) {\mathcal Y}_\pm(E',t,\theta)\bigg) + \frac{1}{2}{\mathcal L}^\star_e(E,t,\theta)\,.\label{eq:pmdiffregular}\end{aligned}$$ The evolution equation of the total electron/positron cascade ${\mathcal Y}_e = {\mathcal Y}_+ + {\mathcal Y}_-$ can then be written as $$\begin{aligned}
\nonumber
\dot {\mathcal Y}_e(E,t,\theta) =&\frac{1}{R^2_L(E)}\int\limits_0^t{\rm d}t'e^{-\Gamma_e(E)(t-t')}\partial^2_\theta {\mathcal Y}_e(E,t',\theta) - \Gamma_e(E) {\mathcal Y}_e(E,t,\theta) + \mathcal{L}_e^{\rm eff}(E,t,\theta)\\&+\int\limits_0^t{\rm d}t'e^{-\Gamma_e(E)(t-t')}\int_E{\rm d}E'\gamma_{ee}(E',E)\bigg[\dot {\mathcal Y}_e(E',t',\theta)+ \Gamma_e(E') {\mathcal Y}_e(E',t',\theta) -\mathcal{L}_e^{\rm eff}(E',t',\theta)\bigg]\,,\label{eq:masterregular}\end{aligned}$$ with an effective source term $$\mathcal{L}_e^{\rm eff}(E,t,\theta) = \sum_{\alpha=e,\gamma}\int_E{\rm d}E' \gamma_{\alpha e}(E',E) {\mathcal Y}_\alpha(E',t,\theta) + {\mathcal L}^\star_e(E,t,\theta)\,.$$ For $t\Gamma_{e}\gg 1$ we can make the replacement $\Gamma_e\exp(-\Gamma_e(t-t'))\to \delta(t-t')$ and Eq. (\[eq:masterregular\]) reduces to $$\label{eq:master2}
\frac{\partial^2_\theta {\mathcal Y}_e(E,t,\theta)}{R_L^2(E)} \simeq\int_E{\rm d}E'\bigg(\Gamma_e(E')\delta(E-E')-\gamma_{ee}(E',E)\bigg)\bigg[\dot {\mathcal Y}_e(E',t,\theta)+ \Gamma_e(E') {\mathcal Y}_e(E',t,\theta) -\mathcal{L}_e^{\rm eff}(E',t,\theta)\bigg]\,.$$ We can further simplify Eq. (\[eq:master2\]) by introducing the mean inelasticity, $$\langle x\rangle= 1 - \int{\rm d}E'\frac{E'}{E}\frac{\gamma_{ee}(E,E')}{\Gamma_{e}(E)}\,.$$ The inelasticity of ICS off CMB photons for electron/positron energies below about $100$ TeV is small and we can hence approximate the differential interaction rate by $\gamma_{ee}(E',E) \simeq \Gamma_e(E')\delta(E-E'(1-\langle x\rangle))$. Using this in Eq. (\[eq:master2\]) and taking the limit $\langle x\rangle\ll1$ we arrive at $$\frac{\partial^2_\theta {\mathcal Y}_e(E,t,\theta)}{\langle x\rangle(E) R_L^2(E)} \simeq-\partial_E\left(E\,\Gamma_e(E)\bigg[\dot {\mathcal Y}_e(E,t,\theta)+ \Gamma_e(E) {\mathcal Y}_e(E,t,\theta) -\mathcal{L}_e^{\rm eff}(E,t,\theta)\bigg]\right)\,.$$ Integrating this equation gives $$\frac{1}{E\,\Gamma_e(E)}\int_E^\infty{\rm d}E'\frac{\partial^2_\theta {\mathcal Y}_e(E',t,\theta)}{\langle x\rangle(E')R_L^2(E')} \simeq\dot {\mathcal Y}_e(E,t,\theta)+ \Gamma_e(E) {\mathcal Y}_e(E,t,\theta) -\mathcal{L}_e^{\rm eff}(E,t,\theta)\,.$$ We hence arrive at the diffusion term (\[eq:diffregular\]) with diffusion matrix (\[eq:Dregular\]) for a regular magnetic field.
If the coherence length $\lambda_B$ of the magnetic field is smaller than the distant to the source we can not neglect the spatial dependence of the diffusion velocity $R_L^{-1}$. Generalizing to two angular variables $\vec{\theta} = (\theta_1,\theta_2)$ in the plane orthogonal to the line-of-sight we start with $$\begin{aligned}
\nonumber
\dot {\mathcal Y}_{\pm}(E,t,\vec\theta) =&\pm\frac{1}{R_L(E)} \vec{n}_{L}(t)\vec\nabla_{\theta} {\mathcal Y}_{\pm}(E,t,\vec\theta) - \Gamma_e(E) {\mathcal Y}_{\pm}(E,t,\vec\theta)\\ &+ \int_E{\rm d}E' \bigg(\frac{1}{2}\gamma_{\gamma e}(E',E) {\mathcal Y}_\gamma(E',t,\vec\theta) + \gamma_{ee}(E',E) {\mathcal Y}_\pm(E',t,\vec\theta)\bigg) + \frac{1}{2}{\mathcal L}^\star_e(E,t,\vec\theta)\,,\label{eq:pmdiffrandom}\end{aligned}$$ where $\vec{n}_L$ is the direction of the Lorentz force projected into the $\theta_1\theta_2$-plane. From here we arrive at $$\begin{aligned}
\nonumber
\dot {\mathcal Y}_e(E,t,\vec\theta) =&\frac{1}{R^2_L(E)}\int_0^t{\rm d}t'e^{-\Gamma_e(E)(t-t')} \vec{n}_{L}(t)\vec\nabla_{\theta}\left[\vec{n}_{L}(t')\vec\nabla_{\theta}{\mathcal Y}_e(E,t',\vec\theta)\right] - \Gamma_e(E) {\mathcal Y}_e(E,t,\vec\theta) + \mathcal{L}_e^{\rm eff}(E,t,\vec\theta)\\&+\int_0^t{\rm d}t'e^{-\Gamma_e(E)(t-t')}\int_E{\rm d}E'\gamma_{ee}(E',E)\bigg[\dot {\mathcal Y}_e(E',t',\vec\theta)+ \Gamma_e(E') {\mathcal Y}_e(E',t',\vec\theta) -\mathcal{L}_e^{\rm eff}(E',t',\vec\theta)\bigg]\,.\label{eq:masterrandom}\end{aligned}$$ For the evaluation of the second time integral in Eq. (\[eq:masterrandom\]) we can proceed as in the case of a regular magnetic field. However, in the first integral we have to account for the fluctuations of $\vec{n}_L(t')$ over the inverse Compton scattering length. These will average to zero except for $\Delta t\lesssim\lambda_B$ and we hence substitute $\Gamma_e\exp(-\Gamma_e(t-t'))\to \min(1,\lambda_B\Gamma_e)\delta(t-t')$. Averaging over the orientation of the magnetic field can be accounted for by an additional factor $1/3$. Proceeding now along the same steps as in the case of a regular field and replacing the angles $\theta_{1/2}$ by spherical coordinates with radius $\theta$ we arrive at the diffusion term (\[eq:diffrandom\]) with diffusion matrix (\[eq:Drandom\]).
Diffusion-Cascade Equations {#sec:appII}
===========================
We start from the Boltzmann equations (\[eq:boltzmann\]) and (\[eq:Thetanevol\]) and define discrete values ${Y}^{(n)}_{e,i} \simeq \Delta E_i {Y}^{(n)}_{e}(E_i)$, $Q_{e,i} \simeq \Delta E_iQ_e(E_i)$, etc. The combined effect of transitions and deflections within the cascade during a sufficiently small time-step $\Delta t$ can be described by $$\begin{aligned}
\label{eq:ThetaCas0}
\begin{pmatrix}{Y}_\gamma({t}+\Delta{t})\\{Y}_e({t}+\Delta{t})\end{pmatrix}^{(0)}_i
&\simeq \sum_j\begin{pmatrix}T_{\gamma\gamma}(\Delta
t)&T_{e\gamma}(\Delta t)\\T_{\gamma e}(\Delta
t)&T_{ee}(\Delta
t)\end{pmatrix}_{ji}\begin{pmatrix}{Y}_\gamma({t})\\{Y}_e({t})\end{pmatrix}^{(0)}_j + \Delta t\begin{pmatrix}Q_\gamma\\Q_e\end{pmatrix}_i\,,\\\label{eq:ThetaCasn}
\begin{pmatrix}{Y}_\gamma({t}+\Delta{t})\\{Y}_e({t}+\Delta{t})\end{pmatrix}^{(n)}_i
&\simeq \sum_j\begin{pmatrix}T_{\gamma\gamma}(\Delta
t)&T_{e\gamma}(\Delta t)\\T_{\gamma e}(\Delta
t)&T_{ee}(\Delta
t)\end{pmatrix}_{ji}\begin{pmatrix}{Y}_\gamma({t})\\{Y}_e({t})\end{pmatrix}^{(n)}_j + \Delta t\begin{pmatrix}0&0\\0&{\mathcal D}\end{pmatrix}_{ji}\begin{pmatrix}{Y}_\gamma({t})\\{Y}_e({t})\end{pmatrix}^{(n-1)}_j\quad(n>0)\,,
\end{aligned}$$ The full cascade solution is then given by $$\begin{pmatrix}{Y}_\gamma({t}')\\{Y}_e({t}')\end{pmatrix}^{(n)}_i
\simeq
\sum\limits_{m=0}^n\sum_j\mathcal{A}^{(m)}_{ji}(t'-t)\begin{pmatrix}{Y}_\gamma({t})\\{Y}_e({t})\end{pmatrix}^{(n-m)}_j + \Delta t\sum_j\mathcal{B}^{(n)}_{ji}(t'-t)\begin{pmatrix}Q_\gamma\\Q_e\end{pmatrix}_j\,.$$ The $2n$ matrizes $\mathcal{A}^{(m)}$ and $\mathcal{B}^{(m)}$ follow the recursive relation $$\begin{aligned}
\label{eq:rec1}
\mathcal{A}^{(n)}(2^p \Delta t) &= \sum\limits_{i=0}^n\mathcal{A}^{(i)}(2^{p-1}\Delta t)\cdot\mathcal{A}^{(n-i)}(2^{p-1}\Delta t)\,,\\
\mathcal{B}^{(n)}(2^p\Delta t) &= \mathcal{B}^{(n)}(2^{p-1}\Delta t) + \sum\limits_{i=0}^n\mathcal{A}^{(i)}(2^{p-1}\Delta t)\cdot\mathcal{B}^{(n-i)}(2^{p-1}\Delta t)\,,\label{eq:rec2}\end{aligned}$$ where the non-zero initial conditions are $\mathcal{A}^{(0)}(\Delta t) = \mathcal{T}(\Delta t)$, $\mathcal{A}^{(1)}_{ij} = {\rm diag}(0,\Delta t \mathcal{D}_{ij})$ and $\mathcal{B}^{(0)}(\Delta t) =\mathbf{1}$. The matrices $\mathcal{A}^{(0)}$ and $\mathcal{B}^{(0)}$ are the familiar transfer matrices for electro-magnetic cascades in the presence of a source term. Using the recursion relations (\[eq:rec1\]) and (\[eq:rec2\]) we can efficiently calculate the matrices $\mathcal{A}^{(n)}$ and $\mathcal{B}^{(n)}$ via matrix-doubling [@Protheroe:1992dx].
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[^1]: We can relax this condition by requiring that the $\gamma$-ray emission is into a jet with sufficiently large jet opening-angle.
[^2]: To be more precise, $\theta$ is an element of the covering space $\mathbf{R}$ of the circle $\mathbf{S}^1$. The distribution along the circle is then obtained by ${\mathcal Y}_{\mathrm{S}^1}(E,\theta) = \sum_{n\in\mathrm{Z}}{\mathcal Y}_{\mathrm{R}}(E,\theta+2\pi n)$. However, we are only interested in small scattering angles $\theta\ll 1^\circ$ and hence ${\mathcal Y}_{\mathrm{S}^1}(E,\theta) \simeq {\mathcal Y}_{\mathrm{R}}(E,\theta)$.
[^3]: We consider only small deflections and can hence approximate the sphere as two-dimensional flat space.
|
---
abstract: 'For systems which contain both superselection structure and constraints, we study compatibility between constraining and superselection. Specifically, we start with a generalisation of Doplicher-Roberts superselection theory to the case of nontrivial centre, and a set of Dirac quantum constraints and find conditions under which the superselection structures will survive constraining in some form. This involves an analysis of the restriction and factorisation of superselection structures. We develop an example for this theory, modelled on interacting QED.'
author:
- |
[Hellmut Baumgaertel]{}\
[ Mathematical Institute, University of Potsdam,]{}\
[ Am Neuen Palais 10, Postfach 601 553,]{}\
[ D–14415 Potsdam, Germany.]{}\
[ E-mail: [email protected]]{}\
[FAX: +49-331-977-1299]{}\
- |
[Hendrik Grundling]{}\
[Department of Mathematics,]{}\
[University of New South Wales,]{}\
[Sydney, NSW 2052, Australia.]{}\
[[email protected]]{}\
[FAX: +61-2-93857123]{}
date: 'RUNNING TITLE: Superselection and Constraints.'
title: '**Superselection in the presence of constraints.**'
---
Introduction
============
In heuristic quantum field theory, there are many examples of systems which contain global charges (hence superselection structure) as well as a local gauge symmetry (hence constraints). Most of these systems cannot currently be written in a consistent mathematical framework, due to the presence of interactions. Nevertheless, the mathematical structure of superselection by itself has been properly developed (cf. [@DR; @Bg; @BW]), as well as the mathematical structure of quantum constraints [@Grundling85; @Lledo; @lands], hence one can at least abstractly consider systems which contain both. This will be the focus of our investigations in this paper. We will address the natural intertwining questions for the two structures, as well as compatibility issues.
There is a choice in how the problem of superselection with constraints is posed mathematically. We will be guided by the most important physics example in this class, which is that of a quantized local gauge field, acting on a fermion field. It has a Gauss law constraint (implementing the local gauge transformations) as well as a set of global charges (leading to superselection).
The architecture of the paper is as follows, in Sect. 2 we give a brief summary of the superselection theory which we intend to use here. We include recent results concerning the case of an observable algebra with nontrivial centre (cf. [@BL]), and some new results on the field algebra. In Sect. 3 we give a summary of quantum constraints, and in Sect. 4 we collect our main results. The proofs for these are in Sect. 6, and in Sect. 5 we present an example.
Fundamentals of superselection
==============================
In this section we summarize the structures from superselection theory which we need. For proofs, we refer to the literature if possible.
The superselection problem in algebraic quantum field theory, as stated by the Doplicher–Haag–Roberts (DHR) selection criterion, led to a profound body of work, culminating in the general Doplicher–Roberts (DR) duality theory for compact groups. The DHR criterion selects a distinguished class of “admissible” representations of a quasilocal algebra $\al A.$ of observables, where the centre is trivial, i.e. $Z(\al A.)=\C\un,$ or even $\al A.$ is assumed to be simple. This corresponds to the selection of a DR–category $\al T.$ of “admissible” endomorphisms of $\al A.\,.$ Furthermore, from this endomorphism category $\al T.$ the DR–analysis constructs a C\*–algebra $\al F.\supset\al A.$ together with a compact group action $\alpha:\al G.\ni g\to\alpha_{g}\in\aut\al F.$ such that
- $\al A.$ is the fixed point algebra of this action,
- $\al T.$ coincides with the category of all “canonical endomorphisms" of $\al A.$ (cf. Subsection \[CanEnd\]).
$\al F.$ is called a Hilbert extension of $\al A.$ in [@BW]. Physically, $\al F.$ is identified as a field algebra and $\al G.$ with a global gauge group of the system. $\{\al F.,\alpha_{\al G.}\}$ is uniquely determined by $\al T.$ up to $\al A.$-module isomorphisms. Conversely, $\{\al F.,\alpha_{\al G.}\}$ determines uniquely its category of all canonical endomorphisms. Therefore one can state the equivalence of the “selection principle", given by $\al T.$ and the “symmetry principle", given by $\al G.\,$. This duality is one of the crucial theorems of the Doplicher-Roberts theory.
In contrast to the original theory of Doplicher and Roberts, we allow here a nontrivial centre for $\al A..$ The reason for this is that when there are constraints present, the system contains nonphysical information, so there is no physical reason why $\al A.$ should be simple. Only after eliminating the constraints should one require the final observable algebra to be simple, hence having trivial centre. Now a duality theorem for a C\*–algebra with nontrivial centre has been proven recently [@BL; @B2], establishing a bijection between distinguished categories of endomorphisms of $\al A.$ and Hilbert extensions of $\al A.$ satisfying some additional conditions, of which the most important is: ${\al A.'\cap\al F.=Z(\al A.)}$ (i.e. the relative commutant is assumed to be minimal). This will be properly explained below. This condition has already been used by Mack and Schomerus [@MS] as a “new principle".
Basic properties of Hilbert systems
-----------------------------------
Below $\al F.$ will always denote a unital C\*–algebra. A Hilbert space $\al H.\subset
\al F.$ is called [**algebraic**]{} if the scalar product $\langle\cdot,\cdot
\rangle$ of $\al H.$ is given by $\langle A,B\rangle\un := A^{\ast}B$ for $A,\; B\in\al H.\,.$ Henceforth we consider only finite-dimensional algebraic Hilbert spaces. The support $\hbox{supp}\,\al H.$ of $\al H.$ is defined by $\hbox{supp}\,\al H.:=\sum_{j=1}^{d}\Phi_j\Phi_{j}^{\ast}$ where $\{\Phi_j\,\big|\,
j=1,\ldots,\,d\}$ is any orthonormal basis of $\al H..$ [*Unless otherwise specified, we assume below that each algebraic Hilbert space $\al H.$ considered, satisfies ${\rm supp}\,\al H.
=\un.$*]{}
We also fix a compact C\*-dynamical system $\{\al F.,\al G.,\alpha\}$, i.e. $\al G.$ is a compact group and $\alpha:\al G.\ni g\to\alpha_{g}\in\aut\al F.$ is a pointwise norm-continuous morphism. For $\gamma\in\wh{\al G.}$ (the dual of $\al G.$) its [**spectral projection**]{} $\Pi_{\gamma}\in\al L.(\al F.)$ is defined by $$\begin{aligned}
\Pi_\gamma(F)&:=&\int_{\al G.}\ol\chi_\gamma(g).\,\alpha_{g}(F)\,dg
\quad\hbox{for all}\quad F\in\al F., \\[1mm]
\hbox{where:}\quad\qquad
\chi_\gamma(g)&:=&\dim\gamma\cdot\tr\pi(g),\quad\pi\in\gamma\,\end{aligned}$$ and its [**spectral subspace**]{} $\Pi_\gamma\al F.$ satisfies $\Pi_\gamma\al F.=\csp\{\al L.\subset\al F.\}$ where $\al L.$ runs through all invariant subspaces of $\al F.$ which transform under $\alpha\s{\al G.}.$ according to $\gamma$ (cf. [@ES]). Define the [**spectrum**]{} of $\alpha_{\al G.}$ by $$\spec\alpha_{\al G.}:=\set \gamma\in\wh{\al G.}, \Pi_\gamma\not=0.$$
Our central object of study is:
The C\*-dynamical system $\{\al F.,\al G.,\alpha\}$ is called a [**Hilbert system**]{} if for each $\gamma\in\wh{\al G.}$ there is an algebraic Hilbert space $\al H._\gamma\subset\al F.,$ such that $\alpha_{\al G.}$ acts invariantly on $\al H._\gamma,$ and the unitary representation $\al G.\rest\al H._\gamma$ is in the equivalence class $\gamma\in\wh{\al G.}$.
Note that for a Hilbert system $\{\al F.,\al G.,\alpha\}$ we have necessarily that the algebraic Hilbert spaces satisfy $\al H._\gamma\subset\Pi_\gamma\al F.$ for all $\gamma,$ and hence that $\spec\alpha\s{\al G.}.=\wh{\al G.}$ i.e. the spectrum is [*full*]{}. The morphism $\alpha:\al G.\to \hbox{Aut}\,\al F.$ is necessarily faithful. So, since $\al G.$ is compact and $\hbox{Aut}\,\al F.$ is Hausdorff w.r.t. the topology of pointwise norm-convergence, $\alpha$ is a homeomorphism of $\al G.$ onto its image. Thus $\al G.$ and $\alpha_{\al G.}$ are isomorphic as topological groups.
We are mainly interested in Hilbert systems whose fixed point algebras coincide such that they appear as extensions of it.
A Hilbert system $\{\al F.,\al G.,\alpha\}$ is called a [**Hilbert extension**]{} of a C\*–algebra $\al A.\subset\al F.$ if $\al A.$ is the fixed point algebra of ${\al G.}.$ Two Hilbert extensions $\{\al F._i,\,\al G.\,,\alpha^{i}\},\;i=1,\,2$ of $\al A.$ (w.r.t. the same group $\al G.$) are called $\al A.\hbox{\bf--module isomorphic}$ if there is an isomorphism $\tau:\al F._1\to\al F._2$ such that $\tau(A)=A$ for $A\in\al A.,$ and $\tau$ intertwines the group actions, i.e. $\tau\circ\alpha^{1}_g=\alpha^{2}_g\circ\tau.$
- Group automorphisms of $\al G.$ lead to $\al A.$-module isomorphic Hilbert extensions of $\al A.$, i.e. if $\{\al F.,\al G.,\alpha\}$ is a Hilbert extension of $\al A.$ and $\xi$ an automorphism of $\al G.$, then the Hilbert extensions $\{\al F.,\al G.,\alpha\}$ and $\{\al F.,\al G.,\alpha\circ\xi\}$ are $\al A.$-module isomorphic. So the Hilbert system $\{\al F.,\al G.,\alpha\}$ depends, up to $\al A.$-module isomorphisms, only on $\alpha_{\al G.}$, which is isomorphic to $\al G.$. In other words, up to $\al A.$-module isomorphism we may identify $\al G.$ and $\alpha_{\al G.}\subset\aut\al F.$ neglecting the action $\alpha$ which has no relevance from this point of view. Therefore in the following, unless it is otherwise specified, we use the notation $\{\al F.,\al G.\}$ for a Hilbert extension of $\al A.$, where $\al G.\subset\aut\al F.$.
- As mentioned above, examples of Hilbert systems arise in DHR–superselection theory cf. [@BW; @Bg]. There are also constructions by means of tensor products of Cuntz algebras (cf. [@DR2]). In these examples the relative commutant of the fixed point algebra $\al A.$, hence also its center, is trivial. Another construction for $\al G.=\T$, by means of the loop group $C^{\infty}(S^{1},\,\T)$ is in [@BC], and for this $Z(\al A.)$ is nontrivial.
\[remark1\] A Hilbert system $\{\al F.,\,\al G.\}$ is a highly structured object;- we list some important facts and properties (for details, consult [@Bg; @BW]):
- The spectral projections satisfy: $$\begin{aligned}
\Pi_{\gamma_1}\Pi_{\gamma_2} &=&
\Pi_{\gamma_2}\Pi_{\gamma_1}=\delta\s\gamma_1\gamma_2.\Pi_{\gamma_1} \\[1mm]
\|\Pi_\gamma\| &\leq& d(\gamma)^{3/2}\;,
\qquad d(\gamma):=\dim(\al H._\gamma)\;, \\[1mm]
\Pi_\gamma\al F. &=& \spa(\al AH._\gamma)\;, \quad \Pi_{\iota}\al F.=\al A.\;,\end{aligned}$$ where $\iota\in\wh{\al G.}$ denotes the trivial representation of $\al G..$
- Each $F\in\al F.$ is uniquely determined by its projections $\Pi_\gamma F,$ $\gamma\in\wh{\al G.},$ i.e. $F=0$ iff $\Pi_\gamma F =0$ for all $\gamma\in\wh{\al G.},$ cf. Corollary 2.6 of [@B2].
- A useful \*-subalgebra of $\al F.$ is $$\al F._{\rm fin}:= \set F\in\al F.,\Pi_\gamma F\not=0\quad\hbox{for
only finitely many $\gamma\in\wh{\al G.}$}.$$ which is dense in $\al F.$ w.r.t. the C\*–norm (cf. [@S]).
- In $\al F.$ there is an $\al A.\hbox{--scalar product}$ given by ${\langle F,\, G\rangle_{\al A.}:=\Pi_\iota FG^*},$ w.r.t. which the spectral projections are symmetric, i.e. $\bra\Pi_\gamma F,G.=\bra F,\Pi_\gamma G.$ for all $F,\; G\in\al F.,$ $\gamma\in\wh{\al G.}$. Using the $\al A.$-scalar product one can define a norm on $\al F.,$ called the $\al A.\hbox{-norm}$ $$\vert F\vert_{\al A.}:=\Vert\langle F,F\rangle\s{\al A.}.\Vert^{1/2},\quad
F\in \al F..$$ Note that $\vert F\vert_{\al A.}\leq \Vert F\Vert$ and that $\al F.$ in general is not closed w.r.t. the $\al A.$-norm. Then for each $F\in\al F.$ we have that $F=\sum_{\gamma\in\wh{\al G.}}\Pi_\gamma F$ where the sum on the right hand side is convergent w.r.t. the $\al A.\hbox{--norm}$ but not necessarily w.r.t. the C\*–norm $\|\cdot\|\,.$ We also have Parseval’s equation: $\langle F,F\rangle_{\al A.}
=\sum_{\gamma\in\wh{\al G.}}\langle\Pi_\gamma F,\Pi_\gamma F\rangle_{\al A.}
\;,$ cf. Proposition 2.5 in [@B2]. Moreover $\big|\Pi_\gamma\big|_{\al A.}=1$ for all $\gamma\in\wh{\al G.}$, where $\vert\cdot\vert_{\al A.}$ denotes the operator norm of $\Pi_\gamma$ w.r.t. the norm $\vert\cdot\vert_{\al A.}$ in $\al F.$.
- Generally for a Hilbert system, the assignment $\gamma\to\al H._\gamma$ is not unique. If $U\in \al A.$ is unitary then also $U\al H._\gamma\subset\Pi_{\gamma}\al F.$ is an $\al G.$-invariant algebraic Hilbert space carrying the representation $\gamma\in\wh{\al G.}.$ Each $\al G.$-invariant algebraic Hilbert space $\al K.$ which carries the representation $\gamma\,$ is of this form, i.e. there is a unitary $V\in\al A.$ such that $\al K.=V\al H._{\gamma}.$ For a general $\al G.\hbox{--invariant}$ algebraic Hilbert space $\al H.\subset\al F.,$ we may have that $\al G.\rest\al H.$ is not irreducible, i.e. it need not be of the form $\al K.=V\al H._{\gamma}.$ Below we will consider further conditions on the Hilbert system to control the structure of these.
- Given two $\al G.\hbox{--invariant}$ algebraic Hilbert spaces $\al H.,\al K.\subset\al F.,$ then $\spa(\al H.\cdot\al K.)$ is also a $\al G.\hbox{--invariant}$ algebraic Hilbert space which we will briefly denote by $\al H.\cdot\al K..$ It is a realization of the tensor product $\al H.\otimes\al K.$ within $\al F.$ and carries the tensor product of the representations of $\al G.$ carried by $\al H.$ and $\al K.$ in the obvious way.
- Let $\al H.,\al K.$ be two $\al G.\hbox{--invariant}$ algebraic Hilbert spaces, but not necessarily of support 1. Then there is a natural isometric embedding $\al J.:\al L.(\al H.,\al K.)\to\al F.$ given by $$\al J.(T):=\sum_{j,k}
t_{j,k}\Psi_{j}\Phi^{\ast}_{k},\quad t_{j,k}\in \C,
\quad T\in\al L.(\al H.,\al K.)$$ where $\{\Phi_{k}\}_{k},\{\Psi_{j}\}_{j}$ are orthonormal bases of $\al H.$ and $\al K.$ respectively, and where $$T(\Phi_{k})=\sum_{j}t_{j,k}\Psi_{j},$$ i.e. $(t_{j,k})$ is the matrix of $T$ w.r.t. these orthonormal bases. One has $$T(\Phi)=\al J.(T)\cdot\Phi,\quad \Phi\in\al H..$$ This implies: if $T_{j}\in\al L.(\al H._{j},\al K._{j}),j=1,2,$ hence $T_{1}\otimes T_{2}\in\al L.(\al H._{1}\al H._{2},\al K._{1}\al K._{2}),$ then ${\al J.(T_{1}\otimes T_{2})\Phi_{1}\Phi_{2}}=\al J.(T_{1})\Phi_{1}\al J.(T_{2})\Phi_{2}$ for $\Phi_{j}\in\al H._{j}$.
Moreover $\al J.(T)\in\al A.$ iff $T\in\al L._{\al G.}(\al H.,\al K.),$ where $\al L._{\al G.}(\al H.,\al K.)$ denotes the linear subspace of $\al L.(\al H.,\al K.)$ consisting of all intertwining operators of the representations of $\al G.$ on $\al H.$ and $\al K.$ (cf. p. 222 [@BW]).
The category of $\al G.$-invariant algebraic Hilbert spaces
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The $\al G.$-invariant algebraic Hilbert spaces $\al H.$ of $\{\al F.,\al G.\}$ form the objects of a category $\al T._{\al G.}$ associated to $\{\al F.,\al G.\}$ whose arrows are given by the elements of $(\al H.,\,\al K.):=\al J.(\al L._{\al G.}(\al H.,\al K.))\subset\al A.$. It is already large enough to carry all tensor products of the representations of $\al G.$ on its objects by Remark \[remark1\](vi) (though not necessarily subrepresentations and direct sums). First, let us state some of its rich structure (cf. [@BW; @DR2]):
\[perm&conj\] For $\{\al F.,\al G.\}$ the category $\al T._{\al G.}$ is a [**tensor C\*-category**]{}, i.e. the arrow spaces $(\al H.,\,\al K.)$ are Banach spaces such that $\bullet$ w.r.t. composition of arrows $R,\, S$ we have $\|R\circ S\|\leq\|R\|\|S\|,$$\bullet$ there is an antilinear involutive contravariant functor $*:\al T._{\al G.}\to\al T._{\al G.}$ such that $\|R^*\circ R\|=\|R\|^2$ for all arrows $R$ with the same range and domain (here the functor $*$ is given by the involution in $\al F.)$$\bullet$ There is an associative product $\cdot$ on $\ob\al T._{\al G.}$ and an identity object $1\in\ob\al T._{\al G.}$ (i.e. $1\cdot\al H.=\al H.
=\al H.\cdot 1)$ and there is an associative bilinear product $\times$ of the arrows, such that if $R\in(\al H.,\al K.)$ and $R'\in(\al H.',\al K.')$ then $R\times R'\in(\al H.\cdot\al H.',\,
\al K.\cdot\al K.').$ Moreover we require that for $R,\;R'$ as above: $$1_\iota\times R=R\times 1_\iota=R\;,\quad (R\times R')^*=R^*\times R'^*\,,$$ where $1_\iota\in(1,1)$ is the identity arrow, as well as the interchange law $$(S\circ R)\times (S'\circ R')=(S\times S')\circ (R\times R')\,,$$ whenever the left hand side is defined.Here in $\al T._{\al G.},$ the product $\cdot$ is given by the product of $\al F.,$ the identity object is $1:=\C\un$ and the product $\times$ is defined by $$R\times R':= \al J.(T\otimes T'),$$ for $R=\al J.(T),\,R'=\al J.(T'),$ where $T\in\al L._{\al G.}(\al H.,\al K.),\,
T'\in\al L._{\al G.}(\al H.',\al K.').$ Note that $(1,1)=(\C\un,\C\un)=\C\un$, i.e. $1_{\iota}=\un.$
$\al T._{\al G.}$ has additional important structures (permutation and conjugation), which we will consider below in Subsection \[PermConj\].
We need to examine conditions to require of $\{\al F.,\,
\al G.\}$ to ensure that $\al T._{\al G.}$ carries subrepresentations and direct sums.
Let $\al H.,\,\al K.\in\ob\al T._{\al G.},$ and define $\al H.<\al K.$ to mean that there is an orthoprojection $E$ on $\al K.$ such that $E\al K.$ is invariant w.r.t. $\al G.$ and the representation $\al G.\rest\al H.$ is unitarily equivalent to $\al G.\rest E\al K..$ Call $\al H.$ a [**subobject**]{} of $\al K.\,.$
It is easy to see that $<$ is a partial order. Note that $\al H.<\al K.$ iff there is an isometry $V\in\al L._{\al G.}(\al H.,\al K.)$ such that $VV^{\ast}=:E$ is a projection of $\al K.$, i.e. $V\al H.=E\al K..$ Then $\al J.(V)\in\al A.$ and $E\al K.=\al J.(V)\cdot\al H..$
If $E\in\al L._{\al G.}(\al K.)$ is an orthoprojection $0<E<\un,$ i.e. $E$ is a reducing projection for the representation of $\al G.$ on $\al K.$, then the question arises whether there is an object $\al H.$ such that the representations on $\al H.$ and $E\al K.$ are unitarily equivalent. This suggests the concept of [*closedness*]{} of $\al T._{\al G.}$ w.r.t. subobjects.
The category $\al T._{\al G.}$ is [**closed w.r.t. subobjects**]{} if to each $\al K.\in\hbox{Ob}\,\al T._{\al G.}$ and to each nontrivial orthoprojection $E\in\al L._{\al G.}(\al K.)$ there is an isometry $\wh{V}\in\al A.$ with $\wh{V}\wh{V}^{\ast}=\al J.(E).$ In this case $\al H.:=\wh{V}^{\ast}\cdot\al K.$ is a subobject $\al H.<\al K.$ assigned to $E$, where $\wh{V}=\al J.(V)$ for some isometry $V\in\al L._{\al G.}(\al H.,\al K.)$ with $VV^*=E.$
Next, we consider when an object of $\al T._{\al G.}$ carries the direct sum of the representations of two other objects. If $V,W\in\al A.$ are isometries with $VV^{\ast}+WW^{\ast}=\un$ and $\al H.,\al K.\in\hbox{Ob}\,\al T._{\al G.}$ then we call the algebraic Hilbert space $V\al H.+W\al K.$ of support 1 a [**direct sum**]{} of $\al H.$ and $\al K.$. It is $\al G.$-invariant and carries the direct sum of the representations on $\al H.$ and $\al K.$ but in general depends on the choice of isometries $V,\;W.$ We define
- The category $\al T._{\al G.}$ is [**closed w.r.t. direct sums**]{} if to each $\al H._{1},\al H._{2}\in\ob\,\al T._{\al G.}$ there is an object $\al K.\in\ob\,\al T._{\al G.}$ and there are isometries $V_{1},V_{2}\in\al A.$ with $V_{1}V_{1}^{\ast}+V_{2}V_{2}^{\ast}=\un$ such that $\al K.=V_{1}\al H._{1}+V_{2}\al H._{2}$ (then $V_{1}\in(\al H._{1},\al K.)$ and $V_{2}\in(\al H._{2},\al K.)$ follow).
- A C\*-algebra $\al A.$ satisfies [**Property B**]{} if there are isometries $V_{1},V_{2}\in\al A.$ such that $V_{1}V_{1}^{\ast}+V_{2}V_{2}^{\ast}=\un.$ A Hilbert system $\{\al F.,\al G.\}$ is said to satisfy [**Property B**]{} if its fixed point algebra $\al A.:=\Pi_{\iota}\al F.$ satisfies Property B.
For a Hilbert system $\{\al F.,\al G.\}$ we have:
- It satisfies Property B iff $\al T._{\al G.}$ is closed w.r.t. direct sums.
- For nonabelian $\al G.,$ the category $\al T._{\al G.}$ is closed w.r.t. subobjects iff it is closed w.r.t. direct sums iff it has Property B cf. Prop. 3.5 of [@B2].
- In the case that $\al G.$ is abelian, the theory simplifies. This is because we already have Pontryagin’s duality theorem, hence it is not necessary to consider closure under subobjects and direct sums to obtain a duality theory.
The category of canonical endomorphisms {#CanEnd}
---------------------------------------
The main aim of DR–theory is to obtain an intrinsic structure on $\al A.$ from which we can reconstruct the Hilbert system $\HS$ in an essentially unique way. Here we want to transport the rich structure of $\al T._{\al G.}$ to $\al A.\,.$
To each $\al G.\hbox{--invariant}$ algebraic Hilbert space $\al H.\subset
\al F.$ there is assigned a corresponding [**inner endomorphism**]{} $\rho\s{\al H.}.\in\endo\al F.$ given by $$\rho\s{\al H.}.(F):=\sum_{j=1}^{d(\al H.)}\Phi_jF\Phi_j^*\;,$$ where $\{\Phi_j\,\big|\,j=1,\ldots,\,d(\al H.)\}$ is any orthonormal basis of $\al H..$ Note that $\rho\s{\al H.}.$ preserves $\al A.\,.$ A [**canonical endomorphism**]{} is the restriction of an inner endomorphism to $\al A.,$ i.e. it is of the form $\rho_{\al H.}\rest\al A.\in\mbox{End}\,\al A.\,.$
\[remark2\]
- The definition of the canonical endomorphisms uses $\al F.$ explicitly. The question arises whether the canonical endomorphisms can be characterised by intrinsic properties within $\al A.$. This interplay between the $\rho_{\al H.}$ and the $\rho_{\al H.}\rest\al A.$ plays an essential role in the DR-theory. Below, we omit the restriction symbol and regard the $\rho_{\al H.}$ also as endomorphisms of $\al A.$.
- If the emphasis is only on the representation $\gamma$ and not on its corresponding algebraic Hilbert space $\al H._{\gamma},$ we will write $\rho_{\gamma}$ instead of $\rho_{\al H._{\gamma}}$.
- Note that $\Phi A=\rho\s{\al H.}.(A)\Phi$ for all $\Phi\in\al H.$ and $A\in\al A..$
- Note that the identity endomorphism $\iota$ is assigned to $\al H.=\C\un,$ i.e. $\rho\s{\C\un}.:=\iota.$
- Let $\al H.,\al K.$ be as before, then $\rho\s{\al H.}.\circ\rho\s{\al K.}.=\rho\s{\al H.\cdot\al K.}..$
- The map $\rho$ from $\ob\al T._{\al G.}$ to the canonical endomorphisms is in general not injective. In fact we have: if $\al H.,\al K.\in\ob\al T._{\al G.},$ then $\rho_{\al H.}\rest\al A.=\rho_{\al K.}\rest\al A.$ iff $\Psi^{\ast}\Phi\in\al A.'\cap\al F.$ for all $\Phi\in\al H.,\Psi\in\al K.\,,$ cf. Prop. 3.9 in [@B2].
Define $\al T.$ to be the category with objects the canonical endomorphisms, and arrows the intertwiner spaces, where the [**intertwiner space**]{} of canonical endomorphisms $\sigma,\;\tau\in\endo\al A.$ is: $$(\sigma,\,\tau):=\set X\in\al A.,X\sigma(A)=\tau(A)X\quad\hbox{for all}
\;{A\in\al A.}.\;.$$ and this is a complex Banach space. For $A\in(\sigma,\sigma'),$ $B\in(\tau,\tau'),$ we define ${A\times B}:=A\sigma(B)\in{(\sigma\tau,\,\sigma'\tau')\,.}$ We will say that $\sigma,\;\tau\in\endo\al A.$ are [**mutually disjoint**]{} if ${(\sigma,\,\tau)}=\{0\}$ when $\sigma\not=\tau.$
\[remark3\]
- We have $(\iota,\,\iota)=Z(\al A.):=$ centre of $\al A..$
- The composition of two canonical endomorphisms (which corresponds to products of the generating Hilbert spaces, see Remark 2.13 (v), i.e. to tensor products of representations) satisfies the correct compatibility conditions with the product $\times$ of intertwiners to ensure that $\al T.$ is a C\*-tensor category cf. Prop. \[perm&conj\] and [@DR]. The identity object is $\iota.$
- Recall the isometry $\al J.:\al L.\s{\al G.}.(\al H.,\al K.)
\longrightarrow \al A.$ encountered in Remark \[remark1\](vii). We claim that its image is in fact contained in ${\left(\rho\s{\al H.}.,\,
\rho\s{\al K.}.\right)}.$ To see this, let $\Phi\in\al H.,\;
A\in\al A.$ and $T\in\al L._{\al G.}(\al H.,\al K.).$ Then $$\al J.(T)\rho\s{\al H.}.(A)\Phi=\al J.(T)\Phi\cdot A=
T(\Phi)\cdot A=\rho\s{\al K.}.(A)T(\Phi)=
\rho\s{\al K.}.(A)\al J.(T)\cdot\Phi$$ hence $$\al J.(T)\rho\s{\al H.}.(A)=\rho\s{\al K.}.(A)\al J.(T)$$ i.e. $\al J.(T)\in {\left(\rho\s{\al H.}.,\,
\rho\s{\al K.}.\right)}$ or $$\label{nonful}
(\al H.,\,\al K.)=
\al J.(\al L._{\al G.}(\al H.,\al K.))\subseteq
(\rho_{\al H.},\,\rho_{\al K.}).$$ In general, the inclusion is proper. Note that for $R\in(\al H.,\al K.),\; R'\in(\al H.',\al K.')$ we have $\al J.(R\otimes R')=R\rho_{\al H.}(R'),$ i.e. $\times$ restricted to the $(\al H.,\al K.)'s$ coincides with the definition of $\times$ in Proposition \[perm&conj\] of the category $\al T._{\al G.}.$
Next we would like to define the concepts of subobject and direct sums on $\ob\al T.$ compatibly with those on $\ob\al T._{\al G.}$ under the morphism $\rho.$ Recall that $\al H.<\al K.$ iff we have an isometry $V\in\al L.\s{\al G.}.(\al H.,\,\al K.)$ and a projection $E\in
\al L.\s{\al G.}.(\al K.)$ with $V\al H.=E\al K.=\al J.(E)\cdot\al K.=\al J.(V)\cdot\al H..$ Then by (\[nonful\]) we get that $\al J.(V)\in
(\rho_{\al H.},\rho_{\al K.})$ and $\al J.(E)\in
(\rho_{\al K.},\rho_{\al K.}).$
Note that if $\al L.=V\al H.+W\al K.$ for isometries $V,\,W\in\al A.$ with $VV^*+WW^*=\un,$ then $V\in(\rho\s{\al H.}.,\rho\s{\al L.}.)$ and $W\in(\rho\s{\al K.}.,
\rho\s{\al L.}.).$
- $\tau\in\ob\al T.$ is a [**subobject**]{} of $\sigma\in\ob\al T.,$ denoted ${\tau<\sigma,}$ if there is an isometry $V\in (\tau,\sigma)$. In this case $\tau(\cdot)=V^{\ast}\sigma(\cdot)V$ and $VV^{\ast}=:E\in (\sigma,\sigma)$ follow.
- $\lambda\in\ob\,\al T.$ is a [**direct sum**]{} of $\sigma,\tau\in\ob\,\al T.,$ if there are isometries $V\in(\sigma,\lambda),\,W\in(\tau,\lambda)$ with $VV^{\ast}+WW^{\ast}=\un$ such that $$\lambda(\cdot)=V\sigma(\cdot)V^{\ast}+W\tau(\cdot)W^{\ast}.$$
\[Tsubobj\]
- The subobject relation $\tau<\sigma$ is again a partial order, because $\tau<\sigma$ and $\sigma<\mu$ imply the existence of isometries $V\in (\tau,\sigma),\,W\in (\sigma,\mu)$. Then $WV\in (\tau,\mu)$ is also an isometry, i.e. $\tau<\mu$.
- A direct sum as defined above is only unique up to unitary equivalence, i.e. if $\lambda,\,\lambda'$ are direct sums of $\sigma,\tau\in\ob\,\al T.,$ then there is a unitary $U\in(\lambda,\lambda').$
- We have $\rho\s{V\al H.+W\al K.}.(\cdot)=V\rho_{\al H.}(\cdot)V^{\ast}+
W\rho_{\al K.}(\cdot)W^{\ast}$ where the isometries $V,\,W\in\al A.$ satisfy $VV^*+WW^*=\un\,.$ Also, if $\al H.<\al K.,$ then $\tau:=\rho_{\al H.}<\rho_{\al K.}=:\sigma.$ However, this does not mean that the partial order $\tau<\sigma$ can be [*defined*]{} by $\al H.<\al K.$ because the transitivity can be violated for some choices of $\al H.,\,\al K.$ cf. Remark \[remark2\](vi).
The closedness of $\al T.$ w.r.t. direct sums is defined by the closedness of $\al T._{\al G.}$ w.r.t. direct sums. The closedness w.r.t. subobjects for $\al T.$ is defined by the closedness w.r.t. subobjects for $\al T._{\al G.}$ in the following sense: If $$\label{endoH}
\lambda=\rho_{\al H.}\in\ob\al T.$$ is given then for all $\al H.$ satisfying (\[endoH\]) and to each nontrivial projection $E\in\al J.(\al L._{\al G.}(\al H.))$ there is an isometry $V\in\al A.$ with $VV^{\ast}=E.$ Then
If $\{\al F.,\al G.\}$ is nonabelian and satisfies Property B then $\al T.$ is closed w.r.t. direct sums and subobjects.
Connection between $\al T._{\al G.}$ and $\al T.$ and further structures. {#PermConj}
-------------------------------------------------------------------------
In the following we assume that $\{\al F.,\al G.\}$ satisfies Property B.
There is a very important relation between the two categories $\al T._{\al G.}$ and $\al T.,$ obtained as follows. The two assignments $\rho:\hbox{Ob}\,\al T._{\al G.}\to\al T.$ (by $\al H.\to\rho_{\al H.})$ and $\al J.:\al L._{\al G.}((\al H.,\al K.))\to(\rho_{\al H.},\rho_{\al K.})$ combine into a faithful categorial morphism from $\al T._{\al G.}$ to $\al T.$ which is compatible with direct sums and subobjects (cf. Remark \[Tsubobj\](iii)) but is not full in general, i.e. the inclusion in Equation (\[nonful\]) is improper for some $\al H.$ and $\al K.$. If $\al A.'\cap\al F.=\C\un,$ then this categorial morphism becomes an isomorphism, cf. Prop. 3.12 in [@B2].
The category $\al T._{\al G.}$ has the following additional structures ([@BW; @DR2]):
\[perm&conj2\] For $\{\al F.,\al G.\}$ the category $\al T._{\al G.}$ satisfies:
- it has a [**permutation structure**]{}, i.e. a map $\epsilon$ from $\ob\al T._{\al G.}\times\ob\al T._{\al G.}$ into the arrows such that
- $\epsilon(\al H.,\al K.)\in (\al H.\al K.,\al K.\al H.)$ is a unitary.
- $\epsilon(\al H.,\al K.)\epsilon(\al K.,\al H.)=\un$.
- $\epsilon(1,\,\al H.)=\epsilon(\al H.,1)=\un$.
- $\epsilon(\al H.\al K.,\al L.)=\epsilon(\al H.,\al L.)\rho_{\al H.}(\epsilon
(\al K.,\al L.))$.
- $\epsilon(\al H.',\al K.')A\times B=B\times A\epsilon(\al H.,\al
K.)$for all $A\in (\al H.,\al H.'),\,B\in (\al K.,\al K.')$.
For $\al T._{\al G.}$ the permutation structure is given by $$\epsilon(\al H.,\al K.):=\al J.(\Theta(\al H.,\al K.))
=\sum_{j,k}\Psi_{j}\Phi_{k}\Psi_{j}^{\ast}\Phi_{k}^{\ast}$$ where $\Theta$ is the flip operator $\al H.\otimes\al K.\rightarrow\al K.\otimes\al H.,$ and where $\{\Phi_{k}\}_{k},\,\{\Psi_{j}\}_{j}$ are orthonormal bases of $\al H.$ and $\al K.$, respectively.
- It has a [**conjugation structure**]{} i.e. for each $\al H.\in\ob\al T._{\al G.}$ there is a conjugated object $\overline{\al H.}\in\ob\al T._{\al G.}$, carrying the conjugated representation of $\al G.$ and there are conjugate arrows $R_{\al H.}\in(1,\overline{\al H.}\al H.),$ $S_{\al H.}=\epsilon(\overline{\al H.},\al H.)R_{\al H.}$ such that $$S_{\al H.}^{\ast}\rho\s{\al H.}.(R_{\al H.})=\un,\quad
R_{\al H.}^{\ast}\rho\s{\overline{\al H.}}.(S_{\al H.})=\un.$$ For $\al T._{\al G.}$ we have $R_{\al H.}:=\sum_{j}\overline{\Psi}_{j}\Psi_{j}$, where $\{\overline{\Psi}_{j}\}_{j}$ is an orthonormal basis of $\overline{\al H.}$. If $\al H.$ carries the representation $\oplus_{j}\gamma_{j},\,\gamma_{j}\in\wh{\al G.},$ then $\overline{\al H.}$ is given by a direct sum of $\al H._{\overline{\gamma_{j}}},$ where $\overline{\gamma_{j}}\in\wh{\al G.}$ represents the conjugated representation of $\gamma_{j}.$
Using the categorial morphism from $\al T._{\al G.}$ to $\al T.$ we equip $\al T.$ with the image permutation and conjugation structures of those on $\al T._{\al G.}$. Note that for the image permutation structure in $\al T.,$ property (v) need not hold for [*all*]{} arrows (cf. Remark \[remark3\](iii)).
For the next definition, observe first that from the operations defined for an abstract tensor category (cf. Prop. \[perm&conj\]), we can define isometries and projections in its arrow spaces, i.e. an arrow $V\in(\lambda,\tau)$ is an isometry if $V^*\circ V=1_\lambda,$ and an arrow $E\in(\lambda,\lambda)$ is a projection if $E=E^*=E\circ E\,.$
An [**(abstract) DR-category**]{} is an (abstract) tensor C\*-category $\al C.$ with $(1,1)=\C\un$ which has a permutation and a conjugation structure, and has direct sums and subobjects, i.e. to all objects $\lambda,\sigma$ there is an object $\tau$ and isometries $V\in(\lambda,\tau),\,W\in(\sigma,\tau)$ such that $VV^{\ast}+WW^{\ast}=1_{\tau}$, and to each nontrivial projection $E\in(\lambda,\lambda)$ there is an object $\sigma$ and an isometry $V\in(\sigma,\lambda)$ such that $E=VV^{\ast}.$
If the Hilbert system $\HS$ satisfies Property B then $\al T._{\al G.}$ is an example of a DR-category, but not necessarily $\al T.$ (since property (v) in Prop. \[perm&conj2\] need not hold for all arrows). However, if additionally $\al A.'\cap\al F.=\C\un$ holds then also $\al T.$ is a DR-category.
Duality Theorems
----------------
Unless otherwise specified, in the following we assume Property B for $\HS$ when $\al G.$ is nonabelian. The DR-theorem produces a bijection between pairs $$\{\al A.,\al T.\} \quad\hbox{and}\quad \HS\,,$$ where $\al T.$ is a DR-category of unital endomorphisms of the unital C\*-algebra $\al A.$ with $Z(\al A.)=\C\un,$ and $\HS$ is a Hilbert extension of $\al A.$ having trivial relative commutant, i.e. $\al A.'\cap\al F.=\C\un$ (see [@DR; @DR5; @B3]). The DR-theorem says that in the case of Hilbert extensions of $\al A.$ with trivial relative commutant, the category $\al T. $ of all canonical endomorphisms can indeed be characterized intrinsically by their abstract algebraic properties as endomorphisms of $\al A.$ and a corresponding bijection can be established.
In this subsection we want to state how to obtain such a bijection for C\*-algebras $\al A.$ with nontrivial center $\al Z.\supset\C\un.$ A first problem is that the category $\al T._\al G.$ and $\al T.$ [*need not*]{} be isomorphic anymore, cf. Remark \[remark3\](iv) and Remark \[remark2\](vi), since now we have $$\C\un\not=\al Z.\subseteq\al A.'\cap\al F.\,.$$ We will investigate in the following the class of Hilbert extensions $\HS$ with compact group $\al G.$ and where the relative commutant satisfies the following [*minimality*]{} condition
A Hilbert system $\{\al F.,\al G.\}$ is called [**minimal**]{} if the condition
$$\label{CZ}
\al A.'\cap\al F.=Z(\al A.)\;.$$
is satisfied.
Then we have cf. Prop. 4.3 of [@B2]:
\[disjZ\] Let $\HS$ be a given Hilbert system. Then $\al A.'\cap\al F.=Z(\al A.)$ iff $(\rho_{\gamma},\rho_{\gamma'})=\{0\}$ for $\gamma\neq\gamma'$, i.e. iff the set $\{\rho_{\gamma}\;\big|\;\gamma\in\wh{\al G.}\}$ is mutually disjoint.
Observe that in any Hilbert system, for each $\tau\in
\ob\al T.$ the space ${\got h}_\tau:=
\al H._\tau Z(\al A.),$ (where $\al H._\tau $ is a $\al G.\hbox{--invariant}$ algebraic Hilbert space) is a $\al G.\hbox{--invariant}$ right Hilbert $Z(\al A.)\hbox{--module}$ i.e. there is a nondegenerate inner product taking its values in $Z(\al A.)$ and it is $\langle A,B\rangle=A^*B\,.$ Now we have cf. [Prop. 3.1 [@BL2]]{}:
\[prop0\] Let $\HS$ be a given minimal Hilbert system, then the correspondence ${\tau\leftrightarrow{\got h}_\tau}$ is a bijection. Thus ${\got h}_\tau=
\al H._\tau Z(\al A.)$ is independent of the choice of $\al H._\tau,$ providing that $\tau=\rho\s{\al H._\tau}..$ This bijection satisfies the conditions $$\begin{aligned}
\sigma\circ\tau &\longleftrightarrow&
{\got h}_\sigma\cdot{\got h}_\tau \\[1mm]
\lambda=(\Ad V)\circ\sigma+(\Ad W)\circ\tau
&\longleftrightarrow&
{\got h}\s\lambda.=V{\got h}_\sigma+W{\got h}_\tau\;.\end{aligned}$$
Thus for minimal Hilbert systems, the $Z(\al A.)\hbox{--modules}$ ${\got h}_\tau$ are uniquely determined by their canonical endomorphisms $\tau,$ even though the choice of $\al H._{\tau}$ is not unique. We are now interested in those choices of $\al H._{\tau}$ which are compatible with products:
A Hilbert system $\HS$ is called [**regular**]{} if there is an assignment $\sigma\rightarrow\al H._{\sigma}$ from $\ob\,\al T.$ to $\al G.$-invariant algebraic Hilbert spaces in $\al F.$ such that
- $\sigma=\rho\s{\al H._\sigma}.,$ i.e. $\sigma$ is the canonical endomorphism of $\al H._\sigma,$
- $\sigma\circ\tau\to\al H._\sigma\cdot\al H._\tau\;.$
In a minimal Hilbert system regularity means that there is a “representing" Hilbert space $\al H._{\tau}\subset{\got h}_{\tau}$ for each $\tau$ with ${\got h}_{\tau}=\al H._{\tau}Z(\al A.)$ such that the compatibility relation (ii) holds.
If a Hilbert system is minimal and $Z(\al A.)=\C\un$ then it is necessarily regular. Thus a class of examples which are trivially minimal and regular, is provided by DHR–superselection theory. A nontrivial example of a minimal and regular Hilbert system is constructed in [@B2].
Then we obtain, cf. Theorem 4.9 of [@B2]:
\[Teo1\] Let $\HS$ be a minimal and regular Hilbert system, then: $\al T.$ contains a C\*–subcategory $\al T._\C$ with the same objects, $\ob\al T._\C=\ob\al T.,$ and arrows $(\sigma,\,\tau)_\C:=(\al H._\sigma,\al H._\tau)=
\al J.\big(\al L.\s{\al G.}.(\al H._\sigma,\al H._\tau)\big)\subset(\sigma,\tau)$ such that:
- $\quad\al T._{\C}$ is a DR-category (in particular $(\iota,\iota)_{\C}
=\C\un)$.
- $\quad(\sigma,\tau)=(\sigma,\tau)\s\C.\sigma(Z(\al A.))
=\tau(Z(\al A.))(\sigma,\tau)\s\C.\sigma(Z(\al A.))\;.$
<!-- -->
- The conditions P.1-P.2 imply that each basis of $(\sigma,\tau)_{\C}$ is simultaneously a module basis of $(\sigma,\tau)$ modulo $\sigma(\al Z.(\al A.))$ as a right module, i.e. the module $(\sigma,\tau)$ is free.
- We will call the DR-subcategory $\al T._\C$ in Theorem \[Teo1\] [**admissible**]{}. If “minimality" is omitted from the hypotheses of Theorem \[Teo1\], then property P.1 remains valid, but not P.2. In this case $\al T._\C$ is a DR-subcategory only. A construction of an example with admissible subcategory can be found in [@B2].
The converse of Theorem \[Teo1\] is also true, and states the main duality result cf. [@BL]:
\[Teo2\] Let $\al T.$ be a C\*–tensor category of unital endomorphisms of $\al A.$ and let $\al T.\s\C.$ be an admissible (DR-)subcategory. Then there is a minimal and regular Hilbert extension $\HS$ of $\al A.$ such that $\al T.$ is isomorphic to the category of all canonical endomorphisms of $\HS.$ Moreover, if $\al T.\s\C.,\;\al T.'_\C$ are two admissible subcategories of $\al T.,$ then the corresponding Hilbert extensions are $\al A.\hbox{--module}$ isomorphic iff $\al T.\s\C.$ is [**equivalent**]{} to $\al T.'_\C$ i.e. iff there is a map $V$ from $\ob\al T.$ to the arrows such that: $$\begin{aligned}
V_{\lambda}&\in& (\lambda,\lambda),\quad
V_{\lambda} \quad\hbox{is unitary, and}\quad V_{\lambda\circ\sigma}=V_{\lambda}\times
V_{\sigma}, \\[1mm]
(\lambda,\sigma)_{\C}'&=&V_{\sigma}(\lambda,\sigma)_{\C}V_{\lambda}^{\ast}
\subset (\lambda,\sigma)\end{aligned}$$ and we have the following compatibility relations for the corresponding permutators $\epsilon,\,\epsilon'$ and conjugates $R_\lambda,\,R'_\lambda:$ $$\begin{aligned}
\epsilon'(\lambda,\sigma)
&=&(V_\sigma\times V_{\lambda})\cdot\epsilon(\lambda,\sigma)\cdot(V_\lambda\times
V_\sigma)^{\ast} \\[1mm]
R'_\lambda&=&V_{\ol\lambda.\circ
\lambda}R_\lambda,\qquad S'_\lambda
=\epsilon'(\lambda,\ol\lambda.)R'_\lambda.\end{aligned}$$
Thus, in minimal and regular Hilbert systems there is an intrinsic characterization of the category of all canonical endomorphisms in terms of $\al A.$ only. Moreover, up to $\al A.\hbox{--module}$ isomorphisms, there is a bijection between minimal and regular Hilbert extensions and C\*-tensor categories $\al T.$ of unital endomorphisms of $\al A.$ with admissible subcategories.
Note that Theorem \[Teo2\] is a generalization of the DR-theorem for the case of nontrivial centre $Z(\al A.)\supset\C\un,$ i.e. it contains the case of the DR-theorem, in that if $Z(\al A.)=\C\un$ then $\al T.$ itself is admissible (hence a DR-category) and the corresponding Hilbert extensions have trivial relative commutant.
Hilbert systems with abelian groups {#abelianHS}
-----------------------------------
If $\al G.$ is abelian the preceding structure simplifies radically. Specifically, $\wh{\al G.}$ is a discrete abelian group (the character group), each $\al H._\gamma,\,\gamma\in\wh{\al G.}$ is one–dimensional with a generating unitary $U_\gamma,$ hence the canonical endomorphisms $\rho_{\al H._{\gamma}}$ (denoted by $\rho_{\gamma},)$ are in fact automorphisms, necessarily outer on $\al A..$ Since $\rho_{\gamma_{1}}\circ\rho_{\gamma_{2}}=\rho_{\gamma_{1}\gamma_{2}}$ in this case the set $\Gamma$ of all canonical endomorphisms $\rho_{\al H._{\gamma}}$ is a group with the property $\wh{\al G.}\cong\Gamma/\hbox{int}\,\al A..$ Hence it is not necessary to consider direct sums, i.e. Property B for $\al A.$ can be dropped.
In the case $Z(\al A.)=\C\un$ the permutators $\epsilon$ (restricted to $\wh{\al G.}\times\wh{\al G.}$) are elements of the second cohomology group $H^{2}(\wh{\al G.})$ and $$U_{\gamma_{1}}\cdot U_{\gamma_{2}}=\omega(\gamma_{1},\gamma_{2})
U_{\gamma_{1}\circ\gamma_{2}},$$ where $$\epsilon(\gamma_{1},\gamma_{2})=\frac{\omega(\gamma_{1},\gamma_{2})}
{\omega(\gamma_{2},\gamma_{1})}$$ and $\omega$ is a corresponding 2-cocycle. The field algebra $\al F.$ is just the $\omega\hbox{--twisted}$ discrete crossed product of $\al A.$ with $\wh{\al G.}$ (see e.g. p.86 ff. [@Bg] for details). For the case $Z(\al A.)\supset\C\un$ see [@BL3] (though the minimal case is not mentioned there).
Kinematics for Quantum Constraints. {#TProcedure}
===================================
In this section we give a brief summary of the method of imposing quantum constraints, developed by Grundling and Hurst [@Grundling85; @Grundling88b; @Lledo]. There are quite a number of diverse quantum constraint methods available in the literature at various levels of rigour (cf. [@lands]). The one we use here is the most congenial from the point of view of C\*–algebraic methods. Our starting point is:
A [**quantum system with constraints**]{} is a pair $(\al B.,\;\al C.)$ where the [**system algebra**]{} $\al B.$ is a unital [C\*]{}–algebra containing the [**constraint set**]{} $\al C.=\al C.^*.$ A [**constraint condition**]{} on $(\al B.,\,\al C.)$ consists of the selection of the physical state space by: $${\got S}_D:=\Big\{ \omega\in{\got S}({\al B.})\mid\pi_\omega(C)
\Omega_\omega=0\quad {\forall}\, C\in {\al C.}\Big\}\,,$$ where ${\got S}({\al B.})$ denotes the state space of $\al B.$, and $(\pi_\omega,\al H._\omega,\Omega_\omega)$ denotes the GNS–data of $\omega$. The elements of ${\got S}_D$ are called [**Dirac states**]{}. The case of [**unitary constraints**]{} means that $\al C.=\al U.-\EINS$, $\;\al U.\subset\al B._u$, and for this we will also use the notation $(\al B.,\,\al U.)$.
The assumption is that all physical information is contained in the pair $(\al B.,{\got S}_D)$. Examples of constraint theories as defined here, have been worked out in detail for various forms of electromagnetism cf. [@Grundling85; @Grundling88c; @Lledo].
For the case of unitary constraints we have the following equivalent characterizations of the Dirac states (cf. [@Grundling85 Theorem 2.19 (ii)]): $$\begin{aligned}
\label{DiracU1}{\got S}_{{D}}&=&\Big\{ \omega\in{\got S}({\al B.})\mid
\omega(U)=1 \quad {\forall}\, U\in {\al U.}\Big\} \\[1mm]
\label{DiracU2} &=&\Big\{ \omega\in{\got S}({\al B.})\mid
\omega(FU)=\omega(F)=\omega(UF) \quad {\forall}\,
F\in\al B.,\; U\in {\al U.}\Big\}.\end{aligned}$$ Moreover, the set $\{\alpha_U:= {\rm Ad}(U)\mid U\in\al U.\}$ of automorphisms of $\al B.$ leaves every Dirac state invariant, i.e. we have $\omega\circ\alpha_U=\omega$ for all $\omega\in {\got S}_{{D}}$, $U\in{\al U.}$.
For a general constraint set $\al C.$, observe that we have: $$\begin{aligned}
{\got S}_D &=& \Big\{ \omega\in{\got S}({\al B.})\mid\omega(C^*C)=0
\quad {\forall}\, C\in\al C.\Big\} \\[1mm]
&=& \Big\{ \omega\in{\got S}({\al B.})\mid \al C.\subseteq
N_\omega\Big\}\kern2mm=\kern2mm\al N.^\perp\cap{\got S}(\al B.)\;.\end{aligned}$$ Here $N_\omega:=\{F\in\al B.\mid\omega(F^*F)=0\}$ is the left kernel of $\omega$ and $\al N.:=\cap\; \{N_{\omega}\mid\omega\in{\got S}_D \}$, and $\al N.^\perp$ denotes the annihilator of $\al N.$ in the dual of $\al B.$. Now $\al N.=\csp(\al BC.)$ because every closed left ideal is the intersection of the left kernels which contains it (cf. 3.13.5 in [@bPedersen89]). Thus $\al N.$ is the left ideal generated by $\al C.$. Since $\al C.$ is selfadjoint and contained in $\al N.$ we conclude $$\al C.\subset {\rm C}^*(\al C.)\subset
\al N.\cap\al N.^*=\csp(\al BC.)\,\cap\,\csp(\al CB.)\;,$$ where ${\rm C}^*(\cdot)$ denotes the C\*–algebra in $\al B.$ generated by its argument. Then we have (cf. [@Lledo]):
\[Teo.3.1\] For the Dirac states we have:
- ${\got S}_{{D}}\neq\emptyset\;$ iff $\;\EINS\not\in {\rm C}^*(\al C.)$ iff $\;\EINS\not\in \al N.\cap\al N.^*=:\al D.$.
- $\omega\in {\got S}_D\;$ iff $\; \pi_{\omega}({\al D.})\Omega_{\omega}=0$.
- An extreme Dirac state is pure.
We will call a constraint set $\al C.$ [**first class**]{} if $\EINS\not\in {\rm C}^*(\al C.)$, and this is the nontriviality assumption which we henceforth make [@Grundling88a Section 3].
Now define $${\al O.} := \{ F\in {\al B.}\mid [F,\, D]:= FD-DF \in {\al D.}\quad
{\forall}\, D\in{\al D.}\}.$$ Then ${\al O.}$ is the C$^*$–algebraic analogue of Dirac’s observables (the weak commutant of the constraints) [@bDirac64]. Then (cf. [@Lledo]):
\[Teo.2.2\] With the preceding notation we have:
- $\al D.=\al N.\cap \al N.^*$ is the unique maximal [C]{}$^*$–algebra in $\, \cap\; \{ {\rm Ker}\,\omega\mid \omega\in
{\got S}_{{D}} \}$. Moreover $\al D.$ is a hereditary [C]{}$^*$–subalgebra of $\al B.$.
- ${\al O.} = {M}_{\al B.}({\al D.})
:=\{ F\in{\cal B}\mid FD\in{\cal D}\ni DF\quad\forall\, D\in{\cal D}\}$, i.e. it is the relative multiplier algebra of ${\al D.}$ in ${\al B.}$.
- $\al O.=\{F\in\al B.\mid\; [F,\,\al C.]\subset\al D.\},$ hence $\al C.'\cap\al B.\subseteq\al O.\;.$
- $\al D.=\csp(\al OC.)=\csp(\al CO.)$.
- For the case of unitary constraints, i.e. $\al C.=\al U.-\EINS$, we have $\al U.\subset\al O.$ and $\al O.={\{F\in\al B.\mid\alpha_U(F)-F\in\al D.
\quad\forall\; U\in\al U.\}}$ where $\alpha_U:={\rm Ad}\,U$.
Thus $\al D.$ is a closed two-sided ideal of $\al O.$ and it is proper when ${\got S}_D\not=\emptyset$ (which we assume here by $\EINS\not\in {\rm C}^*(\al C.)$). Since the traditional observables are $\al C.'\cap\al B.,$ by (iii) we see that these are in $\al O.\,.$ In general $\al O.$ can be much larger than $\al C.'\cap\al B..$
Define the [*maximal [C]{}$^*$–algebra of physical observables*]{} as $${\al R.}:={\al O.}/{\al D.}.$$ The factoring procedure is the actual step of imposing constraints. This method of constructing $\al R.$ from $(\al B.,\,\al C.)$ is called the [**T–procedure**]{} in [@Grundling85], and it defines a map $T$ from first class constraint pairs $(\al B.,\,\al C.)$ to unital C\*–algebras by ${T(\al B.,\,\al C.)}:=\al R.=\al O./\al D..$ We require that after the T–procedure all physical information is contained in the pair $({\al R.}\kern.4mm ,{\got S}
({\al R.}))$, where ${\got S}({\al R.})$ denotes the set of states on $\al R.$. Now, it is possible that $\al R.$ may not be simple [@Grundling85 Section 2], and this would not be acceptable for a physical algebra. So, using physical arguments, one would in practice choose a C$^*$–subalgebra $\al O._c\subset \al O.$ containing the traditional observables $\al C.'$ such that $$\al R._c :=\al O._c / (\al D.\cap\al O._c )\subset \al R.\,,$$ is simple. The following result justifies the choice of $\al R.$ as the algebra of physical observables (cf. Theorem 2.20 in [@Grundling85]):
\[Teo.2.6\] There exists a ${\sl w}^*\hbox{--continuous}$ isometric bijection between the Dirac states on ${\al O.}$ and the states on ${\al R.}$.
Insofar as the physics is now specified by $\al R.,$ this suggests that we call two constraint sets equivalent if they produce the same $\al R..$ More precisely two constraint sets $\al C._1\subset\al B.\supset\al C._2$ are called [**equivalent**]{}, denoted $\al C._1\sim\al C._2,$ if they select the same set of Dirac states, cf. [@Lledo]. In fact $$\al C._1\sim\al C._2\quad\hbox{iff}\quad
\csp(\al BC._1)=\csp(\al BC._2)\quad\hbox{iff}\quad\al D._1=\al D._2\;.$$
The hereditary property of $\al D.$ can be further analyzed, and we list the main points (the proofs are in Appendix A of [@Lledo]).
Denote by $\pi_u$ the universal representation of $\al B.$ on the universal Hilbert space $\al H._u$ [@bPedersen89 Section 3.7]. $\al B.''$ is the strong closure of $\pi_u(\al B.)$ and since $\pi_u$ is faithful we make the usual identification of $\al B.$ with a subalgebra of $\al B.''$, i.e. generally omit explicit indication of $\pi_u$. If $\omega\in{\got S}(\al B.)$, we will use the same symbol for the unique extension of $\omega$ from $\al B.$ to $\al B.''$.
\[Teo.2.7\] For a constrained system $(\al B.,\al C.)$ there exists a projection $P\in\al B.''$ such that
- $\al N.=\al B.''\,P\cap \al B.$,
- $\al D.=P\,\al B.''\,P \cap \al B.$
- ${\got S}_D=\{\omega\in{\got S}(\al B.)\mid\omega(P)=0\}$
- $\al O.=\{ A\in\al B. \mid PA(\EINS-P)=0=
(\EINS-P)AP \}=P'\cap \al B.\;.$
A projection satisfying the conditions of Theorem \[Teo.2.7\] is called [*open*]{} in [@bPedersen89].
What this theorem means, is that with respect to the decomposition $$\al H._u=P\,\al H._u\oplus (\EINS-P)\,\al H._u$$ we may rewrite $$\begin{aligned}
\al D.&=& \Big\{ F\in\al B.\;\Big|\; F=
{\left(\kern-1.5mm\begin{array}{cc}
D \kern-1.6mm & 0 \\ 0 \kern-1.6mm & 0
\end{array} \kern-1.5mm\right)},\;
D\in P\al B.P \Big\}\;\;{\rm and} \\
\al O.&=& \Big\{ F\in\al B.\;\Big|\; F=
{\left(\kern-1.5mm\begin{array}{cc}
A \kern-1.6mm & 0 \\ 0 \kern-1.6mm & B
\end{array} \kern-1.5mm\right)},\;
A\in P\al B.P,\; B\in(\EINS-P)\al B.(\EINS-P) \Big\}\,.\end{aligned}$$ It is clear that in general $\al O.$ can be much greater than the traditional observables $\al C.'\cap\al B.$. Next we show how to identify the final algebra of physical observables $\al R.$ with a subalgebra of $\al B.''$.
\[Teo.2.11\] For $P$ as above we have: $$\al R.\cong\Big\{ F\in\al B.\;\Big|\; F=
\left(\kern-1.5mm\begin{array}{cc}
0 \kern-1.6mm & 0 \\ 0 \kern-1.6mm & A
\end{array} \kern-1.5mm\right) \Big\}=
(\EINS-P)\,(P'\cap\al B.) \subset \al B.''.$$
Below we will need to consider a constraint system contained in a larger algebra, specifically, $\al C.\subset\al A.\subset\al F.$ where $\al C.$ is a first–class constraint set, and $\al A.,\;\al F.$ are unital C\*–algebras. Now there are two constrained systems to consider;- $(\al A.,\,\al C.)$ and $(\al F.,\,\al C.).$ The first one produces the algebras $\al D.\subset\al O.\subseteq\al A.,$ and the second produces $\al D.\s{\al F.}.\subset\al O.\s{\al F.}.\subseteq\al F..$ where as usual, $$\begin{aligned}
\al N.&=& \csp(\al AC.),\qquad \al D.=\al N.\cap\al N.^*,\qquad
\al O.=M\s{\al A.}.(\al D.)
\qquad\hbox{and} \\
\al N.\s{\al F.}.&=&\csp(\al FC.),\qquad
\al D.\s{\al F.}.=\al N.\s{\al F.}.\cap
\al N.^*\s{\al F.}.,\qquad
\al O.\s{\al F.}.=M\s{\al F.}.(\al D.\s{\al F.}.).\end{aligned}$$ Then we have (cf. Theorem 3.2 of [@Grundling88b]):
\[Teo.2.12\] Given as above $\al C.\subset\al A.\subset\al F.$ then $$\al N.\s{\al F.}.\cap\al A.=\al N.,\qquad
\al D.\s{\al F.}.\cap\al A.=\al D.,\qquad
\qquad\hbox{and}\qquad
\al O.\s{\al F.}.\cap\al A.=\al O.\;.$$ Hence $\al R.=\al O./\al D.=(\al O.\s{\al F.}.\cap\al A.)\big/
(\al D.\s{\al F.}.\cap\al A.)\;.$
Superselection with constraints.
================================
Next we would like to consider systems containing both constraints and superselection. There is a choice in how to define this problem mathematically, so let us consider the physical background. Perhaps the most important example, is that of a local gauge theory. It usually has a set of global charges (leading to superselection) as well as a Gauss law constraint (implementing the local gauge symmetry), and possibly also other constraints associated with the field equation. Only if the gauge group is abelian will the Gauss law constraint commute with the global charge, since the Gauss law constraint takes its values in the Lie algebra of the gauge group. Thus, for nonabelian local gauge theories we do not expect the constraints to be in the algebra of gauge invariant observables $\al A.$ of the superselection theory of the global charge. This problem is however not as serious as it looks. The reason is that whilst the global gauge group does not preserve the individual Gauss law constraints, it does preserve the set of these, hence it also preserves the set of Dirac states selected by them. Thus we can replace the original constraint set by an equivalent constraint (i.e. selecting the same set of Dirac states) which is invariant under the global gauge group. Such an equivalent constraint is given by the projection in Theorem \[Teo.2.7\]. It comes at the cost of slightly enlarging the system algebra $\al B.,$ since $P$ is in the universal Von Neumann algebra of $\al B.$. We can avoid this cost if $\csp(\al C.)$ is separable, since then there is an equivalent constraint in $\al B.$ itself, cf. Theorem 3.4 of [@Lledo].
We therefore will assume below that the constraints are in in $\al A..$ This will include the situation where there are two or more local gauge symmetries which mutually commute (e.g. isospin and electromagnetism), in which case the Gauss law constraint of one symmetry will commute with the global charges of the other. We can also easily find constraints which are independent of the gauge symmetries, e.g. restriction to a submanifold, or enforcing a dynamical law.
Let now $(\al A.,\,\al C.)$ be a first–class constraint system, hence we have the associated algebras $\al D.\subset\al O.\subseteq\al A.,$ and $\al R.=\al O./\al D..$ In addition, let $\al A.$ have a superselection structure i.e. there is a given Hilbert extension $\HS$ of $\al A..$ Thus the category $\al T.$ of canonical endomorphisms of $\al A.$ defines a selection criterion of unital endomorphisms of $\al A..$ In the case that the Hilbert extension is minimal and regular, the superselection structure of $\al T.$ is given within $\al A.$ without any reference to the Hilbert extension.
Then the following natural questions arise:
- what compatibility conditions should be satisfied in order to pass the superselection structure through $T,$ thus obtaining a superselection structure on $T(\al A.,\,\al C.)=\al R.?$
- what is the relation between $T(\al A.,\,\al C.)$ and $T(\al F.,\,\al C.)$ where $\al F.$ is the field algebra generated from $\al T.?$
An inverse question also arises, i.e.
- if $\al R.$ has a superselection structure, what is the weakest structure one can expect on $\al A.$ which would produce this superselection structure on $\al R.$ via $T?$ (One should call this a [*weak*]{} superselection structure.)
To address (1) and (2), recall that the map $T$ consists of a restriction (of $\al A.$ to $\al O.)$ followed by a factoring ($\al O.\to\al O./\al D.$). So, we first work out the compatibility conditions involved with restrictions and factoring maps.
Since $\al C.\subset\al A.\subset\al F.\,,$ there are two constrained systems to consider;- $(\al A.,\,\al C.)$ and $(\al F.,\,\al C.).$ The first one produces the algebras $\al D.\subset\al O.\subseteq\al A.,$ and the second produces $\al D.\s{\al F.}.\subset\al O.\s{\al F.}.\subseteq\al F.$ (cf. Theorem \[Teo.2.12\]). Now since $\al C.\subset\al A.,$ the $\al G.\hbox{--invariant}$ part of $\al F.,$ it follows that $\al G.$ preserves the set of Dirac states, hence $\al G.$ preserves both $\al D._{\al F.}$ and $\al O._{\al F.}$, i.e. $g\al D._{\al F.}=\al D._{\al F.}$ and $g\al O._{\al F.}=\al O._{\al F.}$ for all $g\in\al G.$.
We denote the restriction of $\al G.$ to $\al O._{\al F.}$ by $\beta_{g}:={g}\rest\al O._{\al F.}.$ The homomorphism $\beta:\al G.\ni g\rightarrow \beta_{g}\in\aut\,\al
O._{\al F.}$ is not necessarily injective but $\beta$ is again pointwise norm-continuous, hence $\al G./\al K.$ is compact where $\al K.:=\ker\beta.$ The isomorphism $\tilde{\beta}:\al G./\al K.\to
\beta_{\al G.}$ by $\tilde{\beta}(g\al K.):=\beta_{g}$ is also a topological one (cf. p.58 [@HR]). Note that $\wh{(\al G./\al K.)}=\{\gamma\in\wh{\al G.}\,\bigm|\,\gamma(k)=1\quad
\hbox{for all}\;k\in\al K.\}\supseteq\spec\,\beta_{\al G.}\,.$
The spectral projections $\Pi^{\beta}_{\gamma}$ of $\beta\s{\al G.}.$ are given by the restriction to $\al O._{\al F.}$ of the spectral projections $\Pi_{\gamma}$ of $\al G.$, i.e. $\Pi^{\beta}_{\gamma}X=\Pi_{\gamma}X$ for $X\in\al O._{\al F.}.$
We now have the:
- Restriction problem. Find conditions to guarantee that the C\*–dynamical system $\{\al O._{\al F.},\al G.,\beta\}$ is a Hilbert system $\{\al O._{\al F.},\beta_{\al G.}\}$. Thus we have to find conditions to ensure there are algebraic Hilbert spaces in $\Pi^{\beta}_{\gamma}\al O._{\al F.}$ for $\gamma\in\wh{(\al G./\al K.)}$. (Note that this is stronger than what we need;- we only need a Hilbert system on $\al R.\s{\al F.}.$ after factoring out by $\al D.\s{\al F.}..)$
- Factoring problem. Find conditions to guarantee that under the map $\al O.\s{\al F.}.\to\al R.\s{\al F.}.:={\al O.\s{\al F.}.\big/
\al D.\s{\al F.}.}$ the factoring through of the action of $\al G.$ to $\al R.\s{\al F.}.$ is a Hilbert system corresponding to a DR–category. This is of course a special case of the general problem for homomorphic images of Hilbert systems under factoring by invariant ideals. The reason why we require $Z(\al A.)=\C\un$ for $\al R.\s{\al F.}.$ is because after implementing constraints, the final physical algebra should be simple.
Below we list our major results;- since some proofs are lengthy, we defer these to Section \[Proofs\] to preserve the main flow of ideas.
Restricting a superselection structure. {#restrict}
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We consider now for the system above the restriction problem (I), i.e. we are given a Hilbert extension $\HS$ of $\al A.,$ containing constraints $\al C.\subset\al A.,$ and we need to examine when $\{\al O._{\al F.},\,\beta\s{\al G.}.\}$ is a Hilbert system.
\[Teo.4.1\]
- $\{\al O._{\al F.},\al G.,\beta\}$ has fixed point algebra $\al O..$ Moreover, $Z(\al A.)\subseteq
Z(\al O.).$
- For any $\al G.$-invariant algebraic Hilbert space $\al H._\gamma\subset\Pi_\gamma\al F.$ we have either $\al H._\gamma\cap\of=\{0\},$ or $\al H._\gamma\subset\of\,.$ In the latter case we have $\gamma\in
\wh{\al G./\al K.}$ where $\al K.=\ker\beta\,,$ and $$\al H._\gamma\subset\Pi_{\gamma}\of=\csp(\al O.\al H._{\gamma}).$$
- Let $\sigma\in\ob\al T.,$ with $\al H._\sigma\subset\al F.$ a $\al G.$-invariant algebraic Hilbert space such that $\sigma=\rho\s{\al H._\sigma}..$ If $\al H._\sigma\subset\of,$ then $\sigma(\al D.)\subseteq\al D.$ and $\sigma(\al O.)\subseteq\al O..$ Thus $\sigma$ restricts to $\al O.,$ $\sigma\rest\al O.\in\endo
\al O..$
The central condition for $\{\of,\al G.,\beta\}$ to be a Hilbert system $\{\of,\beta_{\al G.}\}$ w.r.t. the factor group $\al G./\al K.$ is $\al H._{\gamma}\subset\of$, i.e. $\al H._{\gamma}\subset\Pi_{\gamma}^{\beta}\of$ for all $\gamma\in\wh{\al G./\al K.}$.
Next, we develop an internal criterion on $\al A.$ to guarantee that a given $\al H.\in\ob\al T._{\al G.}$ is contained in $\of$.
\[Teo.4.2\]
- Given the Hilbert extension $\HS$ of the constrained system $\al C.\subset\al A.$ assumed here, we have for any $\al G.\hbox{--invariant}$ algebraic Hilbert space $\al H.$ that $$\begin{aligned}
\al H.\subset\of\qquad &\hbox{iff}&\qquad
\al D.\sim\rho\s{\al H.}.(\al D.)\\[1mm]
&\hbox{i.e.}&\qquad
\al D.=\csp\big(\al A.\rho\s{\al H.}.(\al D.)\big)\,\cap
\,\csp\big(\rho\s{\al H.}.(\al D.)\al A.\big)\,.\end{aligned}$$
- For all $\sigma,\;\tau\in\ob\al T.$ with $\al H._\sigma,\al H._\tau\subset\of$ we have $$(\sigma,\,\tau)\s{\al A.}.\subseteq\big(\sigma\rest\al O.,\,\tau
\rest\al O.\big)\s{\al O.}.\;.$$
Observe that $\al D.\sim\rho\s{\al H.}.(\al D.)$ implies that $\rho\s{\al H.}.(\al D.)\subseteq\al D..$
\[cor4.1\] We have that $\{\of,\al G./\al K.,\tilde{\beta}\}$ is a Hilbert system $\{\of,\beta_{\al G.}\}$ w.r.t. $\al G./\al K.$ iff $\al D.\sim\rho_\gamma(\al D.)$ holds for all $\gamma\in\wh{\al G./\al K.}.$ In particular, if $\al D.\sim\rho_{\gamma}(\al D.)$ holds for all $\gamma\in\wh{\al G.}$ then $\al G./\al K.\cong\al G.$ i.e. $\al K.$ is trivial.
Whilst the condition $\al D.\sim\rho\s\gamma.(\al D.)$ is exact for $\al H._\gamma\subset\of,$ it may not be in practice that easy to verify. We therefore consider alternative conditions which will allow the main structures involved with Hilbert extensions to survive the restriction of $\HS$ to $\{\of,\,\beta\s{\al G.}.\}.$
Recalling the definition of subobjects, introduce the notation $E\simeq \un({\rm mod}\,\al A.)$ for a projection $E\in\al A.$ to mean that there is an isometry $V\in\al A.,$ $V^*V=\un$ such that $VV^*=E$ (i.e. Murray–Von Neumann equivalence of $E$ and $\un).$
We say the constraint set $\al C.\subset\al A.$ is an [**E–constraint set**]{} if for each projection $E\in\al O.$ such that $E\simeq \un({\rm mod}\,\al A.),$ we have that $E\simeq \un({\rm mod}\,\al O.).$
The E-constraint condition will ensure the survival of decomposition relations of restrictable canonical endomorphisms:
\[pro.4.7\] Let $\HS$ be a Hilbert system and let $\al C.\subset\al O.$ be an E–constraint set, $\sigma\in\ob\al T.$ and $\al H._\sigma\subset\of$ a $\al G.$-invariant algebraic Hilbert space. Then
- to each decomposition $$\sigma(\cdot)=\sum_jV_j\rho\s\gamma_j.(\cdot)
V_j^*\;,\qquad V_j\in(\rho\s\gamma_j.,\,\sigma)\s{
\al A.}.\;,$$ where $\gamma_j\in\wh{\al G.}$ and $V_j\in\al A.$ are isometries, there corresponds a decomposition on $\al O.,$ i.e. there are $\al G.$-invariant algebraic Hilbert spaces $\al K._j\subset\of,$ which carry the representation $\gamma_j$ and with canonical endomorphisms $\tau_j
:=\rho_{\al K._{j}}\rest\al O.\in\endo\al O.$ such that on $\al O.:$ $$\sigma(\cdot)=\sum_jW_j\tau_j(\cdot)
W_j^*\;,\qquad W_j\in(\tau_j,\,\sigma)\s{
\al O.}.\;,$$ where $W_j\in\al O.$ are isometries.
- Let $\{\al F.,\al G.\}$ in addition satisfies Property B and let $\tau<\sigma\in\al T.$ in the sense of $\al A.,$ i.e. there is an isometry $V\in{(\tau,\,\sigma)\s{\al A.}.},$ and let $\al H._\sigma\subset\of.$ Then there is a corresponding Hilbert space $\al H._\tau\subset \of$ i.e. $\tau\rest\al O.<\sigma\rest\al O.\in\endo\al O.$ also in the sense of $\al O..$
\[tensrep\] Let the Hilbert system $\HS$ satisfy Property B where $\al G.$ is a group with a distinguished irreducible representation $\gamma_0\in\wh{\al G.}$ such that every irreducible representation of $\al G.$ is contained in a tensor representation of $\gamma_0.$ Let $\al C.\subset\al A.$ be an E–constraint set then $\al H.\s\gamma_0.\subset\of$ implies that $\{\of,\beta\s{\al G.}.\}$ is a Hilbert system.
This follows from Proposition \[pro.4.7\], by making use of the obvious fact that $\al H._\tau\subset\of\supset\al H._\sigma$ implies that $\al H._\tau\cdot\al H._\sigma\subset\of$ for $\sigma,\;\tau\in\ob\al T.\,.$
If the group $\al G.$ is isomorphic to $U(N)$ then it satisfies the condition of Theorem \[tensrep\].
The property of being an E–constraint set can be characterized in terms of the open projection $P\in\al A.''$ corresponding to the constraints (cf. Theorem \[Teo.2.7\]). Observe that if there is an $E\in\al O.$ with $E\simeq\un({\rm mod}\,\al A.),$ then the set of isometries $$\al V._E:=\set V\in\al A., VV^*=E,\quad V^*V=\un.$$ is nonempty. We have:
Let $E\in\al O.$ with $E\simeq\un({\rm mod}\,\al A.),$ then $\al V._E\cap\al O.\not=
\emptyset$ iff for each $V\in\al V._E$ there is a $$U\in \al U._E:=\set U\in\al A., U^*U=E=UU^*.$$ such that $VPV^*=UPU^*.$
Morphisms of general Hilbert systems. {#morphsec}
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Recall that the second step in the enforcement of constraints, is the factoring $\of\to\rf:=\of/\df.$ We now consider problem [(II)]{}, the factoring problem, first in a general context. Consider a morphism of C\*–algebras $\xi:\al F.\to\al L.=\xi(\al F.).$ This specifies the subgroup of automorphisms $$\autx\al F.:=\set\alpha\in\aut\al F.,\alpha(\ker\xi)\subseteq
\ker\xi.$$ and a homomorphism $\autx\al F.\to\aut\al L.$ by $\alpha\to\alpha^\xi$ where $\alpha^\xi(\xi(F)):=
\xi(\alpha(F))$ for all $F\in\al F..$ Henceforth let $\HS$ be a Hilbert system with Property B and ${\al G.}\subset\autx\al F..$ Our task will be to find the best conditions to ensure that $\{\al L.,\,\al G.^{\xi}\}$ is a Hilbert system associated with a category described in Theorem \[Teo1\]. We will denote the spectral projections of $\al G.$ (resp. $\al G.^\xi)$ by $\Pi_\gamma$ (resp. $\Pi^\xi_\gamma).$ (Recall that in the context of the T–procedure, we have that $\al G.$ preserves $\df$ due to the invariance of the constraints under $\al G..$ So the current analysis applies).
\[Teo.4.3\] Given a Hilbert system $\HS$ and a unital morphism $\xi:\al F.\to\al L.=\xi(\al F.),$ such that ${\al G.}\subset\autx\al F.,$ then we have:
- $\{\al L.,\,\al G.^\xi\}$ is a Hilbert system and $\al G.\cong\al G.^{\xi}.$
- If $\al H._\gamma\subset\Pi_\gamma\al F.$ is an invariant algebraic Hilbert space for $\al G.,$ then so is $\xi(\al H._\gamma)\subset\Pi^\xi_\gamma\al L.$ for $\al G.^\xi.$
- Let $\al N._\gamma$ be any orthonormal basis for $\xi(\al H._\gamma),$ then $\bigcup\set\al N._\gamma,\gamma\in\wh{\al G.}.$ is a left module basis of $\xi(\al F._{\rm fin})$ w.r.t. $\xi(\al A.),$ i.e. the “essential part” of $\xi$ is its action on $\al A..$
- The fixed point algebra of $\al L.$ w.r.t. $\al G.^{\xi}$ is exactly $\xi(\al A.),$ and $\xi(\al F._{\rm fin})=\al L._{\rm fin}\;.$
- If $\HS$ has Property B, so does $\{\al L.,\,\al G.^\xi\}.$
Thus corresponding to the two Hilbert systems $\HS$ and $\{\al L.,\al G.^{\xi}\}$ we now have the two categories $\al T.$ and $\al T.^\xi$ respectively. Moreover:
\[Cor.4.9\] Under the conditions of Theorem \[Teo.4.3\] we have that
- for any canonical endomorphism $\lambda\in\ob\al T.,$ $$\lambda(\ker\xi\cap\al A.)\subseteq\ker\xi\cap\al A.\;.$$ Hence there is a well–defined map $\ob\al T.\ni\lambda\to\lambda^\xi\in\ob\al T.^\xi,$ given by $\lambda^\xi(\xi(A)):=\xi(\lambda(A))$ for all $A\in\al A..$
- the map $\ob\al T.\ni\lambda\to\lambda^\xi\in\ob\al T.^\xi$ is compatible with products, direct sums and subobjects. It also preserves unitary equivalence.
We have that $\big(\ob\al T.\big)^\xi\subseteq\ob\al T.^\xi,$ and we now claim that up to unitary equivalence, we have in fact equality:
\[Teo.4.4\] Under the conditions of Theorem \[Teo.4.3\] we have that
- if $\sigma\in\ob\al T.^\xi,$ then there is always a $\lambda\in\ob\al T.$ such that $\lambda^\xi$ is unitarily equivalent to $\sigma,$ i.e. each unitary equivalence class in $\ob\al T.^\xi$ contains at least one element of the form $\lambda^\xi.$
- the map $\ob\al T.\ni\lambda\to\lambda^\xi\in\ob\al T.^\xi$ produces an isomorphism between the sets of unitary equivalence classes of $\ob\al T.$ and $\ob\al T.^\xi$ which is compatible with products direct sums and subobjects.
The relation between the arrows of the two categories is however less direct:
\[arrow1\] Under the conditions of Theorem \[Teo.4.3\] we have $$\xi\big((\sigma,\,\tau)_{\al A.}\big)\subseteq
\big(\sigma^\xi,\,\tau^\xi\big)\s{\xi(\al A.)}.\;.$$
Next we show that $\ker\xi$ is uniquely determined by $\ker\xi\cap\al F._{\rm fin}.$
\[Prop.4.12\] Under the conditions of Theorem \[Teo.4.3\] we have that
- $\ker\xi\cap\al F._{\rm fin}=
{\rm Span}\set(\ker\xi\cap\al A.)\al H._\gamma,
\gamma\in\wh{\al G.}.\;,$
- $\ker\xi=\clo\s{\vert\cdot\vert_{\al A.}}.
(\ker\xi\cap\al F._{\rm fin})
\cap\al F.\;.$
Thus $\ker\xi$ is in fact uniquely determined by $\ker\xi\cap\al A.,$ as is already suggested by Theorem \[Teo.4.3\](iii). Since $\al F.$ is in general not complete w.r.t. $|\cdot|_{\al A.},$ the intersection with $\al F.$ in \[Prop.4.12\](ii) is necessary.
Theorem \[Teo.4.3\] suggests that we consider the following subcategory of $\al T.^{\xi}$.
\[CatXT\] The subcategory $\xi(\al T.)$ of $\al T.^{\xi}$ is defined by the objects $$\ob\xi(\al T.):=(\ob\al T.)^{\xi}$$ and the arrows $$(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}^{(0)}:=
\xi((\sigma,\tau)_{\al A.}).$$
By Theorem \[Teo.4.4\] the sets of all unitary equivalence classes of $\hbox{Ob}\,\xi(\al T.)$ and $\hbox{Ob}\,\al T.^{\xi}$ coincide, each equivalence class of $\hbox{Ob}\,\xi(\al T.)$ is a subset of the corresponding equivalence class of $\hbox{Ob}\,\al T.^{\xi}$, but in general these equivalence classes are much larger.
Lemma \[arrow1\] says that the arrow sets $(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}$ of the objects of $\hbox{Ob}\,\xi(\al T.)$ considered as objects of $\hbox{Ob}\,\al T.^{\xi}$ are in general larger than the corresponding arrow sets in $\xi(\al T.)$. The reason is that an element $X=\xi(Y),\,Y\in\al A.,$ belongs to $(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}$ iff $Y\sigma(A)-\tau(A)Y\in\ker\,\xi$ for all $A\in\al A.$. The arrow sets coincide only if this relation already implies $Y\sigma(A)-\tau(A)Y=0.$
Morphisms of minimal and regular Hilbert systems
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Recall now that by Theorems \[Teo1\] and \[Teo2\] we have an equivalence between minimal and regular Hilbert systems with Property B and the endomorphism category $\al T.$ with an admissible subcategory $\al T._\C.$ We called a subcategory $\al T._\C$ [*admissible*]{} if it satisfies conditions P.1–P.2 in Theorem \[Teo1\].
As in the last subsection, we consider a unital morphism $\xi:\al F.\to\al L.=\xi(\al F.),$ and recall by Proposition \[Prop.4.12\] that $\xi$ is determined by its action on $\al A..$ Now whilst it is obvious that $\xi(Z(\al A.))\subseteq Z(\xi(\al A.)),$ we require below the stronger condition: $$\label{ZZ}
\xi(Z(\al A.))=Z(\xi(\al A.))\;.$$ When $\xi(\al A.)$ is a simple C\*–algebra (as we require for the final observables after a T–procedure), the condition (\[ZZ\]) will be satisfied.
\[Teo.4.13\] Given a minimal and regular Hilbert system $\HS$ with Property B, and a unital morphism $\xi:\al F.\to\al L.=\xi(\al F.)$ such that ${\al G.}\subset\autx\al F.$ and condition (\[ZZ\]) holds, then:
- there is a DR- subcategory $\al T.^\xi_\C$ of $\xi(\al T.)$,
- property P.2 is satisfied for $\al T.^\xi_\C$ iff $\xi(\al A.)'\cap\xi(\al F.)=\xi(Z(\al A.)).$ In this case the subcategory $\al T.^\xi_\C$ is admissible.
- If $\xi(\al A.)'\cap\xi(\al F.)=\xi(Z(\al A.)),$ then $$\big(\sigma^\xi,\,\tau^\xi\big)\s{\xi(\al A.)}.
= \xi\big((\sigma,\,\tau)\s{\al A.}.\big)$$ for all $\sigma,\;\tau\in\ob\al T.,$ where we made use of the notation and result in Corollary \[Cor.4.9\].
- In this case choose $\al H._{\gamma}\in\ob\al T._{\al G.}\,.$ Then $$\al M.^{\xi}:=\set{\rho\s{\xi(\al H._{\gamma})}.},{\gamma\in\wh{\al G.}}.
\subset\ob\xi(\al T.)$$ is a complete system of (irreducible) and mutually disjoint objects of $\ob\xi(\al T.).$
The inverse problem.
--------------------
\[Teo.4.14\]
Let $\al A.$ be a unital C\*–algebra with Property B, and let $\al T.$ be a C\*-tensor category of unital endomorphisms of $\al A.$ . Let $\al T.$ have an admissible subcategory $\al T._{\C}$ whose arrow spaces are denoted by $(\sigma,\tau)_{\C}$. Furthermore, let $\xi$ be a unital morphism of $\al A.$ such that
- $\xi(Z(\al A.))=Z(\xi(\al A.)),$
- $\lambda(\ker\xi)\subseteq\ker\xi$ for all $\lambda\in\ob\al T..$ Thus we can define endomorphisms $\lambda^\xi\in\endo\xi(\al A.)$ by $\lambda^\xi\big(\xi(A)\big):=\xi\big(\lambda(A)\big)$ for all $A\in\al A.$ and a category $\xi(\al T.)$ with objects $$\label{Obxi}
\ob\,\xi(\al T.):=\set{\lambda^\xi},{\lambda\in\ob\,\al T.}.$$ and arrows $(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}$, which is closed w.r.t. direct sums and products.
- $\xi\big((\sigma,\,\tau)_{\al A.}\big)=
\big(\sigma^\xi,\,\tau^\xi\big)\s{\xi(\al A.)}.$ for all $\sigma,\;\tau\in\ob\al T..$
Then there is a subcategory $\al T.^{\xi}_{\C}$ of $\xi(\al T.)$ with $\ob\al T.^{\xi}_{\C}=\ob\xi(\al T.)$ which is admissible for $\xi(\al T.).$
Thus by Theorem \[Teo2\] there are Hilbert extensions $\al F.$ and $\al F.^\xi$ corresponding to $\al T.$ and $\xi(\al T.)$ repectively. Moreover, the Hilbert extension $\al F.^\xi$ of $\xi(\al A.)$ can be chosen in such a way that it is the homomorphic image of $\al F.$ under a morphism which is an extension of $\xi.$ That is, $\al F.^\xi=\wt\xi(\al F.)$ where $\wt\xi$ is a morphism of $\al F.$ such that $\wt\xi(A)=\xi(A)$ for all $A\in\al A..$
A posteriori, the set of objects $\ob\xi(\al T.)$ defined in (\[Obxi\]) could be enlarged by filling up the unitary equivalence classes of each $\lambda^\xi$ by [*all*]{} $\tau$ with $\tau=\Ad V\circ\xi(\lambda),$ where $V\in\xi(\al A.)$ is unitary. This corresponds to the objects of the category $\al T.^\xi$ of Definition \[CatXT\] In this case we also have to add additional arrows, so if $\tau_i=\Ad V_i\circ\xi(\lambda_i),$ $i=1,2,$ then we also need $$(\tau_1,\,\tau_2)\s{\xi(\al A.)}.:=
V_2\big(\tau_1^\xi,\,\tau_2^\xi\big)\s{\xi(\al A.)}.V_1^{-1}\;.$$ However, for the application of Theorem \[Teo2\] this is not necessary.
Superselection structures left after constraining.
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Recall that the enforcement of constraints by T–procedure produces a final physical algebra $\al R..$ This algebra is usually assumed to be simple;- if it is not, then the physics is not fully defined, and one should extend the constraint set $\al C.\subset\al A.$ to make $\al R.$ simple (the choice of the extension needs to be physically motivated).
In the previous subsections we examined which conditions need to be satisfied by a Hilbert extension ${\{\al F.,\,\al G.\}}$ of $\al A.$ for its structure to pass through the two parts of the T–procedure. Here we combine these to produce conditions on the initial system which will ensure that we obtain a Hilbert extension of $\al R..$ We will also examine when this final Hilbert extension is regular (and this produces then a DR–category via simplicity of $\al R.).$
\[HExR\] Let ${\{\al F.,\,\al G.\}}$ be a Hilbert extension of $\al A.,$ and let $\al C.\subset\al A.$ be a first-class constraint set such that $\al D.\sim\rho_\gamma(\al D.)$ holds for all $\gamma\in\wh{\al G./\al K.}.$ Then ${\{\al R.\s{\al F.}.,
\,\beta\s{\al G.}.\}}$ is a Hilbert extension of $\al R.,$ where $\al R.\s{\al F.}.
=\xi(\al O.\s{\al F.}.),$ and $\xi$ is the factor map $\al O.\s{\al F.}.
\to \al O.\s{\al F.}.\big/\al D.\s{\al F.}..$
By Corollary \[cor4.1\] it follows from the hypotheses that ${\{\al O.\s{\al F.}.,\,\beta\s{\al G.}.\}}$ is a Hilbert extension of $\al O..$ Since the constraint set $\al C.\subset\al A.$ is $\al G.\hbox{--invariant,}$ we have that $\alpha(\al D.\s{\al F.}.)=\al D.\s{\al F.}.$ for all $\alpha\in\beta\s{\al G.}.\subset\aut\al O.\s{\al F.}.,$ i.e. $\beta\s{\al G.}.\subset\autx\al O.\s{\al F.}..$ (Recall the discussion in the introductory part of Section 4). Thus by Theorem \[Teo.4.3\] it follows that $(\beta\s{\al G.}.)^\xi\cong\beta\s{\al G.}.,$ and that $$\left\{\xi\big(\al O.\s{\al F.}.),\,(\beta\s{\al G.}.)^\xi\right\}
=\left\{\al R.\s{\al F.}.,\,\beta\s{\al G.}.\right\}$$ is a Hilbert extension of $\al R.=\xi(\al O.).$
Next, we would like to examine when a Hilbert extension as in Theorem \[HExR\] will produce a minimal and regular Hilbert extension of $\al R.$ (with Property B).
First recall the requirement for a Hilbert system $\{\al F.,\,\al G.\}$ to be regular: there is an assignment $\sigma\to\al H._\sigma$ from $\ob\al T.$ to $\al G.$-invariant algebraic Hilbert spaces in $\al F.$ such that
- $\sigma=\rho\s{\al H._\sigma}.,$ i.e. $\sigma$ is the canonical endomorphism of $\al H._\sigma,$
- $\sigma\circ\tau\to\al H._\sigma\cdot\al H._\tau\;,$
that is, the assignment is compatible with products.
We now want to check whether this property also survives the map $T:\{\al F.,\,\al G.\}\longrightarrow
{\{\al R.\s{\al F.}.,\,\beta\s{\al G.}.\}}. $
\[Tsat\] Let $\al T.$ satisfy regularity. Let $\al D.\sim\rho_\gamma(\al D.)$ for all $\gamma\in\wh{\al G./\al K.},$ then ${\{\al R.\s{\al F.}.,\,\beta\s{\al G.}.\}}$ satisfies regularity, i.e. there is an assignment $\sigma\to\al H._\sigma$ such that
- $\sigma=\rho\s{\al H._\sigma}.,$ i.e. $\sigma$ is the canonical endomorphism of $\al H._\sigma,$
- $\sigma\circ\tau\to\al H._\sigma\cdot\al H._\tau\;.$
Given the assignment $\sigma\to\al H._\sigma$ in $\al F.,$ then whenever $\sigma=\rho_{\gamma},$ $\gamma\in\wh{\beta\s{\al G.}.}$ we have $$\sigma\to\al H._\sigma\subset\al O.\s{\al F.}.\longrightarrow
\al R.\s{\al F.}.$$ where the last map is $\xi,$ so the assignment which we take for this proposition is $\sigma\to\xi(\al H._\sigma).$ Then (i) and (ii) are automatic.
Second, we consider Property B.
Let $\{\al F.,\al G.\}$ satisfy Property B, let $\al G.$ be nonabelian and $\al C.\subset\al A.$ be an E-constraint set. If $\al D.\sim\rho_{\gamma}(\al D.)$ for all $\gamma\in \wh{(\al G./\al K.)},$ then $\{\al O._{\al F.},\beta_{\al G.}\}$ satisfies Property B.
First $\{\al O._{\al F.},\beta_{\al G.}\}$ is a Hilbert extension of $\al O.$ w.r.t. $\al G./\al K.$ because of Corollary \[cor4.1\]. Choose an $\al G.\hbox{-invariant}$ Hilbert space $\al H.\subset\al O._{\al F.}\subset\al F.$ which is not irreducible, i.e. there is a projection $E\in\al J.(\al L._{\al G.}(\al H.)),\,0<E<\un.$ Then one has $E\in (\rho_{\al H.},\rho_{\al H.})_{\al A.}\subset
(\rho_{\al H.}\rest\al O.,\rho_{\al H.}\rest\al O.)_{\al O.}\subset\al O.$ by Theorem \[Teo.4.2\](ii). By Property B we get closure under subobjects, so there is a $V\in\al A.,\,V^{\ast}V=\un,\,VV^{\ast}=E.$ In other words, $E\cong\un(\hbox{mod}\,\al A.).$ Similarly we obtain $\un-E\cong\un(\hbox{mod}\,\al A.).$ Since $\al C.$ is an E-constraint set and $E\in\al O.$ we get that $E\cong\un(\hbox{mod}\,\al O.)$ and $\un-E\cong\un(\hbox{mod}\,\al O.)$ and this is the assertion.
Finally, we need to consider whether the requirement $$\al A.'\cap\al F.=Z(\al A.)$$ passes through the T-procedure. In full generality, this is a very hard problem, because both stages of the T-procedure can eliminate or create elements of $\al A.'.$ In fact, since $\al A.'\cap\al F.\subset\al D.'\cap\al F.\subset
\al O.\s{\al F.}.$ and $Z(\al A.)\subset Z(\al O.),$ we can only deduce from $\al A.'\cap\al F.=Z(\al A.)$ that $\xi(\al A.'\cap\al F.)=\xi(Z(\al A.)).$ On the other hand, $\al R.'\cap\al R.\s{\al F.}.=Z(\al R.)$ iff $$A\in\al O.\s{\al F.}.\quad\hbox{and}\quad [A,\,\al O.]
\subset\al D.\s{\al F.}.\quad\hbox{implies}\quad
A\in \al O.+\al D.\s{\al F.}.\,$$ which can be true in general for more elements than those in $\xi(\al A.'\cap\al F.).$
We do have from Theorem \[Teo.4.2\] and Proposition \[disjZ\] the following condition:
Let $\HS$ be a minimal Hilbert extension of $\al A.,$ and let $\al C.\subset\al A.$ be a first-class constraint set such that $\al D.\sim\rho_\gamma(\al D.)$ holds for all $\gamma\in\wh{\al G./\al K.}.$ If the disjointness of canonical endomorphisms survives the restriction to $\al O.$ then the Hilbert system $\{\of,\beta_{\al G.}\}$ is minimal, i.e. $\al O.'\cap\of=\al Z.(\al O.)$.
Example
=======
It is difficult to produce interesting worked examples in the current state of the theory. The problem is that in almost all theories of physical significance, the canonical endomorphisms $\rho_\gamma$ are not known explicitly, and so one cannot check the compatibility conditions with the constraints explicitly (cf. Corollary \[cor4.1\]). Here we give an example which is extracted from QED, so it may have some physical interest. It consists of a fermion in an Abelian gauge potential. Since the global gauge group $\al G.$ is abelian, the superselection theory simplifies radically. However, we have explicit endomorphisms $\rho_\gamma$ and can check the compatibility conditions with the constraints. Nevertheless, even at this simple level, it is not possible to verify all the conditions of regularity. We will not treat the issue of dynamics.
Constraint structure of QED
---------------------------
We start with a discussion of the set–up of QED in order to motivate our subsequent example. The starting point for QED, is a fermion field $\psi$ in $\R^4$ satisfying the free CARs, and a $U(1)\hbox{--gauge}$ potential $A$ in $\R^4$ satisfying free CCRs, and initially these are assumed to commute. So the appropriate C\*-algebraic framework at this initial level is $$\al B.:={\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)$$ where $\al H.=L^2(\R^4,\,\C^4),$ $S=\al S.(\R^4,\,\R^4)\big/\ker B,$ and $B$ denotes the symplectic form for QEM, coming from the Jordan–Wigner distribution, cf. Sect 5 of [@Lledo]. (Note that the tensor product $\al B.$ is unique because ${\rm CAR}(\al H.)$ is a nuclear algebra.) There is a global charge $Q$ acting on ${\rm CAR}(\al H.)$ and there are constraints in the heuristic theory: $$\begin{aligned}
{A_\mu}^{,\,\mu}(x)&:=&0 \quad\qquad\hbox{(Lorentz condition)} \\[1mm]
\hbox{and}\quad\qquad
\Box A_\mu&:=&j_\mu \quad\qquad\hbox{(Maxwell equation)}\end{aligned}$$ where $j_\mu:=-e\wt\psi\gamma_\mu\psi$ is the electron current, and we denote $\wt\psi:=\psi^*\gamma_0.$ The Lorentz condition has been treated in the C\*–algebra context (cf. [@Lledo]) and it needs special treatment, e.g. indefinite metric or nonregular states, but it is not very interesting for us, since it only affects the electromagnetic field ${\rm CCR}(S,\,B),$ hence is independent of the charge $Q.$ The Maxwell equation is more interesting, since it involves both factors of $\al B.$ and it expresses the interaction between the two fields. It is however very difficult to enforce in the C\*–algebra context (and ultimately leads to the conclusion that $\al B.$ is too small an algebra to do this in). Naively, it seems that we can easily realise both sides of the Maxwell equation in the present C\*–setting: smear the left-hand side over $\al S.(\R^4,\,\R^4)$ $$\int\Box A_\mu(x)\,f^\mu(x)\,dx=\int A_\mu\Box f^\mu dx
=A(\Box f)$$ then this is realised in ${\rm CCR}(S,\,B)$ through the identification of the generating Weyl unitaries $\delta_h$ with the heuristic $\exp iA(h)$ where $h=\Box f.$ If we smear the right-hand side of the Maxwell equation: $$j(f)=-e\int\wt\psi(x)\gamma_\mu\psi(x)f^\mu(x)\, dx\;,$$ then $j(f)$ generates a Bogoliubov transformation $T_f$ on $L^2(\R^4,\,\C^4)$ by: $$\begin{aligned}
\Ad\big(\exp ij(f)\big)\psi(g)&:=&\left(\exp i\ad j(f)\right)\big(\psi(g)\big)\\[1mm]
&=&\psi(T_fg)=:\alpha\s T_f.(\psi(g))\end{aligned}$$ where $\alpha\s T_f.$ is its associated automorphism on ${\rm CAR}(\al H.)$ (we will calculate $T_f$ explicitly in a simplified setting below). Let $G\subset\aut\al B.$ be the discrete group generated in $\aut\al B.$ by $$\set\beta_f:=\alpha\s T_f.\otimes\iota, f\in{\al S.(\R^4,\,\R^4)}.$$ and let $\nu$ denote its action on $\al B..$ Define the crossed product $$\al E.:=G\cross\nu.\al B.={\rm C}^*\set{\al B.,\,U_g},
U^*_g=U_g^{-1},\; \nu_g=\Ad U_g,\; U_gU_h=U_{gh},\;g,h\in G.$$ then we identify the heuristic objects $\exp ij(f)$ with the implementing unitaries $U\s\beta_f..$ So each side of the Maxwell equation has a C\*–realisation, and we only need to decide how to impose the constraint equation. Heuristically, the Maxwell equations are imposed as state conditions: $A(\Box f)\phi = j(f)\phi$ for vectors $\phi$ in the representing Hilbert (or Krein) space. If we take instead the stronger condition $A(\Box f)^n\phi = j(f)^n\phi$ for $n\in\N,$ then we can rewrite the constraint conditions in the form ${e^{iA(\Box f)}\phi}=
{e^{ij(f)}\phi}.$ This suggests that we choose constraint unitaries $V_f:=U\s -\beta_f.\cdot\delta\s\Box f.$ in $\al E.$ and thus select our Dirac states $\omega$ on $\al E.$ by $$\omega(V_f)=1\qquad\forall\; f\in {\al S.(\R^4,\,\R^4)}.$$ As one expects from the interaction, this program encounters problems:
- We always have that $\Box f\in\ker B,$ hence $\Box f$ corresponds to zero in $S$ (since we factor out by $\ker B).$ This can be remedied by changing $S$ to ${\al S.(\R^4,\,\R^4)},$ in which case $(S,\,B)$ is a degenerate symplectic space. This problem is connected to the fact that the heuristic smearing formula $$A(f)=\int_{C_+}\left(a_\mu(\b p.)\wh{f}^\mu(p)
+a_\mu^+(\b p.)\bar{\wh{f}}^\mu(p)\right){d^3p\over p_0}$$ cannot be correct for the interacting theory, since it implies that $A(\Box f)=0\,,$ in contradiction with the Maxwell equation.
- Interaction mixes the fermions and bosons, so it is unrealistic to expect that the interacting fermion and boson fields will commute (as in the tensor product structure of $\al B.).$ Even worse, perturbation theory suggests that the interacting fields need not be canonical, so the assumption of the CCR and CAR relations for the interacting bosons and fermions is problematic.
Model for the interacting Maxwell constraint
--------------------------------------------
Inspired by the observations above, we now propose an example which is a simplified version of the Maxwell constraint. Heuristically, we want to impose a constraint of the form $$a^*(x)\,a(x)=LA(x)$$ where $a(x)$ is a fermion field on $\R^4,$ $A$ is a boson field and $L$ is a linear differential operator on $\al S.(\R^4).$ To realise this, together with a superselection structure in a suitable C\*–algebra setting, we present our construction in six steps.STEP 1.For the fermion field, let $\al H.=L^2(\R^4)$ and define ${\rm CAR}(
\al H.)$ in Araki’s self-dual form (cf. [@Ar]) as follows. On $\al K.:=\al H.\oplus\al H.$ define an antiunitary involution $\Gamma$ by $\Gamma(h_1\oplus h_2):=\ol h._2\oplus\ol h._1\;.$ Then ${\rm CAR}(\al H.)$ is the unique simple C\*–algebra with generators $\set\Phi(k),k\in{\al K.}.$ such that $k\to\Phi(k)$ is antilinear, $\Phi(k)^*=\Phi(\Gamma k)\,,$ and $$\left\{\Phi(k_1),\,\Phi(k_2)^*\right\}=(k_1,\,k_2)\un\;,\qquad
k_i\in\al K.\;.$$ The correspondence with the heuristic creators and annihilators of fermions is given by $\Phi(h_1\oplus h_2)=a(h_1)+a^*(\ol h._2)\;,$ where $$a(h)=\int a(x)\,\ol h(x).\,d^4x\;,\qquad
a^*(h)=\int a^*(x)\, h(x)\, d^4x\;.$$ STEP 2.For the boson field, let $S=\al S.(\R^4,\R),$ and let $K:S\to L^2(M,\mu)$ be a linear map, where $(M,\mu)$ is a fixed measure space. Define a symplectic form on $S$ by $B(f,g):={\rm Im}(Kf,\,Kg),$ where $(\cdot,\cdot)$ is the inner product of $L^2(M,\mu).$ Note that $B$ is degenerate if $\ker K$ is nonzero. Define then ${\rm CCR}(S,\,B)=C^*\set\delta_f,f\in S.$ where the $\delta_f$ are unitaries satisfying the Weyl relations: $$\delta_f\cdot\delta_g=\delta\s f+g.\exp[iB(f,g)/2]$$ i.e. ${\rm CCR}(S,\,B)$ is the $\sigma\hbox{--twisted}$ discrete group algebra of $S$ w.r.t. the two–cocycle $\sigma(f,g):=\exp[iB(f,g)/2]\,.$ STEP 3.To combine the bosons and fermions in one C\*–algebra, we want to allow for the possibility that they may not commute with each other, hence we will not take the tensor algebra ${\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\,.$ However, we don’t know what form their commutators should take, so we start with the free C\*–algebra $\al E.$ generated by ${\rm CAR}(\al H.)$ and ${\rm CCR}(S,\,B)\,.$ The free C\*–algebra $\al E.$ seems to be big enough to allow for possible interactions, but it is also likely to contain redundant elements.
To be explicit, let $\al L.$ be the linear space spanned by all monomials of the form $A_0B_0A_1B_1\cdots A_nB_n$ where $A_i\in{\rm CAR}(\al H.)$ and $B_i\in{\rm CCR}(S,\,B)\,.$ Note that $\al L.$ is an algebra w.r.t. concatenation. Factor out by the ideal generated by $\un\s{\rm CAR}.-
\un\s{\rm CCR}.$ and replace concatenation by multiplication for any two elements in a monomial which are in the same algebra (either ${\rm CAR}$ or ${\rm CCR}$) after the factorisation. Note that this will now produce all possible monomials of elements in ${\rm CAR}(\al H.)$ and ${\rm CCR}(S,\,B)$ - just consider those monomials in $\al L.$ with $A_0$ or $B_n$ the identity to obtain all other monomials. Now the resultant algebra $\al N.$ is a \*–algebra with the involution given by $$(A_0B_0\cdots A_nB_n)^*=B_n^*A_n^*\cdots B_0^*A_0^*\;.$$ Form the enveloping C\*–algebra $\al E.$ of $\al N.,$ i.e. let $$\al I._0:=\bigcap\set\ker\pi,\pi\in
\hbox{Hilbert space representations of $\al N.$}.$$ and set $\al E.:=\ol{\al N.\big/\al I._0}.$ where the closure is w.r.t. the enveloping C\*–norm, i.e. $$\|A\|:=\sup\set\|\pi(A)\|,\pi\in
\hbox{Hilbert space representations of $\al N.$}.\;.$$ That $\al E.$ is nontrivial, follows from the fact that any tensor product representation of ${\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)$ defines a Hilbert space representation of $\al N.,$ hence it follows that $\al E.$ is nonzero and that ${\rm CAR}(\al H.)$ and ${\rm CCR}(S,\,B)$ are faithfully embedded in $\al E.$ (as the images under the factorisation maps of the original generating algebras in the construction). Note that we have a surjective homomorphism $\zeta:\al E.\to
{{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)}$ given by $$\zeta(A_0B_0\cdots A_n B_n):= (A_0\cdots A_n)\otimes(B_0\cdots B_n)\;,
\quad A_i\in {\rm CAR}(\al H.),\; B_i\in{\rm CCR}(S,\,B)\,.$$ Clearly the ideal $\al I._T$ of $\al E.$ generated by the commutators $\big[{\rm CAR}(\al H.),\,{\rm CCR}(S,\,B)\big]$ is in $\ker\zeta.$ Since $\al E.$ probably contains redundant elements, we do not require it to be simple. $\zeta$ will be important in proofs below. STEP 4.Next, we would like to model in the curent C\*–setting, the global and local heuristic charges: $$Q=\int a^*(x)\,a(x)\,d^4x\,,\qquad
Q(f)=\int a^*(x)\,a(x)\,f(x)\,d^4x\,,\quad
f\in\al S.(\R^4,\R)\;.$$ Let us calculate the Bogoliubov transformations which they induce: $$\begin{aligned}
&&\big[Q(f),\Phi(h_1\oplus h_2)\big] \\[1mm]
&=& \int\int\left[a^*(x)a(x)f(x),\,a(y)\ol h_1(y).+
a^*(y)\ol h_2(y).\right]\,d^4x\,d^4y \\[1mm]
&=& \int\int f(x)\bigg\{\Big(a^*(x)a(x)a(y)-a(y)a^*(x)a(x)\Big)
\ol h_1(y).\\[1mm]
&&\qquad+\Big(a^*(x)a(x)a^*(y)- a^*(y)a^*(x)a(x)\Big)\ol h_2(y).
\bigg\}d^4x\,d^4y \\[1mm]
&=& \int\int f(x)\bigg\{-\big\{a^*(x),\,a(y)\big\}a(x)\,\ol h_1(y).\\[1mm]
&&\quad \left(a^*(x)\big(\delta(x-y)-a^*(y)a(x)\big)
-a^*(y)a^*(x)a(x)\right)\ol h_2(y).\bigg\}\,d^4x\,d^4y\\[1mm]
&=&\int\int f(x)\bigg\{-\delta(x-y)\ol h_1(y).a(x) +
\delta(x-y)a^*(x)\ol h_2(y).\bigg\}\,d^4x\,d^4y\\[1mm]
&=& -a(\ol f.\cdot h_1)+a^*(f\cdot\ol h_2.)\\[1mm]
&=&\Phi(-\ol f.\cdot h_1\oplus\ol f.\cdot h_2)
=\Phi\big(f(-h_1\oplus h_2)\big)\end{aligned}$$ since $f$ is real. For the global charge $Q,$ just put $f=1$ in the last calculation. Thus $$\begin{aligned}
\big(\ad Q(f)\big)^n\big(\Phi(h_1\oplus h_2)\big)
&=&\Phi\left(f^n\cdot\big((-1)^nh_1\oplus h_2\big)\right)\qquad
\hbox{hence:}\\[1mm]
\left(\Ad\big(\exp\,iQ(f)\big)\right)\left(\Phi(h_1\oplus h_2)\right)
&=& \Big(\exp\,i\,\ad Q(f)\Big)\left(\Phi(h_1\oplus h_2)\right)\\[1mm]
&=& \sum_{n=0}^\infty{\big(i\,\ad\,Q(f)\big)^n\over n!}
\left(\Phi(h_1\oplus h_2)\right)\\[1mm]
&=& \sum_{n=0}^\infty{i^n\over n!}\,\Phi\Big(f^n\big((-1)^n
h_1\oplus h_2\big)\Big)\\[1mm]
&=& \Phi\big(e^{-if}h_1\oplus e^{if}h_2\big)=:\Phi\big(T_f(h_1\oplus h_2)
\big)\;.\end{aligned}$$ Now $T_f$ is unitary on $\al K.,$ and satisfies ${[T_f,\,\Gamma]}=0$ hence it is a Bogoliubov transformation (cf. p43 in [@Ar]), and so we can define automorphisms on ${\rm CAR}(\al H.)$ by $$\wt\gamma_f\big(\Phi(k)\big):=\Phi(T_fk)\;.$$ It is clear that $T_fT_g=T_{f+g}\,,$ hence that $\wt\gamma:\al S.(\R^4)+\R\to\aut({\rm CAR}(\al H.))$ is a homomorphism. We extend these automorphisms to maps $\gamma_f$ on $\al E.$ by setting $$\gamma_f\restriction {\rm CAR}(\al H.) =\wt\gamma_f\;,\qquad
\hbox{and}\qquad
\gamma_f\restriction {\rm CCR}(S,B) = \iota$$ where $\iota$ is the identity map. The only relations between ${\rm CAR}(\al H.)$ and ${\rm CCR}(S,B)$ in the free construction of $\al E.,$ is $\un\s{\rm CAR}.=\un\s{\rm CCR}.,$ so since the definition of $\gamma_f$ preserves this relation, it will extend to a well-defined map on the free \*–algebra $\al N..$ In fact, since $\gamma_f$ replaces ${\rm CAR}(\al H.)$ by an isomorphic one in a free construction, it will be an automorphism on $\al N.,$ and so will define an automorphism on the enveloping algebra $\al E..$
Let $G$ denote the Abelian group generated in $\aut\al E.$ by $\set\gamma_f,f\in{\al S.}(\R^4)\cup\R.$ and equip it with the discrete topology. Denote its action by $\beta:G\to\aut\al E.,$ and define the algebra $$\al A.:=G\cross\beta.\al E.\;,$$ then we identify the implementing unitaries $U_{\gamma_f}\in\al A.$ of $\gamma_f\in\aut\al E.$ with the heuristic objects $\exp iQ(f),$ $f\in {\al S.}(\R^4)\cup\R$ (in the case that $f=t\in\R,$ we denote $Q(t)=tQ).$ Now $\gamma$ is a surjective homomorphism $\gamma:{\al S.(\R^4)+\R}
\to G$ and from the definitions above, it is clear that its kernel is $2\pi\Z\subset\R,$ hence the discrete group $G$ is isomorphic to $\al S.(\R^4)\times\T.$ Of course $\T$ will be our global gauge group below. STEP 5.Next, we would like to realize in $\al E.$ the heuristic constraints $$Q(f)^n\psi=A(Lf)^n\psi\qquad\forall\;f\in{\al S.}(\R^4),\;\;
n\in\N$$ where $L:S\to\ker K\subseteq\ker B$ is a given linear map. First write the heuristic constraints in bounded form: $$e^{iQ(f)}\psi=e^{iA(Lf)}\psi\;,\quad\hbox{i.e.}\quad
e^{-iA(Lf)}e^{iQ(f)}\psi=\psi\;.$$ So, given the identifications with heuristic objects above, we define our constraint unitaries to be: $$\al U.:=\set\delta\s-Lf.\cdot U_{\gamma_f}=:V_f,f\in{\al S.}(\R^4,\R).
\subset\al A.\;.$$
\[FCU\] $\al U.$ is first–class.
The proof is in the next section. The heuristic constraint conditions now correspond to the application of the T–procedure to $\al U..$ STEP 6.Now we will specify the superselection structure associated with the global charge $Q$ using the fact that $Q$ must take integer values on the vacuum state. Recall that the global gauge transformations $\gamma\s t.\,,$ $t\in\R$ are implemented by the unitaries $U\s\gamma_t.\in \al A.$ which we identify with the heuristic objects $\exp itQ$ (cf. Step 3). For the superselection sectors we need to find cyclic representations ${(\pi,\,\Omega)}$ such that $$\pi(U\s\gamma_t.)\Omega=e^{itn}\Omega\qquad\forall\;t\in\R$$ and some $n\in\Z$ (the heuristic corresponding conditions are $Q\Omega=n\Omega).$ We recognise these as constraint conditions for Dirac states of the constraint unitaries: $$\al V._n:=\set V_t^{(n)}:=e^{-itn}U\s\gamma_t.,t\in\R.\;.$$ Denote the sets of these Dirac states by $${\got S}_D^{(n)}:=\set\omega\in{\got S}(\al A.),
\omega(V_t^{(n)})=1\quad\forall\;t\in\R.\,.$$ These folia of states will be our superselection sectors.
\[SectDisj\] With notation as above, we have:
- ${\got S}_D^{(n)}\cap{\got S}_D^{(m)}=\emptyset$ if $n\not=m\;,$
- ${\got S}_D^{(0)}\not=\emptyset\;.$
\(i) If there is an $\omega\in{\got S}_D^{(n)}\cap{\got S}_D^{(m)}$ for $n\not=m,$ then $$\begin{aligned}
\omega\big(e^{-itn}U\s\gamma_t.\big)&=& 1\; =\;\omega\big(e^{-itm}U\s\gamma_t.\big) \\[1mm]
\hbox{so:}\qquad\omega\big(U\s\gamma_t.\big) &=& e^{itn}=e^{itm}\qquad\forall\;t\end{aligned}$$ which contradicts $n\not=m.$ (ii) In the proof of Lemma \[FCU\] we constructed a state $\omega_3\in{\got S}(\al A.)$ satisfying $\omega_3(U_g)=1$ for all $g\in G.$ If we take $g=\gamma\s t.,$ then this implies that $\omega_3\in{\got S}_D^{(0)}.$
To connect with the usual machinery for superselection used above, we need to exhibit the canonical endomorphisms (automorphisms in the abelian case). We construct an action $\rho:\Z\to\aut\al A.$ such that its dual action on $\al A.^*$ satisfies $\rho^*_k({\got S}_D^{(n)})={\got S}_D^{(n+k)}.$
\[RhDf\] For each $k\in\Z$ define a \*-automorphism $\rho_k$ of $\al A.$ by: $$\begin{aligned}
\rho_k(A)=A\quad\forall\;A\in\al E.;\qquad
\rho_k(U\s\gamma_t.)=e^{itk}U\s\gamma_t.\quad\forall\,t\in\R;\\[1mm]
\rho_k(U\s\gamma_f.)=U\s\gamma_f.\quad\forall\;f\in\al S.(\R^4)\,.\end{aligned}$$
\[RkWD\] $\rho_k$ is well–defined, and $\rho_k\in\aut\al A.\,.$
The proof is in the next section. Recall that for any $\alpha\in\aut\al A.$ we define its dual $\alpha^*:\al A.^*\to\al A.^*$ by $\alpha^*(f):=f\circ\alpha$ for all functionals $f\in\al A.^*.$
With notation as above, we have $\rho^*_k({\got S}_D^{(n)})
={\got S}_D^{(n+k)}$ and ${\got S}_D^{(n)}\not=\emptyset$ for all $n\in\Z\;.$
Let $\omega\in\rho^*_k\big({\got S}_D^{(n)}\big),$ i.e. $\omega=\omega_n\circ\rho_k$ for some $\omega_n\in{\got S}_D^{(n)}.$ Thus $$\omega\left(e^{-it(n+k)}U_{\gamma\s t.}\right)
=\omega_n\left(e^{-it(n+k)}\rho_k\big(U_{\gamma\s t.}\big)\right)
=\omega_n\left(e^{-itn}U_{\gamma\s t.}\right) =1$$ i.e. $\omega\in{\got S}_D^{(n+k)}\,.$ Conversely, for any $\omega\in{\got S}_D^{(n+k)}$ there is an $\omega_n\in{\got S}_D^{(n)}$ for which $\omega=\omega_n\circ\rho_k$ and it is obviously $\omega_n=\omega\circ\rho_{-k}\,.$ Thus $\rho^*_k({\got S}_D^{(n)})
={\got S}_D^{(n+k)}.$ Since we have that ${\got S}_D^{(0)}\not=\emptyset,$ it is now immediate that ${\got S}_D^{(n)}=\rho^*_k({\got S}_D^{(0)})
\not=\emptyset\;.$
Recall from our earlier discussions that the canonical automorphisms (Abelian case) must necessarily be outer on $\al A..$
\[RhOut\] With notation as above, $\rho_k\in\out\al A.$ if $k\not=0\;.$
The proof of this is long, and is in the next section.
From the action $\rho:\Z\to{\rm Out}\,\al A.$ we construct a Hilbert extension (cf. Subsection \[abelianHS\]). First set $$\Lambda:=\set\Ad U\circ\rho_k,{U\in\al A.\;\hbox{unitary,}\;k\in\Z}.$$ so $\Z\cong\Lambda\big/{\rm Inn}\,\al A.\;.$ So the class of $k\in \Z$ in $\Lambda\big/{\rm Inn}\,\al A.$ is $\chi\s k.:=\set\Ad U\circ\rho_k,{U\in\al A._u}.\,.$ Take the monomorphic section $\chi\s k.\to k,$ then it has a trivial cocycle $\sigma(n,\,m)=1$ for all $n,\,m\in\Z\,.$ Define $\al F.:=\Z\cross\rho.\al A.,$ then it has the dense \*–algebra $$\al F._0:=\set\sum_{n\in F}A_nU^n,{A_n\in\al A.\,,\;
F\subset\Z\;\hbox{finite}}.$$ where $U\in\al F.$ is the unitary which implements $\rho_1,$ i.e. $\rho_1=\Ad U\restriction\al A.\,.$ Fix $t\in\T=\wh\Z$ and define an action $\alpha:\T\to\aut\al F.$ by $$\alpha_t\left(\sum_{n\in F}A_nU^n\right)
:=\sum_{n\in F}A_nt^nU^n\qquad\hbox{on $\al F._0\;.$}$$ Then the fixed point algebra of $\alpha$ is $\al A.\,.$ We verify the compatibility condition in Corollary \[cor4.1\]:
$\rho_k(\al D.)\sim\al D.$ for all $k\in\Z\,.$
The constraint unitaries from which we define $\al D.$ are $V_f:=\delta\s-Lf.\cdot U_{\gamma_f}\,,$ $f\in\al S.(\R^4)\,.$ By definition (\[RhDf\]) we have $\rho_k\restriction\al E.
=\iota\,,$ hence $\rho_k(\delta_{-Lf})=\delta_{-Lf}\,.$ Also $\rho_k(U_{\gamma\s f.}) = U_{\gamma\s f.}$ for all $f\in\al S.(\R^4)\,,$ hence $\rho_k(V_f)=V_f$ for all $f\in\al S.(\R^4)\,.$ Thus $\rho_k$ preserves the Dirac states ${\got S}_D$ and hence $\rho_k(\al D.)=\al D.$ for all $k\in\Z\,.$
It remains to show that this Hilbert system is regular and minimal. However, at this stage we do not have a proof because little is known about the ideal $\al I._0$ factored out in Step 3.
Proofs {#Proofs}
======
Proof of Theorem \[Teo.4.1\] {#proof-of-theoremteo.4.1 .unnumbered}
----------------------------
[**(i)**]{} We have that $\beta_g:=g\rest\of.$ The pointwise norm-continuity of $\beta_{\al G.}$ follows from the pointwise norm-convergence topology of $\al G..$ So $\{\of,\al G.,\beta\}$ is a C\*–dynamical system. Since $\al A.$ is the fixed point algebra of $\al G.$, the fixed point algebra of $\beta\s{\al G.}.$ is $\of\cap\al A..$ By Theorem \[Teo.2.12\] we have that $\of\cap\al A.= \al O..$If $A\in Z(\al A.),$ then ${[A,\al C.]}=0,$ hence $A\in\al O.$ by Theorem \[Teo.2.2\](iii), and from this it follows that $Z(\al A.)\subseteq Z(\al O.).$ Now let $\al H._{\gamma}\subset\Pi_{\gamma}\al F.$. If there is a unit vector $\Phi\in\al H._\gamma\cap\of,$ then by invariance of $\al H._\gamma\cap\of$ under $\al G.,$ we also have $\al H._\gamma\cap\of\supset\spa(\al G.\Phi)
=\al H._\gamma,$ where the last equality follows from irreducibility of the action of $\al G.$ on $\al H._\gamma.$ Thus $\al H._\gamma\cap\of \not=\{0\}$ implies that $\al H._\gamma\subset\of,$ hence $\al H._\gamma\subset\Pi_\gamma\of.$
To prove that $\Pi_{\gamma}\of=\hbox{clo-span}\,(\al O.\al H._{\gamma})$ we follow the proof of Lemma 10.1.3 in [@BW]. First, since $\al O.=\Pi_\iota\of$, it follows that $\al O.\al H._\gamma\subseteq\Pi_\gamma\of$ by Remark \[remark1\](v).
By Evans and Sund [@ES], $\Pi_\gamma\of$ is the closed span of all the $\al G.\hbox{--invariant}$ subspaces $\al E.\subset\of$ such that $\beta_\al G.$ acts on $\al E.$ as an element of $\gamma\in\wh{\al G./\al K.}.$ So for the reverse inclusion, $\Pi_{\gamma}\of\subseteq\csp\{\al OH._\gamma\},$ it suffices to show that $\spa\{\al OH._\gamma\}$ contains all $\al G.\hbox{--invariant}$ subspaces $\al E.\subset\of$ such that $\al G.$ acts on $\al E.$ as an element of $\gamma.$ Let $\{\Psi_1,\ldots,\,\Psi_d\},$ $d={\rm dim}\,\gamma$ be a basis of such an $\al E.$ under which the matrix representation of the action of $\al G.$ is an element of $\gamma,$ i.e. $$g\Psi_i=\sum_j\lambda_{ji}(g)\,\Psi_j$$ where the matrix $(\lambda_{ji}(g))$ is a unitary matrix representation of $\al G.$ of the type $\gamma.$ Choose an orthonormal basis $\{\Phi_1,\ldots,\,\Phi_d\}$ of $\al H._\gamma$ which also transforms under $\al G.$ according to $(\lambda_{ji}(g)).$ Consider now the element $A:=\sum\limits_j\Psi_j\Phi^*_j\in\of.$ Then $$\begin{aligned}
g(A)&=&\sum_jg\left(\Psi_j\Phi^*_j\right)=
\sum_{i,\,k}\Big(\sum_j\lambda_{ij}(g)\,\ol\lambda_{kj}(g).\Big)
\Psi_i\Phi^*_k \\[1mm]
&=& \sum_{i,\,k}\delta_{ik}\Psi_i\Phi^*_k =\sum_j\Psi_j\Phi^*_j=A \,.\end{aligned}$$ Thus $A\in\al O.,$ and hence all $\Psi_i=A\Phi_i\in\al OH._\gamma$ i.e. $\al E.\subset\spa(\al OH._\gamma).$ Let $\al H._\sigma$ have an orthonormal basis $\{\Phi_1,\ldots\Phi_d\}$ hence $\rho_\sigma(F)=\sum\limits_{j=1}^d\Phi_jF\Phi_j^*$ for $F\in\al F.,$ $\rho_\sigma\rest\al A.=\sigma.$ Since $\{\Phi_j\}\subset\of=M(\df)$ it is clear that $\rho_\sigma$ preserves both $\df$ and $\of.$ Since $\rho_\sigma$ also preserves $\al A.,$ it preserves $\al D.=\df\cap\al A.$ and $\al O.=\of\cap\al A.,$ where these equalities come from Theorem \[Teo.2.12\].
Proof of Theorem \[Teo.4.2\] {#proof-of-theoremteo.4.2 .unnumbered}
----------------------------
Let $\al H.\subset\of$ have an orthonormal basis $\{\Phi_j\}.$ By the same proof as for Theorem \[Teo.4.1\](iii) we have that $\rh(\al D.)\subseteq\al D..$ Since $\of$ is a \*–algebra and the relative multiplier algebra of $\df\supset\al D.,$ we have that $$\begin{aligned}
[\Phi^*_j,\, D]&\in &\df\quad\hbox{for all}\;D\in\al D.,\;j. \\[1mm]
\hbox{Thus:}\qquad
\Phi_j[\Phi^*_j,\, D]&\in &\Phi_j\df=\rh(\df)\Phi_j
\subset\csp(\rh(\df)\al F.) \\[1mm]
\hbox{i.e.}\qquad
D-\rh(D)&=&\sum_j\left(\Phi_j\Phi^*_jD-\Phi_jD\Phi_j^*\right)
\in\csp(\rh(\df)\al F.\,.) \\[1mm]
\hbox{So}\qquad
D&\in &\csp(\rh(\df)\al F.)\quad\hbox{for all}\; D\in\al D.\,.\end{aligned}$$ Thus we have shown that $\al D.\subset\csp(\rh(\df)\al F.),$ and now we would like to show that ${\csp(\rh(\df)\al F.)}={\csp(\rh(\al D.)\al F.)}.$ We have that $$\csp(\al CF.)=\csp(\al DF.)=\csp(\df\al F.),$$ so if we apply $\rh$ to both sides of the last equation, multiply by $\al F.$ on the right and take closed span, we get:
$$\begin{aligned}
\csp\big(\rh(\al D.)\rh(\al F.)\al F.\big) &=&
\csp\big(\rh(\df)\rh(\al F.)\al F.\big) \\[1mm]
\hbox{i.e.}\qquad
{\csp(\rh(\df)\al F.)} &=&
\csp(\rh(\al D.)\al F.)\;. \\[1mm]
\hbox{Thus:}\qquad\al D. \subset \csp(\rh(\al D.)\al F.) & &\quad
\hbox{and since $\al D.$ is a *--algebra in $\al A.,$} \\[1mm]
\al D.\; \subseteq \;\csp(\rh(\al D.)\al F.)\!&\cap&\!
\csp(\al F.\rh(\al D.))\cap\al A. \\[1mm]
&\subseteq &\csp(\al DF.)\cap\csp(\al FD.)\cap\al A.=\al D. \end{aligned}$$
where we used $\df\cap\al A.=\al D..$ Thus $$\al D.=\csp(\rh(\al D.)\al F.)\cap\csp(\al F.\rh(\al D.))\cap\al A.
=\csp(\rh(\al D.)\al A.)\cap\csp(\al A.\rh(\al D.))$$ which also follows from Theorem \[Teo.2.12\], treating $\rh(\al D.)\subseteq\al D.$ as a second constraint set. Thus $\al D.\sim\rh(\al D.)$ in $\al A..$
For the converse, let $\al D.\sim\rh(\al D.)$ and take $\Phi\in\al H..$ From the equation $\Phi D=\rh(D)\Phi$ for all $D\in\al D.,$ we conclude that $$\Phi\cdot\csp(\al DF.) \subset \csp(\rh(\al D.)\al F.)=\csp(\al DF.)$$ using $\al D.\sim\rh(\al D.).$ Since we have trivially that $\Phi\cdot\csp(\al FD.)\subset\csp(\al FD.),$ it follows that $$\Phi\df=\Phi\left(\csp(\al FD.)\cap\csp(\al DF.)\right)
\subset\df$$ so $\Phi$ is in the left multiplier of $\df.$ We also have that $$\csp(\al FD.)\Phi=\csp(\al F.\rh(\al D.))\Phi=\csp(\al F.\Phi\al D.)
\subseteq\csp(\al FD.)\;.$$ Since trivially $\csp(\al DF.)\Phi\subseteq\csp(\al DF.),$ it follows that $$\df\Phi=\left(\csp(\al FD.)\cap\csp(\al DF.)\right)\Phi
\subset\df$$ and hence $\Phi$ is in the relative multiplier algebra of $\df,$ i.e. $\Phi\in\of$ by Theorem \[Teo.2.2\](ii). Let $\al H._\sigma\subset\of\supset\al H._\tau,$ hence by (i) $\al D.\sim\sigma(\al D.)\sim\tau(\al D.).$
First let $X\in (\sigma,\,\tau)_{\al A.}\cap\al O.,$ i.e. $X\in\al O.$ and $X\sigma(A)=\tau(A)X$ for all $A\in\al A..$ By letting $A$ range over only $\al O.\subset\al A.,$ we immediately get that $X\in{(\sigma\rest\al O.,\,\tau\rest\al O.)_{\al O.},}$ making use of Theorem 4.1(iii). Therefore, it suffices to prove that ${(\sigma,\,\tau)_{\al A.}}\subset\al O..$
Let $X\in{(\sigma,\,\tau)_{\al A.}},$ i.e. $X\in\al A.$ and $X\sigma(A)=\tau(A)X$ for all $A\in\al A..$ Thus $$\begin{aligned}
X\cdot\csp(\al DA.)&=& X\cdot\csp(\sigma(\al D.)\al A.)
=\csp(X\sigma(\al D.)\al A.) \\[1mm]
&\subseteq&
\csp(\tau(\al D.)X\al A.) \subseteq
\csp(\tau(\al D.)\al A.)=\csp(\al DA.)\;.\end{aligned}$$ Since we have trivially that $X\cdot\csp(\al AD.)\subseteq\csp(\al AD.),$ it follows that $$X\al D.\subseteq\csp(\al AD.)\cap\csp(\al DA.)=\al D.\,,$$ i.e. $X$ is in the left multiplier of $\al D..$ Likewise: $$\begin{aligned}
\csp(\al AD.)\cdot X &=& \csp(\al A.\tau(\al D.))\cdot X
=\csp(\al A.\tau(\al D.)X) \\[1mm]
&\subseteq&
\csp(\al A.X\sigma(\al D.))\subseteq\csp(\al A.\sigma(\al D.)) \\[1mm]
&=& \csp(\al AD.)\,.\end{aligned}$$ Since trivially $\csp(\al DA.)\cdot X\subseteq\csp(\al DA.),$ we have: $$\al D.\cdot X\subseteq\csp(\al AD.)\cap\csp(\al DA.)=\al D.\,,$$ and hence $X$ is in the relative multiplier of $\al D.,$ i.e. $X\in\al O..$
Proof of Proposition \[pro.4.7\] {#proof-of-propositionpro.4.7 .unnumbered}
--------------------------------
\(i) According to the decomposition $$\sigma(\cdot)=\sum_{j}V_{j}\rho_{\gamma_{j}}(\cdot)V_{j}^{\ast},\quad
V_{j}\in (\rho_{\gamma_{j}},\sigma)_{\al A.}$$ we have $\al H._{\sigma}=\sum_{j}V_{j}\al K.'_{j}$ where $\rho_{\gamma_{j}}=\rho_{\al K.'_{j}}$ and the $\al K._{j}'$ are irreducible w.r.t. $\al G.$ carrying the representation $\gamma_j\in\wh{\al G.}.$ Moreover $\hbox{supp}\,\al K.'_{j}=1.$
Put $E_{j}:=V_{j}V_{j}^{\ast}.$ Then $\sum_{j}E_{j}=1.$ Since $V_{j}\al K.'_{j}\subset\al H._{\sigma}\subset\of$ it follows that $$E_{j}=\hbox{supp}\,V_{j}\al K.'_{j}\in\al O.$$ for all $j$. Therefore, by assumption, there are isometries $W_{j}\in\al O.$ with $E_{j}=W_{j}W_{j}^{\ast}.$ Now we put $$\al K._{j}:=W_{j}^{\ast}V_{j}\al K.'_{j}\subset\of.$$ Then $\al K._{j}$ is an algebraic Hilbert space with $\hbox{supp}\,\al K._{j}=1$, carrying the representation $\gamma_{j}$ and we have $V_{j}\al K.'_{j}=W_{j}\al K._{j}$. Hence $\al H._{\sigma}=\sum_{j}W_{j}\al K._{j}$ and $$\sigma(\cdot)=\sum_{j}W_{j}\rho_{\al K._{j}}(\cdot)W_{j}^{\ast},\quad
W_{j}\in (\rho_{\al K._{j}},\sigma)_{\al O.}$$ follows.
\(ii) This follows from (i) using the existence of subobjects.
Proof of Theorem \[Teo.4.3\] {#proof-of-theoremteo.4.3 .unnumbered}
----------------------------
Let $\al H.\subset\al F.$ be an arbitrary algebraic Hilbert space. Then $\xi(\al H.)\subset\al L.$ is also an algebraic Hilbert space. with support 1. To see this, let $\{\Phi_{j}\}_{j}$ be an orthonormal basis of $\al H.$, i.e. $\Phi_{j}^{\ast}\Phi_{k}=\delta_{j,k}1$ and $\sum_{j}\Phi_{j}\Phi_{j}^{\ast}=1$ then the same relations are true for the system $\{\xi(\Phi_{j})\}_{j}.$ In particular $\xi$ is injective on $\al H..$ Moreover, if $\al H.$ is $\al G.$-invariant and ${g}(\Phi_{j})=\sum_{k}u_{k,j}(g)\Phi_{k}$ then ${g}^{\xi}\big(\xi(\Phi_{j})\big)=\sum_{k}u_{k,j}(g)\xi(\Phi_{k}),$ i.e. $\xi(\al H.)$ carries the same representation as $\al H..$ In particular, if $\al H._{\gamma}$ carries $\gamma$, i.e. $\al H._{\gamma}\subset\Pi_{\gamma}\al F.$ then $\xi(\al H._{\gamma})\subset\Pi_{\gamma}^{\xi}\al L.$. This proves (ii) and (i).Let $\al N._{\gamma}$ be an orthonormal basis for $\xi(\al H._\gamma),$ then by the first part it is the image under $\xi$ of an orthonormal basis $\{\Phi_{\gamma,j}\}_{j}$ of $\al H._\gamma.$ Let $F=
\sum A_{\gamma,j}\Phi_{\gamma,j}\in\al F._{\rm fin}$ such that $\xi(F)=0=\sum\xi(A_{\gamma,j})\xi(\Phi_{\gamma,j}).$ By applying $\al G.^\xi$ to this equality, and using the relation $g^\xi(\xi(\Phi_{\gamma,j}))=\sum\limits_{k}u_{k,j}(g)\xi(\Phi_{\gamma,k})$ we get $\sum\limits_{\gamma,j,k}u_{k,j}^{\gamma}(g)\xi(A_{\gamma,j})\xi
(\Phi_{\gamma,k})=0$ for all $g\in\al G..$ Now the orthogonality relations for the matrix elements of the irreducible representations of $\al G.$ imply $\xi(A_{\gamma,j})\xi(\Phi_{\gamma,k})=0$ for all $\gamma\in\wh{\al G.},j,k$. Hence $\xi(A_{\gamma,j})=0$ follows. This proves (iii). From $\xi\circ\Pi_{\gamma}=\Pi_{\gamma}^{\xi}\circ\xi$ (iv) follows.For (v) observe that the homomorphic images of isometries $V_i\in\al A.$ with $V_1V_1^*+V_2V_2^*=\un$ produces a pair of isometries in $\xi(\al A.)$ satisfying the same relation. So Property B for $\al A.$ implies Property B for $\xi(\al A.)\,.$
Proof of Corollary \[Cor.4.9\] {#proof-of-corollarycor.4.9 .unnumbered}
------------------------------
\(i) Let $\lambda$ be generated by $\al H.$, i.e. let $\lambda=\rho_{\al H.}$ such that $\lambda(A)=\sum_{j}\Phi_{j}A\Phi_{j}^{\ast}$ where $\{\Phi_{j}\}_{j}$ is an orthonormal basis of $\al H.$. Then $\xi(\lambda(A))=\sum_{j}\xi(\Phi_{j})\xi(A)\xi(\Phi_{j})^{\ast}$ and $\xi(A)=0$ implies $\xi(\lambda(A))=0.$ Furthermore, $\lambda^{\xi}(\xi(A))=\rho_{\xi(\al H.)}(\xi(A)).$(ii)$\lambda(\cdot)=\sum_{j}W_{j}\lambda_{j}(\cdot)W_{j}^{\ast}$ implies $\lambda^{\xi}(\cdot)=\sum_{j}\xi(W_{j})\lambda_{j}^{\xi}(\cdot)\xi(W_{j})^{\ast}$ and $(\lambda\circ\sigma)^{\xi}=\lambda^{\xi}\circ\sigma^{\xi}.$ Further, if $\sigma(\cdot)=V^{\ast}\lambda(\cdot)V$ where $V\in (\sigma,\lambda)$, i.e. $V\sigma(\cdot)=\lambda(\cdot)V$ then $\xi(V)\sigma^{\xi}(\cdot)=\lambda^{\xi}(\cdot)\xi(V)$ and $\sigma^{\xi}(\cdot)=\xi(V)^{\ast}\lambda^{\xi}(\cdot)\xi(V).$
In particular, if $\lambda\cong\sigma$ then $\lambda^{\xi}\cong\sigma^{\xi}.$
Proof of Theorem \[Teo.4.4\] {#proof-of-theoremteo.4.4 .unnumbered}
----------------------------
\(i) Let $\sigma\in\hbox{Ob}\,\al T.^{\xi}$. Then there is a $\al G.$-invariant algebraic Hilbert space $\al H.\subset\al L.$ such that $\sigma (X)=\sum_{j}\Psi_{j}X\Psi_{j}^{\ast},\;X\in\xi(\al A.),$ where $\{\Psi_{j}\}_{j}$ denotes an orthonormal basis of $\al H.$. On the other hand, there is a corresponding $\al G.$-invariant Hilbert space $\al K.\subset\al F.$ such that $\al H.$ and $\al K.$ carry unitarily equivalent representations of $\al G.$. In $\al K.$ we choose an orthonormal basis $\{\Phi_{j}\}_{j}$ such that the representation matrix of $\al G.$ in $\al H.$ w.r.t. $\{\Psi_{j}\}_{j}$ coincides with that in $\al K.$ w.r.t. $\{\Phi_{j}\}_{j}.$ Then $\xi(\al K.)$ transforms under $\al G.$ w.r.t. $\{\xi(\Phi_{j})\}_{j}$ with the same representation matrix. Now we put $$V:=\sum_{j}\Psi_{j}\xi(\Phi_{j})^{\ast}\in\al L..$$ Obviously, $V$ is unitary and ${g}^{\xi}(V)=V$ for all $g\in\al G.$, i.e. $V\in\xi(\al A.)$. Then $V\xi(\Phi_{j})=\Psi_{j}$ or $\al H.=V\xi(\al K.)$ and $\sigma=\hbox{Ad}\,V\circ\rho_{\al K.}^{\xi}.$
\(ii) According to Corollary \[Cor.4.9\] and $(\hbox{Ob}\,\al T.)^{\xi}\subseteq\hbox{Ob}\,\al T.^{\xi}$ the image $\al C.^{\xi}$ of an equivalence class $\al C.\subset\hbox{Ob}\,\al T.$ is contained in a unique equivalence class of $\hbox{Ob}\,\al T.^{\xi}$. But (i) says that every equivalence class $\al E.$ of $\hbox{Ob}\,\al T.^{\xi}$ is an image $\al E.=\al C.^{\xi}$.
Proof of Lemma \[arrow1\] {#proof-of-lemmaarrow1 .unnumbered}
-------------------------
Let $A\in(\sigma,\,\tau)_{\al A.},$ then it follows immediately from $A\sigma(B)=\tau(B)A,\;B\in\al A.$ that $\xi(A)\sigma^{\xi}(\xi(B))=\tau^{\xi}(\xi(B))\xi(A)$ for all $B\in\al A..$ Recall that $\xi(\al A.)$ is the fixed point algebra of $\al G.^\xi.$
Proof of Proposition \[Prop.4.12\] {#proof-of-propositionprop.4.12 .unnumbered}
----------------------------------
\(i) This is obvious because the union $\bigcup_{\gamma}\al N._{\gamma}$ of orthonormal bases $\al N._{\gamma}$ of $\al H._{\gamma}$ is an $\al A.$-left module basis of $\al F._{\rm fin}$.
\(ii) By a straightforward calculation one obtains for all $F\in\al F.$ that: $$\langle\xi(F),\xi(F)\rangle_{\xi(\al A.)}=
\xi(\langle F,F\rangle_{\al A.})$$ and $$\vert\xi(F)\vert_{\xi(\al A.)}=\Vert\xi(\langle F,F\rangle_{\al A.})
\Vert^{1/2}\leq\Vert\langle F,F\rangle_{\al A.}\Vert^{1/2}
=\vert F\vert_{\al A.},$$ i.e. $\xi$ is continuous w.r.t. the norm $\vert\cdot\vert_{\al A.}.$ Now let $F\in\clo\s{\vert\cdot\vert_{\al A.}}.(\ker\xi\cap
\al F._{\rm fin}),$ hence there is a sequence $\{F_n\}\subset
\ker\xi\cap\al F._{\rm fin}$ such that $\vert F_{n}-F\vert_{\al A.}\to 0.$ Then $\xi(F)=0$ follows. Conversely, let $F\in\ker\xi$. Recall $\xi\circ\Pi_{\gamma}=\Pi_{\gamma}^{\xi}\circ\xi$ which implies $\Pi_{\gamma}F\in\ker\xi$. Now, according to Remark \[remark1\] (iv) we have $F=\sum_{\gamma}\Pi_{\gamma}F$ w.r.t. the $\vert\cdot\vert_{\al A.}$-norm convergence. This implies $$F\in\hbox{clo}_{\vert\cdot\vert_{\al A.}}(\ker\xi\cap
\al F._{\rm fin}).$$
Proof of Theorem \[Teo.4.13\] {#proof-of-theoremteo.4.13 .unnumbered}
-----------------------------
\(i) Since $\HS$ is minimal and regular, there exists an assignment $\sigma\to\al H._{\sigma}$ such that an admissible (DR-)subcategory $\al T._{\C}$ can be defined by $$(\sigma,\tau)_{\al A.,\C}:=(\al H._{\sigma},\al H._{\tau}),$$ cf. Theorem \[Teo1\]. Now we use the morphism $\xi$ to define a corresponding subcategory $\al T.^{\xi}_{\C}$ for $\xi(\al T.)$. Recall $\hbox{Ob}\,\xi(\al T.)=(\hbox{Ob}\,\al T.)^{\xi}\subset\hbox{Ob}\,
\al T.^{\xi}.$ We put $$\hbox{Ob}\,\al T.^{\xi}_{\C}:=\hbox{Ob}\,\xi(\al T.).$$ Let $\lambda,\sigma\in\hbox{Ob}\,\al T.$. Then $\lambda^{\xi},\sigma^{\xi}\in\hbox{Ob}\,\xi(\al T.)$ and the arrows are defined by $$(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.),\C}:=
\xi((\sigma,\tau)_{\al A.,\C})=(\xi(\al H._{\sigma}),\xi(\al H._{\tau})).$$ Then $$(\iota^{\xi},\iota^{\xi})_{\xi(\al A.),\C}
=\xi((\iota,\iota)_{\al A.,\C})=\xi(\C\un)=\C\xi(\un).$$ It is straightforward to show that $\al T.^{\xi}_{\C}$ has direct sums and subobjects (in the latter case note that if $F$ is a nontrivial projection from $(\sigma^{\xi},\sigma^{\xi})_{\xi(\al A.),\C}$ then there is a nontrivial projection $E\in (\sigma,\sigma)_{\al A.,\C}$ such that $F=\xi(E)$ because the ($\al G.$-invariant) matrix $\{p_{j,k}\}_{j,k}$ of $F$ w.r.t. $\{\xi(\Phi_{\sigma,j})\}_{j}$, where the $\Phi_{\sigma,j}$ form an orthonormal basis of $\al H._{\sigma}$, can be used to define a corresponding $E$ in $(\sigma,\sigma)_{\al A.,\C}$. Furthermore, the permutation and conjugation structures of $\al T._{\C}$ survive the morphism $\xi$. Thus $\al T.^{\xi}_{\C}$ is a DR-category. We use the notation $\al T.^{\xi}_{\C}=\xi(\al T._{\C}).$ (This result means: The Hilbert system $\{\xi(\al F.),\al G.\}$ is regular.)
\(ii) First let $\xi(\al A.)'\cap\xi(\al F.)=\xi(Z(\al A.)).$ Then, according to Theorem \[Teo1\], property P.2 can be fulfilled by an appropriate subcategory of the form described before. Second, let property P.2 be satisfied. Then $\xi(\al T._{\C})$ is an admissible (DR-)subcategory of $\xi(\al T.)$. Therefore, according to Theorem \[Teo2\] there is a corresponding minimal and regular Hilbert extension $\tilde{\al F.}$ of $\xi(\al A.).$ The uniqueness part of Theorem \[Teo2\] gives that $\tilde{\al F.}$ and $\xi(\al F.)$ are $\al A.$-module isomorphic, hence $\xi(\al A.)'\cap\xi(\al F.)=Z(\xi(\al A.))$ is also true.
\(iii) The inclusion $\supseteq$ is obvious (see Lemma \[arrow1\]). The assertion is $$\label{inclusion}
(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}\subseteq\xi((\sigma,\tau)_{\al A.}).$$ First we prove this inclusion for the admissible subcategory, i.e. we assert $$(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.),\C}\subseteq
\xi((\sigma,\tau)_{\al A.}).$$ This is obvious by $$(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.),\C}=
\xi((\sigma,\tau)_{\C})\subset\xi((\sigma,\tau)_{\al A.}).$$ Second, recall that $\xi((\sigma,\tau)_{\al A.
})$ is a right module w.r.t. $\sigma^{\xi}(\xi(Z(\al A.)))$ and a left module w.r.t. $\tau^{\xi}(\xi(Z(\al A.)))$. On the other hand, $(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}$ is a right module w.r.t. $\sigma^{\xi}(Z(\xi(\al A.)))$ and a left module w.r.t. $\tau^{\xi}(Z(\xi(\al A.))).$ Further, according to P.2. $(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.),\C}$ is a generating subset for this module. Since, by assumption, $Z(\xi(\al A.))$ and $\xi(Z(\al A.))$ coincide, the inclusion (\[inclusion\]) follows.
\(iv) This follows directly from $\xi(\al A.)'\cap\xi(\al F.)=Z(\xi(\al A.))$ and the fact that the unitary equivalence classes of $\al T.^{\xi}$ and $\xi(\al T.)$ coincide.
Proof of Theorem \[Teo.4.14\] {#proof-of-theoremteo.4.14 .unnumbered}
-----------------------------
Since $\al T._{\C}$ is an admissible (DR-)subcategory of $\al T.$ we can apply Theorem\[Teo2\], i.e. there is a corresponding minimal and regular Hilbert extension $\HS$ of $\al A.$. Therefore the arrows of the category $\al T._{\C}$ are given by $$\label{arrowAdm}
(\sigma,\tau)_{\al A.,\C}=(\al H._{\sigma},\al H._{\tau}),$$ where the Hilbert spaces $\al H._{\sigma},\al H._{\tau}$ generate the endomorphisms $\sigma,\tau$ respectively.
Now it is not hard to show that the morphism $\xi$ can be extended to a morphism of $\al F.$ by putting $$\label{MorAdm}
\xi(\Phi_{\lambda,j}):=\Phi_{\lambda,j}$$ where $\lambda$ runs through a complete system of irreducible and mutually disjoint endomorphisms and $\{\Phi_{\lambda,j}\}$ denotes an orthonormal basis of the Hilbert space $\al H._{\lambda}$ which generates $\lambda$ (recall and use Proposition \[Prop.4.12\]). This morphism satisfies the assumptions of Theorem \[Teo.4.3\]. The corresponding Hilbert system is denoted by $\{\al F.^{\xi},\al G.\}$ (recall that $\al G.^{\xi}\cong\al G.$). Equations (\[arrowAdm\]) and (\[MorAdm\]) imply $$\sigma^{\xi}(\xi(\al A.))=\sum_{j}\Phi_{\lambda,j}\xi(A)\Phi_{\lambda,j}^{\ast}
\qquad\hbox{and}\qquad
\xi((\sigma,\tau)_{\al A.,\C})=(\sigma,\tau)_{\al A.,\C}.$$ By assumption (iii) we have $\xi((\sigma,\tau)_{\al A.}=(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.)}$. Since $(\sigma,\tau)_{\al A.,\C}\subset(\sigma,\tau)_{\al A.}$ we have $\xi((\sigma,\tau)_{\al A.,\C})\subset(\sigma^{\xi},\tau^{\xi})_
{\xi(\al A.)}.$ Therefore the subcategory $\al T._{\C}^{\xi}$ of $\xi(\al T.)$ defined by $$(\sigma^{\xi},\tau^{\xi})_{\xi(\al A.),\C}:=\xi((\sigma,\tau)_
{\al A.,\C}=(\sigma,\tau)_{\al A.,\C}$$ is a DR-category.
Now we prove property P.2 for $\al T._{\C}^{\xi}$. We have to show $$\sigma^{\xi}(Z(\xi(\al A.)))(\lambda^{\xi},\sigma^{\xi})_{\xi(\al A.),\C}
\lambda^{\xi}(Z(\xi(\al A.)))=(\lambda^{\xi},\sigma^{\xi})_{\xi(\al A.)}.$$ The left hand side equals $$\sigma^{\xi}\xi(Z(\al A.))(\lambda^{\xi},\sigma^{\xi})_{\xi(\al A.),\C}
\lambda^{\xi}(\xi(Z(\al A.)))=
\xi(\sigma(Z(\al A.)))(\lambda^{\xi},\sigma^{\xi})_{\xi(\al A.),\C}
\xi(\lambda(Z(\al A.)))=$$ $$\xi(\sigma(Z(\al A.)))\xi((\lambda,\sigma)_{\al A.,\C})\xi(\lambda(Z(\al A.)))=
\xi(\sigma(Z(\al A.))(\lambda,\sigma)_{\al A.,\C}\lambda(Z(\al A.)))=
\xi((\lambda,\sigma)_{\al A.})$$ and this coincides, by assumption, with the right hand side.
Now we can apply Theorem \[Teo2\] to obtain a further Hilbert extension $\{\tilde{\al F.}^{\xi},\al G.^{\xi}\}$ where again $\al G.^{\xi}\cong\al G.$. Using the uniqueness part of Theorem \[Teo2\] we obtain that both Hilbert extensions are $\xi(\al A.)$-module isomorphic.
Proof of Proposition \[FCU\] {#proof-of-propositionfcu .unnumbered}
----------------------------
It suffices to show that there is one Dirac state, i.e. a state $\omega\in{\got S}(\al A.)$ with $\omega(\al U.)=1\;.$ Recall the homomorphism $\zeta:\al E.\to{\rm CAR}(\al H.)
\otimes{\rm CCR}(S,B)\;.$ Let $\omega_0\in{\got S}({\rm CAR}(\al H.))$ be that quasi–free state which is zero on any normal–ordered monomial of $a(f)$ and $a^*(h)$ of degree greater or equal to 1. Then $\omega_0$ is invariant w.r.t. $\wt\gamma_f$ for all $f\in\al S.(\R^4)\cup\R.$ Moreover, since $L(S)\subset\ker B,$ there is a state $\omega_1\in {\got S}({\rm CCR}(S,B))$ such that $\omega_1(\delta\s Lf.)=1$ for all $f\in S.$ Then $\omega_2:=\omega_0\otimes\omega_1\in
{\got S}\big({\rm CAR}(\al H.)\otimes{\rm CCR}(S,B)\big)$ is a $\wt\gamma\s f.\otimes\iota\hbox{--invariant}$ state on ${\rm CAR}(\al H.)\otimes{\rm CCR}(S,B)$ such that $\omega_2(\un\otimes\delta\s Lf.)=1$ for all $f\in S.$ From this we define a state on $\al E.$ by $\wt\omega_2:=\omega_2\circ\zeta$ and since $\zeta\circ\beta\s\gamma_f.=\wt\gamma\s f.\otimes\iota,$ it follows that $\wt\omega_2$ is $\beta\s G.\hbox{--invariant}$ on $\al E..$ Thus $\wt\omega_2$ extends to a state $\omega_3$ on $\al A.=G\cross\beta.\al E.$ by $\omega_3(U_g)=1$ for all $g\in G,$ where $U_g$ denotes the unitary implementer for $\beta_g.$ So $\omega_3\in{\got S}(\al A.)$ is a Dirac state w.r.t. the unitaries $U_G\cup\delta\s LS..$ Since the maximal set of constraint unitaries for a Dirac state is a group, it follows that for the products $V_f=\delta\s-Lf.\cdot U_{\gamma_f}$ we have $\omega_3(V_f)=1$ for all $f,$ i.e. $\omega_3$ is a Dirac state w.r.t. $\al U.,$ hence $\al U.$ is first–class.
Proof of Lemma \[RkWD\] {#proof-of-lemmarkwd .unnumbered}
-----------------------
Note that $\rho_k$ on the unitary implementers $\rho_k:U_G\to\al A.$ is a faithful group homomorphism. This is because it is the pointwise product of the identity map $\iota$ with the character $\chi_k:U_G\to\C$ given by $\chi\s k.\big(U\s\gamma_{f+t}.\big):=e^{itk},$ $t\in\R,$ $f\in\al S.(\R^4).$ Furthermore: $\al A.=C^*\big(\rho_k(U_G)\cup\al E.\big).$ Thus the pair ${\{\rho_k(U_G),\;\al E.\}}$ is also a covariant system for the action $\beta:G\to\aut\al E.$ (cf. Step 3), hence by the universal property of cross–products (cf. [@PR]) there is a \*-homomorphism $\theta:\al A.\to\al A.$ such that $\theta(A)=A$ for $A\in\al A.,$ and $\theta(U_g)=\rho_k(U_g)\equiv\hbox{implementing unitary of the second system}.$ Then $\theta$ coincides with the definition of $\rho_k$ on the generating elements, so it follows that $\rho_k$ extends uniquely to a homomorphism. Since it is clear that $\rho_k$ is bijective (its inverse is $\rho_{-k})$ it follows that $\rho_k$ is an automorphism of $\al A..$
Proof of Proposition \[RhOut\] {#proof-of-propositionrhout .unnumbered}
------------------------------
Proof by contradiction. Let $k\not=0$ and assume $\rho_k\in{\rm Inn}\,\al A.,$ i.e. $\rho_k=\Ad V$ for some unitary $V\in\al A.\,.$ Recall the homomorphism $$\zeta:\al E.\to
{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)$$ encountered in Step 3. Since $(S,\,B)$ is degenerate, $\zeta(\al E.)$ is not simple which will be inconvenient in the proof below. Choose therefore a maximal ideal $\al I.$ of ${\rm CCR}(S,\,B)$ (necessarily associated with a character of the centre $Z\big({\rm CCR}(S,\,B)\big)),$ and let $$\eta:{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\to
{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.$$ be the factorisation by the ideal $\un\otimes\al I.\,.$ Then the composition $$\xi:=\eta\circ\zeta :\al E.\to{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.$$ is a homomorphism of which the image is a simple algebra.Now the action $\beta:G\to\aut\al E.$ (Step 4) only affects ${\rm CAR}(\al H.)$ in $\al E.,$ so preserves the ideal generated by the commutators ${[{\rm CAR}(\al H.),\,{\rm CCR}(S,\,B)]}$ in $\al E.$ as well as the ideal $\un\otimes\al I.\,.$ Thus each $\beta_g$ can be taken through the homomorphism $\xi$ to define an action $\beta^\xi:G\to{\aut\big({\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.\big)}$ and it is just $\beta^\xi\s\gamma_f.=\wt\gamma\s f.\otimes \iota\,.$ Thus we can extend $\xi$ from $\al E.$ to $\al A.=
{G\cross\beta.\al E.}$ to get a surjective homomorphism $$\xi:\al A.\to G\cross\beta^\xi.\left({\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.\right)\;.$$ Now $\rho_k\in\aut\al A.$ only affects $U_G,$ leaving $\al E.$ invariant, hence it preserves $\ker\xi\subset\al A..$ Thus $\rho_k$ can be factored through $\xi$ to obtain the automorphisms $\rho^\xi_k\in\aut\xi(\al A.)$ by $$\label{XVinn}
\rho^\xi_k\big(\xi(A)\big):= \xi\big(\rho_k(A)\big)\quad\forall\;A\in\al A.\,,\qquad
\hbox{and so}\qquad \rho^\xi_k=\Ad\xi(V)\;.$$ Recall now that each element of the discrete crossed product $\xi(\al A.)= {G\cross\beta^\xi.\xi(\al E.)}$ (with $\xi(\al E.)={\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.)$ can be written as a C\*-norm convergent series $\sum\limits_{n=1}^\infty B_nU\s g_n.$ where $B_n\in\xi(\al E.)$ and $g_n\in G,$ (with $g_n$ distinct for different $n)$ and that the unitaries $U_G$ form a left $\xi(\al E.)\hbox{--module}$ basis. In particular, for the implementing unitaries $\xi(V)$ of $\rho^\xi_k$ we have a series $\xi(V)=\sum\limits_{n=1}^\infty B_nU\s g_n.\,,$ $B_n\in\xi(\al E.)\backslash 0.$ Since $\rho_k\restriction\al E.=\iota\,,$ it follows from equation (\[XVinn\]) that $\xi(V)A=A\xi(V)$ for all $A\in\xi(\al E.)\,,$ i.e. $$\begin{aligned}
A\xi(V)&=& \sum_{n=1}^\infty AB_nU\s g_n. \\[1mm]
&=&\xi(V)A=\sum_{n=1}^\infty B_nU\s g_n.A
=\sum_{n=1}^\infty B_n\beta^\xi\s g_n.(A)U\s g_n.\end{aligned}$$ for all $A\in\xi(\al E.)\,.$ So by the basis property of $U_G$ we have $$\label{BnIntertw}
AB_n=B_n\beta^\xi\s g_n.(A)\quad\forall\;A\in\xi(\al E.)=
{\rm CAR}(\al H.)\otimes{\rm CCR}(S,\,B)\big/\al I.\,.$$ Since $\beta^\xi\s g_n.\restriction{\rm CCR}(S,\,B)\big/\al I.
=\iota,$ this implies that $B_n\in\left({\rm CCR}(S,\,B)\big/\al I.\right)'.$ From the fact that ${\rm CCR}(S,\,B)\big/\al I.$ is simple (hence has trivial centre) this means that $B_n\in{\rm CAR}(\al H.)\otimes\un,$ and hence equation (\[BnIntertw\]) claims that $B_n$ is a nonzero intertwiner between $\iota$ and $\beta^\xi\s g_n.$ in ${\rm CAR}(\al H.).$ We next prove that $B_n$ is invertible, in which case $\beta^\xi\s g_n.$ becomes inner on ${\rm CAR}(\al H.).$
Let $\pi:{\rm CAR}(\al H.)\to\al B.(\al L.)$ be any faithful irreducible representation of ${\rm CAR}(\al H.)$ on a Hilbert space $\al L.$ e.g. the Fock representation), and let $\psi\in\ker\pi(B_n)\,.$ Then by (\[BnIntertw\]) $$\pi(B_nA)\psi=\pi(\beta\s g_n^{-1}.(A))\,\pi(B_n)\psi=0\qquad
\forall\;A\in{\rm CAR}(\al H.)\;.$$ Thus $\pi({\rm CAR}(\al H.))\psi\subseteq\ker\pi(B_n)\,.$ However, in an irreducible representation every nonzero vector is cyclic, so either $\psi=0$ or $\pi(B_n)=0,$ and the latter case is excluded by $B_n\not=0,$ $\pi$ faithful. Thus $\psi=0,$ i.e. we’ve shown that $\ker\pi(B_n)=\{0\}\,.$ Moreover by equation (\[BnIntertw\]) we have $$\pi(A)\pi(B_n)\varphi=\pi(B_n)\pi(\beta\s g_n.(A))\varphi
\quad\forall\;\varphi\in\al L.\backslash 0\,,\;A\in{\rm CAR}(\al H.)$$ hence $\pi\big({\rm CAR}(\al H.)\big)\big(\pi(B_n)\varphi\big)
\subseteq{\rm Ran}\,\pi(B_n)$ for all $\varphi\in\al L.\backslash 0\,.$ Now $\pi(B_n)\varphi\not=0$ (by $\ker\pi(B_n)=\{0\})$ and so by Dixmier 2.8.4 [@Di] we have that $\pi\big({\rm CAR}(\al H.)\big)\big(\pi(B_n)\varphi\big)=\al L.$ (no closure is necessary). Thus ${\rm Ran}\,\pi(B_n)=\al L.,$ i.e. $\pi(B_n)$ is invertible, and so since $\pi$ is faithful (hence preserves the spectrum of an element) it follows that $B_n$ is also invertible in ${\rm CAR}(\al H.).$
Using the fact that $B_n$ is invertible, equation (\[BnIntertw\]) becomes $\beta\s g_n.(A)=B_n^{-1}AB_n$ for all $A\in{\rm CAR}(\al H.)\,.$ Since $\beta\s g_n.$ is a \*-homomorphism, this implies that $B_n^{-1}A^*B_n=B_n^*A^*(B_n^{-1})^*,$ i.e. $B_nB_n^*A^*=A^*B_nB_n^*$ for all $A\in{\rm CAR}(\al H.)\,,$ and since ${\rm CAR}(\al H.)$ has trivial centre, this means $B_nB_n^*\in\C\un\,.$ Put $B_nB_n^*=:t_n$ (necessarily $t_n>0)$ then $U_n:=B_n\big/\sqrt{t_n}$ satisfies $U_nU_n^*=\un.$ By substituting $A$ by $\beta\s g_n^{-1}.(A)$ in (\[BnIntertw\]) we also obtain $B_n^*B_n\in\C\un$ by the above argument, then using $t_n=\|B_nB_n^*\|=\|B_n\|^2=\|B_n^*B_n\|=B_n^*B_n$ we get also $U_n^*U_n=\un\,.$ Thus $$\beta\s g_n.(A) =B_n^{-1}AB_n=\left({B_n\over\sqrt{t_n}}\right)^{-1}
A\left({B_n\over\sqrt{t_n}}\right)=U_n^*AU_n$$ for $A\in {\rm CAR}(\al H.)\,,$ i.e. $\beta\s g_n.$ is inner on ${\rm CAR}(\al H.)\,.$ Recall however, that on ${\rm CAR}(\al H.)$ $\beta\s g_n.$ is just an automorphism $\wt\gamma\s f_n.$ for some $f_n\in\al S.(\R^4)+\R,$ coming from a Bogoliubov transformation: $\wt\gamma\s f_n.\big(\Phi(k)\big):=\Phi(T_{f_n}k)$ (cf. Step 4). So for $\beta\s g_n.$ to be inner on ${\rm CAR}(\al H.),$ this means that either of $I\pm T_{f_n}$ must be trace–class (cf. Theorem 4.1, p48 of Araki [@Ar] or Theorem 4.1.4 in [@PlR]). However $$T_{f_n}(h_1\oplus h_2):=e^{-if_n}h_1\oplus e^{if_n}h_2\quad\forall\;
h_i\in\al H.=L^2(\R^4)\,.$$ Now for any $f_n$ such that $T_{f_n}\not=I\,,$ it is clear that the multiplication operators on $L^2(\R^4)$ by $(I\pm e^{\pm if_n})$ cannot be trace–class. This contradicts our finding that $\beta\s g_n.$ is inner if $g_n\not=e,$ hence only $g_n=e$ is possible in the series $\xi(V)=\sum\limits_{n=1}^\infty B_nU\s g_n.$ i.e. $\xi(V)= B\cdot U_e\,,$ $B\in\xi(\al E.)\backslash 0\,.$ But in this case equation (\[BnIntertw\]) becomes $AB=BA$ for all $A\in\xi(\al E.)$ and so since $\xi(\al E.)$ is simple, $B\in\C\un\,.$ This however implies that $\iota=\Ad\xi(V)=\rho^\xi_k$ which cannot be because $\rho_k(U\s\gamma_t.)=e^{ikt}U\s\gamma_t.$ factors unchanged through $\xi.$ From this contradiction, it follows that our initial assumption $\rho_k\in {\rm Inn}\,\al A.$ is false.
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---
abstract: 'We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_{\sigma}(D),\>\>\sigma>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_{\sigma}(D)$ to known sampling results for regular functions of exponential type $\sigma$.'
author:
- 'Isaac Z. Pesenson'
title: 'Sampling formulas for one-parameter groups of operators in Banach spaces'
---
[^1]
[**Keywords:**]{}[ One-parameter groups of operators, Exponential and Bernstein vectors, regular and irregular sampling theorems for entire functions of exponential type]{}
[**Subject classifications:**]{} \[2000\] [Primary: 47D03, 44A15; Secondary: 4705 ]{}
Introduction
============
The goal of the paper is to extend some theorems about sampling of entire functions of exponential type to Banach spaces. Our framework starts with considering a generator $D$ of one-parameter strongly continuous group of operators $e^{tD}$ in a Banach space $E$. The operator $D$ is used to define analogs of Bernstein subspaces $\mathbf{B}_{\sigma}(D)$. The main property of vectors $f$ in $\mathbf{B}_{\sigma}(D)$ is that corresponding trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This fact allows to apply known sampling theorems to every function of the form $\left<e^{tD}f, \>g^{*}\right>\in \mathbb{B}_{\sigma}^{\infty}(\mathbb{R}),\>\>\>g^{*}\in E^{*},$ or of the form $t^{-1}\left<e^{tD}f-f, \>g^{*}\right>\in \mathbb{B}_{\sigma}^{2}(\mathbb{R})$, where $ \mathbb{B}_{\sigma}^{\infty}(\mathbb{R}),\>\> \mathbb{B}_{\sigma}^{2}(\mathbb{R})$ are classical Bernstein spaces on $\mathbb{R}$.
In remark \[rem\] we demonstrate that the assumption that $D$ generates a group of operators is somewhat essential if one wants to have non-trivial Bernstein spaces. In section 2 we give a few different descriptions of these spaces and one of them explores resent results in [@BSS] which extend classical Boas formulas [@B1], [@B] for entire functions of exponential type. Note, that in turn, Boas formulas are generalizations of Riesz formulas [@R], [@R1] for trigonometric polynomials.
Section \[reg\] contains two sampling-type formulas which explore regularly spaced samples and section \[irreg\] is about two results in the spirit of irregular sampling. Section \[appl\] contains an application to inverse Cauchy problem for abstract Schrödinger equation.
Note, that if $e^{tD} ,\>\>\>t\in \mathbb{R}, $ is a group of operators in a Banach space $E$ then any trajectory $e^{tD}f,\>\>\>f\in E,$ is completely determined by any (single) sample $e^{\tau D}f,$ because for any $t\in \mathbb{R}$ $$e^{tD}f=e^{(t-\tau)D} \left(e^{\tau D}f\right).$$
Our results in sections \[reg\] and \[irreg\] have, however, a different nature. They represent a trajectory $e^{tD}f$ as a “linear combination” of a countable number of its samples. Such kind results can be useful when the entire group of operators $e^{tD}$ is unknown and only samples $e^{t_{k}D}f$ of a trajectory $e^{tD}f$ are given.
It seems to be very interesting that not matter how complicated one-parameter group can be (think, for example, about a Schrödinger operator $D=-\Delta+V(x)$ and the corresponding group $e^{itD}$ in $L_{2}(\mathbb{R}^{d})$) the formulas (\[s1\]), (\[l0\]), (\[s3\]), (\[l3000\]), (\[s4\]) are universal in the sense that they contain the same coefficients and the same sets of sampling points.
It was my discussions with Paul Butzer and Gerhard Schmeisser during Sampta 2013 in Jacobs University in Bremen of their beautiful work with Rudolf Stens [@BSS], that stimulated my interest in the topic of the present paper. I am very grateful to them for this.
Bernstein vectors in Banach spaces
==================================
We assume that $D$ is a generator of one-parameter group of isometries $e^{ tD}$ in a Banach space $E$ with the norm $\|\cdot
\|$ (for precise definitions see [@BB], [@K]). The notations $\mathcal{D}^{k}$ will be used for the domain of $D^{k}$, and notation $\mathcal{D}^{\infty}$ for $\bigcap_{k\in \mathbb{N}}\mathcal{D}^{k}$.
The subspace of exponential vectors $\mathbf{E}_{\sigma}(D), \>\>\sigma\geq 0,$ is defined as a set of all vectors $f$ in $\mathcal{D}^{\infty}$ for which there exists a constant $C(f,\sigma)>0$ such that $$\label{Exp}
\|D^{k}f\|\leq C(f,\sigma)\sigma^{k}, \>\>k\in \mathbb{N}.$$
Note, that every $\mathbf{E}_{\sigma}(D)$ is clearly a linear subspace of $E$. What is really important is the fact that union of all $\mathbf{E}_{\sigma}(D)$ is dense in $E$ (Corollary \[Density\]).
\[rem\] It is worth to stress that if $D$ generates a strongly continuous bounded semigroup then the set $\bigcup_{\sigma\geq 0}\mathbf{E}_{\sigma}(D)$ may not be dense in $E$.
Indeed, consider a strongly continuous bounded semigroup $T(t)$ in $L_{2}(0,\infty)$ defined for every $f\in L_{2}(0,\infty)$ as $T(t)f(x)=f(x-t),$ if $\>x\geq t$ and $T(t)f(x)=0,$ if $\>\>0\leq x<t$. Inequality (\[Exp\]) implies that if $ f\in \mathbf{E}_{\sigma}(D)$ then for any $g\in L_{2}(0,\infty)$ the function $\left<T(t)f,\>g\right>$ is analytic in $t$. Thus if $g$ has compact support then $\left<T(t)f,\>g\right>$ is zero for all $t$ which implies that $f$ is zero. In other words in this case every space $\mathbf{E}_{\sigma}(D)$ is trivial.
The Bernstein subspace $\mathbf{B}_{\sigma}(D), \>\>\sigma\geq 0,$ is defined as a set of all vectors $f$ in $E$ which belong to $\mathcal{D}^{\infty}$ and for which $$\label{Bernstein}
\|D^{k}f\|\leq \sigma^{k}\|f\|, \>\>k\in \mathbb{N}.$$
\[basic\] Let $D$ be a generator of an one parameter group of operators $e^{tD }$ in a Banach space $E$ and $\|e^{tD}f\|=\|f\|$. If for some $f\in E$ there exists an $\sigma
>0$ such that the quantity
$$\sup_{k\in N} \|D^{k}f\|\sigma ^{-k}=R(f,\sigma)$$ is finite, then $R(f,\sigma)\leq
\|f\|.$
By assumption $\|D^{r}f\|\leq R(f,\sigma)\sigma ^{r}, r\in \mathbb{N}$. Now for any complex number $z$ we have
$$\left\|e^{zD}f\right\|=\left\|\sum ^{\infty}_{r=0}(z^{r}D^{r}g)/r!\right\|\leq R(f,\sigma) \sum
^{\infty}_{r=0}|z|^{r}\sigma^{r}/r!=R(f,\sigma)e^{|z|\sigma}.$$
It implies that for any functional $h^{*}\in E^{*}$ the scalar function $\left<e^{zD}f,h^{*}\right>$ is an entire function of exponential type $\sigma $ which is bounded on the real axis $\mathbb{R}$ by the constant $\|h^{*}\| \|f\|$. An application of the Bernstein inequality gives
$$\left\|<e^{tD}D^{k}f,h^{*}>\right\|_{C(\mathbb{R})}=\left\|\left(\frac{d}{dt}\right)^{k}\left<e^{tD}f,h^{*}\right>\right\|_{
C(\mathbb{R})} \leq\sigma^{k}\|h^{*}\| \|f\|.$$
The last one gives for $t=0$
$$\left|\left<D^{k}f,h^{*}\right>\right|\leq \sigma ^{k} \|h^{*}\| \|f\|.$$
Choosing $h^{*}$ such that $\|h^{*}\|=1$ and $\left<D^{k}f,h^{*}\right>=\|D^{k}f\|$ we obtain the inequality $\|D^{k}f\|\leq
\sigma ^{k} \|f\|, k\in N$, which gives
$$R(f,\sigma)=\sup _{k\in \mathbb{N} }(\sigma ^{-k}\|D^{k}f\|)\leq \|f\|.$$
Lemma is proved.
\[t1\] Let $D$ be a generator of one-parameter group of operators $e^{tD }$ in a Banach space $E$ and $\|e^{tD}f\|=\|f\|$. Then for every $\sigma\geq 0$ $$\mathbf{B}_{\sigma}(D)= \mathbf{E}_{\sigma}(D) , \>\>\>\sigma\geq 0,$$
The inclusion $
\mathbf{B}_{\sigma}(D)= \mathbf{E}_{\sigma}(D) , \>\>\>\sigma\geq 0,
$ is obvious. The opposite inclusion follows from the previous Lemma.
Motivated by results in [@BSS] we introduce the following bounded operators
$$\label{b1}
\mathcal{B}_{D}^{(2m-1)}(\sigma)f=\left(\frac{\sigma}{\pi}\right)^{2m-1}\sum_{k\in \mathbb{Z}}(-1)^{k+1}A_{m,k}e^{\frac{\pi}{\sigma}(k-1/2)D}f,\>\> f\in E, \>\>\sigma>0,\>\>\>m\in \mathbb{N},$$
$$\label{b2}
\mathcal{B}_{D}^{(2m)}(\sigma)f=\left(\frac{\sigma}{\pi}\right)^{2m}\sum_{k\in \mathbb{Z}}(-1)^{k+1}B_{m,k}e^{\frac{\pi k}{\sigma}D}f, \>\>f\in E,\> \sigma>0,\>m\in \mathbb{N},$$
where $$\label{A}
A_{m,k}=(-1)^{k+1} \rm sinc ^{(2m-1)}\left(\frac{1}{2}-k\right)=
$$
$$
\frac{(2m-1)!}{\pi(k-\frac{1}{2})^{2m}}\sum_{j=0}^{m-1}\frac{(-1)^{j}}{(2j)!}\left(\pi(k-\frac{1}{2})\right)^{2j},\>\>\>m\in \mathbb{N},$$ for $k\in \mathbb{Z}$ and $$\label{B}
B_{m,k}=(-1)^{k+1} \rm sinc ^{(2m)}(-k)=\frac{(2m)!}{\pi k^{2m+1}}\sum_{j=0}^{m-1}\frac{(-1)^{j}(\pi k)^{2j+1}}{(2j+1)!},\>\>\>m\in \mathbb{N},\>\>\>k\in \mathbb{Z}\setminus {0},$$ and $$\label{B0}
B_{m,0}=(-1)^{m+1} \frac{\pi^{2m}}{2m+1},\>\>\>m\in \mathbb{N}.$$
Both series converge in $E$ due to the following formulas (see [@BSS])
$$\left(\frac{\sigma}{\pi}\right)^{2m-1}\sum_{k\in \mathbb{Z}}\left|A_{m,k}\right|=\sigma^{2m-1},\>\>\>\>\>
\left(\frac{\sigma}{\pi}\right)^{2m}\sum_{k\in \mathbb{Z}}\left|B_{m,k}\right|=\sigma^{2m}\label{id-2}.$$
Since $\|e^{tD}f\|=\|f\|$ it implies that
$$\label{norms}
\|\mathcal{B}_{D}^{(2m-1)}(\sigma)f\|\leq \sigma^{2m-1}\|f\|,\>\>\>\>\>\|\mathcal{B}_{D}^{(2m)}(\sigma)f\|\leq \sigma^{2m}\|f\|,\>\>\>f\in E.$$
For the following theorem see [@Pes00], [@Pes08], [@Pes11], [@Pes14].
If $D$ generates a one-parameter strongly continuous bounded group of operators $e^{tD}$ in a Banach space $E$ then the following conditions are equivalent:
1. $f$ belongs to $\mathbf{B}_{\sigma}(D)$.
2. The abstract-valued function $e^{tD}f$ is entire abstract-valued function of exponential type $\sigma$ which is bounded on the real line.
3. For every functional $g^{*}\in E^{*}$ the function $\left<e^{tD}f,\>g^{*}\right>$ is entire function of exponential type $\sigma$ which is bounded on the real line.
4. The following Boas-type interpolation formulas hold true for $r\in \mathbb{N}$
$$\label{B1}
D^{r}f=\mathcal{B}_{D}^{(r)}(\sigma)f,\>\>\>\>\>f\in \mathbf{B}_{\sigma}(D).$$
Every $\mathbf{B}_{\sigma}(D)$ is a closed linear subspace of $E$.
Let’s introduce the operator $\>\>
\Delta^{m}_{s}f=(I-e^{sD})^{m}f, \>\>m\in \mathbb{N},
$ and the modulus of continuity [@BB] $$\Omega_{m}(f,s)=\sup_{|\tau|\leq
s}\left\|\Delta^{m}_{\tau}f\right\|\label{dif}.$$
The following theorem is proved in [@Pes09], [@Pes11], [@Pes14].
There exists a constant $C>0$ such that for all $\sigma>0$ and all $f\in \mathcal{D}^{k}$ $$inf_{ g\in \mathbf{B}_{\sigma}(D) }\|f-g\|\leq
C\sigma^{-k}\Omega_{m-k}\left(D^{k}f, \sigma^{-1}\right),
0\leq k\leq m.\label{J}$$
\[Density\] The set $\bigcup_{\sigma\geq 0}\mathbf{B}_{\sigma}(D)$ is dense in $E$.
For a given $f\in E$ the notation $\sigma_{f}$ will be used for the smallest finite real number (if any) for which $$\|D^{k}f\|\leq \sigma_{f}^{k}\|f\|,\>\>\>k\in\mathbb{N}.$$ If there is no such finite number we assume that $\sigma_{f}=\infty$.
Now we are going to prove another characterization of Bernstein spaces. In the case of Hilbert spaces corresponding result was proved in [@Pes08a], [@PZ].
\[new\] Let $f\in E$ belongs to a space $\mathbf{B}_{\sigma}(D),$ for some $0<\sigma<\infty.$ Then the following limit exists $$d_f=\lim_{k\rightarrow \infty} \|D^k
f\|^{1/k} \label{limit}$$ and $d_f=\sigma_f.$
Conversely, if $f\in \mathcal{D}^{\infty}$ and $d_f=\lim_{k\rightarrow \infty}
\|D^k f\|^{1/k},$ exists and is finite, then $f\in\mathbf{B}_{d_f}(D)$ and $d_f=\sigma_f .$
Let us introduce the Favard constants (see [@Akh], Ch. V) which are defined as $$K_{j}=\frac{4}{\pi}\sum_{r=0}^{\infty}\frac{(-1)^{r(j+1)}}{(2r+1)^{j+1}},\>\>\>
j,\>\>r\in \mathbb{N}.$$ It is known [@Akh], Ch. V, that the sequence of all Favard constants with even indices is strictly increasing and belongs to the interval $[1,4/ \pi)$ and the sequence of all Favard constants with odd indices is strictly decreasing and belongs to the interval $(\pi/4, \pi/2],$ i.e., $$K_{2j}\in [1,4/ \pi), \; K_{2j+1}\in (\pi/4, \pi/2].\label{Fprop}$$ We will need the following generalization of the classical Kolmogorov inequality. It is worth noting that the inequality was first proved by Kolmogorov for $L^\infty (\mathbb{R})$ and later extended to $L^p(\mathbb{R})$ for $1\leq p < \infty$ by Stein [@Stein] and that is why it is known as the Stein-Kolmogorov inequality.
Let $f\in \mathcal{D}^{\infty} .$ Then, the following inequality holds $$\left\|D^{k}f\right\|^n \leq C_{k,n}\|D^{n}f\|^{k}\|f\|^{n-k},\>\>\>
0\leq k \leq n,\label{KS}$$ where $C_{k,n}= (K_{n-k})^n/(K_{n})^{n-k}.$ \[LKS\]
Indeed, for any $h^{*}\in E^{*}$ the Kolmogorov inequality [@Stein] applied to the entire function $\left<e^{tD}f,h^{*}\right>$ gives $$\left\|\left(\frac{d}{dt}\right)^{k}\left<e^{tD}f,h^{*}\right>\right\|^n_{
C(\mathbb{R}^{1})}\leq
C_{k,n}\left\|\left(\frac{d}{dt}\right)^{n}\left<e^{tD}f,h^{*}\right>\right\|_{
C(\mathbb{R}^{1})}^{k}\times$$ $$\left\|\left<e^{tD}f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{n-k},\quad
0<k< n,$$ or $$\left\|\left<e^{tD}D^{k}f,h^{*}\right>\right\|_{ C(\mathbb{R}^{1})}^n\leq
C_{k,n}\left\|\left<e^{tD}D^{n}f,h^{*}\right>\right\|_{
C(\mathbb{R}^{1})}^{k}\left\|\left<e^{tD}f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{n-k}.$$ Applying the Schwartz inequality to the right-hand side, we obtain $$\begin{aligned}
\left\|\left<e^{tD}D^{k}f,h^{*}\right>\right\|_{ C(\mathbb{R}^{1})}^n & \leq
C_{k,n}\|h^{*}\|^{k}\|D^{n}f\|^{k}\|h^{*}\|^{n-k}\|f\|^{n-k}\\
&\leq C_{k,n}\|h^{*}\|^n \|D^{n}f\|^{k}\|f\|^{n-k},\end{aligned}$$ which, when $t=0,$ yields $$\left|\left<D^{k}f,h^{*}\right>\right|^n\leq C_{k,n}\|h^{*}\|^n
\|D^{n}f\|^{k}\|f\|^{n-k}.$$ By choosing $h$ such that $\left|<D^{k}f,h^{*}>\right|=\|D^{k}f\|$ and $\|h^{*}\|=1$ we obtain (\[KS\]).
[**Proof of Theorem \[new\].**]{}
From Lemma \[LKS\] we have $$\left\|D^{k}f\right\|^n \leq
C_{k,n}\|D^{n}f\|^{k}\|f\|^{n-k}, \quad 0\leq k \leq n.$$ Without loss of generality, let us assume that $\|f\|=1.$ Thus, $$\left\|D^{k}f\right\|^{1/k} \leq
(\pi/2)^{1/kn}\|D^{n}f\|^{1/n}, \quad 0\leq k \leq n.$$ Let $k$ be arbitrary but fixed. It follows that $$\left\|D^{k}f\right\|^{1/k} \leq
(\pi/2)^{1/kn}\|D^{n}f\|^{1/n}, \mbox{ for all } n\geq k,$$ which implies that $$\left\|D^{k}f\right\|^{1/k}\leq \underline{\lim}_{n\rightarrow
\infty}\|D^{n}f\|^{1/n}.$$ But since this inequality is true for all $k>0,$ we obtain that $$\overline{\lim}_{k\rightarrow\infty}\|D^{k}f\|^{1/k}\leq
\underline{\lim}_{n\rightarrow \infty}\|D^{n}f\|^{1/n},$$ which proves that $d_f=\lim_{k\rightarrow}\|D^{k}f\|^{1/k}$ exists.
Since $f\in \mathbf{B}_{\sigma}(D) $ the constant $\sigma_{f}$ is finite and we have $$\|D^{k}f\|^{1/k}\leq \sigma_f \|f\|^{1/k},$$ and by taking the limit as $k\rightarrow\infty$ we obtain $d_f\leq \sigma_f.$ To show that $d_f= \sigma_f,$ let us assume that $d_f< \sigma_f.$ Therefore, there exist $M>0$ and $\sigma $ such that $0<d_f<\sigma
< \sigma_f$ and $$\|D^k f\|\leq M \sigma^k, \quad \mbox{for all }
k>0 .$$ Thus, by Lemma \[basic\] we have $f\in \mathbf{B}_\sigma(D) ,$ which is a contradiction to the definition of $\sigma_f.$
Conversely, suppose that $d_f=\lim_{k\rightarrow \infty} \|D^k
f\|^{1/k}$ exists and is finite. Therefore, there exist $M>0$ and $\sigma
>0$ such that $d_f<\sigma $ and $$\|D^k f\|\leq M \sigma^k, \quad \mbox{for all } k>0 ,$$ which, in view of Lemma \[basic\], implies that $f\in
\mathbf{B}_{\sigma}(D) .$ Now by repeating the argument in the first part of the proof we obtain $d_f=\sigma_f ,$ where $\sigma_f=\inf\left\{
\sigma : f\in \mathbf{B}_{\sigma}(D)\right\}.$
Theorem \[new\] is proved.
Now consider the following abstract Cauchy problem for the operator $D$. $$\frac{d u(t)}{d t}=Du(t), \>\>u(0)=f,\label{Cauchy2}$$ where $u: \mathbb{R} \rightarrow E$ is an abstract function with values in $E.$ Since solutions of this problem given by the formula $u(t)=e^{tD}f$ we obtain the following result.
A vector $f\in E,$ belongs to $\mathbf{B}_{\sigma}(D)$ if and only if the solution $u(t)$ of the corresponding Cauchy problem (\[Cauchy2\]) has the following properties:
1\) as a function of $t,$ it has an analytic extension $u(z), z\in
\mathbb{C}$ to the complex plane $\mathbb{C}$ as an entire function;
2\) it has exponential type $\sigma$ in the variable $z$, that is $$\|u(z)\|_{E}\leq e^{\sigma|z|}\|f\|_{E}.$$ \[MainThm\] and it is bounded on the real line. \[PW2\]
Sampling-type formulas for one-parameter groups {#reg}
===============================================
We assume that $D$ generates one-parameter strongly cononuous bounded group of operators $e^{tD}, \>\>t\in \mathbb{R},$ in a Banach space $E$. In this section we prove explicit formulas for a trajectory $e^{tD}f$ with $f\in \mathbf{B}_{\sigma}(D)$ in terms of a countable number of equally spaced samples.
If $f\in \mathbf{B}_{\sigma}(D)$ then the following sampling formulas hold for $t\in \mathbb{R}$ $$\label{s1}
e^{tD}f=f+tDf \rm sinc\left(\frac{\sigma t}{\pi}\right)+t\sum_{k\neq 0}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right),$$
$$\label{l0}
f=e^{tD}f-t \left(e^{t D}Df\right)\rm sinc\left(\frac{\sigma t}{\pi}\right)-t\sum_{k\neq 0}\frac{e^{\left( \frac{k\pi}{\sigma}+t \right)D}f-e^{tD}f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}+k\right).$$
It is worth to note that if $\>\>\>t\neq 0,\>\>$ then right-hand side of (\[l0\]) does not contain vector $f$ and we obtain a “linear combination” of $f$ in terms of vectors $e^{\left( \frac{k\pi}{\sigma}+t \right)D}f,\>\>\>k\in \mathbb{Z},$ and $e^{tD}Df$.
If $f\in \mathbf{B}_{\sigma}(D)$ then for any $g^{*}\in E^{*}$ the function $F(t)=\left<e^{tD}f,\>g^{*}\right>$ belongs to $B_{\sigma}^{\infty}(\mathbb{R})$.
We consider $F_{1}\in B_{\sigma}^{2}( \mathbb{R}),$ which is defined as follows. If $t\neq 0$ then $$F_{1}(t)=\frac{F(t)-F(0)}{t}=\left<\frac{e^{tD}f-f}{t},\>g^{*}\right>,$$ and if $t=0$ then $
F_{1}(t)=\frac{d}{dt}F(t)|_{t=0}=\left<Df,\>g^{*}\right>.
$ We have $$F_{1}(t)=\sum_{k}F_{1}\left(\frac{k\pi}{\sigma}\right)\> \rm sinc\left(\frac{\sigma t}{\pi}-k\right),$$ which means that for any $g^{*}\in E^{*}$ $$\left< \frac{e^{tD}f-f}{t},\>g^{*} \right>=\sum_{k}\left<\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}},\>g^{*}\right>\> \rm sinc\left(\frac{\sigma t}{\pi}-k\right).$$ Since $$\rm sinc ^{(n)} x=\sum_{j=0}^{n}C_{n}^{j}(\sin\>\pi x)^{(j)}\left(\frac{1}{\pi x}\right)^{(n-j)}=$$ $$\frac{(-1)^{n}n!}{\pi x^{n+1}}\sum_{j=0}^{n}\sin\left(\pi x+\frac{jx}{2}\right)\frac{(-1)^{j}(\pi x)^{j}}{j!}$$ one has the estimate $$|\rm sinc^{(n)}\>x|\leq \frac{C}{|x|},\>\>\>n=0,1,....$$ which implies convergence in $E$ of the series $$\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right).$$ It leads to the equality for any $g^{*}\in E^{*}$ $$\left< \frac{e^{tD}f-f}{t},\>g^{*} \right>=\left<\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right),\>g^{*}\right>,\>\>\>t\neq 0,$$ and if $t= 0$ it gives the identity $
\left<Df,\>g^{*}\right>=\left<Df,\>g^{*}\right>\sum_{k} \rm sinc \>k.
$ Thus, $$\frac{e^{tD}f-f}{t}=\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right),\>\>\>t\neq 0,$$ or for every $t\in \mathbb{R}$ $$e^{tD}f=f+tDf \rm sinc\left(\frac{\sigma t}{\pi}\right)+t\sum_{k\neq 0}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right).$$
Thus, (\[s1\]) is proved.
If in (\[s1\]) we replace $f$ by $\left(e^{\tau D}f\right)$ for a $\tau\in \mathbb{R}$ we will have
$$\label{s100}
e^{tD}\left(e^{\tau D}f\right)=
$$
$$
\left(e^{\tau D}f\right)+t\sum_{k\neq 0}\frac{e^{\frac{k\pi}{\sigma}D}\left(e^{\tau D}f\right)-\left(e^{\tau D}f\right)}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma t}{\pi}-k\right)+tD\left(e^{\tau D}f\right)\rm sinc\left(\frac{\sigma t}{\pi}\right).$$
For $t=-\tau $ we obtain the next formula which holds for any $\tau\in \mathbb{R},\>\>f\in \mathbf{B}_{\sigma}(D),$ $$\label{l1}
f=e^{\tau D}f-\tau \sum_{k\neq 0}\frac{e^{\left( \frac{k\pi}{\sigma}+\tau \right)D}f-e^{\tau D}f}{\frac{k\pi}{\sigma}} \rm sinc\left(\frac{\sigma \tau}{\pi}+k\right)-\tau D\left(e^{\tau D}f\right)\rm sinc\left(\frac{\sigma \tau}{\pi}\right),$$ which is the formula (\[l0\]). Theorem is proved.
The next Theorem is a generalization of what is known as Valiron-Tschakaloff sampling/interpolation formula [@BFHSS].
For $f\in \mathbf{B}_{\sigma}(D),\>\>\>\sigma>0,$ we have for all $z \in \mathbb{C}$ $$\label{VT}
e^{zD}f=z \>\rm sinc\left(\frac{\sigma z}{\pi}\right)Df+\rm sinc\left(\frac{\sigma z}{\pi}\right)f+\sum_{k\neq 0}\frac{\sigma z}{k\pi}\rm sinc\left(\frac{\sigma z}{\pi}-k\right)e^{\frac{k\pi}{\sigma}D}f$$
If $F\in \mathbf{B}_{\sigma}(D),\>\>\>\sigma>0,$ then for all $z \in \mathbb{C}$ the following Valiron-Tschakaloff sampling/interpolation formula holds [@BFHSS]
$$F(z)=z \>\rm sinc\left(\frac{\sigma z}{\pi}\right)F^{'}(0)+\rm sinc\left(\frac{\sigma z}{\pi}\right)F(0)+\sum_{k\neq 0}\frac{\sigma z}{k\pi}\rm sinc\left(\frac{\sigma z}{\pi}-k\right)F\left(\frac{k\pi}{\sigma}\right)$$
For $f\in \mathbf{B}_{\sigma}(D),\>\>\>\sigma>0$ and $g^{*}\in E^{*}$ we have $F(z)=\left<e^{zD}f,\>g^{*}\right>\in \mathbf{B}_{\sigma}(D)$ and thus
$$\left<e^{zD}f,\>g^{*}\right>=z \>\rm sinc\left(\frac{\sigma z}{\pi}\right)\left<Df,\>g^{*}\right>+
$$
$$
\rm sinc\left(\frac{\sigma z}{\pi}\right)\left<f,\>g^{*}\right>+\sum_{k\neq 0}\frac{\sigma z}{k\pi}\rm sinc\left(\frac{\sigma z}{\pi}-k\right)\left<e^{\frac{k\pi}{\sigma}D}f,\>g^{*}\right>.$$
Since the series $$\sum_{k\neq 0}\frac{\sigma z}{k\pi}\rm sinc\left(\frac{\sigma z}{\pi}-k\right)e^{\frac{k\pi}{\sigma}D}f$$ converges in $E$ for every fixed $z$ we obtain the formula (\[VT\]).
If $f\in \mathbf{B}_{\sigma}(D)$ then the following sampling formula holds for $t\in \mathbb{R}$ and $n\in \mathbb{N}$ $$\label{s2}
e^{tD}D^{n}f=\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}}\left\{ n \>\rm sinc^{(n-1)}\left(\frac{\sigma t}{\pi}-k\right) +\frac{\sigma t}{\pi} \rm sinc^{(n)}\left(\frac{\sigma t}{\pi}-k\right)
\right\}.$$ In particular, for $n\in \mathbb{N}$ one has $$\label{Q}
D^{n}f=\mathcal{Q}_{D}^{n}(\sigma)f,$$ where the bounded operator $\mathcal{Q}_{D}^{n}(\sigma)$ is given by the formula $$\mathcal{Q}_{D}^{n}(\sigma)f=n\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}}\left[ \rm sinc^{(n-1)}\left(-k\right) +\rm sinc^{(n)}\left(-k\right)
\right].$$
Because $F_{1}\in B_{\sigma}^{2}(\mathbb{R})$ we have $$\left(\frac{d}{dt}\right)^{n}F_{1}(t)=\sum_{k}F_{1}\left(\frac{k\pi}{\sigma} \right) \rm sinc^{(n)}\left(\frac{\sigma t}{\pi}-k\right)$$ and since $$\left(\frac{d}{dt}\right)^{n}F(t)=n\left(\frac{d}{dt}\right)^{n-1}F_{1}(t)+t\left(\frac{d}{dt}\right)^{n}F_{1}(t)$$ we obtain $$\left(\frac{d}{dt}\right)^{n}F(t)=n\sum_{k}F_{1}\left(\frac{k\pi}{\sigma}\right)\> \rm sinc^{(n-1)}\left(\frac{\sigma t}{\pi}-k\right)+\frac{\sigma t}{\pi}\sum_{k}F_{1}\left(\frac{k\pi}{\sigma}\right)\> \rm sinc^{(n)}\left(\frac{\sigma t}{\pi}-k\right)$$ Because $
\left(\frac{d}{dt}\right)^{n}F(t)=\left<D^{n}e^{tD}f, g^{*}\right>,
$ and $$F_{1}\left(\frac{k\pi}{\sigma}\right)=\left<\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}}, g^{*}\right>$$ we obtain that for $t\in \mathbb{R},\>\>n\in \mathbb{N},$ $$D^{n}e^{tD}f=\sum_{k}\frac{e^{\frac{k\pi}{\sigma}D}f-f}{\frac{k\pi}{\sigma}}\left[ n\> \rm sinc^{(n-1)}\left(\frac{\sigma t}{\pi}-k\right) +\frac{\sigma t}{\pi} \rm sinc^{(n)}\left(\frac{\sigma t}{\pi}-k\right)
\right].$$ Theorem is proved.
Irregular sampling theorems {#irreg}
===========================
In [@Hig] the following fact was proved.
\[Hig\] Let $\{t_{n}\}_{n\in \mathbb{Z}}$ be a sequence of real numbers such that $$\sup_{n\in\mathbb{Z}}|t_{n}-n|<1/4.$$ Define the entire function $$G(z)=(z-t_{0})\prod_{n=1}^{\infty}\left(1-\frac{z}{t_{n}}\right)\left(1-\frac{z}{t_{-n}}\right).$$ Then for all $f\in \mathbf{B}_{\pi}^{2}(\mathbb{R})$ we have $$f(t)=\sum_{n\in \mathbb{Z}}f(t_{n})\frac{G(t)}{G^{'}(t_{n})(t-t_{n})}$$ uniformly on all compact subsets of $\mathbb{R}$.
An analog of this result for Banach spaces is the following.
If $D$ generates a one-parameter strongly continuous group $e^{tD}$ of isometries in a Banach space $E$. Suppose that assumptions of Theorem \[Hig\] are satisfied and $t_{0}\neq 0$. Then for all $f\in \mathbf{B}_{\pi}(D),\>\>g^{*}\in E^{*}$ and every $t\in \mathbb{R}$ the following formulas hold $$\label{s3}
\left<e^{tD}f,\>g^{*}\right>=
\left<f,\>g^{*}\right>+t\sum_{n\in \mathbb{Z}}\frac{\left<e^{t_{n}D}f,\>g^{*}\right>-\left<f,\>g^{*}\right>}{t_{n}}\frac{G(t)}{G^{'}(t_{n})(t-t_{n})},$$ and $$\label{l3000}
\left<f,g^{*}\right>=\left<e^{t D}f,\>g^{*}\right>+t\sum_{n\in \mathbb{Z}}\frac{\left<e^{(t_{n}-t)D}f,\>g^{*}\right>-\left<e^{t D}f,\>g^{*}\right>}{t_{n}}\frac{G(-t)}{G^{'}(t_{n})(t+t_{n})}$$ uniformly on all compact subsets of $\mathbb{R}$.
The last formula represents a “measurement” $\left<f,\>g^{*}\right>$ through “measurements” $\left<e^{(t_{n}-t)D}f,\>g^{*}\right>$ and $\left<e^{t D}f,\>g^{*}\right>$ which are different from $\left<f,\>g^{*}\right>$.
If $f\in \mathbf{B}_{\sigma}(D)$ then for any $g^{*}\in E^{*}$ the function $F(t)=\left<e^{tD}f,\>g^{*}\right>$ belongs to $B_{\sigma}^{\infty}(\mathbb{R})$. We consider $F_{1}\in B_{\sigma}^{2}( \mathbb{R}),$ which is defined as follows. If $t\neq 0$ then $$F_{1}(t)=\frac{F(t)-F(0)}{t}=\left<\frac{e^{tD}f-f}{t},\>g^{*}\right>,\>\>g^{*}\in E^{*},$$ and if $t=0$ then $$F_{1}(t)=\frac{d}{dt}F(t)|_{t=0}=\left<Df,\>g^{*}\right>.$$ We have $$F_{1}(t)=\sum_{n\in \mathbb{Z}}F_{1}(t_{n})\frac{G(t)}{G^{'}(t_{n})(t-t_{n})}$$ or $$\left<\frac{e^{tD}f-f}{t},\>g^{*}\right>=\sum_{n\in \mathbb{Z}}\left<\frac{e^{t_{n}D}f-f}{t_{n}},\>g^{*}\right>\frac{G(t)}{G^{'}(t_{n})(t-t_{n})}$$ uniformly on all compact subsets of $\mathbb{R}$.
If we pick a non-zero $\tau$ such that $\tau \neq t_{n}$ for all $t_{n}$ and set $f$ in (\[s3\]) to $e^{\tau D}f$ then for $t=-\tau$ we will have the following formula which does not have vector $f$ on the right-hand side $$\label{l3}
\left<f,g^{*}\right>=\left<e^{\tau D}f,\>g^{*}\right>+\tau\sum_{n\in \mathbb{Z}}\frac{\left<e^{(t_{n}-\tau)D}f,\>g^{*}\right>-\left<e^{\tau D}f,\>g^{*}\right>}{t_{n}}\frac{G(-\tau)}{G^{'}(t_{n})(\tau+t_{n})}.$$ Theorem is proved.
In [@S] the following result can be found.
\[S\] Let $\{t_{n}\}_{n\in \mathbb{Z}}$ be a sequence of real numbers such that $$\sup_{n\in\mathbb{Z}}|t_{n}-n|<1/4.$$ Define the entire function $$G(z)=(z-t_{0})\prod_{n=1}^{\infty}\left(1-\frac{z}{t_{n}}\right)\left(1-\frac{z}{t_{-n}}\right).$$ Let $\delta$ be any positive number such that $0<\delta<\pi$. Then for all $f\in \mathbf{B}_{\pi-\delta}(\mathbb{R})$ we have $$f(z)=\sum_{n\in \mathbb{Z}}f(t_{n})\frac{G(z)}{G^{'}(t_{n})(z-t_{n})}$$ uniformly on all compact subsets of $\mathbb{C}$.
From here we obtain the next fact.
\[irregth\] Suppose that $D$ generates a one-parameter strongly continuous group $e^{tD}$ of isometries in a Banach space $E$. Then in the same notations as in Theorem \[S\] one has that for all $f\in \mathbf{B}_{\pi-\delta}(D)$ and all $g^{*}\in E^{*}$ the following formula holds $$\label{s4}
\left<e^{zD}f, g^{*}\right>=\sum_{n\in \mathbb{Z}}\left<e^{t_{n}D}f,\>g^{*}\right>\frac{G(z)}{G^{'}(t_{n})(z-t_{n})}$$ uniformly on all compact subsets of $\mathbb{C}$.
Proof follows from Theorem \[S\] since for any $g^{*}\in E^{*}$ the function $ \left<e^{zD}f, g^{*}\right>$ belongs to $ \mathbf{B}_{\pi-\delta}(\mathbb{R})$.
Note that if in the last formula we will set $z$ to zero and assume that $t_{0}\neq 0$ we will have a representation of $\left<f,\>g^{*}\right>$ in terms of samples $\left<e^{t_{n}D}f,\>g^{*}\right>\neq \left<f,\>g^{*}\right>$ i.e. $$\label{l2}
\left<f, g^{*}\right>= -\sum_{n\in \mathbb{Z}}\left<e^{t_{n}D}f,\>g^{*}\right>\frac{G(0)}{G^{'}(t_{n})t_{n}}.$$
An application to abstract Schrödinger equation {#appl}
===============================================
We now assume that $E$ is a Hilbert space and $D$ is a selfadjoint operator. Then $e^{itD}$ is one-parameter group of isometries of $E$. By the spectral theory [@BS], there exist a direct integral of Hilbert spaces $A=\int
A(\lambda )dm (\lambda )$ and a unitary operator $\mathcal{F}_{D}$ from $E$ onto $A$, which transforms the domain $\mathcal{D}_{k}$ of the operator $ D^{k}$ onto $A_{k}=\{a\in A|\lambda^{k}a\in A
\}$ with norm
$$\|a(\lambda )\|_{A_{k}}= \left (\int^{\infty}_{-\infty} \lambda ^{2k}
\|a(\lambda )\|^{2}_{A(\lambda )} dm(\lambda ) \right )^{1/2}$$ and $\mathcal{F}_{D}(Df)=\lambda (\mathcal{F}_{D}f), f\in
\mathcal{D}_{1}. $
In this situation one can prove [@Pes00] the following.
A vector $f\in E$ belongs to $ \mathbf{B}_{\sigma}(D)$ if and only if support of $\mathcal{F}_{D}f$ is in $[-\sigma, \>\sigma]$.
We consider an abstract Cauchy problem for a self-adjoint operator $D$ which consists of funding an abstract-valued function $u: \mathbf{R}\rightarrow E$ which satisfies Schrödinger equation and has bandlimited initial condition $$\label{C1}
\frac{du(t)}{dt}=iDu(t),\>\>\>u(0)=f\in \mathbf{B}_{\sigma}(D)$$ (see [@BB], [@K] for more details).
In this case formula (\[l2\]) can be treated as a solution to inverse problem associated with (\[C1\]).
If conditions of Theorem \[irregth\] are satisfied and $t_{0}\neq 0$ then initial condition $f\in \mathbf{B}_{\pi-\delta}(D),\>\>0<\delta<\pi,$ in (\[C1\]) can be reconstructed (in weak sense) from the values of the solution $u(t_{n})$ by using the formula $$\left<f, g^{*}\right>=-\sum_{n\in \mathbb{Z}}\left<u(t_{n}), g^{*}\right>\frac{G(0)}{G^{'}(t_{n})t_{n}},\>\>\>g^{*}\in E^{*}.$$
Similar results can be formulated by using formulas (\[l1\]) and (\[l3\]).
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[^1]: Department of Mathematics, Temple University, Philadelphia, PA 19122; [email protected]
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abstract: 'We investigate the distribution of large positive (and negative) values of the Euler-Kronecker constant ${\gamma_{\mathbb{Q}(\sqrt D)}}$ of the quadratic field $\mathbb{Q}(\sqrt{D})$ as $D$ varies over fundamental discriminants $|D|\leq x$. We show that the distribution function of these values is very well approximated by that of an adequate probabilistic random model in a large uniform range. The main tools are an asymptotic formula for the Laplace transform of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ together with a careful saddle point analysis.'
address: 'Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3 Canada'
author:
- Youness Lamzouri
title: 'The distribution of Euler-Kronecker constants of quadratic fields'
---
[^1]
Introduction
============
Let $K$ be an algebraic number field, $\mathcal{O}_K$ be its ring of integers and $N(\mathfrak{a})$ denote the norm of an ideal $\mathfrak{a}$ in $\mathcal{O}_K$. The Dedekind zeta function of $K$ is defined for ${\textup{Re}}(s)>1$ by $$\zeta_K(s)=\sum_{\mathfrak{a}}\frac{1}{N(\mathfrak{a})^s}= \prod_{\mathfrak{p}}\left(1-\frac{1}{N(\mathfrak{p})^s}\right)^{-1},$$ where $\mathfrak{a}$ ranges over non-zero ideals and $\mathfrak{p}$ ranges over the prime ideals in $\mathcal{O}_K$. It is known that $\zeta_K(s)$ has an analytic continuation to $\mathbb{C}\setminus\{1\}$ and a simple pole at $s=1$ with residue $\alpha_K$. The well-known class number formula relates $\alpha_K$ to several algebraic invariants of $K$, including the discriminant, class number and regulator of $K$.
The *Euler-Kronecker* constant (or invariant) of $K$ is defined by $$\gamma_K=\lim_{s\to 1} \left(\frac{\zeta'_K(s)}{\zeta_K(s)}+\frac{1}{s-1}\right).$$ Moreover, if the Laurent series expansion of $\zeta_K(s)$ is $$\zeta_K(s)= \frac{\alpha_K}{s-1} + c_0(K)+ c_1(K) (s-1)+ c_2(K)(s-1)^2\cdots,$$ then $\gamma_K=c_0(K)/\alpha_K.$ Note that when $K=\mathbb{Q}$, we have $\gamma_{K}=\gamma$, where $\gamma=0.577...$ is the Euler-Mascheroni constant.
The Euler-Kronecker constant was first introduced and studied by Ihara in [@Ih1] and [@Ih2]. In particular, Ihara proved in [@Ih1] that if $d_K$ is the discriminant of $K$ then $$-\frac12\log {|d_k|}\leq \gamma_K\leq 2\log\log |d_K|,$$ where the upper bound is conditional on the Generalized Riemann hypothesis GRH. Tsafsman [@Ts] showed that the lower bound is optimal up to a constant, and hence that the maximal order of $|\gamma_K|$ is $\asymp \log |d_K|.$ However, Ihara [@Ih1] proved that this order is much smaller if the degree of $K$ is small.
When $K$ is the cyclotomic field $K(q):=\mathbb{Q}\big(e^{2\pi i/q}\big)$, Ihara [@Ih1] showed that $\gamma_{K(q)}=O(\log^2 q)$ assuming GRH, and this bound was improved to $O(\log q\log\log q)$ by Badzyan [@Ba]. Murty [@Mu] proved an upper bound for the first moment of $\gamma_{K(q)}$, which was refined to an asymptotic formula by Fouvry [@Fo], who showed that the average order of $\gamma_{K(q)}$ is $\log Q$. In the case where $q$ is prime, Ford, Luca and Moree [@FLM] studied $\gamma_{K(q)}$ and showed that it appears in the asymptotic expansion of the number of integers $n\leq x$ for which $\varphi(n)$ is not divisible by $q$, where $\varphi$ is the Euler $\varphi$-function.
In the special case where $K=\mathbb{Q}(\sqrt{D})$ is a quadratic field, we know that the corresponding Dedekind zeta function factorizes as $\zeta_K(s)=\zeta(s) L(s,\chi_D)$, where $\chi_D(n)=(D/n)$ is the Kronecker symbol. Therefore $${\gamma_{\mathbb{Q}(\sqrt D)}}= \gamma+ \frac{L'(1,\chi_D)}{L(1,\chi_D)}.$$ When $\mathbb{Q}(\sqrt{D})$ is imaginary, the Kronecker limit formula expresses ${\gamma_{\mathbb{Q}(\sqrt D)}}$ in terms of special values of the Dedekind $\eta$-function (see Section 2.2 of [@Ih1]).
In [@Ih1], Ihara proved that under GRH we have $$|{\gamma_{\mathbb{Q}(\sqrt D)}}|\leq (2+o(1))\log\log |D|.$$ Using a zero density result of Heath-Brown [@HB], we show in Corollary \[ASBound\] below that this bound is attained for almost all fundamental discriminants. More precisely, we prove that for all but at most $O(x^{\epsilon})$ fundamental discriminants $D$ with $|D|\leq x$ we have $${\gamma_{\mathbb{Q}(\sqrt D)}}\ll_{\epsilon}\log\log |D|.$$ On the other hand, Mourtada and Murty [@MoMu] proved that there are infinitely many $D$ for which $$\pm {\gamma_{\mathbb{Q}(\sqrt D)}}\geq \log\log |D| + O(1).$$ They also showed that this bound can be improved to $\log\log |D|+\log\log\log |D|+O(1)$ under GRH.
In analogy to $L(1,\chi_D)$, we expect that for all fundamental discriminants $D$ with $|D|\leq x$ we have $$\label{TrueRange}
|{\gamma_{\mathbb{Q}(\sqrt D)}}| \leq \log\log x +\log\log\log x+O(1),$$ so that the true order of extreme values of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ is closer to the omega results of Mourtada-Murty rather than the conditional $O$-result of Ihara. Our Theorem \[AsympDistrib\] below gives strong support for this conjecture (see Remark \[Support\] below).
To investigate the distribution of the Euler-Kronecker constant ${\gamma_{\mathbb{Q}(\sqrt D)}}$, our strategy consists in constructing an adequate probabilistic random model for these values. Let $\{X(p)\}_{p \text{ prime}}$ be a sequence of independent random variables, indexed by the primes, and taking the values $1, -1$ and $0$ with the following probabilities $${\mathbb{P}}(X(p)=a)= \begin{cases} \frac{p}{2(p+1)} & \text{ if } a=\pm 1,\\
\frac{1}{p+1} & \text{ if } a=0.\\
\end{cases}$$ We extend the $X(p)$ multiplicatively to all positive integers by setting $X(1)=1$ and $ X(n):= X(p_1)^{a_1}\cdots X(p_k)^{a_k}, $ if $n= p_1^{a_1}\cdots p_k^{a_k}.$ These random variables were first introduced by Granville and Soundararajan [@GrSo] to study the distribution of $L(1,\chi_D)$. The reason for this choice over the simpler $\pm 1$ with probability $1/2$ is that for odd primes $p$, fundamental discriminants $D$ lie in one of $p^2-1$ residue classes mod $p^2$ so that $\chi_D(p)=0$ for $p-1$ of these classes, and the remaining $p(p-1)$ residue classes split equally into $\pm 1$ values (for $p=2$ one can check that the values $0,\pm 1$ occur equally often). We shall compare the distribution of ${\gamma_{\mathbb{Q}(\sqrt D)}}$, as $D$ varies among fundamental discriminants $|D|\leq x$, to that of the following probabilistic random model: $${\gamma_{\textup{rand}}(X)}:= \gamma-\sum_{n=1}^{\infty} \frac{\Lambda(n) X(n)}{n}=\gamma-\sum_{p}\frac{(\log p)X(p)}{p-X(p)}.$$ Since ${\mathbb{E}}(X(n))=0$ unless $n$ is a square (see below), and $\sum_{n\geq 2}(\log n)^2/n^2<\infty$, then it follows from Kolmogorov three series theorem that ${\gamma_{\textup{rand}}(X)}$ is almost surely convergent.
Here and throughout, we denote by ${\mathcal{F}(x)}$ the set of all fundamental discriminants $D$ with $|D|\leq x$. Note that $|{\mathcal{F}(x)}|= 6 x/\pi^2 +O(\sqrt{x}).$ Our main result shows that the distribution of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ is very well approximated by that of the random variable ${\gamma_{\textup{rand}}(X)}$ uniformly in nearly the whole conjectured range .
\[MainTheorem\] Let $x$ be large. There exists a positive constant $C$ such that uniformly in the range $1\leq \tau\leq \log\log x-2\log\log\log x-C$, we have $$\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: {\gamma_{\mathbb{Q}(\sqrt D)}}>\tau \}\big|= {\mathbb{P}}\big({\gamma_{\textup{rand}}(X)}>\tau\big)\left(1+O\left(\frac{e^{\tau}(\log\log x)^3}{\tau\log x}\right)\right),$$ and $$\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: {\gamma_{\mathbb{Q}(\sqrt D)}}<-\tau \}\big|= {\mathbb{P}}\big({\gamma_{\textup{rand}}(X)}<-\tau\big)\left(1+O\left(\frac{e^{\tau}(\log\log x)^3}{\tau\log x}\right)\right).$$
Since $L'/L(1,\chi_D)={\gamma_{\mathbb{Q}(\sqrt D)}}-\gamma$, Theorem 1.1 can be rephrased in terms of the logarithmic derivative of quadratic Dirichlet $L$-functions at $s=1$. The values of logarithmic derivatives of $L$-functions have been studied by Ihara and Matsumoto [@IhMa], and Ihara, Murty and Shimura [@IMS] in the case of Dirichlet $L$-functions, and by Cho and Kim [@ChKi] in the case of Artin $L$-functions. In particular, Ihara and Matsumoto [@IhMa] showed that as $\chi$ varies over non principal characters modulo a prime $q$, $L'/L(1,\chi)$ has a limiting distribution as $q\to\infty$. However, Theorem \[MainTheorem\] is the first result that gives precise information on the distribution of logarithmic derivatives of $L$-functions at $s=1$ with such a great uniformity. We should also note that with a slight modification of our method we can obtain similar results for the distribution of $|\zeta'/\zeta(1+it)|$, and that of $|L'/L(1, \chi)|$ as $\chi$ varies over non-principal characters modulo a large prime $q$. To construct the probabilistic random model in these cases we take the $\{X(p)\}_p$ to be uniformly distributed on the unit circle.
Our next task is to study the asymptotic behavior of the distribution functions ${\mathbb{P}}\big({\gamma_{\textup{rand}}(X)}>\tau\big)$ and ${\mathbb{P}}\big({\gamma_{\textup{rand}}(X)}<-\tau\big)$ in terms of $\tau$, when $\tau$ is large. We achieve this by a careful saddle point analysis. In particular, we show that these distribution functions are double exponentially decreasing in $\tau$.
\[ExponentialDecay\] For large $\tau$ we have $${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)=\exp\left(-\frac{e^{\tau-A_1}}{\tau}\left(1+ O\left(\frac{\log\tau}{\tau}\right)\right)\right),$$ and $${\mathbb{P}}({\gamma_{\textup{rand}}(X)}<-\tau)=\exp\left(-\frac{e^{\tau-A_2}}{\tau}\left(1+O\left(\frac{\log\tau}{\tau}\right)\right)\right),$$ where $$A_1:=A_0+2\frac{\zeta'(2)}{\zeta(2)}, \text{ and }
A_2:=A_0- 2\gamma,$$ and $$A_0:= \int_0^1\frac{\tanh(t)}{t}dt + \int_1^{\infty}\frac{\tanh(t)-1}{t}dt.$$
Combining Theorems \[MainTheorem\] and \[ExponentialDecay\] we deduce that the same asymptotic estimate holds for the distribution function of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ uniformly for $\tau$ in the range $1\ll \tau \leq \log\log x-2\log\log\log x-C.$
\[AsympDistrib\] Let $x$ be large. There exists a positive constant $C$ such that uniformly in the range $1\ll \tau\leq \log\log x-2\log\log\log x-C$, we have $$\label{AsymptoticEstimate1}
\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: {\gamma_{\mathbb{Q}(\sqrt D)}}>\tau \}\big|= \exp\left(-\frac{e^{\tau-A_1}}{\tau}\left(1+ O\left(\frac{\log\tau}{\tau}\right)\right)\right),$$ and $$\label{AsymptoticEstimate2}
\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: {\gamma_{\mathbb{Q}(\sqrt D)}}<-\tau \}\big|= \exp\left(-\frac{e^{\tau-A_2}}{\tau}\left(1+ O\left(\frac{\log\tau}{\tau}\right)\right)\right).$$
\[Support\] Note that the asymptotic estimate on the right hand side of (or ) becomes $<1/|{\mathcal{F}(x)}|$ if $\tau>\log\log x+\log\log\log x+C_0$ for some constant $C_0$. Therefore, if the asymptotic estimates in and were to persist in this full viable range, then one would deduce that $|{\gamma_{\mathbb{Q}(\sqrt D)}}|\leq \log\log |D|+\log\log\log |D|+O(1).$
In [@GrSo], Granville and Soundararajan investigated the distribution of $L(1,\chi_D)$ and proved that uniformly for $\tau$ in the range $1\ll \tau\leq \log\log x +O(1)$ we have $$\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: L(1,\chi_D) >e^{\gamma}\tau \}\big|= \exp\left(-\frac{e^{\tau-A_0}}{\tau}\left(1+O\left(\frac{1}{\tau}\right)\right)\right).$$ Their method relies upon careful analysis of large complex moments of $L(1,\chi_D)$. In her thesis, Mourtada [@Mo] remarked that it is a difficult problem to compute complex moments of ${\gamma_{\mathbb{Q}(\sqrt D)}}$. Instead, our approach relies on computing the Laplace transform of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ (defined as the average of $\exp(s\cdot {\gamma_{\mathbb{Q}(\sqrt D)}})$ over $D\in {\mathcal{F}(x)}$) using only asymptotics for integral moments of ${\gamma_{\mathbb{Q}(\sqrt D)}}$. We should also note that in comparison to the treatment for $L(1, \chi_D)$, there is an additional technical difficulty in our case which comes from the fact that $\exp({{\gamma_{\mathbb{Q}(\sqrt D)}}})$ grows much faster than $L(1,\chi_D)$. To overcome this difficulty, we compute the Laplace transform of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ after first removing the contribution of a small set of “bad” discriminants $D$, namely those for which ${\gamma_{\mathbb{Q}(\sqrt D)}}$ might be large.
\[AsympLaplace\] Given $0<\epsilon\leq 1/2$ there exists a constant $C_{\epsilon}>0$ and a set of fundamental discriminants ${\mathcal{E}(x)}\subset {\mathcal{F}(x)}$ with $|{\mathcal{E}(x)}|=O\left(x^{\epsilon}\right)$, such that for all complex numbers $s$ with $|s|\leq C_{\epsilon}\log x/(\log\log x)^2$ we have $$\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)}\exp\left(s \cdot {\gamma_{\mathbb{Q}(\sqrt D)}}\right) = {\mathbb{E}}\Big(\exp\big(s\cdot{\gamma_{\textup{rand}}(X)}\big) \Big)+
O\left(\exp\left(-\frac{\log x}{50\log\log x}\right)\right).$$
To prove this result we show that large integral moments of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ are very close to those of the random model ${\gamma_{\textup{rand}}(X)}$. For a fixed natural number $k$, asymptotic formulae for the $k$-th moment of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ have been obtained by Mourtada and Murty in [@MoMu], building on an earlier work of Ihara, Murty and Shimura [@IMS]. However, the significant feature of our result is the uniformity in the range of moments.
\[MomentsGamma\] For all positive integers $k$ with $k\leq \log x/(50\log\log x)$ we have $$\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in \mathcal{F}^*(x)} \big({\gamma_{\mathbb{Q}(\sqrt D)}}\big)^k= {\mathbb{E}}\left(\big({\gamma_{\textup{rand}}(X)}\big)^k\right) +O\left(x^{-1/30}\right),$$ where $ \mathcal{F}^*(x)$ denotes the set of fundamental discriminants $D\in {\mathcal{F}(x)}$ such that $L(s,\chi_D)$ has no Siegel zeros.
Note that if $L(s, \chi_D)$ has a Siegel zero, we could have $
{\gamma_{\mathbb{Q}(\sqrt D)}}$ as large as $q^{\epsilon}$, so that when $k$ is large, the $k$-th moment of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ would be heavily affected by the contribution of this particular character. This justifies the condition $D\in {\mathcal{F}(x)}^*$ in Theorem \[MomentsGamma\]. Furthermore, it is known that these characters if they exist must be very rare, in particular we have $|{\mathcal{F}(x)}|-|{\mathcal{F}(x)}^*|\ll \log x$ (see for example [@Da]).
The paper is organized as follows: In Section 2 we investigate the moments of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ and prove Theorem \[MomentsGamma\]. This result is then used to study the Laplace transform of ${\gamma_{\mathbb{Q}(\sqrt D)}}$ and prove Theorem \[AsympLaplace\] in Section 3. In Section 4 we study the Laplace transform of the random model ${\gamma_{\textup{rand}}(X)}$ and prove an asymptotic estimate for it. We then relate the distribution function of ${\gamma_{\textup{rand}}(X)}$ to its Laplace transform and prove Theorem \[ExponentialDecay\] in Section 5. Finally, in Section 6 we combine all these results to derive Theorem \[MainTheorem\].
Large moments of ${\gamma_{\mathbb{Q}(\sqrt D)}}$: proof of Theorem \[MomentsGamma\]
====================================================================================
For any positive integer $k$, we define $$\Lambda_{k}(n)=\sum_{\substack{n_1,n_2,\dots,n_k\geq 1\\ n_1n_2\cdots n_k=n}}\Lambda(n_1)\Lambda(n_2)\cdots\Lambda(n_k).$$ Then for all complex numbers $s$ with ${\textup{Re}}(s)>1$ we have $$\left(-\frac{L'}{L}(s,\chi_D)\right)^k=\sum_{n=1}^{\infty} \frac{\Lambda_k(n)}{n^s}\chi_D(n).$$ Moreover, note that $$\label{boundLambda}
\Lambda_{k}(n) \leq \left(\sum_{m|n}\Lambda(m)\right)^k= (\log n)^k.$$ We shall extract Theorem \[MomentsGamma\] from the following result, which gives an asymptotic formula for large integral moments of $-L'/L(1,\chi_D)$.
\[MomentsEuler\] For all positive integers $k$ with $k\leq \log x/(50\log\log x)$ we have $$\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in \mathcal{F}^*(x)} \left(-\frac{L'}{L}(1,\chi_D)\right)^k= \sum_{m=1}^{\infty}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right) +O\left(x^{-1/20}\right).$$
First, we need the following lemma, which provides a bound for $L'/L(s,\chi_D)$ when $s$ is far from a zero of $L(z, \chi_D)$.
\[BoundLD\] Let $t$ be a real number and suppose that $L(z,\chi_D)$ has no zero for ${\textup{Re}}(z)>\sigma_0$ and $|{\textup{Im}}(z)|\leq |t|+1$, then for any $\sigma>\sigma_0$ we have $$\frac{L'}{L}(\sigma+it,\chi_D)\ll \frac{\log(D(|t|+2))}{\sigma-\sigma_0}.$$
Let $\rho$ runs over the non-trivial zeros of $L(s,\chi)$. Then it follows from equation (4) of Chapter 16 of Davenport [@Da] that $$\begin{aligned}
\frac{L'}{L}(\sigma+it,\chi_D)
&= \sum_{\substack{\rho\\ |t-{\textup{Im}}(\rho)|<1}}\frac{1}{\sigma+it-\rho}+ O\big(\log(D(|t|+2))\big)\\
&\ll \frac{1}{\sigma-\sigma_0}\left(\sum_{\substack{\rho\\ |t-{\textup{Im}}(\rho)|<1}} 1\right) + \log(D(|t|+2))\\
&\ll \frac{\log(D(|t|+2))}{\sigma-\sigma_0},\end{aligned}$$ as desired.
The key ingredient in the proof of Theorem \[MomentsEuler\] is the following result which shows that we can approximate large powers of $-L'/L(1,\chi_D)$ by short Dirichet polynomials, if $L(s,\chi_D)$ has no zeros in a certain region to the left of the line ${\textup{Re}}(s)=1$.
\[ApproximationLarge\] Let $0<\delta<1/2$ be fixed, and $D$ be a fundamental discriminant with $|D|$ large. Let $y\geq (\log |D|)^{10/\delta}$ be a real number and $k\leq 2\log |D|/\log y$ be a positive integer. If $L(s,\chi_D)$ is non-zero for ${\textup{Re}}(s)>1-\delta$ and $|{\textup{Im}}(s)|\leq y^{k\delta}$, then we have $$\left(-\frac{L'}{L}(1,\chi_D)\right)^k=\sum_{n\leq y^k} \frac{\Lambda_k(n)}{n}\chi_D(n)+O_{\delta}\Big(y^{-k\delta/4}\Big).$$
Without loss of generality, suppose that $y^k\in \mathbb{Z}+1/2$. Let $c=1/(k\log y)$, and $T$ be a large real number to be chosen later. Then by Perron’s formula, we have $$\frac{1}{2\pi i}\int_{c-iT}^{c+iT} \left(-\frac{L'}{L}(1+s,\chi_D)\right)^k \frac{y^{ks}}{s}ds=
\sum_{n\leq y^k} \frac{\Lambda_k(n)}{n}\chi_D(n)+ O\left(\frac{y^{kc}}{T}\sum_{n=1}^{\infty} \frac{\Lambda_k(n)}{n^{1+c}|\log(y^k/n)|}\right).$$ To bound the error term of this last estimate, we split the sum into three parts: $n\leq y^k/2$, $y^k/2<n<2y^k$ and $n\geq 2y^k$. The terms in the first and third parts satisfy $|\log(y^k/n)|\geq \log 2$, and hence their contribution is $$\ll \frac{1}{T} \sum_{n=1}^{\infty}\frac{\Lambda_k(n)}{n^{1+c}}=\frac{1}{T} \left(\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{1+c}}\right)^k\ll \frac{(2k\log y)^k}{T},$$ by the prime number theorem. To handle the contribution of the terms $y^k/2<n<2y^k$, we put $r=n-y^k$, and use that $|\log(y^k/n)|\gg |r|/y^k$. In this case, we have $\Lambda_k(n)\leq (\log n)^k\leq (2k\log y)^k$, and hence the contribution of these terms is $$\ll\frac{(2k\log y)^{k}}{Ty^{k}}\sum_{|r|\leq y^{k}}\frac{y^k}{|r|}\ll \frac{(2k\log y)^{k+1}}{T}.$$ We now choose $T=y^{k\delta/2}$ and move the contour to the line ${\textup{Re}}(s)=-\delta/2$. By our assumption, we only encounter a simple pole at $s=0$ which leaves a residue $(-L'/L(1,\chi_D))^k$. Therefore, we deduce that $$\frac{1}{2\pi i}\int_{c-iT}^{c+iT} \left(-\frac{L'}{L}(1+s,\chi_D)\right)^k \frac{y^{ks}}{s}ds= \left(-\frac{L'}{L}(s,\chi_D)\right)^k+ E_1,$$ where $$\begin{aligned}
E_1&=\frac{1}{2\pi i} \left(\int_{c-iT}^{-\delta/2-iT}+ \int_{-\delta/2-iT}^{-\delta/2+iT}+ \int_{-\delta/2+iT}^{c+iT}\right) \left(-\frac{L'}{L}(1+s,\chi_D)\right)^k \frac{y^{ks}}{s}ds\\
& \ll_{\delta} \frac{(\log (|D|T))^k}{T}+ y^{-k\delta/2}\left(\frac{\log (|D|T)}{\delta}\right)^{k+1}\\
&\ll_{\delta} y^{-k\delta/4},\end{aligned}$$ by Lemma \[BoundLD\]. Finally, since $(2k\log y)^{k+1}/T\ll y^{-k\delta/4}$, the result follows.
Now, using a zero density estimate due to Heath-Brown (see equation below), we deduce from Proposition \[ApproximationLarge\] that large powers of $-L'/L(1,\chi_D)$ can be approximated by short Dirichlet polynomials for almost all fundamental discriminants $D$ with $|D|\leq x$.
\[AAApproximation\] Let $k$ be a positive integer such that $k\leq \log x/(50(\log\log x))$. For all except $O(x^{3/4})$ fundamental discriminants $D$ with $|D|\leq x$ we have $$\left(-\frac{L'}{L}(1,\chi_D)\right)^k=\sum_{n\leq x} \frac{\Lambda_k(n)}{n}\chi_D(n)+O\Big(x^{-1/20}\Big).$$
Let $N(\sigma, T, \chi_D)$ denote the number of zeros of $L(s, \chi_D)$ in the rectangle $ \sigma<{\textup{Re}}(s)\leq 1$ and $|{\textup{Im}}(s)|\leq T$. Health-Brown [@HB] showed that $$\label{ZeroDensity}
\sum_{D\in {\mathcal{F}(x)}} N(1-\delta, T, \chi_D)\ll_{\epsilon} (xT)^{\epsilon} x^{3\delta/(1+\delta)} T^{(1+2\delta)/(1+\delta)}.$$ Choosing $\delta=1/5$, we deduce that for all except $O(x^{3/4})$ fundamental discriminants $D$ with $|D|\leq x$, $L(s, \chi_D)$ does not vanish in the region ${\textup{Re}}(s)>1-\delta$ and $|{\textup{Im}}(s)|\leq x^{\delta}$. We now take $y=x^{1/k}$ in Proposition \[ApproximationLarge\], to obtain that for all except $O(x^{3/4})$ fundamental discriminants $D$ with $\sqrt{x}\leq |D|\leq x$ we have $$\left(-\frac{L'}{L}(1,\chi_D)\right)^k=\sum_{n\leq x} \frac{\Lambda_k(n)}{n}\chi_D(n)+O\Big(x^{-1/20}\Big),$$ as desired.
We also deduce from Proposition \[ApproximationLarge\] that ${\gamma_{\mathbb{Q}(\sqrt D)}}\ll \log\log |D|$ for almost all fundamental discriminants $|D|\leq x$.
\[ASBound\] Let $\epsilon>0$. Then for all but $O(x^{\epsilon})$ fundamental discriminants $|D|\leq x$ we have $${\gamma_{\mathbb{Q}(\sqrt D)}}\ll_{\epsilon} \log\log D.$$
Taking $\delta=\epsilon/5$, $k=1$ and $y=(\log |D|)^{50/\epsilon}$ in Proposition \[ApproximationLarge\] and using as in the proof of Corollary \[AAApproximation\] we deduce that for all except $O(x^{\epsilon})$ fundamental discriminants $D$ with $|D|\leq x$, we have $$\begin{aligned}
{\gamma_{\mathbb{Q}(\sqrt D)}}&=\gamma+ \frac{L'}{L}(1,\chi_D)=\gamma-\sum_{n\leq y} \frac{\Lambda(n)}{n}\chi_D(n)+O\Big(y^{-\epsilon/20}\Big)\\
&\ll_{\epsilon} \log\log |D|.\end{aligned}$$
Let ${\mathcal{E}(x)}$ be the exceptional set in Corollary \[AAApproximation\]. Then it follows from this result that $$\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}\setminus {\mathcal{E}(x)}} \left(-\frac{L'}{L}(1,\chi_D)\right)^k
= \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}\setminus {\mathcal{E}(x)}} \sum_{n\leq x} \frac{\Lambda_k(n)}{n}\chi_D(n)+O\Big(x^{-1/20}\Big).$$ Note that $$\label{BoundPowerLambda}
\sum_{n\leq x}\frac{\Lambda_k(n)}{n}\leq \left(\sum_{n\leq x}\frac{\Lambda(n)}{n}\right)^k\leq (2\log x)^k \ll x^{1/40},$$ if $x$ is large enough. Hence, we deduce that $$\label{MomentsMain} \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}\setminus {\mathcal{E}(x)}} \left(-\frac{L'}{L}(1,\chi_D)\right)^k
= \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}} \sum_{n\leq x} \frac{\Lambda_k(n)}{n}\chi_D(n)+O\Big(x^{-1/20}\Big).$$ To evaluate the sum on the right hand side of this estimate, we first consider the contribution of perfect squares, which gives the main term. In this case, we use the following standard estimate (see for example [@GrSo]) $$\sum_{D\in {\mathcal{F}(x)}} \chi_D(m^2)=\sum_{\substack{D\in {\mathcal{F}(x)}\\ (D,m)=1}}1=\frac{6}{\pi^2}x\prod_{p|m}\left(\frac{p}{p+1}\right)+ O\big(x^{1/2} d(m)\big),$$ where $d(m)$ is the divisor function. Therefore the contribution of the terms $n=m^2$ to the right hand side of equals $$\label{diagonalChar}
\sum_{m\leq \sqrt{x}}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right)+O\left(x^{-1/2}\sum_{m\leq \sqrt{x}}
\frac{\Lambda_{k}(m^2)}{m^2}d(m)\right).$$ By , the error term in the last estimate is $$\label{Error2.6}
\ll x^{-1/2}\sum_{m\leq \sqrt{x}}\frac{(2\log m)^k d(m)}{m^2}\leq x^{-1/2} (\log x)^k\sum_{m=1}^{\infty}\frac{d(m)}{m^2}\ll x^{-1/4}.$$ Further, since the function $(\log t)^k/\sqrt{t}$ is decreasing for $t\geq e^{2k}$, we obtain $$\sum_{m> \sqrt{x}}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right)\leq \sum_{m> \sqrt{x}}
\frac{(2\log m)^k}{m^2} \ll \frac{(\log x)^{k}}{\sqrt{x}}\sum_{m>\sqrt{x}}\frac{1}{m^{3/2}}\ll \frac{(\log x)^{k}}{x}\ll x^{-1/2}.$$ Thus, combining this bound with and we deduce that the contribution of the squares to the right hand side of is $$\label{squares}
\sum_{m=1}^{\infty}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right) +O\left(x^{-1/4}\right).$$
To bound the contribution of the non-squares, we use the following simple application of the Pólya-Vinogradov inequality, which corresponds to Lemma 4.1 of [@GrSo] and states that $$\sum_{D\in {\mathcal{F}(x)}} \chi_D(n)\ll x^{1/2}n^{1/4}\log n,$$ if $n$ is not a perfect square. Using this bound along with , we deduce that the contribution of the non-squares to the right hand side of is $$\label{nonsquares}
\ll x^{-1/4}\log x\sum_{n\leq x}\frac{\Lambda_{k}(n)}{n}\ll x^{-1/4}(2\log x)^{k+1}\ll x^{-1/6}.$$ Furthermore, it follows from Lemma \[BoundLD\] along with the classical zero free region for $L(s, \chi_D)$ that for $D\in {\mathcal{F}^*(x)}$ we have $$\label{ClassicalBound}
\frac{L'}{L}(1,\chi_D)\ll (\log |D|)^2.$$ Therefore, combining this bound with equations , and we derive $$\begin{aligned}
\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}} \left(-\frac{L'}{L}(1,\chi_D)\right)^k
&= \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}\setminus {\mathcal{E}(x)}} \left(-\frac{L'}{L}(1,\chi_D)\right)^k +O\left(x^{-1/4}(\log x)^{2k}\right)\\
&=\sum_{m=1}^{\infty}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right)+ O \big(x^{-1/20}\big).\end{aligned}$$
We are now ready to prove Theorem \[MomentsGamma\].
Note that for any prime $p$ and positive integer $k$ we have $${\mathbb{E}}\left((X(p)^k\right)= \frac{p}{2(p+1)}+ (-1)^k\frac{p}{2(p+1)}.$$ Therefore, by the independence of the random variables $X(p)$ we deduce that $$\label{ortho}
{\mathbb{E}}\big(X(n)\big)=\begin{cases} \prod_{p|n} \left(\frac{p}{p+1}\right) & \text{if } n \text{ is a square},\\
0 & \text{otherwise}.\\
\end{cases}$$ Hence, we obtain $$\mathbb{E} \left(\left( \sum_{n=1}^{\infty} \frac{\Lambda(n)X(n)}{n} \right)^{k}\right)={\mathbb{E}}\left( \sum_{n=1}^{\infty} \frac{\Lambda_k(n)X(n)}{n}\right)=\sum_{m=1}^{\infty}
\frac{\Lambda_{k}(m^2)}{m^2}\prod_{p|m}\left(\frac{p}{p+1}\right).$$ Therefore, it follows from Theorem \[MomentsEuler\] that $$\begin{aligned}
\frac{1}{|{\mathcal{F}(x)}|} \sum_{D\in {\mathcal{F}^*(x)}} ({\gamma_{\mathbb{Q}(\sqrt D)}})^k&= \frac{1}{|{\mathcal{F}(x)}|} \sum_{D\in {\mathcal{F}^*(x)}}\left(\gamma+
\frac{L'}{L}(1,\chi_D)\right)^k\\
&= \sum_{j=0}^k \binom{k}{j} \gamma^{k-j}\frac{1}{|{\mathcal{F}(x)}|} \sum_{D\in {\mathcal{F}^*(x)}} \left(\frac{L'}{L}(1,\chi_D)\right)^j\\
&= \sum_{j=0}^k \binom{k}{j} \gamma^{k-j} (-1)^j {\mathbb{E}}\left(\left( \sum_{n=1}^{\infty} \frac{\Lambda(n)X(n)}{n} \right)^j\right)+O\left(x^{-1/30}\right)\\
&= {\mathbb{E}}\left(({\gamma_{\textup{rand}}(X)})^k\right)+O\left(x^{-1/30}\right).\end{aligned}$$
The Laplace transform of ${\gamma_{\mathbb{Q}(\sqrt D)}}$: proof of Theorem \[AsympLaplace\]
============================================================================================
In order to obtain Theorem \[AsympLaplace\] from Theorem \[MomentsGamma\], we need a uniform bound for the moments of ${\gamma_{\textup{rand}}(X)}$. We prove
\[BoundMomRand\] There exists a constant $c>0$ such that for all positive integers $k\geq 8$ we have $${\mathbb{E}}\left(\left|{\gamma_{\textup{rand}}(X)}\right|^k\right)\leq \big(c\log k\big)^k.$$
Let $y>2$ be a real number to be chosen later. By Minkowski’s inequality we have $$\label{Minkowski}
\begin{aligned}
{\mathbb{E}}\left(\left|{\gamma_{\textup{rand}}(X)}\right|^k\right)^{1/k}
&\leq {\mathbb{E}}\left(\left|\gamma-\sum_{n\leq y}\frac{\Lambda(n)X(n)}{n}\right|^k\right)^{1/k} +{\mathbb{E}}\left(\left|\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right|^k\right)^{1/k}\\
&\leq \gamma+\sum_{n\leq y}\frac{\Lambda(n)}{n}+{\mathbb{E}}\left(\left|\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right|^k\right)^{1/k}.\\
&\ll \log y+{\mathbb{E}}\left(\left|\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right|^k\right)^{1/k}.\\
\end{aligned}$$ Furthermore, by the Cauchy-Schwarz inequality we have $$\label{Cauchy}
{\mathbb{E}}\left(\left|\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right|^k\right) \leq {\mathbb{E}}\left(\left(\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right)^2\right)^{1/2} {\mathbb{E}}\left(\left(\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right)^{2(k-1)}\right)^{1/2}$$ Let $$\Lambda_{\ell,y}(n):=\sum_{\substack{n_1,n_2,\dots, n_{\ell}>y\\ n_1n_2\cdots n_{\ell}=n}} \Lambda(n_1)\Lambda(n_2)\cdots \Lambda(n_{\ell}).$$ Then, for every positive integer $m$ we have $$\begin{aligned}
{\mathbb{E}}\left(\left(\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right)^{2m}\right)
&= {\mathbb{E}}\left(\sum_{n>y^{2m}}\frac{\Lambda_{2m,y}(n)X(n)}{n}\right)\\
&=\sum_{n>y^{m}}\frac{\Lambda_{2m,y}(n^2)}{n^2}\prod_{p|n}\left(\frac{p}{p+1}\right)\\
&\leq
\sum_{n>y^{m}}\frac{(2\log n)^{2m}}{n^2},\end{aligned}$$ since $\Lambda_{\ell,y}(n)\leq \Lambda_{\ell}(n)\leq (\log n)^{\ell}$. Moreover, since $(\log n)^{2m}/\sqrt{n}$ is decreasing for $n>e^{4m}$, we deduce that if $y\geq e^4 $ then $${\mathbb{E}}\left(\left(\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right)^{2m}\right)\leq \frac{(2m\log y)^{2m}}{y^{m/2}}\sum_{n>y^{m/2}}\frac{1}{n^{3/2}}\ll\frac{(2m\log y)^{2m}}{y^{m}}.$$ Thus if $y\geq e^4 $ then by we obtain that $${\mathbb{E}}\left(\left|\sum_{n>y}\frac{\Lambda(n)X(n)}{n}\right|^k\right)^{1/k}\ll \frac{k\log y}{\sqrt{y}}.$$ Choosing $y=k^2$ and inserting this estimate in completes the proof.
Given $\epsilon>0$, it follows from Corollary \[ASBound\] that there exists a constant $B_{\epsilon}>0$ such that $$|{\gamma_{\mathbb{Q}(\sqrt D)}}|\leq B_{\epsilon} \log\log x,$$ for all fundamental discriminants $D\in {\mathcal{F}(x)}$ except for a set ${\mathcal{E}(x)}$ with $|{\mathcal{E}(x)}|=O\left(x^{\epsilon}\right).$ Let $N=\lfloor \log x/(50\log\log x)\rfloor$. Then we obtain $$\label{TaylorLaplace}
\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)}\exp\left(s \cdot {\gamma_{\mathbb{Q}(\sqrt D)}}\right)
= \sum_{k=0}^N \frac{s^k}{k!} \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)} \left({\gamma_{\mathbb{Q}(\sqrt D)}}\right)^k+E_2\\$$ where $$E_2\ll \sum_{k>N} \frac{|s|^k}{k!} (B_{\epsilon}\log\log x)^k\leq \sum_{k>N} \left(\frac{3B_{\epsilon}|s|\log\log x}{N}\right)^k \ll e^{-N}$$ by Stirling’s formula, if $|s|\leq C_{\epsilon}\log x/(\log\log x)^2$ for some small constant $C_{\epsilon}>0$. Furthermore, it follows by Theorem \[MomentsGamma\] and equation that for all integers $0\leq k\leq N$ we have $$\begin{aligned}
\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)} \left({\gamma_{\mathbb{Q}(\sqrt D)}}\right)^k&=\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}^*(x)}} \left({\gamma_{\mathbb{Q}(\sqrt D)}}\right)^k +O\left(x^{-1+\epsilon}(\log x)^{2k}\right)\\
&={\mathbb{E}}\left({\gamma_{\textup{rand}}(X)}^k\right) +O\left(x^{-1/20}\right).\end{aligned}$$ Moreover, it follows from Proposition \[BoundMomRand\] and Stirling’s formula that for some positive constant $C$ we have $$\sum_{k>N} \frac{s^k}{k!} {\mathbb{E}}\left({\gamma_{\textup{rand}}(X)}^k\right) \ll \sum_{k>N} \left(\frac{C|s|\log k}{k}\right)^k\ll \sum_{k>N} \left(\frac{C|s|\log N}{N}\right)^k\ll e^{-N},$$ if $C_{\epsilon}$ is suitably small. Finally, inserting these estimates in , we derive $$\begin{aligned}
\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)}\exp\left(s \cdot {\gamma_{\mathbb{Q}(\sqrt D)}}\right)&= \sum_{k=0}^N \frac{s^k}{k!} {\mathbb{E}}\left({\gamma_{\textup{rand}}(X)}^k\right) +O\left(e^{-N}+ x^{-1/20}e^{|s|}\right)\\
&= {\mathbb{E}}\Big(\exp\big( s \cdot {\gamma_{\textup{rand}}(X)}\big)\Big)+ O\left(e^{-N}\right),\end{aligned}$$ as desired.
The Laplace transform of ${\gamma_{\textup{rand}}(X)}$
======================================================
For any $s\in \mathbb{C}$ we define $$M(s):= \log\left({\mathbb{E}}\Big(\exp\big(s\cdot{\gamma_{\textup{rand}}(X)}\big) \Big)\right).$$ Since the $X(p)$ are independent and $ {\gamma_{\textup{rand}}(X)}= \gamma-\sum_{p}(\log p)X(p)/(p-X(p))$ we deduce that $$\label{LaplaceProduct}
M(s)= \gamma s+ \sum_{p}\log h_p(s),$$ where $$h_p(s):= {\mathbb{E}}\left(\exp\left(-\frac{s(\log p) X(p)}{p-X(p)}\right)\right).$$ Note that $$\label{expectation}
h_p(s)= \frac{p}{2(p+1)}\exp\left(\frac{s\log p}{p+1}\right)+ \frac{p}{2(p+1)}\exp\left(\frac{-s\log p}{p-1}\right)+ \frac{1}{p+1}.$$
The main purpose of this section is to investigate the asymptotic behavior of $M(r)$ and its derivatives, where $r$ is a large real number. We establish the following proposition.
\[LaplaceRand\] For any real number $r\geq 4$ we have $$\label{LaplaceRand1}
M(r)=r\left(\log r+\log\log r+A_1-1+O\left(\frac{\log\log r}{\log r}\right)\right),$$ $$\label{LaplaceRand2}
M(-r)=r\left(\log r+\log\log r+A_2-1+O\left(\frac{\log\log r}{\log r}\right)\right),$$ $$\label{LaplaceRand3}
M'(r)= \log r+\log\log r +A_1+ O\left(\frac{\log\log r}{\log r}\right),$$ and $$\label{LaplaceRand4}
M'(-r)= -\log r-\log\log r -A_2+ O\left(\frac{\log\log r}{\log r}\right).$$ Moreover, for all real numbers $y, t$ such that $|y|\geq 3$ we have $$\label{LaplaceRand5}
M''(y)\asymp \frac{1}{|y|}, \text{ and } M'''(y+it)\ll \frac{1}{|y|^2}.$$
To prove this result we first need some preliminary lemmas.
\[estimate1\] Let $r\geq 4$ be a real number. Then we have
$$\label{approxlarge}
\log h_p(r)=\begin{cases}
r\frac{\log p}{p+1} +O(1) & \text{ if } p\leq r^{2/3}\\
\log\cosh\left(\frac{r\log p}{p+1}\right)+O\left(\frac{r\log p}{p^2}\right) & \text{ if } p>r^{2/3}.
\end{cases}$$
and $$\label{approxlarge2}
\log h_p(-r)=\begin{cases}
r\frac{\log p}{p-1} +O(1) & \text{ if } p\leq r^{2/3}\\
\log\cosh\left(\frac{r\log p}{p-1}\right)+O\left(\frac{r\log p}{p^2}\right) & \text{ if } p>r^{2/3}.
\end{cases}$$
We only prove since can be obtained similarly. First, if $p<r^{2/3}$ then $$\label{Smallprimes}
h_p(r)= \frac{p}{2(p+1)}\exp\left(\frac{r\log p}{p+1}\right)
\left(1+O\left(\exp\left(-r^{1/3}\right)\right)\right),$$ from which the desired estimate follows in this case.
Now, if $p>r^{2/3}$ then $$\label{locallargep}
\begin{aligned}
h_p(r)
&= \frac{p}{(p+1)} \cosh\left(\frac{r\log p}{p+1}\right) \left(1+O\left(\frac{r\log p}{p^2}\right)\right)+\frac{1}{p+1}\\
&= \cosh\left(\frac{r\log p}{p+1}\right) \left(1+O\left(\frac{r\log p}{p^2}\right)\right),
\end{aligned}$$ since $ \cosh(t)-1 \ll t\cosh(t)$, for all $t\geq 0$. This completes the proof.
\[estimatelogarithmicderivative\] Let $r\geq 4$ be a real number. Then we have
$$\label{approxlarge3}
\frac{h'_p(r)}{h_p(r)}=\begin{cases} \frac{\log p}{p+1} \left(1+O\left(e^{-r^{1/3}}\right)\right)& \text{ if } p\leq r^{2/3}\\
\frac{\log p}{p+1}\tanh\left(r\frac{\log p}{p+1}\right) + O\left(\frac{\log p}{p^2}+\frac{r\log^2p}{p^3}\right) & \text{ if } p>r^{2/3}.
\end{cases}$$
and $$\label{approxlarge4}
\frac{h'_p(-r)}{h_p(-r)}=\begin{cases} -\frac{\log p}{p-1} \left(1+O\left(e^{-r^{1/3}}\right)\right)& \text{ if } p\leq r^{2/3}\\
-\frac{\log p}{p-1}\tanh\left(r\frac{\log p}{p-1}\right) + O\left(\frac{\log p}{p^2}+\frac{r\log^2p}{p^3}\right) & \text{ if } p>r^{2/3}.
\end{cases}$$
We only prove since the proof of is similar. By we have $$h_p'(r)=
\frac{p\log p}{2(p+1)^2}\exp\left(\frac{r\log p}{p+1}\right)-\frac{p\log p}{2(p^2-1)}\exp\left(\frac{-r\log p}{p-1}\right)$$ First, for $p<r^{2/3}$ we have by $$h_p'(r)= \frac{\log p}{p+1} h_p(r)\left(1+O\left(\exp\left(-r^{1/3}\right)\right)\right).$$ On the other hand, if $p>r^{2/3}$ then $$h_p'(r)= \frac{\log p}{p+1}\left(\sinh\left(\frac{r\log p}{p+1}\right) + O\left(\frac{1}{p}\cosh\left(\frac{r\log p}{p+1}\right)+\frac{r\log p}{p^2}\right)\right).$$ Therefore, by we obtain $$\frac{h_p'(r)}{h_p(r)}= \frac{\log p}{p+1}\tanh\left(\frac{r\log p}{p+1}\right)+ O\left(\frac{\log p}{p^2}+\frac{r\log^2p}{p^3}\right).$$
\[sumprimes\] We have $$\label{sumprimes1}
\sum_{p\leq y} \frac{\log p}{p-1}= \log y-\gamma+O\left(\frac{1}{\log y}\right),$$ and $$\label{sumprimes2}
\sum_{p\leq y} \frac{\log p}{p+1}= \log y - \gamma +2\frac{\zeta'(2)}{\zeta(2)}+ O\left(\frac{1}{\log y}\right).$$
We have $$\sum_{p\leq y} \frac{\log p}{p-1}= \sum_{p\leq y} \log p\sum_{a=1}^{\infty}\frac{1}{p^a}=\sum_{n\leq y} \frac{\Lambda(n)}{n}+ O\big(y^{-1/2}\big).$$ The first assertion follows from the classical estimate $$\label{LambdaEstimate}
\sum_{n\leq y} \frac{\Lambda(n)}{n}=\log y-\gamma +O\left(\frac{1}{\log y}\right).$$ Moreover, the second assertion follows from the first upon noting that $$\sum_{p\leq y} \frac{\log p}{p+1}
= \sum_{p\leq y} \frac{\log p}{p-1}-2\sum_{p\leq y}\frac{\log p}{p^2-1}= \sum_{p\leq y} \frac{\log p}{p-1} +2\frac{\zeta'(2)}{\zeta(2)}+O\left(\frac{1}{y}\right).$$
Let $$f(t):= \begin{cases} \log \cosh(t) & \text{ if } 0\leq t <1 \\
\log \cosh(t)- t & \text{ if } t \geq 1.\end{cases}$$ Then we prove
\[logcosh\] $f$ is bounded on $[0,\infty)$ and $f(t)=t^2/2+O(t^4)$ if $0\leq t <1.$ Moreover we have $$\label{asympldcosh}
f'(t)=\begin{cases} t +O(t^2) & \text{ if } 0< t<1 \\
O(e^{-2t}) & \text{ if } t > 1.\end{cases}$$
Since $e^t/2\leq \cosh(t)\leq e^t$, it follows that $f$ is bounded on $[0,\infty)$. Now, for $t\in [0,1)$ we have $\cosh(t)=1+t^2/2+O(t^4)$ and hence $f(t)=t^2/2+O(t^4)$.
Moreover, if $0< t< 1$ then $f'(t)=\tanh(t)=t+O(t^2)$. Now, if $t> 1$ then $$f'(t)=\tanh(t)-1= \frac{e^t-e^{-t}}{e^t+e^{-t}}=O(e^{-2t}).$$
We are now ready to prove Proposition \[LaplaceRand\].
We only prove and , since , and follow along the same lines. By Lemma \[estimate1\] and the prime number theorem we obtain $$M(r)=\gamma r+\sum_{p\leq r^{2/3}}\frac{r\log p}{p+1}+\sum_{p>r^{2/3}}\log\cosh\left(\frac{r\log p}{p+1}\right)+ O\left(r^{2/3}\right).$$ Let $R$ be the unique solution to $r\log R=R+1$. Then we have $$R= r\log r\left(1+O\left(\frac{\log\log r}{\log r}\right)\right).$$ Since $(\log t)/(t+1)$ is decreasing for $t\geq 4$ we deduce $$M(r)=\gamma r+\sum_{p\leq R}\frac{r\log p}{p+1}+\sum_{p>r^{2/3}}f\left(\frac{r\log p}{p+1}\right)+ O\left(r^{2/3}\right).$$ Moreover, by we have $$\sum_{p\leq R}\frac{\log p}{p+1}= \log R-\gamma+ 2\frac{\zeta'(2)}{\zeta(2)}+ O\left(\frac{1}{\log R}\right)= \log r+\log\log r -\gamma+ 2\frac{\zeta'(2)}{\zeta(2)}+ O\left(\frac{\log\log r}{\log r}\right).$$ Now, by Lemma \[logcosh\] and the prime number theorem in the form $\pi(t)-\text{Li}(t)\ll t/(\log t)^3$ we derive $$\label{sumintegral}
\begin{aligned}
\sum_{p>r^{2/3}}f\left(\frac{r\log p}{p+1}\right) &=\int_{r^{2/3}}^{\infty} f\left(\frac{r\log t}{t+1}\right)d\pi(t)\\
&= \int_{r^{2/3}}^{\infty} f\left(\frac{r\log t}{t+1}\right)\frac{dt}{\log t} +E_3,
\end{aligned}$$ where $$E_3\ll r^{2/3} + r\int_{r^{2/3}}^{\infty} \left|f'\left(\frac{r\log t}{t+1}\right)\right|\frac{1}{t(\log t)^2}dt\ll \frac{r}{\log r},$$ since $f'(t)$ is bounded by Lemma \[logcosh\]. To evaluate the main term on the right hand side of we make the change of variables $u= r(\log t)/(t+1).$ Since $t\geq r^{2/3}$ we obtain that $$du= r \left(\frac{1}{t(t+1)}-\frac{\log t}{(t+1)^2}\right)dt
= -r\frac{(\log t)dt}{(t+1)^2}\left(1+O\left(\frac{1}{\log r}\right)\right)
= -\frac{u^2}{r}\frac{dt}{\log t}\left(1+O\left(\frac{1}{\log r}\right)\right).$$ Putting $r_1= r(\log(r^{2/3}))/(r^{2/3}+1)$, we deduce by Lemma \[logcosh\] that $$\label{sumintegral2}
\sum_{p>r^{2/3}}f\left(\frac{r\log p}{p+1}\right) = r \int_0^{r_1} \frac{f(u)}{u^2}du + O\left(\frac{r}{\log r}\right) = r \int_0^{\infty} \frac{f(u)}{u^2}du + O\left(\frac{r}{\log r}\right).$$ Moreover, by a simple integration by parts we have $$\int_0^{\infty}\frac{f(u)}{u^2}du=\int_0^{\infty}\frac{f'(u)}{u}du-1.$$ Collecting the above estimates yields .
Now, we prove . First, note that $$M'(r)= \gamma+\sum_{p} \frac{h_p'(r)}{h_p(r)}.$$ Using Lemma \[estimatelogarithmicderivative\] we obtain $$\begin{aligned}
M'(r)&= \gamma+ \sum_{p<r^{2/3}}\frac{\log p}{p+1}+\sum_{p>r^{2/3}}\frac{\log p}{p+1}\tanh\left(\frac{r\log p}{p+1}\right)+ O\left( r^{-1/3}\log r\right)\\
& = \gamma+ \sum_{p<R}\frac{\log p}{p+1}+\sum_{p>r^{2/3}}\frac{\log p}{p+1}f'\left(\frac{r\log p}{p+1}\right)+ O\left( r^{-1/3}\log r\right)\\
&= \log r+\log\log r+ 2\frac{\zeta'(2)}{\zeta(2)}+\sum_{p>r^{2/3}}\frac{\log p}{p+1}f'\left(\frac{r\log p}{p+1}\right)+ O\left( \frac{\log\log r}{\log r}\right).\end{aligned}$$ Finally, using the prime number theorem and partial integration as in , one can deduce that $$\sum_{p>r^{2/3}}\frac{\log p}{p-1}f'\left(\frac{r\log p}{p-1}\right)=\int_0^{\infty}\frac{f'(u)}{u}du+O\left(\frac{1}{\log r}\right).$$
The distribution function of ${\gamma_{\textup{rand}}(X)}$: proof of Theorem \[ExponentialDecay\]
=================================================================================================
To shorten our notation, we define ${\mathcal{L}_{\textup{rand}}}(s):= {\mathbb{E}}\left(\exp\left(s\cdot {\gamma_{\textup{rand}}(X)}\right)\right)$. Let $\phi(y)=1$ if $y>1$ and equals $0$ otherwise. To relate the distribution function of ${\gamma_{\textup{rand}}(X)}$ (or that of ${\gamma_{\mathbb{Q}(\sqrt D)}}$) to its Laplace transform, we use the following smooth analogue of Perron’s formula, which is a slight variation of a formula of Granville and Soundararajan (see [@GrSo]).
\[SmoothPerron\] Let $\lambda>0$ be a real number and $N$ be a positive integer. For any $c>0$ we have for $y>0$ $$0\leq \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N \frac{ds}{s} -\phi(y)\leq
\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N \frac{1-e^{-\lambda N s}}{s}ds.$$
For any $y>0$ we have $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}
= \frac{1}{\lambda^N}\int_{0}^{\lambda}\cdots \int_0^{\lambda} \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}
\left(ye^{t_1+ \cdots+ t_N}\right)^s\frac{ds}{s} dt_1\cdots dt_N$$ so that by Perron’s formula we obtain $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}
= \begin{cases} = 1 & \text{ if } y\geq 1, \\ \in [0,1] & \text{ if } e^{-\lambda N } \leq y < 1,\\
=0 & \text{ if } 0<y< e^{-\lambda N }. \end{cases}$$ Therefore we deduce that $$\label{indicator}
\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s e^{-\lambda N s} \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N \frac{ds}{s} \leq \phi(y)\leq \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^s \left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N \frac{ds}{s}$$ which implies the result.
Let $\tau$ be a real number and consider the equation $M'(r)=\tau$ (recall that $M(r)=\log{\mathcal{L}_{\textup{rand}}}(r)$). By Proposition \[LaplaceRand\] it follows that $\lim_{r\to\infty} M'(r)=\infty$ and $\lim_{r\to-\infty} M'(r)=-\infty$. Moreover, a simple calculation shows that $h_p''(r)h_p(r)>(h_p'(r))^2$ for all primes $p$, and hence that $M''(r)>0$. Thus, it follows that the equation $M'(r)=\tau$ has a unique solution $\kappa$. Using a carefull saddle point analysis we obtain an asymptotic formula for ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)$ in terms of the Laplace transform of ${\gamma_{\textup{rand}}(X)}$ evaluated at the saddle point $\kappa$.
\[SaddlePoint\] Let $\tau$ be large and $\kappa$ denote the unique solution to $M'(r)=\tau$. Then, we have $${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)= \frac{{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}}{\kappa \sqrt{2\pi M''(\kappa)}}\left(1+O\left(\frac{1}{ \sqrt{\kappa}} \right)\right).$$ Similarly, if $\widetilde{\kappa}$ is the unique solution to $M'(-r)=-\tau$ then $${\mathbb{P}}({\gamma_{\textup{rand}}(X)}<-\tau)= \frac{{\mathcal{L}_{\textup{rand}}}(-\widetilde{\kappa})e^{-\tau \widetilde{\kappa}}}{\widetilde{\kappa} \sqrt{2\pi M''(-\widetilde{\kappa})}}\left(1+O\left(\frac{1}{ \sqrt{\widetilde{\kappa}}} \right)\right).$$
Before proving this theorem, we need to show that ${\mathcal{L}_{\textup{rand}}}(r+it)$ is rapidly decreasing in $t$.
\[DecayLaplace\] Let $s=r+it \in \mathbb{C}$ where $|r|$ is large. Then, in the range $|t|\geq |r|$ we have
$$|{\mathcal{L}_{\textup{rand}}}(s)|\leq \exp\left(- \frac{|t|}{4\log|t|} \right) {\mathcal{L}_{\textup{rand}}}(r).$$
For simplicity we suppose that $r$ and $t$ are both positive. Since $|h_p(s)|\leq h_p(r)$ we obtain that for any $y\geq 2$ $$\label{decay1}
\frac{\left|{\mathcal{L}_{\textup{rand}}}(s)\right|}{{\mathcal{L}_{\textup{rand}}}(r)}\leq \prod_{p>y}\frac{|h_p(s)|}{h_p(r)}.$$ Moreover, the same argument leading to shows that for primes $p>|s|^{2/3}$ we have $$h_p(s)= \cosh\left(\frac{s\log p}{p}\right) \left(1+O\left(\frac{|s|\log p}{p^2}\right)\right).$$ Let $y=t(\log t)^2$. Since $\log\cosh(z)=z^2/2+ O(|z|^4)$ for $|z|\leq 1$, we deduce that for all primes $p>y$ $$\frac{h_p(s)}{h_p(r)}
= \exp\left(\frac{(s^2-r^2)(\log p)^2}{2p^2}+ O\left(\frac{t\log p}{p^{2}}+\frac{t^4(\log p)^4}{p^4}\right)\right).$$ Since $\text{Re}(s^2-r^2)=-t^2$, it follows from the prime number theorem and equation that $$\begin{aligned}
\frac{\left|{\mathcal{L}_{\textup{rand}}}(s)\right|}{{\mathcal{L}_{\textup{rand}}}(r)}
&\leq \exp\left(-\frac{t^2}{2}\sum_{p>y}\frac{(\log p)^2}{p^{2}}+ O\left(t\sum_{p>y}\frac{\log p}{p^2}+t^4\sum_{p>y}\frac{(\log p)^4}{p^4}\right)\right)\leq \exp\left(- \frac{t}{4\log t} \right).\end{aligned}$$
We only prove the estimate for ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)$ since the corresponding asymptotic for ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}<-\tau)$ requires only minor modifications.
Let $0<\lambda<1/(2\kappa)$ be a real number to be chosen later. Note that ${\gamma_{\textup{rand}}(X)}>\tau$ if and only if $\exp({\gamma_{\textup{rand}}(X)}-\tau)>1$. Therefore, using Lemma \[SmoothPerron\] with $N=1$ we obtain $$\label{approximation1}
\begin{aligned}
0&\leq \frac{1}{2\pi i}\int_{\kappa-i\infty}^{\kappa+i\infty}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s}\frac{ds}{s}-{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\\
&\leq \frac{1}{2\pi i}\int_{\kappa-i\infty}^{\kappa+i\infty} {\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s} \frac{\left(e^{\lambda s}-1\right)}{\lambda s} \frac{\left(1-e^{-\lambda s}\right)}{s}ds.
\end{aligned}$$ Since $\lambda\kappa<1/2$ we have $|e^{\lambda s}-1|\leq 3 \text{ and } |e^{-\lambda s}-1|\leq 2$. Hence, by Lemma \[DecayLaplace\] we obtain $$\label{error12}
\int_{\kappa-i\infty}^{\kappa-i\kappa}+ \int_{\kappa+i\kappa}^{\kappa+i\infty}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s}\frac{ds}{s} \ll \frac{e^{-\kappa/(4\log \kappa)}}{\lambda \kappa} {\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa},$$ and similarly $$\label{error2}
\int_{\kappa-i\infty}^{\kappa-i\kappa}+ \int_{\kappa+i\kappa}^{\kappa+i\infty}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s} \frac{\left(e^{\lambda s}-1\right)}{\lambda s} \frac{\left(1-e^{-\lambda s}\right)}{s}ds \ll \frac{e^{-\kappa/(4\log \kappa)}}{\lambda \kappa} {\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa}.$$ Let $s=\kappa+it$. If $|t|\leq \kappa$ then $\left|(1-e^{-\lambda s})(e^{\lambda s}-1)\right|\ll \lambda^2|s|^2$. Since $|{\mathcal{L}_{\textup{rand}}}(s)|\leq |{\mathcal{L}_{\textup{rand}}}(\kappa)$ we derive $$\int_{\kappa-i\kappa}^{\kappa+i\kappa} {\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s} \frac{\left(e^{\lambda s}-1\right)}{\lambda s} \frac{\left(1-e^{-\lambda s}\right)}{s}ds \ll \lambda\kappa{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}.$$ Therefore, combining this estimate with equations , and we deduce that $$\label{approximation2}
{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau) - \frac{1}{2\pi i}\int_{\kappa-i\kappa}^{\kappa+i\kappa}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s^2} ds \ll \left(\lambda\kappa+\frac{e^{-\kappa/(4\log \kappa)}}{\lambda \kappa}\right){\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa}.$$ On the other hand, it follows from equation that for $|t|\leq \kappa$ we have $$M(\kappa+it)= M(\kappa)+it M'(\kappa)-\frac{t^2}{2}M''(\kappa)+ O\left(|t|^3\frac{1}{\kappa^2}\right).$$ Also, note that $$\frac{e^{\lambda s}-1}{\lambda s^2}=\frac{1}{s}\big(1+O(\lambda \kappa)\big)= \frac{1}{\kappa}\left(1-i\frac{t}{\kappa}+ O\left(\lambda \kappa+\frac{t^2}{\kappa^2}\right)\right).$$ Hence, using that ${\mathcal{L}_{\textup{rand}}}(s)=\exp(M(s))$ and $M'(\kappa)=\tau$ we obtain $$\begin{aligned}
&{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s^2}\\
= &\frac{1}{\kappa}{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa}\exp\left(-\frac{t^2}{2}M''(\kappa)\right)
\left(1-i\frac{t}{\kappa}+O\left(\lambda\kappa+ \frac{t^2}{\kappa^2}+ |t|^3\frac{1}{\kappa^2}\right)\right).\\\end{aligned}$$ Thus, we get $$\label{TaylorSaddle}
\begin{aligned}
&\frac{1}{2\pi i}\int_{\kappa-i\kappa}^{\kappa+i\kappa}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s^2} ds\\
=& \frac{1}{\kappa}{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa} \frac{1}{2\pi} \int_{-\kappa}^{\kappa}\exp\left(-\frac{t^2}{2}M''(\kappa)\right)
\left(1+ O\left(\lambda\kappa+ \frac{t^2}{\kappa^2}+ |t|^3\frac{1}{\kappa^2}\right)\right)dt
\end{aligned}$$ since the integral involving $it/{\kappa}$ vanishes. Further, since $M''(\kappa)\asymp 1/\kappa$ by we derive $$\frac{1}{2\pi} \int_{-\kappa}^{\kappa}\exp\left(-\frac{t^2}{2}M''(\kappa)\right)dt= \frac{1}{\sqrt{2\pi M''(\kappa)}}\left(1+O\left(e^{-\sqrt{\kappa}}\right)\right),$$ and $$\int_{-\kappa}^{\kappa}|t|^n\exp\left(-\frac{t^2}{2}M''(\kappa)\right)dt\ll \frac{1}{M''(\kappa)^{(n+1)/2}}\ll \frac{\kappa^{n/2}}{\sqrt{M''(\kappa)}}.$$ Inserting these estimates in we deduce that $$\label{main}
\begin{aligned}
&\frac{1}{2\pi i}\int_{\kappa-i\kappa}^{\kappa+i\kappa}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\frac{e^{\lambda s}-1}{\lambda s^2} ds\\
=& \frac{{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau\kappa}}{\kappa\sqrt{2\pi M''(\kappa)}}
\left(1+ O\left(\lambda\kappa+ \frac{1}{\sqrt{\kappa}}\right)\right).
\end{aligned}$$ Finally, combining the estimates and and choosing $\lambda= \kappa^{-2}$ completes the proof.
Again we only prove the estimate for ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)$, as the corresponding estimate for ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}<-\tau)$ can be obtained similarly. By Theorem \[SaddlePoint\] and equation , we have $${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)= \frac{{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}}{\kappa \sqrt{2\pi M''(\kappa)}}\left(1+O\left(\frac{1}{ \sqrt{\kappa}} \right)\right)= \exp\Big(M(\kappa)-\tau\kappa+O(\log \kappa)\Big),$$ where $\kappa$ is the unique solution to $M'(\kappa)=\tau$. Furthermore, by we have $$\label{EstSaddle1}
\tau= \log\kappa+\log\log\kappa+A_1+O\left(\frac{\log\log \kappa}{\log\kappa}\right),$$ and hence we deduce from that $$\label{EstSaddle2}
{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)= \exp\left(-\kappa +O\left(\frac{\kappa \log\log \kappa}{\log \kappa}\right)\right).$$ Now, implies that $\log\kappa=\tau+O(\log \tau)$ and $$\kappa\log \kappa= e^{\tau-A_1}\left(1+O\left(\frac{\log \tau}{\tau}\right)\right).$$ Thus, we obtain $$\label{OrderSaddle}
\kappa= \frac{e^{\tau-A_1}}{\tau}\left(1+O\left(\frac{\log \tau}{\tau}\right)\right).$$ The result follows upon inserting the estimate in .
The distribution of extreme values of ${\gamma_{\mathbb{Q}(\sqrt D)}}$: proof of Theorem \[MainTheorem\]
=========================================================================================================
By Theorem \[AsympLaplace\] there exists a constant $B>0$ and a set of fundamental discriminants ${\mathcal{E}(x)}\subset {\mathcal{F}(x)}$ with $|{\mathcal{E}(x)}|=O\left(\sqrt{x}\right)$, such that for all complex numbers $s$ with $|s|\leq \log x/(B(\log\log x)^2)$ we have $$\label{Theorem1.5}
\frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)}\exp\left(s \cdot {\gamma_{\mathbb{Q}(\sqrt D)}}\right) = {\mathcal{L}_{\textup{rand}}}(s)+
O\left(\exp\left(-\frac{\log x}{50\log\log x}\right)\right).$$ To shorten our notation we let $${\mathbb{P}}_x({\gamma_{\mathbb{Q}(\sqrt D)}}\in S):=\frac{1}{|{\mathcal{F}(x)}|}\big|\{D\in {\mathcal{F}(x)}: {\gamma_{\mathbb{Q}(\sqrt D)}}\in S \}\big|,$$ and $${\mathcal{L}_x}(s)= \frac{1}{|{\mathcal{F}(x)}|}\sum_{D\in {\mathcal{F}(x)}\setminus \mathcal{E}(x)}\exp\left(s \cdot {\gamma_{\mathbb{Q}(\sqrt D)}}\right).$$
As before, $\kappa$ denotes the unique solution to $M'(r)=\tau$. Let $N$ be a positive integer and $0<\lambda<\min\{1/(2\kappa), 1/N\}$ be a real number to be chosen later.
Let $Y=\log x/(2B(\log\log x)^2)$. If $x$ is large enough then equation insures that $\kappa\leq Y$. Also, note that holds for all complex numbers $s=\kappa+it$ with $|t|\leq Y$. We consider the integrals $$I(\tau)= \frac{1}{2\pi i}\int_{\kappa-i\infty}^{\kappa+i\infty}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}$$ and $$J_x(\tau)= \frac{1}{2\pi i}\int_{\kappa-i\infty}^{\kappa+i\infty}{\mathcal{L}_x}(s)e^{-\tau s}\left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}.$$ Then, using equation we obtain $$\label{Mellin1}
{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\leq I(\tau)\leq {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau-\lambda N),$$ and $$\label{Mellin2}
{\mathbb{P}}_x\Big({\gamma_{\mathbb{Q}(\sqrt D)}}>\tau\Big)+O\left(x^{-1/2}\right)\leq J_x(\tau)\leq {\mathbb{P}}_x\Big({\gamma_{\mathbb{Q}(\sqrt D)}}>\tau-\lambda N\Big)+O\left(x^{-1/2}\right),$$ since $|{\mathcal{E}(x)}|/|{\mathcal{F}(x)}|\ll x^{-1/2}.$
Further, using that $|e^{\lambda s}-1|\leq 3$ and $|{\mathcal{L}_{\textup{rand}}}(s)|\leq {\mathcal{L}_{\textup{rand}}}(\kappa)$ we obtain $$\label{tail1}
\int_{\kappa-i\infty}^{\kappa-iY}+ \int_{\kappa+iY}^{\kappa+i\infty}{\mathcal{L}_{\textup{rand}}}(s)e^{-\tau s}\left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}\ll \frac{1}{N}\left(\frac{3}{\lambda Y}\right)^N{\mathcal{L}_{\textup{rand}}}(\kappa)
e^{-\tau \kappa}.$$ Similarly, using that $|{\mathcal{L}_x}(s)|\leq {\mathcal{L}_x}(\kappa)$ along with Theorem \[AsympLaplace\] we get $$\label{tail2}
\int_{\kappa-i\infty}^{\kappa-iY}+ \int_{\kappa+iY}^{\kappa+i\infty}{\mathcal{L}_x}(s) e^{-\tau s}\left(\frac{e^{\lambda s}-1}{\lambda s}\right)^N\frac{ds}{s}
\ll \frac{1}{N}\left(\frac{3}{\lambda Y}\right)^N{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}.$$ Moreover, note that $|(e^{\lambda s}-1)/\lambda s|\leq 3$, which is easily seen by looking at the cases $|\lambda s|\leq 1$ and $|\lambda s|>1.$ Therefore, combining equations , and we obtain $$\label{difference1}
J_x(\tau)- I(\tau)\ll \frac1N\left(\frac{3}{\lambda Y}\right)^N{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}+ \frac{Y}{\kappa}3^N e^{-\tau\kappa} \exp\left(-\frac{\log x}{50\log\log x}\right).$$ Furthermore, it follows from Theorem \[SaddlePoint\] and equation that $$\label{order}
{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\asymp\frac{{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}}{k\sqrt{M''(\kappa)}}
\asymp\frac{{\mathcal{L}_{\textup{rand}}}(\kappa)e^{-\tau \kappa}}{\sqrt{\kappa}}.$$ Thus, choosing $N=[\log\log x]$ and $\lambda= e^{10}/Y$ we deduce that $$\label{difference2}
J_x(\tau)- I(\tau)\ll \frac{1}{(\log x)^{5}} {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau).$$ On the other hand, it follows from Theorem \[ExponentialDecay\] that $$\label{shift}
\begin{aligned}
{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau\pm \lambda N)&= {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\exp\left(O\left(\lambda N \frac{e^{\tau}}{\tau}\right)\right)\\
&= {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\left(1+O\left(\frac{e^{\tau}(\log\log x)^3}{\tau\log x}\right)\right).\\
\end{aligned}$$ Combining this last estimate with , , and we obtain $$\begin{aligned}
{\mathbb{P}}_x({\gamma_{\mathbb{Q}(\sqrt D)}}>\tau)&
\leq J_x(\tau)+O\big(x^{-1/2}\big) \\
&\leq I(\tau)+ O\left(\frac{{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)}{(\log x)^{5}}+x^{-1/2}\right)\\
&\leq {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\left(1+O\left(\frac{e^{\tau}(\log\log x)^3}{\tau\log x}\right)\right)+ O\big(x^{-1/2}\big),\end{aligned}$$ and $$\begin{aligned}
{\mathbb{P}}_x({\gamma_{\mathbb{Q}(\sqrt D)}}>\tau)&
\geq J_x(\tau+\lambda N)+O\big(x^{-1/2}\big) \\
&\geq I(\tau+\lambda N)+ O\left(\frac{{\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)}{(\log x)^{5}}+ x^{-1/2}\right)\\
&\geq {\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\left(1+O\left(\frac{e^{\tau}(\log\log x)^3}{\tau\log x}\right)\right)+ O\big(x^{-1/2}\big).\\\end{aligned}$$ The result follows from these estimates together with the fact that ${\mathbb{P}}({\gamma_{\textup{rand}}(X)}>\tau)\gg x^{-1/4}$ in our range of $\tau$, by Theorem \[ExponentialDecay\].
[DDDD]{}
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[^1]: The author is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
|
---
abstract: |
Mahlmann and Schindelhauer (2005) defined a Markov chain which they called $k$-Flipper, and showed that it is irreducible on the set of all connected regular graphs of a given degree (at least 3). We study the 1-Flipper chain, which we call the flip chain, and prove that the flip chain converges rapidly to the uniform distribution over connected $2r$-regular graphs with $n$ vertices, where $n\geq 8$ and $r = r(n)\geq 2$. Formally, we prove that the distribution of the flip chain will be within $\varepsilon$ of uniform in total variation distance after $\text{poly}(n,r,\log(\varepsilon^{-1}))$ steps. This polynomial upper bound on the mixing time is given explicitly, and improves markedly on a previous bound given by Feder et al. (2006). We achieve this improvement by using a direct two-stage canonical path construction, which we define in a general setting.
This work has applications to decentralised networks based on random regular connected graphs of even degree, as a self-stabilising protocol in which nodes spontaneously perform random flips in order to repair the network.
*Keywords:* Markov chain; graph; connected graph; regular graph
author:
- |
Colin Cooper[^1]\
[Department of Computer Science]{}\
[King’s College London]{}\
[London WC2R 2LS, U.K.]{}\
[`[email protected]`]{}\
- |
Martin Dyer\
[School of Computing]{}\
[University of Leeds]{}\
[Leeds LS2 9JT, U.K.]{}\
[`[email protected]`]{}\
- |
Catherine Greenhill[^2]\
[School of Mathematics and Statistics]{}\
[UNSW Australia]{}\
[Sydney, NSW 2052, Australia]{}\
[`[email protected]`]{}\
- |
Andrew Handley[^3]\
[School of Computing]{}\
[University of Leeds]{}\
[Leeds LS2 9JT, U.K.]{}\
[`[email protected]`]{}
date: 13 June 2018
title: |
The flip Markov chain\
for connected regular graphs [^4] [^5]
---
Introduction
============
Markov chains which walk on the set of random regular graphs have been well studied. The switch chain is a very natural Markov chain for sampling random regular graphs. A transition of the switch chain is called a *switch*, in which two edges are deleted and replaced with two other edges, without changing the degree of any vertex. The switch chain was studied by Kannan et al. [@kannan99markovbipartite] for bipartite graphs, and in [@cooper05sampling] for regular graphs. Although a switch changes only a constant number of edges per transition, it is not a local operation since these edges may be anywhere in the graph.
Random regular graphs have several properties which make them good candidates for communications networks: with high probability, a random $\Delta$-regular graph has logarithmic diameter and high connectivity, if $\Delta\geq 3$. (See, for example, [@Wormald99].) Bourassa and Holt [@bourassa03swan] proposed a protocol for a decentralised communications network based on random regular graphs of even degree. Since then, Markov chains which sample regular graphs using local operations have been applied to give *self-stabilising* or *healing* protocols for decentralised communications networks which have a regular topology. To help the network to recover from degradation in performance due to arrivals and departures, clients in a decentralised network can spontaneously perform local transformations to re-randomize the network, thereby recovering desirable properties.
Mahlmann and Schindelhauer [@schindelhauer05kflipper] observed that the switch operation is unsuitable for this purpose, since a badly-chosen switch may disconnect a connected network. Furthermore, since the switch move is non-local there are implementation issues involved in choosing two random edges in a decentralised network. (If the network is an expander then this problem can be solved by performing two short random walks in the network and taking the final edge of each for the next operation [@bourassa03swan].)
To overcome these problems, Mahlmann and Schindelhauer proposed an alternative operation which they called a $k$-Flipper. When $k=1$, the 1-Flipper (which we call the *flip*) is simply a restricted switch operation, in which the two edges to be switched must be at distance one apart in the graph. Figure \[fig:switchAndFlip\] illustrates a switch and a flip.
(-1,0) circle (0.1); at (-1.2,0) [$b$]{}; (-1,1) circle (0.1); at (-1.2,1.0) [$a$]{}; (0,0) circle (0.1); at (0.2,0) [$c$]{}; (0,1) circle (0.1); at (0.2,1) [$d$]{}; (-1,0) – (-1,1); (0,0) – (0,1); at (1.5,0.5) [[$\Rightarrow_S$]{}]{}; (3,0) circle (0.1); at (2.8,0) [$b$]{}; (3,1) circle (0.1); at (2.8,1.0) [$a$]{}; (4,0) circle (0.1); at (4.2,0) [$c$]{}; (4,1) circle (0.1); at (4.2,1) [$d$]{}; (3,1) – (4,0); (3,0) – (4,1); (7,0) circle (0.1); at (6.8,0) [$b$]{}; (7,1) circle (0.1); at (6.8,1.0) [$a$]{}; (8,0) circle (0.1); at (8.2,0) [$c$]{}; (8,1) circle (0.1); at (8.2,1) [$d$]{}; (7,1) – (7,0) – (8,0) – (8,1); at (9.5,0.5) [[$\Rightarrow_F$]{}]{}; (11,0) circle (0.1); at (10.8,0) [$b$]{}; (11,1) circle (0.1); at (10.8,1.0) [$a$]{}; (12,0) circle (0.1); at (12.2,0) [$c$]{}; (12,1) circle (0.1); at (12.2,1) [$d$]{}; (11,1) – (12,0) – (11,0) – (12,1);
By design, the flip operation preserves the degree of all vertices and preserves connectivity. Indeed, it is the smallest such randomising operation: no edge exchange on fewer than four vertices preserves the degree distribution, and the only shallower tree than the 3-path is the 3-star, which has no degree-preserving edge exchange. In order to use flips as a self-stabilising protocol, each peer waits for a random time period using a Poisson clock, after which it instigates a new flip. This involves communication only with peers at distance at most two away in the network.
We will show that flips have asymptotic behaviour comparable to switches, in that both operations randomise a network in polynomial time. Simulations also suggest that they operate equally fast, up to a constant factor (see [@AHthesis Section 2.3.6]). A discussion of a distributed implementation of the flip operation is given in [@CDH].
Our approach will be to relate the mixing time of the flip chain ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ to an intermediate chain ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$, which is the switch chain restricted to connected graphs. In turn, the mixing time of the connected switch chain ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ will be related to the mixing time of the (standard) switch chain ${\ensuremath{\mathcal{M}_\mathrm{S}}}$, using the analysis from [@cooper05sampling]. Each of these two steps will be performed using a technique which we call “two-stage direct canonical path construction”, described in Section \[sec:two-stage-direct\]. Specifically, we prove the following result. This is a corrected version of the bound given in [@CDH], which failed on certain rare graphs.
For each $n\geq 8$ let $\Delta=\Delta(n)\geq 4$ be a positive even integer such that $n\geq \Delta +1$. The mixing time of the flip Markov chain ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ on the set of $\Delta$-regular graphs with $n$ vertices is at most $$480\, \Delta^{35}\, n^{15}\, \left(\Delta n\log(\Delta n) + \log(\varepsilon^{-1})\right),$$ of which the two-stage direct construction is responsible for a factor of $480\, \Delta^{12} n^7$. \[thm:flip-mixing-time\]
Section \[sec:solution\] relates the flip chain and the connected switch chain, while Section \[sec:disconnected\] relates the connected switch chain and the switch chain. The proof of Theorem \[thm:flip-mixing-time\] can be found at the end of Section \[sec:disconnected\]. The rest of this section contains some background material, some helpful graph theory results, precise definitions of the flip and switch Markov chains, and some definitions regarding canonical paths.
We remark that a “one-stage direct analysis” of the flip chain seems difficult. Furthermore, the multicommodity flow defined for the switch chain in [@cooper05sampling] does not apply directly to the connected switch chain, since the flow may pass through disconnected graphs, even when both end-states are connected. It may be possible to circumvent this problem by using an alternative multicommodity flow argument, but we do not explore that option here.
Notation, terminology and history
---------------------------------
We will use the notation ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ to denote the switch operation as illustrated in Figure \[fig:switchAndFlip\].
The switch chain ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ applies a random switch at each step, and the flip chain ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ applies a random flip at each step. (We define both chains formally in Section \[s:chains\].) The state space $\Omega_S$ for the switch chain is the set of all $\Delta$-regular graphs on the vertex set $[n]=\{1,2,\ldots, n\}$, while the state space $\Omega_F$ for the flip chain is the set of all connected graphs in $\Omega_S$.
The switch chain is known to be ergodic, and has uniform stationary distribution on $\Omega_S$. Mahlmann and Schindelhauer [@schindelhauer05kflipper] showed that the flip chain is ergodic on $\Omega_F$. They also proved that the transitions of the flip chain are symmetric, so that the stationary distribution of the flip chain is uniform on $\Omega_F$. (We use a slightly different transition procedure, described in Section \[s:chains\], which is also symmetric.)
The mixing time of a Markov chain is a measure of how many steps are required before the Markov chain has a distribution which is close to stationary. Kannan et al. [@kannan99markovbipartite] considered the mixing time of the switch Markov chain on “near regular” bipartite graphs. Cooper et al. [@cooper05sampling] gave a polynomial bound on the mixing time ${\ensuremath{\tau\ifthenelse{\equal{CDG}{}}{}{_{\mathrm{CDG}}}(\varepsilon)}}$ of the switch chain for all regular simple graphs. The result of [@cooper05sampling] allows the degree $\Delta=\Delta(n)$ to grow arbitrarily with $n$: it is not restricted to regular graphs of constant degree.
In Section \[sec:solution\] we extend the result of [@cooper05sampling] from switches to flips, using a method which we call *two-stage direct canonical path construction*. We restrict our attention to even degree, since this allows the network to have any number of vertices (even or odd). So for the rest of the paper we assume that $\Delta=2r$ for some $r\geq 2$, unless otherwise stated. It happens that even-degree graphs have some other desirable properties, as will we see in Section \[s:useful\], which will simplify parts of our argument. However, we see no reason why the flip chain would fail to be rapidly mixing for odd degrees.
Mahlmann and Schindelhauer [@schindelhauer05kflipper] do not provide a bound for the mixing time of ${\ensuremath{\mathcal{M}_\mathrm{F}}}$, though they do show [@schindelhauer05kflipper Lemma 9] that a similar chain with the flip edges at a distance of ${{\ensuremath{\mathrm{\Theta}\!\left(\Delta^2 n^2 \log{\varepsilon^{-1}}\right)}}}$ apart will give an $\varepsilon$-approximate expander in time ${{\ensuremath{\mathrm{O}\!\left(\Delta n\right)}}}$. However, these moves are highly non-local and therefore are unsuitable for the application to self-stabilisation of decentralised networks.
An upper bound for the mixing time of ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ was given by Feder et al. [@saberi06switchflip] using a comparison argument with the switch Markov chain, ${\ensuremath{\mathcal{M}_\mathrm{S}}}$. Applying a comparison argument to relate these chains is a difficult task, since a switch may disconnect a connected graph and a flip cannot. Feder et al. [@saberi06switchflip] solve this difficulty by embedding ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ into a restricted switch chain which only walks on connected graphs. They prove that this can be done in such a way that a path in ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ is only polynomially lengthened. Their result relaxes the bound in [@cooper05sampling] by ${{\ensuremath{\mathrm{O}\!\left(\Delta^{41} n^{45}\right)}}}$. By employing a two-stage direct construction, we give a much tighter result, relaxing the bound on ${\ensuremath{\tau\ifthenelse{\equal{CDG}{}}{}{_{\mathrm{CDG}}}(\varepsilon)}}$ by a factor of ${{\ensuremath{\mathrm{O}\!\left(r^{12} n^7\right)}}}$ in the case of $2r$-regular graphs with $r\geq 2$.
Recently, Allen-Zhu et al. [@ABLMO] proved a result related to the work of Mahlmann and Schindelhauer [@schindelhauer05kflipper Lemma 9]. They proved that for $\Delta=\Omega(\log n)$, after at most $O(n^2\Delta^2\sqrt{\log n})$ steps of the flip chain, the current graph will be an algebraic expander in the sense that the eigenvalue gap of the adjacency matrix is $\Omega(\Delta)$. The proof is based on tracking the change of the graph Laplacian using a potential function rather than analysing the flip chain directly, and does not address the distribution of the output graph, or the case $\Delta = o(\log n)$, which may be the most convenient in practical applications.
Indeed, it may be possible to reduce the mixing time bound by combining our analysis with the results of [@ABLMO]. If $\Delta=o(\log n)$ then the bound from Theorem \[thm:flip-mixing-time\], is $O(n^{16}\operatorname{poly}(\log n))$. Otherwise, when $\Delta = \Omega(\log n)$, the first $O(n^2\Delta^2\sqrt{\log n})$ steps of the flip chain may be viewed as a preprocessing phase. As shown by [@ABLMO], at the end of this preprocessing phase the current graph is an expander with high probability, and hence has logarithmic diameter. The diameter appears in our analysis, for example in the proof of Lemma \[lem:lF\] below, and in the analysis of the switch chain [@cooper05sampling]. Therefore, it is possible that this preprocessing phase could result in an improvement in the mixing time bound of at least one factor of the order $\frac{\log n}{n}$. To achieve this improvement, it appears necessary to show that a canonical path between two expanders only visits graphs with logarithmic diameter. We do not pursue this approach here.
Graph-theoretical preliminaries {#s:useful}
-------------------------------
For further details see, for example, Diestel [@diestel]. Unless otherwise stated, $G$ denotes a simple graph $G = (V,E)$ with $V=\{1,2,\ldots,n\}$. The set of neighbours in $G$ of a vertex $v$ is denoted by $N_G(v)$.
A vertex set $S\subseteq V$ has *edge boundary* $\partial_E{S} = \{uv\,|\,u \in S, v \notin S\}$, which contains the edges crossing from inside to outside of $S$.
A graph $G$ is 2-edge-connected if it is connected and every edge of $G$ lies in a cycle. More generally, an edge-cut in $G$ is a set $F\subseteq E$ of edges of $G$ such that the graph $G - F$ (which is $G$ with the edges in $F$ deleted) is disconnected. If $|F|=k$ then $F$ is a $k$-edge-cut. If vertices $v$, $w$ belong to distinct components of $G-F$ then we say that $F$ *separates* $v$ and $w$ in $G$.
A *cut vertex* in a graph $G$ is a vertex $v$ such that deleting $v$ disconnects the graph.
If $p$ is a path from $v$ to $w$ then we say that $p$ is a $(v,w)$-path. Further, if $p=v\cdots a\cdots b\cdots w$ then we write $p[a:b]$ to denote the (inclusive) subpath of $p$ with endvertices $a$ and $b$.
We make use of the following structural properties of $2r$-regular graphs with $r\geq 2$.
Let $r\geq 2$ be an integer and let $G=(V,E)$ be a $2r$-regular graph on $n\geq \max\{ 8,\, 2r+1\}$ vertices. Then $G$ satisfies the following properties:
1. For any $S \subseteq V$, the edge boundary $\partial_E{S}$ of $S$ is even.
2. Every edge of $G$ lies in a cycle; that is, every connected component of $G$ is 2-edge-connected.
3. $G$ has fewer than $n/(2r)$ connected components.
4. Let $u,v$ be distinct vertices in the same connected component of $G$. The number of $2$-edge-cuts in $G$ which separate $u$ and $v$ is at most $n^2/(15 r^2)$.
\[lem:even-d\]
For (i), we argue by contradiction. Assume that there is an odd number $k = 2k' + 1$ of edges incident upon some set $S$. Let $G[S]$ be the subgraph induced by $S$. Let $|S| = s$ and suppose that $G[S]$ has $m$ edges. Denote the sum of degrees of the vertices in $S$ by $d_G(S)$. Clearly $d_G(S) = 2rs$. Exactly $k$ edges incident with $S$ have precisely one endvertex in $S$, so $$2m = d_{G[S]}(S) = 2r s - k = 2rs - 2k' - 1 = 2(rs-k') - 1,$$ but no integer $m$ satisfies this equation. Hence our assumption is false, proving (i).
For (ii), it suffices to observe that every component of $G$ is Eulerian, as all vertices have even degree. Next, (iii) follows since the smallest $2r$-regular graph is $K_{2r+1}$, and $G$ can have at most $n/(2r+1) < n/(2r)$ components this small. The proof of (iv) is deferred to the Appendix, as it is somewhat lengthy.
Canonical paths {#s:canonical}
---------------
We now introduce some Markov chain terminology and describe the canonical path method. For further details, see for example Jerrum [@jerrum03lectures] or Sinclair [@Sincla92].
Let $\mathcal{M}$ be a Markov chain with finite state space $\Omega$ and transition matrix $P$. We suppose that $\mathcal{M}$ is ergodic and has unique stationary distribution $\pi$. The graph underlying the Markov chain $\mathcal{M}$ is given by $\mathcal{G}=(\Omega,E(\mathcal{M}))$ where the edge set of $\mathcal{G}$ corresponds to (non-loop) transitions of $\mathcal{M}$; specifically, $$E(\mathcal{M}) = \{ xy \mid x,y\in\Omega,\, x\neq y,\, P(x, y) > 0\}$$ We emphasise that $\mathcal{G}$ has no loops.
Let $\sigma$, $\rho$ be two probablity distributions defined on a finite set $\Omega$. The *total variation distance* between $\sigma$ and $\rho$, denoted ${\ensuremath{\mathrm{d_\mathrm{TV}}}}(\sigma,\rho)$, is defined by $${\ensuremath{\mathrm{d_\mathrm{TV}}}}(\sigma,\rho) = \nfrac{1}{2}\sum_{x\in\Omega} |\sigma(x) - \rho(x)|.$$ The *mixing time* ${\ensuremath{\tau\ifthenelse{\equal{}{}}{}{_{\mathrm{}}}(\varepsilon)}}$ of a Markov chain is given by $${\ensuremath{\tau\ifthenelse{\equal{}{}}{}{_{\mathrm{}}}(\varepsilon)}}= \max_{x\in{}X} \,\min\,\{T \ge 0\ |\ {\ensuremath{\mathrm{d_\mathrm{TV}}}}(P^t_x,
\pi) \le \varepsilon \mbox{ for all } t \ge T\},$$ where $P^t_x$ is the distribution of the random state $X_t$ of the Markov chain after $t$ steps with initial state $x$. Let the eigenvalues of $P$ be $$1 = \lambda_0 > \lambda_1 \geq \lambda_2 \cdots \geq \lambda_{N-1} \geq -1.$$ Then the mixing time satisfies $$\label{mixing}
\tau(\varepsilon) \leq (1-\lambda_\ast)^{-1}\left(\ln(1/\pi^\ast) +
\ln(\varepsilon^{-1}) \right)$$ where $\lambda_\ast = \max\{ \lambda_1,\, |\lambda_{N-1}|\}$ and $\pi^\ast = \min_{x\in\Omega}\pi(x)$ is the smallest stationary probability. (See [@Sincla92 Proposition 1].)
The *canonical path method*, introduced by Jerrum and Sinclair [@jerrum89conductancepermanent], gives a method for bounding $\lambda_1$. Given any two states $x,y\in\Omega$, a “canonical” directed path $\gamma_{xy}$ is defined from $x$ to $y$ in the graph $\mathcal{G}$ underlying the Markov chain, so that each step in the path corresponds to a transition of the chain. The *congestion* $\bar{\rho}(\Gamma)$ of the set $\Gamma = \{ \gamma_{xy} \mid x,y\in\Omega \}$ of canonical paths is given by $$\bar{\rho}(\Gamma) = \max_{e\in E({\ensuremath{\mathcal{M}}})} Q(e)^{-1} \sum_{\substack{x,y\in\Omega\\ e\in\gamma_{xy}}}\, \pi(x)\pi(y)|\gamma_{xy}|$$ where $Q(e) = \pi(u)\, P(u,v)$ when $e=uv\in E({\ensuremath{\mathcal{M}}})$. If the congestion along each transition is low then the chain should mix rapidly. This can be made precise as follows: if $\lambda^\ast = \lambda_1$ then $$\label{canonicalpath}
(1-\lambda_1)^{-1} \le\ \bar{\rho}(\Gamma).$$ (See [@Sincla92 Theorem 5] or [@jerrum03lectures] for more details.) In many applications the chain $\mathcal{M}$ is made *lazy* by replacing its transition matrix $P$ by $(I+P)/2$: this ensures that the chain has no negative eigenvalues and hence that $\lambda_\ast = \lambda_1$. However, in many cases it is easy to see that $(1+\lambda_{N-1})^{-1}$ is smaller than the best-known upper bound on $(1-\lambda_1)^{-1}$, such as that provided by the canonical path method (\[canonicalpath\]). In this situation, combining (\[mixing\]) and (\[canonicalpath\]) gives an upper bound on the mixing time without the need to make the chain lazy.
The *multicommodity flow* method is a generalisation of the canonical path method. Rather than send $\pi(x)\pi(y)$ units down a single “canonical” path from $x$ to $y$, a set $\mathcal{P}_{xy}$ of paths is defined, and the flow from $x$ to $y$ is divided among them. The congestion can be defined similarly, and low congestion leads to rapid mixing: see Sinclair [@Sincla92].
Cooper et al. [@cooper05sampling] gave a multicommodity flow argument to bound the mixing time of the switch chain ${\ensuremath{\mathcal{M}_\mathrm{S}}}$. For every pair of distinct states $x,y$, a set of paths from $x$ to $y$ was defined, one for each vector $(\phi_v)_{v\in [n]}$, where $\phi_v$ is a “pairing” of the edges of the symmetric difference of $x$ and $y$ incident with the vertex $v$. Then the flow between $x$ and $y$ was shared equally among all these paths. In our analysis below, we assume that a pairing has been *fixed* for each distinct pair of states $x,y$, leading to one canonical path $\gamma_{xy}$ between them. We will work with the set of these canonical paths for the switch chain, $\Gamma_S = \{ \gamma_{xy}\}$. Since our congestion bounds will be independent on the choice of pairing, the same bound would hold if the original multicommodity flow from [@cooper05sampling] was used.
The switch and flip Markov chains {#s:chains}
---------------------------------
Given $\Delta\geq 3$ and $n\geq \Delta+1$, the *switch chain* ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ has state space $\Omega_S$ given by the set of all $d$-regular graphs on $n$ vertices. The flip chain ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ has state space $\Omega_F$ given by the set of all connected $\Delta$-regular graphs on $n$ vertices. We restrict our attention to regular graphs of even degree, setting $\Delta=2r$ for some integer $r\geq 2$.
Let $P_S$ (respectively, $P_F$) denote the transition matrix of the switch (respectively, flip) Markov chain. The stationary distribution of the switch (respectively, flip) chain is denoted by $\pi_S$ (respectively, $\pi_F$). The graph underlying the Markov chain will be denoted by $\mathcal{G}_S=(\Omega_S,E({\ensuremath{\mathcal{M}_\mathrm{S}}}))$ and $\mathcal{G}_F=(\Omega_F,E({\ensuremath{\mathcal{M}_\mathrm{F}}}))$, respectively.
The transition procedure for the switch Markov chains is given in Figure \[fig:MarkovSwitchChain\].
The switch chain is irreducible [@petersen; @taylor]. Furthermore, as $P_S(x,x) \geq 1/3$ for all $x\in\Omega_S$, the switch chain is aperiodic, and hence ergodic. Using an approach of Diaconis and Saloff-Coste [@DiaSal93 p. 702], this also implies that the smallest eigenvalue $\lambda_{N-1}$ of ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ satisfies $$(1+\lambda_{N-1})^{-1}\leq
\nfrac{1}{2}\, \max_{x\in\Omega_S} P(x,x)^{-1}\leq 3/2.$$ Since the upper bound on $(1-\lambda_1)^{-1}$ proved in [@cooper05sampling] is (much) larger than constant, this proves that we can apply the canonical path method to the switch chain without making the switch chain lazy.
If $x, y$ are distinct elements of $\Omega_S$ which differ by a switch then $P_S(x,y) = 1/(3 a_{n,2r})$ where (see [@cooper05sampling]) $$\label{an2r}
a_{n,2r} = \binom{rn}{2} - n\, \binom{2r}{2},$$ Hence the transition matrix $P_S$ is symmetric and the stationary distribution of the switch chain is uniform over $\Omega_S$.
Figure \[fig:MarkovFlipChain\] gives the transition procedure of the flip chain. Here we start from a randomly chosen vertex $a$ and perform a random walk from $a$ of length 3, allowing backtracking. There are $(2r)^3n$ choices for the resulting walk $(a,b,c,d)$, and we choose one of these uniformly at random.
Mahlmann and Schindelhauer proved that the flip chain is irreducible on $\Omega_F$ [@schindelhauer05kflipper]. For all states $x\in\Omega_F$ we have $P(x,x) \geq 1/(2r)$, since in particular the proposed transition is rejected when $c=a$. Therefore the flip chain is aperiodic, and hence ergodic. Again, by [@DiaSal93 p. 702], this implies that the smallest eigenvalue $\lambda_{N-1}$ of ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ satisfies $(1+\lambda_{N-1})^{-1} \leq r$. This in turn allows us to apply the canonical path method without making the chain lazy, since the upper bound we obtain on $(1-\lambda_1)^{-1}$ will be (much) bigger than $r$.
Now suppose that $x, y$ are distinct elements of $\Omega_F$ which differ by a flip, and that this flip replaces edges $ab, cd$ with $ac, bd$. Then if exactly one of the edges $bc$, $ad$ is present in $x$, say $bc$, then the transition from $x$ to $y$ takes place if and only if the chosen 3-walk is $(a,b,c,d)$ or $(d,c,b,a)$. Hence in this case $$P_F(x,y) = \frac{2}{(2r)^3 n} = P_F(y,x).$$ If both of the edges $bc$ and $ad$ are present then either of them could take the role of the “hub edge” of the flip, and $$P_F(x,y) = \frac{4}{(2r)^3 n} = P_F(y,x).$$ Hence $P_F$ is a symmetric matrix and the stationary distribution of the flip chain is uniform, as claimed. (Mahlmann and Schindelhauer use a slightly different transition procedure, and must argue about the number of triangles which contain a given edge in order to show that their transitions are symmetric: see [@schindelhauer05kflipper Lemma 3].)
Two-stage direct canonical path construction {#sec:two-stage-direct}
============================================
Suppose ${\ensuremath{\mathcal{M}}}$ is an ergodic Markov chain whose mixing time we would like to study, and ${\ensuremath{\mathcal{M}}}'$ is a chain for which a canonical path set $\Gamma'$ has been defined, with congestion bounded by $\bar{\rho}(\Gamma')$. Let $P$, $\pi$ and $\Omega$ (respectively, their primed counterparts) be the transition matrix, stationary distribution and state space of ${\ensuremath{\mathcal{M}}}$ (respectively, ${\ensuremath{\mathcal{M}}}'$). Guruswami [@guruswami] proved that under mild conditions on ${\ensuremath{\mathcal{M}}}'$, a bound on the mixing time of ${\ensuremath{\mathcal{M}}}'$ implies the existence of an appropriate set $\Gamma'$ with bounded congestion.
We wish to “simulate” transition in ${\ensuremath{\mathcal{M}}}'$ using sequences of transitions in ${\ensuremath{\mathcal{M}}}$, and showing that not too many transitions in ${\ensuremath{\mathcal{M}}}$ can be involved in any one such simulation. Together with $\bar{\rho}(\Gamma')$ this results on a bound of the congestion of $\Gamma$, and hence on the mixing time of ${\ensuremath{\mathcal{M}}}$. Here we assume that $\Omega\subseteq\Omega'$.
If $\Omega$ is a *strict* subset of $\Omega'$ then canonical paths in $\Gamma'$ that begin and end in $\Omega$ may travel through states in $\Omega'\setminus\Omega$. (This will be the case in Section \[sec:disconnected\].) To deal with this, we must define a surjection $h\colon \Omega'\longrightarrow\Omega$ such that $h(u)=u$ for any $u\in\Omega$, and such that the preimage of any $y\in\Omega$ is not too large. This means that each state in $\Omega$ “stands in” for only a few states in $\Omega'$. If $\Omega=\Omega'$ then $h$ is the identity map, by definition.
Suppose that $\gamma'_{wz}\in\Gamma'$ is a canonical path from $w$ to $z$ in $\mathcal{G}'$. Write $$\gamma'_{wz} = (Z_0, Z_1,\dotsc, Z_j,\, Z_{j+1},\dotsc, Z_\ell),$$ with $Z_0=w,Z_\ell=z$ and $(Z_j,Z_{j+1})\in E({\ensuremath{\mathcal{M}}}')$ for all $j\in[\ell-1]$. We now define the *two-stage direct canonical path* $\gamma_{w,z}$ with respect to the fixed surjection $h:\Omega'\rightarrow \Omega$.
For each $(Z_j,Z_{j+1})\in\gamma'_{wz}$, construct a path $$\sigma_{Z_jZ_{j+1}} = (X_0,X_1,\dotsc,X_i, X_{i+1},\dotsc,X_\kappa),$$ with $X_0=h(Z_j)$ and $X_{\kappa}=h(Z_{j+1})$, such that $X_i\in\Omega$ for $i=0,\ldots, \kappa$ and $(X_i,X_{i+1})\in E({\ensuremath{\mathcal{M}}})$ for $i=0,\ldots, \kappa-1$. We call $\sigma_{Z_j Z_{j+1}}$ a $({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')$-*simulation path* (or just a *simulation path*, if no confusion can arise), since this path simulates a single transition in ${\ensuremath{\mathcal{M}}}'$ using edges of ${\ensuremath{\mathcal{M}}}$.
These simulation paths are concatenated together to form a canonical path $\gamma_{wz}$ from $w$ to $z$ in $\mathcal{G}$, as follows: $$\gamma_{wz} = (\sigma_{Z_0Z_1}, \sigma_{Z_1Z_2}, \dotsc,
\sigma_{Z_{\ell-1}Z_\ell}).$$ The following algorithmic interpretation of $\gamma_{wz}$ may be useful as an illustration. Begin by querying $\gamma'_{wz}$ for the first transition $(Z_0,Z_1)$ to simulate. Beginning from $h(Z_0)$, perform transitions in the corresponding simulation path until state $h(Z_1)$ is reached. Now query $\gamma'_{wz}$ for the next transition $(Z_1,Z_2)$ and simulate that; and so on, until you have simulated all of $\gamma'_{wz}$, completing the simulation path $\gamma_{wz}$.
Denote the set of all $({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')$-simulation paths as $$\Sigma = {\ensuremath{\left\{\sigma_{xy}\mid (x,y)\in E({\ensuremath{\mathcal{M}}}')\right\}}}.$$ For these simulation paths to give rise to canonical paths for ${\ensuremath{\mathcal{M}}}$ with low congestion, no transition in ${\ensuremath{\mathcal{M}}}$ can feature in too many simulation paths. For each $t\in E({\ensuremath{\mathcal{M}}})$, let $$\Sigma(t) = {\ensuremath{\left\{\sigma\in\Sigma\mid t\in\sigma\right\}}}$$ be the number of simulation paths containing the transition $t$. The measures of quality of $\Sigma$ are the maximum number of simulation paths using a given transition, $$B(\Sigma) = \max_{t\in E({\ensuremath{\mathcal{M}}})} |\Sigma(t)|,$$ and the length of a maximal simulation path, $$\ell(\Sigma) = \max_{\sigma\in\Sigma} |\sigma|.$$ Note that the simulation paths all depend on the fixed surjection $h$, and so the same is true of the bounds $B(\Sigma)$ and $\ell(\Sigma)$. In particular, the quality of the surjection $h$ is measured by $\max_{y\in\Omega}\, |\{ z\in\Omega' \mid h(z)=y\}$ and this quantity will be needed when calculating $B(\Sigma)$ in particular (as we will see in Section \[sec:disconnected\]).
Some chains are ill-suited to simulation. For instance, a transition in ${\ensuremath{\mathcal{M}}}'$ with large capacity might necessarily be simulated in ${\ensuremath{\mathcal{M}}}$ by a simulation path that contained an edge with small capacity. Hence we define two quantities that measure the gap between $\pi$ and $\pi'$, and $P$ and $P'$. We define the *simulation gap* $D({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')$ between ${\ensuremath{\mathcal{M}}}$ and ${\ensuremath{\mathcal{M}}}'$ by $$D({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')=\max_{\substack{uv\in E({\ensuremath{\mathcal{M}}})\\zw\in E({\ensuremath{\mathcal{M}}}')}}\,
\frac{\pi'(z)\, P'(z,w)}{\pi(u)\, P(u,v)}.$$
We combine these ingredients in the following lemma. The key point is that the congestion across transitions in $E({\ensuremath{\mathcal{M}}}')$ (and hence across the simulation paths) is already bounded. If the number of simulation paths making use of any given transition in $E({\ensuremath{\mathcal{M}}})$ is bounded, and the graphs are not too incompatible then the congestion of each two-stage direct canonical path in $\Gamma$ will also be bounded.
\[thm:twostagedirect\] Let ${\ensuremath{\mathcal{M}}}$ and ${\ensuremath{\mathcal{M}}}'$ be two ergodic Markov chains on $\Omega$ and $\Omega'$. Fix a surjection $h:\Omega'\rightarrow\Omega$. Let $\Gamma'$ be a set of canonical paths in ${\ensuremath{\mathcal{M}}}'$ and let $\Sigma$ be a set of $({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')$-simulation paths defined with respect to $h$. Let $D=D({\ensuremath{\mathcal{M}}},{\ensuremath{\mathcal{M}}}')$ be the simulation gap defined above. Then there exists a set $\Gamma$ of canonical paths in ${\ensuremath{\mathcal{M}}}$ whose congestion satisfies $$\bar{\rho}(\Gamma)\leq D\, \ell(\Sigma)\, B(\Sigma)\, \bar{\rho}(\Gamma').$$
We begin with the definition of congestion of the set of paths $\Gamma$ for ${\ensuremath{\mathcal{M}}})$, and work towards rewriting all quantities in terms of the set of canonical paths $\Gamma'$ for ${\ensuremath{\mathcal{M}}}'$, as follows: $$\begin{aligned}
\bar{\rho}(\Gamma)&=
\max_{uv\in E({\ensuremath{\mathcal{M}}})}\, \frac1{\pi(u)P(u,v)}
\sum_{\substack{x,y\in\Omega\\uv\in\gamma_{xy}}}
\pi(x)\pi(y)\,|\gamma_{xy}|\notag\\
&=\max_{uv\in E({\ensuremath{\mathcal{M}}})}\, \frac1{\pi(u)P(u,v)}
\sum_{\substack{zw\in E({\ensuremath{\mathcal{M}}}')\\ \sigma_{zw}\in\Sigma(uv)}}
\:
\sum_{\substack{x,y\in\Omega\\ \sigma_{zw}\in\gamma_{xy}}}
\pi(x)\pi(y)\,|\gamma_{xy}|\label{line:path-split}\\
&\leq \max_{uv\in E({\ensuremath{\mathcal{M}}})}\, \frac1{\pi(u)P(u,v)}
\sum_{\substack{zw\in E({\ensuremath{\mathcal{M}}}')\\ \sigma_{zw}\in\Sigma(uv)}}
\:
\sum_{\substack{x,y\in\Omega\\zw\in\gamma'_{xy}}}
\pi(x)\pi(y)\,|\gamma_{xy}|.
\label{line:sigmas-switch}\end{aligned}$$ Line (\[line:path-split\]) splits the two-stage canonical paths into simulation paths. By a slight abuse of notation, we write $\sigma_{zw}\in \gamma_{xy}$ to mean that $\sigma_{zw}$ is one of the simulation paths which was concatenated together to make $\gamma_{xy}$. Here we assume that each edge $uv$ occurs at most once in any given simulation path $\sigma\in\Sigma$. Otherwise, the second use of $uv$ in any $\sigma$ forms a cycle which may be pruned: continue pruning until no edge appears more than once in any simulation path. We also assume that no self-loops appear in any simulation path (these may also be pruned.) Line (\[line:sigmas-switch\]) uses the one-to-one correspondence between $\gamma_{xy}$ and $\gamma'_{xy}$ for all $x,y\in\Omega$. Note that this correspondence is still one-to-one even when $\Omega\subset\Omega'$, since the endpoints are identical (as $h$ acts as the identity map on $\Omega$) and each canonical path is determined by its endpoints.
Now we observe that for all $uv\in E({\ensuremath{\mathcal{M}}})$ and $zw\in E({\ensuremath{\mathcal{M}}}')$, by definition of the simulation gap $D$, $$\frac{1}{\pi(u)P(u,v)} \leq D\, \frac{1}{\pi'(z)P'(z,w)}.$$ Similarly, for all $x,y\in\Omega'$, $$|\gamma_{xy}| \leq \ell(\Sigma)\, |\gamma'_{xy}|.$$ Therefore, as each summand of (\[line:sigmas-switch\]) is nonnegative and $\Omega\subseteq \Omega'$, we obtain $$\begin{aligned}
\bar{\rho}(\Gamma)&\leq D\, \ell(\Sigma)\, \max_{uv\in E({\ensuremath{\mathcal{M}}})} \,
\sum_{\substack{zw\in E({\ensuremath{\mathcal{M}}}')\\\sigma_{zw}\in \Sigma(uv)}}
\frac1{\pi'(z)P'(z,w)}
\sum_{\substack{x,y\in\Omega'\\zw\in\gamma'_{xy}}}
\pi'(x)\pi'(y)\,|\gamma'_{xy}|\notag\\
&\leq D\, \ell(\Sigma)\, \max_{uv\in E({\ensuremath{\mathcal{M}}})} \,
\sum_{\substack{zw\in E({\ensuremath{\mathcal{M}}})\\\sigma_{zw}\in \Sigma(uv)}}
\bar{\rho}(\Gamma')\notag\\
&\leq D\, \ell(\Sigma)\, B(\Sigma)\, \bar{\rho}(\Gamma'),\notag\end{aligned}$$ as required.
We remark that in the above theorem, the surjection $h$ is only needed to construct the set of simulation paths (which are given as input to the theorem).
Analysis of the flip chain {#sec:solution}
==========================
Our aim is to define canonical paths in the flip chain that simulate canonical paths defined in the switch chain. There are two main differences between the switch and flip chains which cause difficulties. The state space of the flip chain is restricted to connected graphs, and no flip can disconnect a connected graph. However, a switch can increase or decrease the number of components in a graph by one. Hence, from a connected graph, the set of available flips may be strictly smaller than the set of available switches. Following [@saberi06switchflip] we overcome these difficulties in two steps, using an intermediate chain to bridge the gap.
Define the *connected switch chain* ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ to be the projection of the switch chain ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ onto state space ${\ensuremath{\Omega_\mathrm{F}}}$. A transition of ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ is a *connected switch*, which is a switch on a connected graph that preserves connectedness. To be precise, if ${\ensuremath{P_\mathrm{SC}}}$ denotes the transition matrix of ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ and ${\ensuremath{P_\mathrm{S}}}$ denotes the transition matrix of ${\ensuremath{\mathcal{M}_\mathrm{S}}}$, then for all $x\neq y\in{\ensuremath{\Omega_\mathrm{S}}}$, $$\label{conn-switch}
{\ensuremath{P_\mathrm{SC}}}(x,y) = \begin{cases} {\ensuremath{P_\mathrm{S}}}(x,y)&\text{if }x,y\in{\ensuremath{\Omega_\mathrm{F}}},\\
0&\text{otherwise}.
\end{cases}$$ Any transition of ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ which produces a disconnected graph is replaced by a self-loop in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$. Hence, for all $x\in{\ensuremath{\Omega_\mathrm{F}}}$, $${\ensuremath{P_\mathrm{SC}}}(x,x) = {\ensuremath{P_\mathrm{S}}}(x,x)+\sum_{y\in{\ensuremath{\Omega_\mathrm{S}}}\setminus{\ensuremath{\Omega_\mathrm{F}}}} {\ensuremath{P_\mathrm{S}}}(x,y).$$ Transitions are symmetric, and so the stationary distribution ${\ensuremath{\pi_\mathrm{SC}}}$ of ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ is the uniform stationary distribution on $\Omega_F$ (that is, ${\ensuremath{\pi_\mathrm{SC}}}= {\ensuremath{\pi_\mathrm{F}}}$).
We proceed as follows. In Section \[sec:long-flip\] we show that a connected switch can be simulated by sequences of flips, which we will call a *long-flip*. This gives a set of $({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$-simulation paths. Here the surjection $h$ is the identity map, since these chains share the same state space. We apply Theorem \[thm:twostagedirect\] to this set of simulation paths.
Then in Section \[sec:disconnected\] we apply the theorem again to relate ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ and ${\ensuremath{\mathcal{M}_\mathrm{S}}}$. We use a construction of Feder et al. [@saberi06switchflip] to define a surjection $h:\Omega_S\rightarrow \Omega_F$. We then define $({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$-simulation paths with respect to this surjection and apply Theorem \[thm:twostagedirect\]. Theorem \[thm:flip-mixing-time\] is proved by combining the results of these two sections.
The two-stage direct method has lower cost, in the sense of relaxation of the mixing time, than the comparison method used in [@saberi06switchflip]. Both approaches have at their heart the construction of paths in ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ to simulate transitions in ${\ensuremath{\mathcal{M}_\mathrm{S}}}$, but the two-stage canonical path analysis is more direct and yields significantly lower mixing time bounds than those obtained in [@saberi06switchflip].
Simulating a connected switch: the long-flip {#sec:long-flip}
--------------------------------------------
We must define a set of $({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$-simulation paths, as defined in Section \[sec:two-stage-direct\]. The two Markov chains have the same state space $\Omega_F$, and so we use the identity map as the surjection.
Let $S = (Z,Z')$ be a connected switch which we wish to simulate using flips. Then $Z$ and $Z'$ are connected graphs which differ by exactly four edges, two in $Z\setminus Z'$ and two in $Z'\setminus Z$. We say that the two edges $ab, cd\in Z\setminus Z'$ have been *switched out* by $S$, and the two edges $ac, bd\in Z'\setminus Z$ have been *switched in* by $S$. Given an arbitrary labelling of one switch edge as $ab$ and the other as $cd$, these four permutations of the vertex labels $$(\, ),\quad (a\, b)(c\, d),\quad (a\, c)(b\, d),\quad (a\, d)(b\, c)
\label{perms}$$ preserve the edges switched in and out by ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$.
To simulate the connected switch $S=(Z,Z')$ we will define a simulation path called a *long-flip*, which consists of a sequence of flips $${\ensuremath{\sigma}}_{Z Z'} = (X_0, X_1, \dotsc, X_{i-1},X_{i} \dotsc, X_\kappa)$$ with $X_0=Z$ and $X_\kappa = Z'$, such that $(X_{i-1},X_i)\in E({\ensuremath{\mathcal{M}_\mathrm{F}}})$ for $i=1,\ldots, \kappa$. We define these simulation paths in this section, and analyse their congestion in Section \[sec:longflip-congestion\].
A long-flip uses a *hub path* $p=(b,p_1,p_2,\ldots,p_{\nu-1},c)$ from $b$ to $c$. In the simplest case (in which $p$ does not contain $a$ or $d$ and no internal vertex of $p$ is adjacent to either of these vertices) we will simply “reverse” $p$ using flips, as illustrated later in Figure \[fig:switchAsFlips\]. Here by “reversing $p$” we imagine removing the path $p$ with vertex $b$ at the “left” and $c$ at the “right”. Now remove $p$ from the graph and replacing it in the reverse orientation, but with $a$ reattached to the left-most vertex of $p$, which is now $c$, and with $d$ reattached to the right-most vertex of $p$, which is now $b$. This has the effect of deleting the edges $ab,cd$ and replacing them with $ac,bd$, as desired. This process is described in more detail in Section \[case1\] below The choice of hub path is important, and we now explain how the hub path will be (deterministically) selected.
Suppose that ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ is the connected switch which takes $Z$ to $Z'$, which we wish to simulate using flips. We say that a path $p$ is a *valid hub path* for the switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ in $Z$ if it satisfies the following *$(a,b,c,d)$ path conditions*:
1. $p$ is a $(b,c)$-path in $Z$;
2. $a$ and $d$ do not lie on $p$; and
3. the vertices of $p$ are only adjacent through path edges; that is, there are no edges in $Z$ of the form $p_ip_j$ with $|i-j|\neq 1$ for $i,j\in{\ensuremath{\left\{0,\dotsc,|p|\right\}}}$.
We refer to condition (iii) as the *no-shortcut property*. Given a path $p$ which satisfies conditions (i) and (ii), it is easy to find a path with the no-shortcut property, by simply following the shortcuts. A shortest $(b,c)$-path is a valid $(a,b,c,d)$ path provided it avoids $a$ and $d$.
Given $Z$ and the switch edges $ab,cd$, we perform the following procedure:
- Let $p$ be the lexicographically-least shortest $(b,c)$-path in $Z$ which avoids $a$ and $d$, if one exists;
- If not, let $p$ be the lexicographically-least shortest $(a,d)$-path in $Z$ which avoids $b$ and $c$, if one exists, and apply the vertex relabelling $(a\, b)(c\, d)$. (The result of this relabelling is that now $p$ is the lexicographically-least shortest $(b,c)$-path in $Z$ which avoids $a$ and $d$.)
- If a path $p$ was found, return the path $p$. Otherwise, return FAIL.
If a path $p$ was found by the above procedure then we will say that we are in Case 1. Otherwise the output is “FAIL”, and we say that we are in Case 2.
Suppose that if $p$ is a lexicographically-least shortest path between vertices $\alpha$ and $\beta$ in a graph $Z$. We treat $p$ as a directed path from $\alpha$ to $\beta$, with $\alpha$ on the “left” and $\beta$ on the “right”. Suppose that vertices $\gamma$, $\delta$ lie on $p$, with $\gamma$ to the “left” of $\delta$. Then the lexicographically-least shortest $(\gamma,\delta)$-path in $Z$ is a subpath of $p$ (or else we contradict the choice of $p$). This important property, satisfied by hub paths, will be called the *recognisability property*. It will enable us to reconstruct the hub path cheaply, as we will see in Section \[sec:longflip-congestion\].
We now define the simulation path for Case 1 and Case 2 separately. Recall that since flips cannot disconnect a connected graph, the simulation paths always remain within $\Omega_F$.
### Case 1: the long-flip, and triangle-breaking {#case1}
We must define a simulation path $\sigma_{ZZ'}$ for the connected switch from $Z$ to $Z'$ given by ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$. The path $p$ selected above (the lexicographically-least shortest $(b,c)$-path which avoids $a$ and $d$) is a valid hub path for this switch, and we refer to it as the hub path. If $p$ has length 1 then this switch is a flip, and so we assume that $p$ has length at least two.
We simulate the connected switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ by reversing the path $p$, as shown in Figure \[fig:switchAsFlips\]. In (A), the switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ is shown with alternating solid and dashed edges, with solid edges belong to $Z$ and dashed edges belonging to $Z'$. Write the hub path as $p = (b, p_1, \dots, p_{\nu-1},c)$, where $b=p_0$ and $c=p_{\nu}$.
(0,1) circle (0.1); at (-0.2,1.0) [$a$]{}; (0,0.1) – (0.1,0) – (0,-0.1) – (-0.1,0) – (0,0.1); at (-0.2,0) [$b$]{}; (5,0) circle (0.1); at (5.2,0) [$c$]{}; (5,1) circle (0.1); at (5.2,1) [$d$]{}; at (2.5,1.0) [$(A)$]{}; (0,0.1) – (0,1); (5,0) – (5,1); (0,1) – (5,0.071); (5,1) – (0.071,0.071); (0.1,0) – (2,0); (4,0) – (5,0); (2,0) – (4,0); (1,0) circle (0.1); at (1.0,-0.2) [$p_1$]{}; (2,0) circle (0.1); at (2.0,-0.2) [$p_2$]{}; (4,0) circle (0.1); at (4.0,-0.2) [$p_{\nu-1}$]{}; at (6.5,0.5) [[$\Rightarrow_F$]{}]{}; (8,1) circle (0.1); at (7.8,1.0) [$a$]{}; (8,0) circle (0.1); at (7.8,0) [$p_1$]{}; (13,0) circle (0.1); at (13.2,0) [$c$]{}; (13,1) circle (0.1); at (13.2,1) [$d$]{}; at (10.5,1.0) [$(B)$]{}; (8,0) – (8,1); (13,0) – (13,1); (8,0) – (8.9,0); (9.1,0) – (10,0); (12,0) – (13,0); (10,0) – (12,0); (9,0.1) – (9.1,0) – (9,-0.1) – (8.9,0) – (9,0.1); at (9.0,-0.2) [$b$]{}; (10,0) circle (0.1); at (10.0,-0.2) [$p_2$]{}; (12,0) circle (0.1); at (12.0,-0.2) [$p_{\nu-1}$]{}; at (0,-2.5) [[$\Rightarrow_F^\ast$]{}]{}; (1.5,-2) circle (0.1); at (1.3,-2.0) [$a$]{}; (1.5,-2.9) – (1.6,-3) – (1.5,-3.1) – (1.4,-3) – (1.5,-2.9); at (1.3,-3) [$p_1$]{}; (6.5,-3.0) circle (0.1); at (6.7,-3.0) [$b$]{}; (6.5,-2) circle (0.1); at (6.7,-2) [$d$]{}; at (4.0,-2.0) [$(C)$]{}; (1.5,-2.9) – (1.5,-2); (6.5,-3) – (6.5,-2); (1.6,-3) – (3.5,-3); (5.5,-3) – (6.5,-3); (3.5,-3) – (5.5,-3); (2.5,-3) circle (0.1); at (2.5,-3.2) [$p_2$]{}; (3.5,-3) circle (0.1); at (3.5,-3.2) [$p_3$]{}; (5.5,-3) circle (0.1); at (5.5,-3.2) [$c$]{}; at (8.0,-2.5) [[$\Rightarrow_F^\ast$]{}]{}; (9.5,-2) circle (0.1); at (9.3,-2.0) [$a$]{}; (9.5,-3) circle (0.1); at (9.3,-3) [$c$]{}; (14.5,-3.0) circle (0.1); at (14.7,-3.0) [$b$]{}; (14.5,-2) circle (0.1); at (14.7,-2) [$d$]{}; at (12.0,-2.0) [$(D)$]{}; (9.5,-3) – (9.5,-2); (14.5,-3) – (14.5,-2); (9.5,-3) – (11.5,-3); (13.5,-3) – (14.5,-3); (11.5,-3) – (13.5,-3); (10.5,-3) circle (0.1); at (10.5,-3.2) [$p_{\nu-1}$]{}; (11.5,-3) circle (0.1); at (11.5,-3.2) [$p_{\nu-2}$]{}; (13.5,-3) circle (0.1); at (13.5,-3.2) [$p_1$]{};
The reversal of $p$ proceeds in *runs*, in which successive vertices are moved along the path one at a time into their final position. The run that moves vertex $p_i$ into place is called the *$p_i$-run*, during which $p_i$ is referred to as the *bubble* due to the way it percolates along path $p$. The first stage of the path reversal is the $b$-run, which moves $b$ adjacent to $d$. Next is the $p_1$-run, and then the $p_2$-run, until the $p_{\nu-2}$-run, which puts $c$ adjacent to $a$ and completes the reversal of $p$. Specifically, the $p_i$-run consists of the sequence of flips $$(a,p_{i},p_{i+1},p_{i+2}), (p_{i+1},p_{i},p_{i+2},p_{i+3}),
(p_{i+2},p_{i},p_{i+3},p_{i+4}), \dotsc, (p_{\nu-1},p_{i},c,p_{i-1}),$$ where $p_0=b$, $p_\nu=c$ (and in the case of the $b$-run, $p_{-1}$ is understood to denote $d$). After the $p_i$ run, $c$ has moved one step closer to $a$ on path $p$. See Figure \[fig:switchAsFlips\], where the current bubble is shown as a diamond. The result of the first flip of the $b$-run is shown in (B), and the situation at the end of the $b$-run is shown in (C), At this stage, $b$ is in its final position, adjacent to $d$. Finally in (D) we see the graph after all runs are complete, with the path reversed. (Note that $\Rightarrow_F$ denotes a single flip while $\Rightarrow_F^\ast$ denotes a sequence of zero or more flips.)
Since the distance from $c$ to $a$ decreases by one after each run, the total number of flips used to reverse the path $p$ is $\frac12\nu(\nu+1)$. The no-shortcut property of hub paths is required so that no flip in any run is blocked by parallel edges within $p$.
However, the first flip of the $b$-run is blocked if $p_1$ is adjacent to $a$. Before we can reverse the path $p$, we must break this *triangle* $[a,b,p_1]$ using the following procedure. Choose the least-labelled vertex $x$ that is adjacent to $b$ but not $p_1$. The $(a,b,c,d)$ path conditions imply that $p_2$ is adjacent to $p_1$, but not to $b$, so the existence of $x$ follows from $2r$-regularity. Then the triangle $[a,b,p_1]$ can be removed by performing a flip on the path $(x,b,p_1,p_2)$, as shown below.
; (0,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$x$\] (x) (1,1) node \[circle,draw,fill=black,label=left:
------------------------------------------------------------------------
$a$\] (a) (1,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$b$\] (b) (2,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_1$\] (p\_1) (3,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_2$\] (p\_2) (3.5,0) node (c); (x)–(b)–(p\_1)–(p\_2) (b)–(a)–(p\_1); (p\_2)–(c);
; (0,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$x$\] (x) (2,1) node \[circle,draw,fill=black,label=right:
------------------------------------------------------------------------
$a$\] (a) (1,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_1$\] (p\_1) (2,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$b$\] (b) (3,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_2$\] (p\_2) (3.5,0) node (c); (x)–(b)–(p\_1)–(p\_2) (b)–(a)–(p\_1); (p\_2)–(c);
Now $p_1$ is still adjacent to $a$, but it is no longer on the $(b,c)$-path, and the length of the $(b,c)$-path has been reduced by $1$. However, it is possible that $p_2$ is also adjacent to $a$, so we may still have a triangle. If so, we must repeat the process, as indicated below, successively shifting $b$ to the right on the $(b,c)$-path, and reducing its length. To break the triangle $[a,b,p_2]$ we flip on $(p_1,b,p_2,p_3)$.
; (0,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$x$\] (x) (2,1) node \[circle,draw,fill=black,label=above:
------------------------------------------------------------------------
$a$\] (a) (1,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_1$\] (p\_1) (2,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$b$\] (b) (3,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_2$\] (p\_2) (4,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_3$\] (p\_3) (4.5,0) node (c); (x)–(b)–(p\_1)–(p\_2)–(p\_3) (b)–(a)–(p\_1) (a)–(p\_2); (p\_3)–(c);
; (0,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$x$\] (x) (3,1) node \[circle,draw,fill=black,label=above:
------------------------------------------------------------------------
$a$\] (a) (1,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_1$\] (p\_1) (2,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_2$\] (p\_2) (3,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$b$\] (b) (4,0) node \[circle,draw,fill=black,label=below:
------------------------------------------------------------------------
$p_3$\] (p\_3) (4.5,0) node (c); (x)–(b)–(p\_1)–(p\_2)–(p\_3) (b)–(a)–(p\_1) (a)–(p\_2); (p\_3)–(c);
More generally, if the edges $a p_1,\ldots, a p_k$ are all present but the edge $a p_{k+1}$ is not then the $k$’th triangle $[a,b,p_k]$ is broken with the flip ($p_{k-1},b,p_k,p_{k+1}$). After this flip, $b$ is adjacent to $p_k$ and $p_{k+1}$. We call these *triangle-breaking flips*.
The no-shortcut property of the $(b,c)$-path guarantees that all the triangle-breaking flips after the first one are valid. The triangle-breaking process must end when $bc$ becomes an edge, at the very latest. This implies that $k\leq \nu-1$. Furthermore, since $a$ has degree $2r$ we also have $k\leq 2r-1$. When the process terminates, the edge $ap_{k+1}$ is absent and $bp_{k+1}$ is the first edge on the $(b,c)$-path. Now the long-flip can be performed on this new $(b,c)$ hub path. If $k=\nu-1$ then $p_{k+1}=c$ and this long-flip is a single flip.
After the path is fully reversed we must undo the triangle removals, by reversing each triangle-breaking flip, in reverse order. That is, if $k$ triangle-breaking flips were performed, then performing these flips in order will reinstate these triangles: $$(p_{k-1},p_k,b,p_{k+1}),\,\, (p_{k-2}, p_{k-1}, b, p_k),\ldots
(p_1,p_2,b,p_3),\,\,\, (x, p_1,b,p_2).$$ (When $k=1$, only the last flip is performed.) We call these *triangle-restoring flips*. These flips are not blocked because the only edges whose presence could block them are guaranteed not to exist by the no-shortcut property of $p$, and by the fact that these edges were removed by the triangle-breaking flips.
### Case 2: The detour flip {#case2}
In Case 2, we know that there are no paths from $b$ to $c$ in $Z$ which avoid both $a$ and $d$, and that there are no paths from $a$ to $d$ in $Z$ which avoid $b$ and $c$. It follows that $Z$ must have the structure illustrated in Figure \[fig:Case2structure\].
at (0.3,3.7) [$Z_{ab}$]{}; at (3.7,3.7) [$Z_{ac}$]{}; at (3.7,0.3) [$Z_{cd}$]{}; at (0.3,0.3) [$Z_{bd}$]{};
(3,3) ellipse (1.30cm and 0.6cm);
(3,1) ellipse (1.30cm and 0.6cm);
(1,1) ellipse (1.30cm and 0.6cm);
(1,3) ellipse (1.30cm and 0.6cm);
(0.0,2) – (2.0,4.0); (2.0,0) – (4.0,2.0); (2.0,0.0) circle (0.13); at (2.0,-0.2) [$d$]{}; (0.0,2) circle (0.13); at (-0.2,2) [$b$]{}; (2,4) circle (0.13); at (2,4.2) [$a$]{}; (4.0,2.0) circle (0.13); at (4.2,2.0) [$c$]{};
In this figure, each $Z_{ij}$ (shown as a grey ellipse) is an induced subgraph of $Z$, with $i,j\in Z_{ij}$. There are no edges from $Z_{ij}\setminus \{ i,j\}$ to $Z_{k\ell}\setminus \{ k,\ell\}$ whenever $\{i,j\}\neq \{k,\ell\}$, by the assumption that $Z$ is in Case 2. Furthermore, $Z_{ab}$ and $Z_{cd}$ are both connected, since $Z$ is connected and $ab, cd\in E(Z)$.
At most one of $Z_{ac}$ or $Z_{bd}$ may be empty or disconnected, since $Z$ is connected. A deterministic procedure will now be described for constructing a 3-path $(u,d,w,z)$ in $Z$ which will be used for the first flip in the simulation path.
There are two subcases to consider for the choice of $w$:
- First suppose that, after applying the relabelling $(a\, b)(c\, d)$ if necessary, $Z_{ac}$ is empty or disconnected. Since $cd$ is not a bridge in $Z$, by Lemma \[lem:even-d\](ii), there exists a cycle in $Z_{cd}$ which passes through the edge $cd$. It follows that $d$ has at least one neighbour in $Z_{cd}\setminus c$ which is connected to $c$ by a path in $Z_{cd}\setminus cd$. Let $w$ be the least-labelled such neighbour of $d$ in $Z_{cd}\setminus cd$.
- Secondly, suppose that both $Z_{ac}$ and $Z_{bd}$ are nonempty and connected. (So far, we have not performed any relabelling.) Then at most one of $Z_{ab} \setminus ab$, $Z_{cd}\setminus cd$ may be disconnected, or else the graph $Z'$ will be disconnected, a contradiction. Applying the relabelling $(a\, c)(b\, d)$ if necessary, we can assume that $Z_{cd} \setminus cd$ is connected. Hence there is a path from $c$ to $d$ in $Z_{cd}\setminus cd$, so $d$ has a neighbour in $Z_{cd}\setminus c$ which is joined to $c$ by a path in $Z_{cd}\setminus cd$. Let $w$ be the least-labelled such neighbour of $d$ in $Z_{cd}\setminus c$.
In both subcases, we have established the existence of a suitable vertex $w$. Next, observe that since (in both the above subcases) $Z_{bd}$ is nonempty and connected, there is a path from $b$ to $d$ in $Z_{bd}$ of length at least two (as $bd\not\in E(Z)$). This implies that $d$ has a neighbour in $Z_{bd}$ which is joined to $b$ by a path in $Z_{bd}\setminus d$. Let $u$ be the least-labelled such neighbour of $d$ in $Z_{bd}\setminus b$. Finally, since all neighbours of $w$ belong to $Z_{cd}$ and $u\not\in Z_{cd}$, it follows that $\{ u,w\}\subseteq N_Z(d)\setminus N_Z(w)$. Since $Z$ is regular, this implies that $$|N_Z(w)\setminus N_Z(d)| = |N_Z(d)\setminus N_Z(w)| \geq 2.$$ Hence there is some vertex other than $d$ in $N_Z(w)\setminus N_Z(d)$. Let $z$ be the least-labelled such vertex. Then $z\in Z_{cd}\setminus \{c,d\}$, since $c\in N_Z(d)$ and every neighbour of $w$ belongs to $Z_{cd}$.
Now we can define the simulation path $\sigma_{ZZ'}$ from $X_0=Z$ to $X_\kappa = Z'$. The first step in the simulation path is the flip on the path $(u,d,w,z)$, giving the graph $X_1$. We call this the *detour flip*, as it will give us a way to avoid $d$. See the first line of Figure \[fig:Case2\].
(0.5,2.5) circle (0.1); at (0.3,2.5) [$a$]{}; (0.5,1) circle (0.1); at (0.3,1) [$b$]{}; (4.5,1) circle (0.1); at (4.7,1) [$c$]{}; (4.5,2.5) circle (0.1); at (4.7,2.5) [$d$]{}; (2.0,1) circle (0.1); at (2.0,0.8) [$u$]{}; (3.0,1) circle (0.1); at (2.9,0.8) [$w$]{}; (5.5,0) circle (0.1); at (5.7,0) [$z$]{}; at (2.6,-0.3) [$X_0$]{}; (0.5,1) – (0.5,2.5); (4.5,1) – (4.5,2.5); (0.5,1) – (2.0,1); (3.0,1) – (4.5,1); (2.0,1) – (4.5,2.5); (3.0,1) – (4.5,2.5); (3.0,1) – (5.5,0); at (6.5,1.7) [[$\Rightarrow_F$]{}]{}; (8,2.5) circle (0.1); at (7.8,2.5) [$a$]{}; (8,1) circle (0.1); at (7.8,1) [$b$]{}; (12.0,1) circle (0.1); at (12.0,0.8) [$c$]{}; (12.0,2.5) circle (0.1); at (12.2,2.5) [$d$]{}; (9.5,1) circle (0.1); at (9.5,0.8) [$u$]{}; (10.5,1) circle (0.1); at (10.5,0.8) [$w$]{}; (13.0,0) circle (0.1); at (13.2,0) [$z$]{}; at (10.0,-0.3) [$X_1$]{}; (8,1) – (8,2.5); (12.0,1) – (12.0,2.5); (8,1) – (9.5,1); (10.5,1) – (12.0,1); (9.5,1) – (10.5,1); (10.5,1) – (12.0,2.5); (12.0,2.5) – (13.0,0); at (0,-2.2) [[$\Rightarrow_F^\ast$]{}]{}; (1.5,-1.5) circle (0.1); at (1.3,-1.5) [$a$]{}; (1.5,-3) circle (0.1); at (1.3,-3) [$c$]{}; (5.5,-3) circle (0.1); at (5.5,-3.2) [$b$]{}; (5.5,-1.5) circle (0.1); at (5.7,-1.5) [$d$]{}; (3.0,-3) circle (0.1); at (3.0,-3.2) [$w$]{}; (4.0,-3) circle (0.1); at (4.0,-3.2) [$u$]{}; (6.5,-4) circle (0.1); at (6.7,-4) [$z$]{}; at (3.5,-4.3) [$X_{\kappa-1}$]{}; (1.5,-3) – (1.5,-1.5); (5.5,-3) – (5.5,-1.5); (1.5,-3) – (3.0,-3); (4.0,-3) – (5.5,-3); (3.0,-3) – (4.0,-3); (3.0,-3) – (5.5,-1.5); (5.5,-1.5) – (6.5,-4); at (8.0,-2.5) [[$\Rightarrow_F$]{}]{}; (9.5,-1.5) circle (0.1); at (9.3,-1.5) [$a$]{}; (9.5,-3) circle (0.1); at (9.3,-3) [$c$]{}; (13.5,-3) circle (0.1); at (13.5,-3.2) [$b$]{}; (13.5,-1.5) circle (0.1); at (13.7,-1.5) [$d$]{}; (11.0,-3) circle (0.1); at (11.0,-3.2) [$w$]{}; (12.0,-3) circle (0.1); at (12.2,-3.2) [$u$]{}; (14.0,-4.3) circle (0.1); at (14.2,-4.3) [$z$]{}; at (11.5,-4.3) [$X_{\kappa}$]{}; (9.5,-3) – (9.5,-1.5); (13.5,-3) – (13.5,-1.5); (9.5,-3) – (11.0,-3); (12.0,-3) – (13.5,-3); (12.0,-3) – (13.5,-1.5); (11.0,-3) – (13.5,-1.5); (11.0,-3) – (14.0,-4.3);
After the detour flip, there is a path from $b$ to $u$ in $X_1\setminus \{ a, d\}$, by choice of $u$, and there is a path from $c$ to $w$ in $X_1\setminus \{ a,d\}$, by choice of $w$. It follows that there exists a $(b,c)$-path in $X_1$ which avoids $\{ a,d\}$: let $p$ be the lexicographically-least shortest such path. Note that $p$ necessarily contains the edge $uw$. (See $X_1$ in Figure \[fig:Case2\]: the path $p$ is indicated by the dotted lines together with the edge from $u$ to $w$.) We proceed as in Case 1, simulating the switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ from $X_1$ using a long-flip with hub path $p$. At the end of the long-flip the path $p$ is reversed and we reach the graph $X_{\kappa-1}$ shown at the bottom left of Figure \[fig:Case2\].
For the final step of the simulation path, we reverse the detour flip by performing a flip on the path $(u,w,d,z)$. This produces the final graph $X_\kappa = Z'$ shown at the bottom right of Figure \[fig:Case2\], completing the description of the simulation path in Case 2.
Bounding the congestion {#sec:longflip-congestion}
-----------------------
The previous section defines a set ${\ensuremath{\Sigma_\mathrm{F}}}$ of $({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$-simulation paths. First, we prove an upper bound on the maximum length $\ell({\ensuremath{\Sigma_\mathrm{F}}})$ of a simulation path in ${\ensuremath{\Sigma_\mathrm{F}}}$.
If $r\geq 2$ then $\ell({\ensuremath{\Sigma_\mathrm{F}}})\leq \nfrac{1}{2} n^2$. \[lem:lF\]
Since hub paths are simple and do not include $a$ or $d$, every hub path contains at most $n-2$ vertices. Therefore, at most $\frac12(n-2)(n-3)$ path-reversing flips will be required. Furthermore, we may have up to $2r-1$ triangle-breaking flips, and up to $2r-1$ triangle-restoring flips. Additionally, if we are in Case 2 then we also have a detour flip and its inverse. Therefore $$\ell({\ensuremath{\Sigma_\mathrm{F}}}) \leq \frac{(n-2)(n-3)}{2}+4r,$$ and this is bounded above by $\nfrac{1}{2} n^2$ whenever $r\geq 2$, since $n \geq 2r+1$.
For the remainder of this subsection we work towards an upper bound on $B({\ensuremath{\Sigma_\mathrm{F}}})$, the maximum number of simulation paths containing a given flip. A given flip $t=(X,X')$ could perform one of a number of roles in a simulation path for a connected switch $S = (Z,Z')$:
- a triangle-breaking flip;
- a triangle-restoring flip;
- a flip on the path reversal (that is, part of the long-flip);
- in Case 2, the detour flip $(u,w,d,z)$;
- in Case 2, the inverse detour flip $(u,d,w,z)$.
We bound the number of simulation paths that could use $t$ in each of these roles separately, in Lemmas \[lem:detour\]–\[lem:pathreversal\] below. Summing over these bounds over all roles yields an upper bound for $B({\ensuremath{\Sigma_\mathrm{F}}})$. In our exposition, we attempt to uniquely determine a given simulation path ${\ensuremath{\sigma}}_{Z Z'}$ which contains $t$, by “guessing” from a number of alternatives. Since the number of possible alternatives is precisely the number of simulation paths ${\ensuremath{\sigma}}_{ZZ'}$ that use $t$, this approach gives the desired bound.
The simulation path is determined once we have identified the switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ and either $Z$ or $Z'$ (since each of these can be obtained from the other using the switch or its inverse). Observe that in order to uniquely identify the switch it is not enough to just identify the two edges $e_1$, $e_2$ which are switched out (or to just identify the two edges that are switched in). This follows since there are up to two possible switches on these edges: if $e_1 = ab$ and $e_2=cd$ then we could switch in either the edges $ac$, $bd$ or the edges $ad$, $bc$. We pay a factor of 2 to distinguish between these two possibilities, and will describe this as *orienting* one of the switch edges.
Finally, suppose that the current flip $t = (X,X')$ switches out the edges $\alpha\beta$, $\gamma\delta$ and replaces them by $\alpha\gamma$, $\beta\delta$. If both $\alpha\delta$ and $\beta\gamma$ are present in $X$ (and hence, both present in $X'$) then either one of these could have been the middle edge of the 3-path chosen for the flip. In this case, if we wish to identify $\beta$, say, then there are 4 possibilities. However, if we know a priori that the edge $\alpha\delta$ is not present in $X$, say, then we can guess $\beta$ from two possibilities.
If $t=(X,X')$ is a detour flip then we can guess the connected switch $S=(Z,Z')$ from at most $8r^2 n$ possibilities. The same bound holds if $t$ is an inverse detour flip. \[lem:detour\]
First suppose that $t$ is a detour flip. This is the first flip in the simulation path, so $Z=X$.
- The flip is on the 3-path $(u,w,d,z)$, and $uz\not\in E(X)$ by construction, so we can guess $d$ from 2 possibilities.
- Next, we can guess $c\in N_X(d)$ from at most $2r$ possibilities.
- Finally, guess the edge $ab\in E(X)$ and orient it, from $2rn$ possibilities.
From the switch edges and $Z$ we can obtain $Z'$. Hence we can guess the switch $S$ from at most $8r^2 n$ possibilities overall when $t$ is a detour flip. The same argument holds when $t$ is an inverse detour flip, replacing $X$ and $Z$ by $X'$ and $Z'$ and exchanging the roles of $b$ and $c$.
Suppose that $t=(X,X')$ is a triangle-breaking flip. Then we can guess the connected switch $S=(Z,Z')$ from at most $4(2r)^8 n$ possibilities. The same bound holds if $t$ is a triangle-restoring flip. \[lem:trianglebreaking\]
Suppose that $t$ is a triangle-breaking flip.
- One of the vertices involved in the flip $t$ is $b$, so we can guess it from $4$ alternatives. (If $t$ is not the first triangle-breaking flip then by the no-shortcuts property, we can guess $b$ from 2 alternatives. But in the first triangle-breaking flip $xp_2$ may be an edge of $Z$, in which case there are 4 possibilities for $b$.)
- We can guess $a$ by adjacency to $b$, from $2r$ possibilities.
- We can guess the edge $cd\in E(X)$ and orient it, from $2rn$ possibilities.
We now know the switch vertices, so recovering $Z$ will give us the rest of the path. There may have been some other triangle-breaking flips which took place before $t$, and if we are in Case 2 then there was also a detour flip at the start of the simulation path. We continue our accounting below.
- First suppose that we are in Case 1, and that $t$ is the $k$th triangle-breaking flip which has been performed (so far). Then $k\geq 1$. Let $p'=(x,p_1,\ldots,p_{k},b)$ denote the path containing vertices which were removed from the hub path by the first $k$ triangle-breaking flips. Now $(a,p_1,x)$ is a 2-path in $X$, so we can guess $x$ and $p_1$ from $2r(2r-1)\leq (2r)^2$ alternatives.
- Next, $p_k$ belongs to $N_X(b)\setminus a$, so we can guess $p_k$ from at most $2r$ alternatives. (Here we allow the possibility that $p_1=p_k$, which covers the case that $t$ is the first triangle-breaking flip.) Then we can reconstruct $p'$ using the fact that $(p_1,\ldots, p_k)$ is the lexicographically-least shortest $(p_1,p_k)$-path in $X\setminus \{a,d\}$, by the recognisability property. Using $p'$, we can reverse these $k$ triangle-breaking flips to obtain $Z$ from $X$.
- Finally, we must guess whether we are in Case 1 (1 possibility) or Case 2. If we are in Case 2 then we must also reverse the detour flip before we obtain $Z$. Since $u$ and $w$ are neighbours of $d$ and $z$ is a neighbour of $w$, in this step we guess from at most $1 + (2r)^2\, (2r-1) < (2r)^3$ possibilities.
Overall, we can guess the switch $S$ from a total of at most $ 4 (2r)^8 n$ possibilities, when $t$ is a triangle-breaking flip.
The argument is similar when $t$ is a triangle-restoring flip:
- First, suppose that we are in Case 1. Guess $b$, then guess $d$ from the neighbourhood of $b$, then guess and orient the edge $ac\in E(X)$. This gives the switch edges, guessed from at most $4\,(2r)^3$ alternatives.
- Recover $p'=(x,p_1,\ldots, p_k,b)$ as in the triangle-breaking case, from at most $(2r)^3$ alternatives. Now we can $Z'$ uniquely by performing the remaining triangle-restoring flips to move $b$ past $p_1$.
- Again, there at most $(2r)^3$ possibilities for deciding whether or not we are in Case 2 and if so, guessing the inverse detour flip and applying it to produce $Z'$.
Hence there are at most $4(2r)^8 n$ possibilities for the connected switch $S$ when $t$ is a triangle-restoring flip.
Suppose that $t=(X,X')$ is a flip on the path-reversal. We can guess the connected switch $S=(Z,Z')$ from at most $4(2r)^8\, \left((2r)^2 - 1\right) n^3$ possibilities. \[lem:pathreversal\]
Suppose that $t$ is a flip on the path-reversal with respect to the hub path $p = (b,p_1,\ldots, p_{\nu-1},c)$, where $b=p_0$ and $c=p_{\nu}$. Then $t$ is part of a $p_i$ run, for some $i$. The path-reversal may have been preceded by some triangle-breaking flips, and (if we are in Case 2) a detour flip. We need to guess the switch edges and reconstruct $Z$.
First suppose that we are in Case 1. The typical situation is shown in Figure \[fig:p-prime\], which shows that graph $X'$ produced by the flip $t$. The current bubble $p_i$ is shown as a diamond, and the flip $t$ has just been performed on the 3-path $(p_j,p_i,p_{j+1},p_{j+2})$, moving $p_i$ past $p_{j+1}$.
(1.0,0) circle (0.1); at (1.0,-0.2) [$x$]{}; (2.0,0) circle (0.1); at (2.0,-0.2) [$p_1$]{}; (3.0,0) circle (0.1); at (3.0,-0.2) [$p_2$]{}; (3.0,0) – (4.5,0); (4.5,0) circle (0.1); at (4.4,-0.3) [$p_{k}$]{}; (5.5,1.5) circle (0.1); at (5.7,1.5) [$a$]{}; (5.5,0) circle (0.1); at (5.3,0.0) [$a'$]{}; (5.5,0) – (7,0); (7,0) circle (0.1); at (7,0.2) [$p_j$]{}; (8,0) circle (0.1); at (8,0.2) [$p_{j+1}$]{}; (9,0.1) – (9.1,0) – (9,-0.1) – (8.9,0) – (9,0.1); at (9,0.2) [$p_{i}$]{}; (10,0) circle (0.1); at (10,0.2) [$p_{j+2}$]{}; (10,0) – (12,0); (12,0) circle (0.1); at (12,0.2) [$c$]{}; (13,0) circle (0.1); at (13,0.2) [$c'$]{}; (14.5,0) circle (0.1); at (14.5,0.2) [$p_{k+1}$]{}; (15.5,0) circle (0.1); (13,0) – (14.5,0); (14.5,0) – (15.5,0); at (15.7,0) [$b$]{}; (15.5,1.5) circle (0.1); at (15.7,1.5) [$d$]{}; (1.0,0) – (3.0,0) – (5.5,1.5) – (3.0,0); (2.0,0) – (5.5,1.5) – (5.5,0); (4.5,0) – (5.5,1.5); (7,0) – (8.9,0); (9.1,0) – (10,0); (12,0) – (13,0); (15.5,0) – (15.5,1.5); (4.5,0) – (4.5,-1.8) – (15.5,-1.8) – (15.5,0); (5.6,-0.3) – (7.9,-0.3); at (6.75,-0.55) [$p[a' : p_{j+1}]$]{}; (10.1,-0.3) – (11.9,-0.3); at (11,-0.55) [$p[p_{j+2} : c]$]{}; (13.1,-0.3) – (14.4,-0.3); at (13.8,-0.55) [$\operatorname{rev} p[p_{k+1} : c']$]{};
- The current bubble $p_i$ is one of the four vertices involved in the flip, so it can be guessed from four possibilities.
- Next we guess $a$, $c$ and $d$ from $n^3$ alternatives.
- We recover $b$ as follows: if the current run is the $b$-run then $b$ is the bubble; otherwise, $b$ is already adjacent to $d$, and we can guess it from the $2r$ neighbours of $d$. This gives $2r+1$ possibilities for $b$.
- Vertices $a'$ and $c'$ can be guessed from at most $2r$ alternatives each, by their adjacency to $a$ and $c$.
Now we know the switch ${\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$, and it remains to recover $Z$. In order to do that, we must recover the path $p$, reverse the path-reversal flips and then reverse the triangle-breaking flips, if any.
Observe that having identified $p_i$, the identities of $p_{j+1}$ and $p_{j+2}$ can be inferred. To see this, note that $p_ip_{j+2}$ is one of the two edges switched in by the flip $t$, which identifies $p_{j+2}$, while $p_{j+1}p_{j+2}$ is one of the two edges switched out by the flip $t$, which identifies $p_{j+1}$.
Next, observe that $a'=p_{i+1}$, and so the subpath $p_{i+1}\cdots p_{j+1}$ is the lexicographically-least shortest $(p_{i+1},p_{j+1})$-path in $X' \setminus \{ a,d\}$, by the recognisability property. This holds since the flips made on the path-reversal so far, between $Z$ and $X'$, have not altered this section of the original path. Similarly, the subpath $p_{j+2}\cdots c$ can be recovered as the lexicographically-least shortest $(p_{j+2},c)$-path in $X'\setminus \{a,d\}$.
If the current bubble is $b$ then $c'=c$. Otherwise, observe that $c'=p_{i-1}$, since $c'$ is the most recent bubble that successfully completed its run. We continue our accounting below.
- Guess $p_{k+1}$ from $N_{X'}(b)\setminus d$, from at most $2r-1$ possibilities. Reconstruct the subpath $p_{k+1}\cdots p_{i-1}$ as the lexicographically-least shortest $(p_{k+1},p_{i-1})$-path in $X'\setminus \{a,d\}$.
- By now we have recovered all sections of the path between $a'$ and $b$ as shown in Figure \[fig:p-prime\], and we can reverse all path-reversing flips. Now there may have been no triangle-restoring flips, in which case we have constructed $Z$, or else we may need to reverse the triangle-breaking flips as explained earlier, after guessing the identities of $x$, $p_1$ and $p_{k-1}$ from at most $(2r)^2(2r-1)$ possibilities. This gives a factor of at most $1 + (2r)^2(2r-1) \leq (2r)^3$.
- Finally, multiply by $1 + (2r)^2(2r-1)\leq (2r)^3$ to account for the fact that we may be in Case 1 or Case 2, as explained in Lemma \[lem:trianglebreaking\].
Overall, we have guessed the switch $S$ from at most $$4(2r+1)(2r-1)(2r)^8\, n^3 = 4 (2r)^{8}\left((2r)^2-1\right) n^3$$ possibilities, when $t$ is a flip on a path-reversal.
Now we apply the two-stage direct canonical path theorem to our $({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$-simulation paths. Let $\bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}})$ denote the congestion of some set of canonical paths for the connected switch ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$. We will define such a set of canonical paths in Section \[sec:disconnected\] below.
\[lem:BF\] The $({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$-simulation paths defined in Section \[sec:long-flip\] define a set of canonical paths for ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ with congestion $\bar{\rho}({\ensuremath{\Gamma_\mathrm{F}}})$ which satisfies $$\bar{\rho}({\ensuremath{\Gamma_\mathrm{F}}}) \leq 8\,(2r)^{11} n^4\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}}).$$
Firstly, we bound the maximum number $B({\ensuremath{\Sigma_\mathrm{F}}})$ of simulation paths which contain a given flip $t$, by adding together the number of possibilities from each of the five roles that $t$ may play. Using Lemmas \[lem:detour\]– \[lem:pathreversal\], we obtain $$\begin{aligned}
B({\ensuremath{\Sigma_\mathrm{F}}}) &\leq 4(2r)^{8} \left((2r)^2-1\right) n^3 {} + 8 (2r)^8 n {} + 16 r^2 n \\
&\leq 4\left( (2r)^{10} - (2r)^8 + 2 (2r)^6 + 1\right) n^3\\
&\leq 4(2r)^{10}\, n^3,\end{aligned}$$ since $n\geq 2r\geq 4$. Next, we calculate the simulation gap ${\ensuremath{D_\mathrm{F}}}= D({\ensuremath{\mathcal{M}_\mathrm{F}}},{\ensuremath{\mathcal{M}_\mathrm{SC}}})$. Since ${\ensuremath{\mathcal{M}_\mathrm{F}}}$ and ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ share the same state space $\Omega_F$, and both have uniform stationary distribution, $${\ensuremath{D_\mathrm{F}}}= \max_{\substack{uv\in E({\ensuremath{\mathcal{M}_\mathrm{F}}})\\zw\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})}}\,
\frac{|{\ensuremath{\Omega_\mathrm{F}}}|\, {\ensuremath{P_\mathrm{SC}}}(z,w)}{|{\ensuremath{\Omega_\mathrm{F}}}|\, {\ensuremath{P_\mathrm{F}}}(u,v)} =
\frac{{\ensuremath{P_\mathrm{SC}}}(z,w)}{{\ensuremath{P_\mathrm{F}}}(u,v)}.$$ Let $uv\in E({\ensuremath{\mathcal{M}_\mathrm{F}}})$ and $zw\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})$. Then $u\neq v$, as self-loop transitions are not included in $E({\ensuremath{\mathcal{M}_\mathrm{F}}})$ or $E({\ensuremath{\mathcal{M}_\mathrm{SC}}})$. Hence $$\frac{1}{{\ensuremath{P_\mathrm{F}}}(u,v)}\leq \frac{(2r)^3 n}{2} \quad \text{ and } \quad
{\ensuremath{P_\mathrm{SC}}}(z,w) = \frac{1}{3 a_{n,2r}} \leq \frac{2}{r^2n^2}$$ using (\[an2r\]), since $n\geq 8$. Therefore ${\ensuremath{D_\mathrm{F}}}\leq 8r/n$.
Substituting these quantities into Theorem \[thm:twostagedirect\], using Lemma \[lem:lF\], we obtain $$\begin{aligned}
\bar{\rho}({\ensuremath{\Gamma_\mathrm{F}}})&\leq
D_{F}\, \ell(\Sigma_{F})\, B(\Sigma_{F})\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}}) \\
&\leq 8\,(2r)^{11} n^4\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}}),\end{aligned}$$ as claimed.
Disconnected graphs {#sec:disconnected}
===================
Next, we must define a set of $({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$-simulation paths and apply the two-stage direct canonical path method. The process is similar to the approach introduced by Feder et al. [@saberi06switchflip]: the chief difference is in the analysis.
We begin by defining a surjection ${\ensuremath{h}}\colon {\ensuremath{\Omega_\mathrm{S}}}\to{\ensuremath{\Omega_\mathrm{F}}}$ such that $\max_{G\in{\ensuremath{\Omega_\mathrm{F}}}}|{\ensuremath{h}}^{-1}(G)|$ is polynomially bounded. Using this we construct short paths in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ to correspond to single transitions in ${\ensuremath{\mathcal{M}_\mathrm{S}}}$. In [@saberi06switchflip] this step is responsible for a factor of ${{\ensuremath{\mathrm{O}\!\left(r^{34} n^{36}\right)}}}$ in the overall mixing time bound. In contrast, by using the two-stage direct method we construct canonical paths with a relaxation of only a factor of $120r n^3$.
First we must define the surjection ${\ensuremath{h}}$. Let $G_S\in{\ensuremath{\Omega_\mathrm{S}}}$ and let $\{ H_1,\ldots, H_k\}$ be the set of components in $G_S$. Let $v_i$ the vertex in $H_i$ of highest label, for $i=1,\ldots, k$. By relabelling the components if necessary, we assume that $$v_1 < v_2 < \cdots < v_k.$$ We call each $v_i$ an *entry vertex*. By construction, $v_k$ is the highest-labelled vertex in $G_S$.
In each component we will identify a *bridge edge* $e_i=v_iv'_i$, where $v'_i$ is the neighbour of $v_i$ with the highest label. We call edge $v_i'$ an *exit vertex*. The graph $G=h(G_S)$ is formed from $G_S$ by removing the bridge edges $e_1,\ldots, e_k$ and replacing them with *chain edges* $e'_1,\ldots, e_k'$, where $e_i = v'_i v_{i+1}$ for $1\leq i\leq k-1$, and $e_k' = v_k' v_1$. We call $e_k'$ the *loopback edge* and we call this procedure *chaining*. (When discussing the chain, arithmetic on indices is performed cyclically; that is, we identify $v_{k+1}$ with $v_1$ and $v'_{k+1}$ with $v'_1$.)
By construction we have $v_i' < v_i < v_{i+1}$ for $i=1,\ldots, k-1$. Hence the “head” $v_{i+1}$ of the chain edge $e_i'$ is greater than the “tail” $v_i'$, except for possibly the loopback edge: $v_1$ may or may not be greater than $v_k'$. We prove that $G=H(G_S)$ is connected in Lemma \[lem:H-range-biconnected\] below.
An example of a chained graph $G$ is shown in Figure \[fig:chain\] in the case $k=3$. Here $G=H(G_S)$ for a graph $G_S$ with three components $H_1,H_2,H_3$. The shaded black vertices are the entry vertices $v_1$, $v_2$ and $v_3$, and the white vertices are the exit vertices $v_1'$, $v_2'$ and $v_3'$. After chaining, $H_1$ contains two cut edges (shown as dashed edges), because $e_1 = v_1 v'_1$ was part of a cut cycle in $H_1$. Similarly, $H_3$ contains one cut edge. Note that these dashed edges are not chain edges. The loopback edge is $e'_3$ and $v_1 < v_2 < v_3$.
(1,1) circle (1); (3.5,1) circle (1); (6,1) circle (1); (9,1) circle (1); (12,1) circle (1); (14.5,1) circle (1); (1.7,1) – (2.8,1); (4.2,1) – (5.3,1); (12.7,1) – (13.8,1); (0.3,1) circle (0.1); at (0.5,0.8) [$v_1$]{}; (6.7,1) circle (0.1); at (6.55,0.8) [$v_1'$]{}; (8.3,1) circle (0.1); at (8.5,0.7) [$v_2$]{}; (9.7,1) circle (0.1); at (9.55,0.8) [$v_2'$]{}; (11.3,1) circle (0.1); at (11.5,0.8) [$v_3$]{}; (15.2,1) circle (0.1); at (15.05,0.8) [$v_3'$]{}; (6.8,1) – (8.3,1); at (7.5,1.1) [$e_1'$]{}; (9.8,1) – (11.3,1); at (10.5,1.1) [$e_2'$]{};
(-0.5,-1) rectangle (16.0,4.2); (0.3,1) to \[out=150,in=30\] (15.27,1.07);
at (7.5,3.9) [$e_3'$]{}; (0,-0.2) – (7,-0.2); at (3.5,-0.8) [$H_1$]{}; at (9.0,-0.3) [$H_2$]{}; (11,-0.2) – (15.5,-0.2); at (13.25,-0.8) [$H_3$]{};
\[lem:H-range-biconnected\] Let ${\ensuremath{h}}$ be the map defined above. Then $h(G_S)$ is connected for all $G_S\in {\ensuremath{\Omega_\mathrm{S}}}$, and ${\ensuremath{h}}\colon {\ensuremath{\Omega_\mathrm{S}}}\to{\ensuremath{\Omega_\mathrm{F}}}$ is a surjection.
Let $G_S\in{\ensuremath{\Omega_\mathrm{S}}}$ and suppose $G={\ensuremath{h}}(G_S)$. Lemma \[lem:even-d\](ii) implies that each connected component $H_i$ remains at least $1$-connected after removal of bridge edges. The chain edges then connect these components in a ring, forming a connected graph. Additionally, ${\ensuremath{h}}$ preserves degrees since each vertex $v_i$, $v'_i$ is adjacent to exactly one bridge edge that is removed and exactly one chain edge that is added. Hence, $G\in{\ensuremath{\Omega_\mathrm{F}}}$. Finally, observe that $h(G)=G$ whenever $G$ is connected, which proves that $h$ is a surjection since ${\ensuremath{\Omega_\mathrm{F}}}\subset {\ensuremath{\Omega_\mathrm{S}}}$.
\[lem:roughly-injective\] Let ${\ensuremath{h}}\colon {\ensuremath{\Omega_\mathrm{S}}}\to{\ensuremath{\Omega_\mathrm{F}}}$ be defined as above. Then $$\max_{G\in{\ensuremath{\Omega_\mathrm{F}}}}\,|{\ensuremath{h}}^{-1}(G)|\leq 2rn.$$
Let $G_F\in\Omega_F$ be given. First we prove that given an edge $e = \{ v,w\}$ of $G_F$ and an orientation $(v,w)$ of $e$, there is a unique disconnected graph $G_S\in h^{-1}(G_F)$ such that when the chaining procedure is performed with input $G_S$ (producing $G_F$), the loopback edge $e_k'$ equals $e$ and $v_1=v$. We recover $G_S$ from $G_F$ and the oriented edge $e$ using the *unchaining algorithm* which we now describe.
To identify $e'_1$ given $e'_{k}$ and $v_1$, begin by searching the graph $G_F\setminus e'_{k}$, starting from $v_1$, until a vertex is reached with a label greater than $v_1$. By construction, this vertex must be $v_2$, as $v_1$ has the greatest label of any vertex in $H_1$. The edge traversed to reach $v_2$ must be $e'_1$. Repeat this process until the chain edges $e_1',\ldots, e_{k-1}'$ have been discovered. Then $G_S$ is recovered by deleting all chain edges (including the loopback edge) from $G_F$ and replacing them with the bridge edges $e_1,\ldots, e_k$.
For future reference, note that this unchaining algorithm will still correctly recover $G_S$ so long as $v'_i$ is any neighbour of $v_i$ in $G_S$ (that is, $v_i'$ does not need to be the neighbour with largest label).
Hence the number of choices for the oriented loopback edge gives an upper bound on $|h^{-1}(G_S)|-1$. We use the naive estimate of $2r(n-1)$ as an upper bound on this quantity, since the loopback edge could be any oriented edge which does not originate in $v_k$ (the highest-labelled vertex in $G_F$). Since $G_F$ is the only connected graph in $h^{-1}(G_F)$, we obtain $|h^{-1}(G_S)|\leq 2r(n-1) + 1 \leq 2rn$. (It should be possible to obtain a tighter bound with a little more work, though we do not attempt that here.)
Simulating general switches {#ss:simgen}
---------------------------
Now we must construct a set of $({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$-simulation paths ${\ensuremath{\Sigma_\mathrm{SC}}}$ with respect to the surjection $h$ defined above. Given a switch $S = (W,W')\in E({\ensuremath{\mathcal{M}_\mathrm{S}}})$, with $W,W'\in{\ensuremath{\Omega_\mathrm{S}}}$, we must simulate it by a path $\sigma_{WW'}$ in ${\ensuremath{\Gamma_\mathrm{SC}}}$, consisting only of connected switches. We then give an upper bound on the congestion parameters for the set ${\ensuremath{\Sigma_\mathrm{SC}}}$ of simulation paths, and apply Theorem \[thm:twostagedirect\]. The main difficulty is that switches in ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ may create or merge components, whereas connected switches in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ cannot.
In this section we refer to transitions in $E({\ensuremath{\mathcal{M}_\mathrm{S}}})$ as *general switches*, to distinguish them from connected switches. We say that a general switch $S=(W,W')$ is
- *neutral* if $W$ and $W'$ have the same number of components;
- *disconnecting* if $W'$ has one more component than $W$;
- *reconnecting* if $W'$ has one fewer component than $W$.
The general switch $S={\ensuremath{(ab,cd\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}ac,bd)}}$ is disconnecting if $ab,cd$ form a 2-edge-cut in the same component of $W$ and $ac,bd$ do not. If $S$ is a reconnecting switch then $ab$ and $cd$ below to distinct components of $W$. Finally, $S$ is a neutral switch if $ab$ and $cd$ both belong to the same component of $W$, but do not form a 2-edge-cut, or if $ab$ and $cd$ form a 2-edge-cut in a component of $W$, and $ac$ and $bd$ form another 2-edge-cut in the same component.
To simulate the general switch $S=(W,W')$, we must define a path $$\sigma_{W,W'} = (Z_0, Z_1,\dotsc,Z_q),$$ in $E({\ensuremath{\mathcal{M}_\mathrm{SC}}})$, where $Z_0={\ensuremath{h}}(W)$ and $Z_q={\ensuremath{h}}(W')$, such that $Z_i\in{\ensuremath{\Omega_\mathrm{F}}}$ for $i=0,\ldots, q$ and $(Z_i,Z_{i+1})\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})$ for $i = 0,\ldots, q-1$. Here are the kinds of connected switch that we will need in our simulation paths.
- Suppose that a bridge edge $v_j v_j'$ of $W$ is one of the edges to be removed by the switch $S$. This is a problem, since all bridge edges have been deleted and replaced by chain edges during the chaining process used to construct $Z_0=h(W)$. To deal with this, we first perform a connected switch called a *bridge change* switch. The bridge change switch reinstates the edge $e_je_j'$ (so that the desired switch $S$ can be performed) and changes the choice of exit vertex (previously $e_j'$) in $H_j$. Importantly, the resulting chain structure can still be unwound using the algorithm in Lemma \[lem:roughly-injective\]. Specifically, to reinstate the bridge edge $e_j=v_j v'_j$ in component $H_j$, let $v^*_j\in N_{Z_0}(v_j)\setminus{\ensuremath{\left\{v'_{j-1}\right\}}}$ be the highest-labelled neighbour of $v_j$ in $Z_0[H_j]$ and perform the connected switch ${\ensuremath{(v_jv^*_j,v'_jv_{j+1}\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}v_jv'_j,v^*_jv_{j+1})}}$. (Clearly $v_j^\ast\neq v_j'$ since the edge $v_jv_j'$ is absent in $Z_0$, and $v_{j+1}v_j^\ast$ is not an edge of $Z_0$ due to the component structure of $W$, so this connected switch is valid.) The new chain edge is $v^*_j v_{j+1}$ and the new bridge edge is $v_j v_j^*$. We denote the bridge change switch by [B$\Delta$]{}. See Figure \[fig:bridge-change\].
(2.5,2.5) circle (1.5); at (2.5, 0.8) [$H_j$]{}; (6.5,2.5) circle (1.5); at (6.5, 0.8) [$H_{j+1}$]{}; (1.6,2.5) circle (0.1); at (1.6,2.7) [$v_j$]{}; (3.4,2.5) circle (0.1); at (3.4,2.7) [$v_j'$]{}; (2.5,1.5) circle (0.1); at (2.3,1.4) [$v_j^*$]{}; (5.6,2.5) circle (0.1); at (5.6,2.7) [$v_{j+1}$]{}; (7.4,2.5) circle (0.1); at (7.4,2.7) [$v_{j+1}'$]{}; (3.5,2.5) – (5.6,2.5); (3.3,2.5) – (1.6,2.5); (1.6,2.5) – (2.5,1.5); (2.5,1.5) – (5.6,2.5); (1.5,2.5) – (0.5,2.5); (0.0,2.5) – (0.5,2.5); (7.5,2.5) – (8.5,2.5); (8.5,2.5) – (9.0,2.5);
- We will need a connected switch to simulate the general switch $S$. We denote this switch as [NS]{}, [DS]{} or [RS]{} if $S$ is neutral, disconnecting or reconnecting, respectively, and call it the *neutral* (respectively, *disconnecting* or *reconnecting*) *simulation flip*. It removes the same two edges that $S$ removes, and inserts the same two edges that $S$ inserts.
- Let $Z$ be the current graph after the [DS]{}, [RS]{} or [NS]{} has been performed. We say that component $H_j$ is *healthy* in $Z$ if there is no neighbour of the entry vertex $v_j$ in $Z$ with an label higher than the exit vertex $v_j'$. We must ensure that all components of $Z$ are healthy before performing the unchaining algorithm, or else we lose the guarantee that $Z=h(W')$.
A component $H_j$ can only become unhealthy if some switch along the simulation path involves the entry vertex $v_j$ and introduces a new neighbour of $v_j$, say $v_j^*$, which has a higher label than the (current) exit vertex $v_j'$. In this case, $v_j^*$ also has the highest label among all current neighbours of $v_j$. (In fact, the switch that makes $H_j$ unhealthy will be the switch which simulates $S$, namely, the [DS]{}, [RS]{} or [NS]{}.) Note, if there has been a bridge change in component $H_j$ at the start of the simulation path, then $v_j'$ might not be the exit vertex originally produced in $H_j$ by the chaining procedure. But this will not cause any problems, as we will see.
We can make $H_j$ healthy by performing the bridge change switch ${\ensuremath{(v_jv^*_j,v'_jv_{j+1}\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}v_jv'_j,v^*_jv_{j+1})}}$, precisely as described above. (Now the main purpose of this switch is not to reinstate an edge, but to ensure the correct vertex of $H_j$ becomes the exit vertex before unchaining.) When this bridge change is performed at the end of the simulation path we will refer to it as a *bridge rectification* switch, denoted [BR]{}.
- If $S$ is a disconnecting switch then performing this switch will produce a disconnected graph (and destroy the chain structure) by splitting $H_j$ into two components, one of which is not chained. In order to avoid this, we perform a *disconnected housekeeping switch*, denoted [DHK]{}, before the disconnecting simulation flip ([DS]{}). The [DHK]{} is defined in detail in Section \[ss:disconn\] below.
- Similarly, if $S$ is a reconnecting switch then performing this switch destroys the chain structure, by merging $H_i$ and $H_j$ together, forming one component which is incident with four chain edges instead of two. To fix the chain structure we perform a *reconnecting housekeeping switch*, denoted [RHK]{}, after the reconnecting simulation flip ([RS]{}). More detail is given in Section \[ss:conn\] below.
We now define simulation paths for the three types of general switch $S=(W,W')$. The paths are summarised in Table \[tab:switch-decompositions\], with optional steps denoted by square brackets. Here “optional” means that these steps may or may not appear in the simulation path for a general switch of that type. However, given a particular switch $S$, the definition of the simulation path complete determines whether or not these “optional” switches are required.
---------------------- -- ------------------- -- ---------------------------------------------------------------
Neutral switch $\longrightarrow$ \[[B$\Delta$]{}\] [NS]{} \[[BR]{}\]
Disconnecting switch $\longrightarrow$ \[[B$\Delta$]{}\] [DHK]{} [DS]{} \[[BR]{}\] \[[BR]{}\]
Reconnecting switch $\longrightarrow$ \[[B$\Delta$]{}\] \[[B$\Delta$]{}\] [RS]{} [RHK]{} \[[BR]{}\]
---------------------- -- ------------------- -- ---------------------------------------------------------------
: Simulation path for a general switch $S\in{\ensuremath{\Omega_\mathrm{S}}}$ using connected switches.[]{data-label="tab:switch-decompositions"}
### The simulation path for a neutral switch
Suppose that the general switch $S$ is a *neutral switch*. The simulation path for $S$ is formed using the following procedure.
- Let $H_i$ be the component of $W$ that contains all vertices involved in $S$.
- If the bridge edge $v_iv'_i$ is switched away by $S$ then perform a bridge change.
- After the bridge change, if any, perform a connected switch [NS]{} which switches in and out the the edges specified by $S$.
- If a bridge edge is unhealthy after this switch then perform a bridge rectification.
### The simulation path for a disconnecting switch {#ss:disconn}
Suppose that the general switch $S$ is a *disconnecting switch*. The simulation path for $S$ is formed using the following procedure.
- Let $H_j$ contain the 2-edge-cut $ab,cd$ and let $B_1,B_2$ be the subgraphs of $H_j$ separated by this cut. Without loss of generality, suppose that $B_1$ contains $v_j$.
- If a bridge change is required in $H_j$ then perform the bridge change. In that case, $v_j v_j'$ is one of the edges forming the 2-edge-cut, so all other neighbours of $v_j$ must lie in $B_1$. Therefore the new exit vertex $v_j^*$ also belongs to $B_1$.
- After the bridge change, if any, notice that switching in and out the edges specified by $S$ would split $H_j$ into two components $B_1$ and $B_2$. Of these, $B_1$ is already in the correct place in the chain, but $B_2$ is not connected to the chain at all. This is not allowed, since we are restricted to connected switches. Therefore we must first perform the *disconnected housekeeping switch* ([DHK]{}) to place $B_2$ into the correct position in the chain. This switch is described in more detail below.
- After the [DHK]{} has been performed, we perform the disconnecting switch specified by $S$. (This is now safe, as the [DHK]{} ensures that the resulting graph is not disconnected.)
- Up to two bridge edges may now be unhealthy. If so, then perform up to two bridge rectification switches. For definiteness, if two [BR]{} switches are required, perform the one with the lower component index first.
It remains to specify the disconnected housekeeping switch [DHK]{}. See Figure \[fig:dhk\]. Let $v_+$ be the vertex in $B_2$ with the highest label and let $v'_+$ be its greatest neighbour. Then $e_+ = v_+ v'_+$ must become the bridge edge of $B_2$. Find $i$ such that $v_i < v_+ < v_{i+1}$, so $B_2$ belongs in the chain between $H_i$ and $H_{i+1}$ in the chain. (Note that $v_k$ is the highest-labelled vertex in the graph, so $v_+ \leq v_k$, but it is possible that $v_+ < v_1$, in which case $B_2$ must become the first component in the chain and $v_k' v_+$ must become the new loopback edge.) Then [DHK]{} is the switch ${\ensuremath{(v_+v'_+,v'_iv_{i+1}\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}v_+v'_i,v'_+v_{i+1})}}$, which inserts $B_2$ into the correct position in the chain as required.
(0.5,4) circle (1.5); at (0.5, 2.2) [$H_i$]{}; (4,4) circle (1.5); at (4, 2.2) [$H_{i+1}$]{}; (-0.5,4.5) circle (0.1); at (-0.4,4.25) [$v_i$]{}; (1.5,4.5) circle (0.1); at (1.4,4.3) [$v_i'$]{}; (3,4.5) circle (0.1); at (3.15,4.25) [$v_{i+1}$]{}; (5,4.5) circle (0.1); at (4.8,4.4) [$v_{i+1}'$]{}; (10,2.95) circle (2.0 and 2.25); (9,4.5) circle (0.1); at (9,4.3) [$v_j$]{}; (11,4.5) circle (0.1); at (11,4.3) [$v_j'$]{}; (9,1.7) circle (0.1); at (9.2,1.45) [$v_+$]{}; (11,1.7) circle (0.1); at (10.7,1.55) [$v_+'$]{}; (9.5,2.5) circle (0.1); at (9.3,2.5) [$b$]{}; (9.5,3.5) circle (0.1); at (9.3,3.5) [$a$]{}; (10.5,2.5) circle (0.1); at (10.7,2.5) [$d$]{}; (10.5,3.5) circle (0.1); at (10.7,3.5) [$c$]{}; (8,3) – (12,3); (9.0,1.7) – (10.9,1.7); (9.5,2.5) – (9.5,3.5); (10.5,2.5) – (10.5,3.5); (1.6,4.5) – (3.0,4.5); (-0.5,4.5) – (-1.5,4.5); (-1.5,4.5) – (-2,4.5); (5.1,4.5) – (6,4.5); (6,4.5) – (6.7,4.5); (9,4.5) – (8,4.5); (8,4.5) – (7.3,4.5); (11.1,4.5) – (12,4.5); (12,4.5) – (12.5,4.5); at (8.3, 1.4) [$H_j$]{};
### The simulation path for a reconnecting switch {#ss:conn}
Suppose that the general switch $S$ is a reconnecting switch. The simulation path for $S$ is defined using the following procedure.
- Suppose that $ab\in H_i$ and $cd\in H_j$ where, without loss of generality $v_i < v_j$. Up to two bridge changes may be necessary, as this switch involves two components. If both are necessary, perform the bridge change in $H_i$ first.
- Then perform the *reconnecting switch* [RS]{} that switches in and out the edges specified by $S$. At this point, the component formed by linking $H_i$ and $H_j$ is incident with four chain edges, so the chain structure is not correct. In effect, this new component is occuping both position $i$ and position $j$ in the chain. Since $v_i < v_j$, vertex $v_j$ must be the entry vertex of the new component, so the correct position for this component is between $H_{j-1}$ and $H_{j+1}$.
- The *reconnecting housekeeping* switch ([RHK]{}) ${\ensuremath{(v_iv'_{i-1},v'_iv_{i+1}\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}v_iv'_i,v'_{i-1}v_{i+1})}}$ restores the chain by placing the new component in position between $H_{j-1}$ and $H_{j+1}$, linking $H_{i-1}$ directly to $H_{i+1}$ (which is now the next component in the chain) and reinstating the bridge edge $v_iv_i'$. Here $v_i'$ and $v_{i-1}'$ refer to endvertices of the *current* chain edges, *after* the at most two bridge changes have been performed. (These may differ from the endvertices of the original chain edges, before the bridge rectification steps.)
- Finally, at most one bridge rectification may be required to ensure that the single bridge edge $v_jv_j'$ is healthy.
In the special case when the reconnecting switch $S=(W,W')$ results in a connected graph $W'$, we must have $i=1$ and $j=2$: here the reconnecting housekeeping switch reinstates *both* bridge edges, and no bridge rectification is required.
Bounding the congestion {#bounding-the-congestion}
-----------------------
Let us calculate upper bounds for $B({\ensuremath{\Sigma_\mathrm{SC}}})$ and $\ell({\ensuremath{\Sigma_\mathrm{SC}}})$. From Table \[tab:switch-decompositions\], the longest possible simulation path consists of five connected switches, so $$\label{ellSSC}
\ell({\ensuremath{\Sigma_\mathrm{SC}}})=5.$$ We spend the rest of this subsection bounding $B({\ensuremath{\Sigma_\mathrm{SC}}})$.
Each connected switch $t=(Z,Z')$ can play a number of roles in a simulation path. We consider each kind of general switch separately (neutral, disconnecting, reconnecting). As in Section \[sec:longflip-congestion\], the number of simulation paths (of a given kind) containing $t$ is equal to the amount of extra information required to identify a general switch $S$ of this kind uniquely, given $t$ (and assuming that $t$ belongs to the simulation path for $S$): we speak of “guessing” this information. Again, it is enough to identify one of $W$ or $W'$, together with the edges switched in/out by $S$, and then the other graph ($W'$ or $W$) can be inferred.
### Neutral switches {#neu}
A connected switch $t=(Z,Z')$ in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ could take on three possible roles when simulating a neutral switch $S=(W,W')$. We consider each separately below.
([**[B$\Delta$]{}**]{}) First, suppose that the connected switch $t$ is a bridge change in a simulation path for a neutral (general) switch $S$.
- Since $t$ is a bridge change, one of the edges which is switched in by $t$ is also an edge which is switched out by $S$. Guess which one, from $2$ alternatives.
- The other edge which is switched out by $S$ is an edge of $Z'$, so this edge and its orientation can be guessed from fewer than $2rn$ alternatives.
- Also guess whether $W$ is connected and if not, guess the oriented loopback edge from at most $2rn$ alternatives (as in the proof of Lemma \[lem:roughly-injective\]). If $W$ is connected then $W=Z$. Otherwise, we must apply the unchaining algorithm from Lemma \[lem:roughly-injective\] to the graph $Z$ with respect to the chosen oriented loopback edge. This produces the graph $W$, and then using the switch edges we can determine $W'$.
In all we guessed $S$ from $2\cdot (2rn)^2 = 8r^2n^2$ alternatives.
([**[NS]{}**]{})Next, suppose that the connected switch $t$ is a neutral simulating switch for a neutral (general) switch $S$.
- The edges switched in/out by $t$ are precisely the edges switched in/out by $S$. This gives us the vertices $\{a,b,c,d\}$ involved in the switch. Guess the (current) oriented loopback edge in $Z$ from at most $2rn$ alternatives, and perform the unchaining algorithm to reveal the chain structure of $Z$. In particular, we now know the entry vertex and (current) exit vertex of every component.
- If $t$ was not preceded by a bridge change then $Z=Z_0$. Otherwise, a bridge change has been performed before $t$, which implies that $Z=Z_1$. In this case, one of the edges switched out by $t$ was originally a bridge edge in $W$, and hence is incident with $v_i$ for some component $H_i$ of $W$. Since $S$ is a neutral switch, all vertices involved in the switch belong to the same component $H_i$, and hence $v_i$ has the largest label among $a,b,c,d$. Without loss of generality, if $a=v_i$ then $b=v_i'$, while our knowledge of the chain structure of $Z$ gives us the current exit vertex $v_i^\ast$. Hence we can uniquely reverse the bridge change switch to determine $Z_0$. Therefore we can find $Z_0$ by guessing between just two alternatives, namely, whether or not a bridge change was performed.
From $Z_0$ we obtain $W$ by deleting the chain edges and reinstating the bridge edges (taking care to adjust the exit vertex of $H_i$ if a bridge change has been performed). Since we know $W$ and the switch edges, we can determine $W'$.
We have guessed $S$ from $4rn$ alternatives, when $t$ is a neutral simulating switch. (The above analysis holds even if the switch $S$ replaces one 2-edge-cut in $W$ by another.)
([**[BR]{}**]{})Finally, suppose that $t$ is a bridge rectification switch in a simulation path for a neutral (general) switch $S$. Then $t$ is the final switch in the simulation path.
- Guess the oriented loopback edge in $Z'$, from at most $2rn$ alternatives, and perform the unchaining algorithm on $Z'$ to produce $W'$. (Again, this includes the possibility that $W'$ is connected, in which case $W'=Z'$.)
- Since $t$ is a bridge rectification, the switch $S$ switches in an edge which is incident with the entry vertex $v_i$ of $H_i$, giving it a new neighbour with a higher label than the current exit vertex in $H_i$. This edge is one of the two edges switched out by $t$: guess which one, from 2 alternatives.
- The other edge which is switched in by $S$, together with its orientation, can be guessed from the (oriented) edges of $Z$: at most $2rn$ possibilities. Now $W$ can be inferred from $W'$ and the switch edges.
We have guessed $S$ from at most $2rn\cdot 2\cdot 2rn = 8r^2 n^2$ alternatives.
Adding the contributions from these three roles, a connected switch $t$ may be part of at most $$\label{neutral}
8r^2n^2 + 4rn + 8r^2n^2 \leq (8r^2+1)n^2$$ simulation paths for neutral general switches.
Before we move on to consider disconnecting and reconnecting switches, we make a couple of comments. The analysis of the neutral simulation switch ([NS]{}) shows that once the edges of the general switch $S$ have been guessed, we have enough information to decide whether a bridge rectification switch is required and if so, its specification. Since these bridge rectification switches do not affect the number of choices for $S$ in each subcase below, we only mention them when $t$ itself is playing the role of a bridge rectification, and otherwise we omit them from our accounting.
In every situation we must guess the oriented loopback edge, so that we can perform the unchaining algorithm. In some cases we must also be careful to specify which graph is given as input to the unchaining algorithm. In particular, we must not perform the unchaining algorithm immediately after a disconnecting housekeeping switch or immediately before a reconnecting housekeeping switch, since in those graphs the chaining structure has been temporarily compromised.
### Disconnecting switches {#s:discon}
A connected switch $t=(Z,Z')$ in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ could take on five possible roles when simulating a disconnecting switch $S=(W,W')$. We consider each separately below.
([**[B$\Delta$]{}**]{}) If $t$ is a bridge change then we can guess $S$ from at most $8r^2n^2$ alternatives, as explained in Section \[neu\].
([**[DHK]{}**]{}) Next, suppose that $t$ is a disconnecting housekeeping switch.
- Guess the (current) oriented loopback edge as an edge of $Z$ from at most $2rn$ possibilities. Perform the unchaining algorithm on $Z$ to reveal the chain structure of $Z$. Again, this tells us the entry and (current) exit vertex of every component of $W$.
- Decide whether or not a bridge change was performed before $t$, from 2 possibilities. As described in Section \[neu\], we can uniquely determine the bridge change from the chain structure of $Z$, so in both cases we can find $Z_0$ (note that $Z=Z_0$ if there was no bridge change, and $Z=Z_1$ if a bridge change was performed before $t$). By reversing the bridge change if necessary we obtain $Z_0$, and then from our knowledge of the chain structure we can determine $W$. (If a bridge change was performed then we must use the original exit vertex for that component: this vertex is revealed by reversing the bridge change.)
Since $t$ is a disconnected housekeeping switch, it switches out a chain edge $v'_jv_{j+1}$, for some $j$, and an edge $v_+v'_+$ which is entirely contained within a component $H_i$, for some $i\neq j$. Hence we can distinguish between these two edges of $t$ and identify which is the chain edge and which is $v_+v_+'$, also determining $i$ and $j$. Furthermore, we can identify $v_+$ uniquely as it has a higher label than $v_+'$.
- Let $v_i$ be the entry vertex in $H_i$. A disconnected housekeeping switch is only performed when the disconnected simulation flip ([DS]{}) representing $S$ would switch out the two edges of a 2-edge-cut separating $v_+$ and $v_i$, say. By Lemma \[lem:even-d\](iv), we can guess which 2-edge-cut is switched out by $S$ from at most $n^2/(15 r^2)$ possibilities. The orientation of the switch $S$ is uniquely determined, since there is only one possibility which would disconnect $H_i$. Now we know $W$ and the switch edges, we can determine $W'$.
We have guessed $S$ from at most $2rn\cdot 2\cdot n^2/(15 r^2) = 4n^3/(15r)$ possibilities.
([**[DS]{}**]{}) Now suppose that $t$ is a disconnecting simulation switch. The edges switched in/out by $t$ are precisely the edges switched in/out by $S$, so we do not need to guess these. Note that $Z$ does not have a valid chain structure, but $Z'$ does.
- We know that $W'$ is not connected, so we guess the oriented loopback edge from the (oriented) edges of $Z'$: at most $2rn$ possibilities. Perform the unchaining algorithm on $Z'$, revealing the chain structure of $Z'$. Suppose that $H_i$ and $H_j$ are the two components of $Z'$ which contain an edge which was switched in by $t$, and suppose that $i < j$. Then the disconnecting housekeeping switch ([DHK]{}) is uniquely determined: it is the switch ${\ensuremath{(v_iv'_i,v'_{i-1}v_{i+1}\ifthenelse{\equal{}{}}{\Rightarrow}{\Rightarrow_{}}v_iv'_{i-1},v'_iv_{i+1})}}$.
- Guess whether or not a bridge change was performed before the disconnecting housekeeping switch: if so, it is uniquely specified (as explained earlier), so we guess from 2 possibilities. Reverse this bridge change to obtain $Z_0$ and hence find $W$. From $W$ and the switch edges we can determine $W'$.
We have guessed the general switch $S$ from at most $4rn$ possibilities.
([**The first [BR]{}**]{})Now suppose that $t$ is the first bridge rectification switch in the simulation path for a disconnecting (general) switch $S$. There may or may not be a second bridge rectification switch after $t$. The calculations are similar to the bridge rectification case in Section \[neu\].
- Guess the oriented loopback edge in $Z'$, from at most $2rn$ alternatives, and perform the unchaining algorithm to reveal the chain structure of $Z'$. (Note, we know that $W'$ is disconnected when $S$ is a disconnecting switch.)
- Since $t$ is a bridge rectification switch, one of the edges switched in by $S$ is $v_iv_i^\ast$, where $v_i$ is the entry vertex of some component $H_i$ and $v_i^\ast$ has a higher label than the current exit vertex $v_i'$ in that component. This edge is then switched out by $t$. (It is possible that this condition is satisfied by both edges of $S$, but if so, we know that the first bridge rectification will act on the component $H_i$ with the lower index first.) Guess which edge of $t$ is $v_iv_i^\ast$, for a factor of 2.
- Guess the other edge switched out by $S$, and the orientation of the switch, from at most $2rn$ possibilities. (The other edge switched out by $S$ must lie in a distinct component from $H_i$, but for an upper bound we ignore this.) Whether or not a second bridge change will be needed after $t$ is completely determined by the switch edges and the chain structure, as described in Section \[neu\]. If a second bridge rectification is needed, perform it to produce $Z_{q}$. From $Z_q$ we may determine $W'$ using our knowledge of the chain structure of $Z'$ (and adjusting the exit vertex of the component in which the second bridge rectification was performed, if necessary). From $W'$ and the switch edges we may deduce $W$.
We have guessed $S$ from at most $2rn\cdot 2\cdot 2rn = 8 r^2n^2$ alternatives.
([**The second [BR]{}**]{})Here we assume that one bridge rectification has been already performed to produce $Z$. The connected switch $t$ is the last one in the simulation path.
- Again, we guess the oriented loopback edge for $Z'$, from at most $2rn$ possibilities. Apply the unchaining algorithm to produce $W'$.
- Choose which edge switched out by $t$ is an edge switched in by $S$, out of 2 possibilities. The endvertex of this edge with the higher label is $v_i$, for some $i$.
- Since this is the second bridge rectification, the other edge switched in by $S$ was $v_jv_j^*$ for some other component $H_j$, and this edge has been made into the bridge edge in $H_j$, using the first bridge rectification. Therefore we can identify and orient this edge once we guess $H_j$, from at most $n/r$ possibilities (using Lemma \[lem:even-d\](iii)). This determines the edges switched in by $S$. From $W'$ and the switch edges, we can determine $W$.
We have guessed $S$ from at most $2rn\cdot 2\cdot n/r = 4n^2$ possibilities.
Summing over these five roles, a connected switch $t$ can be included in at most $$\label{disconnecting}
8r^2 n^2 + 4n^3/(15r) + 4rn + 8r^2n^2 + 4n^2 \leq 9rn^3$$ simulation paths for disconnecting general switches. (The upper bound follows since $2\leq r\leq n/2$.)
### Reconnecting switches {#recon}
A connected switch $t=(Z,Z')$ in ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ could take on give possible roles when simulation a reconnecting (general) switch $S=(W,W')$. We consider each separately below.
([**The first [B$\Delta$]{}**]{})When $t$ is the first bridge change we can guess $S$ from at most $8 r^2 n^2$ alternatives, using arguments very similar to those given in Section \[neu\].
([**The second [B$\Delta$]{}**]{})Now suppose that $t$ is the second of two bridge changes in the simulation path for some reconnecting (general) switch $S$.
- Guess the oriented loopback edge in $Z$, from at most $2rn$ possibilities. Perform the unchaining algorithm to reveal the chain structure of $Z$.
- Since $t$ is a bridge change, one of the edges switched in by $t$ is an edge which will be switched out by $S$. Choose which, from 2 possibilities. This edge is incident with the entry vertex $v_j$ of some component $H_j$.
- Since $t$ is the second bridge change, the other edge which will be switched out by $S$ must be incident with the entry vertex of some other component $H_i$, with $i<j$. (It has been put back into $Z$ by the first bridge change.) By Lemma \[lem:even-d\](iii) there are at most $n/(2r)$ components, so we choose one to determine $v_i$.
- Choose a non-chain edge incident with $v_i$ and orient it, from $2(2r-1)$ possibilities. This determines the edges switched in and out by $S$. From these edges and $W$, we may determine $W'$.
We have guessed $S$ from at most $2rn\cdot 2\cdot n/(2r)\cdot 2(2r-1) < 8r n^2$ possibilities.
([**[RS]{}**]{})Suppose that $t$ is a reconnecting simulation switch. The edges switched in/out by $t$ are precisely the edges switched in/out by $S$.
- Guess the oriented loopback edge for $Z$ from at most $2rn$ possibilities. Perform the unchaining algorithm on $Z$ to reveal the chain structure of $Z$.
- Up to two bridge changes may have been performed before $t$, corresponding to the up to two edges switched out by $S$ which are incident with entry vertices in some component of $Z$. If two bridge changes have been performed then their order is uniquely determined. As argued in Section \[neu\], we must just decide whether or not a bridge change has been performed in each component, and then the rest is specified, so there are $2^2=4$ possiblities for the bridge changes before $t$, including the possibility that there were none. Once the bridge change switches are known, they can be reversed, which reveals $W$. Together with the switch edges, this determines $W'$ and hence $S$.
We have guessed $S$ from at most $8rn$ possibilities.
([**[RHK]{}**]{})Suppose that $t$ is a reconnecting housekeeping switch. Then $Z'$ has a valid chain structure, though $Z$ does not.
- Guess the oriented loopback edge for $Z'$ from at most $2rn$ possibilities. Perform the unchaining algorithm on $Z'$ to reveal the chain structure of $Z'$.
- The switch $t$ involves entry vertices $v_i$ and $v_j$, where $i<j$. Since $t$ is a reconnecting housekeeping switch, the edges switched in by $S$ form a 2-edge-cut in $Z'$ which separate $v_i$ and $v_j$. By Lemma \[lem:even-d\](iv) we can guess this 2-edge-cut from at most $n^2/(15 r^2)$ possibilities. This specifies the edges switched out by $S$. The orientation of the switch is determined by the fact that one edge switched out by $S$ is contained in $H_i$ and the other is contained in $H_j$.
- Finally, we must decide whether or not a bridge rectification is needed to produce $W'$, giving 2 possibilities. If a bridge rectification is needed then it is uniquely determined: perform it to obtain $W'$, and then $W$ can be obtained using the switch edges.
We have guessed $S$ from a total of at most $2rn\cdot n^2/(15 r^2)\cdot 2 = 4n^3/(15 r)$ possibilities.
([**[BR]{}**]{})Suppose that $t$ is a bridge rectification for a reconnecting (general) switch $S$. Then $t$ is the final switch in the simulation path. In particular, $W'$ is disconnected, since otherwise the reconnecting housekeeping switch produces $W'$, and no [BR]{} or unchaining is necessary.
- Guess the oriented loopback edge and perform the unchaining algorithm in $Z'$ to produce $W'$, from at most $2rn$ alternatives.
- One of the edges switched in by $S$ is an edge which was switched out by $t$: choose one, from 2 possibilities, and call it $e$.
- Let $\widehat{e}$ be the other edge which is switched in by $S$. Recall that the edges removed by the [RHK]{} are $e_{i-1}=v_iv_{i-1}'$ and $e_i=v_i'v_{i+1}$, where $e_i'$, $e_{i-1}'$ are the chain edges in the graph $Z_{q}$ obtained *after* the (at most two) bridge changes were performed. (See Section \[ss:conn\].) Let $Z_{q+1}$ be the result of performing the [RS]{} simulating $S$, starting from $Z_q$. Since the edges switched in by the [RS]{} must be disjoint from the edges of $Z_q$, we conclude that $\widehat{e}\not\in\{ e_{i-1}',\, e_i'\}$: the three edges $\widehat{e}$, $e_{i-1}'$, $e_i'$ are distinct and all belong to $Z_{q+1}$. Next, the [RHK]{} $(Z_{q+1},Z_{q+2})$ removes precisely two edges, namely $e_{i-1}'$, $e_{i}'$, and hence $\widehat{e}$ is still present in $Z=Z_{q+2}$. Therefore we can guess and orient $\widehat{e}$ from among the $2rn$ oriented edges of $Z$. (In fact, the two edges switched in by $S$ form a 2-edge-cut in $Z$, but we do not use that fact here.) Now we know $W'$ and the switch edges, we can determine $W$.
We have guessed $S$ from at most $2rn\cdot 2\cdot 2rn = 8r^2 n^2$ possibilities.
Summing the contribution from these five roles, a connected switch $t$ can be included in at most $$\label{reconnecting}
8 r^2 n^2 + 8 r n^2 + 8rn + 4n^3/(15 r) + 8 r^2 n^2 \leq 11 r n^3$$ simulation paths for reconnecting general switches. (The upper bound follows since $2\leq r\leq n/2$.)
Completing the analysis
-----------------------
Now we apply the two-stage direct canonical path construction to our set of $({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$-simulation paths. Here $\bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}})$ denotes the congestion of any set of canonical paths, or multicommodity flow, for the switch chain. We will use the bound given in [@cooper05sampling].
The $({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$-simulation paths defined in Section \[ss:simgen\] define a set of canonical paths for ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ with congestion $\bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}})$ which satisfies $$\bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}}) \leq 120\, r n^3\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}}).$$ \[lem:BSSC\]
First, by adding together (\[neutral\]) – (\[reconnecting\]), the maximum number $B({\ensuremath{\Sigma_\mathrm{SC}}})$ of simulation paths containing a given connected switch satisfies $$B({\ensuremath{\Sigma_\mathrm{SC}}}) \leq (8r^2+1)n^2 + 9r n^3 + 11 rn^3
\leq 24 r n^3,$$ using the fact that $r\geq 2$ and $n\geq 2r+1\geq 5$. Next we calculate the simulation gap ${\ensuremath{D_\mathrm{SC}}}= D({\ensuremath{\mathcal{M}_\mathrm{SC}}},{\ensuremath{\mathcal{M}_\mathrm{S}}})$. Recall that for all $uv\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})$ and $zw\in E({\ensuremath{\mathcal{M}_\mathrm{S}}})$ we have $u\neq v$ and $z\neq w$, and hence $${\ensuremath{P_\mathrm{SC}}}(u,v) = {\ensuremath{P_\mathrm{S}}}(z,w) = \frac{1}{3 a_{n,r}},$$ by definition of both chains. The state space of ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ is ${\ensuremath{\Omega_\mathrm{S}}}$ and the state space of ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ is ${\ensuremath{\Omega_\mathrm{F}}}$, with ${\ensuremath{\Omega_\mathrm{F}}}\subseteq {\ensuremath{\Omega_\mathrm{F}}}$. Since both ${\ensuremath{\mathcal{M}_\mathrm{S}}}$ and ${\ensuremath{\mathcal{M}_\mathrm{SC}}}$ have uniform stationary distribution, $${\ensuremath{D_\mathrm{SC}}}= \max_{\substack{uv\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})\\zw\in E({\ensuremath{\mathcal{M}_\mathrm{S}}})}}
\frac{|{\ensuremath{\Omega_\mathrm{F}}}|\, {\ensuremath{P_\mathrm{S}}}(z,w)}{|{\ensuremath{\Omega_\mathrm{S}}}|\, {\ensuremath{P_\mathrm{SC}}}(u,v)} \leq
\max_{\substack{uv\in E({\ensuremath{\mathcal{M}_\mathrm{SC}}})\\ zw\in E({\ensuremath{\mathcal{M}_\mathrm{S}}})}}\,
\frac{{\ensuremath{P_\mathrm{S}}}(z,w)}{{\ensuremath{P_\mathrm{SC}}}(u,v)} =
1.$$ Substituting these values and (\[ellSSC\]) into Theorem \[thm:twostagedirect\] gives $$\begin{aligned}
\bar{\rho}({\ensuremath{\Gamma_\mathrm{SC}}})&\leq
D_{SC}\, \ell(\Sigma_{SC})\, B(\Sigma_{SC})\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}}) \\
&\leq 120\, r n^3\, \bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}}),\end{aligned}$$ as required.
Finally, we may prove Theorem \[thm:flip-mixing-time\].
Combining the bounds of Lemma \[lem:BF\] and Lemma \[lem:BSSC\] gives $$\label{relaxation}
\frac{\bar{\rho}({\ensuremath{\Gamma_\mathrm{F}}})}{\bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}})} \leq
480\,(2r)^{12} n^7 .$$ The upper bound on the mixing time of the switch chain given in [@cooper05sampling; @corrigendum] has (an upper bound on) $\bar{\rho}({\ensuremath{\Gamma_\mathrm{S}}})$ as a factor. Therefore, multiplying this bound by the right hand side of (\[relaxation\]), we conclude that the mixing time of the flip chain is at most $$480\, (2r)^{35}\, n^{15}\, \left(2rn\log(2rn) + \log(\varepsilon^{-1})\right),$$ completing the proof of Theorem \[thm:flip-mixing-time\]. (This uses the fact that $|\Omega_S| \leq (rn)^{rn}$, which follows from known asymptotic enumeration results [@bendercanfield].)
It is unlikely that the bound of Theorem \[thm:flip-mixing-time\] is tight. Experimental evidence was presented in an earlier version of this work [@CDH] which provides support for the conjecture that the true mixing time for the flip chain is ${{\ensuremath{\mathrm{O}\!\left(n\log n\right)}}}$, when $r$ is constant. Perhaps $O(rn\log n)$ is a reasonable conjecture when $r$ grows with $n$.
Maximising 2-edge-cuts separating two vertices
==============================================
We now present the deferred proof of Lemma \[lem:even-d\](iv). Our aim here is to present a simple upper bound on $\Lambda(u,v)$ which holds for all relevant values of $n$ and $r$, though our proof establishes tighter bounds than the one given in the statement of Lemma \[lem:even-d\](iv).
Let $\Lambda(u,v)$ denote the number of 2-edge-cuts in $G$ which separate $u$ and $v$. By considering the connected component of $G$ which contains $u$, if necessary, we may assume that $G$ is connected. Consider the binary relation $\sim$ on $V$ defined by $w_1\sim w_2$ if and only if there are at least three edge-disjoint paths between $w_1$ and $w_2$ in $G$. Then $\sim$ is an equivalence relation [@tutte] which partitions the vertex set $V$ into equivalence classes $U_0,\ldots, U_k$. For $j=0,\ldots, k$ let $H_j = G[U_j]$ be the subgraph of $G$ induced by $U_j$. Each $H_j$ is either a maximal 3-edge-connected induced subgraph of $G$, or a single vertex. Without loss of generality, suppose that $u\in H_0$ and $v\in H_k$.
Note that the number of edges from $H_i$ to $H_j$ is either 0, 1 or 2 for all $i\neq j$, by construction. Define the *node-link* multigraph $\widetilde{G}$ of $G$ by replacing each $H_j$ by a single vertex $h_j$, which we call a *node*, and replacing each edge from a vertex of $H_i$ to a vertex of $H_j$ by an edge from $h_i$ to $h_j$, which we call a *link*. In particular, if there is a 2-edge-cut from $H_i$ to $H_j$ in $G$ then the link $h_ih_j$ has multiplicity 2 in $\widetilde{G}$. Each node has even degree in $\widetilde{G}$, by Lemma \[lem:even-d\](i). Furthermore, every link in $\widetilde{G}$ belongs to a cycle (possibly a 2-cycle, which is a double link), by Lemma \[lem:even-d\](ii), and these cycles in $\widetilde{G}$ must be edge-disjoint, or the corresponding edge of $G$ cannot be part of a 2-edge-cut. Therefore $\widetilde{G}$ is a planar multigraph (with edge multiplicity at most two) which has a tree-like structure, as illustrated in Figure \[fig:two-cuts\]: the black squares represent the nodes of $\widetilde{G}$.
(1,2.2) rectangle node\[black\] ++(0.25,0.25) ; (5.5,2.2) rectangle node\[black\] ++(0.25,0.25) ; (4,2.2) rectangle node\[black\] ++(0.25,0.25) ; (-1,1) rectangle node\[black\] ++(0.25,0.25) ; (2.75,1) rectangle node\[black\] ++(0.25,0.25) ; (6.5,1) rectangle node\[black\] ++(0.25,0.25) ; (6.55,3.4) rectangle node\[black\] ++(0.25,0.25) ; (4.6,3.4) rectangle node\[black\] ++(0.25,0.25) ; (1,4.2) rectangle node\[black\] ++(0.25,0.25) ; (-3,1) rectangle node\[black\] ++(0.25,0.25) ; (8.5,1) rectangle node\[black\] ++(0.25,0.25) ; (10,1.95) rectangle node\[black\] ++(0.25,0.25) ; (10,-0.2) rectangle node\[black\] ++(0.25,0.25) ; (11.45,1) rectangle node\[black\] ++(0.25,0.25) ; (1,-0.45) rectangle node\[black\] ++(0.25,0.25) ; (4.75,-0.45) rectangle node\[black\] ++(0.25,0.25) ; (1.1,3.35) circle(1) (7.6,1) circle(1) (5.7,3.35) circle(1) (-1.9,1) circle(1) ; (1,1) node\[state,draw\] (4.75,1) node\[state,draw\] (10.1,1) node\[state,draw,scale=0.8\] ; (1.1,-0.8) node [$h_0$]{}; (12.2,1.1) node [$h_k$]{};
A node of $\widetilde{G}$ which belongs to more than one cycle is a *join node*. The join nodes each have degree 4, and all other nodes of $\widetilde{G}$ have degree two.
Since we seek an upper bound on $\Lambda(u,v)$ for a given number of vertices $n$, we may assume that all nodes of $\widetilde{G}$ lie on some path from $h_0$ to $h_k$ in $\widetilde{G}$. (This corresponds to trimming all “branches” of $\widetilde{G}$ which do not lie on the unique “path” from $h_0$ to $h_k$ in $\widetilde{G}$, viewed as a tree.) This only removes 2-edge-cuts of $G$ which do not separate $u$ and $v$. Hence for an upper bound, by relabelling the nodes if necessary, we can assume that $\widetilde{G}$ consists of $\ell$ cycles $C_1,\ldots, C_\ell$, where $h_0$ belongs only to $C_1$ and $h_k$ belongs only to $C_\ell$, such that $C_j$ and $C_{j+1}$ intersect at a single join node $h_j$, for $j=1,\ldots, \ell-1$, while all other cycles are disjoint. This situation is illustrated in Figure \[fig:cuts-path\]. (In particular, $h_0$ and $h_k$ are not join nodes.)
(1,2.2) rectangle node\[black\] ++(0.25,0.25) ; (5.5,2.2) rectangle node\[black\] ++(0.25,0.25) ; (4,2.2) rectangle node\[black\] ++(0.25,0.25) ; (-1,1) rectangle node\[black\] ++(0.25,0.25) ; (2.75,1) rectangle node\[black\] ++(0.25,0.25) ; (6.5,1) rectangle node\[black\] ++(0.25,0.25) ; (8.5,1) rectangle node\[black\] ++(0.25,0.25) ; (10,1.95) rectangle node\[black\] ++(0.25,0.25) ; (10,-0.2) rectangle node\[black\] ++(0.25,0.25) ; (11.45,1) rectangle node\[black\] ++(0.25,0.25) ; (1,-0.45) rectangle node\[black\] ++(0.25,0.25) ; (4.75,-0.45) rectangle node\[black\] ++(0.25,0.25) ; (7.6,1) circle(1); (1,1) node\[state,draw\] (4.75,1) node\[state,draw\] (10.1,1) node\[state,draw,scale=0.8\] ; (12.1,1.1) node [$h_k$]{}; (1.1,-0.8) node [$h_0$]{}; (2.9,-0.2) node [$h_1$]{}; (6.7,-0.2) node [$h_2$]{}; (8.7,-0.2) node [$h_3$]{}; (1,1.1) node [$C_1$]{} (4.9,1.1) node [$C_2$]{} (7.5,1.1) node [$C_3$]{} (10.1,1.1) node [$C_4$]{};
Each cycle $C_j$ can be considered as the disjoint union of two paths between $h_{j-1}$ and $h_j$, or between $h_{\ell-1}$ and $h_k$ if $j=\ell$. Denote by $a_j$, $a_j'$ the lengths of these two paths around $C_j$. Any 2-edge-cut which separates $u$ from $v$ in $G$ corresponds to a pair of links in $\widetilde{G}$ which both belong to some cycle $C_j$, with one link from each of the two paths which comprise $C_j$. Hence $\Lambda(u,v) = \sum_{j=1}^\ell a_j a_j'$. If the length $a_j+a_j'$ of each cycle is fixed then the quantity $a_j a'_j$ is maximised when $|a_j-a_j'|\leq 1$. Therefore $\Lambda(u,v)\leq \sum_{i=1}^\ell s_i^2$, where $s_j = |C_j|/2\geq 1$. (This upper bound holds even when some cycles have odd length.) Now, the total number of nodes in $\widetilde{G}$ is $N=2\left(\sum_{j=1}^\ell s_j\right) -(\ell-1)$, by inclusion-exclusion. Thus, for a given number of nodes $N$ and a given value of $\ell$, the bound $S$ is maximised by setting $s_j=1$ for $j=2,\ldots,\ell$, and $s_1 = (N - \ell+1)/2$. Therefore $$\label{Tdef}
\Lambda(u,v)\leq T = \frac{(N - \ell + 1)^2}{4} + \ell-1.$$
First suppose that $r\geq 3$. Then every subgraph $H_j$ must contain a vertex which is only adjacent to other vertices of $H_j$, since every node in $\widetilde{G}$ has degree at most 4. Therefore each $H_j$ must contain at least $2r+1$ vertices, so there are $N\leq n/(2r+1)$ nodes in $\widetilde{G}$. Note that ${\textrm{d}T/\textrm{d}\ell} = -\ell + 1$ which equals zero when $\ell=1$ and is negative when $\ell >1$. So $T$ is largest when $\ell=1$, giving $\Lambda(u,v)\leq T\leq N^2/4 \leq n^2/(4r+2)^2$, as claimed. An extremal graph $G$ is shown in Figure \[fig:maxnumber\]: each grey block is $K_{2r+1}$ minus an edge. This construction gives an extremal example for any positive $n=0\bmod (4r+2)$, so this upper bound is asymptotically tight.
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (a1) (-.4,0) node [$u$]{} ; (1,0) circle (0.1) node (a2) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (a3) ; (1,0) circle (0.1) node (a4) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (a5) ; (1,0) circle (0.1) node (a6) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (a7) ; (1,0) circle (0.1) node (a8) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (a9) ; (1,0) circle (0.1) node (a10) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (b10) (-.4,0) node [$v$]{} ; (1,0) circle (0.1) node (b9) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (b8) ; (1,0) circle (0.1) node (b7) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (b6) ; (1,0) circle (0.1) node (b5) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (b4) ; (1,0) circle (0.1) node (b3) ;
(0,0)–(-0.1,1)–(0.5,1.1)–(1.1,1)–(1,0)–(0.5,0.25)–cycle; (0,0) circle (0.1) node (b2) ; (1,0) circle (0.1) node (b1) ;
(a2)–(a3) (a4)–(a5) (a6)–(a7) (a8)–(a9) (a10)–(b10) (b9)–(b2) (b1)–(b4) (b3)–(b6) (b5)–(b8) (b7)–(a1) ;
Certainly the weaker bound $\Lambda(u,v)\leq n^2/(15 r^2)$ also holds when $r\geq 3$.
Now suppose that $r=2$ and consider the induced subgraphs $H_0,\ldots, H_k$. Again we may assume that $s_2 = \cdots = s_\ell = 1$, giving (\[Tdef\]). If $h_j$ is a join node of $\widetilde{G}$ then $H_j$ may be as small as a single vertex, since each join node of $\widetilde{G}$ has degree 4. We call such a join node $h_j$ a *singleton*. Otherwise, if $H_j$ contains a vertex whose neighbourhood is contained within $H_j$ then $H_j$ contains at least $2r+1=5$ vertices. In particular, this must hold for $h_k$ and for all nodes in $C_1$ except $h_1$. A final option (only available when $r=2$) is that a join node $h_j$ corresponds to to a subgraph $H_j$ which is isomorphic to $K_4$, and hence has precisely four vertices: two of these vertices are joined to a vertex in $H_{j-1}$ and two are joined to $H_{j+1}$ (or $H_k$, if $j=\ell-1$). Here we use the fact that any part $U_j$ of the partition of $V$ with more than one vertex must have at least 4 vertices, as $G[U_j]$ must be 3-edge-connected. Since $G$ has no repeated vertices, at most every second join node can be a singleton, with the remaining join nodes contributing at least 4 vertices to $G$.
It follows that $\Lambda(u,v) \leq T = s_1^2 + \ell-1$ and $$n \geq 10s_1 + \left\lceil\frac{\ell-1}{2}\right\rceil
+ 4\left\lceil\frac{\ell-1}{2}\right\rceil
= \begin{cases} 10s_1 + 5(\ell-1)/2 & \text{ if $\ell$ is odd,}\\
10s_1 + 5\ell/2 - r & \text{ if $\ell$ is even.}
\end{cases}$$ For an upper bound on $\Lambda(u,v)$ we take equality the above expression for $n$ and assume that $\ell$ is even, leading to $\Lambda(u,v) \leq T = s_1^2 - 4 s_1 + 2(n+4)/5$. (If $\ell$ is odd then only the term which is independent of $s_1$ changes.) Hence ${\textrm{d}T/\textrm{d}s_1}$ is negative when $s_1=1$, zero when $s_1=2$ and positive for $s_1\geq 3$. It follows that for a given value of $n$, the maximum possible value of $T$ occurs when $s_1=1$ or when $s_1$ is as large as possible.
When $s_1=1$ we have $\Lambda(u,v)\leq \ell\leq 2(n-6)/5$, at least when $\ell$ is even. A graph $G$ on $n$ vertices which meets this bound can be constructed whenever $n\geq 11$ and $n=1\bmod 5$. The example shown in Figure \[fig:n=21\] has $n=21$ vertices and $\ell=6$ 2-edge-cuts between $u$ and $v$.
(-1,1) circle (0.1) node (a0) ; (0,0) circle (0.1) node (a1) (-1.4,1) node [$u$]{} ; (2,0) circle (0.1) node (a2) ; (2,2) circle (0.1) node (a3) ; (0,2) circle (0.1) node (a4) ; (4,1) circle (0.1) node (a5) ; (a4)–(a0)–(a1) (a2)–(a0)–(a3) ; (a1)–(a2) (a3)–(a4)–(a1)–(a3) (a2)–(a4) (a2)–(a5)–(a3); (6,0) circle (0.1) node (b1) ; (8,0) circle (0.1) node (b2) ; (8,2) circle (0.1) node (b3) ; (6,2) circle (0.1) node (b4) ; (10,1) circle (0.1) node (b5) ; (b1)–(b2)–(b3)–(b4)–(b1)–(b3) (b2)–(b4) (b2)–(b5)–(b3) (b1)–(a5)–(b4); (12,0) circle (0.1) node (c1) ; (14,0) circle (0.1) node (c2) ; (14,2) circle (0.1) node (c3) ; (12,2) circle (0.1) node (c4) ; (16,1) circle (0.1) node (c5) ; (c1)–(c2)–(c3)–(c4)–(c1)–(c3) (c2)–(c4) (c2)–(c5)–(c3) (c1)–(b5)–(c4); (18,0) circle (0.1) node (d1) ; (20,0) circle (0.1) node (d2) ; (20,2) circle (0.1) node (d3) ; (18,2) circle (0.1) node (d4) (21.4,1) node [$v$]{} ; (d1)–(d2)–(d3)–(d4) (d1)–(d3) (d2)–(d4) (d1)–(c5)–(d4); (21,1) circle (0.1) node (d0) ; (d4)–(d0)–(d1) (d2)–(d0)–(d3) ;
Since $2(n-6)/5\leq n^2/60$ for all values of $n$, this establishes the required bound.
Next, consider the subcase when $s_1$ is as large as possible, for a fixed value of $n$. This is achieved by setting $\ell=1$, so there are no join nodes. Then every $H_j$ has at least 5 vertices, so for an upper bound we take $s_1 = n/10$ and $\Lambda(u,v)\leq s_1^2 = n^2/100 < n^2/60$. This completes the proof.
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[^1]: Supported by EPSRC Research Grant EP/M004953/1.
[^2]: Supported by the Australian Research Council Discovery Project DP140101519.
[^3]: Supported by EPSRC Research Grant EP/D00232X/1.
[^4]: An earlier version of this paper appeared as an extended abstract in PODC 2009 [@CDH].
[^5]: This research was partly performed while the first three authors were visiting the Simons Institute for the Theory of Computing.
|
---
abstract: 'Let $f$ be the ${{\mathbb F}_q}$-linear map over ${{\mathbb F}}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajbók et al. in “A new family of MRD-codes” (2018). For $n$ big enough, e.g. $n\geq5$ when $s=1$, we classify the values of $b/a$ such that the kernel of $f$ has dimension at most $1$. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of $f$; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.'
author:
- 'Olga Polverino, Giovanni Zini and Ferdinando Zullo[^1]'
title: On certain linearized polynomials with high degree and kernel of small dimension
---
11T06, 11G20, 51E20, 51E22
linearized polynomial, algebraic curve, linear set, MRD code, Hasse-Weil bound
Introduction
============
Let $q$ be a prime power and let $m$ be a positive integer. A $q$-*polynomial*, or *linearized polynomial*, over ${{\mathbb F}}_{q^m}$ is a polynomial of the form $$f(x)=\sum_{i=0}^t a_i x^{q^i},$$ where $a_i\in {{\mathbb F}}_{q^m}$, $t$ is a positive integer. If $a_t \neq 0$, we say that $t=\deg_q f(x)$ is the $q$-*degree* of $f$. We denote by $\mathcal{L}_{m,q}$ the set of all $q$-polynomials over ${{\mathbb F}}_{q^m}$ and by $\tilde{\mathcal{L}}_{m,q}$ the following quotient $\mathcal{L}_{m,q}/(x^{q^m}-x)$. The ${{\mathbb F}}_q$-linear maps of ${{\mathbb F}}_{q^m}$ can be identified with the polynomials in $\tilde{\mathcal{L}}_{m,q}$. This shows the relevance of linearized polynomials in the theory of finite fields and their algebraic and geometric applications. A fundamental problem in the theory of linearized polynomials is to characterize precisely the dimension of the kernel of the given polynomial in terms of its coefficients. Results in this direction are given in [@qres; @teoremone; @GQ2009; @McGuireSheekey; @PZ2019; @wl; @Zanella].
Let $n,s$ be positive integers such that $s<2n$ $\gcd(s,n)=1$. First in [@CMPZ], and later in [@PZ2019], the following polynomials are investigated $$\label{eq:form} f_{a,b,s}(x)=x+ax^{q^s}+bx^{q^{s+n}} \in \tilde{{{\mathcal L}}}_{2n,q}.$$ The following results are known from [@CMPZ] and [@PZ2019]:
- if ${\mathrm{N}}_{q^{2n}/q^n}(a)={\mathrm{N}}_{q^{2n}/q^n}(b)$, then $\dim_{{{\mathbb F}}_q} \ker f_{a,b,s}(x)\leq 1$;
- if ${\mathrm{N}}_{q^{2n}/q^n}(a)\neq {\mathrm{N}}_{q^{2n}/q^n}(b)$, then $\dim_{{{\mathbb F}}_q} \ker f_{a,b,s}(x)\leq 2$;
where ${\mathrm{N}}_{q^{2n}/q^n}(x)=x^{1+q^n}$.
Our main result is Theorem \[th:mainmain\] and concerns the existence, for every $\delta\in{{\mathbb F}}_{q^{2n}}$ with $\mathrm{N}_{q^{2n}/q^n}(\delta)\notin\{0,1\}$, of an element $a\in{{\mathbb F}}_{q^{2n}}$ such that the kernel of $f_{a,\delta a,s}$ has dimension $2$, providing $n$ is large enough.
\[th:mainmain\] Let $q$ be a prime power and $n,s$ be two relatively prime positive integers. Suppose that $$n\geq\begin{cases} 4s+2 & \textrm{if}\; q=3\textrm{ and }s>1,\,\textrm{or}\;q=2\textrm{ and }s>2; \\ 4s+1 & \textrm{otherwise}. \end{cases}$$ For every $\delta \in {{\mathbb F}}_{q^{2n}}^*$ with ${\mathrm{N}}_{q^{2n}/q^n}(\delta)\neq 1$ there exists $a \in {{\mathbb F}}_{q^{2n}}^*$ such that $$\dim_{{{\mathbb F}}_q} \ker (f_{a,b,s}(x))=2,$$ where $b=\delta a$.
In Remark \[rem:adjoint\] we show that we can always suppose $n>2s$, up to considering the adjoint polynomial.
The first step in the proof of Theorem \[th:mainmain\] is to manipulate the shape of $f_{a,b,s}(x)$ to translate the condition on the dimension of the kernel into the existence of $\mathbb{F}_{q^n}$-rational points in the intersection of certain ${{\mathbb F}}_{q^n}$-rational hypersurfaces, which are described in Theorem \[th:main\]. Then we prove that this intersection is described by means of an ${{\mathbb F}}_{q^{2n}}$-rational curve $\mathcal{X}$. Using intersection theory and function field theory, the curve $\mathcal{X}$ is shown to be absolutely irreducible of genus $q^{2s}-q^s-1$; Theorem \[th:mainmain\] now follows by Hasse-Weil bound.
Theorem \[th:mainmain\] also has applications in the theory of scattered polynomials. A polynomial $f(x)\in\tilde{{{\mathcal L}}}_{m,q}$ is said to be *scattered* if $$\dim_{{{\mathbb F}}_q}\ker(f(x)-\lambda x)\leq 1, \quad\textrm{for all}\;\; \lambda\in{{\mathbb F}}_{q^m}.$$ Scattered polynomials have been widely investigated, especially after the paper [@Sheekey2016], where Sheekey builds a bridge between scattered polynomials and rank metric codes. The family of linearized binomials $f_{\delta,s}(x)=x^{q^s}+\delta x^{q^{n+s}}\in\tilde{{{\mathcal L}}}_{2n,q}$ with $\delta\ne0$ contains a large number of scattered polynomials when $n$ is $3$ or $4$, as proved in [@CMPZ] and [@PZ2019]. The question arises whether there exist other values of $n$, possibly infinitely many, for which $f_{\delta,s}(x)$ is scattered. Many authors have considered the problem of classifying *exceptional* scattered polynomials $f(x)\in\tilde{{{\mathcal L}}}_{m,q}$, i.e. scattered polynomials which remain scattered over infinitely many extensions ${{\mathbb F}}_{q^{\ell m}}$ of ${{\mathbb F}}_{q^m}$; partial classification results have been provided by Bartoli and Zhou [@BZ], Bartoli and Montanucci [@BM], Ferraguti and Micheli [@FM]. Their results rely on the fact that the order of ${{\mathbb F}}_{q^{\ell m}}$ is much larger than the degree of $f(x)$; as a matter of fact, the key role in [@BZ; @BM] is played by the application of the Hasse-Weil bound to a curve whose degree has the same order of magnitude as $\deg f(x)$, and hence is small with respect to $q^{\ell m}$ (see [@BZ Lemma 2.1]). The aforementioned binomial $f_{\delta,s}(x)$ is not taken into account by their results, because $\deg f_{\delta,s}(x)=q^{n+s}$ is high with respect to the order of ${{\mathbb F}}_{q^{2n}}$. As a byproduct of Theorem \[th:mainmain\], we prove in Theorem \[th:noscatt\] that $f_{\delta,s}(x)$ is not scattered when $n$ is large enough with respect to $s$; for instance, when $s=1$ it is enough to choose $n\geq5$.
Finally, in Theorem \[th:applMRD\] we use Theorem \[th:mainmain\] to give an asymptotic classification of the family of rank-metric codes defined by the binomials $f_{\delta,s}(x)$.
The paper is organized as follows. Section \[sec:preliminaries\] contains preliminary results about algebraic curves and function function fields which are used in Section \[sec:proof\]. Section \[sec:proof\] is devoted to the proof of Theorem \[th:mainmain\]; the cases $q$ odd and $q$ even are studied separately, respectively in Section \[sec:qodd\] and in Section \[sec:qeven\]. Section \[sec:appl\] provides the applications of Theorem \[th:mainmain\]; namely, Section \[sec:linearsets\] shows the applications to scattered polynomials and linear sets, while Section \[sec:MRD\] shows the applications to rank metric codes.
Preliminaries on algebraic curves {#sec:preliminaries}
=================================
Let ${{\mathcal C}}$ be a projective, absolutely irreducible, algebraic curve over the algebraically closed field $\mathbb{K}=\overline{\mathbb{F}}_q$, embedded in a projective space ${\mathrm{PG}}(r,\mathbb{K})$ with homogeneous coordinates $(X_1\colon\ldots\colon X_{r+1})$ and not contained in the hyperplane at infinity $H_{\infty}:X_{r+1}=0$. Let $I(\mathcal{C})$ be the ideal of $\mathcal{C}$. Denote by $\mathbb{K}(\mathcal{C})$ the field of ($\mathbb{K}$-)rational functions on $\mathcal{C}$, briefly the function field of $\mathcal{C}$. Clearly, $\mathbb{K}(\mathcal{C})$ is generated over $\mathbb{K}$ by the coordinate functions $x_1,\ldots,x_r$ with $x_i=\frac{X_i+I(\operatorname{\mathcal{C}})}{X_{r+1}+I(\operatorname{\mathcal{C}})}$, and $\mathbb{K}(\operatorname{\mathcal{C}})\colon\mathbb{K}$ is a field extension of transcendence degree $1$. We denote by $\mathbb{P}(\mathcal{C})$ the set of places of $\mathcal{C}$, that is, the set of places of its function field $\mathbb{K}(\mathcal{C})$. For every $P\in\mathbb{P}(\mathcal{C})$ and every nonzero $z\in\mathbb{K}(\operatorname{\mathcal{C}})$, we denote by $v_P(z)\in\mathbb{Z}$ the valuation of $z$ at $P$; $P$ is said to be a zero (resp. a pole) of $z$ if $v_P(z)>0$ (resp. $v_P(z)<0$).
Suppose that $\operatorname{\mathcal{C}}$ is defined over $\mathbb{F}_q$, i.e. $I(\operatorname{\mathcal{C}})$ is generated by polynomials over $\mathbb{F}_q$. Then $\mathbb{F}_q(\operatorname{\mathcal{C}})$ denotes the $\mathbb{F}_q$-rational function field of $\operatorname{\mathcal{C}}$, i.e. the field of $\mathbb{F}_q$-rational functions on $\mathcal{C}$. The $\mathbb{F}_q$-rational places of $\operatorname{\mathcal{C}}$ are those places $P\in\mathbb{P}(\operatorname{\mathcal{C}})$ which are defined over $\mathbb{F}_q$; that is, $\mathbb{F}_q$-rational places of $\operatorname{\mathcal{C}}$ are the places of degree $1$ in $\mathbb{F}_q(\operatorname{\mathcal{C}})$, which are exactly the restriction to $\mathbb{F}_q(\operatorname{\mathcal{C}})$ of the places of $\mathbb{K}(\operatorname{\mathcal{C}})$ in the constant field extension $\mathbb{K}(\operatorname{\mathcal{C}})\colon\mathbb{F}_q(\operatorname{\mathcal{C}})$. The center of an $\mathbb{F}_q$-rational place is an $\mathbb{F}_q$-rational point of ${{\mathcal C}}$; conversely, if $P$ is a simple $\mathbb{F}_q$-rational point of ${{\mathcal C}}$, then the only place centered at $P$ is $\mathbb{F}_q$-rational, and may be identified with $P$.
Let $\varphi:\operatorname{\mathcal{C}}^\prime\to\operatorname{\mathcal{C}}$ be a covering of curves, i.e. a non-constant rational map from the curve $\operatorname{\mathcal{C}}^\prime$ to the curve $\operatorname{\mathcal{C}}$, of degree $\deg(\varphi)=[\mathbb{K}(\operatorname{\mathcal{C}}^\prime)\colon\mathbb{K}(\operatorname{\mathcal{C}})]$. We denote by $\varphi$ also the induced map $\mathbb{P}(\operatorname{\mathcal{C}}^\prime)\to\mathbb{P}(\operatorname{\mathcal{C}})$; if $\varphi$ is $\mathbb{F}_q$-rational, then $\varphi$ maps $\mathbb{F}_q$-rational places of $\operatorname{\mathcal{C}}^\prime$ to $\mathbb{F}_q$-rational places of $\operatorname{\mathcal{C}}$. The pull-back of $\varphi$ is denoted by $\varphi^*:\mathbb{K}(\operatorname{\mathcal{C}})\to\mathbb{K}(\operatorname{\mathcal{C}}^\prime)$. When $P\in\mathbb{P}(\operatorname{\mathcal{C}})$ and $P^\prime\in\mathbb{P}(\operatorname{\mathcal{C}}^\prime)$ satisfy $\varphi(P^\prime)=P$, we write $P^\prime|P$ and say that $P^\prime$ lies over $P$ in $\varphi$. We denote by $e(P^\prime|P)$ the ramification index of $P^\prime|P$, that is the unique positive integer such that $v_{P^\prime}(\varphi^*(w))=e(P^\prime|P)\cdot v_P(w)$ for all $w\in\mathbb{K}(\operatorname{\mathcal{C}})$; we have $\sum_{P^\prime:P^\prime|P}e(P^\prime|P)=\deg(\varphi)$. We say that $P^\prime$ is ramified over $P$ if $e(P^\prime|P)>1$, and totally ramified if $e(P^\prime|P)=\deg(\varphi)$; otherwise it is unramified. A ramified place $P^\prime$ is wildly ramified (resp. tamely ramified) if $e(P^\prime|P)$ is divisible (resp. not divisible) by $p$. We refer to [@HKT; @Sti] for further details on algebraic curves and function fields.
\[th:hurwitz\][(Hurwitz genus formula, [@Sti Theorem 3.4.13])]{} Let $\mathcal{C},\mathcal{C}^\prime$ be two absolutely irreducible curves over $\mathbb{K}=\overline{\mathbb{F}}_q$ and $\varphi:\mathcal{C}^\prime\to\mathcal{C}$ be a covering. For every place $P$ of $\mathcal{C}$ and every place $P^\prime$ of $\mathcal{C}^\prime$ lying over $P$ in $\varphi$, let $t\in\mathbb{K}(\mathcal{C})$ be a local parameter at $P$, $t^\prime\in\mathbb{K}(\mathcal{C}^\prime)$ be a local parameter at $P^\prime$, and $\varphi^*(t)\in\mathbb{K}(\mathcal{C}^\prime)$ be the pull-back of $t$ with respect to $\varphi$. Then $$2g(\mathcal{C}^\prime)-2=\deg(\varphi)\cdot(2g(\mathcal{C})-2)+\sum_{P^\prime\in\mathbb{P}(\mathcal{C}^\prime)}v_{P^\prime}\left(\frac{d\varphi^*(t)}{d t^\prime}\right).$$
If $P^\prime$ is not wildly ramified, then $v_{P^\prime}\left(\frac{d\varphi^*(t)}{d t^\prime}\right)=e(P^\prime|P)-1$. We now recall two important types of coverings. The following results are the application of [@Sti Corollary 3.7.4] and [@Sti Theorem 3.7.10] in the case of an algebraically closed constant field $\mathbb{K}$.
\[th:kummer\][[@Sti Corollary 3.7.4]]{} Let $\operatorname{\mathcal{C}}\colon F(X,Y)=0$ be an absolutely irreducible plane curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$, and $m$ be a positive integer with $\gcd(m,p)=1$. Let $f(X,Y)\in\mathbb{F}_q[X,Y]$ be such that there exists an $\overline{\mathbb F}_q$-rational place $Q$ of $\mathcal{C}$ at which the valuation of the rational function $f(x,y)$ is coprime with $m$, i.e. $\gcd(v_Q(f(x,y)),m)=1$. Let $\mathcal{C}^\prime$ be the curve given by the two affine equations $F(X,Y)=0$ and $Z^m=f(X,Y)$. Then the following holds.
- $\mathcal{C}^\prime$ is absolutely irreducible and defined over $\mathbb{F}_q$; $\operatorname{\mathcal{C}}'$ is called a *Kummer cover* of $\mathcal{C}$.
- The $\mathbb{F}_q$-rational covering $\varphi:\operatorname{\mathcal{C}}^\prime\to\operatorname{\mathcal{C}}$, $(X,Y,Z)\mapsto(X,Y)$, has degree $m$.
- For every place $P$ of $\mathcal{C}$ and every place $P^\prime$ of $\operatorname{\mathcal{C}}'$ lying over $P$ in $\varphi$, we have $e(P^\prime| P)=m/r_P$, where $r_P=\gcd(v_P(f(x,y)),m)>0$.
- The Hurwitz genus formula reads $$g(\operatorname{\mathcal{C}}')=1+m(g(\operatorname{\mathcal{C}})-1)+\frac{1}{2}\sum_{P\in\mathbb{P}(\mathcal{C})}(m-r_P).$$
If $\operatorname{\mathcal{C}}^\prime$ is an absolutely irreducible curve over $\mathbb{F}_q$ defined by the two affine equations $F(X,Y)=0$ and $L(Z)=f(X,Y)$, for some $f(X,Y),F(X,Y)\in\mathbb{F}_q[X,Y]$ and some separable $p$-polynomial $L(T)\in\mathbb{F}_q[T]$, then $\mathcal{C}^\prime$ is said to be a *generalized Artin-Schreier cover* of the curve $\mathcal{C}:F(X,Y)=0$, with generalized Artin-Schreier covering $\varphi:\operatorname{\mathcal{C}}^\prime\to\operatorname{\mathcal{C}}$, $(X,Y,Z)\mapsto(X,Y)$.
\[th:artinschreier\][[@Sti Theorem 3.7.10]]{} Let $\mathcal{C}:F(X,Y)=0$ be an absolutely irreducible plane curve defined over a finite field $\mathbb{F}_q$ of charateristic $p$. Let $L(T)\in\mathbb{F}_q[T]$ be a separable $p$-polynomial of degree $\bar{q}$ with all its roots in $\mathbb{F}_q$. Let $f(X,Y)\in\mathbb{F}_q[X,Y]$ be such that for every place $P\in\mathbb{P}(\mathcal{C})$ there exists a rational function $\omega$ on $\mathcal{C}$ (depending on $P$) satisfying either $v_P(f(x,y)-L(\omega))\geq0$ or $v_P(f(x,y)-L(\omega))=-m$ with $m>0$ and $p\nmid m$. Define $m_P=-1$ in the former case and $m_P=m$ in the latter case. Let $\mathcal{C}^\prime$ be the space curve given by the two affine equations $F(X,Y)=0$ and $L(Z)=f(X,Y)$. If there exists a place $Q\in\mathbb{P}(\mathcal{C})$ with $m_Q>0$, then $\mathcal{C}^\prime$ is a generalized Artin-Schreir cover of $\operatorname{\mathcal{C}}$, defined over $\mathbb{F}_q$.
With the above notation, the following holds for generalized Artin-Schreier curves.
- The $\mathbb{F}_q$-rational covering $\varphi:\operatorname{\mathcal{C}}^\prime\to\operatorname{\mathcal{C}}$, $(X,Y,Z)\mapsto(X,Y)$, has degree $\bar{q}$.
- For every place $P$ of $\mathcal{C}$ and every place $P^\prime$ of $\mathcal{C}^\prime$ lying over $P$ in $\varphi$, $e(P^\prime|P)$ is equal either to $1$ or to $\bar{q}$ according to $m_P=-1$ or $m_P>0$, respectively.
- The Hurwitz genus formula reads $$g(\mathcal{C}^\prime)=\bar{q}\cdot g(\mathcal{C})+\frac{\bar{q}-1}{2}\cdot\left(-2+\sum_{P\in\mathbb{P}(\mathcal{C})}(m_P+1)\right).$$
We now recall the well-known Hasse-Weil bound.
\[th:hasseweil\][[@Sti Theorem 5.2.3] (Hasse-Weil bound)]{} Let $\mathcal{C}$ be an absolutely irreducible curve defined over $\mathbb{F}_q$ and with genus $g$. Then the number $N_{q}$ of $\mathbb{F}_q$-rational places of $\mathcal{C}$ satisfies $$q+1-2g\sqrt{q}\leq N_{q} \leq q+1+2g\sqrt{q}.$$
Proof of Theorem \[th:mainmain\] {#sec:proof}
================================
In this section we prove Theorem \[th:mainmain\]. First we determine necessary and sufficient conditions on $a$ and $b$ for $f_{a,b,s}(x)$ having kernel of dimension $2$; cf. Theorem \[th:main\]. Then we investigate such conditions by means of algebraic-geometric tools.
The first remark shows that different choices of $a,b$ with the same norm of $b/a$ over $\mathbb{F}_{q^n}$ provide polynomials $f_{a,b,s}(x)$ with the same behaviour.
\[rk:normdelta\] Assume that the linearized polynomial $f_{a,b,s}(x)=x+ax^{q^s}+bx^{q^{s+n}} \in {{\mathbb F}}_{q^{2n}}[x]$, with $\gcd(s,n)=1$ and $b=\delta a$, has kernel of dimension two. Clearly, for each $\lambda \in {{\mathbb F}}_{q^{2n}}^*$ we have $$\dim_{{{\mathbb F}}_q} \ker(\lambda^{-1}f_{a,b,s}(\lambda x))=2,$$ where $$\lambda^{-1}f_{a,b,s}(\lambda x)=x+a \lambda^{q^s-1}x^{q^s}+a\lambda^{q^s-1}\delta \lambda^{q^s(q^n-1)}x^{q^{s+n}}=f_{a',b',s}(x),$$ with $a'=a\lambda^{q^s-1}$, $\delta'=\lambda^{q^s(q^n-1)}\delta$ and $b'=a'\delta'$. Note that for each element $\delta' \in {{\mathbb F}}_{q^{2n}}$ with ${\mathrm{N}}_{q^{2n}/q^n}(\delta')={\mathrm{N}}_{q^{2n}/q^n}(\delta)$ there exists $\lambda \in {{\mathbb F}}_{q^{2n}}$ such that $\delta'=\delta \lambda^{q^s(q^n-1)}$. Therefore, if $\dim_{{{\mathbb F}}_q} \ker(f_{a,b,s}(x))=2$, with $b=\delta a$, then for each $\delta' \in {{\mathbb F}}_{q^{2n}}$ with ${\mathrm{N}}_{q^{2n}/q^n}(\delta')={\mathrm{N}}_{q^{2n}/q^n}(\delta)$ there exists $a' \in {{\mathbb F}}_{q^{2n}}$ such that $\dim_{{{\mathbb F}}_q} \ker(f_{a',b',s}(x))=2$, with $b'=\delta' a'$.
The second remark shows that we may assume $s<n/2$.
\[rem:adjoint\] The *adjoint* of a $q$-polynomial $f(x)=\sum_{i=0}^{n-1}a_i x^{q^i}$, with respect to the bilinear form $\langle x,y\rangle=\mathrm{Tr}_{q^n/q}(xy)$, is given by $$\hat{f}(x)=\sum_{i=0}^{n-1}a_{i}^{q^{n-i}} x^{q^{n-i}}.$$ In particular, if $f(x)$ is a $q$-polynomial of shape , then $$f_{a,b,s}(x)=x+ax^{q^s}+bx^{q^{n+s}}\in \tilde{\mathcal{L}}_{2n,q},$$ with $\gcd(s,n)=1$ and its adjoint is $$\hat{f}_{a,b,s}(x)=x+a^{q^{2n-s}}x^{q^{2n-s}}+b^{q^{n-s}}x^{q^{n-s}}.$$ Therefore, choosing $s^\prime=2n-s$, $a^\prime=a^{q^{2n-s}}$, $b^\prime=b^{q^{n-s}}$, we get $$\hat{f}_{a,b,s}(x)=f_{a',b',s'}(x),$$ while choosing $s^{\prime\prime}=n-s$, $a^{\prime\prime}=b^{q^{n-s}}$, $b^{\prime\prime}=a^{q^{2n-s}}$, we get $$\hat{f}_{a,b,s}(x)=f_{a^{\prime\prime},b^{\prime\prime},s^{\prime\prime}}(x),$$ i.e. $\hat{f}_{a,b,s}(x)$ is of shape . Therefore, the family of $q$-polynomials we are studying is closed by the adjoint operation. Furthermore, we underline that by [@BGMP2015 Lemma 2.6], the kernels of $f_{a,b,s}$ and $\hat{f}_{a,b,s}$ have the same dimension (see also [@CsMP pages 407–408]). Thus, we can assume $s< n/2$.
We now prove that the shape of $\delta$ can be chosen as in .
\[th:deltachoice\] Let $f_{a,b,s}(x) \in {{\mathbb F}}_{q^{2n}}[x]$, with $b=a\delta$. Then $\dim_{{{\mathbb F}}_q}\ker(f_{a,b,s}(x))=2$ if and only if $\dim_{{{\mathbb F}}_q}\ker(f_{\overline{a},\overline{b},s}(x))=2$, with $$\label{eq:deltachoice}
\overline{\delta}=\frac{\xi^{q^{s+n}}-\xi^{q^n}}{\xi^{q^n}-\xi^{q^s}},$$ for some $\xi \in {{\mathbb F}}_{q^{2n}}\setminus{{\mathbb F}}_{q^n}$ and some $\overline{a}\in {{\mathbb F}}_{q^{2n}}$, $\overline{b}=\overline{\delta}\overline{a}$.
Assume that $\dim_{{{\mathbb F}}_q}\ker(f_{a,b,s}(x))=2$, i.e. there exist $x_0 \in {{\mathbb F}}_{q^{2n}}^*$ and $y_0\in {{\mathbb F}}_{q^{2n}}\setminus{{\mathbb F}}_q$ such that $x_0/y_0 \notin {{\mathbb F}}_q$ and $$\frac{x_0^{q^s}+\delta x_0^{q^{s+n}}}{x_0}=\frac{y_0^{q^s}+\delta y_0^{q^{s+n}}}{y_0},$$ which may be rewritten as follows $$\delta(y_0 x_0^{q^{s+n}}-x_0y_0^{q^{s+n}})=x_0y_0^{q^s}-y_0x_0^{q^s}.$$ If $y_0 x_0^{q^{s+n}}-x_0y_0^{q^{s+n}}$ would be zero, than $x_0/y_0 \in {{\mathbb F}}_{q^{2n}}\cap{{\mathbb F}}_{q^{s+n}}={{\mathbb F}}_q$, a contradiction. Hence, $$\delta=\frac{x_0y_0^{q^s}-y_0x_0^{q^s}}{y_0 x_0^{q^{s+n}}-x_0y_0^{q^{s+n}}},$$ and, since $y_o=\xi x_0$ for some $\xi \in {{\mathbb F}}_{q^{2n}}\setminus {{\mathbb F}}_q$, we have $$\delta= \frac{1}{-x_0^{q^{s+n}-q^s}} \frac{\xi^{q^s}-\xi}{\xi^{q^{s+n}}-\xi}.$$ By Remark \[rk:normdelta\], $\dim_{{{\mathbb F}}_q}\ker(f_{a,b,s}(x))=2$ if and only if there exists $\overline{a},\overline{b}$ as in the claim such that $\dim_{{{\mathbb F}}_q}\ker(f_{\overline{a},\overline{b},s}(x))=2$. If $\xi\in\mathbb{F}_{q^n}$, then $\overline{\delta}=-1$, and hence $\dim_{{{\mathbb F}}_q}\ker(f_{\overline{a},\overline{b},s}(x))\leq1$. The claim follows.
As a consequence of Theorem \[th:deltachoice\] we get the following result.
There exist $\delta \in {{\mathbb F}}_{q^{2n}}^*$ for which $\dim_{{{\mathbb F}}_q}\ker(f_{a,b,s}(x))\leq 1$, with $b=\delta a$, for each $a\in {{\mathbb F}}_{q^{2n}}^*$ if and only if $$\left| \left\{ {\mathrm{N}}_{q^{2n}/q^n}\left( \frac{\xi^{q^{n+s}}-\xi^{q^n}}{\xi^{q^n}-\xi^{q^s}} \right) \colon \xi \in {{\mathbb F}}_{q^{2n}}\setminus{{\mathbb F}}_{q^n} \right\} \right|< q^n-1.$$
Since $\xi \notin {{\mathbb F}}_{q^n}$, we have that $\xi$ is the root of an irreducible polynomial $X^2-SX-T \in {{\mathbb F}}_{q^n}[X]$, where ${\mathrm{N}}_{q^{2n}/q^n}(\xi)=-T$ and $\mathrm{Tr}_{q^{2n}/q^n}(\xi)=S$. Also, $\{1,\xi\}$ is an ${{\mathbb F}}_{q^n}$-basis of ${{\mathbb F}}_{q^{2n}}$ and so there exist $A,B \in {{\mathbb F}}_{q^n}$ such that $\xi^{q^s}=A+B\xi$. In the next we give some relations involving $A,B,S$ and $T$.
The following holds:
1. $S^{q^s}=2A+BS$;
2. $-T^{q^s}=A^2+B(AS-BT)$.
In particular, $\mathrm{Tr}_{q^{2n}/q^n}(\xi^{q^s+1})=2BT+AS+BS^2$ and $\mathrm{Tr}_{q^{2n}/q^n}(\xi^{q^s+q^n})=AS-2BT$.
As $$\xi^{q^s+q^n}=(A+B\xi)(S-\xi)=AS-BT-A\xi$$ and $$\xi^{1+q^{n+s}}=-BT+(S^{q^s}-A-BS)\xi,$$ we have that $$\mathrm{Tr}_{q^{2n}/q^n}(\xi^{q^s+q^n})=AS-2BT+(S^{q^s}-2A-BS)\xi.$$ Since $\mathrm{Tr}_{q^{2n}/q^n}(\xi^{q^s+q^n})\in {{\mathbb F}}_{q^n}$, we get the first relation. Also, $$-T^{q^s}={\mathrm{N}}_{q^{2n}/q^n}(\xi^{q^s})=A^2+ABS-B^2T,$$ i.e. the second relation.
Let $\alpha \in {{\mathbb F}}_{q^n}^*$ with $\alpha \neq 1$. Then $${\mathrm{N}}_{q^{2n}/q^n}\left( \frac{\xi^{q^{n+s}}-\xi^{q^n}}{\xi^{q^n}-\xi^{q^s}} \right)=\frac{\xi^{q^n+1}+\xi^{q^s+q^{n+s}}-(\xi^{1+q^{n+s}}+\xi^{q^s+q^n})}{\xi^{q^n+1}+\xi^{q^s+q^{n+s}}-(\xi^{q^n+q^{n+s}}+\xi^{q^s+1})}=\alpha,$$ which can be written as $$(1-\alpha)(T+T^{q^s})-\alpha S^{q^s+1}+(1+\alpha)(AS-2BT)=0.$$
Hence, we have the following result.
\[th:main\] Let $\alpha \in {{\mathbb F}}_{q^n}^*$ with $\alpha \ne 1$ and $s$ a positive integer with $\gcd(s,n)=1$. If there exist $T,S,A,B \in {{\mathbb F}}_{q^n}$ such that
1. $(1-\alpha)(T+T^{q^s})-\alpha S^{q^s+1}+(1+\alpha)(AS-2BT)=0$;
2. $X^2-SX-T \in {{\mathbb F}}_{q^n}[X]$ is irreducible over ${{\mathbb F}}_{q^n}$;
3. $S^{q^s}=2A+BS$;
4. $-T^{q^s}=A^2+B(AS-BT)$,
then for every $\delta\in {{\mathbb F}}_{q^{2n}}$, with ${\mathrm{N}}_{q^{2n}/q^n}(\delta)=\alpha$, there exists $a\in\mathbb{F}_{q^{2n}}^*$ such that $\dim_{\mathbb{F}_q}\ker(f_{a,b,s}(x))=2$, where $b=\delta a$.
In the rest of this section $q=p^h$ with $p$ prime. We will show that the existence of the parameters $T,S,A,B\in\mathbb{F}_{q^n}$ satisfying the hypothesis of Theorem \[th:main\] is equivalent to the existence of a suitable affine $\mathbb{F}_{q^n}$-rational point of the algebraic plane curve with equation or , for $q$ odd or $q$ even respectively.
Proof of Theorem \[th:mainmain\] for $q$ odd {#sec:qodd}
--------------------------------------------
Denote by $\Delta=S^2+4T$. By 3. and 4. of Theorem \[th:main\], we get $$B=\epsilon\Delta^{\frac{q^s-1}{2}},\quad A=\frac{1}{2}(S^{q^s}-\epsilon S \Delta^{\frac{q^s-1}{2}}),$$ where $\epsilon\in\{1,-1\}$. Hence we get $AS-2BT=\frac{1}{2}\epsilon\Delta^{\frac{q^s-1}{2}}-\frac{1}{2}S^{q^s+1}$. Replacing such values in 1. of Theorem \[th:main\], we get $$\label{eq:qodd1}
2(T+T^{q^s})(1-\alpha)+(1-\alpha)S^{q^s+1}=\epsilon(\alpha+1)\Delta^{\frac{q^s+1}2}.$$ Also, the irreducibility of $X^2-SX-T$ over ${{\mathbb F}}_{q^n}$ is equivalent to the existence of a nonsquare element $\eta$ of ${{\mathbb F}}_{q^n}$ and a nonzero element $Z$ of ${{\mathbb F}}_{q^n}$ such that $\Delta=\eta Z^2$. Therefore, becomes $$2(T+T^{q^s})+S^{q^s+1}=\beta \eta^{\frac{q^s+1}2} Z^{q^s+1},$$ where $\beta=\epsilon\frac{\alpha+1}{1-\alpha}$. Using that $T=\frac{\eta Z^2-S^2}{4}$, we get the following equation: $$\label{eq:curveqodd} -(S^{q^s}-S)^2+\eta Z^2 + \eta^{q^s} Z^{2q^s} - 2\beta\eta^{\frac{q^s+1}{2}}Z^{q^s+1}=0.$$
\[th:qodd\] Let $\beta\in\mathbb{F}_{q^n}\setminus\{1,-1\}$ and $\eta$ a non-square in $\mathbb{F}_{q^n}$. The plane curve $\mathcal{C}$ with affine equation is absolutely irreducible and has genus $g(\mathcal{C})=q^{2s}-q^s-1$.
Let $G(Z)=\eta Z^2+\eta^{q^s} Z^{2q^s}-2\beta\eta^{\frac{q^s+1}{2}}Z^{q^s+1}\in\mathbb{F}_{q^n}[Z]$, and let $\mathcal{C}_1$ be the plane curve with affine equation $F_1(U,Z)=0$, where $F_1(U,Z)=U^2-G(Z)$. By direct computation using the assumption $\beta\ne\pm1$ follows that $0$ is the unique multiple root of $G(Z)$, with multiplicity $2$; the other $2q^s-2$ roots $\lambda_1,\ldots,\lambda_{2q^s-2}$ of $G(Z)$ are simple. Then $G(Z)$ is not a square in $\mathbb{K}[Z]$, whence $F_1(U,Z)$ is irreducible over $\mathbb{K}=\overline{\mathbb{F}}_{q^n}$, i.e. $\mathcal{C}_1$ is absolutely irreducible.
The genus of the quadratic Kummer cover $\mathcal{C}_1$ of the projective line is computed as follows. Let $z,u$ be the coordinate functions of $\mathcal{C}_1$, so that the function field of $\mathcal{C}_1$ is $\mathbb{K}(\mathcal{C}_1)=\mathbb{K}(z,u)$. The valuation of $G(z)$ at the zero of $z-\lambda_i$ in $\mathbb{K}(\mathbb{P}_z^1)=\mathbb{K}(z)$ is $1$, for every $i=1,\ldots,2q^s-2$. The valuation of $G(z)$ at any other place of $\mathbb{P}_z^1$ is even; namely, it is $2$ at the zero of $z$, $-2q^s$ at the pole of $z$, and $0$ at the zero of $z-\mu$ whenever $G(\mu)\ne0$. By Theorem \[th:kummer\], the only ramified places in $\mathcal{C}_1\to\mathbb{P}_z^1$ are the zeros of $z-\lambda_1,\ldots,z-\lambda_{2q^s-2}$; hence, $$g(\mathcal{C}_1)= 1+2(g(\mathbb{P}_z^1)-1)+\frac{1}{2}(2q^s-2)(2-1)=q^s-2.$$
Since $\mathcal{C}$ has equation $(S^{q^s}-S)^2=G(Z)$, it is enough to show that $\mathcal{C}$ is an Artin-Schreier cover of $\mathcal{C}_1$, with covering $\varphi:\mathcal{C}\to\mathcal{C}_1$, $(Z,S)\mapsto(Z,U=S^{q^s}-S)$, of degree $q^s$. To this aim, consider the two poles $P_{\infty}$ and $Q_\infty$ of $u$ on $\mathcal{C}_1$; the rational function $1/z$ is a local parameter at each of them, i.e. $v_{P_{\infty}}(1/z)=v_{Q_{\infty}}(1/z)=1$. By direct computation, the Laurent series of $u$ at $P_\infty$ with respect to $1/z$ is $$u=\sqrt{\eta^{q^s}}\left(1/z\right)^{-q^s}-\beta\sqrt{\eta}\left(1/z\right)^{-1}+\frac{\eta-\beta^2\eta}{2\sqrt{\eta^{q^s}}}(1/z)^{q^s-2}+w,$$ for some $w\in\mathbb{K}(\mathcal{C}_1)$ with $v_{P_\infty}(w)>q^s-2$. By choosing $\omega_{P_{\infty}}=\sqrt{\eta}z$ one has that $u-(\omega_{P_{\infty}}^{q^s}-\omega_{P_{\infty}})$ has valuation $-1$ at $P_\infty$, because $\beta\ne1$. Analogously, there exists $\omega_{Q_\infty}$ such that $v_{Q_{\infty}}(u-(\omega_{Q_{\infty}}^{q^s}-\omega_{Q_{\infty}}))=-1$. Hence, by Theorem \[th:artinschreier\], $\mathcal{C}$ is an absolutely irreducible Artin-Schreier extension $\mathcal{C}_1$ of degree $q^s$.
The ramified places in $\mathcal{C}\to\mathcal{C}_1$ are exactly $P_\infty$ and $Q_\infty$, which are totally ramified; any other place of $\mathcal{C}_1$ is unramified under $\mathcal{C}$. Therefore, $$g(\mathcal{C})=q^s\cdot g(\mathcal{C}_1)+\frac{q^s-1}{2}\left(-2+2\cdot2\right)=q^{2s}-q^s-1.$$
\[prop:HWqodd\] Let $\mathcal{C}$ be the plane curve with affine equation . If $$n\geq\begin{cases} 4s+1 & \textrm{if }\,q>3, \\
4s+2 & \textrm{if }\,q=3,s>1,\\
5 & \textrm{if }\,q=3,s=1;
\end{cases}$$ then there exists an $\mathbb{F}_{q^n}$-rational affine point $(\bar{z},\bar{s})$ of $\mathcal{C}$ such that $\bar{t}=\frac{\eta \bar{z}^2-\bar{s}^2}{4}$ is different from zero.
By Theorem \[th:qodd\], $\mathcal{C}$ is absolutely irreducible with genus $g(\mathcal{C})=q^{2s}-q^s-1$. By Theorem \[th:hasseweil\], the number $N_{q^n}$ of $\mathbb{F}_{q^n}$-rational places of $\mathcal{C}$ satisfies $$N_{q^n}\geq q^n+1-2(q^{2s}-q^s-1)\sqrt{q^n}.$$ From the proof of Theorem \[th:qodd\] the following facts follow.
- $z$ has exactly $2$ poles on $\mathcal{C}$, which coincide with the poles of $s$, namely the places lying over $P_\infty$ and $Q_\infty$.
- Using the equation of $\mathcal{C}$, the zeros of $t=\frac{\eta z^2-s^2}{4}=\frac{(\sqrt{\eta}z-s)(\sqrt{\eta}z+s)}{4}$ on $\mathcal{C}$ are also zeros of $(\beta-1)z^{q^s+1}$ and hence of $z$ as $\beta\ne1$; thus, they are the common zeros of $z$ and $s$ on $\mathcal{C}$, and there are exactly $2$ of them.
Altogether, there are $4$ places of $\mathcal{C}$ which are either poles of $s$ or $z$ or $t$, or zeros of $t$. The assumption on $n$ implies that $$q^n+1-2(q^{2s}-q^s-1)\sqrt{q^n}>4,$$ whence $N_{q^n}>4$. Then there exists an $\mathbb{F}_{q^n}$-rational place $P$ which is not a pole of $z$, $s$, or $t$, and is not a zero of $t$. Then the point $(\bar{z},\bar{s})=(z(P),s(P))$ yields the claim.
From Theorem \[th:main\] and Proposition \[prop:HWqodd\] follows Corollary \[cor:mainmain\_qodd\], which is our main result Theorem \[th:mainmain\] when $q$ is odd.
\[cor:mainmain\_qodd\] Let $q$ be an odd prime power, $s\geq1$ be such that $\gcd(s,n)=1$. Suppose that $$n\geq\begin{cases} 4s+2 & \textrm{if}\; q=3\textrm{ and }s>1; \\ 4s+1 & \textrm{otherwise}. \end{cases}$$ Then for every $\delta\in\mathbb{F}_{q^{2n}}$ satisfying $\mathrm{N}_{q^{2n}/q^n}(\delta)\notin\{0,1\}$ there exists $a\in\mathbb{F}_{q^{2n}}^*$ such that $\dim_{\mathbb{F}_q}\ker(f_{a,b})=2$, where $b=\delta a$.
Proof of Theorem \[th:mainmain\] for $q$ even {#sec:qeven}
---------------------------------------------
Let $q$ be a power of $2$. The conditions of Theorem \[th:main\] read:
1. $T+T^{q^s}+\beta S^{q^{s}+1}+AS=0$, with $\beta=\frac{\alpha}{1+\alpha}\notin\{0,1\}$;
2. $S\ne0$ and $\mathrm{Tr}_{q^n/2}(T/S^2)=1$;
3. $B=S^{q^s-1}$;
4. $A^2+A S^{q^s}+S^{2q^s-2}T+T^{q^s}=0$.
By 1. we get $$A= \beta S^{q^s}+\frac{T+T^{q^s}}S,$$ which can be replaced in 4. obtaining $$\label{eq:curveqeven} (\beta^2+\beta) S^{2(q^s+1)}+S^{q^s+1}(T^{q^s}+T)+S^{2q^s}T+S^2T^{q^s}+T^{2q^s}+T^2=0.$$ Set $T=S^2 Y$. Then reads $H(S,Y)=0$, where $$\label{eq:miserve}
\begin{array}{lll} H(S,Y)=Y^2+S^{4(q^s-1)}Y^{2q^s}+\beta^2 S^{2(q^s-1)}+S^{q^s-1}Y+ \\ \\
S^{3(q^s-1)}Y^{q^s}+\beta S^{2(q^s-1)}+S^{2(q^s-1)}Y+S^{2(q^s-1)}Y^{q^s}.
\end{array}$$ Straightforward computation using $\mathrm{Tr}_{q^s/2}(Y)+\mathrm{Tr}_{q^s/2}(Y)^2=Y^{q^s}+Y$ shows that the polynomial $H(S,Y)$ in splits as follows.
We have $H(S,Y)=G(S,Y)\cdot G^\prime(S,Y)$, where $$G(S,Y)= S^{2(q^s-1)}Y^{q^s}+S^{q^s-1}(1+\beta+ \mathrm{Tr}_{q^s/2}(Y))+Y,$$ $$G^\prime(S,Y)= S^{2(q^s-1)}Y^{q^s}+S^{q^s-1}(\beta + \mathrm{Tr}_{q^s/2}(Y))+Y.$$
The condition 2. is equivalent to the existence of an element $Z\in {{\mathbb F}}_{q^n}$ such that $$\label{eq:T}
T=S^2(Z^2+Z+\epsilon),$$ for some fixed $\epsilon\in\mathbb{F}_{q^n}$ such that $\mathrm{Tr}_{q^{n}/2}(\epsilon)=1$.
Let $\mathcal{C}$ be the plane curve with affine equation $F(S,Z)=G^\prime(S,Z^2+Z+\epsilon)$.
In order to prove Theorem \[th:mainmain\] when $q$ is even, by Theorem \[th:main\] and the arguments at the beginning of Section \[sec:qeven\], it is enough to prove the existence of an ${{\mathbb F}_{q^n}}$-rational affine point $(\bar{s},\bar{z})$ of $\mathcal{C}$ such that $\bar{s}\ne0$ and $\bar{z}^2+\bar{z}+\epsilon\ne0$. This is done by showing that $\mathcal{C}$ is absolutely irreducible, computing its genus, and applying the Hasse-Weil lower bound. To this aim, we consider the following subcovers: $$\varphi_2:\operatorname{\mathcal{C}}\to\operatorname{\mathcal{C}}_2,\quad (S,Z)\mapsto(X=S^{q^s-1},Z),$$ $$\varphi_1:\operatorname{\mathcal{C}}_2\to\operatorname{\mathcal{C}}_1,\quad (X,Z)\mapsto(X,Y=Z^2+Z+\epsilon).$$ The curves $\operatorname{\mathcal{C}}_2,\operatorname{\mathcal{C}}_1$ have equation $\operatorname{\mathcal{C}}_2\colon F_2(X,Z)=0$ and $\operatorname{\mathcal{C}}_1\colon F_1(X,Y)=0$, where $$F_2(X,Z)= X^2(Z^2+Z+\epsilon)^{q^s}+X(\beta + \mathrm{Tr}_{q^s/2}(Z^2+Z+\epsilon))+Z^2+Z+\epsilon,$$ $$F_1(X,Y)= X^2Y^{q^s}+X(\beta + \mathrm{Tr}_{q^s/2}(Y))+Y.$$
We first prove that $\mathcal{C}_2$ is absolutely irreducible by direct inspection, and that $\mathcal{C}$ is absolutely irreducible being a Kummer cover of $\mathcal{C}_2$. To compute the genus of $\mathcal{C}$, we start by the genus of the absolutely irreducible subcover $\mathcal{C}_1$ of $\mathcal{C}_2$, which is computed with the Hurwitz genus formula. Then we compute the genus of $\mathcal{C}_2$ as an Artin-Schreier cover of $\mathcal{C}_1$. Finally, the genus of the Kummer cover $\mathcal{C}$ of $\mathcal{C}_2$ is computed.
Let $\gamma,\gamma+1$ be the roots of $Z^2+Z+\epsilon\in\mathbb{F}_{q^n}[Z]$. Hence, $\mathrm{Tr}_{q^{2n}/q^n}(\gamma)=1$ and $\mathrm{N}_{q^{2n}/q^n}(\gamma)=\epsilon$; also, $\mathrm{Tr}_{q^{n}/2}(\epsilon)=1$ implies $\gamma\in\mathbb{F}_{q^{2n}}\setminus\mathbb{F}_{q^n}$.
\[lemma:C’2irr\] The curve $\mathcal{C}_2$ is absolutely irreducible.
By contradiction, suppose $F_2(X,Z)=\hat{F}(X,Z)\cdot \tilde{F}(X,Z)$ for some non-constant polynomials $\hat{F},\tilde{F}\in\mathbb{K}[X,Z]$. Then $\hat{F}(X,Z)=X\cdot A(Z)+B(Z)$ and $\tilde{F}(X,Z)=X\cdot C(Z)+D(Z)$; up to scalar multiplication, $$A(Z)=(Z+\gamma)^a(Z+\gamma+1)^b,\qquad C(Z)=(Z+\gamma)^c(Z+\gamma+1)^d,$$ where $a,b,c,d\geq0$ satisfy $a+c=b+d=q^s$. Also, $$(Z+\gamma)^a(Z+\gamma+1)^bD(Z)+(Z+\gamma)^c(Z+\gamma+1)^dB(Z)=\beta + \mathrm{Tr}_{q^s/2}(Z^2+Z+\epsilon).$$ Clearly, $a,b \in \{0,q^s\}$ as $\beta\ne0$. If $a=0$, then $b=c=q^s$ and $d=0$, since $B(Z)D(Z)=Z^2+Z+\epsilon$; hence $$\label{eq:Zirr} (Z+\gamma+1)^{q^s}D(Z)+(Z+\gamma)^{q^s}B(Z)=\beta + \mathrm{Tr}_{q^s/2}(Z^2+Z+\epsilon).$$ This implies $D(\gamma)\ne0$ and $B(\gamma+1)\ne0$, whence $D(Z)=\lambda(Z+\gamma+1)$ and $B(Z)=\lambda^{-1}(Z+\gamma)$ for some $\lambda\in\mathbb{K}^*$. With $Z=\gamma$ in , we get $\beta=1$, a contradiciton. If $a=q^s$, the same arguments yield a contradiction.
\[prop:C’1\] The curve $\operatorname{\mathcal{C}}_1$ is absolutely irreducible with genus $q^s/2$.
As $\operatorname{\mathcal{C}}_1$ is a subcover of $\operatorname{\mathcal{C}}_2$, the absolute irreducibility of $\operatorname{\mathcal{C}}_1$ follows from Lemma \[lemma:C’2irr\]. Let $(X\colon Y\colon V)$ be the homogeneous coordinates of the affine point $(X,Y)$. By direct computation, the affine points of $\mathcal{C}_1$ are simple, and hence we identify each of them with the unique place of $\mathcal{C}_1$ centered at it; the points at infinity $E_0=(1\colon 0\colon 0)$ and $E_1=(0\colon 1\colon 0)$ of $\mathcal{C}_1$ are singular. The point $E_0$ is $q^s$-fold with unique tangent line $\ell_Y : Y=0$ having intersection multiplicity $q^s+1$ with $\mathcal{C}_1$ at $E_0$; the point $E_1$ is double with unique tangent line $\ell_X: X=0$ having intersection multiplicity $q^s+1$ with $\mathcal{C}_1$ at $E_1$.
We compute the genus of $\operatorname{\mathcal{C}}_1$ by applying Theorem \[th:hurwitz\] to the covering $\varphi_0\colon\operatorname{\mathcal{C}}_1\to\mathbb{P}_y^1$, $(X\colon Y\colon V)\mapsto(Y\colon V)$, where $y$ is the coordinate function of $Y$; $\varphi_0$ has degree $2$. We describe the places of $\mathbb{P}_y^1$ which ramify in $\varphi_0$; we denote by $P_\mu$, $\mu\in\mathbb{K}$, the zero of $y-\mu$ on $\mathbb{P}_y^1$, and by $P_\infty$ the pole of $y$ on $\mathbb{P}_y^1$.
- If $\bar{y}\in\mathbb{K}$ satisfies $\bar{y}\ne0$ and $\beta+\mathrm{Tr}_{q^s/2}(\bar{y})\ne0$, then $F_1(X,\bar{y})$ has two distinct roots $\bar{x}_1,\bar{x}_2\in\mathbb{K}$, and hence $P_{\bar y}$ does not ramify in $\varphi_0$.
- If $\bar{y}\in\mathbb{K}$ is one of the $q^s/2$ distinct roots of $\beta+\mathrm{Tr}_{q^s/2}(Y)$, then $(\bar{x},\bar{y})\in\varphi_0^{-1}(P_{\bar{y}})$, with $\bar{x}=\sqrt{\bar{y}^{1-q^s}}$. The tangent line to $\operatorname{\mathcal{C}}_1$ at $(\bar{x},\bar{y})$ is $\ell_{\bar y}:Y-\bar{y}=0$, having multiplicity intersection $2$ with $\operatorname{\mathcal{C}}_1$ at $(\bar{x},\bar{y})$. Hence, $$e((\bar{x},\bar{y})\mid P_{\bar y})= e((\bar{x},\bar{y})\mid P_{\bar y})\cdot v_{P_{\bar y}}(y-\bar{y})=v_{(\bar{x},\bar{y})}(y-\bar{y})=2,$$ so that $P_{\bar y}$ totally ramifies in $\varphi_0$.
- The change of coordinates $(X\colon Y\colon V)\mapsto(X\colon V\colon Y)$ maps $E_1$ to the origin $E_2=(0\colon 0\colon 1)=(0,0)$ and $\mathcal{C}_1$ to the curve $\overline{\mathcal{C}}_1$ with affine equation $\overline{F}_1(X,Y)=0$ where $$\overline{F}_1(X,Y)=X^2+X(\beta Y^{q^s+1}+\sum_{i=0}^{sh-1}Y^{q^s+1-2^i}) + Y^{q^s+1}.$$ The point $E_2$ is double for $\overline{\mathcal{C}}_2$ with tangent line $\ell_X\colon X=0$; while this holds, we apply iteratively the quadratic transformation $(X\colon Y\colon V)\mapsto(XV\colon Y^2\colon YV)$ which maps $\overline{\mathcal{C}}_1$ to the curve with equation $\overline{F}_1(XY,Y)/Y^2=0$. After $k$ times, the curve has equation $$X^2+X(\beta Y^{q^s+1-k}+\sum_{i=0}^{sh-1}Y^{q^s+1-2^i-k})+Y^{q^s+1-2k}=0.$$ Hence, after $q^s/2$ times, the curve has only one affine point on the line $\ell_Y\colon Y=0$, which is a simple point. This means that $\mathcal{C}_1$ has exactly one place centered at $E_1$, which we identify with $E_1$. Since the intersection multiplicity of $\operatorname{\mathcal{C}}_1$ at $E_1$ with $\ell_\infty\colon V=0$ and $\ell_Y$ is $2$ and $0$ respectively, we have that $v_{E_1}(y)=-2$; see [@HKT Theorem 4.36]. Thus, the pole of $y$ in $\mathbb{P}^1_y$ is totally ramified in $\varphi_0$.
- The unique place centered at $E_2=(0,0)$ is clearly a zero of $y$. The only place of $\mathbb{P}^1_y$ which can be covered by $E_0$ is the zero of $y$. Therefore, the zero of $y$ in $\mathbb{P}^1_y$ is not ramified in $\varphi_0$, and $E_0$ and $E_2$ are the simple zeros of $y$ on $\mathcal{C}_1$.
The $q^s/2+1$ ramification places $P'$, namely $E_1$ and $(\bar{x},\bar{y})$ with $\beta+\mathrm{Tr}_{q^2/2}(\bar{y})=0$, are wildly ramified. For each of them, we choose a local parameter $t^\prime$ at $P^\prime$, a local parameter $t$ at the place $P$ lying under $P^\prime$ in $\varphi_0$, and compute $v_{P^\prime}(d\varphi_0^*(t)/dt^\prime)$, where the pull-back $\varphi_0^*$ of $\varphi_0$ is the identity on $\mathbb{K}(\mathbb{P}_y^1)=\mathbb{K}(y)$.
- Let $P^\prime=E_1$, lying over $P=P_\infty$. We choose $t=1/y$. From $v_{E_1}(y)=-2$ we get $v_{E_1}(x)=q^s-1$; hence we can choose $t^\prime=1/(xy^{q^s/2})$. By direct computation, the Laurent series of $t$ at $E_1$ with respect to $t^\prime$ is $t= (t^\prime)^2 +(t^\prime)^3 + w$, with $v_{E_1}(w)\geq4$. Thus, $\frac{dt}{dt^\prime}=(t^\prime)^2+\frac{dw}{dt^\prime}$ has valuation $2$ at $E_1$.
- Let $P^\prime=(\bar{x},\bar{y})$, lying over $P=P_{\bar y}$ with $\beta+\mathrm{Tr}_{q^s/2}(\bar y)=0$. We choose $t=y-\bar{y}$ and $t^\prime=x-\bar{x}$. By direct computation, the Laurent series of $t$ at $P^\prime$ with respect to $t^\prime$ is $t=\frac{\bar{y}^{q^s}}{\bar{x}+1}(t^\prime)^2+\frac{\bar{y}^{q^s}}{\bar{x}^2+1}(t^\prime)^3+w$, with $v_{P^\prime}(w)\geq4$. Thus, $\frac{dt}{dt^\prime}=\frac{\bar{y}^{q^s}}{\bar{x}^2+1}(t^\prime)^2+\frac{dw}{dt^\prime}$ has valuation $2$ at $P^\prime$.
Theorem \[th:hurwitz\] now yields $$2g(\operatorname{\mathcal{C}}_1)-2=\deg(\varphi_0)\cdot(2g(\mathbb{P}_y^1)-2)+\left(\frac{q^s}{2}+1\right)\cdot2,$$ whence $g(\operatorname{\mathcal{C}}_1)=\frac{q^s}{2}$.
\[prop:C’2\] The curve $\operatorname{\mathcal{C}}_2$ has genus $q^s-1$.
By Lemma \[lemma:C’2irr\], $\operatorname{\mathcal{C}}_2$ is absolutely irreducible; hence, the covering $\varphi_1:\operatorname{\mathcal{C}}_2\to\operatorname{\mathcal{C}}_1$, $(X,Z)\mapsto(X,Y=Z^2+Z+\epsilon)$, is an Artin-Schreier covering of degree $2$. Every place of $\operatorname{\mathcal{C}}_1$ which is not a pole of $y-\epsilon$ is unramified in $\varphi_1$. We consider the unique pole of $y-\epsilon$ on $\operatorname{\mathcal{C}}_1$, namely $E_1$. By direct computation, the Laurent series of $y$ at $E_1$ with respect to the local parameter $t^\prime=1/(xy^{q^s/2})$ is $y-\epsilon=(t^\prime)^{-2}+(t^\prime)^{-1}+w$, with $v_{E_1}(w)\geq0$. Choosing $\omega=(t^\prime)^{-1}$, we have $v_{E_1}((y-\epsilon)-(\omega^2+\omega))=v_{E_1}(w)\geq0$. Thus, by Theorem \[th:artinschreier\], $E_1$ is unramified in $\varphi_1$. Altogether, the covering $\varphi_1:\operatorname{\mathcal{C}}_2\to\operatorname{\mathcal{C}}_1$ is unramified. This implies $$2g(\operatorname{\mathcal{C}}_2)-2 = \deg(\varphi_1)\cdot(2g(\operatorname{\mathcal{C}}_1)-2),$$ whence $g(\operatorname{\mathcal{C}}_2)=q^s-1$.
\[th:C’\] For the curve $\mathcal{C}$ the following holds.
- $\operatorname{\mathcal{C}}$ is absolutely irreducible with genus $q^{2s}-q^s-1$.
- Let $s$ and $z$ be the coordinate functions of $\mathcal{C}$, and let $t=s^2(z^2+z+\epsilon)$. The number of $\mathbb{F}_{q^n}$-rational places of $\operatorname{\mathcal{C}}$ which are zeros of $t$, or poles of either $s$ or $z$ or $t$, is at most $2q^s+2$.
We compute the valuation of $x$ at the places of $\operatorname{\mathcal{C}}_2$. From the proofs of Propositions \[prop:C’1\] and \[prop:C’2\] follows that $x$ has exactly $2$ zeros on $\operatorname{\mathcal{C}}_1$, namely $E_2$ with $v_{E_2}(x)=1$ and $E_1$ with $v_{E_1}(x)=q^s-1$ ; also, $x$ has exactly $4$ zeros on $\operatorname{\mathcal{C}}_2$, namely the two places $Q_1,Q_2$ lying over $E_2$ and the two places $Q_3,Q_4$ over $E_1$. Using the ramification indices, this implies $v_{Q_1}(x)=v_{Q_2}(x)=1$ and $v_{Q_3}(x)=v_{Q_4}(x)=q^s-1$. The unique pole of $x$ on $\operatorname{\mathcal{C}}_1$ is $E_0$; since $v_{E_0}(y)=1$, this implies $v_{E_0}(x)=-q^s$. The poles of $x$ on $\operatorname{\mathcal{C}}_2$ are the two places $R_1,R_2$ lying over $E_0$, with $v_{R_1}(x)=v_{R_2}(x)=-q^s$.
Therefore, by Theorem \[th:kummer\], $\operatorname{\mathcal{C}}$ is absolutely irreducible and $\varphi_2:\operatorname{\mathcal{C}}\to\operatorname{\mathcal{C}}_2$ is a Kummer covering of degree $q^s-1$. The places of $\operatorname{\mathcal{C}}_2$ which ramify in $\varphi_2$ are exactly $Q_1,Q_2,R_1,R_2$, and they are totally ramified; any other place is unramified in $\varphi_2$. The genus of $\operatorname{\mathcal{C}}$ is $$g(\operatorname{\mathcal{C}})=1+\deg(\varphi_2)\cdot(g(\operatorname{\mathcal{C}}_2)-1)+\frac{1}{2}\cdot 4\cdot(\deg(\varphi_2)-1)=q^{2s}-q^s-1.$$ Using the proofs of Propositions \[prop:C’1\] and \[prop:C’2\], we obtain that:
- $s$ has exactly $2$ poles on $\operatorname{\mathcal{C}}$, namely the places over $R_1$ or $R_2$;
- $z$ has exactly $2(q^s-1)$ poles on $\operatorname{\mathcal{C}}$, namely the places over $Q_3$ or $Q_4$;
- $s^2$ has exactly $2(q^s-1)+2$ zeros on $\operatorname{\mathcal{C}}$; namely, two of them lie over $Q_1$ or $Q_2$, while $2(q^s-1)$ of them lie over $Q_3$ or $Q_4$ (and have been been already considered above);
- $z^2+z+\epsilon=y$ has exactly $4$ zeros on $\operatorname{\mathcal{C}}$, namely the places over $Q_1,Q_2,R_1,R_2$ (which have been already considered above).
Altogether, the number of $\mathbb{F}_{q^n}$-rational places of $\operatorname{\mathcal{C}}$ which are poles of $s$, $z$, $t$, or are zeros of $t$, is smaller than or equal to $2q^s+2$.
From Theorems \[th:main\] and \[th:C’\] follows Corollary \[cor:mainmain\_qeven\], which is our main result Theorem \[th:mainmain\] when $q$ is even.
\[cor:mainmain\_qeven\] Let $q$ be an even prime power, $s\geq1$ be such that $\gcd(s,n)=1$. Suppose that $$n\geq\begin{cases} 4s+2 & \textrm{if}\;q=2\textrm{ and }s>2; \\ 4s+1 & \textrm{otherwise}. \end{cases}$$ Then for every $\delta\in\mathbb{F}_{q^{2n}}$ satisfying $\mathrm{N}_{q^{2n}/q^n}(\delta)\notin\{0,1\}$ there exists $a\in\mathbb{F}_{q^{2n}}^*$ such that $\dim_{\mathbb{F}_q}\ker(f_{a,b,s})=2$, where $b=\delta a$.
By Theorems \[th:hasseweil\] and \[th:C’\][*[(a)]{}*]{}, the number $N_{q^n}$ of $\mathbb{F}_{q^n}$-rational places of $\mathcal{C}$ satisfies $$N_{q^n}\geq q^n+1 - 2(q^{2s}-q^s-1)\sqrt{q^n} > 2q^s+2.$$ By Theorem \[th:C’\][*[(b)]{}*]{}, there exists an $\mathbb{F}_{q^n}$-rational affine poin $(\bar{s},\bar{z})$ of $\mathcal{C}$ such that $\bar{t}=\bar{s}^2 (\bar{z}^2+\bar{z}+\epsilon)$ is different from zero. The claim follows.
Applications to linear sets and rank metric codes {#sec:appl}
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Linear sets {#sec:linearsets}
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Let $\Lambda={\mathrm{PG}}(V,{{\mathbb F}}_{q^m})={\mathrm{PG}}(1,q^m)$, where $V$ is a vector space of dimension $2$ over ${{\mathbb F}}_{q^m}$. A point set $L$ of $\Lambda$ is said to be an *${{\mathbb F}}_q$-linear set* of $\Lambda$ of rank $k$ if it is defined by the non-zero vectors of a $k$-dimensional ${{\mathbb F}}_q$-vector subspace $U$ of $W$, i.e. $$L=L_U=\{{\langle}{\bf u} {\rangle}_{\mathbb{F}_{q^m}} \colon {\bf u}\in U\setminus \{{\bf 0} \}\}.$$ We say that two linear sets $L_U$ and $L_W$ of $\Lambda={\mathrm{PG}}(1,q^m)$ are $\mathrm{P}\Gamma \mathrm{L}$-*equivalent* if there exists $\varphi \in \mathrm{P}\Gamma \mathrm{L} (2,q^m)$ such that $\varphi(L_U)=L_W$.
We start by pointing out that if the point $\langle (0,1) \rangle_{{{\mathbb F}}_{q^m}}$ is not contained in a linear set $L_U$ of rank $m$ of ${\mathrm{PG}}(1,q^m)$ (which we can always assume after a suitable projectivity), then $U=U_f=\{(x,f(x))\colon x\in {{\mathbb F}}_{q^m}\}$ for some $q$-polynomial $\displaystyle f(x)=\sum_{i=0}^{m-1}a_ix^{q^i}\in \tilde{\mathcal{L}}_{m,q}$. In this case we will denote the associated linear set by $L_f$. Also, recall that the *weight of a point* $P=\langle \mathbf{u} \rangle_{{{\mathbb F}}_{q^m}}$ is $w_{L_U}(P)=\dim_{{{\mathbb F}}_q}(U\cap\langle \mathbf{u} \rangle_{{{\mathbb F}}_{q^m}})$.
One of the most studied classes of linear sets of the projective line, especially because of its applications (see e.g. [@Polverino; @Sheekey2016]), is the family of maximum scattered linear sets. A [*maximum scattered*]{} ${{\mathbb F}}_q$-linear set of ${\mathrm{PG}}(1,q^m)$ is an ${{\mathbb F}}_q$-linear set of rank $m$ of ${\mathrm{PG}}(1,q^m)$ of size $(q^m-1)/(q-1)$, or equivalently a linear set of rank $m$ in ${\mathrm{PG}}(1,q^m)$ all of whose points have weight one. If $L_f$ is a maximum scattered linear set in ${\mathrm{PG}}(1,q^m)$, we also say that $f$ is a *scattered polynomial*. The known scattered polynomials of ${{\mathbb F}}_{q^m}$ are
1. $f_1(x)=x^{q^s}\in \tilde{\mathcal{L}}_{m,q}$, with $\gcd(s,m)=1$, see [@BL2000];
2. $f_2(x)= x^{q^s}+\alpha x^{q^{m-s}}\in\tilde{\mathcal{L}}_{m,q}$, with $m\geq 4$, $\gcd(s,m)=1$, ${\mathrm{N}}_{q^m/q}(\alpha) \notin\{0,1\}$, see [@LMPT2015; @LP2001; @Sheekey2016];
3. $f_3(x)= x^{q^s}+\alpha x^{q^{s+\frac{m}2}}\in\tilde{\mathcal{L}}_{m,q}$, $m \in \{6,8\}$, $\gcd(s,\frac{m}2)=1$ and some conditions on $\alpha$, see [@CMPZ] and below;
4. $f_4(x)=x^q+x^{q^3}+\alpha x^{q^5}\in \tilde{\mathcal{L}}_{6,q}$, $q$ odd and $\alpha^2+\alpha=1$, see [@CsMZ2018; @MMZ];
5. $f_5(x)=h^{q-1}x^q-h^{q^2-1}x^{q^2}+x^{q^4}+x^{q^5}\in \tilde{\mathcal{L}}_{6,q}$, $q$ odd, $h^{q^3+1}=-1$, see [@BZZ; @ZZ].
In [@CMPZ], the authors introduced the family of linear sets $L_{\delta,s}$ of rank $2n$ in ${\mathrm{PG}}(1,q^{2n})$ mentioned in 3., i.e. those linear sets defined by the ${{\mathbb F}}_q$-subspace $$\label{eq:Ud,s}
U_{\delta,s}=\{ (x,f_{\delta,s}(x)) \colon x \in {{\mathbb F}}_{q^{2n}} \}\subset {{\mathbb F}}_{q^{2n}}\times {{\mathbb F}}_{q^{2n}},$$ where $$f_{\delta,s}(x)=x^{q^s}+\delta x^{q^{n+s}}\in{\tilde {{\mathcal L}}}_{2n,q},$$ with $\mathrm{N}_{q^{2n/q^n}}(\delta) \notin\{0,1\}$, $1 \leq s \leq 2n-1$ and $\gcd(s,n)=1$. The relevance of this family relies on the property that each point of $L_{\delta,s}$ has weight at most two; see [@CMPZ Proposition 4.1]. In [@CMPZ Section 7] the authors proved that for $n=3$ and $q>4$ there exists $\delta \in {{\mathbb F}}_{q^2}$ such that $L_{s,\delta}$ is scattered; for $n=4$, $q$ odd and $\delta^2=-1$ the linear set $L_{\delta,s}$ is scattered. In [@PZ2019 Theorem 7.3] the authors completely determined for $n=3$ necessary and sufficient conditions on $\delta$ ensuring $L_{\delta,s}$ to be scattered. Note that for $n=3$ we may restrict to the case $s=2$, since every linear set $L_{\delta,s}$ is equivalent to $L_{\delta',2}$ for some $\delta'\in {{\mathbb F}}_{q^{2n}}^*$. More precisely, if ${\mathrm{N}}_{q^6/q^3}(\delta)\notin\{0,1\}$ and we denote $A=-\frac{1}{\delta^{q^3+1}-1}$, one has that $L_{\delta,2}$ is scattered if and only if the equation $$\label{eq:eq2degree} Y^2-(\mathrm{Tr}_{q^3/q}(A)-1)Y+{\mathrm{N}}_{q^3/q}(A)=0$$ admits two distinct roots in ${{\mathbb F}}_q$.
\[th:noscatt\] Let $q$ be a prime power and $n,s$ be two relatively prime positive integers. Suppose that $$n\geq\begin{cases} 4s+2 & \textrm{if}\; q=3\textrm{ and }s>1,\,\textrm{or}\;q=2\textrm{ and }s>2; \\ 4s+1 & \textrm{otherwise}. \end{cases}$$ Then, for every $\delta\in\mathbb{F}_{q^{2n}}^*$, the $\mathbb{F}_q$-linear set $L_{\delta,s}$ in ${\mathrm{PG}}(1,q^{2n})$ is not scattered.
For every $m\in\mathbb{F}_{q^{2n}}$, the weight of the point $\langle(1,m)\rangle_{\mathbb{F}_{q^{2n}}}$ in $L_{\delta,s}$ coincides with the dimension over $\mathbb{F}_q$ of the kernel of $f_{\delta,s}(x)-mx$.
If ${\mathrm{N}}_{q^{2n}/q^n}(\delta)=1$, then the point $\langle(1,0)\rangle_{\mathbb{F}_{q^{2n}}}$ has weight $n$ in $L_{\delta,s}$. Let ${\mathrm{N}}_{q^{2n}/q^n}(\delta)\ne1$. By Theorem \[th:mainmain\], there exists $a\in{{\mathbb F}}_{q^{2n}}^*$ such that $\dim_{{{\mathbb F}}_q}\ker(f_{a,\delta a,s}(x))=2$, whence $$\dim_{{{\mathbb F}}_q}\ker\left(a\left(f_{\delta,s}(x)+\frac{1}{a}x\right)\right)=2.$$ This implies that the point $\langle\left(1,-\frac{1}{a}\right)\rangle_{{{\mathbb F}}_{q^{2n}}}$ has weight $2$ in $L_{\delta,s}$. The claim is proved.
Hence, we have the following description for the linear set $L_{\delta,s}$.
\[cor:classbin\] Let $q$ be a prime power and $n,s$ be two relatively prime positive integers.
- If $n=3$, then $L_{\delta,s}$ is a scattered linear set if and only if Equation \[eq:eq2degree\] admits two distinct roots in ${{\mathbb F}}_q$.
- If $n=4$, $q$ is odd and $\delta^2=-1$ then $L_{\delta,s}$ is scattered.
- If $$n\geq\begin{cases} 4s+2 & \textrm{if}\; q=3\textrm{ and }s>1,\,\textrm{or}\;q=2\textrm{ and }s>2, \\ 4s+1 & \textrm{otherwise}, \end{cases}$$ then, for every $\delta\in\mathbb{F}_{q^{2n}}^*$, $L_{\delta,s}$ is not scattered.
The claim follows from [@PZ2019 Theorem 7.3], [@CMPZ Theorem 7.2], and Theorem \[th:noscatt\].
Among the known scattered polynomials listed above, the families in 3., 4. and 5. provide scattered polynomials for infinitely many $q$’s, but only over a specific extension of ${{\mathbb F}}_q$, namely either ${{\mathbb F}}_{q^6}$ or ${{\mathbb F}}_{q^8}$. Unlike this situation, the families in 1. and 2. provide scattered polynomials over infinitely many extensions ${{\mathbb F}}_{q^m}$ of ${{\mathbb F}}_{q}$; they are named respectively as scattered polynomials of pseudoregulus type, and as scattered polynomials of LP type (after Lunardon and Polverino).
The scattered polynomials of pseudoregulus or LP type have raised the following question: which polynomials over $\mathbb{F}_{q^m}$ are scattered over infinitely many extensions of $\mathbb{F}_{q^m}$?
[[@BZ Section 1]]{} Let $f(x)\in\tilde{{{\mathcal L}}}_{m,q}$, $0\leq t\leq m-1$, $\ell\geq1$, and $U_{\ell}=\{(x^{q^t},f(x))\colon x\in{{\mathbb F}}_{q^{m\ell}}\}$. We say that $f(x)$ is an exceptional scattered polynomial of index $t$ if $L_{U_{\ell}}$ is a scattered ${{\mathbb F}}_q$-linear set in ${\mathrm{PG}}(1,q^{m\ell})$ for infinitely many $\ell$’s.
Clearly, the scattered polynomials of pseudoregulus type are exceptional scattered of index $0$. Also, for the scattered polynomial $f_2(x)$ of LP type, $$U_{f_2}=\{(x^{q^s}, x^{q^{2s}}+\alpha x)\colon x\in{{\mathbb F}}_{q^m}\};$$ thus, the polynomial $x^{q^{2s}}+\alpha x$ is exceptional scattered of index $s$.
For a scattered polynomial $f(x)\in\tilde{{{\mathcal L}}}_{m,q}$ of index $t$, we say that $f(x)$ is $t$-normalized if the following properties hold: $f(x)$ is monic; the coefficient of $x^{q^t}$ in $f(x)$ is zero; if $t>0$, the coefficient of $x$ in $f(x)$ is nonzero. Up to ${\rm PGL}$-equivalence of the corresponding scattered linear set, we may always assume that $f(x)$ is $t$-normalized.
Let $f(x)\in\tilde{{{\mathcal L}}}_{m,q}$ be a $t$-normalized exceptional scattered polynomial of index $t$. Then the following holds.
- If $t=0$, then $f(x)$ is of pseudoregulus type; see [@BZ Corollary 3.4] for $q>5$, [@BM Section 4] for $q\leq5$.
- If $t=1$ or $t=2$, then $f(x)$ is either of pseudoregulus type or of LP type; see [@BZ Corollary 3.7] for $t=1$, [@BM Corollary 1.4] for $t=2$.
- If $t\geq3$, $q$ is odd, and $\max\{\deg_q f(x),t\}$ is an odd prime, then $f(x)=x$; see [@FM Theorem 1.2].
Recall that the polynomials $f_3(x)$ of family 3. in the list above are scattered under certain assumptions for $m\in\{6,8\}$; even when $f_3(x)$ is not scattered, still all the points of $L_{f_3}$ have weight at most $2$. Thus, one may conjecture that family 3. contains scattered polynomials over ${{\mathbb F}}_{q^m}$ for every even $m$. Note that, even if this is the case, the arising scattered polynomials are not exceptional: not only the coefficients but also the degree depend heavily on the underlying field ${{\mathbb F}}_{q^m}$.
Our asymptotic result Theorem \[th:mainmain\] shows that the family of scattered polynomial in 3. cannot be extended to any higher extension $\mathbb{F}_{q^m}$ when $m$ is large enough with respect to $s$.
Rank metric codes {#sec:MRD}
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Rank metric codes were introduced by Delsarte [@Delsarte] in 1978 and they have been intensively investigated in recent years because of their applications; we refer to [@sheekey_newest_preprint] for a recent survey on this topic. The set of $m \times n$ matrices ${{\mathbb F}_q}^{m\times n}$ over ${{\mathbb F}_q}$ may be endowed with a metric, called *rank metric*, defined by $$d(A,B) = \mathrm{rk}\,(A-B).$$ A subset $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}_q}^{m\times n}$ equipped with the rank metric is called a *rank metric code* (shortly, a *RM*-code). The minimum distance of $\operatorname{\mathcal{C}}$ is defined as $$d = \min\{ d(A,B) \colon A,B \in \operatorname{\mathcal{C}},\,\, A\neq B \}.$$ Denote the parameters of a RM-code $\operatorname{\mathcal{C}}\subseteq{{\mathbb F}_q}^{m,n}$ with minimum distance $d$ by $(m,n,q;d)$. We are interested in ${{\mathbb F}_q}$-*linear* RM-codes, i.e. ${{\mathbb F}_q}$-subspaces of ${{\mathbb F}_q}^{m\times n}$. Delsarte showed in [@Delsarte] that the parameters of these codes must obey a Singleton-like bound, i.e. $$|\operatorname{\mathcal{C}}| \leq q^{\max\{m,n\}(\min\{m,n\}-d+1)}.$$ When equality holds, we call $\operatorname{\mathcal{C}}$ a *maximum rank distance* (*MRD* for short) code. Examples of ${{\mathbb F}_q}$-linear MRD-codes were first found in [@Delsarte; @Gabidulin]. We say that two ${{\mathbb F}_q}$-linear RM-codes $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ are equivalent if there exist $X \in \mathrm{GL}(m,q)$, $Y \in \mathrm{GL}(n,q)$, and $\sigma\in{\rm Aut}({{\mathbb F}_q})$ such that $$\operatorname{\mathcal{C}}'=\{XC^\sigma Y \colon C \in \operatorname{\mathcal{C}}\}.$$
The *left* and *right* idealisers of $\operatorname{\mathcal{C}}$ are defined in [@LN2016] as $L(\operatorname{\mathcal{C}})=\{A \in \mathrm{GL}(m,q) \colon A \operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}\}$ and $R(\operatorname{\mathcal{C}})=\{B \in \mathrm{GL}(n,q) \colon \operatorname{\mathcal{C}}B \subseteq \operatorname{\mathcal{C}}\}$. They are invariant under the equivalence of rank metric codes, and have been investigated in [@LTZ2]; further invariants have been introduced in [@GZ; @NPH2].
Much of the focus on MRD-codes of ${{\mathbb F}_q}^{m\times m}$ to date has been on codes which are ${{\mathbb F}}_{q^m}$-*linear*, i.e. codes in which the left (or right) idealiser contains a field isomorphic to ${{\mathbb F}}_{q^m}$, since for such codes a fast decoding algorithm has been developed in [@Gabidulin]. Very few examples of such codes are known, see [@BZZ; @CMPZ; @CsMPZh; @CsMZ2018; @Delsarte; @Gabidulin; @LP2001; @MMZ; @Sheekey2016; @ZZ].
In [@Sheekey2016 Section 5] Sheekey showed that scattered ${{\mathbb F}}_q$-linear sets of ${\mathrm{PG}}(1,q^m)$ of rank $m$ yield ${{\mathbb F}}_q$-linear MRD-codes with parameters $(m,m,q;m-1)$ with left idealiser isomorphic to ${{\mathbb F}}_{q^m}$; see [@CsMPZ2019; @CSMPZ2016; @ShVdV] for further details on such kind of connections. We briefly recall here the construction from [@Sheekey2016]. Let $U_f=\{(x,f(x))\colon x\in {{\mathbb F}}_{q^m}\}$, where $f(x)$ is a scattered $q$-polynomial. The choice of an ${{\mathbb F}}_q$-basis for ${{\mathbb F}}_{q^m}$ defines a canonical ring isomorphism between $\mathrm{End}({{\mathbb F}}_{q^m},{{\mathbb F}}_q)$ and ${{\mathbb F}}_q^{m\times m}$. Thus, the set $$\operatorname{\mathcal{C}}_f=\{x\mapsto af(x)+bx \colon a,b \in {{\mathbb F}}_{q^m}\}\subset \mathrm{End}({{\mathbb F}}_{q^m},{{\mathbb F}}_q)$$ corresponds to a set of $m\times m$ matrices over ${{\mathbb F}}_q$ forming an ${{\mathbb F}}_q$-linear MRD-code with parameters $(m,m,q;m-1)$. Also, as $\operatorname{\mathcal{C}}_f$ is an ${{\mathbb F}}_{q^m}$-subspace of $\mathrm{End}({{\mathbb F}}_{q^m},{{\mathbb F}}_q)$, its left idealiser $L(\operatorname{\mathcal{C}}_f)$ is isomorphic to ${{\mathbb F}}_{q^m}$; see also [@CMPZ Section 6].
Now consider the set $$\operatorname{\mathcal{C}}_{f_{\delta,s}}=\{ x\mapsto a(x^{q^s}+\delta x^{q^{s+n}})+bx \colon a,b \in {{\mathbb F}}_{q^{2n}} \},$$ which corresponds to a set of $2n\times 2n$ matrices over ${{\mathbb F}}_q$ forming an ${{\mathbb F}}_q$-linear rank metric code with parameters $(2n,2n,q;2n-i)$, where $$i=\max\{ w_{L_{\delta,s}}(P) \colon P \in {\mathrm{PG}}(1,q^{2n}) \}.$$
The following theorem is a consequence of Corollary \[cor:classbin\] and states that, when $n$ is large enough, $\operatorname{\mathcal{C}}_{f_{\delta,s}}$ is not an MRD-code.
\[th:applMRD\] Let $q$ be a prime power and $n,s$ be two relatively prime positive integers.
- If $n=3$, then $\operatorname{\mathcal{C}}_{f_{\delta,s}}$ is an MRD-code if and only if Equation [\[eq:eq2degree\]]{} admits two distinct roots in ${{\mathbb F}}_q$; see [[@CMPZ]]{} and [[@PZ2019]]{}.
- If $n=4$, $q$ odd and $\delta^2=-1$ then $\operatorname{\mathcal{C}}_{f_{\delta,s}}$ is an MRD-code; see [[@CMPZ]]{}.
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Olga Polverino, Giovanni Zini and Ferdinando Zullo\
Dipartimento di Matematica e Fisica,\
Università degli Studi della Campania “Luigi Vanvitelli”,\
Viale Lincoln 5,\
I–81100 Caserta, Italy\
[[*{olga.polverino,giovanni.zini,ferdinando.zullo}@unicampania.it*]{}]{}
[^1]: This research was supported by the project “VALERE: VAnviteLli pEr la RicErca" of the University of Campania “Luigi Vanvitelli”, and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).
|
---
abstract: 'In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.'
address: 'Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan'
author:
- Tatsuya Miura
bibliography:
- 'bibliography.bib'
title: Polar tangential angles and free elasticae
---
Introduction {#sectintroduction}
============
The purpose of this note is to develop a geometric approach to elastic curve problems, i.e., variational problems involving the total squared curvature, also known as the bending energy. The variational study of elastic curves is originated with D. Bernoulli and L. Euler in the 1740’s, but it is still ongoing even in view of the original (clamped) boundary value problem. In particular, the properties of solutions such as uniqueness or stability are not fully understood and sensitively depend on the parameters in the constraints; see e.g. [@Linner1998a; @Singer2008; @Sachkov2008; @Miura2020] and references therein.
In order to study boundary conditions involving the tangent vector, it would be helpful to precisely understand the so-called polar tangential angle for a plane curve, which is the angle formed between the position vector and the tangent vector. In this note we give a geometric characterization of the first-derivative sign of the polar tangential angle, and then deduce the monotonicity of the angle for certain curves of monotone curvature with the help of the classical Tait-Kneser theorem.
Our geometric aspect would be useful for studying an elastic curve, since its curvature is represented by using the Jacobi elliptic function, and hence a monotone-curvature curve naturally appears as (a part of) an elastic curve. As a concrete example, we apply our monotonicity result to a free boundary problem involving the bending energy. More precisely, we minimize the bending energy among graphical curves $u:[-1,1]\to\mathbb{R}$ subject to the boundary condition $u(\pm1)=0$ such that $u\geq\psi$, where $\psi$ is a given obstacle function. The existence of graph minimizers is a somewhat delicate issue [@DallAcqua2018; @Mueller2019; @Yoshizawa2019] in contrast to the confinement-type obstacle problem in [@Dayrens2018]. In this paper, choosing $\psi$ to be a symmetric cone, we prove (non)existence results depending on the height of the cone. Our results reprove the nonexistence result of Müller [@Mueller2019] by a novel geometric approach, and also provide a new uniqueness result in the class of symmetric graphs; very recently, the same uniqueness result is independently obtained by Yoshizawa [@Yoshizawa2019] in a different way. See Remark \[rem:obstacle\] for a precise comparative review.
This paper is organized as follows. Section \[sec:polar\] is devoted to understanding the polar tangential angle. In Section \[sec:obstacle\] we apply a monotonicity result (Corollary \[cor:monotoneangle\]) to the aforementioned obstacle problem (Theorem \[thm:obstacle\]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
When attending a mini-symposium in the OIST hosted by , the author was informed by that he has also obtained the same kind of results [@Yoshizawa2019]. The author would like to thank them for encouraging publication of this paper. The author is also grateful to for giving helpful comments to an earlier version of this manuscript. This work is in part supported by JSPS KAKENHI Grant Number 18H03670 and 20K14341, and by Grant for Basic Science Research Projects from The Sumitomo Foundation.
Geometry of polar tangential angles {#sec:polar}
===================================
Throughout this section we consider a smooth plane curve $\gamma$ parameterized by the arclength parameter $s$, namely, $\gamma\in C^\infty([0,L),\mathbb{R}^2)$, where $L\in(0,\infty]$, and $|\gamma_s(s)|=1$ for any $s$, where the subscript of $s$ denotes the arclength derivative. Let $T$ denote the unit tangent $\gamma_s$, and $N:=R_{\pi/2}T$ the unit normal, where $R_\theta$ stands for the counterclockwise rotation matrix through angle $\theta\in\mathbb{R}$.
Our main object is the polar tangential angle. To define it rigorously, we call a plane curve $\gamma$ [*generic*]{} if the curve does not pass the origin $O\in\mathbb{R}^2$ except for $s=0$. For such a curve we denote the normalized position vector by $X:=\gamma/|\gamma|$.
For a generic plane curve $\gamma:[0,L)\to\mathbb{R}^2$, the [*polar tangential angle function*]{} $\omega:(0,L)\to\mathbb{R}$ is defined as a smooth function such that $R_{\omega}X=T$ holds on $(0,L)$.
![Polar tangential angle.[]{data-label="fig:polartangentialangle"}](polartangentialangle.eps){width="25mm"}
The value of $\omega$ coincides with the angle between $X$ and $T$, i.e., $\arccos(X\cdot T)$, where $\cdot$ denotes the inner product, as long as $\omega\in[0,\pi]$; see Figure \[fig:polartangentialangle\]. The polar tangential angle function $\omega$ is smooth everywhere and unique up to addition by a constant in $2\pi\mathbb{Z}$. Unless a curve is generic, the function $\omega$ need to be discontinuous.
The polar tangential angle is a classical notion and often used in the literature. For example, the logarithmic spiral ($r=e^{a\theta}$ in the polar coordinate) is also known as the [*equiangular spiral*]{} since its polar tangential angle is constant. In this section we gain more insight into the behavior of the polar tangential angle.
A general derivative formula for the polar tangential angle
-----------------------------------------------------------
We first give a general formula of the derivative of the polar tangential angle function $\omega$. Let $\kappa$ denote the [*(signed) curvature*]{}, i.e., $\gamma_{ss}=\kappa N$. Recall that the [*evolute*]{} ${\varepsilon}$ of a plane curve $\gamma$ is defined as the locus of the centers of osculating circles, that is, for points where $\kappa\neq0$, $${\varepsilon}:=\gamma+\kappa^{-1}N.$$ Furthermore, in order to state our theorem in a unified way, we introduce $${\bar{\varepsilon}}:=\kappa\gamma+N.$$ This is same as $\kappa{\varepsilon}$ with the understanding that $\kappa{\varepsilon}=N$ when $\kappa=0$, and in particular defined everywhere as opposed to the original evolute. Obviously, the direction of ${\bar{\varepsilon}}$ is same as (resp. opposite to) that of ${\varepsilon}$ if $\kappa>0$ (resp. $\kappa<0$).
Here is the key identity for the polar tangential angle.
\[thmcharacterization\] The identity $\omega_s=|\gamma|^{-2}(\gamma\cdot{\bar{\varepsilon}})$ holds for any generic plane curve $\gamma$.
Since $N_s=-\kappa T$ and also $X_s=|\gamma|^{-1}[T-(X\cdot T)X]$, we have $$\begin{aligned}
(X\cdot N)_s= X_s\cdot N+X\cdot N_s =-(X\cdot T)[(\gamma\cdot{\bar{\varepsilon}})/|\gamma|^2].
\end{aligned}$$ Inserting $X\cdot T=\cos\omega$ and $X\cdot N=-\sin\omega$ to the above identity, we obtain $$\begin{aligned}
\label{eqn:mainthm1}
(-\cos\omega)\omega_s=(-\cos\omega)[(\gamma\cdot{\bar{\varepsilon}})/|\gamma|^2],
\end{aligned}$$ which ensures the assertion as long as $\cos\omega\neq0$. Similarly, we also have $$\begin{aligned}
(X\cdot T)_s =|\gamma|^{-1}(1-(X\cdot T)^2)+\kappa(X\cdot N) =(X\cdot N)[(\gamma\cdot{\bar{\varepsilon}})/|\gamma|^2],
\end{aligned}$$ where the identity $1-(X\cdot T)^2=(X\cdot N)^2$ is used, and hence $$\begin{aligned}
\label{eqn:mainthm2}
(-\sin\omega)\omega_s=(-\sin\omega)[(\gamma\cdot{\bar{\varepsilon}})/|\gamma|^2].
\end{aligned}$$ Combining (\[eqn:mainthm1\]) and (\[eqn:mainthm2\]), we complete the proof.
As Theorem \[thmcharacterization\] is not geometrically intuitive, we now characterize the first-derivative sign of the polar tangential angle from a geometric point of view. To this end we separately consider two cases, depending on whether the curvature vanishes.
We first think of the simpler case that the curvature vanishes. For a generic plane curve $\gamma$ we let $L(s)$ denote the half line from the origin and in the same direction as the vector $\gamma(s)$. Then we have the following geometric characterization, the proof of which is safely omitted; see Figure \[fig:sign1\].
\[cor:sign1\] Let $\gamma$ be a generic plane curve and assume that $\kappa(s)=0$ at some $s$. Then the signs of $\omega_s(s)$ and $\gamma(s)\cdot N(s)$ coincide. In other words, $\omega_s(s)$ is positive (resp. negative) if and only if the curve $\gamma$ transversally passes $L(s)$ from left to right (resp. right to left) at $s$. In addition, $\omega_s(s)=0$ if and only if $\gamma$ touches $L(s)$ at $s$.
![The sign of $\omega_s$ when $\kappa=0$.[]{data-label="fig:sign1"}](sign1.eps){width="60mm"}
We now turn to the more interesting case that the curvature does not vanish. Since $\omega$ is an oriented notion, it is suitable to consider the sign of $\kappa\omega_s$ rather than $\omega_s$. For a given generic curve, we let $C(s)$ denote the osculating circle at $s$, namely, the circle of radius $1/|\kappa(s)|$ centered at ${\varepsilon}(s)$ ($=\gamma(s)+(\kappa(s))^{-1}N(s)$) provided that $\kappa(s)\neq0$. In addition, let $\hat{c}(s)$ be the circle whose diameter is attained by the points $\gamma(s)$ and ${\varepsilon}(s)$. Then we have the following geometric characterization, the proof of which is again safely omitted; see Figure \[fig:sign2\].
\[cor:sign2\] Let $\gamma$ be a generic plane curve and assume that $\kappa(s)\neq0$ at some $s$. Then the signs of $\kappa(s)\omega_s(s)$ and $\gamma(s)\cdot{\varepsilon}(s)$ coincide. In other words, $\kappa(s)\omega_s(s)$ is positive (resp. negative) if and only if the origin is outside (resp. inside) the circle $\hat{c}(s)$. In addition, $\kappa(s)\omega_s(s)=0$ if and only if the origin lies on $\hat{c}(s)$. In particular, $\kappa\omega_s$ is positive as long as the osculating circle $C$ does not enclose the origin.
![The sign of $\omega_s$ when $\kappa>0$.[]{data-label="fig:sign2"}](sign2.eps){width="90mm"}
Sufficient conditions for monotonoicity
---------------------------------------
In this subsection we provide simple sufficient conditions for the monotonicity of the polar tangential angle.
The key assumption is monotonicity of the curvature, which controls the global behavior of osculating circles. Indeed, the classical [*Tait-Kneser theorem*]{} [@Tait1896; @Kneser1912] (see also [@Ghys2013]) states that if the curvature of a plane curve is strictly monotone, then the osculating circles are pairwise disjoint. In addition, if the curvature has no sign change, then subsequent circles are nested. Moreover, it is not difficult to observe that if the curvature and its derivative has same sign (resp. different sign), then the osculating circles become smaller (resp. larger) as the arclength parameter increases.
The monotonicity of the polar tangential angle is somewhat delicate, and in fact the monotonicity of curvature is still not sufficient. Here we impose an additional assumption on the initial state $s=0$. Let $D(s)$ denote the open disk enclosed by the osculating circle $C(s)$. For a generic plane curve $\gamma$ such that $(\kappa^2)_s=2\kappa\kappa_s>0$ for any $s>0$, we define the [*initial osculating disk*]{} $\widetilde{D}(0)$ as the open set given by $$\widetilde{D}(0):=\lim_{s\downarrow0}D(s),$$ where the limit is well defined since $0<s_1<s_2\Rightarrow D(s_1)\subset D(s_2)$ thanks to Tait-Kneser. Notice that if $\kappa(0)\neq0$, then $\widetilde{D}(0)$ is nothing but the open disk $D(0)$ enclosed by the osculating circle $C(0)$. If $\kappa(0)=0$, then $\widetilde{D}(0)$ is a limit half-plane: $$\widetilde{D}(0)=\gamma(0)+\{ p\in\mathbb{R}^2 \mid p\cdot \widetilde{N}(0)>0 \},\quad \mbox{where}\ \widetilde{N}(0):=\lim_{s\downarrow0}\gamma_{ss}(s)/|\gamma_{ss}(s)|.$$ Then we have the following
\[cor:monotone1\] Let $\gamma$ be a generic plane curve. If the derivative of $\kappa^2$ is positive at any $s>0$, and if the initial osculating disk $\widetilde{D}(0)$ does not include the origin $O$, then $\kappa(s)\omega_s(s)>0$ for any $s>0$. In particular, $\omega$ is strictly monotone.
\[cor:monotone2\] By the Tait-Kneser theorem and by the positivity of the derivative of $\kappa^2$, we have $\overline{D(s_1)}\subset D(s_0)$ for any $0<s_0<s_1$, where $\overline{D(s_0)}$ denotes the closure of $D(s_0)$. Since $D(s_0)\subset\widetilde{D}(0)$ holds by definition of $\widetilde{D}(0)$, the assumption $O\not\in\widetilde{D}(0)$ implies that $O\not\in\overline{D(s)}$ for any $s>0$. Therefore, Corollary \[cor:sign2\] implies that $\kappa\omega_s>0$ for any $s>0$.
In particular, the condition $O\not\in\widetilde{D}(0)$ is obviously satisfied if the above curve starts from the origin. We conclude this section by stating
\[cor:monotoneangle\] Let $\gamma:[0,L]\to\mathbb{R}^2$ be a plane curve such that $\gamma(0)=0$ and $\gamma(s)\neq0$ for $s>0$. If $\kappa\kappa_s$ is positive on $(0,L)$, then so is $\kappa\omega_s$. In particular, $\omega$ is strictly monotone.
Application to an obstacle problem for free elasticae {#sec:obstacle}
=====================================================
We apply our monotonicity result to the following higher order obstacle problem: $$\label{eq:min}
\inf_{u\in X_\psi} B[u], \qquad \mbox{where}\ B[u]:= \int_{\operatorname{graph}{u}}\kappa^2ds,$$ the admissible function space is given by $$X_\psi=\{u\in W^{2,2}(I)\mid u(\pm1)=0,\ u\geq\psi\}\quad \mbox{with}\ I=(-1,1),$$ and $\psi\in C(\bar{I})$ is a symmetric cone such that $\psi(\pm1)<0$. Here we call $\psi$ a symmetric cone if $\psi(x)=\psi(-x)$ and $\psi$ is affine on $(0,1)$. The functional $B[u]$ means the total squared curvature (also known as the bending energy) along the graph curve of $u$. For a graphical curve, $B$ can be expressed purely in terms of the height function $u$ via the formula: $B[u]=\int_Iu''^2(1+u'^2)^{-5/2}dx$.
Here we prove that Corollary \[cor:monotoneangle\] implies certain (non)existence results. Let $$\label{eq:height}
c_*:=\int_{0}^{\pi/2}\sqrt{\cos\varphi}d\varphi, \quad h_*:=\frac{2}{c_*} = 1.66925368...$$
\[thm:obstacle\] Let $\psi\in C(\bar{I})$ be a symmetric cone such that $\psi(\pm1)<0$ and $\psi(0)=:h>0$. If $h\geq h_*$, then there is no minimizer of the functional $B$ in $X_\psi$. If $h<h_*$, then there is a unique minimizer $\bar{u}$ of $B$ in the subspace $X_{\psi,\mathrm{sym}}$ of all even symmetric functions in $X_\psi$; namely, $B[\bar{u}]=\inf_{u\in X_{\psi,\mathrm{sym}}}B[u]$.
Our proof of Theorem \[thm:obstacle\] (or more precisely Lemma \[lem:1\] below) immediately implies that the curvature as well as the angle function of our unique symmetric solution for $h\in(0,h_*)$ can be explicitly parameterized in terms of elliptic functions. Note however that to this end we need to use some constants uniquely characterized by $h$, respectively, which solve somewhat complicated transcendental equations involving elliptic functions. In this paper we do not go into the details of completely explicit formulae.
If we allow non-graphical curves to be competitors, then there is no minimizer because an arbitrary large circular arc circumventing the obstacle is admissible so that the infimum is zero, but this infimum is not attained as a straight segment is not admissible due to the obstacle. Thus our graphical minimizers may be regarded as nontrivial critical points in the non-graphical problem.
\[rem:obstacle\] The existence of a minimizer in the symmetric class is already obtained by Dall’Acqua-Deckelnick [@DallAcqua2018] for a more general $\psi$, and hence the novel part is the uniqueness result. The nonexistence result is obtained by Müller [@Mueller2019] except for the critical value $h=h_*$. In addition, very recently, Yoshizawa [@Yoshizawa2019] independently obtained the same results as in Theorem \[thm:obstacle\] by a different approach, which is based on a shooting method and directly deals with a fourth order ODE for $u$. Our method is more geometric and mainly focuses on the curvature, thus being significantly different from the previous methods [@DallAcqua2018; @Mueller2019; @Yoshizawa2019]. We expect that our geometric aspect is also useful for analyzing critical points of other functionals, e.g., including the effect of length, or dealing with non-quadratic exponents.
Free elastica {#subsec:freeelastica}
-------------
For later use, as well as clarifying the reason why $h_*$ appears in Theorem \[thm:obstacle\], we recall some well-known facts about minimizers.
We first recall that if $u\in X_{\psi}$ is a minimizer in , then we have $$\begin{aligned}
\label{eq:BVP}
\begin{cases}
u\ \mbox{is concave},\\
2\kappa_{ss}+\kappa^3=0 \quad\mbox{on}\ \operatorname{graph}{u}\setminus\operatorname{graph}{\psi},\\
u(\pm1)=u''(\pm1)=0.
\end{cases}\end{aligned}$$ In fact, the concavity follows since otherwise taking the concave envelope decreases the energy, cf. [@DallAcqua2018]; the equation in the second line and the last boundary condition follow by standard calculation of the first variation. We remark that the equation is understood first in the sense of distribution, but then in the classical sense by using a standard bootstrapping argument, cf. [@Dayrens2018]. The second order boundary condition means the curvature of the graph vanishes at the endpoints. Notice that by concavity of $u$ we immediately deduce that the coincidence set $\operatorname{graph}{u}\cap\operatorname{graph}{\psi}$ is either empty or the apex of the cone; we will see later that it cannot be empty.
A solution to the equation in the second line of is called a [*free elastica*]{}, which is a specific example of Euler’s elastica. This equation possesses the fine scale invariance in the sense that if a curve $\gamma$ is a solution, then so is every curve similar to $\gamma$. A free elastica is essentially unique and described in terms of the Jacobi elliptic function.
Given $k\in[0,1]$, we let $\operatorname{cn}(x;k)$ denote the elliptic cosine function with elliptic modulus $k$, that is, $\operatorname{cn}(x;k):=\cos\phi$ for a unique value $\phi$ such that $x=\int_0^\phi g(\theta)d\theta$, where $g(\theta)=(1-k^2\sin^2\theta)^{-1/2}$. Recall that $\operatorname{cn}(x;k)$ is $4K(k)$-periodic and symmetric in the sense that $\operatorname{cn}(x;k)=\operatorname{cn}(-x;k)=-\operatorname{cn}(x+2K(k);k)$, and $\operatorname{cn}(x;k)$ strictly decreases from $1$ to $0$ as $x$ varies from $0$ to $K(k)$, where $K(k)$ denotes the complete elliptic integral of the first kind $\int_0^{\pi/2}g(\theta)d\theta$. In addition, the elliptic sine function is similarly defined by $\operatorname{sn}(x;k):=\sin\phi$ by using the above $\phi$, and also the delta amplitude by $\operatorname{dn}(x;k):=(1-k^2\sin^2\phi)^{1/2}$. For later use we recall the basic formulae: $\operatorname{sn}^2+\operatorname{cn}^2=1$, $\frac{d}{dx}\operatorname{sn}=\operatorname{cn}\operatorname{dn}$, and $\frac{d}{dx}\operatorname{cn}=-\operatorname{sn}\operatorname{dn}$.
It is known (cf. [@Linner1993 Proposition 2.3]) that any solution to $2\kappa_{ss}+\kappa^3=0$ is of the form $$\kappa(s)=\sqrt{2}\lambda\operatorname{cn}(\lambda s+\mu;\tfrac{1}{\sqrt{2}}), \quad \mbox{where}\ \lambda,\mu\in\mathbb{R}.$$ If $\lambda=0$, then the solution is a trivial straight segment, while if $\lambda\neq0$, then the solution curve is called a rectangular elastica. Since $\lambda$ is the scaling factor and $\mu$ is just shifting the variable, nontrivial solutions are essentially unique.
Up to similarity, a rectangular elastica is represented by a part of the graph curve of a periodic function $U:\mathbb{R}\to\mathbb{R}$ as in Figure \[fig:freeelastica\].
![The graph of $U$: A rectangular elastica.[]{data-label="fig:freeelastica"}](freeelastica.eps){width="60mm"}
More precisely, we can take $U$ so that
- $U(x)=U(-x)=-U(x+2)$ (and hence $U|_{[0,1]}$ determines the whole shape),
- $U$ is smooth in $(-1,1)$ while having vertical slope at $x=\pm1$,
- $U$ takes the minimum $-h_*$ at $x=0$,
- the curvature $\kappa$ of $\operatorname{graph}{U}$ is positive for $x\in[0,1)$, and vanishes for $x=1$,
- the arclength derivative $\kappa_s$ vanishes for $x=0$, and is negative for $x\in(0,1]$.
To verify this fact, letting $\lambda=1$ and $\mu=0$, we only need to investigate the behavior of a curve $\gamma=(x,y)$ such that $\kappa(s)=\sqrt{2}\operatorname{cn}(s;\tfrac{1}{\sqrt{2}})$ in the quarter period $[0,K(\tfrac{1}{\sqrt{2}})]$ (corresponding to the graph curve of $U|_{[0,1]}$ up to similarity). We first notice that, since the primitive function of $\operatorname{cn}(t;k)$ is $\frac{1}{k}\arcsin(k\operatorname{sn}(t;k))$, by normalizing $\theta(0)=0$ we can represent the angle function $\theta$ of $\gamma$ by $$\label{eq01}
\theta(s) = \int_0^s\kappa(s')ds' = 2\arcsin\left(\tfrac{1}{\sqrt{2}}\operatorname{sn}\left(s;\tfrac{1}{\sqrt{2}}\right)\right).$$ In particular, we have $$\label{eq02}
\theta\big(K(\tfrac{1}{\sqrt{2}})\big)-\theta(0)=\tfrac{\pi}{2}.$$ In addition, using , and noting that $0\leq\theta(s)\leq\pi/2$ in the quarter period, we obtain the following representations for $s\in[0,K(\tfrac{1}{\sqrt{2}})]$: $$\begin{aligned}
y(s)-y(0) &= \int_0^s \sin\theta(t)dt= \int_0^s 2\sin\tfrac{\theta(t)}{2}\sqrt{1-\sin^2\tfrac{\theta(t)}{2}}dt\\
&= \int_0^s \sqrt{2}\operatorname{sn}(t;\tfrac{1}{\sqrt{2}})\operatorname{dn}(t;\tfrac{1}{\sqrt{2}}) dt = \sqrt{2}\operatorname{cn}(s;\tfrac{1}{\sqrt{2}}),\\
x(s)-x(0) &= \int_0^s \cos\theta(t)dt = \int_0^s (1-2\sin^2\tfrac{\theta(t)}{2}) dt = \int_0^s (1-\operatorname{sn}^2(t;\tfrac{1}{\sqrt{2}})) dt\\
&= \int_0^s \tfrac{d}{dt}\Big(F\big(\operatorname{sn}(t;\tfrac{1}{\sqrt{2}})\big)\Big) dt = F(\operatorname{sn}(s;\tfrac{1}{\sqrt{2}})),\end{aligned}$$ where $F(r):=\int_0^r(1-\sigma^2)^{1/2}(1-\sigma^2/2)^{-1/2}d\sigma$. In particular, since the change of variables $\sigma=\sqrt{2}\sin(\varphi/2)$ implies that $\sqrt{2}F(1)=c_*$, cf. , we have $$\label{eq03}
y\big(K(\tfrac{1}{\sqrt{2}})\big)-y(0)=h_*\Big(x(K\big(\tfrac{1}{\sqrt{2}})\big)-x(0)\Big).$$ In view of and , renormalizing $\lambda=c_*/\sqrt{2}$, we can easily check that a curve $\gamma$ with $\kappa(s)=c_*\operatorname{cn}(\tfrac{c_*}{\sqrt{2}}s;\tfrac{1}{\sqrt{2}})$ defines the desired function $U$.
From the representation of a free elastica, we can now deduce that the coincidence set in is not empty, since otherwise the graph curve of $u$ would be fully a free elastica that satisfies the vanishing-curvature boundary condition, but this contradicts the fact that $u$ cannot have a vertical slope (as $W^{2,2}(I)\subset C^1(\bar{I})$). Therefore, with the help of concavity we have $$\label{eq:apex}
\operatorname{graph}{u}\cap\operatorname{graph}{\psi}=\{(0,\psi(0))\}.$$ This in particular means that a minimizer $u$ is smooth expect at the origin.
Boundary value problems {#subsec:BVP}
-----------------------
Keeping the facts in Section \[subsec:freeelastica\] in mind, we now turn to the proof of Theorem \[thm:obstacle\]. To this end, given a positive constant $h>0$, we consider the following boundary value problem for a smooth function $u$ on $[0,1]$: $$\begin{aligned}
\label{eq:BVPsym}
\begin{cases}
2\kappa_{ss}+\kappa^3=0,\\
u(0)=u''(0)=0,\ u(1)=h,\ u'(1)=0,
\end{cases}\end{aligned}$$ where the first equation is solved by the whole graph curve of $u$.
\[lem:1\] If $h\geq h_*$, then there is no solution to . If $h<h_*$, then there exists a unique solution to .
Let $U_*:[0,1]\to[0,h_*]$ be an increasing concave function such that $U_*(x)=U(x+1)$, where $U$ is defined in Section \[subsec:freeelastica\]. In view of the uniqueness property of free elasticae, a smooth function $u:[0,1]\to\mathbb{R}$ satisfies if and only if $$\begin{cases}
\exists\lambda>0,\ \exists\phi\in(-\pi/2,0)\ \mbox{such that}\ \operatorname{graph}{u}\subset \lambda R_\phi\operatorname{graph}{U_*},\\
u(1)=h,\ u'(1)=0,
\end{cases}$$ cf. Figure \[fig:BVP\], where $R_\phi$ is the counterclockwise rotation matrix through $\phi$. Therefore, for a given $h>0$, solving is now reduced to the following problem: $$\label{eq:reduced}
\mbox{Find $s\in(0,L_*)$ such that $\tan(-\omega_*(s))=h$},$$ where $L_*$ and $\omega_*$ denote the length and the polar tangential angle of $\operatorname{graph}{U_*}$, respectively. As we have also observed that the curvature $\kappa_*$ of $\operatorname{graph}{U_*}$ satisfies that $\kappa_*<0$ and $(\kappa_*)_s<0$ on $(0,L_*)$, we are able to use Corollary \[cor:monotoneangle\] and thus deduce that $\omega_*$ is strictly decreasing. Since $U_*(1)=h_*$, we in particular have $\omega_*((0,L_*))=(-\theta_*,0)$, where $\theta_*:=\arctan{h_*}$ ($=1.03106...\approx 59.06^\circ$). Therefore, if $h<h_*$, then there is a unique solution $s\in(0,L_*)$ to , while if $h\geq h_*$, no solution exists. Since the number of solutions to is characterized by that of , the proof is now complete.
![The correspondence between $U_*$ and $u$ in Lemma \[lem:1\][]{data-label="fig:BVP"}](BVP.eps){width="90mm"}
We are now in a position to complete the proof of Theorem \[thm:obstacle\].
We first address the case of $h<h_*$. The existence of a minimizer in the symmetric class is already known (cf. [@DallAcqua2018 Lemma 4.2]), so we only need to prove the uniqueness. If a function $u\in X_{\psi,\mathrm{sym}}$ solves , then by , symmetry, and $C^1$-regularity of $u$, we deduce that the restriction $u|_{[-1,0]}$ solves up to the shift $x\mapsto x+1$, and hence such a function $u$ must be unique in view of Lemma \[lem:1\].
We turn to the case of $h\geq h_*$. We prove by contradiction, so suppose that a solution $u$ would exist. Then, up to reflection, $u$ would take its maximum in $(-1,0]$. Letting $\bar{x}\in(-1,0]$ be a maximum point, and noting the scale invariance of free elasticae, we would deduce that the rescaled restriction $\tilde{u}(x):=\frac{1}{a}u|_{[-1,\bar{x}]}(a(x+1)-1)$, where $a=\bar{x}-(-1)$, solves up to the shift as above and replacing $h$ with $\tilde{h}:=\frac{1}{a}u(\bar{x})$. Then we would have $\tilde{h}\geq h_*$ since $a\leq1$ and $u(\bar{x})=\max{u}\geq\max{\psi}=h\geq h_*$, but this contradicts the nonexistence part of Lemma \[lem:1\]. Therefore, no solution exists in this case.
By the nature of the variational inequality corresponding to our obstacle problem, the global regularity of a minimizer $u$ in is improved so that $u\in C^2(\bar{I})$ and $u'''\in BV(I)$, cf. [@DallAcqua2018; @Dayrens2018]. The unique symmetric minimizer obtained here has this regularity, of course, but is not of class $C^3$ by the construction; this fact confirms optimality of the above regularity. Incidentally, we mention that our proof of uniqueness relies on just the trivial $C^1$-regularity at the apex of the cone, not invoking the higher regularity. However, we expect that the higher regularity would play an important role if we tackle the uniqueness problem in the general (nonsymmetric) class $X_{\psi}$ by our approach. We finally note that the $C^2$-regularity (or concavity etc.) is no longer true for an obstacle problem with an adhesion effect, cf. [@Miura2016; @Miura2017].
|
---
abstract: 'We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We investigate which of the co-Frobenius path subcoalgebras can be endowed with Hopf algebra structures, in order to produce some quantum groups with non-zero integrals, and we classify all these structures over a field with primitive roots of unity of any order. These turn out to be liftings of quantum lines over certain not necessarily abelian groups.'
address:
- |
${}^1$University of Bucharest, Facultatea de Matematica si Informatica\
Str. Academiei 14, Bucharest 1, RO-010014, Romania
- '${}^2$University of Southern California, 3620 S Vermont Ave, KAP 108, Los Angeles, CA 90089, USA;'
- 'e-mail: [email protected], [email protected],Constantin\[email protected]'
author:
- 'S.Dăscălescu${}^{1,*}$, M.C. Iovanov${}^{1,2}$, C. Năstăsescu${}^1$'
title: 'Path subcoalgebras, finiteness properties and quantum groups'
---
[^1]
Introduction and Preliminaries
==============================
Let $K$ be an arbitrary field. A quadratic algebra is a quotient of a free noncommutative algebra $K<x_1,\dots,x_n>$ in $n$ variables by an ideal $I$ generated by elements of degree $2$. The usual commutative polynomial ring is such an example, with $I$ generated by $x_ix_j-x_jx_i$. Quadratic algebras are important in many places in mathematics, and one relevant class of such objects consists of Koszul algebras and Koszul duals of quadratic algebras. More generally, one can consider quotients $K<x_1,\ldots ,x_n>/I$ for ideals $I$ generated by homogeneous elements. Several algebras occur in this way in topology, noncommutative geometry, representation theory, or theoretical physics (see the examples and references in [@berger2]). Such are the cubic Artin-Schreier regular algebras $\CC<x,y>/(ay^2x+byxy+axy^2+cx^3,ax^2y+bxyx+ayx^2+xy^3)$ in noncommutative projective algebraic geometry (see [@artins]), the skew-symmetrizer killing algebras $\CC<x_1,\dots,x_n>/(\sum\limits_{\sigma\in\Sigma_p}{\rm
sgn}(\sigma)x_{i_{\sigma(1)}}\dots x_{i_{\sigma(p)}})$ (the ideal we factor by has $n\choose p$ generators, each one corresponding to some fixed $1\leq i_1<\ldots <i_p\leq n$) for a fixed $2\leq p\leq
n$, in representation theory (see [@berger]), or the Yang-Mills algebras $\CC<\nabla_0,\dots,\nabla_n>/(\sum_{\lambda,\mu}g^{(\lambda,\mu)}[\nabla_\lambda[\nabla_\mu,\nabla_\nu]])$ (with $(g^{(\lambda,\mu)})_{\lambda,\mu}$ an invertible symmetric real matrix, and the ideal we factor by has $n+1$ generators, as $0\leq \nu\leq n$) in theoretical physics (see [@cdb]), to name a few. More generally, one could start with a quiver $\Gamma$, and define path algebras with relations by taking quotients of the path algebra $K[\Gamma]$ by an ideal (usually) generated by homogeneous elements, which are obtained as linear combinations of paths of the same length. Note that the examples above are of this type: the free algebra with $n$ elements can be thought as the path algebra of the quiver $\Gamma$ with one vertex $1$ (which becomes the unit in the algebra) and $n$ arrows $x_1,\dots,x_n$ starting and ending at $1$; the relations are then given by linear combinations of paths of the same length. This approach, for example, allows the generalization of N-Koszulity to quiver algebras with relations, see [@gmmz].
We aim to study a general situation which is dual to the ones above, but is also directly connected to it. If $\Gamma$ is a quiver, the path algebra $K[\Gamma]$ of $\Gamma$ plays an important role in the representation theory of $\Gamma$. The underlying vector space of the path algebra also has a coalgebra structure, which we denote by $K\Gamma$ and call the path coalgebra of $\Gamma$. One motivation for replacing path algebras by path coalgebras is the following: given an algebra $A$, and its category of finite dimensional representations, one is often lead to considering the category ${{Ind}}(A)$ generated by all these finite dimensional representations (direct limits of finite dimensional representations). ${{Ind}}(A)$ is well understood as the category of comodules over the finite dual coalgebra $A^0$ of $A$ (also called the algebra of representative functions on $A$), and it cannot be regarded as a full category of modules over a ring unless $A$ is finite dimensional. Such situations extend beyond the realm of pure algebra, encompassing representations of compact groups, affine algebraic groups or group schemes, differential affine groups, Lie algebras and Lie groups, infinite tensor categories etc.
Another reason for which the study of path coalgebras is interesting is that any pointed coalgebra embeds into the path coalgebra of the associated Gabriel quiver, see [@ni], [@cm]. On the other hand, if $X$ is a locally finite partially ordered set, the incidence coalgebra $KX$ provides a good framework for interpreting several combinatorial problems in terms of coalgebras, as explained by Joni and Rota in [@jr]. There are several features common to path coalgebras and incidence coalgebras. They are both pointed, the group-like elements recover the vertices of the quiver, respectively the points of the ordered set, the injective envelopes of the simple comodules have similar descriptions, etc. Moreover, as we show later in Section \[snew\], Proposition \[propembedding\], any incidence coalgebra embeds in a path coalgebra, and in many situations, it has a basis where each element is a sum of paths of the same length. We note that this is precisely the dual situation to that considered above: for algebras, one considers a path algebra with homogeneous relations, that is $K[\Gamma]$ quotient out by an ideal generated by homogeneous elements, i.e. sums of paths of the same length, with coefficients. For a coalgebra, one considers subcoalgebras of the path coalgebra of $\Gamma$ such that the coalgebra has a basis consisting of linear combinations of paths of the same length (“homogeneous” elements; more generally, a coalgebra generated by such elements).
In this paper we study Frobenius type properties for path coalgebras, incidence coalgebras and certain subcoalgebras of them. Recall that a coalgebra $C$ is called left co-Frobenius if $C$ embeds in $C^*$ as a left $C^*$-module. Also, $C$ is called left quasi-co-Frobenius if $C$ embeds in a free module as a left $C^*$-module. The (quasi)-co-Frobenius properties are interesting for at least three reasons. Firstly, coalgebras with such properties have rich representation theories. Secondly, for a Hopf algebra $H$, it is true that $H$ is left quasi-co-Frobenius if and only if $H$ is left co-Frobenius, and this is also equivalent to $H$ having non-zero left (or right) integrals. Co-Frobenius Hopf algebras are important since they generalize the algebra of representative functions $R(G)$ on a compact group $G$, which is a Hopf algebra whose integral is the left Haar integral of $G$. Moreover, more recent generalizations of these have been made to compact and locally compact quantum groups (whose representation categories are not necessarily semisimple). Thus co-Frobenius coalgebras may be the underlying coalgebras for interesting quantum groups with non-zero integrals. Thirdly, by keeping in mind the duality with Frobenius algebras in the finite dimensional case, co-Frobenius coalgebras have connections to topological quantum field theory.
We propose an approach leading to similar results for path coalgebras and incidence coalgebras, and which also points out the similarities between these as mentioned above. It will follow from our results that a path coalgebra (or an incidence coalgebra) is left (quasi)-co-Frobenius if and only if the quiver consists only of isolated points, i.e. the quiver does not have arrows (respectively the order relation is the equality). Thus the left co-Frobenius coalgebras arising from path coalgebras or incidence coalgebras are just grouplike coalgebras. In order to discover more interesting left co-Frobenius coalgebras, we focus our attention to classes of coalgebras larger than just path coalgebras and incidence coalgebras. On one hand we consider subcoalgebras of path coalgebras which have a linear basis consisting of paths. We call these [*path subcoalgebras*]{}. On the other hand, we look at subcoalgebras of incidence coalgebras; any such coalgebra has a basis consisting of segments. In Section \[sectionbilforms\] we apply a classical approach to the (quasi)-co-Frobenius property. It is known that a coalgebra $C$ is left co-Frobenius if and only if there exists a left non-degenerate $C^*$-balanced bilinear form on $C$. Also, $C$ is left quasi-co-Frobenius if and only if there exists a family $(\beta_i)_{i\in I}$ of $C^*$-balanced bilinear forms on $C$ such that for any non-zero $x\in C$ there is $i\in I$ with $\beta_i(x,C)\neq 0$. We describe the balanced bilinear forms on path subcoalgebras and subcoalgebras of incidence coalgebras. Such a description was given in [@DNV] for the full incidence coalgebra, and in [@br] for certain matrix-like coalgebras. In Section \[s3\] we use this description and an approach using the injective envelopes of the simple comodules to show that a coalgebra lying in one of the two classes is left quasi-co-Frobenius if and only if it is left co-Frobenius, and to give several equivalent conditions including combinatorial ones (just in terms of paths of the quiver, or segments of the ordered set).
In Section \[s4\] we classify all possible left co-Frobenius path subcoalgebras. We construct some classes of left co-Frobenius coalgebras $K[\AA_{\infty},r]$ and $K[\AA_{0,\infty},r]$ starting from the infinite line quiver $\AA_\infty$, and a class of left co-Frobenius coalgebras $K[\CC_n,s]$ starting from cyclic quiver $\CC_n$. Our result says that any left co-Frobenius path subcoalgebra is isomorphic to a direct sum of coalgebras of types $K[\AA_{\infty},r]$, $K[\AA_{0,\infty},r]$, $K[\CC_n,s]$ or $K$, with special quivers $\AA_{\infty}, \AA_{0,\infty},\CC_n$ and $r,s$ being certain general types of functions on these quivers. For subcoalgebras of incidence coalgebras we do not have a complete classification in the left co-Frobenius case. We show in Section \[snew\] that more complicated examples than the ones in the path subcoalgebra case can occur for subcoalgebras of incidence coalgebras, and a much larger class of such coalgebras is to be expected. Also, we give several examples of co-Frobenius subcoalgebras of path coalgebras, which are not path subcoalgebras, and moreover, examples of pointed co-Frobenius coalgebras which are not isomorphic to any one of the above mentioned classes. In Section \[s6\] we discuss the possibility of defining Hopf algebra structures on the path subcoalgebras that are left and right co-Frobenius, classified in Section \[s4\]. The main reason for asking this question is the interest in constructing quantum groups with non-zero integrals, whose underlying coalgebras are path subcoalgebras. We answer completely this question in the case where $K$ contains primitive roots of unity of any positive order. Thus we determine all possible co-Frobenius path subcoalgebras admitting a Hopf algebra structure. Moreover, we describe up to an isomorphism all such Hopf algebra structures. It turns out that they are liftings of quantum lines over certain not necessarily abelian groups. In particular, this also answers the question of finding the Hopf algebra structures on finite dimensional path subcoalgebras and on quotients of finite dimensional path algebras by ideals spanned by paths. Our results contain, as particular cases, some results of [@chyz], where finite quivers $\Gamma$ and finite dimensional path subcoalgebras $C$ of $K\Gamma$ are considered, such that $C$ contains all vertices and arrows of $\Gamma$. The co-Frobenius coalgebras of this type are determined, and all Hopf algebra structures on them are described in [@chyz]. These results follow from our more general Theorem \[th.qcf\] and Theorem \[teoremastructuriHopf\]. We note that Hopf algebra structures on incidence coalgebras have been of great interest for combinatorics, see for example [@sch], [@af]. We also note that the classification of path coalgebras that admit a graded Hopf algebra structure was done in [@cr], see also [@gs] for a different point of view on Hopf algebra structures on path algebras. In particular, some of the examples in the classification have deep connections with homological algebra: the monoidal category of chain $s$-complexes of vector spaces over $K$ is monoidal equivalent to the category of comodules of $K[\AA_\infty|s]$, a subclass of the Hopf algebras classified here ([@IG; @B]).
We also note that the unifying approach we propose here seems to suggest that in general for pointed coalgebras interesting methods and results could be obtained provided one can find some suitable bases with properties resembling those of paths in quiver algebras or segments in incidence coalgebras.
Throughout the paper $\Gamma=(\Gamma_0,\Gamma_1)$ will be a quiver. $\Gamma_0$ is the set of vertices, and $\Gamma_1$ is the set of arrows of $\Gamma$. If $a$ is an arrow from the vertex $u$ to the vertex $v$, we denote $s(a)=u$ and $t(a)=v$. A path in $\Gamma$ is a finite sequence of arrows $p=a_1a_2\ldots a_n$, where $n\geq 1$, such that $t(a_i)=s(a_{i+1})$ for any $1\leq i\leq n-1$. We will write $s(p)=s(a_1)$ and $t(p)=t(a_n)$. Also the length of such a $p$ is ${\rm length}(p)=n$. Vertices $v$ in $\Gamma_0$ are also considered as paths of length zero, and we write $s(v)=t(v)=v$. If $q$ and $p$ are two paths such that $t(q)=s(p)$, we consider the path $qp$ by taking the arrows of $q$ followed by the arrows of $p$. We denote by $K\Gamma$ the path coalgebra, which is the vector space with a basis consisting of all paths in $\Gamma$, and comultiplication $\Delta$ defined by $\Delta(p)=\sum
_{qr=p}q\otimes r$ for any path $p$, and counit $\epsilon$ defined by $\epsilon(v)=1$ for any vertex $v$, and $\epsilon(p)=0$ for any path of positive length. In particular, the arrows $x$ between two vertices $v$ and $w$, i.e. $s(x)=v, t(x)=w$, are the nontrivial elements of $P_{w,v}$, the space of $(w,v)$-skew-primitive elements: $\Delta(x)=v\otimes x + x\otimes w$. When we use Sweedler’s sigma notation $\Delta(p)=\sum p_1\otimes p_2$ for a path $p$, we always take representations of the sum such that all $p_1$’s and $p_2$’s are paths.\
We also consider partially ordered sets $(X,\leq)$ which are locally finite, i.e. the interval $[x,y]=\{z|\; x\leq z\leq y\}$ is finite for any $x\leq y$. The incidence $K$-coalgebra of $X$, denoted by $KX$, is the $K$-vector space with basis $\{
e_{x,y}|x,y\in X, x\leq y\}$, and comultiplication $\Delta$ and counit $\epsilon$ defined by $$\Delta(e_{x,y})=\sum_{x\leq z\leq y}e_{x,z}\otimes e_{z,y}$$ $$\epsilon (e_{x,y})=\delta_{x,y}$$ for any $x,y\in X$ with $x\leq y$, where by $\delta_{x,y}$ we denote Kronecker’s delta. The elements $e_{x,y}$ are called segments. Again, when we use Sweedler’s sigma notation $\Delta(p)=\sum
p_1\otimes p_2$ for a segment $p$, we always take representations of the sum such that all $p_1$’s and $p_2$’s are segments. Recall that the length of a segment $e_{x,y}$ is the maximum length $n$ of a chain $x=z_0<z_1<\dots<z_n=y$\
For basic terminology and notation about coalgebras and Hopf algebras we refer to [@DNR] and [@mo].
Balanced bilinear forms for path subcoalgebras and for subcoalgebras of incidence coalgebras {#sectionbilforms}
============================================================================================
In the rest of the paper we will be interested in two classes of coalgebras more general than path coalgebras and incidence coalgebras. Thus we will study
$\bullet$ Subcoalgebras of the path coalgebra $K\Gamma$ having a basis $\mathcal{B}$ consisting of paths in $\Gamma$. Such a coalgebra will be called a path subcoalgebra. Note that if $p\in
\mathcal{B}$, then any subpath of $p$, in particular any vertex involved in $p$, lies in $\mathcal{B}$.\
$\bullet$ Subcoalgebras of the incidence coalgebra $KX$. By [@DNV Proposition 1.1], any such subcoalgebra has a basis $\mathcal{B}$ consisting of segments $e_{x,y}$, and moreover, if $e_{x,y}\in \mathcal{B}$ and $x\leq a\leq b\leq y$, then $e_{a,b}\in
\mathcal{B}$.
It is clear that for a coalgebra $C$ of one of these two types, the distinguished basis $\mathcal{B}$ consists of all paths (or segments) which are elements of $C$. Let $C$ be a coalgebra of one of these two types, with basis $\mathcal{B}$ as above. When we use Sweedler’s sigma notation $\Delta (p)=\sum p_1\otimes p_2$ for $p\in
\mathcal{B}$, we always consider representations of the sum such that all $p_1$’s and $p_2$’s are in $\mathcal{B}$.
A bilinear form $\beta:C\times C\rightarrow K$ is $C^*$-balanced if $$\label{eq*}
\sum \beta(p_2,q)p_1=\sum \beta(p,q_1)q_2 \;\;\;\;\;\; {\rm for}\;
{\rm any}\; p,q\in \mathcal{B}$$ It is clear that (\[eq\*\]) is equivalent to the fact that for any $p,q\in \mathcal{B}$, the following three conditions hold.
$$\begin{aligned}
\beta(p_2,q)& = & \beta(p,q_1)\,\,\,\,\,{\rm for\,those\,of\,the\,}p_2{\rm's\,and\,the\,}q_1{\rm's\,such\,that\,}p_1=q_2\label{eq.d1}\\
\beta(p_2,q) & = & 0\,\,\,\,\,{\rm for\,those\,}p_2{\rm 's\,for\,which\,}p_1{\rm\,is\,not\,equal\,to\,any\,}q_2\label{eq.d2}\\
\beta(p,q_1) & = & 0\,\,\,\,\,{\rm for\,those\,}q_1{\rm
's\,for\,which\,}q_2{\rm
\,is\,not\,equal\,to\,any\,}p_1\label{eq.d3}\end{aligned}$$
In the following two subsections we discuss separately path subcoalgebras and subcoalgebras of incidence coalgebras.
Path subcoalgebras
------------------
In this subsection we consider the case where $C$ is a path subcoalgebra. We note that if $\Gamma$ is acyclic, then for any paths $p$ and $q$ there is at most a pair $(p_1,q_2)$ (in (\[eq\*\])) such that $p_1=q_2$.\
Denote by ${\mathcal{F}}$ the set of all paths $d$ satisfying the following three properties\
$\bullet$ $d=qp$ for some $q,p\in \mathcal{B}$.\
$\bullet$ For any representation $d=qp$ with $q,p\in \mathcal{B}$, and any arrow $a\in\Gamma_1$, if $ap\in \mathcal{B}$ then $q$ must end with $a$.\
$\bullet$ For any representation $d=qp$ with $q,p\in \mathcal{B}$, and any arrow $b\in\Gamma_1$, if $qb\in \mathcal{B}$ then $p$ starts with $b$.
Now we are able to describe all balanced bilinear forms on $C$.
\[formebilpath\] A bilinear form $\beta:C\times C\rightarrow K$ is $C^*$-balanced if and only if there is a family of scalars $(\alpha_d)_{d\in{\mathcal{F}}}$ such that for any $p,q\in \mathcal{B}$ $$\begin{aligned}
\beta(p,q) & = & \left\{
\begin{array}{l}
\alpha_d,\,\,\,{\rm if\,}s(p)=t(q){\rm\,and\,}qp=d\in{\mathcal{F}}\\
0,\,\,\,\;\;{\rm otherwise}
\end{array}\right.\end{aligned}$$ In particular the set of all $C^*$-balanced bilinear forms on $C$ is in bijective correspondence to $K^{\mathcal{F}}$.
Assume that $\beta$ is $C^*$-balanced. If $p,q\in \mathcal{B}$ and $t(q)\neq s(p)$, then $\beta(p,q)s(p)$ appears in the left-hand side of (\[eq\*\]), but $s(p)$ does not show up in the right-hand side, so $\beta(p,q)=0$. Let ${\mathcal{P}}$ be the set of all paths in $\Gamma$ for which there are $p,q\in \mathcal{B}$ such that $d=qp$. Let $d\in {\mathcal{P}}$ and let $d=qp=q'p'$, $p,q,p',q'\in \mathcal{B}$ be two different decompositions of $d$, and say that, for example, ${\rm
length}(p')<{\rm length}(p)$. Then there is a path $r$ such that $p=rp'$ and $q'=qr$, and clearly $r\in \mathcal{B}$ since it is a subpath of $q'\in B$. Use (\[eq.d1\]) for $p$ and $q'$, for which there is an equality $p_1=q'_2=r$ (and the corresponding $p_2=p'$ and $q'_1=q$), and find that $\beta(p',q')=\beta(p,q)$. Therefore, for any $d\in {\mathcal{P}}$ (not necessarily in $\mathcal{B}$) and any $p,q\in \mathcal{B}$ such that $d=qp$, the scalar $\beta(p,q)$ depends only on $d$. This shows that there is a family of scalars $(\alpha_d)_{d\in {\mathcal{P}}}$ such that $\beta(p,q)=\alpha_d$ for any $p,q\in \mathcal{B}$ with $qp=d$.\
Let $d\in{\mathcal{P}}$ such that $d=qp$ for some $p,q\in \mathcal{B}$, and there is an arrow $a\in\Gamma_1$ with $ap\in \mathcal{B}$, but $q$ does not end with $a$. That is, $q$ is not of the form $q=ra$ for some path $r\in \mathcal{B}$. We use (\[eq.d2\]) for the paths $ap\in \mathcal{B}$ and $q\in \mathcal{B}$, more precisely, for the term $(ap)_1=a$, which cannot be equal to any of the $q_2$’s (otherwise $q$ would end with $a$), and we see that $\beta(p,q)=0$, i.e. $\alpha_d=0$.\
Similarly, if $d\in{\mathcal{P}}$, $d=qp$ with $p,q\in \mathcal{B}$ and there is $b\in \Gamma_1$ with $qb\in B$ and $p$ not of the form $br$ for some path $r$ (i.e. $p$ does not start with $b$), then we use (\[eq.d3\]) for $p$ and $qb$, and $(qb)_2=b$, and we find that $\beta(p,q)=0$, i.e. $\alpha_d=0$. In conclusion, $\alpha_d$ may be non-zero only for $d\in{\mathcal{F}}$.\
Conversely, assume that $\beta$ is of the form indicated in the statement. We show that (\[eq.d1\]), (\[eq.d2\]) and (\[eq.d3\]) are satisfied. Let $p,q\in \mathcal{B}$ be such that $p_1=q_2=r$ for some $p_1$ and $q_2$ (from the comultiplication $\sum p_1\otimes p_2$ of $p$ and, respectively, the comultiplication $\sum q_1\otimes q_2$ of $q$). Then $p=rp'$ and $q=q'r$ for some $p',q'\in \mathcal{B}$. Let $d=q'rp'$. If $d\in{\mathcal{F}}$, then $\beta(p',q)=\beta(p,q')=\alpha_d$, while if $d\notin{\mathcal{F}}$ we have that $\beta(p',q)=\beta(p,q')=0$ by definition. Thus (\[eq.d1\]) holds. Now let $p,q\in \mathcal{B}$ and fix some $p_2$ (from the comultiplication $\sum p_1\otimes p_2$ of $p$) such that the corresponding $p_1$ is not equal to any $q_2$. If $s(p_2)\neq t(q)$, then clearly $\beta(p_2,q_1)=0$ by the definition of $\beta$. If $s(p_2)=t(q)$, then $d=qp_2\notin{\mathcal{F}}$. Indeed, let $r$ be a maximal path such that $p_1=er$ for some path $e$ and $q$ ends with $r$, say $q=q'r$. Note that $e$ has length at least $1$, since $p_1$ is not equal to any of the $q_2$’s. Then the terminal arrow of $e$ cannot be the terminal arrow of $q'$, and this shows that $d=p_2q=(p_2r)q'\notin{\mathcal{F}}$. Then $\beta(p_2,q)=0$ and (\[eq.d2\]) is satisfied. Similarly, (\[eq.d3\]) is satisfied.
Subcoalgebras of incidence coalgebras
-------------------------------------
In this subsection we assume that $C$ is a subcoalgebra of the incidence coalgebra $KX$. Let $\mathcal{D}$ be the set of all pairs $(x,y)$ of elements in $X$ such that $x\leq y$ and there exists $x'$ with $x\leq x'\leq y$ and $e_{x,x'},e_{x',y}\in \mathcal{B}$. Fix $(x,y)\in \mathcal{D}$. Let $$U_{x,y}=\{ u\;|\; x\leq u\leq y\;{\rm and}\; e_{x,u},e_{u,y}\in
\mathcal{B}\}$$ and define the relation $\sim$ on $U_{x,y}$ by $u\sim v$ if and only if there exist a positive integer $n$, and $u_0=u,u_1,\ldots,u_n=v$ and $z_1,\ldots,z_n$ in $U_{x,y}$, such that $z_i\leq u_{i-1}$ and $z_i\leq u_i$ for any $1\leq i\leq n$. It is easy to see that $\sim$ is an equivalence relation on $U_{x,y}$. Let $U_{x,y}/\sim$ be the associated set of equivalence classes, and denote by $(U_{x,y}/\sim)_0$ the set of all equivalence classes $\mathcal{C}$ satisfying the following two conditions.\
$\bullet$ If $u\in \mathcal{C}$, and $v\in X$ satisfies $v\leq u$ and $e_{v,y}\in \mathcal{B}$, then $x\leq v$.\
$\bullet$ If $u\in \mathcal{C}$, and $v\in X$ satisfies $u\leq v$ and $e_{x,v}\in \mathcal{B}$, then $v\leq y$.
Now we can describe the balanced bilinear forms on $C$.
\[formebilinc\] A bilinear form $\beta:C\times C\rightarrow K$ is $C^*$-balanced if and only if there is a family of scalars $(\alpha_{\mathcal{C}})_{\mathcal{C}\in \bigsqcup\limits_{(x,y)\in
\mathcal{D}}(U_{x,y}/\sim)_0}$ such that for any $e_{t,y},e_{x,z}\in
\mathcal{B}$ $$\begin{aligned}
\beta(e_{t,y},e_{x,z}) & = & \left\{
\begin{array}{l}
\alpha_{\mathcal{C}},\,\,\,{\rm if\,}(x,y)\in\mathcal{D}, z=t\in U_{x,y}{\rm\,and\,}{\rm the\,}{\rm class\,}\\
\hspace{.8cm}\mathcal{C}{\rm\, of\,}z{\rm\, in\,}U_{x,y}/\sim {\rm\, is\,}{\rm\, in\,}(U_{x,y}/\sim)_0\\
0,\,\,\,\;\;{\rm otherwise}
\end{array}\right.\end{aligned}$$ In particular the set of all $C^*$-balanced bilinear forms on $C$ is in bijective correspondence to $K^{\bigsqcup\limits_{(x,y)\in
\mathcal{D}}(U_{x,y}/\sim)_0}$.
Assume that $\beta$ is $C^*$-balanced. Fix some $x\leq y$ such that $U_{x,y}\neq \emptyset$. We first note that if $x\leq z\leq t\leq y$ and $z,t\in U_{x,y}$, then by applying (\[eq.d1\]) for $p=e_{z,y},
q=e_{x,t}$ and $p_1=q_2=e_{z,t}$, we find that $\beta(e_{t,y},e_{x,t})=\beta(e_{z,y},e_{x,z})$. Now let $u,v\in
U_{x,y}$ such that $u\sim v$. Let $u_0=u,u_1,\ldots,u_n=v$ and $z_1,\ldots,z_n$ in $U_{x,y}$, such that $z_i\leq u_{i-1}$ and $z_i\leq u_i$ for any $1\leq i\leq n$. By the above $\beta(e_{u_{i-1},y},e_{x,u_{i-1}})=\beta(e_{u_{i},y},e_{x,u_{i}})=\beta(e_{z_{i},y},e_{x,z_{i}})$ for any $i$, and this implies that $\beta(e_{u,y},e_{x,u})=\beta(e_{v,y},e_{x,v})$. This shows that $\beta(e_{u,y},e_{x,u})$ takes the same value for any $u$ in the same equivalence class in $U_{x,y}/\sim$.\
Now assume that for some $u\in U_{x,y}$ there is $v\in X$, such that $v\leq u$, $x\nleqslant v$ and $e_{v,y}\in \mathcal{B}$. Use (\[eq.d2\]) for $p=e_{v,y}$, $q=e_{x,u}$ and $p_1=e_{v,u}$. Note that $p_1\neq q_2$ for any $q_2$. We get that $\beta(e_{u,y},e_{x,u})=0$.\
Similarly, if $u\in U_{x,y}$, and there is $v\in X$ such that $u\leq
v$, $v\nleqslant y$ and $e_{x,v}\in \mathcal{B}$, then using (\[eq.d3\]) for $p=e_{u,y}, q=e_{x,v}$ and $q_2=e_{u,v}$, we find that $\beta(e_{u,y},e_{x,u})=0$. We have thus showed that $\beta$ has the desired form.\
Conversely, assume that $\beta$ has the indicated form. We show that it satisfies (\[eq.d1\]), (\[eq.d2\]) and (\[eq.d3\]). Let $p,q\in \mathcal{B}$ such that $p_1=q_2$ for some $p_1$ and $q_2$. Then $p=e_{z,y},q=e_{x,t}$ and $p_1=q_2=e_{z,t}$ for some $x\leq
z\leq t\leq y$. Clearly $t\sim z$, and let $\mathcal{C}$ be the equivalence class of $t$ in $U_{x,y}/\sim$. Then $\beta(p_2,q)=\beta(e_{t,y},e_{x,t})$ and $\beta(p,q_1)=\beta(e_{z,y},e_{x,z})$, and they are both equal to $\alpha_{\mathcal{C}}$ if $\mathcal{C}\in (U_{x,y}/\sim)_0$, and to 0 if $\mathcal{C}\notin (U_{x,y}/\sim)_0$. Thus (\[eq.d1\]) is satisfied.\
Let now $p=e_{z,y},p_1=e_{z,t},p_2=e_{t,y}$ and $q=e_{x,u}$ such that $p_1\neq q_2$ for any $q_2$. Then $\beta(p_2,q)=\beta(e_{t,y},e_{x,u})$. If $u\neq t$, this is clearly 0. Let $u=t$. Then $x\nleqslant z$, otherwise $p_1=q_2$ for some $q_2$. We have that $t\in U_{x,y}$, but the equivalence class of $t$ in $U_{x,y}/\sim$ is not in $(U_{x,y}/\sim)_0$, since $e_{z,y}\in
\mathcal{B}$, $z\leq t$, but $x\nleqslant z$. It follows that $\beta(p_2,q)=0$, and (\[eq.d2\]) holds. Similarly we can show that (\[eq.d3\]) holds.
Left quasi-co-Frobenius path subcoalgebras and subcoalgebras of incidence coalgebras {#s3}
====================================================================================
In this section we investigate when a path subcoalgebra of a path coalgebra or a subcoalgebra of an incidence coalgebra is left co-Frobenius. We keep the notation of Section \[sectionbilforms\]. Thus $C$ will be either a path subcoalgebra of a path coalgebra $K\Gamma$, or a subcoalgebra of an incidence coalgebra $KX$. The distinguished basis of $C$ consisting of paths or segments will be denoted by $\mathcal{B}$. We note that in each of the two cases $\mathcal{B}\cap C_n$ is a basis of $C_n$, where $C_0\subseteq
C_1\subseteq \ldots $ is the coradical filtration of $C$. The injective envelopes of the simple left (right) comodules were described in [@simson Lemma 5.1] for incidence coalgebras and in [@chin Corollary 6.3] for path coalgebras. It is easy to see that these descriptions extend to the following.
\[inj\] (i) If $C$ is a path subcoalgebra, then for each vertex $v$ of $\Gamma$ such that $v\in C$, the injective envelope of the left (right) $C$-comodule $Kv$ is (the $K$-span) $E_l(Kv)=<p\in \mathcal{B}|t(p)=v>$ (and $E_r(Kv)=<p\in \mathcal{B}|s(p)=v>$ respectively).\
(ii) If $C$ is a subcoalgebra of the incidence coalgebra $KX$, then for any $a\in X$ such that $e_{a,a}\in C$, the injective envelope of the left (right) $C$-comodule $Ke_{a,a}$ is (the $K$-span) $E_l(Ke_{a,a})=<e_{x,a}|x\in X, e_{x,a}\in C>$ (and $E_r(Ke_{a,a})=<e_{a,x}|x\in X, e_{a,x}\in C>$).
The following shows that we have a good left-right duality for comodules generated by elements of the basis $\mathcal{B}$.
\[l.duals\] (i) Let $C$ be a subcoalgebra of the incidence coalgebra $KX$, and let $e_{a,b}\in C$. Then $(C^*e_{a,b})^*\cong e_{a,b} C^*$ as right $C^*$-modules (or left $C$-comodules).\
(ii) Let $C$ be a path subcoalgebra of $K\Gamma$, and let $p$ be a path in $C$. Then $(C^* p)^*\cong p C^*$ as right $C^*$-modules (or left $C$-comodules).
\(i) Clearly the set of all segments $e_{a,x}$ with $a\leq x\leq b$ is a basis of $C^*e_{a,b}$. Denote by $e_{a,x}^*$ the corresponding elements of the dual basis of $(C^*e_{a,b})^*$. Since for $c^*\in
C^*$ and $a\leq x,y\leq b$ we have $$\begin{aligned}
(e_{a,x}^*c^*)(e_{a,y})&=&\sum_{a\leq z\leq y}c^*(e_{z,y})e_{a,x}^*(e_{a,z})\\
&=&\left\{
\begin{array}{l}
0,\mbox{ if } x\nleqslant y\\ c^*(e_{x,y}), \mbox{ if } x\leq y
\end{array}\right.\end{aligned}$$ we get that $$\label{eq1dual}
e_{a,x}^*c^*=\sum_{x\leq y\leq b}c^*(e_{x,y})e_{a,y}^*$$ On the other hand $e_{a,b}C^*$ has a basis consisting of all segments $e_{x,b}$ with $a\leq x\leq b$, and $$\label{eq2dual}
e_{x,b}c^*=\sum_{x\leq y\leq b}c^*(e_{x,y})e_{y,b}$$ Equations (\[eq1dual\]) and (\[eq2dual\]) show that the linear map $\phi:(C^*e_{a,b})^*{\rightarrow}e_{a,b}C^*$ defined by $\phi
(e_{a,x}^*)=e_{x,b}$, is an isomorphism of right $C^*$-modules.\
(ii) Let $p=a_1\ldots a_n$ and $v=s(p)$. Denote $p_i=a_1\ldots a_i$ for any $1\leq i\leq n$, and $p_0=v$. Then $\{ p_0,p_1,\ldots
,p_n\}$ is a basis of $C^*p$, and let $(p_i^*)_{0\leq i\leq n}$ be the dual basis of $(C^*p)^*$. For any $0\leq t\leq j\leq n$ denote by $\overline{p_{t,j}}$ the path such that $p_j=p_t\overline{p_{t,j}}$. Then a simple computation shows that $p_i^*c^*=\sum_{i\leq j\leq n}c^*(\overline{p_{i,j}})p_j^*$ for any $i$ and any $c^*\in C^*$.\
On the other hand, $\{ \overline{p_{i,n}}\;|\; 0\leq i\leq n\}$ is a basis of $pC^*$, and it is easy to see that $\overline{p_{i,n}}c^*=\sum_{i\leq r\leq
n}c^*(\overline{p_{i,r}})\overline{p_{r,n}}$ for any $i$ and any $c^*\in C^*$. Then the linear map $\phi:(C^*p)^*{\rightarrow}pC^*$ defined by $\phi (p_i^*)=\overline{p_{i,n}}$ for any $0\leq i\leq n$, is an isomorphism of right $C^*$-modules.
For a path subcoalgebra $C$ let us denote by $R(C)$ the set of vertices $v$ in $C$ such that the set $\{p\in C\; |\; p \mbox{ path
and }s(p)=v\}$ is finite (i.e. $E_r(Kv)$ is finite dimensional) and contains a unique maximal path. Note that $v\in R(C)$ if and only if $E_r(Kv)$ is finite dimensional and local. Indeed, if $E_r(Kv)$ is finite dimensional and contains a unique maximal path $p=a_1\ldots a_n$, then keeping the notation from the proof of Lemma \[l.duals\], we have that $E_r(Kv)=C^*p$ and $C^*p_{n-1}=<p_0,\ldots ,p_{n-1}>$ is the unique maximal $C^*$-submodule of $C^*p$. Conversely, if $E_r(Kv)$ is finite dimensional and local with the unique maximal subcomodule $N$, then the set $(\mathcal{B}\cap E_r(Kv))/N$ is nonempty. If $p$ is a path which belongs to this set, $E_r(Kv)=C^*p$. Then clearly $p$ is the unique maximal path in $\{q\in C\; |\; q \mbox{ path and
}s(q)=v\}$.\
Similarly, denote by $L(C)$ the set of vertices $v$ of $C$ such that $E_l(Kv)$ is a finite dimensional local left $C$-comodule. Also, for each vertex $v\in R(C)$ let $r(v)$ denote the endpoint of the maximal path starting at $v$, and for $v\in L(C)$ let $l(v)$ be the starting point of the maximal path ending at $v$.\
Similarly, for a subcoalgebra $C$ of the incidence coalgebra $KX$, let $R(C)$ be the set of all $a\in X$ for which $e_{a,a}\in C$ and the set $\{x\in X\;|\; a\leq x,\,e_{a,x}\in C\}$ is finite and has a unique maximal element, and $L(C)$ be the set of all $a\in X$ for which $e_{a,a}\in C$ and the set $\{x\in X|x\leq a,\,e_{x,a}\in C\}$ is finite and has a unique minimal element. As before, $R(C)$ (respectively $L(C)$) consists of those $a\in X$ for which $E_r(Ke_{a,a})$ (respectively, $E_l(Ke_{a,a})$) are local, hence generated by a segment. Here $r(a)=r(e_{a,a})$ for $a\in R(C)$ denotes the maximum element in the set $\{x\;|\; x\geq
a,\,e_{a,x}\in C\}$ and $l(a)$ for $a\in L(C)$ means the minimum of $\{x\;|\; x\leq a, e_{x,a}\in C\}$.
\[p.qcf\] (I) Let $C$ be a path subcoalgebra of the path coalgebra $K\Gamma$. Then the following are equivalent.\
(a) $C$ is left co-Frobenius.\
(b) $C$ is left quasi-co-Frobenius.\
(c) $R(C)$ consists of all vertices belonging to $C$, $r(R(C))\subseteq L(C)$ and $lr(v)=v$, for any vertex $v$ in $C$.\
(d) For any path $q\in \mathcal{B}$ there exists a path $p\in
\mathcal{B}$ such that $qp\in \mathcal{F}$ (for $\mathcal{F}$ defined in the previous section).\
(II) Let $C$ be a subcoalgebra of the incidence coalgebra $KX$. Then the following are equivalent.\
(a) $C$ is left co-Frobenius.\
(b) $C$ is left quasi-co-Frobenius.\
(c) $R(C)$ consists of all $a\in X$ such that $e_{a,a}\in C$, $r(R(C))\subseteq L(C)$ and $lr(a)=a$, $\forall\,a\in X$ with $e_{a,a}\in C$.\
(d) For any segment $e_{x,z}\in C$ there exists $y\geq z$ such that $e_{z,y}\in C$ and the class of $z$ in $U_{x,y}/\sim$ lies in $(U_{x,y}/\sim)_0$.
$(I)$ (a)$\Rightarrow $(b) is clear.\
(b)$\Rightarrow$(c) We apply the QcF characterization of [@II] and [@IG]. If $C$ is left QcF then for any vertex $v\in C$, there is a vertex $u\in C$ such that $E_r(Kv)\cong E_l(Ku)^*$. Hence $E_r(Kv)$ is finite dimensional and local (by [@IF Lemma 1.4]), so $v\in R(C)$ and $E_r(Kv)=C^*p$ for a path $p$ by the discussion preceding this Proposition. Let $t(p)=w$. Then it is easy to see that the linear map $\phi:C^*p{\rightarrow}Kw$ taking $p$ to $w$, and any other $q$ to 0, is a surjective morphism of left $C^*$-modules. Since $E_r(Kv)\cong E_l(Ku)^*$, there is a surjective morphism of left $C^*$-modules $E_l(Ku)^*{\rightarrow}Kw$, inducing an injective morphism of right $C^*$-modules $(Kw)^*{\rightarrow}E_l(Ku)$. Since $(Kw)^*\cong Kw$ as right $C^*$-modules, and the socle of the comodule $E_l(Ku)$ is $Ku$, we must have $w=u$, and thus $u=r(v)$. By Lemma \[l.duals\], $E_l(Ku)\cong E_r(Kv)^*=(C^* p)^*\cong p C^*$, so $E_l(Ku)$ is generated by $p$, and this shows that $p$ is the unique maximal path ending at $u$. Hence, $u=r(v)\in L(X)$, and $l(u)=v$. Thus $l(r(v))=v$.\
(c)$\Rightarrow$ (d) Let $q\in \mathcal{B}$, and let $v=s(q)$. Since $v\in R(C)$, there exists a unique maximal path $d$ starting at $v$, and $d=qp$ for some path $p$. We show that $d\in \mathcal{F}$. Denote $t(d)=v'$, and let $d=q'p'$ for some paths $q',p'$ in $\mathcal{B}$. Let $u=t(q')=s(p')$. If there is an arrow $b$ (in $\Gamma_1$) starting at $u$, such that $q'b\in \mathcal{B}$, then $q'b$ is a subpath of $d$, since $d$ is the unique maximal path starting at $v$. It follows that $p'$ starts with $b$. On the other hand, $v'=r(v)\in L(C)$ and $l(v')=lr(v)=v$, so $d$ is the unique maximal path in $\mathcal{B}$ ending at $v'$. This shows that if an arrow $a$ (in $\Gamma_1$) ends at $u$, and $ap'\in \mathcal{B}$, then $ap'$ is a subpath of $d$, so the last arrow of $q'$ is $a$. We conclude that $d\in \mathcal{F}$.\
(d)$\Rightarrow$(a) Choose a family $(\alpha_d)_{d\in \mathcal{F}}$ of scalars, such that $\alpha_d\neq 0$ for any $d$. Associate a $C^*$-balanced bilinear form $B$ on $C$ to this family of scalars as in Theorem \[formebilpath\]. Then $B$ is right non-degenerate, so $C$ is left co-Frobenius.
$(II)$ (a)$\Rightarrow$(b) is clear; (b)$\Rightarrow$(c) is proved as the similar implication in $(I)$, with paths replaced by segments.\
(c)$\Rightarrow$(d) Let $e_{x,z}\in C$. If $r(x)=y$, then clearly $z\leq y$ and $e_{x,y}\in \mathcal{B}$, so $U_{x,y}=[x,y]$. Then any two elements in $U_{x,y}$ are equivalent with respect to $\sim$ (since they are both $\geq x$), so there is precisely one equivalence class in $U_{x,y}/\sim$, the whole of $U_{x,y}$. We show that this class lies in $(U_{x,y}/\sim)_0$. Indeed, if $u\in
U_{x,y}$, $v\in X$, $v\leq u$ and $e_{v,y}\in \mathcal{B}$, then $v\in \{ a|e_{a,y}\in \mathcal{B}\}$, and since $l(y)=l(r(x))=x$, we must have $x\leq v$. Also, if $u\in U_{x,y}$, $v\in X$, $u\leq v$ and $e_{x,v}\in \mathcal{B}$, then $v\in \{a|e_{x,a}\in
\mathcal{B}\}$. Then $v\leq y$ since $r(x)=y$.\
(d)$\Rightarrow$(a) follows as the similar implication in $(I)$ if we take into account Theorem \[formebilinc\].
As a consequence we obtain the following result, which was proved for incidence coalgebras in [@DNV].
\[proppathcoFrob\] If $C=K\Gamma$, a path coalgebra, or $C=KX$, an incidence coalgebra, the following are equivalent\
(i) $C$ is co-semisimple (i.e. $\Gamma$ has no arrows for $C=K\Gamma$, and the order relation on $X$ is the equality for $C=KX$).\
(ii) $C$ is left QcF.\
(iii) $C$ is left co-Frobenius.\
(iv) $C$ is right QcF.\
(v) $C$ is right co-Frobenius.
As an immediate consequence we describe the situations where a finite dimensional path algebra is Frobenius. We note that the path algebra of a quiver $\Gamma$ (as well as the path coalgebra $K\Gamma$) has finite dimension if and only if $\Gamma$ has finitely many vertices and arrows, and there are no cycles.
A finite dimensional path algebra is Frobenius if and only if the quiver has no arrows.
It follows from the fact that the dual of a finite dimensional path algebra is a path coalgebra, and by Corollary \[proppathcoFrob\].
Classification of left co-Frobenius path subcoalgebras {#s4}
======================================================
Proposition \[p.qcf\] gives information about the structure of left co-Frobenius path subcoalgebras. The aim of this section is to classify these coalgebras. We first use Proposition \[p.qcf\] to give some examples of left co-Frobenius path subcoalgebras. These examples will be the building blocks for the classification.
\[gn\] Let $\Gamma=\AA_{\infty}$ be the quiver such that $\Gamma_0=\ZZ$ and there is precisely one arrow from $i$ to $i+1$ for any $i\in \ZZ$.
$$\AA_{\infty}: \;\;\;\; \dots\longrightarrow \circ^{-1}\longrightarrow
\circ^{0}\longrightarrow \circ^{1}\longrightarrow
\circ^2\longrightarrow\dots$$
For any $k<l$, let $p_{k,l}$ be the (unique) path from the vertex $k$ to the vertex $l$. Also denote by $p_{k,k}$ the vertex $k$. Let $r:\ZZ {\rightarrow}\ZZ$ be a strictly increasing function such that $r(n)>n$ for any $n\in \ZZ$. We consider the path subcoalgebra $K[\AA_{\infty}, r]$ of $K\AA_{\infty}$ with the basis $$\begin{aligned}
\mathcal{B}&=&\bigcup_{n\in \ZZ}\{ p\; |\; p \mbox{ is a path in
}\AA_{\infty}, s(p)=n \;{\rm and}\;{\rm length}(p)\leq r(n)-n\}\\
&=&\{ p_{k,l}\;|\; k,l\in \ZZ \;{\rm and}\; k\leq l\leq r(k)\}\end{aligned}$$ Note that $K[\AA_{\infty}, r]$ is indeed a subcoalgebra, since $$\Delta(p_{k,l})=\sum\limits_{i=k}^lp_{k,i}\otimes p_{i,l}, \,\,\,k\leq l$$ The counit is given by $$\varepsilon(p_{k,l})=\delta_{k,l}$$ Note that this can also be seen as a subcoalgebra of the incidence coalgebra of $(\NN,\leq)$, consisting of the segments $e_{k,l}$ for $k\leq l\leq r(k)$.\
The construction immediately shows that the maximal path starting from $n$ is $p_{n,r(n)}$. Note that for each $n\in \ZZ$, $p_{n,r(n)}$ is the unique maximal path into $r(n)$. If there would be another longer path $p_{l,r(n)}$ into $r(n)$ in $K[\AA_{\infty},r]$, then $l<n$. Then, since $p_{l,r(n)}$ is among the paths in $K[\AA_{\infty},r]$ which start at $l$ we must have that it is a subpath of $p_{l,r(l)}$, and so $r(l)\geq r(n)$. But since $l<n$, this contradicts the assumption that $r$ is strictly increasing. Therefore, we see that the conditions of Proposition \[p.qcf\] are satisfied: $p_{n,r(n)}$ is the unique maximal path in the (finite) set of all paths starting from a vertex $n$, and it is simultaneously the unique maximal path in the (finite) set of all paths ending at $r(n)$. Therefore if $l:L(C)={\rm Im}(r)\rightarrow
R(C)$ is the function used in Proposition \[p.qcf\] for $C=K[\AA_{\infty},r]$ satisfies $l(r(n))=n$. This means that $K[\AA_{\infty},r]$ is a left co-Frobenius coalgebra.\
$K[\AA_{\infty},r]$ is also right co-Frobenius if and only if there is a positive integer $s$ such that $r(n)=n+s$ for any $n\in \ZZ$. Indeed, if $r$ is of such a form, then $K[\AA_{\infty},r]$ is right co-Frobenius by the right-hand version of Proposition \[p.qcf\]. Conversely, assume that $K[\AA_{\infty},r]$ is right co-Frobenius. If $r$ would not be surjective, let $m\in \ZZ$ which is not in the image of $r$. Then there is $n\in \ZZ$ such that $r(n)<m<r(n+1)$. The maximal path ending at $m$ is $p_{n+1,m}$. Indeed, this maximal path cannot start before $n$ (since then $p_{n,r(n)}$ would be a subpath of $p_{n,m}$ different from $p_{n,m}$), and $p_{n+1,m}$ is a path in $K[\AA_{\infty},r]$, as a subpath of $p_{n+1,r(n+1)}$. Hence $r(l(m))=r(n+1)\neq m$, and then $K[\AA_{\infty},r]$ could not be right co-Frobenius by the right-hand version of Proposition \[p.qcf\], a contradiction. Thus $r$ must be surjective, and then it must be of the form $r(n)=n+s$ for any $n$, where $s$ is an integer. Since $n<r(n)$ for any $n$, we must have $s>0$. For simplicity we will denote $K[\AA_{\infty},r]$ by $K[\AA_{\infty}|s]$ in the case where $r(n)=n+s$ for any $n\in \ZZ$.
\[gn0\] Let $\Gamma=\AA_{0,\infty}$ be the subquiver of $\AA_{\infty}$ obtained by deleting all the negative vertices and the arrows involving them. Thus $\Gamma_0=\NN$, the natural numbers (including 0). $$\AA_{0,\infty}:\;\;\; \circ^{0}\longrightarrow
\circ^{1}\longrightarrow
\circ^2\longrightarrow\circ^3\longrightarrow \dots$$ We keep the same notation for $p_{k,l}$ for $0\leq k\leq l$. Let $r:\NN{\rightarrow}\NN$ be a strictly increasing function with $r(0)>0$ (so then $r(n)>n$ for any $n\in \NN$), and define $K[\AA_{0,\infty},r]$ to be the path subcoalgebra of $K\AA_{0,\infty}$ with basis $\{ p_{k,l}\;|\; k,l\in
\NN, k\leq l\leq r(k)\}$. With the same arguments as in Example \[gn\] we see that $K[\AA_{0,\infty},r]$ is a left co-Frobenius coalgebra. We note that $l(0)=0$, and then $r(l(0))=r(0)>0$. By a right-hand version of Proposition \[p.qcf\], this shows that $K[\AA_{0,\infty},r]$ is never right co-Frobenius.
\[lps\] For any $n\geq 2$ we consider the quiver $\CC_n$, whose vertices are the elements of $\ZZ_n=\{ \overline{0},\ldots,\overline{n-1}\}$, the integers modulo $n$, and there is one arrow from $\overline{i}$ to $\overline{i+1}$ for each $i$.
$\xymatrix{& &\circ^{\overline{1}}\ar[r] & \circ^{\overline{2}}\ar[r] & \circ^{\overline{3}}\ar[dr] & \\
\CC_n: & \circ^{\overline{0}}\ar[ur] & & & & \circ\ar[dl]\\
& & \dots\ar[ul]& \dots & \circ\ar[l] &}$\
We also denote by $\CC_1$ the quiver with one vertex, denoted by $\overline{0}$, and one arrow $\xymatrix{\circ\ar@(ul,ur)[]}$, and by $\CC_0$ the quiver with one vertex and no arrows.\
Let $n\geq 1$ and $s>0$ be integers. Let $K[\CC_n,s]$ be the path subcoalgebra of the path coalgebra $K\CC_n$, spanned by all paths of length at most $s$. Denote by $q_{\overline{k}|l}$ the path (in $\CC_n$) of length $l$ starting at $\overline{k}$, for any $\overline{k}\in \ZZ_n$ and $0< l\leq s$. Also denote by $q_{\overline{k}|0}$ the vertex $\overline{k}$. Since the comultiplication and counit of $K\CC_n$ are given by $$\Delta(q_{\overline{k}|l})=\sum\limits_{i=0}^lq_{\overline{k}|i}\otimes q_{\overline{k+i}|l-i},$$ $$\varepsilon(q_{\overline{k}|l})=\delta_{0,l}$$ we see that indeed $K[\CC_n,s]=<q_{\overline{k}|l}\;|\;
\overline{k}\in \ZZ, 0\leq l\leq s>$ is a subcoalgebra of $K\CC_n$. Clearly $q_{\overline{k}|s}$ is the unique maximal path in $K[\CC_n,s]$ starting at $\overline{k}$, so $\overline{k}\in
R(K[\CC_n,s])$ and $r(\overline{k})=\overline{k+s}$. Also $\overline{k+s}\in L(K[\CC_n,s])$ and the maximal path ending at $\overline{k+s}$ is also $q_{\overline{k}|s}$, thus $lr(\overline{k})=\overline{k}$, and by Proposition \[p.qcf\] we get that $K[\CC_n,s]$ is a left co-Frobenius coalgebra. Since it has finite dimension $n(s+1)$, it is right co-Frobenius, too. This example was also considered in [@chyz 1.6].
For a path subcoalgebra $C\subseteq K\Gamma$, denote by $C\cap
\Gamma$ the subgraph of $\Gamma$ consisting of arrows and vertices of $\Gamma$ belonging to $C$.
\[l.graph\] If $C\subseteq K\Gamma$ is a left co-Frobenius path subcoalgebra, then $C\cap\Gamma=\bigsqcup\limits_i\Gamma_i$, a disjoint union of subquivers of $\Gamma$, where each $\Gamma_i$ is of one of types $\AA_\infty$, $\AA_{0,\infty}$ or $\CC_n$, $n\geq 0$, and $C=\bigoplus\limits_{i}C_i$, where $C_i$, a path subcoalgebra of $K\Gamma_i$, is the subcoalgebra of $C$ spanned by the paths of $\mathcal{B}$ contained in $\Gamma_i$.
Let $v$ be a vertex in $C\cap \Gamma$. By Proposition \[p.qcf\] there is a unique maximal path $p\in \mathcal{B}$ starting at $v$, and any path in $\mathcal{B}$ starting at $v$ is a subpath of $p$. This shows that at most one arrow in $\mathcal{B}$ starts at $v$ (the first arrow of $p$, if $p$ has length $>0$). We show that at most one arrow in $\mathcal{B}$ ends at $v$, too. Otherwise, if we assume that two different arrows $a$ and $a'$ in $\mathcal{B}$ end at $v$, let $s(a)=u$ and $s(a')=u'$ (clearly $u\neq u'$, since at most one arrow starts at $u$), and let $q$ and $q'$ be the maximal paths in $\mathcal{B}$ starting at $u$ and $u'$, respectively. Then $q=az$ and $q'=a'z'$ for some paths $z$ and $z'$ starting at $v$. But then $z$ and $z'$ are subpaths of $p$, so one of them, say $z$, is a subpath of the other one. If $w=t(z)$, then $w=r(u)$, so $w\in
L(C)$ and any path in $\mathcal{B}$ ending at $w$ is a subpath of $q=az$. This provides a contradiction, since $a'z$ is in $\mathcal{B}$ (as a subpath of $q'$) and ends at $w$, but it is not a subpath of $q$.\
We also have that if there is no arrow in $\mathcal{B}$ starting at a vertex $v$, then there is no arrow in $\mathcal{B}$ ending at $v$ either. Indeed, the maximal path in $\mathcal{B}$ starting at $v$ has length zero, so $r(v)=v$, and then $v\in L(C)$ and $l(v)=v$, which shows that no arrow in $\mathcal{B}$ ends at $v$.\
Now taking the connected components of $C\cap \Gamma$ (regarded just as an undirected graph), and then considering the (directed) arrows, we find that $C\cap\Gamma=\bigsqcup\limits_i\Gamma_i$ for some subquivers $\Gamma_i$ which can be of the types $\AA_\infty$, $\AA_{0,\infty}$ or $\CC_n$, and this ends the proof.
\[l.1\] Let $C\subseteq K\Gamma$ be a left co-Frobenius path subcoalgebra. Let $u,v\in C\cap\Gamma$ be different vertices, and denote by $p_u$ and $p_v$ the maximal paths starting at $u$ and $v$, respectively. Then $p_u$ is not a subpath of $p_v$.
Assume otherwise, so $p_u$ is a subpath of $p_v$. We know that $p_u$ and $p_v$ end at $r(u)$ and $r(v)$, respectively. Let $q$ be the subpath of $p_v$ which starts at $v$ and ends at $r(u)$. Since $p_u$ is a subpath of $p_v$, then $q$ contains $p_u$, too. Then both $q$ and $p_u$ end at $r(u)$, and since by Proposition \[p.qcf\] $p_u$ is maximal with this property, we get that $q=p_u$. This means that $u=v$ (as starting points of $p_a$ and $q$), a contradiction.
Now we are in the position to give the classification result for left co-Frobenius path subcoalgebras.
\[th.qcf\] Let $C$ be a path subcoalgebra of the path coalgebra $K\Gamma$, and let $\mathcal{B}$ be a basis of paths of $C$. Then $C$ is left co-Frobenius if and only if $C\cap\Gamma=\bigsqcup\limits_i\Gamma_i$, a disjoint union of subquivers of $\Gamma$ of one of types $\AA_\infty$, $\AA_{0,\infty}$ or $\CC_n$, $n\geq 1$, and the path subcoalgebra $C_i$ of $K\Gamma_i$ spanned by the paths of $\mathcal{B}$ contained in $\Gamma_i$ is of type $K[\AA_{\infty},r]$ if $\Gamma_i=\AA_{\infty}$, of type $K[\AA_{0,\infty},r]$ if $\Gamma_i=\AA_{0,\infty}$, of type $K[\CC_n,s]$ with $s\geq 1$ if $\Gamma_i=\CC_n$, $n\geq 1$, and of type $K$ if $\Gamma_i=\CC_0$. In this case $C=\bigoplus\limits_{i}C_i$, in particular left co-Frobenius path subcoalgebras are direct sums of coalgebras of types $K[\AA_{\infty},r]$, $K[\AA_{0,\infty},r]$, $K[\CC_n,s]$ or $K$.
By Lemma \[l.graph\], $C\cap \Gamma=\bigsqcup\limits_i\Gamma_i$, and any $\Gamma_i$ is of one of the types $\AA_\infty$, $\AA_{0,\infty}$ or $\CC_n$, $n\geq 0$. Moreover, $C=\bigoplus\limits_{i}C_i$, so $C$ is left co-Frobenius if and only if all $C_i$’s are left co-Frobenius (see for example [@DNR Chapter 3]). If all $C_i$’s are of the indicated form, then they are left co-Frobenius by Examples \[gn\], \[gn0\] and \[lps\], and then so is $C$. Assume now that $C$ is left co-Frobenius. Then each $C_i$ is left co-Frobenius, so we can reduce to the case where $\Gamma$ is one of $\AA_\infty$, $\AA_{0,\infty}$ or $\CC_n$, and $C\cap \Gamma=\Gamma$. As before, for each vertex $v$ we denote by $r(v)$ the end-point of the unique maximal path in $C$ starting at $v$, and by $p_v$ this maximal path. Also denote by $m(v)$ the length of $p_v$.\
Case I. Let $\Gamma=\CC_n$. If $n=0$, then $C\cong K$. If $n=1$, then $C\cong K[\CC_1,s]$, since $m(\overline{0})=s>0$ because $\Gamma_1\subset C$, so there must be at least some nontrivial path in $C$. If $n\geq 2$, then $m(\overline{k})\leq m(\overline{k+1})$ for any $\overline{k}\in \ZZ_n$, since otherwise $p_{\overline{k+1}}$ would be a subpath of $p_{\overline{k}}$, a contradiction by Lemma \[l.1\]. Thus $m(\overline{0})\leq
m(\overline{1})\leq \ldots m(\overline{n-1})\leq m(\overline{0})$, so $m(\overline{0})= m(\overline{1})= \ldots m(\overline{n-1})=
m(\overline{0})=s$ for some $s\geq 0$. Since $C\cap \Gamma=\Gamma$, there are non-trivial paths in $C$, so $s>0$, and then clearly $C\cong K[\CC_n,s]$.\
Case II. If $\Gamma=\AA_\infty$ or $\Gamma=\AA_{0,\infty}$, then for any $n$ (in $\ZZ$ if $\Gamma=\AA_\infty$, or in $\NN$ if $\Gamma=\AA_{0,\infty}$) $m(n)\leq m(n+1)$ holds, otherwise $p_{n+1}$ would be a subpath of $p_n$, again a contradiction. Now if we take $r(n)=n+m(n)$ for any $n$, then $r$ is a strictly increasing function. Clearly $r(n)>n$, since $m(n)=0$ would contradict $C\cap
\Gamma=\Gamma$. Now it is obvious that $C\cong K[\Gamma,r]$.
\[clascoFrobenius\] Let $C\subseteq K\Gamma$ be a left and right co-Frobenius path subcoalgebra. Then $C$ is a direct sum of coalgebras of the type $K[\AA_{\infty}|s]$, $K[\CC_n,s]$ or $K$.
It follows directly from Theorem \[th.qcf\] and the discussion at the end of each of Examples \[gn\], \[gn0\] and \[lps\], concerned to the property of being left and right co-Frobenius.
[(1) We have a uniqueness result for the representation of a left co-Frobenius path subcoalgebras as a direct sum of coalgebras of the form $K[\AA_{\infty},r]$, $K[\AA_{0,\infty},r]$, $K[\CC_n,s]$ or $K$. To see this, an easy computation shows that the dual algebra of a coalgebra of any of these four types does not have non-trivial central idempotents, so it is indecomposable as an algebra. Now if $(C_i)_{i\in I}$ and $(D_j)_{j\in J}$ are two families of coalgebras with indecomposable dual algebras such that $\oplus_{i\in
I}C_i\simeq \oplus _{j\in J}D_j$ as coalgebras, then there is a bijection $\phi:J\rightarrow I$ such that $D_j\simeq C_{\phi(j)}$ for any $j\in J$. Indeed, if $f:\oplus_{i\in I}C_i\rightarrow \oplus
_{j\in J}D_j$ is a coalgebra isomorphism, then the dual map $f^*:\prod_{j\in J}D_j^*\rightarrow \prod_{i\in I}C_i^*$ is an algebra isomorphism. Since all $C_i^*$’s and $D_j^*$’s are indecomposable, there is a bijection $\phi:J\rightarrow I$ and some algebra isomorphisms $\gamma_j:D_j^*\rightarrow C_{\phi(j)}^*$ for any $j\in J$, such that for any $(d_j^*)_{j\in J}\in \prod_{j\in
J}D_j^*$, the map $f^*$ takes $(d_j^*)_{j\in J}$ to the element of $\prod_{i\in I}C_i^*$ having $\gamma_j(d_j^*)$ on the $\phi(j)$-th slot. Regarding $C=(C_i)_{i\in I}$ as a left $C^*$-module, and $D=\oplus _{j\in J}D_j$ as a left $D^*$-module in the usual way, with actions denoted by $\cdot$, the relation $f(f^*(d^*)\cdot
c)=d^*\cdot f(c)$ holds for any $c\in C$ and $d^*\in D^*$. This shows that $f$ induces coalgebra isomorphisms $C_{\phi(j)}\simeq D_j$ for any $j\in J$.\
(2) The coalgebras of types $K[\AA_{\infty},r]$, $K[\AA_{0,\infty},r]$, $K[\CC_n,s]$ or $K$ can be easily classified if we take into account that the sets of grouplike elements are just the vertices and the non-trivial skew-primitives are scalar multiples of the arrows. There are no isomorphic coalgebras of two different types among these four types. Moreover: (i) $K[\AA_{\infty},r]\simeq K[\AA_{\infty},r']$ if and only if there is an integer $h$ such that $r'(n)=r(n+h)$ for any integer $n$; (ii) $K[\AA_{0,\infty},r]\simeq K[\AA_{0,\infty},r']$ if and only if $r=r'$; (iii) $K[\CC_n,s]\simeq K[\CC_m,s']$ if and only $n=m$ and $s=s'$.]{}
Examples {#snew}
========
It is known (see [@ni], [@cm]) that any pointed coalgebra can be embedded in a path coalgebra. Thus it is expected that there is a large variety of co-Frobenius subcoalgebras of path coalgebras if we do not restrict only to the class of path subcoalgebras. The aim of this section is to provide several such examples. We first explain a simple construction connecting incidence coalgebras and path coalgebras, and producing examples as we wish.
As a pointed coalgebra, any incidence coalgebra can be embedded in a path coalgebra. However, there is a more simple way to define such an embedding for incidence coalgebras than for arbitrary pointed coalgebras. Indeed, let $X$ be a locally finite partially ordered set. Consider the quiver $\Gamma$ with vertices the elements of $X$, and such that there is an arrow from $x$ to $y$ if and only if $x<y$ and there is no element $z$ with $x<z<y$. With this notation, it is an easy computation to check the following.
\[propembedding\] The linear map $\phi:KX\rightarrow K\Gamma$, defined by $$\phi(e_{x,y})=\sum_{p\; {\rm path}\atop s(p)=x,t(p)=y}p$$ for any $x,y\in X, x\leq y$, is an injective coalgebra morphism.
Note that in the previous proposition $\phi(KX)$ is in general a subcoalgebra of $K\Gamma$ which is not a path subcoalgebra. This suggests that when we deal with left co-Frobenius subcoalgebras of incidence coalgebras, which of course embed themselves in $K\Gamma$ (usually not as path subcoalgebras), structures that are more complicated than those of left co-Frobenius path subcoalgebras can appear. Thus the classification of left co-Frobenius subcoalgebras of incidence coalgebras is probably more difficult. The next example is evidence in this direction.
\[examplediamond\] Let $s\geq 2$ and $X=\{ a_n|n\in \ZZ\} \cup (\cup_{n\in \ZZ}\{
b_{n,i}|1\leq i\leq s\})$ with the ordering $\leq$ such that $a_n<b_{n,i}<a_{n+1}$ for any integer $n$ and any $1\leq i\leq s$, and $b_{n,i}$ and $b_{n,j}$ are not comparable for any $n$ and $i\neq j$.\
Let $C$ be the subcoalgebra of $KX$ spanned by the following elements\
$\bullet$ the elements $e_{x,x}$, $x\in X$.\
$\bullet$ all segments $e_{x,y}$ of length 1.\
$\bullet$ the segments $e_{a_n,a_{n+1}}$, $n\in \ZZ$.\
$\bullet$ the segments $e_{b_{n,i},b_{n+1,i}}$, with $n\in \ZZ$ and $1\leq i\leq s$.\
Then by applying Proposition \[p.qcf\], we see that $C$ is co-Frobenius.\
If we take the subcoalgebra $D$ of $C$ obtained by restricting to the non-negative part of $X$, i.e. $D$ is spanned by the elements $e_{x,y}$ in the indicated basis of $C$ with both $x$ and $y$ among $\{ a_n|n\geq 0\} \cup (\cup_{n\geq 0}\{ b_{n,i}|1\leq i\leq s\})$, we see that $D$ is left co-Frobenius, but not right co-Frobenius.\
Now let $\Gamma$ be the quiver associated to the ordered set $X$ as in the discussion above. [$$\xymatrix{
\dots & b_{0,1} \ar[dr] & & b_{1,1}\ar[dr] & & \dots & b_{n,1}\ar[dr] & \dots \\
\dots \; a_0\ar[ur]\ar[r]\ar[ddr] & b_{0,2}\ar[r] & a_1\ar[ur]\ar[r]\ar[ddr] & b_{1,2}\ar[r] & a_2 & \dots\;\; a_n\ar[ur]\ar[r]\ar[ddr] & b_{n,2}\ar[r] & a_{n+1} \dots \\
& \dots & & \dots & & \dots & \dots & \\
\dots & b_{0,s}\ar[uur] & & b_{1,s}\ar[uur] & & &
b_{n,s}\ar[uur] & \dots }$$]{} If $\phi:KX\rightarrow K\Gamma$ is the embedding described in Proposition \[propembedding\], then $\phi(C)$ is a co-Frobenius subcoalgebra of $K\Gamma$. We see that $\phi(C)$ is the subspace of $K\Gamma$ spanned by the vertices of $\Gamma$, the paths of length 1, the paths $[b_{n,i}a_{n+1}b_{n+1,i}]$ with $n\in \ZZ$ and $1\leq i\leq s$, and the elements $\sum_{1\leq i\leq s}[a_nb_{n,i}a_{n+1}]$ with $n\in
\ZZ$, thus $\phi(C)$ is not a path subcoalgebra. Here we denoted by $[b_{n,i}a_{n+1}b_{n+1,i}]$ and $[a_nb_{n,i}a_{n+1}]$ the paths following the indicated vertices and the arrows between them. By restricting to the non-negative part of $X$, a similar description can be given for $\phi(D)$, a subcoalgebra of $K\Gamma$ which is left co-Frobenius, but not right co-Frobenius.
It is possible to embed some of the co-Frobenius path subcoalgebras in other path coalgebras as subcoalgebras which are not path subcoalgebras.
Consider the quiver $\AA_{\infty}$ with vertices indexed by the integers, with the path from $i$ to $j$ denoted by $p_{i,j}$. Consider the path subcoalgebra $D=K[\AA_{\infty}|2]$, with basis $\{
p_{i,i},p_{i,i+1},p_{i,i+2}|i\in \ZZ\}$. We also consider the quiver $\Gamma$ below [$$\xymatrix{
\dots & b_0 \ar[dr] & & b_1\ar[dr] & & \dots & b_n\ar[dr] & \dots \\
\dots \;\;\; a_0\ar[ur]\ar[rr] & & a_1\ar[ur]\ar[rr] & & a_2 & \dots \;\;\;
a_n\ar[ur]\ar[rr] & & a_{n+1}\;\; \dots }$$ ]{} Then $\AA_{\infty}$ is a subquiver of $\Gamma$ if we identify $a_i$ with $2i$ and $b_i$ with $2i+1$ for any integer $i$. Thus $K\AA_{\infty}$ is a subcoalgebra of $K\Gamma$ in the obvious way, and then so is $D$. However, there is another way to embed $D$ in $K\Gamma$. Indeed, the linear map $\phi:D\rightarrow K\Gamma$, defined such that $$\phi(p_{2i,2i})=a_i, \phi(p_{2i+1,2i+1})=b_i,$$ $$\phi(p_{2i,2i+1})=[a_ib_i], \phi(p_{2i+1,2i+2})=[b_ia_{i+1}],$$ $$\phi(p_{2i,2i+2})=[a_ia_{i+1}]+[a_ib_ia_{i+1}],$$ $$\phi(p_{2i+1,2i+3})=[b_ia_{i+1}b_{i+1}]$$ for any $i\in \ZZ$, is an injective morphism of coalgebras. Here we denoted by $[a_ib_i]$, $[a_ib_ia_{i+1}]$, etc, the paths following the respective vertices and arrows. We conclude that the subcoalgebra $C=\phi(D)$ of $K\Gamma$, spanned by all vertices $a_n,b_n$, all arrows $[a_na_{n+1}],[a_nb_n],[b_na_{n+1}]$ and the elements $[a_nb_na_{n+1}]+[a_na_{n+1}]$ and $[b_na_{n+1}b_{n+1}]$, is co-Frobenius. Note that $D$ is not a path subcoalgebra of $K\Gamma$. This can be also seen as the subcoalgebra of the incidence coalgebra of $\ZZ$ with basis consisting of segments of length at most $2$.
Note that in the above example, we can also consider a similar situation but with all segments $e_{n,n+i}$ of the incidence coalgebra of $\ZZ$ which have length less or equal to a certain positive integer $s$ ($i\leq s$); the same properties as above would then hold for this situation.
We consider the same situation as above, but we restrict the quiver $\Gamma$ to the non-negative part: [$$\xymatrix{
& b_0 \ar[dr] & & b_1\ar[dr] & & \dots & & b_n\ar[dr] & \dots \\
a_0\ar[ur]\ar[rr] & & a_1\ar[ur]\ar[rr] & & a_2 & \dots & a_n\ar[ur]\ar[rr] & & a_{n+1} \;\dots
}$$ ]{} Equivalently, we consider the subcoalgebra of the incidence coalgebra of $\NN$ with a basis of all segments of length less or equal to $2$ (or $\leq s$ for more generality). This coalgebra is left co-Frobenius but not right co-Frobenius, it is a subcoalgebra of an incidence coalgebra, and it can also be regarded as a subcoalgebra of a path coalgebra, but without a basis of paths.
Now we prove a simple, but useful result, which shows that the category of incidence coalgebras is closed under tensor product of coalgebras.
\[tensorincidence\] Let $X,Y$ be locally finite partially ordered sets. Consider on $X\times Y$ the order $(x,y)\leq(x',y')$ if and only if $x\leq y$ and $x'\leq y'$. Then there is an isomorphism of coalgebras $K(X\times Y)\cong KX\otimes KY$.
It is clear that $X\times Y$ is locally finite. We show that the natural isomorphism of vector spaces $\varphi:K(X\times
Y)\rightarrow KX\otimes KY$, $\varphi(e_{(x,y),(x',y')})=e_{x,x'}\otimes e_{y,y'}$ is a morphism of coalgebras. This is well defined by the definition of the order relation on $X\times Y$. For comultiplication we have $$\begin{aligned}
& \sum \varphi(e_{(x,y),(x',y')})_1\otimes (e_{(x,y),(x',y')})_2 = & \\
= & \sum\limits_{x\leq a\leq x'}\sum\limits_{y\leq b\leq y'}e_{x,a}\otimes e_{y,b}\otimes e_{a,x'}\otimes e_{x',b} & \\
= & \sum\limits_{(x,y)\leq (a,b)\leq (x',y')}\varphi(e_{(x,y),(a,b)})\otimes \varphi(e_{(a,b),(x',y')}) & \\
= & \varphi((e_{(x,y),(x',y')})_1)\otimes
\varphi((e_{(x,y),(x',y')})_2) & \end{aligned}$$ and it is also easy to see that $\varepsilon_{KX\otimes
KY}\circ\varphi =\varepsilon_{K(X\times Y)}$.
Consider the ordered set $(\ZZ \times \ZZ,\leq)$, with order given by the direct product of the orders of $(\ZZ,\leq)$ and $(\ZZ,\leq)$. Thus $(i,j)\leq (p,q)$ if and only if $i\leq p$ and $j\leq q$. We know from Proposition \[tensorincidence\] that $\psi:K\ZZ \otimes K\ZZ\rightarrow K(\ZZ\times \ZZ)$, $\psi(e_{i,p}\otimes
e_{j,q})=e_{(i,j),(p,q)}$, is an isomorphism of coalgebras.\
With the notation preceding Proposition \[propembedding\], the quiver $\Gamma$ associated to the locally finite ordered set $(\ZZ
\times \ZZ,\leq)$ is $$\xymatrix{
& \dots & \dots & \dots &\\
\dots \ar[r] & a_{n-1,k+1}\ar[r]\ar[u] & a_{n,k+1}\ar[r]\ar[u] & a_{n+1,k+1}\ar[r]\ar[u] & \dots \\
\dots \ar[r] & a_{n-1,k}\ar[r]\ar[u] & a_{n,k}\ar[r]\ar[u]\ar[r]\ar[u] & a_{n+1,k}\ar[r]\ar[u] & \dots \\
\dots \ar[r] & a_{n-1,k-1}\ar[r]\ar[u] & a_{n,k-1}\ar[r]\ar[u] & a_{n+1,k-1}\ar[r]\ar[u] & \dots \\
& \dots \ar[u] & \dots \ar[u] & \dots \ar[u] & }$$ where we just denoted the vertices by $a_{n,k}$ instead of just $(n,k)$. Let $\phi:K(\ZZ \times \ZZ)\rightarrow K\Gamma$ be the embedding from Proposition \[propembedding\]. If we consider the subcoalgebra $K[\AA_{\infty}|1]$ of $K\ZZ$, then $K[\AA_{\infty}|1]\otimes
K[\AA_{\infty}|1]$ is a subcoalgebra of $K\ZZ\otimes K\ZZ$, so then $C=\phi \psi(K[\AA_{\infty}|1]\otimes K[\AA_{\infty}|1])$, which is the subspace spanned by the vertices of $\Gamma$, the arrows of $\Gamma$, and the elements $[a_{n,k}a_{n+1,k}a_{n+1,k+1}]+[a_{n,k}a_{n,k+1},a_{n+1,k+1}]$, is a subcoalgebra of $K\Gamma$. Since $K[\AA_{\infty}|1]$ is co-Frobenius, and the tensor product of co-Frobenius coalgebras is co-Frobenius (see [@IG Proposition 4.15]), we obtain that $C$ is a co-Frobenius coalgebra. Alternatively, it can be seen that $\psi(K[\AA_{\infty}|1]\otimes K[\AA_{\infty}|1])$, which is the subspace spanned by the elements $e_{(n,k),(n,k)},
e_{(n,k),(n+1,k)}, e_{(n,k),(n,k+1)}, e_{(n,k),(n+1,k+1)}$ with arbitrary $n,k\in \ZZ$, is co-Frobenius by applying Proposition \[p.qcf\]. $C$ can be seen as both a subcoalgebra of an incidence coalgebra and of a path coalgebra, but not with a basis of paths. We note that $C$ is not even isomorphic to a path subcoalgebra. Indeed, if it were so, it should be isomorphic to some $K[\AA_\infty|s]$, since it is infinite dimensional and indecomposable. But in $C$, for any grouplike element $g$ there are precisely two other grouplike elements $h$ with the property that the set of non-trivial $(h,g)$-skew-primitive elements is nonempty, while for any grouplike element g of $K[\AA_\infty|s]$ there is only one such $h$.
With similar arguments, we can give a more general version of the previous example, by considering finite tensor products of coalgebras of type $K[\AA_\infty|s]$, as follows.
Let $D=K[\AA_\infty|s_1]\otimes K[\AA_\infty|s_2]\otimes \ldots
\otimes K[\AA_\infty|s_m]$, where $m\geq 2$ and $s_1,\ldots ,s_m$ are positive integers. Then $D$ is co-Frobenius as a tensor product of co-Frobenius coalgebras, and $D$ embeds in the $m$-fold tensor product $K\ZZ\otimes K\ZZ \otimes \ldots \otimes K\ZZ$. But this last tensor product is isomorphic to the incidence coalgebra of the ordered set $\ZZ^m=\ZZ\times \ZZ\times \ldots \times \ZZ$, with the direct product order. The image of $D$ via this embedding is the subcoalgebra $E$ of $K(\ZZ\times \ZZ\times \ldots \times \ZZ)$ spanned by all the segments $e_{(n_1,\ldots,n_m),(k_1,\ldots,k_m)}$ with $n_1\leq k_1\leq n_1+s_1,\,\ldots \, , n_m\leq k_m\leq n_m+s_m$.\
Now if we consider the quiver $\Gamma$ associated to the ordered set $\ZZ\times \ZZ\times \ldots \times \ZZ$ as in the beginning of this section, we have an embedding of $K(\ZZ\times \ZZ\times \ldots
\times \ZZ)$ in $K\Gamma$. Denote the vertices of $\Gamma$ by $a_{n_1,\ldots,n_m}$. The image of $E$ through this embedding is the subcoalgebra $C$ of $K\Gamma$ spanned by all the elements of the form $S(\Gamma, (n_1,\ldots,n_m),(k_1,\ldots,k_m))$, with $n_1,\ldots,n_m,k_1,\ldots,k_m$ integers such that $n_1\leq k_1\leq
n_1+s_1,\ldots , n_m\leq k_m\leq n_m+s_m$, where by $S(\Gamma,
(n_1,\ldots,n_m),(k_1,\ldots,k_m))$ we denote the sum of all paths in $\Gamma$ starting at $a_{n_1,\ldots,n_m}$ and ending at $a_{k_1,\ldots,k_m}$. Thus $C$ is a co-Frobenius subcoalgebra of $K\Gamma$, which is also isomorphic to a subcoalgebra of an incidence coalgebra. However, $C$ is not a path subcoalgebra, and not even isomorphic to a path subcoalgebra. Indeed, for any grouplike element $g$ of $E$ there are precisely $m$ grouplike elements $h$ for which there are non-trivial $(h,g)$-skew-primitive elements, while in a co-Frobenius path subcoalgebra for any grouplike element $g$ there is at most one such $h$.
We note that the co-Frobenius coalgebra $C$ constructed in Example \[examplediamond\] is not isomorphic to a coalgebra of the form $K[\AA_\infty|s_1]\otimes K[\AA_\infty|s_2]\otimes \ldots \otimes
K[\AA_\infty|s_m]$. Indeed, if $g=b_{n,i}$ there exists exactly one grouplike element $h$ of $C$ such that there are non-trivial $(h,g)$-skew-primitive elements (this is $h=a_{n+1}$), and if $g=a_n$ there exist $s$ such grouplike elements $h$ (these are $b_{n,1},\ldots,b_{n,s}$). On the other hand, in $K[\AA_\infty|s_1]\otimes K[\AA_\infty|s_2]\otimes \ldots \otimes
K[\AA_\infty|s_m]$ for any grouplike element $g$ there exist precisely $m$ such elements $h$.
We end with another explicit example, which shows that there are co-Frobenius subcoalgebras of path coalgebras that are isomorphic neither to a path subcoalgebra nor to a subcoalgebra of an incidence coalgebra.
Let $\Gamma$ be the graph: $$\xymatrix{ \dots\ar[r] & a_0\ar[r]^{y_0}\ar@(ul,ur)[]^{x_0} &
a_1\ar[r]^{y_1}\ar@(ul,ur)[]^{x_1} &
a_2\ar[r]^{y_2}\ar@(ul,ur)[]^{x_2} & \dots \ar[r] &
a_n\ar[r]^{y_n}\ar@(ul,ur)[]^{x_n} &\dots }$$ and let $C$ be the subcoalgebra of the path coalgebra of $\Gamma$ having a basis the elements $a_n,x_n,y_n$ and $y_n+x_ny_n$. This is, in fact, isomorphic to $K[\CC_1|1]\otimes K[\AA_\infty|1]$, so it is co-Frobenius. By the classification theorem for co-Frobenius path subcoalgebras and the structure of the skew-primitive elements of $C$, we see that $C$ is not isomorphic to a path subcoalgebra. We note that it is not isomorphic either to a subcoalgebra of an incidence coalgebra, because in an incidence coalgebra, if $g$ is any grouplike element, there is no $(g,g)$- skew-primitive element, while in $C$ for each grouplike $g=a_n$, $x_n$ is a $(g,g)$- skew-primitive.
Hopf algebra structures on path subcoalgebras {#s6}
=============================================
In this section we discuss the possibility of extending the coalgebra structure of a path subcoalgebra to a Hopf algebra structure. First of all, it is a simple application of Proposition \[proppathcoFrob\] to see when a finite dimensional path coalgebra has a Hopf algebra structure.
If the path coalgebra $K\Gamma$ is finite dimensional, then it has a Hopf algebra structure if and only if it is cosemisimple, i.e. $\Gamma$ has no arrows.
If the finite dimensional $K\Gamma$ has a Hopf algebra structure, then it has non-zero integrals, so it is left (and right) co-Frobenius, and $K\Gamma$ is cosemisimple by Proposition \[proppathcoFrob\]. Conversely, if there are no arrows, then $K\Gamma$ can be endowed with the group Hopf algebra structure obtained if we consider a group structure on the set of vertices.
Next, we are interested in finding examples of Hopf algebra structures that can be defined on some path subcoalgebras. At this point we discuss only cases where the resulting Hopf algebra has non-zero integrals, i.e. it is left (or right) co-Frobenius. Thus the path subcoalgebras that we consider are among the ones in Corollary \[clascoFrobenius\]. We ask the following general question.
[*[**PROBLEM.**]{} Which of the left and right co-Frobenius path subcoalgebras (classified in Corollary \[clascoFrobenius\]) can be endowed with a Hopf algebra structure?*]{}
In the rest of this section we solve the problem in the case where $K$ is a field containing primitive roots of unity of any positive order, in particular $K$ has characteristic zero. We will make this assumption on $K$ from this point on. We just note that some of the constructions can be also done in positive characteristic, if we just require that $K$ contains certain primitive roots of unity and the characteristic of $K$ is large enough.
\(I) Let $s>0$ be an integer. Let $q$ be a primitive $(s+1)$th root of unity in $K$. Let $G$ be a group such that there exist an element $g\in Z(G)$ of infinite order and a character $\chi\in G^*$ such that $\chi(g)=q$. Also let $\alpha \in K$ which may be non-zero only if $\chi^{s+1}=1$. Consider the algebra generated by the elements of $G$ (and preserving the group multiplication on these elements), and $x$, subject to relations $$xh=\chi(h)hx \mbox{ for any }h\in G$$ $$x^{s+1}=\alpha (g^{s+1}-1)$$ (that is, the free or amalgamated product $K[x]*K[G]$, quotient out by the above relations). Then this algebra has a unique Hopf algebra structure such that the elements of $G$ are grouplike elements, $\Delta (x)=1\otimes x+x\otimes g$, and $\varepsilon(x)=0$. We denote this Hopf algebra structure by $H_{\infty}(s,q,G,g,\chi,\alpha)$.\
(II) Let $n\geq 2$ and $s>0$ be integers such that $s+1$ divides $n$. Let $q$ be a primitive $(s+1)$th root of unity in $K$. Consider a group $G$ such that there exist an element $g\in Z(G)$ of order $n$ and a character $\chi\in G^*$ such that $\chi(g)=q$. Also let $\alpha \in K$ which may be non-zero only if $\chi^{s+1}=1$. Consider the algebra generated by the elements of $G$ (and preserving the group multiplication on these elements), and $x$, subject to relations $$xh=\chi(h)hx \mbox{ for any }h\in G$$ $$x^{s+1}=\alpha (g^{s+1}-1)$$ Then this algebra has a unique Hopf algebra structure such that the elements of $G$ are grouplike elements, $\Delta (x)=1\otimes
x+x\otimes g$, and $\varepsilon(x)=0$. We denote this Hopf algebra structure by $H_n(s,q,G,g,\chi,\alpha)$.
We consider an approach similar to the one in [@bdg]. For both (I) and (II) we consider the Hopf group algebra $KG$, and its Ore extension $KG[X,\overline{\chi}]$, where $\overline{\chi}$ is the algebra automorphism of $KG$ such that $\overline{\chi}(h)=\chi(h)h$ for any $h\in G$. Since $g\in Z(G)$, this Ore extension has a unique Hopf algebra structure such that $\Delta(X)=1\otimes X+X\otimes g$ and $\varepsilon(X)=0$, by the universal property for Ore extensions (see for example [@bdg Lemma 1.1]). Since $(1\otimes
X)(X\otimes g)=q(X\otimes g)(1\otimes X)$, the quantum binomial formula shows that $\Delta (X^{s+1})=1\otimes X^{s+1}+X^{s+1}\otimes
g^{s+1}$, so then the ideal $I=(X^{s+1}-\alpha (g^{s+1}-1))$ is in fact a Hopf ideal of $KG[X,\overline{\chi}]$. Then we can consider the factor Hopf algebra $KG[X,\overline{\chi}]/I$, and this is just the desired Hopf algebra $H_{\infty}(s,q,G,g,\chi,\alpha)$ in case (I) and $H_n(s,q,G,g,\chi,\alpha)$ in case (II). The condition that $\alpha=0$ whenever $\chi^{s+1}\neq 1$ guarantees that the map $G\rightarrow KG[X,\overline{\chi}]/I$ taking an element $h\in G$ to its class modulo $I$ is injective, thus $G$ is the group of grouplike elements of this factor Hopf algebra.
In the following example we give examples of co-Frobenius path subcoalgebras that can be endowed with Hopf algebra structures. Moreover, we don’t only introduce one such structure, but a family of Hopf algebra structures on each path subcoalgebra considered in the example.
\[exemplustructuri\] (i) $K[\AA_{\infty}|s]$ can be endowed with a Hopf algebra structure for any $s\geq 1$. Indeed, let $q$ be a primitive $(s+1)$th root of unity in K, and let $\alpha\in K$. We define a multiplication (on basis elements, then extended linearly) on $K[\AA_{\infty}|s]$ by $$p_{i,i+u}p_{j,j+v}=\left\{ \begin{array}{l} q^{ju}\,{{u+v}\choose
{u}}_{q}\,p_{i+j,i+j+u+v},\\ \;\hspace{2cm} {\rm if}\; u+v\leq s\\
\alpha
q^{ju}\frac{(u+v-s-1)_q!}{(u)_q!(v)_q!}(p_{i+j+s+1,u+v+i+j}-p_{i+j,u+v+i+j-s-1}), \\ \;\hspace{2cm}{\rm
if}\; u+v\geq s+1
\end{array} \right.$$ where${{u+v}\choose {u}}_{q}$ denotes the $q$-binomial coefficient. Then this multiplication makes $K[\AA_{\infty}|s]$ an algebra, which together the initial coalgebra structure define a Hopf algebra structure on $K[\AA_{\infty}|s]$. Indeed, we can see this by considering the Hopf algebra $H_{\infty}(s,q,C_{\infty},c,\chi,\alpha)$, where $C_{\infty}$ is the (multiplicative) infinite cyclic group generated by an element $c$, and the character $\chi$ is defined by $\chi(c)=q$. Thus $H_{\infty}(s,q,C_{\infty},c,\chi,\alpha)$ is generated as an algebra by the elements $c$ and $x$, subject to relations $xc=q cx$ and $x^{s+1}=\alpha (c^{s+1}-1)$, and with coalgebra structure such that $\Delta (c)=c\otimes c$, $\varepsilon (c)=1$, and $\Delta
(x)=1\otimes x+x\otimes c$. Since $(1\otimes x)(x\otimes c)=q
(x\otimes c)(1\otimes x)$, we can apply the quantum binomial formula and get that $$\Delta (x^u)=\sum_{0\leq h\leq u}{u\choose
h}_{q}x^{u-h}\otimes c^{u-h}x^h$$ and then $$\Delta \left(\frac{1}{(u)_{q}!}\, c^ix^u\right)=
\sum_{0\leq h\leq u}\frac{1}{(u-h)_{q}!}\, c^ix^{u-h}\otimes
\frac{1}{(h)_{q}!}\, c^{i+u-h}x^h$$ for any $0\leq u\leq s$ and any integer $i$. Therefore if we denote $\frac{1}{(u)_{q}!}\, c^ix^u$ by $P_{i,i+u}$, this means that $\Delta (P_{i,i+u})=\sum_{0\leq h\leq
u} P_{i,i+h}\otimes P_{i+h,i+u}$, showing that the linear isomorphism $\phi:K[\AA_{\infty}|s]\rightarrow
H_{\infty}(s,q,C_{\infty},c,\chi,\alpha)$ taking $p_{i,i+u}$ to $P_{i,i+u}$ for any $0\leq u\leq s$ and $i\in \ZZ$, is an isomorphism of coalgebras. Now we just transfer the algebra structure of $H_{\infty}(s,q,C_{\infty},c,\chi,\alpha)$ through $\phi^{-1}$ and get precisely the multiplication formula given above.\
(ii) Let us consider now the coalgebra $C$ which is a direct sum of a family of copies of (the same) $K[\AA_{\infty}|s]$, indexed by a non-empty set $P$. Then $C$ can be endowed with a Hopf algebra structure. To see this, we extend the example from (i) as follows. Let $G$ be a group such that there exist an element $g\in Z(G)$ of infinite order and a character $\chi\in G^*$ for which $q=\chi(g)$ is a primitive $(s+1)$th root of unity, and moreover the factor group $G/<g>$ is in bijection with the set $P$ (note that such a triple $(G,g,\chi)$ always exists; we can take for instance a group structure on the set $P$, $G=C_{\infty}\times P$, $g$ a generator of $C_{\infty}$, and $\chi$ defined such that $\chi(g)=q$ and $\chi(p)=1$ for any $p\in P$). For simplicity of the notation, we can assume that $P$ is a set of representatives for the $<g>$-cosets of G. Consider the Hopf algebra $A=H_{\infty}(s,q,G,g,\chi,\alpha)$, where $\alpha$ is a scalar which may be non-zero only if $\chi^{s+1}=1$. Then the subalgebra $B$ of $A$ generated by $g$ and $x$ is a Hopf subalgebra isomorphic to $K[\AA_{\infty}|s]$ as a coalgebra, and $A=\oplus_{p\in P}\; pB$ is a direct sum of subcoalgebras, all isomorphic to $K[\AA_{\infty}|s]$. Thus $A$ is isomorphic as a coalgebra to $C$, and we can transfer the Hopf algebra structure of $A$ to $C$.\
(iii) Assume that $n\geq 2$ and $s+1$ divides $n$. Then $K[\CC_n,s]$ can be endowed with a Hopf algebra structure. Indeed, we proceed as for $K[\AA_{\infty}|s]$, but replacing the Hopf algebra $H_{\infty}(s,q,C_{\infty},c,\chi,\alpha)$ by $H_n(s,q,C_n,c,\chi,\alpha)$, where $C_n$ is a cyclic group of order $n$ with a generator $c$ (we have the same relations for $c$ and $x$ as in (i), to which we add $c^n=1$). Thus the multiplication of $K[\AA_{\infty}|s]$ is given by $$q_{\overline{i}|u}q_{\overline{j}|v}=\left\{ \begin{array}{l} q^{ju}\,{{u+v}\choose
{u}}_{q}\,q_{\overline{i+j}|u+v},\\ \hspace{1cm} \;{\rm if}\; u+v\leq s\\
\alpha
q^{ju}\frac{(u+v-s-1)_q!}{(u)_q!(v)_q!}(q_{\overline{i+j+s+1}|u+v-s-1}-q_{\overline{i+j}|u+v-s-1}),\\ \hspace{1cm}\;{\rm
if}\; u+v\geq s+1
\end{array} \right.$$ Also, as in (ii), a direct sum of copies of the same $K[\CC_n,s]$, indexed by an arbitrary non-empty set $P$, can be endowed with a Hopf algebra structure isomorphic to some $H_n(s,q,G,g,\chi,\alpha)$ for some $q,G,g,\chi,\alpha$, where $q$ is a primitive $(s+1)$th root of unity, $G$ is a group, $g\in Z(G)$ is an element of order $n$, $G/<g>$ is in bijection with $P$, $\chi \in G^*$ is a character such that $\chi(g)=q$, and $\alpha\in K$ is a scalar which may be non-zero only if $\chi^{s+1}=1$.\
The examples given in (iii) appear (for finite sets $P$) in [@chyz].
Now we can prove the main result of this section.
\[teoremastructuriHopf\] Assume that $K$ is a field containing primitive roots of unity of any positive order (in particular, $K$ has characteristic $0$). Then a co-Frobenius path subcoalgebra $C\neq 0$ can be endowed with a Hopf algebra structure if and only if it is of one of the following three types:\
(I) A direct sum of copies (indexed by a set $P$) of the same $K[\AA_{\infty}|s]$ for some $s\geq 1$. In this case, any Hopf algebra structure on $C$ is isomorphic to a Hopf algebra of the form $H_{\infty}(s,q,G,g,\chi,\alpha)$ for some $q,G,g,\chi,\alpha$, where $q$ is a primitive $(s+1)$th root of unity, $G$ is a group, $g\in Z(G)$ is an element of infinite order, $G/<g>$ is in bijection with $P$, $\chi \in G^*$ is a character such that $\chi(g)=q$, and $\alpha\in K$ is a scalar which may be non-zero only if $\chi^{s+1}=1$.\
(II) A direct sum of copies (indexed by a set $P$) of the same $K[\CC_n,s]$ for some $n\geq 2$ and $s\geq 1$ such that $s+1$ divides $n$. In this case, any Hopf algebra structure on $C$ is isomorphic to a Hopf algebra of the form $H_n(s,q,G,g,\chi,\alpha)$ for some $q,G,g,\chi,\alpha$, where $q$ is a primitive $(s+1)$th root of unity, $G$ is a group, $g\in Z(G)$ is an element of order $n$, $G/<g>$ is in bijection with $P$, $\chi \in G^*$ is a character such that $\chi(g)=q$, and $\alpha\in K$ is a scalar which may be non-zero only if $\chi^{s+1}=1$.\
(III) A direct sum of copies of $K$. In this case, any Hopf algebra structure on $C$ is isomorphic to a group Hopf algebra $KG$ for some group $G$.
By Example \[exemplustructuri\] we see that a coalgebra of type (I) or (II) has a Hopf algebra structure. Obviously, a coalgebra of type (III) is a grouplike coalgebra $KX$ for some set $X$, so then it has a Hopf algebra structure, obtained if we endow $X$ with a group structure.\
Conversely, let $C$ be a co-Frobenius path subcoalgebra which can be endowed with a Hopf algebra structure. By Corollary \[clascoFrobenius\], $C$ is isomorphic to a direct sum of coalgebras of types $K[\AA_{\infty}|s]$, $K[\CC_n,s]$ or $K$. We have that $G=G(C)$, the set of all vertices of $C$, is a group with the induced multiplication. We look at the identity element 1 of this group and distinguish three cases.\
[*Case 1.*]{} If 1 is a vertex in a connected component of type $K[\AA_{\infty}|s]$, denote the vertices of this connected component by $(v_n)_{n\in \ZZ}$ such that $v_0=1$. Also denote by $a_n$ the arrow from $v_n$ to $v_{n+1}$ for any $n\in \ZZ$. If $g=v_1$, then $\Delta (a_1)=1\otimes a_1+a_1\otimes g$, and $a_1\notin C_0$. Then $\Delta (ga_1)=g\otimes ga_1+ga_1\otimes g^2$, and $ga_1\notin C_0$, so $P_{g^2,g}(C)\nsubseteq C_0$. Since the only $h\in G$ such that $P_{h,g}(C)$ is not trivial (i.e. $\neq K(h-g)$, or equivalently, not contained in $C_0$) is $h=v_2$, we obtain that $v_2=g^2$. Recurrently we see that $v_n=g^n$ for any positive integer $n$, and also for any negative integer $n$.\
Let us take some $h\in G$. Then $\Delta (ha_1)=h\otimes
ha_1+ha_1\otimes hg$ and $ha_1\notin C_0$, so $P_{hg,h}(C)\neq
K(hg-h)$. Hence there is an arrow starting at $h$ and ending at $hg$ in $C$; as before, inductively we get that there are in $C$ arrows as follows $$\;\;\;\; ^{\hspace{-8mm}\ldots\;\longrightarrow}
_{hg^-1}\hspace{-5mm}^{\circ\;\;\longrightarrow\;\;}
\hspace{0.1mm}^{\circ\;\;\longrightarrow}_h
\hspace{1mm}^{\circ\;\longrightarrow\;}_{hg}
\hspace{0.5mm}^{\circ\;\longrightarrow\;\dots}_{\hspace{-1mm}hg^2}$$ which shows that the vertex $h$ belongs to a connected component $D$ of type $K[\AA_{\infty}|s']$ for some $s'\geq 1$. Moreover, $\Delta
(a_1h)=h\otimes a_1h+a_1h\otimes gh$, we also have $P_{gh,h}(C)\neq
K(gh-h)$, so there is an arrow from $h$ to $gh$ in $C$. This shows that $hg=gh$, so then $g$ must lie in $Z(G)$.\
If we denote by $p_{h,g^ih}$ the unique path from $h$ to $g^ih$, for any $h\in G$ and $i\geq 0$, then $$\Delta (p_{1,g^s})-1\otimes p_{1,g^s}-p_{1,g^s}\otimes g^s\in
C_{s-1}\otimes C_{s-1}$$ and $p_{1,g^s}\notin C_{s-1}$. Then $$\Delta (hp_{1,g^s})-h\otimes hp_{1,g^s}-hp_{1,g^s}\otimes hg^s\in
C_{s-1}\otimes C_{s-1}$$ and $hp_{1,g^s}\notin C_{s-1}$. But it is easy to check that in the path coalgebra $K\Gamma$ (whose subcoalgebra is $C$) the relation $\Delta(c)-h\otimes c-c\otimes
hg^s\in (K\Gamma)_{s-1}\otimes (K\Gamma)_{s-1}$ holds if and only if $c\in (K\Gamma)_{s-1}+Kp_{h,hg^s}$. Applying this for $c=hp_{1,g^s}\notin C_{s-1}$, we obtain that $hp_{1,g^s}=c'+\gamma
p_{h,hg^s}$ for some $c'\in (K\Gamma)_{s-1}$ and $\gamma \in K^*$. This shows that $p_{h,hg^s}$ must be in $C$, so it also lies in $D$, which implies that $s'\geq s$ (otherwise $D$ cannot have paths of length $s$).\
Similarly, since $$\Delta (h^{-1}p_{h,hg^{s'}})-1\otimes h^{-1}p_{h,hg^{s'}}-h^{-1}p_{h,hg^{s'}}\otimes g^{s'}\in
C_{s'-1}\otimes C_{s'-1}$$ and $h^{-1}p_{h,hg^{s'}}\notin C_{s'-1}$, we obtain that $s\geq s'$. In conclusion $s'=s$, and $C$ is a direct sum of coalgebras isomorphic to $K[\AA_{\infty}|s]$. Moreover, this direct sum is indexed by a set in bijection with $G/<g>$.\
In order to uncover the Hopf algebra structures on $C$, we use the Lifting Method proposed in [@as]. Since $C_0=K\Gamma$ is a Hopf subalgebra of $C$, the coradical filtration $C_0\subseteq
C_1\subseteq \ldots$ of $C$ is a Hopf algebra filtration, and we can consider the associated graded space ${\rm gr}\, C=C_0\oplus
\frac{C_1}{C_0}\oplus \ldots$, which has a graded Hopf algebra structure. Denote $H=K\Gamma$, the degree 0 component of ${\rm gr}\,
C$, and by $\gamma:H\rightarrow {\rm gr}\, C$ the inclusion morphism. The natural projection $\pi:{\rm gr}\, C\rightarrow H$ is a Hopf algebra morphism. Then the coinvariants $R=({\rm gr}\,
C)^{co\, H}$ with respect to the right $H$-coaction induced via $\pi$, i.e. $$R=\{ z\in {\rm gr}\, C\;|\; (I\otimes \pi)\Delta (z)=z\otimes
1\}$$ is a left Yetter-Drinfeld module over $H$, with left $H$-action defined by $h\cdot r=\sum \gamma(h_1)rS(\gamma(h_2))$ for any $h\in H, r\in R$, and left $H$-coaction $\delta (r)=\sum
r_{(-1)}\otimes r_{(0)}=(\pi \otimes I)\Delta(r)$. Moreover, $R$ is a graded subalgebra of ${\rm gr}\, C$, with grading denoted by $R=\oplus_{n\geq 0}R(n)$, and it also has a coalgebra structure with comultiplication $\Delta_R(r)=\sum r^{(1)}\otimes r^{(2)}=\sum
r_1\gamma\pi (S(r_2))\otimes r_3$, and these make $R$ a braided Hopf algebra in the category $^H_HYD$ of Yetter-Drinfeld modules over $H$. The Hopf algebra ${\rm gr}\, C$ can be reconstructed from $R$ by bosonization, i.e. ${\rm gr}\, C\simeq R\# H$, the biproduct of $R$ and $H$. The multiplication of this biproduct is the smash product given by $(r\# h)(p\# v)=\sum r(h_1\cdot p)\# h_2v$, while the comultiplication is the smash coproduct $\Delta (r\# h)=\sum
(r^{(1)}\# (r^{(2)})_{(-1)}h_1)\otimes
(r^{(2)})_{(0)}\# h_2$.\
Since in our case $C_i$ is the span of all paths of length at most $i$ in $C$, if $z=\hat{c}\in R(n)$, then $c=\sum_i \alpha_ip_i$, a linear combination of paths $p_i$ of length $i$, and $\sum_i\alpha_i\widehat{p_i}\otimes
t(p_i)=\sum_i\alpha_i\widehat{p_i}\otimes 1$. Then $\alpha_i=0$ for any $i$ such that $t(p_i)\neq 1$, showing that $R(i)$ is spanned by the classes of the paths of length $i$ which end at $1$. We conclude that $R(i)$ has dimension 1 for any $0\leq i\leq s$, and ${\rm
dim}(R)=s+1$. By [@as Theorem 3.2] (see also [@cdmm Proposition 3.4]) $R$ is isomorphic to a quantum line, i.e. $R\simeq
R_q(H,v,\chi)$ for some primitive $(s+1)$’th root of unity $q$, an element $v\in G$ and a character $\chi\in G^*$ such that $\chi(v)=q$, and $\chi(h)hv=\chi(h)vh$ for any $h\in G$, i.e. $v\in
Z(G)$ (we use the notation of [@cdmm Section 2]). As an algebra we have $R_q(H,v,\chi)=K[y]/(y^{s+1})$, and the coalgebra structure is such that the elements $d_0=1,d_1=y,
d_2=\frac{y^2}{(2)_q!},\ldots,\frac{y^s}{(s)_q!}$ form a divided power sequence, i.e. $\Delta (d_i)=\sum_{0\leq j\leq i}d_j\otimes
d_{i-j}$ for any $0\leq i\leq s$. The $H$-action on $R_q(H,v,\chi)$ is such that $h\cdot y=\chi(h)y$ for any $h\in G$, and the $H$-coaction is such that $y\mapsto v\otimes y$.\
By [@cdmm Proposition 3.1], there exists a $(1,v)$-skew-primitive $z$ in $C$, which is not in $C_0$, such that $vz=qzv$, $C$ is generated as an algebra by $z$ and $G$, and the class $\hat{z}$ in $\frac{C_1}{C_0}$ corresponds to the element $y\#1$ in $R_q(H,v,\chi)\# H$ via the isomorphism ${\rm gr}\,
C\simeq R_q(H,v,\chi)\# H$. It follows that $v$ must be $g^{-1}$. Since for $h\in G$ both $zh$ and $hz$ are $(h,g^{-1}h)$-skew-primitives, we must have $zh=\lambda hz+\beta
(g^{-1}h-h)$ for some scalars $\lambda$ and $\beta$. But $zhg=(\lambda hz+\beta(g^{-1}h-h))g=q\lambda hgz+\beta (h-hg)$, and on the other hand $zgh=qgzh=q\lambda ghz+q\beta (h-hg)$, showing that $\beta=0$. Thus $zh=\lambda hz$, and passing to ${\rm gr}\, C$, this gives $\hat{z}h=\lambda h\hat{z}$. But in $R_q(H,v,\chi)\# H$ we have that $(1\# h)(y\# 1)=\chi (h) (y\# 1)(1\# h)$, so $\lambda=\chi(h)$. Therefore $zh=\chi (h)hz$. Replace the generator $z$ by $x=gz$, which is a $(g,1)$-skew-primitive. By the quantum binomial formula we see that $\Delta (x^{s+1})=1\otimes
x^{s+1}+x^{s+1}\otimes g^{s+1}$, so then $x^{s+1}=\alpha
(g^{s+1}-1)$ for some scalar $\alpha$. Since $x^{s+1}h=\chi(h)^{s+1}hx^{s+1}$, we see that if $\chi^{s+1}\neq 1$, then $\alpha$ must be zero. Now it is clear that $C\simeq
H_{\infty}(s,q^{-1},G,g,\chi,\alpha)$.
[*Case 2.*]{} If 1 is a vertex in a connected component $D$ of type $K[\CC_1,s]$, with $s\geq 1$, then let $x$ be the arrow from 1 to 1, which is a primitive element, i.e. $\Delta(x)=x\otimes 1+1\otimes
x$. Then $gx\notin C_0$ and $\Delta(gx)=gx\otimes g+g\otimes gx$ for any $g\in G$, so there is an arrow from $gx$ to $gx$. This shows that $C$ must be a direct sum of coalgebras of type $K[\CC_1,s']$ (for possible different values of $s'$). Then looking at $\Delta(x^i)-x^i\otimes 1-1\otimes x^i$, it is easy to show by induction that $x^i$ lies in $D$ for any $i\geq 1$. Since $x$ is a non-zero primitive element, the set $(x^i)_{i\geq 1}$ is linearly independent, a contradiction to the finite dimensionality of $D$. Thus this situation cannot occur.\
If 1 is a vertex in a connected component of type $K[\CC_n,s]$, with $n\geq 2$, the proof goes as in Case 1, and leads us to the conclusion that $C$ is a direct sum of coalgebras isomorphic to $K[\CC_n,s]$, and that $C$ is isomorphic as a Hopf algebra to one of the form $H_n(s,q,G,g,\chi,\alpha)$. The only difference is that instead of using the paths $p_{h,g^ih}$, we deal with paths denoted by $p_{h|l}$, and meaning the path of length $l$ starting at the vertex $h$. Also, since $\chi(g)=q$, a $(s+1)$’th root of unity, and $g^n=1$, $s+1$ must divide $n$.
[*Case 3.*]{} If 1 is a vertex in a connected component of type $K$, then proceeding as in Case 1, we can see that there are no arrows in $C$, so $C$ is a direct sum of copies of $K$. Thus $C$ is a grouplike coalgebra, and Hopf algebra structures on $C$ are just group Hopf algebras.
We note that the above theorem completely classifies finite dimensional Hopf algebras whose underlying algebras are quotients of finite dimensional path algebras by ideals generated by paths, or whose underlying coalgebras are path subcoalgebras. These are the algebras $KG$, $H_n(s,q,G,g,\chi,\alpha)$ and their duals, because a finite dimensional Hopf algebra is Frobenius as an algebra and co-Frobenius as a coalgebra.
Acknowledgment
The research of the first and the third authors was supported by Grant ID-1904, contract 479/13.01.2009 of CNCSIS. For the second author, this work was supported by the strategic grant POSDRU/89/1.5/S/58852, Project “Postdoctoral programe for training scientific researchers” cofinanced by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007-2013.
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[^1]: ${}^*$ corresponding author
|
¶[\_\^(x,)]{} \#1[(\[\#1\])]{} \#1\#2
[ **Huygens’ Principle in Minkowski Spaces and Soliton\
Solutions of the Korteweg-de Vries Equation**]{}\
[**Yuri Yu. Berest**]{} and [**Igor M. Loutsenko**]{}\
[*Université de Montréal, Centre de Recherches Mathématiques,\
C.P. 6128, succ. Centre-Ville, Montreal (Quebec), H3C 3J7, Canada,\
e-mail: [email protected] , [email protected]*]{}
**Abstract**
> A new class of linear second order hyperbolic partial differential operators satisfying Huygens’ principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens’ principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.
[**Mathematics Subject Classification:**]{} 35Q51, 35Q53, 35L05, 35L15, 35Q05.
I. Introduction {#i.-introduction .unnumbered}
===============
The present paper deals with the problem of describing all linear second order partial differential operators for which Huygens’ principle is valid in the sense of “Hadamard’s minor premise”. Originally posed by J.Hadamard in his Yale lectures on hyperbolic equations [@Had], this problem is still far from being completely solved[^1].
The simplest examples of Huygens’ operators are the ordinary wave operators $$\label{1}
\Box_{n+1} = \left( \frac{\partial}{\partial x^{0}}\right)^{2} - \left( \frac{\partial}{\partial x^{1}}\right)^{2} -
\ldots - \left( \frac{\partial}{\partial x^{n}}\right)^{2}$$ in an odd number $ \, n \geq 3\, $ of space dimensions and those ones reduced to (\[1\]) by means of elementary transformations, i.e. by local nondegenerate changes of coordinates $\ x \mapsto f(x)\ $; gauge and conformal transformations of a given operator $ {\cal L} \mapsto \theta(x) \circ {\cal L } \circ \theta(x)^{-1}\ , \ {\cal L} \mapsto
\mu (x) {\cal L} $ with some locally smooth nonzero functions $ \theta(x) $ and $ \mu (x) $. These operators are usually called [*trivial Huygens’ operators*]{}, and the famous “Hadamard’s conjecture” claims that all Huygens’ operators are trivial.
Such a strong assertion turns out to be valid only for (real) Huygens’ operators with a constant principal symbol in $\, n=3\, $ [@Mat]. Stellmacher [@Stell] found the first non-trivial examples of hyperbolic wave-type operators satisfying Huygens’ principle, and thereby disproved Hadamard’s conjecture in higher dimensional Minkowski spaces. Later Lagnese & Stellmacher [@Stell1] extended these examples and even solved [@Lag] Hadamard’s problem for a restricted class of hyperbolic operators, namely $$\label{2}
{\cal L} = \Box_{n+1} + u(x^0)\ ,$$ where $\, u\left( x^0 \right)\, $ is an analytic function (in its domain of definition) depending on a single variable only. It turns out that the potentials $ u(z) $ entering into (\[2\]) are rational functions which can be expressed explicitly in terms of some polynomials[^2] $ {\cal P}_k \left( z \right)$: $$\label{3}
u(z) = 2 \left( \frac{d}{dz}\right)^2 \log {\cal P}_k \left( z \right)\ , \quad k=0,1,2, \ldots ,$$ the latter being defined via the following differential-recurrence relation: $$\label{4}
{\cal P}_{k+1}' {\cal P}_{k-1}- {\cal P}_{k-1}' {\cal P}_{k+1} =
(2 k+1) {\cal P}_{k}^{2}\ , \quad {\cal P}_{0} = 1\ , \ {\cal P}_{1} = z\ .$$ Since the works of Moser et al. [@AM], [@AMM] the potentials (\[3\]) are known as [*rational solutions*]{} of the Korteweg-de Vries equation decreasing at infinity[^3].
A wide class of Huygens’ operators in Minkowski spaces has been discovered recently by Veselov and one of the authors [@BV1], [@BV2] (see also the review article [@BV]). These operators can also be presented in a self-adjoint form $$\label{5}
{\cal L} = \Box_{n+1} + u(x)$$ with a locally analytic potential $ u \left( x \right) $ depending on several variables. More precisely, $ u \left( x \right) $ belongs to the class of so-called [*Calogero-Moser potentials*]{} associated with finite reflection groups ([*Coxeter groups*]{}): $$\label{6}
u(x)= \sum\limits_{\alpha \in \Re_{+}}^{} \frac{m_{\alpha}(m_{\alpha} + 1) (\alpha, \alpha)}
{(\alpha, x)^2}\ .$$ In formula (\[6\]) $ \Re_{+} \equiv \Re_{+} ({\cal G}) $ stands for a properly chosen and oriented subset of normals to reflection hyperplanes of a Coxeter group $ {\cal G}
$. The group $ {\cal G} $ acts on $ {\bf M} ^ {n+1} $ in such a way that the time direction is preserved. The set $ \left\{ m_{\alpha} \right\} $ is a collection of non-negative integer labels attached to the normals $ {\alpha} \in \Re $ so that $
m_{w(\alpha )} = m_{ \alpha } $ for all $ w \in {\cal G} . $ Huygens’ principle holds for (\[5\]), (\[6\]), provided $ n $ is odd, and $$\label{7}
n \geq 3 + 2 \sum\limits_{\alpha \in \Re_{+}}^{} m_{\alpha}\ .$$ In the present work we construct a new class of self-adjoint wave-type operators (\[5\]) satisfying Huygens’ principle in Minkowski spaces. As we will see, this class provides a natural extension of the hierarchy of Huygens’ operators associated to Coxeter groups. On the other hand, it turns out to be related in a surprisingly simple and fundamental way to the theory of solitons.
To present the construction we consider a $(n+1)$-dimensional Minkowski space $ { \bf M }^{ n+1 } \cong { \bf R }^{ 1 , n } $ with the metric signature $ ( + , -, -, \ldots, -) $ and fixed time direction $ \theta \in { \bf M }^{ n+1 } $. We write $ {\bf Gr}_{\perp} (n+1,2) \subset {\bf Gr}(n+1,2) $ for a set of all 2-dimensional space-like linear subspaces in $ { \bf M }^{n+1} $ orthogonal to $ \theta $. Every 2-plane $ E \in {\bf Gr}_{ \perp } (n+1,2) $ is equipped with the usual Euclidean structure induced from $ {\bf M }^{ n+1 } $. To define the potential $ u(x) $ we fix such a plane $ E $ and introduce polar coordinates $ ( r, \varphi ) $ therein.
Let $ (k_{i})^{N}_{i=1} $ be a strictly increasing sequence of integer positive numbers[^4]: $ 0 \leq k_1 < k_2 < \ldots < k_{N-1} < k_N $, and let $ \left\{ \Psi_{i} ( \varphi ) \right\} $ be a set of $ 2 \pi$-periodic functions on $ {\bf R}^{1}$: $$\label{8}
\Psi_{i}(\varphi) := \cos (k_{i} \,\varphi + \varphi_{i})\ , \quad \varphi_{i} \in {\bf R}\ ,$$ associated to $ (k_{i}) $. The Wronskian of this set $$\label{9}
{\cal W}\left[\Psi_{1}, \Psi_{2}, \ldots , \Psi_{N} \right] := \det
\left(
\begin{array}{cccc}
\Psi_{1}(\varphi) & \Psi_{2} (\varphi) & \ldots & \Psi_{N} (\varphi)\\
\Psi_{1}'(\varphi) & \Psi_{2}'(\varphi) & \ldots & \Psi_{N}'(\varphi) \\
\vdots & \vdots & \ddots & \vdots \\
\Psi_{1}^{(N-1)} (\varphi) & \Psi_{2}^{(N-1)}(\varphi) & \ldots &
\Psi_{N}^{(N-1)}(\varphi)
\end{array}
\right)$$ does not vanish indentically since $ \Psi_{i}(\varphi) $ are linearly independent.
Let $$\Xi := \left\{ x \in \M \ | \ r^{|k|}\, {\cal W}\left[\Psi_{1}, \Psi_{2}, \ldots , \Psi_{N} \right] = 0 \ ,
\ |k| := \sum_{i=1}^{N} k_{i} \right\}$$ be an algebraic hypersurface of zeros of the Wronskian in the Minkowski space $ \M $, and let $ \Omega \subset \M \setminus \Xi $ be an open connected part in its complement.
We define $ u(x) $ in terms of cylindrical coordinates in $ \M $ with polar components in $ E $ : [10]{} u = u\_[k]{}(x) := - ( )\^2 . It is easy to see that in a standard Minkowskian coordinate chart $ u(x) $ is a real [*rational*]{} function on $ \M $ having its singularities on $ \Xi $. In particular, it is locally analytic in $ \Omega $.
Our main result reads as follows.
[**Theorem.**]{} [*Let $\ \M \cong {\bf R}^{1,n} $ be a Minkowski space, and let* ]{} [11]{} [L]{}\_[(k)]{} := \_[n+1]{} + u\_[k]{}(x) [*be a wave-type second order hyperbolic operator with the potential associated to an arbitrary strictly monotonic partition $ (k_i) $ of height $ N $ : $$0 \leq k_1< k_2 < \ldots < k_{N}\ , \quad k_{i} \in {\bf Z}\ ,\quad i=1,2, \ldots, N\ .$$ Then operator $ {\cal L}_{(k)} $ satisfies Huygens’ principle at every point $ \xi \in \Omega $, provided $ n $ is odd, and* ]{} [12]{} n 2 k\_[N]{} + 3 .
[*Remark I.*]{} A similar result is also valid if one takes an arbitrary Lorentzian 2-plane $ H \in {\bf Gr}_{\parallel}(n+1,2) $ in the Minkowski space $ \M $ containing the time-like vector $ \theta $. More precisely, in this case the potential $ u_{k}(x) $ associated to the partition $ (k_{i}) $ is introduced in terms of pseudo-polar coordinates $ (\varrho,\vartheta) $ in $ H $: [13]{} u\_[k]{}(x) := - ( )\^2 , where $ x^{0} = \varrho \sinh \vartheta\,$, and, say, $\, x^{1} = \varrho \cosh \vartheta\, $. The functions $ \psi_{i} $ involved in are given by [14]{} \_[i]{} = (k\_[i]{} + \_[i]{}) , \_[i]{} . The theorem formulated above holds when the potential is replaced by .
[*Remark II.*]{} The potentials considerably extend the class of Calogero-Moser potentials related to Coxeter groups of rank $ 2 $. Indeed, in $ {\bf R}^{2} $ any Coxeter group $ {\cal G} $ is a dihedral group $ I_{2}(q) $, i.e. the group of symmetries of a regular $2q$-polygon. It has one or two conjugacy classes of reflections according as $ q $ is odd or even. The corresponding potential can be rewritten in terms of polar coordinates as follows (see [@OP]): $$u(r,\varphi) = \frac{m(m+1)q^2}{r^2 \sin^{2}(q\,\varphi)} \ ,\quad \mbox{\rm when}\ q \ \mbox{\rm odd} \ ,$$ and $$u(r,\varphi) = \frac{m(m+1)\left(q/2\right)^2}{r^2 \sin^{2} (q/2)\,\varphi} +
\frac{m_{1}(m_{1}+1) \left(q/2 \right)^2}{r^2 \cos^{2} (q/2)\,\varphi}
\ , \quad \mbox{\rm when}\ q \ \mbox{\rm even}\ .$$ It is easy to verify that formula boils down to these forms if we fix $\, N := m \, ; \,
\varphi_{i} := (-1)^{i} \pi /2 \, ,\, i =1,2, \ldots, N\, $, and choose $$k := (q, \, 2q, \, 3q, \, \ldots ,\, mq ) \ ,$$ when $ q $ is odd, and $$k := (\frac{q}{2} , \, q, \, \frac{3q}{2}, \, \ldots ,\, (m-m_1)\frac{q}{2} , \, q+ (m-m_1)\frac{q}{2} , \,
2q+ (m-m_1)\frac{q}{2}, \, \ldots\, ,\, (m+m_1)\frac{q}{2} )\ ,$$ when $ q $ is even and $\, m > m_{1} \,$, respectively.
[*Remark III.*]{} Let us set $ \varphi_{i} = 4 k_{i}^{3} t + \varphi_{0i} $ and $ \vartheta_{i} = - 4 k_{i}^{3} t + \vartheta_{0i} \ ,
\ i=1,2, \ldots, N \ ;\ \varphi_{0i} , \vartheta_{0i} \in {\bf R}. $ The angular parts of potentials , , i.e. [15]{} v() = -2 ( )\^2 , [16]{} v() = -2 ( )\^2 , are known (see, e.g., [@Deift], [@MS]) to be respectively singular [*periodic*]{} and proper [*N-soliton*]{} solutions of the Korteweg-de Vries equation [17]{} v\_[t]{} = - v\_ + 6 v v\_ . It is also well-known that $N$-soliton potentials constitute the whole class of so-called [*reflectionless*]{} real potentials for the one-dimensional Schrödinger operator $ L = - \partial^2 / \partial \vartheta^2 + v(\vartheta) $ (see, e.g., [@AC]).
In conclusion of this section we put forward the following conjecture[^5].
[**Conjecture.**]{} The wave-type operators with potentials of the form give [*a complete*]{} solution of Hadamard’s problem in Minkowski spaces $ \M $ within a restricted class of linear second order hyperbolic operators $${\cal L} = \left( \frac{\partial}{\partial x^{0}}\right)^{2} - \left( \frac{\partial}{\partial x^{1}}\right)^{2}
- \left( \frac{\partial}{\partial x^{2}}\right)^{2} -
\ldots - \left( \frac{\partial}{\partial x^{n}}\right)^{2} + u(x^1, x^2)$$ with real locally analytic potentials $ u = u(x^1, x^2) $ depending on [*two*]{} spatial variables and homogeneous of degree $ (- 2) $: $\, u(\alpha x^1, \alpha x^2) = \alpha^{-2} u(x^1, x^2)\,,\, \alpha >0\,$.
One of the authors (Yu. B.) is grateful to Prof. A. Veselov (Loughborough University, UK) and his collaborators Dr. O. Chalykh and M. Feigin who have kindly informed him [@Ves] about their recent results showing the existence of new algebraically integrable Schrödinger operators which are not related to Coxeter root systems. In fact, this observation was a motivation for the present work. We would like also to thank Prof. P. Winternitz (CRM, Université de Montréal) for his encouragement and highly stimulating discussions.
The work of Yu. B. was partially supported by the fellowship from Institut des Sciences Mathématiques (Montréal) which is gratefully acknowledged. The second author (I. L.) is grateful to Prof. L. Vinet for his support.
II. Huygens’ principle and Hadamard-Riesz expansions {#ii.-huygens-principle-and-hadamard-riesz-expansions .unnumbered}
====================================================
The proof of the theorem stated above rests heavily on the Hadamard theory of Cauchy’s problem for linear second order hyperbolic partial differential equations. Here, we summarize briefly some necessary results from this theory following essentially M. Riesz’s approach [@Riesz] (see also [@Fri], [@Gun]).
Let $ \M \cong {\bf R}^{1,n} $ be a Minkowski space, and let $ \Omega $ be an open connected part in $ \M $. We consider a (formally) self-adjoint scalar wave-type operator [18]{} [L]{} = \_[n+1]{} + u(x) , defined in $ \Omega $, the scalar field (potential) $ u(x) $ being assumed to be in $ {\cal C}^{\infty}(\Omega) $. For any $ \xi \in \Omega $, we define a cone of isotropic (null) vectors in $ \M $ with its vertex at $ \xi $: [19]{} (x,) := (x\^0- \^0)\^[2]{} - (x\^1- \^1)\^[2]{} - …- (x\^[n]{} - \^[n]{})\^[2]{} = 0 , and single out the following sets : [20]{}
[lcr]{} C\_() & := & { x | (x,) = 0, \^0 x\^[0]{} } ,\
J\_() & := & { x | (x,) > 0, \^0 x\^[0]{} } .
[**Definition.**]{} [*A (forward) Riesz kernel*]{} of operator $ {\cal L} $ is a holomorphic (entire analytic) mapping $ \lambda \mapsto \Phi_{\lambda}^{\Omega}(x,\xi)\,,\,
\lambda \in \mbox{\bf C} $, with values in the space of distributions[^6] $ {\cal D}'(\Omega) $, such that for any $ \xi \in \Omega $: [21]{}
[lcl]{} & (i) & \_\^(x,) ,\
& (ii) & [L]{}= \_[-1]{}\^(x,) ,\
& (iii) & \_[0]{}\^(x,) = (x -) .
The value of the Riesz kernel $\ \Phi_{1}^{\Omega}(x,\xi) := \Phi_{+}(x,\xi)\ $ at $ \lambda = 1 $ is called [*a (forward) fundamental solution*]{} of the operator $ {\cal L} $: $$\label{22}
{\cal L} [\Phi_{+}(x, \xi)] = \delta(x-\xi) \ , \quad \mbox{supp}\ \Phi_{+}(x, \xi) \subseteq \overline{J_{+}(\xi)} \ .$$ Such a solution is known to exist for any $ u(x) \in {\cal C}^{\infty}(\Omega) $, and it is uniquely determined.
[**Definition.**]{} The operator $ {\cal L} $ defined by satisfies [*Huygens’ principle*]{} in a domain $ \Omega_{0} \subseteq \Omega $ in $ \M $ if $$\label{23}
\mbox{\rm supp} \ \Phi_{+}(x, \xi) \subseteq \overline{C_{+}(\xi)} = \partial J_{+}(\xi) \ .$$ for every point $ \xi \in \Omega_{0} $.
The analytic description of singularities of Riesz kernel distributions (and, in particular, fundamental solutions) for second order hyperbolic differential operators is given in terms of their asymptotic expansions in the vicinity of the characteristic cone by a graded scale of distributions with weaker and weaker singularities. Such “asymptotics in smoothness”, usually called [*Hadamard-Riesz expansions*]{}, turn out to be very important for testing Huygens’ principle for the operators under consideration.
In order to construct an appropriate scale of distributions ([*Riesz convolution algebra*]{}) in Minkowski space $ \M $ we consider (for a fixed $ \xi \in \M $) a holomorphic ${\cal D}'$-valued mapping $\ {\bf C} \to {\cal D}'(\M)\ , \
\lambda \mapsto R_{\lambda}(x,\xi) $, such that $ R_{\lambda}(x,\xi) $ is an analytic continuation (in $ \lambda $) of the following (regular) distribution: [24]{} R\_(x,), g(x) = \_[J\_[+]{}()]{}\^ g(x) dx , > , where $ dx = dx^0 \wedge dx^1 \wedge \ldots \wedge dx^n $ is a volume form in $ \M $, $ g(x) \in {\cal D}(\M) $, and $ H_{n+1}(\lambda) $ is a constant given by [25]{} H\_[n+1]{}() = 2 \^ 4\^[- 1]{} () (- (n-1)/2) .
The following properties of this family of distributions are deduced directly from their definition.
For all $ \lambda \in {\bf C} $ and $ \xi \in \M $ we have [26]{} R\_(x, ) [27]{} \_[n+1]{} R\_ = R\_[- 1]{} , [28]{} R\_ \* R\_ = R\_[+ ]{} , , [29]{} (x - , \_[x]{}) R\_ = (2 - n + 1) R\_ , [30]{} \^ R\_ = 4\^ ()\_ (- (n-1)/2)\_ R\_[+ ]{} ,\_[0]{} , where $ (\kappa)_{\nu} := \Gamma(\kappa + \nu) / \Gamma(\kappa) $ is Pochhammer’s symbol, and $ \gamma = \gamma(x,\xi) $ is a square of the geodesic distance between $ x $ and $ \xi $ in $ \M $.
In addition, when $ n $ is odd, one can prove that [31]{} R\_(x, ) = = 1,2, …, (n-1)/2 , where $ \delta_{+}^{(m)} (\gamma) $ stands for the $m$-th derivative of Dirac’s delta-measure concentrated on the surface of the future-directed characteristic half-cone $ \overline{C_{+}(\xi)} $.
Another important property of Riesz distributions is that [32]{} R\_[0]{}(x, ) = (x - ) . Formulas , , show that $ R_{\lambda}(x,\xi) $ is a Riesz kernel for the ordinary wave operator $ \Box_{n+1} $. The property means precisely that in even-dimensional Minkowski spaces $ \M $ ($ n $ is odd) Huygens’ principle holds for sufficiently low powers of the wave operator $ \Box^{d} \,,\, d \leq (n-1)/2 $.
Now we are able to construct the Hadamard-Riesz expansion for the Riesz kernel of a general self-adjoint wave-type operator on $ \M $.
First, we have to find a sequence of two-point smooth functions $\ U_{\nu} := U_{\nu}(x, \xi) \in \
{\cal C}^{\infty}(\Omega \times \Omega)\ , \ \nu = 0,1,2 \ldots $, as a solution of the following [*transport equations*]{}: [33]{} (x-, \_[x]{} ) U\_(x,) + U\_(x, ) = - [L]{} , 1 . It is well-known (essentially due to [@Had]) that the differential-recurrence system has a [*unique*]{} solution provided each $ U_{\nu} $ is required to be bounded in the vicinity of the vertex of the characteristic cone and $ U_{0}(x,\xi) $ is fixed for a normalization, i.e. $$U_{0}(x,\xi) \equiv 1 \ , \qquad U_{\nu}(\xi,\xi) \sim {\cal O}(1)\ ,\quad \forall\, \nu = 1,2,3, \ldots$$ These functions $ U_{\nu} $ are called [*Hadamard’s coefficients*]{} of the operator $ {\cal L}.$
In terms of $ U_{\nu} $ the required asymptotic expansion can be presented as follows: [34]{} \_\^(x,) \~\_[=0]{}\^ 4\^ ()\_ U\_(x,) R\_[+ ]{}(x,) . One can prove that for a hyperbolic differential operator $ {\cal L} $ with locally analytic coefficients the Hadamard-Riesz expansion is locally uniformly convergent. From now on we will restrict our consideration to this case.
For $ \lambda =1 $ formula provides an expansion of the fundamental solution of the operator $ {\cal L} $ in a neighborhood of the vertex $ x =\xi $ of the characteristic cone: [35]{} \_[+]{}(x,) = \_[=0]{}\^ 4\^ ! U\_(x,) R\_[+1]{}(x,) . When $ n $ is even, we have $\ \mbox{supp}\ R_{\nu+1}(x, \xi) = \overline{J_{+}(\xi)}\ $ for all $\, \nu = 0,1,2, \ldots $, and therefore Huygens’ principle never occurs in odd-dimensional Minkowski spaces $ {\bf M}^{2l+1} $.
On the other hand, in the case of an odd number of space dimensions $ n \geq 3 $, we know due to that for $ \nu = 0, 1, 2, \ldots, (n-3)/2 , \ \mbox{supp}\ R_{\nu+1}(x, \xi) = \overline{C_{+}(\xi)}\ $. Hence, using , we can rewrite the series in following form: [36]{} \_[+]{}(x,) = ( V(x,) \_[+]{}\^[(p-1)]{}() + W(x,) \_[+]{}() ) , where $ p := (n-1)/2 \ ,\ \eta_{+}(\gamma) $ is a regular distribution characteristic for the region $ J_{+}(\xi) $: $$\langle \eta_{+}(\gamma), g(x) \rangle = \int\limits_{J_{+}(\xi)}^{} g(x)\, dx\ , \quad g(x) \in {\cal D}(\M)\ ,$$ and $ V(x,\xi) $ , $ W(x,\xi) $ are analytic functions in a neighborhood of the vertex $ x=\xi $ which admit the following expansions therein: [37]{} V(x,) = \_[=0]{}\^[p-1]{} U\_(x,) \^ , [38]{} W(x,) = \_[=p]{}\^ U\_(x,) \^[- p]{} , p = . The function $ W(x,\xi) $ is usually called [*a logarithmic term*]{} of the fundamental solution[^7].
It follows directly from the representation formula that operator $ {\cal L } $ satisfies Huygens’ principle in a neighborhood of the point $ \xi $, if and only if, the logarithmic term $ W(x,\xi) $ of its fundamental solution vanishes in this neighborhood identically in $ x $: $ W(x,\xi) \equiv 0\, $.
The function $ W(x,\xi) $ is known to be a regular solution of the characteristic Goursat problem for the operator $ {\cal L}\, $: [39]{} [L]{} = 0 with a boundary value given on the cone surface $ \overline{C_{+}(\xi)} \,$. Such a boundary problem has a unique solution, and hence, the necessary and sufficient condition for $ {\cal L } $ to be Huygens’ operator becomes [40]{} W(x,) 0 , where the symbol $ \triangleq $ implies that the equation in hand is satisfied only on $ \overline{C_{+}(\xi)} $. By definition , the latter condition is equivalent to the following one [41]{} U\_[p]{}(x,) 0 ,p = . In this way, we arrive at the important criterion for the validity of Huygens’ principle in terms of coefficients of the Hadamard-Riesz expansion . Equation is essentially due to Hadamard [@Had]. It will play a central role in the proof of our main theorem.
III. Proof of the main theorem {#iii.-proof-of-the-main-theorem .unnumbered}
==============================
We start with some remarks concerning the properties of the one-dimensional Schrödinger operator [42]{} L\_[(k)]{} := -( )\^2 + v\_[k]{}() with a general periodic soliton potential [43]{} v\_[k]{}() := - 2 ( )\^2 . Here, as already discussed in the Introduction, $ {\cal W}\left[\Psi_{1}, \Psi_{2}, \ldots , \Psi_{N} \right] $ stands for a Wronskian of the set of periodic functions on $ {\bf R}^{1} $: [44]{} \_[i]{}() := (k\_[i]{} + \_[i]{}) , \_[i]{} , associated to an arbitrary strictly monotonic sequence of real positive numbers (“soliton amplitudes”): $\ 0 \leq k_1< \ldots < k_{N-1}<k_{N}\, $.
It is well-known (see, e.g., [@MS]) that any such operator $ L_{(k)} $ (as well as its proper solitonic counterpart ) can be constructed by a successive application of [*Darboux-Crum factorization transformations*]{} ([@Darb], [@Crum]) to the Schrödinger operator with the identically zero potential: [45]{} L\_[0]{} := -( )\^2 . To be precise, let $ L $ be a second order ordinary differential operator with a sufficiently smooth potential: [46]{} L := -( )\^2 + v() . We ask for formal factorizations of the operator [47]{} L - I = A\^[\*]{}A , where $ I $ is an identity operator, $\, \lambda \,$ is a (real) constant, and $\, A\, , \, A^{*}\, $ are the first order operators adjoint to each other in a formal sense.
According to Frobenius’ theorem (see, e.g., [@Ince]), the most general factorization is obtained if we take $ \chi(\varphi) $ as a generic element in $\, \mbox{Ker}(L - \lambda \, I )\setminus\left\{ 0 \right\}\, $ and set [48]{} A := ( ) \^[-1]{}, A\^[\*]{} := - \^[-1]{} ( ) . Indeed, $\, A^{*}\circ A\, $ is obviously self-adjoint second order operator with the principal part $\, - \partial^2 / \partial \varphi^2\, $. Hence, it is of the form . Moreover, since $\, A[\chi] = 0 \,$, we have $\, \chi \in \mbox{Ker}\, A^{*}\circ A \,$, so that becomes evident.
Note that for every $\, \lambda \in {\bf R}\, $ we actually get a one-parameter family of factorizations of $\, L - \lambda \, I \,$. This follows from the fact that $\, \dim\, \mbox{Ker}(L - \lambda \, I ) = 2 $, whereas $ \chi(\varphi) $ and $\, C\,\chi(\varphi) $ give rise to the same factorization pair $ (A, A^{*})\, $.
By definition, the Darboux-Crum transformation maps an operator $\, L = \lambda \, I + A^{*}\circ A \,$ into the operator [49]{} := I + A A\^[\*]{} , in which $ A $ and $ A^{*} $ are interchanged. The operator $ \tilde{L} $ is also a (formally) self-adjoint second-order differential operator [50]{} := -( )\^2 + () , where $ \tilde{v}(\varphi) $ is given explicitly by [51]{} () = v() - 2 ( )\^2 () . The initial operator $ L $ and its Darboux-Crum transform $ \tilde{L} $ are obviously related to each other via the following intertwining indentities: [52]{} A = A L , L A\^[\*]{} = A\^[\*]{} . The Darboux-Crum transformation has a lot of important applications in the spectral theory of Sturm-Liouville operators and related problems of quantum mechanics [@IH]. In particular, it is used to insert or remove one eigenvalue without changing the rest of the spectrum of a Schrödinger operator (for details see the monograph [@MS] and references therein).
The explicit construction of the family of operators with periodic soliton potentials is based on the following Crum’s lemma:
[**Lemma**]{}([@Crum]). Let $ L $ be a given second order Sturm-Liouville operator with a sufficiently smooth potential, and let $\, \left\{ \Psi_{1}, \Psi_{2}, \ldots , \Psi_{N} \right\}\,$ be its eigenfunctions corresponding to arbitrarily fixed pairwise different eigenvalues $\, \left\{ \lambda_{1}, \lambda_{2}, \ldots , \lambda_{N} \right\}\,$, i.e. $ \,
\Psi_{i} \in \mbox{Ker} (L - \lambda_{i}\,I ) \, , \, i =1,2, \ldots , N \, $. Then, for arbitrary $\, \Psi \in \mbox{Ker} (L - \lambda\,I ) \, ,\, \lambda \in {\bf R}\,$, the function [53]{} \_[N]{} () := satisfies the differential equation [54]{} \_[N]{} () = \_[N]{} () with the potential [55]{} v\_[N]{}() := v() - 2 ( )\^2 .
Given a sequence of real positive numbers $ \, (k_{i})^{N}_{i=1} $: $\, 0 \leq k_1 < k_2 < \ldots < k_{N}\,$, the Darboux-Crum factorization scheme: [56]{} L\_[i]{} := A\_[i-1]{} A\^[\*]{}\_[i-1]{} + k\_[i]{}\^[2]{}I = A\^[\*]{}\_[i]{} A\_[i]{} + k\_[i+1]{}\^[2]{}I L\_[i+1]{} := A\_[i]{} A\^[\*]{}\_[i]{} + k\_[i+1]{}\^[2]{}I , starting from the Schrödinger operator with a zero potential $$L_{0} \equiv -\left( \frac{\partial}{\partial \varphi}\right)^2 = A^{*}_{0} \circ A_{0} + k_{1}^{2}\,I\ ,$$ produces the required operator $ L_{(k)} \equiv L_{N} $ with the general periodic potential .
Now we proceed to the proof of our main theorem formulated in the Introduction.
When $\, N=0\, $, the statement of the theorem is evident, since the operator $ {\cal L}_{0} $ is just the ordinary wave operator in an odd number $ n $ of spatial variables.
Using the Darboux-Crum scheme as outlined above we will carry out the proof by induction in $ N $.
Suppose that the statement of the theorem is valid for all $ m= 0, 1, 2, \ldots, N $. Consider an arbitrary [*integer*]{} monotonic partition $ (k_i) $ of height $ N $ : $\, 0<k_1< k_2 < \ldots < k_{N}\, , \
k_{i} \in {\bf Z} $.
By our assumption, the wave-type operator [57]{} [L]{}\_[N]{} := [L]{}\_[(k)]{} = \_[n+1]{} + u\_[k]{}(x) , associated to this partition, satisfies Huygens’ principle in the $(n+1)$-dimensional Minkowski space $ \M $ with $ n $ odd, and $\, n \geq 2 \, k_{N} + 3 \,$. We fix the minimal admissible number of space variables, i.e. $ \, n = 2 \, k_{N} + 3 \, $, and denote [58]{} p:= = k\_[N]{} + 1 . By construction, the operator $ {\cal L}_{N} $ can be written explicitly in terms of suitably chosen cylindrical coordinates in $ \M $: [59]{} [L]{}\_[N]{} = \_[n-1]{} - , where $ (r, \varphi) $ are the polar coordinates in some Euclidean $2$-plane $ E $ orthogonal to the time direction in $ \M $, i.e. $\, E \in {\bf Gr}_{ \perp} (n+1,2)\,$; $\, \Box_{n-1}\, $ is a wave operator in the orthogonal complement $ \, E^{\perp} \cong {\bf M}^{n-1} \, $ of $ E $ in $ \M $; and $ v_{N}(\varphi) $ is a $ 2\pi$-periodic potential given by .
Let $ k := k_{N+1} $ be an arbitrary positive integer such that [60]{} k > k\_[N]{} . We apply the Darboux-Crum transformation with the spectral parameter $ k $ to the angular part of the Laplacian in $ E $. For this we rewrite $ {\cal L}_{N} $ in the form [61]{} [L]{}\_[N]{} = \_[n-1]{} - , and set [62]{} [L]{}\_[N+1]{} := \_[n-1]{} - , where $\, A_{N} := A_{N}(\varphi) \,$ and $\, A^{*}_{N} := A^{*}_{N}(\varphi) \,$ are the first order ordinary differential operators of the form .
According to , we have [63]{} [L]{}\_[N+1]{} A\_[N]{} = A\_[N]{} \_[N]{} , \_[N]{} A\_[N]{}\^[\*]{} = A\_[N]{}\^[\*]{} \_[N+1]{} . Let $\, \Phi_{\lambda}^{N}(x,\xi)\, $ and $\, \Phi_{\lambda}^{N+1}(x,\xi)\, $ be the Riesz kernels of hyperbolic operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1} $ respectively. Then, by virtue of we must have the relation [64]{} A\^[\*]{}\_[N]{}() - A\_[N]{}() = 0 , where $ \, A_{N}(\phi) \,$ is the differential operator $\, A_{N} \,$ written in terms of the variable $ \phi $ conjugated to $\, \varphi\, $. Indeed, if identity were not valid, one could define a holomorphic mapping $ \tilde{\Phi}^{N}\,:\, {\bf C} \to {\cal D}',\, \lambda \mapsto \tilde{\Phi}_{\lambda}^{N}(x,\xi)\,$, such that [65]{} \_\^[N]{}(x,) := \_\^[N]{}(x,) + a ( A\^[\*]{}\_[N]{}() - A\_[N]{}() ) . The distribution $\,\tilde{\Phi}_{\lambda}^{N}(x,\xi) \,$, depending on an arbitrary complex parameter $ a \in {\bf C} $, would also satisfy all the axioms in the definition of a Riesz kernel for the operator $ {\cal L}_{N} $. In this way, we would arrive at the contradiction with the uniqueness of such a kernel.
In particular, when $ \lambda = 1 $, the identity gives the relation between the fundamental solutions $\, {\Phi}_{+}^{N}(x,\xi) \equiv {\Phi}_{1}^{N}(x,\xi) \, $ and $\, {\Phi}_{+}^{N+1}(x,\xi) \equiv {\Phi}_{1}^{N+1}(x,\xi) \, $ of operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1} \,$. In accordance with , we have [66]{} \^[N]{}\_[+]{}(x,) = ( V\_[N]{}(x,) \_[+]{}\^[(p-1)]{}() + W\_[N]{}(x,) \_[+]{}() ) and [67]{} \^[N+1]{}\_[+]{}(x,) = ( V\_[N+1]{}(x,) \_[+]{}\^[(p-1)]{}() + W\_[N+1]{}(x,) \_[+]{}() ) , where $ \gamma $ is a square of the geodesic distance between the points $ x $ and $ \xi $ in $ \M $. Substituting , into , we get the relation between the logarithmic terms $ \, W_{N}(x,\xi)\, $ and $\, W_{N+1}(x,\xi)\, $ of operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1}\, $ [68]{} A\^[\*]{}\_[N]{}() - A\_[N]{}() = 0 . By our assumption, $ {\cal L}_{N} $ is a Huygens’ operator in $ \M $, so that $\, W_{N}(x,\xi) \equiv 0 $. Hence, equation implies $\, A^{*}_{N}(\varphi) \, \left[ W_{N+1}(x,\xi) \right] = 0 \, $. On the other hand, as discussed in Sect. II, the logarithmic term $\, W_{N+1}(x,\xi) \,$ is a regular solution of the characteristic Goursat problem for $ {\cal L}_{N+1}\, $, i.e. in particular, [69]{} [L]{}\_[N+1]{} = 0 . Taking into account definition of the operator $ {\cal L}_{N+1}\, $, we arrive at the following equation for $\, W_{N+1}(x,\xi) \,$: [70]{} \_[n-1]{} W\_[N+1]{}(x,) = ( ()\^[2]{} + - )W\_[N+1]{}(x,) . According to , the logarithmic term $ W_{N+1} $ admits the following expansion [71]{} W\_[N+1]{}(x,) = \_[=p]{}\^ U\_(x,) , p = , where $ \, U_{\nu}(x,\xi)\, $ are the Hadamard coefficients of the operator $ \, {\cal L}_{N+1}\,$. Since the potential of the wave-type operator $ \, {\cal L}_{N+1}\,$ depends only on the variables $\, r, \varphi \,$, its Hadamard coefficients $\, U_{\nu}\, $ must depend on the same variables $\, r, \varphi \, $ and their conjugates $\, \rho, \phi \,$ only: [72]{} U\_ = U\_(r, , , ) = 0, 1, 2, … This follows immediately from the uniqueness of solution of Hadamard’s transport equations .
On the other hand, since [73]{} = s\^2 - r\^2 - \^2 + 2 r (- ) , where $ s $ is a geodesic distance in the space $ \, E^{\perp} \cong {\bf M}^{n-1} \, $ orthogonally complementary to the $2$-plane $ E $, we conclude that $ W_{N+1} $ is actually a function of five variables: $ \, W_{N+1} = W_{N+1}(s, r, \rho, \varphi, \phi )\, $. On the space of such functions the wave operator $\, \Box_{n-1} \, $ in $ E^{\perp} $ acts in the same way as its “radial part”, i.e. $$\Box_{n-1} W_{N+1} =
\left( \left(\frac{\partial}{\partial s}\right)^{2} + \frac{n-2}{s} \, \frac{\partial}{\partial s} \right)\,W_{N+1} \ .$$ Hence, equation becomes [74]{} ( ()\^[2]{} - ()\^[2]{} - + - )W\_[N+1]{} = 0 . Now we substitute the expansion [75]{} W\_[N+1]{} = \_[=p]{}\^ U\_(r, , , ) , p = , into the left-hand side of the latter equation and develop the result into the similar power series in $ \gamma \,$, taking into account formula . After simple calculations we obtain [76]{} \_[=p]{}\^ = 0 , $$ where the prime means differentiation with respect to $ r \,$.
Since the functions $ U_{\nu} $ do not depend explicitly on $ \gamma \,$, equation can be satisfied only if each coefficient under the powers of $ \gamma $ vanishes separately. In this way we arrive at the following differential-recurrence relation for the Hadamard coefficients of the operator $ {\cal L}_{N+1}\,$: $$4\,\rho^{2}\,\sin^{2}(\varphi -\phi)\,U_{\nu+2} = \left( U_{\nu}'' + \frac{1}{r}\, U_{\nu}' - \frac{k^2}{r^2} U_{\nu} \right) +$$ [77]{} + (-) ( 2r U\_[+1]{}’ + U\_[+1]{} ) - 4 ( r U\_[+1]{}’ + (+1)U\_[+1]{} ) , where $ \nu $ runs from $ p\, $: $\, \nu = p,\, p+1,\, p+2, \ldots $
To get a further simplification of equation we notice that all the Hadamard coefficients of the operators under consideration , are homogeneous functions of appropriate degrees. More precisely, they have the following specific form [78]{} U\_ (r, , , ) = \_ (, ) , = 0, 1, 2, … , where $\, \sigma_{\nu} (\varphi, \phi) = \sigma_{\nu} (\phi, \varphi) \, $ are symmetric $2\pi$-periodic functions depending on the angular variables only.
In order to prove Ansatz we have to go back to the relation between the Riesz kernels of operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1}\,$: [79]{} A\^[\*]{}\_[N]{}() - A\_[N]{}() = 0 , , If we substitute the Hadamard-Riesz expansions of the kernels $\, \Phi_{\lambda}^{N} (x,\xi)\, $ and $\, \Phi_{\lambda}^{N+1} (x,\xi)\, $ into directly and take into account that $ A_{N} $ and its adjoint $ A^{*}_{N} $ are the first order ordinary differential operators of the following form (cf. ): [80]{} A\_[N]{}() = - f\_[N]{}() , A\_[N]{}\^[\*]{}() = - - f\_[N]{}() , where $\, f_{N}(\varphi) = (\partial/\partial \varphi) \log \chi_{N}(\varphi)\,$, we obtain $$\sum\limits_{\nu = 0}^{\infty}\, 4^{\nu}\, (\lambda)_{\nu}\, \biggl[ 2 r \rho\,\sin(\varphi - \phi)\,
\left( U^{N+1}_{\nu+1} - U^{N}_{\nu+1} \right) -$$ [81]{} - ( + f\_[N]{}()) U\^[N+1]{}\_ - ( - f\_[N]{}()) U\^[N]{}\_ \] R\_[+ ]{} = 0 , where $\, U_{\nu}^{N}(r, \varphi, \rho, \phi)\,$ and $\, U_{\nu}^{N+1}(r, \varphi, \rho, \phi)\,$ are the Hadamard coefficients of operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1} $ respectively; $\, R_{\lambda} :=
R_{\lambda}(x,\xi) $ is the family of Riesz distributions in $ \M \,$.
The same argument as above (see the remark before formula ) shows that all the coefficients of the series under the Riesz distributions of different weights must vanish separately. So we arrive at the recurrence relation between the sequences of Hadamard’s coefficients of operators $ {\cal L}_{N} $ and $ {\cal L}_{N+1}\,$: [82]{} U\^[N+1]{}\_[+1]{} = U\^[N]{}\_[+1]{} + , where $ \, U^{N+1}_{0} = U^{N}_{0} \equiv 1\, $ and $ \nu = 0, 1, 2, \dots $ Now it is easy to conclude from by induction in $ N $ that the Ansatz really holds for Hadamard’s coefficients of all wave-type operators with potentials .
Returning to equation and substituting therein, we obtain the following three-term recurrence relation for the angular functions $ \sigma_{\nu}(\varphi, \phi)\,$: [83]{} 4\^[2]{}(- ) \_[+2]{} = (\^2 - k\^2)\_ - 2 (2+1) (- )\_[+1]{} , where $\, \nu = p,\,p+1,\, p+2, \ldots \,$.
In order to analyze equation it is convenient to introduce a formal generating function for the quantities $ \{\, \sigma_{\nu}\,\}\,$: [84]{} F(t) := \_[= p]{}\^ \_(, ) .
The recurrence relation turns out to be equivalent to the classical hypergeometric differential equation for the function $ F(t) $ [85]{} ( 4(1-\^[2]{}) + 4 t - t\^2 ) + (2 p + 1)(2 - t ) + (k\^2 - p\^2)F = 0 , where $ \omega := \cos(\varphi - \phi)\, $. The general solution to is given in terms of Gauss’ hypergeometric series: [86]{} F(t) = C \_[2]{}[**F**]{}\_[1]{} (p-k; p+k; p+1/2 | z) + C\_[1]{} z\^[-p+1/2]{}\_[2]{}[**F**]{}\_[1]{} (1/2-k; 1/2+k; 3/2 - p | z) , where $ z := (t- 2 \omega +2 )/4 $ and $\, {}_{2}\,{\bf F}_{1}\,$ is defined by [87]{} \_[2]{}[**F**]{}\_[1]{} (a; b; c | z ) := \_[=0]{}\^ .
As discussed in Sect.II, the Hadamard coefficients $ U_{\nu}(x,\xi) $ must be regular in a neighborhood of the vertex of the characteristic cone $ x = \xi\, $. When $\, x \to \xi \,$, we have $\, \omega \to 1\, $ and $\, U_{p}(\xi,\xi) \propto
\sigma_{p}(\phi, \phi) = F(0) |_{\omega =1}\, $ is not bounded unless $\, C_{1} = 0\, $.
In this way, setting $\, C_{1} = 0\, $ in , we obtain [88]{} \_[= p]{}\^ \_(, ) = C \_[2]{}[**F**]{}\_[1]{} ( p-k; p+k; p+1/2 | (t- 2 + 2 )/4 ) . Now it remains to recall that by our assumption $\, k \in {\bf Z}\,$ and $\, k > k_{N} \,$. Since $\,
p = (n-1)/2 = k_{N} +1\,$, we have $\, k \geq p\, $. So the hypergeometric series in the right-hand side of equation is truncated. In fact, the generating function is expressed in terms of the classical Jacobi polynomial $ \, {\bf P}^{(p-1/2 , p+1/2)}_{k-p} (\omega - t/2)\, $ of degree $ \, k-p \,$. Hence, $ \, \sigma_{k+1} (\varphi, \phi) \equiv 0 \,$, and the $(k+1)$-th Hadamard coefficient of the operator $ {\cal L}_{N+1} $ vanishes identically: [89]{} U\_[k+1]{}(x, ) 0 . According to Hadamard’s criterion , it means that the operator $ {\cal L}_{N+1} $ satisfies Huygens’ principle in Minkowski space $ \M \,$, if $ n $ is odd and $$n \geq 2\,k + 3\ ,$$ Thus, the proof of the theorem is completed.
IV. Concluding remarks and examples {#iv.-concluding-remarks-and-examples .unnumbered}
===================================
In the present paper we have constructed a new hierarchy of Huygens’ operators in higher dimensional Minkowski spaces $ \M , n > 3 $. However, the problem of complete description of the whole class of such operators for arbitrary $ n $ still remains open. As mentioned in the Introduction, the famous Hadamard’s conjecture claiming that any Huygens’ operator $ {\cal L} $ can be reduced to the ordinary d’Alembertian $ \, \Box_{n+1} \,$ with the help of trivial transformations is valid only in $ {\bf M}^{3+1}\,$. Recently, in the work [@Ber] one of the authors put forward the relevant modification of Hadamard’s conjecture for Minkowski spaces of arbitrary dimensions. Here we recall and discuss briefly this statement.
Let $ \Omega $ be an open set in Minkowski space $ \M \cong {\bf R}^{n+1}\, $, and let $\, {\cal F}(\Omega)\, $ be a ring of partial differential operators defined over the function space $\, C^{\infty}(\Omega)\,$. For a fixed pair of operators $\, {\cal L}_{0},
{\cal L} \in {\cal F}(\Omega)\, $ we introduce the map [90]{} \_[[L]{}, [L]{}\_[0]{}]{}: [F]{}() () , A \_[[L]{}, [L]{}\_[0]{}]{}\[A\] , such that [91]{} \_[[L]{}, [L]{}\_[0]{}]{}\[A\] := [L]{} A - A \_[0]{} . Then, given $\, M \in {\bf Z}_{>0}\, $, the iterated $ \mbox{\rm ad}_{{\cal L}, {\cal L}_{0}}$-map is determined by [92]{} \_[[L]{}, [L]{}\_[0]{}]{}\^[M]{}\[A\] := \_[[L]{}, [L]{}\_[0]{}]{}= \_[k=0]{}\^[M]{} (-1)\^[k]{} [M k]{} [L]{}\^[M-k]{}A \_[0]{}\^[k]{} .
[**Definition.**]{} The operator $ \, {\cal L} \in {\cal F}(\Omega)\, $ is called [*M-gauge related*]{} to the operator $ \, {\cal L}_{0} \in {\cal F}(\Omega)\, $, if there exists a smooth function $\, \theta(x) \in {\cal C}^{\infty}(\Omega)\,$ non-vanishing in $ \Omega $, and an integer positive number $ M \in {\bf Z }_{>0}\,$, such that [93]{} 0 [F]{}() . In particular, when $ M = 1\, $, the operators $ {\cal L} $ and $ {\cal L}_{0} $ are connected just by the trivial gauge transformation $\, {\cal L} = \theta(x) \circ {\cal L}_{0} \circ \theta(x)^{-1}\,$.
The modified Hadamard’s conjecture claims:
> [*Any Huygens’ operator $ {\cal L} $ of the general form [94]{} [L]{} = \_[n+1]{} + ( a(x), ) + u(x) , in a Minkowski space $ \M $ ($ n $ is odd, $ n \geq 3 $) is $M$-gauge related to the ordinary wave operator $ \Box_{n+1} $ in $ \M $.* ]{}
For Huygens’ operators associated to the rational solutions of the KdV-equation , and to Coxeter groups , this conjecture has been proved in [@Ber] and [@BM]. In these cases the required identities are the following [95]{} \_[[L]{}\_[k]{}, [[L]{}\_[0]{}]{}]{}\^[M\_[k]{}+1]{}\[[P]{}\_[k]{}(x\^0)\] = 0 , M\_[k]{} := , where $ {\cal L}_{k} $ is given by with the potential for $\, k=0,1,2, \ldots $ and [96]{} \_[[L]{}\_[m]{}, [[L]{}\_[0]{}]{}]{}\^[M\_[m]{}+1]{}\[\_[m]{}(x)\] = 0 , M\_[m]{} := \_[\_[+]{}]{}\^ m\_ , where $ {\cal L}_{m} $ is defined by , and $\, \pi_{m}(x) :=
\prod_{\alpha \in \Re_{+}}^{} (\alpha, x)^{m_{\alpha}}\, $.
It is remarkable that for the operators constructed in the present work the modified Hadamard’s conjecture is also verified. More precisely, for a given wave-type operator [97]{} [L]{}\_[(k)]{} = \_[n+1]{} - ( )\^2 , associated to a positive integer partition $ (k_{i})\,$:$\, 0 \leq k_1 < k_2 < \ldots < k_N\,$, we have the identity [98]{} \_[[L]{}\_[(k)]{}, [[L]{}\_[0]{}]{}]{}\^[|k|+1]{}\[ \_[(k)]{}(x)\] = 0 , where $\, \Theta_{(k)}(x) := r^{|k|}\, {\cal W}\left[\Psi_{1}, \Psi_{2}, \ldots , \Psi_{N} \right] \,$ and $\, |k| := \sum\nolimits_{i=1}^{N} k_{i} \,$ is a weight of the partition $\, (k_{i}) \,$.
We are not going to prove in the present paper. A more detailed discussion of this identity and associated algebraic structures will be the subject of our subsequent work. Here, we only mention that such type identities naturally appear [@Ber]–[@Ber1] in connection with a classification of overcomplete commutative rings of partial differential operators [@VCh1], [@VCh2], [@VChS], and with the bispectral problem [@DG].
We conclude the paper with several concrete examples illustrating our main theorem.
[**1.**]{} As a first example we consider the dihedral group $ I_{2}(q) ,\, q \in {\bf Z}_{>0}, $ acting on the Euclidean plane $ E \cong {\bf R^2} \subset {\bf Gr}_{\perp} (n+1,2) $ and fix the simplest partition $ k=(q) $ and the phase $ \varphi = \pi/2 $. According to Remark II, in this case our theorem gives the wave-type operator with the Calogero-Moser potential related to the Coxeter group $ I_{2}(q) $ with $ m=1 $: $${\cal L}_{(k)} = \Box_{n+1} + \frac{2\,q^2}{r^2 \sin^{2}(q\,\varphi)}\ .$$ This operator satisfies Huygens’ principle in $ \M $ if $ n $ is odd and $ n \geq 2\,q +3 $. The Hadamard coefficients of $ {\cal L}_{(k)} $ can be presented in a simple closed form in terms of polar coordinates on $ E $: $$U_{0} =1\ ,$$ $$U_{\nu} = \frac{1}{(2r\rho)^{\nu}}\, \frac{T_{q}^{(\nu)}(\cos(\varphi - \phi))}{\sin(q\, \varphi) \sin(q\, \phi)}\ , \quad \nu \geq 1\ ,$$ where $ T_{q}(z) := \cos(q\, \arccos(z)), \, z \in [-1,1] $, is the $q$-th Chebyshev polynomial, and $ T_{q}^{(\nu)}(z) $ is its derivative of order $ \nu $ with respect to $ z $. These formulas are easily obtained with the help of recurrence relation .
[**2.**]{} Now we fix $ N = 2 ,\, k_{1}=2,\, k_{2}=3 $ and $\, \varphi_{1}= \pi/2 ,\, \varphi_{2}=0\, $. The corresponding wave-type operator $${\cal L}_{(k)} = \Box_{n+1} +
\frac{
10\,\left( x_{1}^{2}+ x_{2}^{2} \right)
\left(15\, x_{2}^{2}- x_{1}^{2}\right)
}
{
\left(5 x_{2}^{2}+ x_{1}^{2} \right)^{2}\, x_{1}^{2}
}\ ,$$ satisfies Huygens’ principle for odd $ n \geq 9 $. The nonzero Hadamard coefficients of this operator are given explicitly by the formulas: $$U_{{0}}=1,$$ $$U_{{1}}={\frac {40\,x_{{
2}}\xi_{{1}}\xi_{{2}}x_{{1}}+15\,{\xi_{{1}}}^{2}{x_{{2}
}}^{2}+75\,{\xi_{{2}}}^{2}{x_{{2}}}^{2}+15\,{\xi_{{2}}}
^{2}{x_{{1}}}^{2}-5\,{\xi_{{1}}}^{2}{x_{{1}}}^{2}}{2\,
\xi_{{1}}x_{{1}}\left (5\,{x_{{2}}}^{2}+{x_{{1}}}^{2}
\right )\left (5\,{\xi_{{2}}}^{2}+{\xi_{{1}}}^{2}
\right )}},$$ $$U_{{2}}={\frac {120\,x_{{2}}\xi_{{1}}\xi_{{2
}}x_{{1}}+15\,{\xi_{{2}}}^{2}{x_{{1}}}^{2}-5\,{\xi_{{1}
}}^{2}{x_{{1}}}^{2}+15\,{\xi_{{1}}}^{2}{x_{{2}}}^{2}+75
\,{\xi_{{2}}}^{2}{x_{{2}}}^{2}}{4\,{\xi_{{1}}}^{2}{x_{{
1}}}^{2}\left (5\,{x_{{2}}}^{2}+{x_{{1}}}^{2}\right )
\left (5\,{\xi_{{2}}}^{2}+{\xi_{{1}}}^{2}\right )}},$$ $$U_{{3}}=-{\frac {15\,x_{{2}}\xi_{{2}}}{{\xi_{{1}}}^{2}{x_{
{1}}}^{2}\left (5\,{x_{{2}}}^{2}+{x_{{1}}}^{2}\right )
\left (5\,{\xi_{{2}}}^{2}+{\xi_{{1}}}^{2}\right )}}\ .$$
[**3.**]{} Now we take $ N=3 $, the partition $ k = (1,\, 3,\, 4)\,$, and the phases $\, \varphi_{1}= \varphi_{2}= \varphi_{3}= \pi/2\, $. The corresponding operator $${\cal L}_{(k)} = \Box_{n+1} +
\frac{
12\, \left(
49 x_{1}^{4}+ 28 x_{1}^{2} x_{2}^{2} - x_{2}^{4} \right)
}
{
x_{2}^{2}\,
\left (7 x_{1}^{2} + x_{2}^{2}
\right)^{2}
},$$ is a Huygens operator in $ \M $ when $ n $ is odd and $ n \geq 11\, $. The nonzero Hadamard’s coefficients are $$U_{{0}}=1,$$ $$U_{{1}}={\frac {-21\,{\xi_{{2}}}^{2}{x_{{1}}}^{2}-42\,x_
{{2}}\xi_{{1}}\xi_{{2}}x_{{1}}-21\,{\xi_{{1}}}^{2}{x_{{
2}}}^{2}+3\,{\xi_{{2}}}^{2}{x_{{2}}}^{2}-147\,{\xi_{{1}
}}^{2}{x_{{1}}}^{2}}{\xi_{{2}}x_{{2}}\left (7\,{x_{{1}}
}^{2}+{x_{{2}}}^{2}\right )\left (7\,{\xi_{{1}}}^{2}+{
\xi_{{2}}}^{2}\right )}},$$ $$U_{{2}}={\frac {735\,{\xi_{{1}
}}^{2}{x_{{1}}}^{2}+504\,x_{{2}}\xi_{{1}}\xi_{{2}}x_{{1
}}+105\,{\xi_{{1}}}^{2}{x_{{2}}}^{2}-21\,{\xi_{{2}}}^{2
}{x_{{2}}}^{2}+105\,{\xi_{{2}}}^{2}{x_{{1}}}^{2}}{4\,{
\xi_{{2}}}^{2}{x_{{2}}}^{2}\left (7\,{x_{{1}}}^{2}+{x_{
{2}}}^{2}\right )\left (7\,{\xi_{{1}}}^{2}+{\xi_{{2}}}^
{2}\right )}},$$ $$U_{{3}}={\frac {-1260\,x_{{2}}\xi_{{1}}
\xi_{{2}}x_{{1}}+21\,{\xi_{{2}}}^{2}{x_{{2}}}^{2}-105\,
{\xi_{{1}}}^{2}{x_{{2}}}^{2}-105\,{\xi_{{2}}}^{2}{x_{{1
}}}^{2}-735\,{\xi_{{1}}}^{2}{x_{{1}}}^{2}}{8\,{\xi_{{2}
}}^{3}{x_{{2}}}^{3}\left (7\,{x_{{1}}}^{2}+{x_{{2}}}^{2
}\right )\left (7\,{\xi_{{1}}}^{2}+{\xi_{{2}}}^{2}
\right )}},$$ $$U_{{4}}={\frac {315\,x_{{1}}\xi_{{1}}}{4\,{
\xi_{{2}}}^{3}{x_{{2}}}^{3}\left (7\,{x_{{1}}}^{2}+{x_{
{2}}}^{2}\right )\left (7\,{\xi_{{1}}}^{2}+{\xi_{{2}}}^
{2}\right )}}\ .$$
[**4.**]{} The last example illustrates Remark I following the theorem (see Introduction). In this case we consider the operator with the potential associated with the proper $N$-soliton solution of the KdV equation. We take $ N=2 $ and fix $\, k_{1}=1,\, k_{2}=2\, $. The real phases are chosen as follows $\, \vartheta_{1}=
\mbox{arctanh}\,(1/2) ,\, \vartheta_{2} = \mbox{arctanh}\,(1/4)
\,$. The corresponding operator $ {\cal L}_{(k)} $ reads $${\cal L}_{(k)} =
\Box_{n+1} +
\frac {
2 \left(2 x_{0} -3 x_{1}\right)
\left (3 x_{1}^{3}- 6 x_{0} x_{1}^{2}+
4 x_{1} x_{0}^{2}+8 x_{0}^{3} \right)
}
{ x_{1}^{2} \left(4 x_{0}^{2}-2 x_{0}x_{1}- x_{1}^{2}
\right)^{2}
}
\ .$$ According to the theorem, it is huygensian provided $ n $ is odd and $ n \geq 7 $. The nonzero Hadamard coefficients are given by the following formulas: $$U_{{0}}=1,$$ $$U_{1}=
\frac {
4 \xi_{0}^{2}x_{1}^{2}+9 \xi_{1}^{2}x_{1}^{2}-
16 \xi_{0}^{2}x_{0}^{2}+8 \xi_{0}^{2} x_{0}x_{1}
}
{
2 x_{1}\,\left(
4 x_{0}^{2}-2 x_{0}x_{1}- x_{1}^{2} \right)
\xi_{1}\left(4 \xi_{0}^{2}-2 \xi_{0} \xi_{1}- \xi_{1}^{2}
\right )
} +$$ $$+
\frac{
8 \xi_{0} \xi_{1} x_{0}^{2}-12 \xi_{1}^{2}x_{0} x_{1}
+ 4 \xi_{1}^{2} x_{0}^{2} -12 \xi_{0}\xi_{1}x_{1}^{2}
+16 \xi_{0}\xi_{1} x_{0} x_{1}
}
{
2 x_{1}\,\left(
4 x_{0}^{2}-2 x_{0}x_{1}- x_{1}^{2} \right)
\xi_{1}\left(4 \xi_{0}^{2}-2 \xi_{0} \xi_{1}- \xi_{1}^{2}
\right )
}\ ,$$
$$U_{{2}}=-{\frac {
5\,\left (2\,\xi_{{0}}- \xi_{{1}}\right )\left (2\,x_{{0
}}-x_{{1}}\right )}{4\,x_{{1}}\left (4\,{x_{{0}}}^{2}-2
\,x_{{0}}x_{{1}}-{x_{{1}}}^{2}\right )\xi_{{1}}\left (4
\,{\xi_{{0}}}^{2}-2\,\xi_{{0}}\xi_{{1}}-{\xi_{{1}}}^{2}
\right )}}\ .$$
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[^1]: Hadamard’s problem, or [*the problem of diffusion of waves*]{}, has received a good deal of attention and the literature is extensive (see, e.g., [@BV], [@McL1], [@Cou], [@Fri], [@PG], [@Gun], [@Hel], [@Ibr], [@McL], and references therein). For a historical account we refer the reader to the articles [@Dui], [@Gun1].
[^2]: This remarkable class of polynomials seems to have been found for the first time by Burchnall and Chaundy [@BCh].
[^3]: The coincidence of such rational solutions of the KdV-hierarchy with the Lagnese-Stellmacher potentials has been observed by Schimming [@Sch1], [@Sch2].
[^4]: Using the terminology adopted in the group representation theory we will call such integer monotonic sequences [*partitions*]{}.
[^5]: [**Note added in the proof.**]{} This conjecture has been proved recently by one of the authors in [@Ber11].
[^6]: By a distribution $ f \in {\cal D}'(\Omega) $ we mean, as usual, a linear continuous form on the space $ {\cal D}(\Omega) $ of ${\cal C}^{\infty}$-functions with supports compactly imbedded in $ \Omega $ (cf., e.g., [@GSh]).
[^7]: Such a terminology goes back to Hadamard’s book [@Had], where the function $ W(x,\xi) $ is introduced as a coefficient under the logarithmic singularity of an elementary solution (see for details [@Cou], pp. 740–743).
|
---
abstract: 'We study *simple wrinkled fibrations*, a variation of the simplified purely wrinkled fibrations introduced in [@Williams1], and their combinatorial description in terms of *surface diagrams*. We show that [simple wrinkled fibrations]{}induce handle decompositions on their total spaces which are very similar to those obtained from Lefschetz fibrations. The handle decompositions turn out to be closely related to surface diagrams and we use this relationship to interpret some cut-and-paste operations on 4-manifolds in terms of surface diagrams. This, in turn, allows us classify all closed 4-manifolds which admit [simple wrinkled fibrations]{}of genus one, the lowest possible fiber genus.'
address: |
Max-Planck-Institute for Mathematics\
Bonn, Germany
author:
- Stefan Behrens
title: 'On 4-manifolds, folds and cusps'
---
Introduction
============
After the pioneering work of Donaldson and Gompf on symplectic manifolds and Lefschetz fibrations [@Donaldson; @GS] (and later Auroux, Donaldson and Katzarkov on near-symplectic manifolds [@ADK]), the study of singular fibration structures on smooth 4-manifolds has drawn a considerable interest among 4-manifold theorists. Among the highlights in the field have been existence results for so called *broken Lefschetz fibrations* over the 2-sphere on all closed, oriented 4-manifolds [@Akbulut-Karakurt; @Baykur1; @Gay-Kirby; @Lekili] as well as a classification of these maps up to homotopy [@Lekili; @Williams1]. Furthermore, the classical observation that Lefschetz fibrations over the 2-sphere are accessible via handlebody theory and can be described more or less combinatorially in terms of collections of simple closed curves on a regular fiber known as the *vanishing cycles* [@Kas; @GS] was extended to the broken Lefschetz setting in [@Baykur2].
Our starting point is the work of Williams [@Williams1] who introduced the closely related notion of *simplified purely wrinkled fibrations*, proved their existence and exhibited a similar combinatorial description of these maps, again by collections of simple closed curves on a regular fiber, which he calls *surface diagrams*. In particular, it follows that all smooth, closed, oriented 4-manifolds can be described by a surface diagram. However, the correspondence between simplified purely wrinkled fibrations and surface diagrams has been somewhat unsatisfactory in that it usually involves arguments using broken Lefschetz fibrations and one has to assume the fiber genus to be sufficiently high.
It is one of our goals to provide a detailed and intrinsic account of this correspondence and to clarify the situation in the lower genus cases. After that we will give some applications. Let us describe the contents of this paper in more detail.
We begin by recalling some preliminaries from the singularity theory of smooth maps and the theory of mapping class groups of surfaces. This section is slightly lengthy because we intend to use it as a reference for future work.
The following two sections form the technical core of this paper. In Section \[S:SWFs over general base\] we introduce *[simple wrinkled fibrations]{}* over a general base surface. In the case when the base is the 2-sphere our definition is almost equivalent to Williams’ simplified purely wrinkled fibrations and our reason for introducing a new name is mainly to reduce the number of syllables. We then explain how the study of [simple wrinkled fibrations]{}reduces to certain fibrations over the annulus which we call *annular [simple wrinkled fibrations]{}*. From these we extract *twisted surface diagram* and establish a correspondence between annular [simple wrinkled fibrations]{}and twisted surface diagrams (Theorem \[T:annular SWFs <-> twisted SDs\]) up to suitable notions of equivalence. Along the way we show that annular [simple wrinkled fibrations]{}induce (relative) handle decompositions of their total spaces which are, in fact, encoded in a twisted surface diagram (Section \[S:handle decompositions\]). These handle decompositions bare a very close resemblance with those obtained from Lefschetz fibrations, the only difference appearing in the framings of certain 2-handles. The section ends with an investigation of the ambiguities for gluing surface bundles to the boundary components of annular [simple wrinkled fibrations]{}.
In Section \[S:SWFs over disk and sphere\] we specialize to the case when the base surface is either the disk or the 2-sphere and recover Williams’ setting. Using our results about annular [simple wrinkled fibrations]{}we obtain a precise correspondence between Williams’ (untwisted) surface diagrams and certain [simple wrinkled fibrations]{}over the disk (Proposition \[T:SDs and descending SWFs\]). In particular, our approach provides a direct way to construct a [simple wrinkled fibration]{}from a given surface diagram circumventing the previously necessary detour via broken Lefschetz fibrations.[^1]
Next, we address the question which surface diagrams describe [simple wrinkled fibrations]{}that extend to fibrations over the sphere and thus describe closed 4-manifolds. Just as in the theory of Lefschetz fibrations the key is to understand the boundary of the associated [simple wrinkled fibration]{}over the disk. We show how to identify this boundary with a mapping torus and describe its monodromy in terms of the surface diagram. Unfortunately, it turns out that the boundary is much harder to understand than in the Lefschetz setting.
We then go on to review the handle decompositions exhibited in Section \[S:SWFs over general base\] when the base is the disk or the sphere and describe a recipe for drawing Kirby diagrams for them. To complete the picture, we compare our decompositions with the ones obtained via simplified broken Lefschetz fibrations.
In the Sections \[S:substitutions\] and \[S:genus 1 classification\] we give some applications. We show that certain substitutions of curve configurations in surface diagrams correspond to cut-and-paste operations on 4-manifolds. In particular, we give a surface diagram interpretation of blow-ups and sum-stabilizations, i.e. connected sums with ${\mathbb{C}P^2}$, ${\overline{\mathbb{C}P^2}}$ and $S^2\times S^2$. Using these we easily obtain a classification of closed 4-manifolds which admit [simple wrinkled fibrations]{}with the lowest possible fiber genus.
\[T:genus 1 classification, intro\] A smooth, closed, oriented 4-manifold admits a [simple wrinkled fibration]{}of genus one if and only if it is diffeomorphic to $k S^2\times S^2$ or $m{\mathbb{C}P^2}\# n{\overline{\mathbb{C}P^2}}$ where $k,m,n\geq1$.
This result should be compared to [@Baykur-Kamada] and [@Hayano1] where a classification of genus one simplified broken Lefschetz fibration is addressed but only partially achieved. However, it should also be noted that the latter class of maps is strictly larger than that of genus one simple wrinkled fibrations and it is thus conceivable that the classification is more complicated.
In the final Section \[S:concluding remarks\] we close this paper by highlighting what we consider as some of the main problems in the field and by outlining some related developments.
Conventions {#conventions .unnumbered}
-----------
By default all manifolds are smooth, compact and orientable and all diffeomorphisms are orientation preserving. When we speak of of submanifolds we always mean tubular . We use the symbol $\nu S$ (resp. $\nubar S$) for an open (resp. closed) tubular of a submanifold $S\subset M$.
For induced orientations on boundaries we use the *outward normal first* convention. Moreover, if $f\colon M\ra N$ is smooth, $M$ and $N$ are connected and $p\in N$ is a regular value, then orientations on two out of the three manifolds $M$, $N$ and $f\inv(p)$ induce an orientation on the third as follows. There is a small ball $D\subset N$ containing $p$ such that $f\inv(D)$ can naturally be identified with $f\inv(p)\times D$ and we choose the third orientation such that this identification preserves orientations where $f\inv(p)\times D$ carries the product orientation.
Finally, (co-)homology is always taken with integral coefficients. Exceptions to these rules will be explicitly stated and we reserve the right to sometimes restate some of the conditions for emphasis.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work is part of the author’s ongoing PhD project carried out at the Max-Planck-Institute for Mathematics in Bonn, Germany. The author would like to thank Inanc Baykur for helpful comments on an early draft of this paper as well as his advisor Prof. Dr. Peter Teichner for letting him work on this project. The author is supported by an IMPRS Scholarship of the Max-Planck-Society.
Preliminaries {#S:preliminaries}
=============
To fix some terminology, let $f\colon M\ra N$ be a smooth map with differential $df\colon TM\ra TN$. A *critical point* (or a *singularity*) of $f$ is a point $p\in M$ such that $df_p$ is not surjective. The set of critical points, called the *critical locus* of $f$, will be denoted by $${\mathcal{C}}_{f}:=\left\{p\in M\middle\vert \rk df_p < \dim N\right\} \subset M.$$ The image of a critical point is called a *critical value* and the set of all critical values is called the *critical image* of $f$.
As customary, we will call the preimage of a point a *fiber*, usually decorated with the adjectives regular or singular indicating whether or not the fiber contains critical points. Note that regular fibers are always smooth submanifolds with trivial normal bundle.
Folds, cusps and Lefschetz singularities {#S:singularity theory}
----------------------------------------
As a warm up, recall that a generic map from any compact manifold to a 1-dimensional manifold has only finitely many critical points on which it is injective and, moreover, all critical points are of *Morse type*, i.e. they are locally modeled[^2] on the maps $$(x_1,\dots,x_n) \mapsto -x_1^2-\dots-x_k^2+x_{k+1}^2+\dots+x_n^2,$$ where the number $k$ is called the *(Morse) index* of the critical point.
A similar statement holds for maps to surfaces. For convenience we will take the source to be 4-dimensional from now on. In this setting the Morse critical points are replaced by two types of singularities known as *folds* and *cusps* which can also be described in terms of local models. The model for a fold singularity is the map $\R^4\ra \R^2$ given by the formula $$\label{E:fold model}
(t,x,y,z)\mapsto (t,-x^2-y^2\pm z^2)$$ and the cusps are locally modeled on $$\label{E:cuspmodel}
(t,x,y,z)\mapsto (t,-x^3 +3tx-y^2\pm z^2).$$ If the sign in either of the above equations is positive (resp. negative), then the singularity is called *indefinite* (resp. *definite*).
An easy calculation shows that the critical loci of the fold and cusp models are given by $\left\{\, (r,0,0,0) \;\middle\vert\; r\in\R \,\right\}$ and $\left\{\, (r^2,r,0,0) \;\middle\vert\; r\in\R \,\right\}$, respectively. As a consequence, the critical image of a smooth map is a smooth 1-dimensional submanifold near fold and cusp points. The critical images of both models are shown in Figure \[F:fold and cusp\].
\[F:fold and cusp\] 
Note that the critical image is smoothly embedded in the fold model where as in the cusp case it is topologically embedded via a smooth homeomorphism whose inverse fails to be smooth only at the cusp point.
It follows directly from the models that folds always come in 1-dimensional families on which the map restricts to an immersion. We will usually be sloppy and refer to such an arc of fold points in the source as well as their image in the target as *fold arcs*. Furthermore, cusps are isolated in the critical locus in the sense that there is a small which contains no other cusps. However, cusps are not isolated singularities. In fact, one can show that any cusp is surrounded by two fold arcs, at least one of which is indefinite.
We can now state the normal form of generic maps from 4-manifolds to surfaces.
\[T:generic maps to surfaces\] A generic map from a 4-manifold to a surface has only fold and cusp singularities, it is injective on the cusps and restricts to an immersion with only transverse intersections between fold arcs.
Note, in particular, that the above discussion shows that the critical locus of a generic map to a surface is a smooth 1-dimensional submanifold of the source. For more details, including a proof of the above theorem for arbitrary source dimension, we refer the reader to [@GG].
\[R:Morse 2-functions\] Recently, these generic maps to surfaces have appeared under the name *Morse 2-functions* in the work of Gay and Kirby [@GK1; @GK2; @GK3].
In what follows we will only deal with indefinite singularities. So from now on, when we speak of folds and cusps, we will always mean the indefinite ones.
Figure \[F:fold and cusp\] contains some further decorations which we will now explain. Both, the fold and the cusp singularity are intimately related to 3-dimensional Morse-Cerf theory. The fold models a trivial homotopy of a Morse functions with one critical point (of index two) on the vertical slices. This means that the model restricted to a small arc transverse to the fold locus is a Morse function with one critical point of index one or two depending on the direction. The arrows in the picture indicate the direction in which the index is two. Note that the topology of the fibers of either side of a fold arc is necessarily different.
Similarly, the cusp is also a homotopy of Morse functions on the vertical slices, although a nontrivial one. It models the cancellation of a pair of critical points (of index one and two). The arrows indicate the index two direction of the fold arcs adjacent to the cusp.
For the moment, this is all we have to say about folds and cusps. Another important type of singularity which has its roots in (complex) algebraic geometry is the *Lefschetz singularity* and its local model is given in complex coordinates by $$L \colon \C^2 \ra \C
\quad ; \quad
(z,w)\mapsto zw.$$ At this point it becomes important whether the charts that we use to model the map are orientation preserving. Although this does not matter for folds and cusps[^3], it makes a surprisingly big difference in the case of Lefschetz singularities. So from now on we will always use orientation preserving charts to model singularities whenever the source or target are oriented.
As stated in the introduction, maps with (indefinite) fold, cusp and Lefschetz singularities have been prominently featured in many research papers over the past decade. Unfortunately, various authors have used various names for various types of maps and there is yet no commonly accepted terminology in the field. For the purpose of this paper we will use the following terminology.
\[D:singular fibrations\] A surjective map $f\colon X\ra B$ from an oriented 4-manifold to an oriented surface is called (a) a *wrinkled fibration*, (b) a *(broken) Lefschetz fibration* or (c) a *broken fibration* if its critical locus contains only
1. indefinite folds and cusps,
2. Lefschetz singularities (and indefinite folds),
3. indefinite folds, cusps and Lefschetz singularities,
all critical points are contained in the interior of $X$ and all intersections in the critical image are transverse intersections of fold arcs.
In accordance with the use of the word fibration we will usually refer to the source as the *total space* and to the target as the *base*. Note that the regular fibers of a broken fibration are (orientable) surfaces. Furthermore, if we assume that $\del X= f\inv(\del B)$, which we will do later on, then the fibers are closed.
It is quite useful to think of broken fibrations as (singular) families of surfaces parametrized by the base. More precisely, the images of the folds and cusps cut the base into several regions which may or may not contain Lefschetz singularities. The regular fibers are (orientable) surfaces whose topological type depends only on the region that it maps into. One thus decorates the base with the topological type of the fibers over each region together with some information about what happens to a fiber if one crosses a fold arc (the little arrows we have indicated above together with the corresponding fold vanishing cycle) or runs into a Lefschetz singularity (the Lefschetz vanishing cycle). Under certain circumstances this data is enough to determine the map as we will see later on (see also [@GK3]).
We finish this section with a short review of the homotopy classification of broken fibrations over $S^2$ that was mentioned in the introduction. An important contribution of Lekili [@Lekili] is that he showed how to pass back and forth between broken Lefschetz fibrations and wrinkled fibrations via two *local homotopies*, i.e. homotopies that are supported in arbitrarily small balls. As portrayed in Figure \[F:Lefschetz vs cusp\] one can *wrinkle* a Lefschetz point into an indefinite triangle (i.e. an indefinite circle with three cusps) and one can exchange a cusp for a Lefschetz singularity, this move is sometimes called *unsinking* a Lefschetz point from a fold. (Moreover, he showed that these modifications work equally well with achiral Lefschetz singularities which, together with the results of [@Gay-Kirby], proves the existence of broken Lefschetz fibrations.)
![(a) Wrinkling and (b) unsinking a Lefschetz singularity.[]{data-label="F:Lefschetz vs cusp"}](homotopies_2.pdf)
As a consequence, one can translate questions about broken fibrations into questions about wrinkled fibrations which are accessible by means of singularity theory. For example, there is a structural result similar to Theorem \[T:generic maps to surfaces\] for generic homotopies between wrinkled fibrations. The basic building blocks include isotopies of the base and total space and three types of modifications (and their inverses) that are realized by local homotopies: the *birth/death*, the *merge* and the *flip*. Figure \[F:basic homotopies\] shows their effect on the critical image.
![The basic local homotopies: (a) birth, (b) merge, (c) flip.[]{data-label="F:basic homotopies"}](homotopies_1.pdf)
In general, such a generic homotopy will pass through maps with definite singularities. However, the main theorem in [@Williams1] states that indefinite singularities can, in fact, be avoided. In other words, any two homotopic wrinkled fibrations are homotopic through wrinkled fibrations.
\[R:reversing homotopy moves\] It has become common to refer to an application of any of the above mentioned modifications as *moves* performed on a broken fibration. It is important to note that most of these moves are not strictly reversible in the following sense. If the critical image of a given broken fibration exhibits a configuration as on the left hand side of any of the pictures, then it is always possible to replace it by the configuration on the right hand side. However, it might not be possible to go into the other direction. The only exception is the birth. In all other cases some extra conditions are needed to go from right to left. This is indicated in our pictures with shaded arrows.
\[R:merging folds and cusps\] There has been some disagreement in the literature about which direction in Figure \[F:basic homotopies\](b) should be called merge and which inverse merge. To avoid this decision we will simply speak of *merging cusps* and *merging folds*, respectively.
Surfaces and simple closed curves {#S:mapping class groups review}
---------------------------------
As we pointed out, the regular fibers of broken fibrations are surfaces and these fibers will be prominently featured later on. Unfortunately, this is yet another field of mathematics in which different authors use different conventions and, in the current author’s experience, it can be confusing to decide whether a statement in some reference actually applies to a situation at hand. For that reason we will give very precise definitions, deliberately risking to be overly precise.
By a *surface* ${\Sigma}$ we mean a compact, orientable, 2-dimensional manifold, possibly with boundary and some marked points in the interior. A *simple closed curve* in ${\Sigma}$ is a closed, connected, 1-dimensional submanifold of ${\Sigma}$ that does not meet the boundary or the marked points. We usually consider [simple closed curves]{}up to ambient isotopy in ${\Sigma}$ relative to $\del{\Sigma}$ and the marked points and will not make a notational distinction between a [simple closed curve]{}and its isotopy class. Note that according our definition [simple closed curves]{}are unoriented objects. However, from time to time it will be convenient to choose orientations on them in order to speak of their homology classes.
Given two [simple closed curves]{}$a,b\subset{\Sigma}$ we define their *geometric intersection number* as $$i(a,b):=\min\left\{ \#(\alpha\cap\beta) \middle\vert \alpha\sim a,\; \beta\sim b,\; \alpha\pitchfork \beta \right\}\in\N$$ where the signs $\sim$ and $\pitchfork$ indicate isotopy and transverse intersection. If the curves as well as the surface are oriented, then we also have an *algebraic intersection number* which is obtained by a signed count of intersections after making the curves transverse. Equivalently, this number can be described as $$\scp{a,b}:=\scp{[a],[b]}_{\Sigma}:=\scp{[a],[b]}_{H_1({\Sigma})}\in\Z$$ where bracket on the right hand side denotes the intersection form on $H_1({\Sigma})$.
Note that the algebraic intersection number is alternating and depends only on the homology classes of the oriented [simple closed curves]{}while the geometric intersection number is symmetric and depends on the isotopy classes. Both intersection numbers have the same parity (i.e. even or odd) and satisfy the inequality $$\label{E:algebraic vs geometric intersections}
|\scp{a,b}|\leq i(a,b).$$ We say that $a$ and $b$ are *geometrically dual* (resp. *algebraically dual*) if their geometric (reap. algebraic) intersection number is one.
A [simple closed curve]{}$a\subset{\Sigma}$ is called *non-separating* if its complement is connected, otherwise it is called *separating*. Note that a [simple closed curve]{}is separating if and only if it is null-homologous (with either orientation) and thus [simple closed curves]{}that have geometric or algebraic duals are automatically non-separating.
### Diffeomorphisms of surfaces
Let us now turn to diffeomorphisms of surfaces. Let $\Diff^+({\Sigma},\del{\Sigma})$ denote the set of orientation preserving diffeomorphisms that restrict to the identity on $\del{\Sigma}$ and preserve the set of marked points. The *mapping class group* of ${\Sigma}$ is defined as $${\mathcal{M}}(\Sigma):=\pi_0(\Diff^+({\Sigma},\del{\Sigma}),\id).$$ Given a [simple closed curve]{}$a\subset{\Sigma}$ there is a well defined mapping class $\tau_a\in{\mathcal{M}}({\Sigma})$ called the (right-handed) *Dehn twist* about $a$. Similarly, any simple arc $r\subset{\Sigma}$ that connects two distinct marked points gives rise to a *half twist* $\bar{\tau}_r\in{\mathcal{M}}({\Sigma})$.
It is well known that ${\mathcal{M}}({\Sigma})$ is generated by the collection of Dehn twist and half twists, where the latter are only needed in the presence of marked points. On the other hand, mapping classes can be effectively studied by their action on (isotopy classes of) simple closed curves. In particular, it is desirable to understand the effect of Dehn twists on [simple closed curves]{}. While this can be tricky, the situation simplifies significantly on the level of homology classes.
\[T:Picard-Lefschetz formula\] Let ${\Sigma}$ be a surface, $a\subset{\Sigma}$ a [simple closed curve]{}and let $x\in H_1({\Sigma})$. Then for any orientation on $a$ we have $$(\tau_a^k)_*x=x+k\scp{[a],x}[a].$$ In particular, if $b$ is an oriented [simple closed curve]{}, then $$[\tau_a^k(b)]=[b]+k\scp{[a],[b]}[a].$$
See [@primer], Proposition 6.3.
\[R:Picard-Lefschetz on torus\] The Picard-Lefschetz formula is particularly useful for the torus since, in that case, mapping classes are completely determined by their action on homology.
Another useful tool is the so called *change of coordinates principle* which roughly states that any two configurations of [simple closed curves]{}on a surface with the same intersection pattern can be mapped onto each other by a diffeomorphism. We will only use the following special cases. For details we refer to [@primer], Chapter 1.3.
\[T:change of coordinates principle\] If $a,b\subset{\Sigma}$ is a pair of non-separating [simple closed curves]{}, then there exists some $\phi\in\Diff^+({\Sigma},\del{\Sigma})$ such that $\phi(a)=b$. Furthermore, if $a,b$ and $a',b'$ are two pairs of geometrically dual curves, then there is some $\phi\in\Diff^+({\Sigma},\del{\Sigma})$ such that $\phi(a)=a'$ and $\phi(b)=b'$.
### Mapping tori and their automorphisms {#S:mapping tori}
Given a surface ${\Sigma}$ and a diffeomorphism $\mu\colon {\Sigma}\ra {\Sigma}$ we can form its *mapping torus* $${\Sigma}(\mu) := \big( {\Sigma}\times[0,1] \big) / \big( (x,1)\sim(\mu(x),0) \big)$$ which is a 3-manifolds that carries a canonical map to $S^1\cong [0,1]/\{0,1\}$ which turns out to be a submersion. In other words, ${\Sigma}(\mu)$ fibers over $S^1$. If ${\Sigma}$ is oriented and $\mu$ is orientation preserving, then our conventions in the introduction induce an orientation on ${\Sigma}(\mu)$. It is well known that all surface bundles over $S^1$ can be described as mapping tori. Indeed, if a 3-manifold fibers over $S^1$, then one chooses a fiber and a lift of a vector field that determines the orientation of $S^1$ and the return map of the flow of this vector field induces a diffeomorphism of the fiber which is usually called the *monodromy*.
Let $Y$ be an oriented 3-manifold that fibers over the circle via a map $f\colon Y\ra S^1$. An *automorphism* of $(Y,f)$ is an orientation and fiber preserving diffeomorphism of $Y$. We denote the group of automorphisms by $\Aut(Y,f)$ or simply by $\Aut(Y)$ when the fibration is clear from the context. If we identify $Y$ with a mapping torus, say ${\Sigma}(\mu)$, then we obtain a description of $\Aut(Y)$ in terms of diffeomorphisms of ${\Sigma}$. Indeed, any element $\phi\in\Aut({\Sigma}(\mu))$ can be considered as a path $(\phi_t)_{t\in[0,1]}$ in $\Diff^+({\Sigma})$ connecting some element $\phi_0\in\Diff^+({\Sigma})$ to $\phi_1=\mu\inv\phi_0\mu$. In particular, $\phi_0$ must be isotopic to $\mu\inv\phi_0\mu$ and thus represents an element of $C_{{\mathcal{M}}({\Sigma})}(\mu)$, the centralizer in ${\mathcal{M}}({\Sigma})$ of (the mapping class represented by) $\mu$. Elaborating on this idea one arrives at the conclusion that $$\label{E:automorphisms of mapping tori}
\pi_0\big(\Aut(Y)\big) \cong \pi_0\big(\Aut({\Sigma}(\mu))\big) \cong
C_{{\mathcal{M}}({\Sigma})}(\mu) \ltimes \pi_1(\Diff({\Sigma}),\id),$$ where the multiplication on the right hand side is given by $$(g,\sigma)\cdot(h,\tau)=(h\circ g, (g\inv\tau g)\ast\sigma ).$$ This means that there are essentially two types of automorphism of mapping tori, the ones that are constant on the fibers coming from $C_{{\mathcal{M}}({\Sigma})}(\mu)$ and the ones coming from $\pi_1(\Diff({\Sigma}),\id)$ that vary with the fibers and restrict to the identity on the reference fiber. Fortunately, there are no non-constant automorphisms most of the time due to the following classical result.
\[T:Earle-Eells\] Let ${\Sigma}$ be a closed, orientable surface of genus $g$ without marked points. Then $$\pi_1(\Diff({\Sigma}),\id)\cong
\begin{cases}
\Z_2 & \text{if $g=0$} \\
\Z\oplus\Z & \text{if $g=1$} \\
1 & \text{if $g\geq2$}.
\end{cases}$$
Hence, as soon as the genus of the fiber of a mapping torus is at least two, all automorphisms are isotopic (through automorphisms) to constant ones.
\[R:Aut ain’t Diff!\] It is important not to confuse the group $\Aut(Y)$ with the group of all (orientation preserving) diffeomorphisms of $Y$. A general diffeomorphism will not even be isotopic to a fiber preserving one!
Theorem \[T:Earle-Eells\] has many important consequences of which we only highlight one.
\[T:surface bundles over the sphere\] Let $P\ra S^2$ be a surface bundle with closed fibers of genus $g$.
1. If $g=0$, then $P$ is diffeomorphic to $S^2\times S^2$ or ${\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}$.
2. If $g=1$, then $P$ is diffeomorphic to $T^2\times S^2$, $S^1\times S^3$ or $S^1\times L(n,1)$.
3. If $g\geq2$, then $P$ is diffeomorphic to ${\Sigma}_g\times S^2$
For the genus one case see [@Baykur-Kamada Lemma 10]. The other cases are well known.
[Simple wrinkled fibrations]{}over general base surfaces {#S:SWFs over general base}
========================================================
We are finally ready to introduce the main objects of study in this paper.
\[D:simple wrinkled fibrations\] Let $X$ be a 4-manifold and $B$ a surface, both oriented. A *[simple wrinkled fibration]{}* with *total space* $X$ and *base* $B$ is a surjective smooth map of pairs $w\colon (X,\del X)\ra (B,\del B)$ with the following properties:
1. $w$ is a wrinkled fibration, i.e. ${\mathcal{C}}_{w}$ contains only indefinite folds and cusps,
2. ${\mathcal{C}}_{w}\cap\del X=\emptyset$,
3. ${\mathcal{C}}_{w}$ is non-empty, connected, and contains a cusp,
4. $w$ is injective on ${\mathcal{C}}_{w}$ and
5. all fibers of $w$ are connected.
Two [simple wrinkled fibrations]{}$w\colon X\ra B$ and $w'\colon X'\ra B'$ are *equivalent* if there are orientation preserving diffeomorphisms $\hat{\phi}\colon X\ra X'$ and $\check{\phi}\colon B\ra B'$ such that $w'\circ\hat{\phi}=\check{\phi}\circ w$.
Since we assume the base and total space of a [simple wrinkled fibration]{}to be oriented, the regular fibers are closed, oriented surfaces (of varying genus as explained below). We can thus define the *genus* of $w$ as the maximal genus among all regular fibers. A of the critical image of a [simple wrinkled fibration]{}is shown in Figure \[F:SWF base\]
![A of the critical image of a [simple wrinkled fibration]{}.[]{data-label="F:SWF base"}](SWF_base.pdf)
Before we continue we make some remarks about the definition.
\[R:difference from SPWFs\] [Simple wrinkled fibrations]{}over $S^2$ are essentially the same as Williams’ simplified purely wrinkled fibrations with two minor differences. One one hand we do not put restrictions on the fiber genus but on the other we require the presence of cusps. Both conditions can always be achieved by applying a *flip-and-slip move* (see Remark \[R:flip+slip\] below) and are thus merely of technical nature. Moreover, the “[simple wrinkled fibrations]{}without cusps” are easily classified (see Example \[eg:ADK sphere\]) so that one does not lose too much by ignoring them.
\[R:flip+slip\] Given a [simple wrinkled fibration]{}over $S^2$ there is an important homotopy to another such [simple wrinkled fibration]{}which has become known as a *flip-and-slip move*. Its effect on the base diagram is shown in Figure \[F:flip+slip\].
![The base diagrams during a flip-and-slip move. (The pictures show the complement of a disk in the lower genus region of the original fibration.)[]{data-label="F:flip+slip"}](flip+slip.pdf)
One first perform two flips on the same fold arc and then chooses an isotopy of the total space (the *slip*) during which the critical image undergoes the changes demonstrated in the picture. A flip-and-slip increases the fiber genus by one and introduces four new cusps.
\[R:SWFs are simple\] In spite of the lengthy definition, [simple wrinkled fibrations]{}are arguably the simplest possible maps from 4-manifolds to surfaces, at least as far as their singularity structure is concerned. As will be explained in detail it is this simplicity which makes it possible to give nice combinatorial descriptions of 4-manifolds.
\[R:homotopy vs. equivalence\] So far [simple wrinkled fibrations]{}have usually been studied up to homotopy instead of equivalence. However, we believe that the former point of view does not interact well with surface diagrams (which will be introduced momentarily) while the latter fits in perfectly. It would be interesting to relate the concepts of homotopy and equivalence but to our knowledge there is no obvious way to do so.
Given the rather specialized nature of [simple wrinkled fibrations]{}one might wonder whether they actually exist. This is indeed the case and we begin by giving some simple constructions.
\[eg:birth on bundles\] Let $\pi\colon X\ra B$ be a surface bundle over a surface $B$ with closed fibers of genus $g$. Then we can perform a birth homotopy on $\pi$ to obtain a genus $g+1$ [simple wrinkled fibration]{}with two cusps.
\[eg:wrinkled Lefschetz fibrations\] If $f\colon X\ra B$ is a Lefschetz fibration (possibly achiral) with closed fibers of genus $g$, then after wrinkling all the Lefschetz singularities we obtain a number of disjoint circles with three cusps in the critical image. By suitably merging cusps we can turn this configuration into a single circle resulting in a [simple wrinkled fibration]{}of genus $g+1$.
\[eg:ADK sphere\] This example includes the broken Lefschetz fibration on $S^4$ from [@ADK] that was mentioned in the introduction. Let $\Omega$ be a cobordism from ${\Sigma}_g$ to ${\Sigma}_{g-1}$ together with a Morse function $\mu\colon \Omega\ra I$ with exactly one critical point of index two. Then $\mu\times\id \colon \Omega\times S^1 \ra I\times S^1$ is a stable map with one circle of indefinite folds which fails to be a [simple wrinkled fibration]{}only because it does not have any cusps. Nevertheless, we can use $\Omega\times S^1$ to build wrinkled fibrations over $S^2$ by suitably filling in the two boundary components with ${\Sigma}_g\times D^2$ and ${\Sigma}_{g-1}\times D^2$ such that the fibration structures on the boundary extends. Using the handle decomposition constructed in [@Baykur2] it is easy to see that this constructions one gives the following total spaces: $P\# S^1\times S^3$ where $P$ is any ${\Sigma}_{g-1}$-bundle over $S^2$ and, if $g=1$, $S^4$ and some other manifolds with finite cyclic fundamental group (see [@Baykur-Kamada; @Hayano1]). Having build these maps one can then apply a flip-and-slip to obtain honest [simple wrinkled fibrations]{}. In particular, we see that $S^4$ carries a [simple wrinkled fibration]{}of genus two.
As a side remark, the above mentioned genus one fibration on $S^4$ already appeared in [@ADK] and is probably the reason why people became interested in constructing broken fibrations on general 4-manifolds.
The above examples show that [simple wrinkled fibrations]{}can be considered as a common generalization of surface bundles and (achiral) Lefschetz fibrations. The vastness of this generalization is indicated by the following remarkable theorem.
\[T:Williams existence\] Let $X$ be a closed, oriented 4-manifold. Then any map $X\ra S^2$ is homotopic to a [simple wrinkled fibration]{}of arbitrarily high genus.
\[R:existence proof\] Williams’ proof builds on results of Gay and Kirby [@Gay-Kirby] which, in turn, depend on deep theorems in 3-dimensional contact topology[^4]. This somewhat unnatural dependence could be removed by refining the singularity theory based approach of [@Baykur1] to produce maps which are injective on their critical points.
Williams [@Williams1] also introduced a combinatorial description of [simple wrinkled fibrations]{}over $S^2$ in terms of what he calls *surface diagrams*. In the remainder of this section we will generalize his construction to the setting of general base surfaces and prove a precise correspondence. Along the way we will see how [simple wrinkled fibrations]{}give rise to handle decompositions. In Section \[S:SWFs over disk and sphere\] we will return to Williams’ surface diagrams and use them to prove some results.
Let $w\colon X\ra B$ be a [simple wrinkled fibration]{}. As explained in Section \[S:singularity theory\], it follows from the definition of [simple wrinkled fibrations]{}that the critical locus ${\mathcal{C}}_w\subset X$ of a [simple wrinkled fibration]{} $w\colon X\ra B$ is a smoothly embedded circle and that $w$ restricts to a topological embedding of ${\mathcal{C}}_w$ into $B$. Furthermore, the critical image $w({\mathcal{C}}_w)$ separates $B$ into two components. Indeed, if its complement were connected, then all regular fibers would be diffeomorphic. But according to the fold model, the topology of the fibers on the two sides of a fold arc must be different. In fact, since we require that all fibers are connected, the genus on one side has to be one higher than on the other side. We will call the two components of $B\setminus w({\mathcal{C}}_w)$ the *higher* (resp. *lower*) *genus region*.
We would like to understand more precisely how the topology of the fibers changes across the critical image. A *reference path* for $w$ is an oriented, embedded arc $R\subset B$ that connects a point $p_+$ in the higher genus region to a point $p_-$ in the lower genus region and intersects $w({\mathcal{C}}_w)$ transversely in exactly one fold point. Then the *reference fibers* ${\Sigma}_\pm(R):=w\inv(p_\pm)$ over the *reference points* $p_\pm$ are closed, oriented surfaces.
\[T:fold vanishing cycles\] A reference path $R\subset B$ induces a nonseparating [simple closed curve]{}${\gamma}(R)\subset{\Sigma}_+(R)$ which depends only on the isotopy class of $R$ relative to its reference points and the cusps.
\[D:fold vanishing cycle\] The curve ${\gamma}(R)\subset{\Sigma}_+(R)$ is called the *(fold) vanishing cycle* associated to $R$.
The fold model implies that $w\inv(R)$ is a cobordism from ${\Sigma}_+(R)$ to ${\Sigma}_-(R)$ on which $w$ restricts to a Morse function with exactly one critical point of index 2. Thus $w\inv(R)$ is diffeomorphic to ${\Sigma}_+(R)\times[0,1]$ with a (3-dimensional) 2-handle attached along a simple closed curve in ${\Sigma}_+(R)\times\set{1}$ which is canonically identified with a [simple closed curve]{}${\gamma}(R)\subset{\Sigma}_+(R)$.
Next, let us look at what happens around the cusp. Let $R_1$ and $R_2$ be two reference paths for $w$ with common reference points and assume that their interiors are disjoint. We call $R_1$ and $R_2$ *adjacent* if their union $R_1\cup R_2$ bounds a disk in $B$ that contains exactly one cusp.
\[T:adjacent reference paths\] Let $R_1$ and $R_2$ be adjacent reference paths. Then the vanishing cycles ${\gamma}(R_1)$ and ${\gamma}(R_2)$ in ${\Sigma}_+:={\Sigma}_+(R_1)={\Sigma}_+(R_2)$ are geometrically dual.
As in the proof of Lemma \[T:fold vanishing cycles\] the preimages $w\inv(R_i)$, $i=1,2$, are both cobordisms from ${\Sigma}_+$ to ${\Sigma}_-$, each consisting of a 2-handle attachment along ${\gamma}(R_i)$. By reversing the orientation of $R_1$ we can consider $w\inv(R_1)$ as a cobordism from ${\Sigma}_-$ to ${\Sigma}_+$, now consisting of a 1-handle attachment. In this process the former attaching sphere of the 2-handle ${\gamma}(R_1)$ becomes the belt sphere of the 1-handle.
Gluing $w\inv(R_1)$ and $w\inv(R_2)$ together along ${\Sigma}_+$ gives a cobordism from ${\Sigma}_-$ to itself consisting of a 1-handle attachment followed by a 2-handle attachment. Now recall that the cusp singularity models the death (or birth) of a canceling pair of critical points. Hence, the attaching sphere of the 2-handle, which is ${\gamma}(R_2)$, intersects the belt sphere of the 1-handle, which is ${\gamma}(R_1)$, in a single point.
Looking a bit ahead, our strategy will be to choose suitable collections of reference paths and to study [simple wrinkled fibrations]{}in terms of the induced collection of vanishing cycles. The only obstacle for doing so is the possibly complicated topology of the base surface. But this can easily be overcome by the following observation. We can cut the base into three pieces $$B=B_+\cup A\cup B_-$$ where $A$ is a regular of the critical image of $w$ (diffeomorphic to an annulus) and $B_\pm$ are the closures of the complement of $A$. The subscript in $B_\pm$ indicates whether the surface is contained in the higher or lower genus region. Note that $w$ restricts to surface bundles over $B_\pm$ and, although complicated, these are a rather well studied class of objects. Thus the interesting new part of $w$ is the restriction $w\inv(A)\ra A$ which is a [simple wrinkled fibration]{}over an annulus. Moreover, this fibration has the property that the critical image does not bound a disk in $A$ or, in other words, it is boundary parallel.
\[D:annulas SWFs\] A [simple wrinkled fibration]{}$w\colon W\ra A$ over an annulus $A$ is called *annular* if its critical image is boundary parallel.
So in order to understand [simple wrinkled fibrations]{}over any base surface, it is enough to understand annular [simple wrinkled fibrations]{}and this is where surface diagrams enter the picture. The remainder of this section is devoted to proving the following theorem.
\[T:annular SWFs <-> twisted SDs\] There is a bijective correspondence between annular [simple wrinkled fibrations]{}up to equivalence and twisted surface diagrams up to equivalence
We will split the proof of the theorem into the two obvious parts. The first part is the subject of Section \[S:SWFs -> SDs\] (see Proposition \[T:annular SWFs -> twisted SDs\]) and the second is treated in Section \[S:SDs -> SWFs\] (see Proposition \[T:annular SWFs <- twisted SDs\]). Along the way, we will see in Section \[S:handle decompositions\] that, just as Lefschetz fibrations, annular [simple wrinkled fibrations]{}are directly accessible via handlebody theory.
\[R:Gay-Kirby reconstruction\] Recently Gay and Kirby have published a result that contains Theorem \[T:annular SWFs <-> twisted SDs\] as a special case [@GK3]. Although their methods are somewhat similar to ours we feel that our approach is of independent interest.
Twisted surface diagrams of annular [simple wrinkled fibrations]{} {#S:SWFs -> SDs}
------------------------------------------------------------------
Consider an annular [simple wrinkled fibration]{} $w\colon W\ra A$. We denote by ${\del_\pm}A$ the boundary components of the base annulus $A$ contained in the higher (resp. lower) genus region and we let ${\del_\pm}W=w\inv({\del_\pm}A)$.
\[D:reference system\] Let $w\colon W\ra A$ be an annular [simple wrinkled fibration]{}. A *reference system* ${\mathcal{R}}=\set{R_1,\dots,R_c}$ for $w$ (where $c$ is the number of cusps) is a collection of reference paths for $w$ such that
1. all reference paths have the same reference points $p_\pm\in{\del_\pm}A$,
2. the interiors of the arcs are pairwise disjoint,
3. with respect to the orientations on ${\del_\pm}A$ the arcs leave ${\del_+}A$ and enter ${\del_-}A$ in order of increasing index (see Figure \[F:reference system\]) and
4. each fold arc is hit by exactly one of the $R_i$.
![A reference system for an annular [simple wrinkled fibration]{}.[]{data-label="F:reference system"}](reference_system.pdf)
As before, we denote the reference fibers by ${\Sigma}_\pm:={\Sigma}_\pm({\mathcal{R}})=w\inv(p_\pm)$. Using the reference fibers we can write ${\del_\pm}W$ as mapping tori $${\del_\pm}W\cong {\Sigma}_\pm(\mu_\pm)$$ where $\mu_\pm\in{\mathcal{M}}({\Sigma}_\pm)$ is the monodromy of $w$ over ${\del_\pm}A$ (in the positive direction). We will refer to $\mu_+$ (resp. $\mu_-$) as the *higher* (resp. *lower*) *monodromy* of $w$.
\[T:reference systems -> circuits\] Let $w\colon W\ra A$ be an annular [simple wrinkled fibration]{}together with a reference system ${\mathcal{R}}=\set{R_1,\dots,R_c}$ and let ${\gamma}_i={\gamma}(R_i)\subset{\Sigma}_+$. Then for $i<c$ the vanishing cycles ${\gamma}_i$ and ${\gamma}_{i+1}$ are geometrically dual and, moreover, so are $\mu_+({\gamma}_c)$ and ${\gamma}_1$.
In the proof of this Lemma and subsequent consideration we will need the following notion. Let $B$ be an oriented surface and let $R\subset B$ be a proper arc which hits a boundary component $\del_i B\subset\del B$ transversely in a single point. We parametrize a small collar of $\del_i B$ by $S^1\times[0,1]$ in such a way that $\del_i B$ corresponds to $S^1\times\set{1}$, $R$ corresponds to $\set{1}\times[0,1]$ and the induced orientations on $\del_i B$ agree. We say that the arc $R'$ which corresponds to $$\left\{ (e^{2\pi t},t) \middle\vert t\in [0,1] \right\}$$ via the parametrization is obtained from $R$ by *swinging once around $\del_i B$*.
\[R:swinging vs. Dehn twists\] At first glance, swinging about a boundary component seems to be the same as applying a boundary parallel Dehn twist and up to isotopy this is indeed the case. However, there is a subtle difference since a boundary parallel Dehn twists is usually assumed to be supported in the interior of the surface and thus fixes a small collar of the boundary point wise while the support of the swinging diffeomorphism goes right up to the boundary. This difference becomes important in the following situation.
Let $S$ be another arc with the same properties as $R$ such that $R$ and $S$ are disjoint in the interior of $B$ and assume that $S$ leaves $\del_i B$ after $R$. If we swing $S$ once around $\del_i B$, then the resulting arc remains disjoint from $R$ but now leaves $\del_i B$ before $R$. On the other hand, if we perform a boundary parallel Dehn twist on $S$, then we keep the exit order at the price of introducing an interior intersection point. In particular, if ${\mathcal{R}}=\set{R_1,\dots,R_c}$ is a reference system for an annular [simple wrinkled fibration]{}, then we obtain a new reference system by swinging the last arc $R_c$ once around each boundary component fo the annulus.
The first statement follows from Lemma \[T:adjacent reference paths\] since for $i<c$ the reference paths $R_i$ and $R_{i+1}$ are clearly adjacent. The second statement needs an additional arguments. We first swing $R_c$ once around the boundary of $A$ so that the resulting reference path $R_c'$ is adjacent to $R_1$ and thus ${\gamma}(R_c')$ is geometrically dual to ${\gamma}(R_1)$. Next we observe that $R_c'$ is homotopic to $R_c$ precomposed with the boundary curve. Thus the parallel transport along $R_c'$ is the composition of the parallel transport along $R_c$ and the higher genus monodromy. In particular, we have ${\gamma}(R_c')=\mu_+({\gamma}_c)$.
\[R:other side of reference paths\] Note that in the above proof we did not actually need the whole reference system but only the parts of the arcs contained in the higher genus region.
Let us isolate the combinatorial structures encountered in the above Lemma.
\[D:circuits\] Let ${\Sigma}$ be a surface. A *circuit* (of *length* $c$) on ${\Sigma}$ is an ordered collection of [simple closed curves]{}$\Gamma=({\gamma}_1,\dots,{\gamma}_c)$ such that any two adjacent curves ${\gamma}_i$ and ${\gamma}_{i+1}$ are geometrically dual for $i<c$. A *switch* for ${\Gamma}$ is a mapping class $\mu\in{\mathcal{M}}({\Sigma})$ such that $\mu({\gamma}_c)$ and ${\gamma}_1$ are geometrically dual. We say that ${\Gamma}$ is *closed* if ${\gamma}_c$ and ${\gamma}_1$ are geometrically dual, i.e. if the identity works as a switch.
\[D:surface diagrams\] A *twisted surface diagram* is a triple ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$ where ${\Sigma}$ is a closed, oriented surface, ${\Gamma}$ is a circuit in ${\Sigma}$ and $\mu\in{\mathcal{M}}({\Sigma})$ is a switch for ${\Gamma}$.
\[R:chains of curves\] There is no restriction on the intersections of non-adjacent curves in a circuit. Circuits in which non-adjacent curves are disjoint, so called *chains of curves*, are well known objects in the theory of mapping class groups of surfaces where they play an important role.
\[R:orienting circuits\] Sometimes it will be convenient to choose orientations on the curves in a circuit ${\Gamma}=({\gamma}_1,\dots,{\gamma}_c)$ in order to speak of their homology classes. If the ambient surface is oriented, we will always choose orientations such that the intersection of ${\gamma}_i$ and ${\gamma}_{i+1}$, $i<c$, has positive sign.
With this terminology we can rephrase Lemma \[T:reference systems -> circuits\] as stating that an annular [simple wrinkled fibration]{}$w\colon W\ra A$ together with a reference system ${\mathcal{R}}$ induces a twisted surface diagram $${\mathfrak{S}}_{w,{\mathcal{R}}}:=({\Sigma}_+,{\Gamma}_{w,{\mathcal{R}}},\mu_+)$$ where the higher monodromy works as a switch.
Note that when the higher monodromy is trivial we obtain a closed circuit and recover Williams’ surface diagrams for which we shall reserve this name, i.e. in the following the term surface diagram will always mean a triple $({\Sigma},{\Gamma},\id)$ which we simply denote by $({\Sigma},{\Gamma})$ or sometimes even $({\Sigma};{\gamma}_1,\dots,{\gamma}_c)$. Whenever we allow nontrivial higher monodromy we will explicitly speak of twisted surface diagrams.
Not surprisingly, the twisted surface diagrams constructed in Lemma \[T:reference systems -> circuits\] depend on the choice of the reference system. To understand this dependence we observe that a reference system is uniquely determined (up to isotopy relative to the boundary and the cusps) by specifying the first reference path – this follows directly from the definition. Furthermore, it is easy to see that any two reference paths which have the same reference points and hit the same fold arc become isotopic after suitably swinging around the boundary components of $A$.
Now let ${\mathcal{R}}=\set{R_1,\dots,R_c}$ and ${\mathcal{S}}=\set{S_1,\dots,S_c}$ be two reference systems with common reference points and let $S_k$ hit the same fold arc as $R_1$. As in the proof of Lemma \[T:reference systems -> circuits\] we successively swing the arcs $S_c,S_{c-1},\dots,S_k$ once around each boundary component to obtain a new reference system ${\mathcal{S}}'$ in which the first reference path hits the same fold arc as $R_1$. Now, by further swinging all of ${\mathcal{S}}'$ simultaneously, but this time independently around the boundary components, we can match the two first reference paths and thus the whole reference systems.
Let us analyze the effect of this matching procedure on the twisted surface diagram. For brevity of notation let ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$ be the twisted surface diagram associated to an annular [simple wrinkled fibration]{}$w\colon W\ra A$ together with a reference system ${\mathcal{R}}$. Since the surface ${\Sigma}$ and the switch $\mu$ only depend on the reference points, only the circuit ${\Gamma}=({\gamma}_1,\dots,{\gamma}_c)$ will be affected by swinging some reference paths. Moreover, note again that the vanishing cycles ${\gamma}_i$ only depend on the part of the reference paths contained in the higher genus region. Thus swinging around the lower genus boundary does not change the circuit.
Now, as we have already observed, if we swing the last reference path in ${\mathcal{R}}$ once around both boundary components, we obtain a new reference system ${\mathcal{R}}'$ and which induces the circuit $${\Gamma}_\mu^{[1]}:=\big( \mu({\gamma}_c),{\gamma}_1,\dots,{\gamma}_{c-1} \big) .$$ This operation of going from ${\mathfrak{S}}$ to ${\mathfrak{S}}^{[1]}:=({\Sigma},{\Gamma}_\mu^{[1]},\mu)$ makes sense in the abstract setting of twisted surface diagrams and we call it (and its obvious inverse) *switching*. Note that if the higher monodromy $\mu$ is trivial, then switching amounts to a cyclic permutation of the vanishing cycles.
Since we can relate any two reference systems for a given annular [simple wrinkled fibration]{}by suitably swinging reference paths, we see that the twisted surface diagram is well defined up to switching.
Next we want to compare the twisted surface diagrams of two equivalent annular [simple wrinkled fibrations]{}as in the commutative diagram below. $$\xymatrix{X\ar[d]_w \ar[r]^{\hat{\phi}} & X'\ar[d]^{w'} \\ A\ar[r]^{\check{\phi}} & A'}$$ If ${\mathcal{R}}$ is a reference system for $w$, then ${\mathcal{R}}':=\check{\phi}({\mathcal{R}})$ is a reference system for $w'$. Let ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$ and ${\mathfrak{S}}'=({\Sigma}',{\Gamma}',\mu')$ be the associated twisted surface diagrams. Then $\hat{\phi}$ induces an orientation preserving diffeomorphism $\phi\colon{\Sigma}\ra {\Sigma}'$ and clearly the higher monodromies satisfy $\mu'=\phi\mu\phi\inv$. It is also easy to see that $${\Gamma}'=\phi({\Gamma}):=\big( \phi({\gamma}_1),\dots,\phi({\gamma}_c) \big)$$ where, as usual, ${\Gamma}=({\gamma}_1,\dots,{\gamma}_c)$. Again, the effect of an equivalence of annular [simple wrinkled fibrations]{}makes sense for abstract twisted surface diagrams and we say that ${\mathfrak{S}}$ and ${\mathfrak{S}}'$ are *diffeomorhpic* via $\phi$. Putting this together with switching we end up with the following definition.
\[D:equivalence of SDs\] Two twisted surface diagrams ${\mathfrak{S}}$ and ${\mathfrak{S}}'$ called *equivalent* if, for some integer $k$, ${\mathfrak{S}}'$ is diffeomorphic to ${\mathfrak{S}}^{[k]}$.
Summing up the content of this section we have proved the first half of Theorem \[T:annular SWFs <-> twisted SDs\]:
\[T:annular SWFs -> twisted SDs\] To an annular [simple wrinkled fibration]{}$w\colon W\ra A$ we can assign a twisted surface diagram $${\mathfrak{S}}_w=({\Sigma}_+,{\Gamma}_w,\mu_+)$$ which is well defined up to switching. Moreover, equivalent annular [simple wrinkled fibrations]{}have equivalent twisted surface diagram.
\[R:picturing surface diagrams\] We would like to point out that it is very convenient that only the equivalence class of the surface diagram plays a role. Indeed, in order to actually visualize the twisted surface diagram of an annular [simple wrinkled fibration]{}one has to identify the higher genus reference fiber with some model surface and there is no canonical way to do so. However, any two such identifications will differ by a diffeomorphism of the model surface and thus be equivalent. So we can safely forget about the choice of identification whenever we are only interested in the equivalence class of the [simple wrinkled fibrations]{}or the diffeomorphism type of its total space.
Handle decompositions for annular [simple wrinkled fibrations]{} {#S:handle decompositions}
----------------------------------------------------------------
As a next step we relate the twisted surface diagrams associated to annular [simple wrinkled fibrations]{}to the topology of their total spaces. We will see that the situation is very similar to Lefschetz fibrations
\[T:handle decompositions from SWFs\] Let $w\colon W\ra A$ be an annular [simple wrinkled fibration]{}. Then $W$ has a relative handle decomposition on ${\del_+}W$ with one 2-handle for each fold arc. Such a handle decomposition is encoded in any twisted surface diagram for $w$.
In the following we will refer to the 2-handles associated to the fold arcs as *fold handles*.
The rough idea is to parametrize $A$ by the model annulus $S^1\times [0,1]$ such that the composition of $w$ and the projection $p\colon S^1\times [0,1]\ra [0,1]$ becomes a Morse function. The details go as follows.
We equip $S^1\times [0,1]$ with coordinates $(\theta,t)$ refer to the direction in which $t$ increases as *right*. We say that a parametrization $\kappa\colon A\ra S^1\times [0,1]$ is *$w$-regular* if the critical image $C_\kappa:=\kappa\circ w({\mathcal{C}}_w)$ is in the following *standard position*:
- all cusps point to the right
- each $R_\theta:=\{\theta\}\times[0,1]$ meets $C_\kappa$ in exactly one point, either in a cusp or transversely in a fold point and
- the projection $p$ restricted to $C_\kappa$ has exactly one minimum on each fold arc.
We claim that for any $w$-regular parametrization $\kappa$, the map $$p_\kappa:=p\circ\kappa\circ w\colon W\ra [0,1]$$ is a Morse function. Clearly, the critical points of $p_\kappa$ are contained in ${\mathcal{C}}_w$. Thus we have to understand how the projection $p$ interacts with the critical image $C_\kappa$. By the standard position assumption there are three ways how a level set $S_t:=S^1\times\set{t}$ can intersect $C_\kappa$ (see Figure \[F:slices\]):
1. $S_t$ intersects $C_\kappa$ transversely in a fold point,
2. $S_t$ meets $C_\kappa$ in a cusp and the fold arcs surrounding the cusp are on the left side of $S_t$ or
3. $S_t$ is tangent to a fold arc which is located on the right side of $S_t$. We will refer to this phenomenon as a *concave tangency*.
![Level sets intersecting the critical image.[]{data-label="F:slices"}](level_intersections.pdf)
It turns out that only the concave tangencies contribute critical points of $p_\kappa$. In fact, from the models for the fold an cusp we immediately see that $p_\kappa$ is modeled on the compositions $$(t,x,y,z)\mapsto(t,-x^3+3tx-y^2+z^2)\mapsto t$$ in case of a cusp intersection and $$(t,x,y,z)\mapsto(t,-x^2-y^2+z^2)\mapsto \pm t$$ for a transverse fold intersection[^5] which shows that these are regular points of $p_\kappa$.
It remains to treat the concave tangencies. These occur precisely at the minima of $p_\kappa|_{C_\kappa}$. This minimum can be modeled by $t\mapsto t^2$ and it is easy to see that $p_\kappa$ is modeled on $$(t,x,y,z)\mapsto(-x^2-y^2+z^2+t^2)$$ which is a Morse singularity of index 2. By assumption there is exactly one concave tangency for each fold arc and, using the correspondence between Morse functions and handle decompositions, we obtain the desired handle decomposition.
In order to understand how the fold handles are attached consider the arcs $R_i:=R_{\theta_i}\subset S^1\times [0,1]$ where $\theta_1,\dots,\theta_c\in S^1$ is a sequence of numbers ordered according to the orientation of $S^1$ (e.g. the $c$-th roots of unity). The $w$-regular parametrization $\kappa$ can be chosen in such a way that each $R_i$ is a reference path for precisely one fold arc and $C_\kappa$ is contained in the open annulus $S^1\times (\epsilon,1-\epsilon)$ for some $\epsilon>0$. For each $R_i$ we obtain a vanishing cycle ${\gamma}_i$ in the fiber of $w$ over $(\theta_i,0)\in{\del_+}A$ and the local model for the fold singularity implies that the fold handles are attached to ${\del_+}W\times[0,\epsilon]$ along the vanishing cycles ${\gamma}_i$ pushed off into the fiber over $(\theta_i,\epsilon)$ with respect to the canonical framing induced by the fiber.
The relation to twisted surface diagrams now becomes obvious. There is a canonical way to turn the reference paths $\Theta_1,\dots,\Theta_c$ into a reference system by fixing $\Theta_1$ and successively sliding the endpoints of the remaining arcs along the boundary onto $\Theta_1$ against the orientation. Thus the vanishing cycles record the attaching curves of the fold handles.
\[R:why we need cusps\] The above proposition is one of the reasons that made us require the presence of cusps in the critical loci of [simple wrinkled fibrations]{}. If there were no cusps, then it would not be possible to avoid *convex tangencies* which correspond to 3-handles instead of 2-handles. Thus the presence of cusps guarantees that the total spaces of annular [simple wrinkled fibrations]{}are (relative) 2-handlebodies.
\[R:vertical tangencies\] The observation that fold tangencies correspond to Morse singularities also appears in [@GK1] in their more general setting of Morse 2-functions. The fact that the real part of the Lefschetz model is also a Morse function allows to include Lefschetz singularities in the discussion. Proceeding this way, one can recover Baykur’s result about handle decompositions from broken Lefschetz fibrations (see [@Baykur2]).
\[R:similarity with Lefschetz fibrations\] The reader familiar with Lefschetz fibrations will have noticed the strong resemblance of the handle decompositions described above with the ones induces by Lefschetz fibrations. In fact, the handle decompositions have exactly the same structure except that the fold handles are attached with respect to the fiber framing while the framing of the *Lefschetz handles* differs by $-1$.
Annular [simple wrinkled fibrations]{}from twisted surface diagrams {#S:SDs -> SWFs}
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Using the handle decompositions exhibited in the previous section as a stepping stone we can now build annular [simple wrinkled fibrations]{}out of twisted surface diagrams and thus complete the proof of Theorem \[T:annular SWFs <-> twisted SDs\].
\[T:annular SWFs <- twisted SDs\] A twisted surface diagram ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$ determines an annular [simple wrinkled fibration]{}$w_{\mathfrak{S}}\colon W_{\mathfrak{S}}\ra S^1\times[0,1]$ with higher genus fiber ${\Sigma}$ and higher monodromy $\mu$.
To make the construction of $w_{\mathfrak{S}}$ more transparent we begin with some preliminary considerations.
One important ingredient is the mapping cylinder ${\Sigma}(\mu)$ which is equipped with a canonical fibration $p\colon{\Sigma}(\mu)\ra S^1$. Given the construction of ${\Sigma}(\mu)$ it is convenient to consider $S^1$ as the quotient $[0,1]/\{0,1\}$ and we will identify ${\Sigma}$ with the fiber over the point $0\sim1$.
We will now describe a collection of arcs ${\mathcal{R}}=\{R_1,\dots,R_c\}$ in $S^1\times[0,1]$, which we consider as $$S^1\times[0,1]=[0,1]\times[0,1] / (0,t)\sim(1,t),$$ that will serve as a reference system for $w_{\mathfrak{S}}$ (see Figure \[F:builsing SWFs\] (a)).
Let $r\colon[0,1]\ra [0,1]$ be a smooth function that has the constant value $1$ on the interval $[\tfrac{1}{3},\tfrac{2}{3}]$, satisfies $r(0)=r(1)=0$ and is strictly increasing (resp. decreasing) for $t\leq\tfrac{1}{3}$ (resp. $t\geq\tfrac{2}{3}$). If the length of ${\Gamma}$ is $c$, then for $i=1,\dots,c$ we let $\theta_i:=\tfrac{i-1}{c}$ and define $$R_i:=
\big\{
\big(\theta_i r(t),t\big)/\sim
\;\big\vert \;
t\in[0,1] \subset S^1\times[0,1]
\big\}$$
![Building a [simple wrinkled fibration]{}from a surface diagram. (bold: critical image, dashed: reference path)[]{data-label="F:builsing SWFs"}](building_SWFs.pdf)
We these remarks in place we can now begin with the construction of $W_{\mathfrak{S}}$ and $w_{\mathfrak{S}}$. This will be done in three steps.
*Step 1:* We begin by taking the product $$W_1:={\Sigma}(\mu)\times[0,\tfrac{1}{3}]$$ and define a map $w_1\colon W_1\ra S^1\times[0,\tfrac{1}{3}]$ by sending $(x,t)$ to $(p(x),t)$.
*Step 2:* Next, we construct $W_2$ by attaching 2-handles to $W_1$ in the following way. Let ${\Gamma}=({\gamma}_1,\dots,{\gamma}_c)$. Using the arc $R_i\subset S^1\times[0,1]$ described above we can parallel transport the curve ${\gamma}_i\subset{\Sigma}$ to the fiber of $w_1$ over $(\theta_i,\tfrac{1}{3})$. We attach a 2-handle to the resulting curve with respect to the fiber framing.
This choice of framing allows us to extend $w_1$ over each 2-handle. Indeed, we can consider attaching the $i$-th (4-dimensional) 2-handle as a 1-parameter family of 3-dimensional 2-handle attachments parametrized by a small of $(\theta_i,1)$ in $S^1\times\set{1}$. (Of course, these are pairwise disjoint.) For each point $\theta$ in such a , the restriction of $w_1$ to the *$\theta$-ray* $\set{\theta}\times[0,\tfrac{1}{3}]$ extends to a Morse function (with one critical point of index 2) over a slightly longer ray, say $\set{\theta}\times[0,\tfrac{2}{3}]$, in the standard way. Using these 1-parameter families of Morse functions we can extend $w_1$ to map from $W_2$ to an annulus with “bumps” on one side as shown in Figure \[F:builsing SWFs\] (b) and this map has an arc of indefinite folds on each bump. We can then smooth out the bumps by standard techniques from differential topology to obtain a map $w_2\colon W_2\ra S^1\times[0,\tfrac{2}{3}]$ in which each 2-handle attachment has created an arc of indefinite folds whose endpoints hit the boundary of $W_2$ transversely in the component that was affected by the handle attachment (Figure \[F:builsing SWFs\] (c)), let us call this component $\del_2 W_2$
*Step 3:* For the final step we first note that the restriction of $w_2$ over $S^1\times\{\tfrac{2}{3}\}$ is a circle valued Morse function with a pair of critical points of index 1 and 2 for each fold arc of $w_2$. The crucial observation is that the condition that ${\Gamma}$ is a circuit with switch $\mu$ implies that all these pairs of critical points cancel! Thus there is a standard homotopy, which we parametrize by $[\tfrac{2}{3},1]$, from $w_2|_{\del_{2}W_2}$ to a submersion that realizes this cancellation. We let $$W_{\mathfrak{S}}:=W_2\cup_{\del_{2}W_2} \del_{2}W_2\times[\tfrac{2}{3},1]$$ and extend $w_2$ by tracing out the homotopy over the newly added collar of $\del_{2}W_2$ to obtain a map $w_{\mathfrak{S}}\colon W_{\mathfrak{S}}\ra S^1\times[0,1]$. This last step removes all critical points from the boundary and introduces an interior cusp for any canceling pair. Clearly $w_{\mathfrak{S}}$ is an annular [simple wrinkled fibration]{}with base diagram as in Figure \[F:builsing SWFs\] (d).
Note that $W_{\mathfrak{S}}$ is diffeomorphic to $W_2$ and thus has the same relative handle decomposition. Moreover, it follows directly from the construction that ${\mathcal{R}}$ is a reference system for $w_{\mathfrak{S}}$ with ${\mathfrak{S}}$ as its twisted surface diagram.
In order to finish the proof of Theorem \[T:annular SWFs <-> twisted SDs\] we have to show that equivalent twisted surface diagram give equivalent annular [simple wrinkled fibrations]{}. Recall that an equivalence of surface diagram is a combination of two things: switching and a diffeomorphism. We will treat these separately.
\[T:diffeomorphis SDs -> equivalent SWFs\] If ${\mathfrak{S}}$ and ${\mathfrak{S}}'$ are diffeomorphic, then $w_{\mathfrak{S}}$ and $w_{{\mathfrak{S}}'}$ are equivalent.
Let ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$, ${\mathfrak{S}}'=({\Sigma}',{\Gamma}',\mu')$ and let $\phi\colon{\Sigma}\ra {\Sigma}'$ be a diffeomorphism such that ${\Gamma}'=\phi({\Gamma})$ and $\mu'=\phi\mu\phi\inv$. We will extend $\phi$ to a diffeomorphism $\hat{\phi}\colon W_{\mathfrak{S}}\ra W_{{\mathfrak{S}}'}$ which fits in the commutative diagram $$\xymatrix{
W_{\mathfrak{S}}\ar[dr]_{w_{\mathfrak{S}}} \ar[rr]^{\hat{\phi}} & & W_{{\mathfrak{S}}'}\ar[dl]^{w_{{\mathfrak{S}}'}} \\
& S^1\times[0,1] &
}$$ This will be done by going through the steps in the proof of Proposition \[T:annular SWFs <- twisted SDs\]. Let $W_i$ and $W_i'$, $i=1,2$, denote the 4-manifolds built in each step.
From the identity $\mu'=\phi\mu\phi\inv$ we see that $\phi$ induces a fiber preserving diffeomorphism ${\Sigma}(\mu)\ra {\Sigma}'(\mu')$. Taking the product with the identity, we obtain $\hat{\phi}_1\colon W_1\ra W_1'$.
In the second step, where the 2-handles are attached to the curves in ${\Gamma}$, we simply note that $\hat{\phi}_1$ maps the attaching regions into each other and can thus be extended over the 2-handles to $\hat{\phi}_2\colon W_2\ra W_2'$. Note that the smoothing of the bumpy annulus does not cause any trouble since it does not involve the total space.
For the third step observe that, given a homotopy from $w_2|_{\del_2 W_2}$ to a submersion, we can push it forward via $\hat{\phi}_2|_{\del_2 W_2}$ to obtain such a homotopy for $w_2'|_{\del_2 W_2'}$.
\[T:switching -> equivalent SWFs\] If ${\mathfrak{S}}$ is a twisted surface diagram, then $w_{\mathfrak{S}}$ and $w_{{\mathfrak{S}}^{[1]}}$ are equivalent.
If we take the canonical reference system for $w_{\mathfrak{S}}$ and swing the last reference path once around the boundary, we obtain a reference system which induces ${\mathfrak{S}}^{[1]}$. Thus $w_{\mathfrak{S}}$ and $w_{{\mathfrak{S}}^{[1]}}$ can be considered as the same annular [simple wrinkled fibration]{}.
Combining these two lemmas we obtain
\[T:equ SDs -> equ SWFs\] If ${\mathfrak{S}}$ and ${\mathfrak{S}}'$ are equivalent, then so are $w_{\mathfrak{S}}$ and $w_{{\mathfrak{S}}'}$.
Gluing ambiguities {#S:Gluing ambiguities}
------------------
Now that we know how to study annular [simple wrinkled fibrations]{}in terms of their twisted surface diagrams, recall that [simple wrinkled fibrations]{}over arbitrary base surfaces can be obtained from an annular ones by gluing suitable surface bundles to the boundary components. To be precise, let $w_0\colon W\ra A$ be an annular [simple wrinkled fibration]{}and let $\pi_\pm\colon Y_\pm\ra B_\pm$ be surface bundles over surfaces $B_\pm$ such that there are boundary components $C_\pm\subset B_\pm$ and fiber preserving diffeomorphisms $\psi_\pm\colon \pi_\pm\inv(C_\pm)\ra {\del_\pm}W$. Then we can form a [simple wrinkled fibration]{} $$w\colon Y_+\cup_{\psi_+} W \cup_{\psi_-} Y_- \lra B_+ \cup_{C_+} A \cup_{C_-} B_-.$$ Of course, different choices of gluing diffeomorphisms may lead to inequivalent [simple wrinkled fibrations]{}. If we fix a pair $\psi_\pm$ of gluing maps, then we can obtain any other such pair by composing with automorphisms (in the sense of Section \[S:mapping tori\]) of the boundary fibrations $w_0\colon{\del_\pm}W\ra S^1$. Obviously, isotopic gluing maps give rise to equivalent [simple wrinkled fibrations]{}and the gluing ambiguities are a priori parametrized by $$\pi_0\big(\Aut({\del_+}W, w)\big) \times \pi_0\big(\Aut({\del_-}W,w)\big)$$ However, it turns out that the first factor can be eliminated.
\[T:boundary diffeomorphisms extend\] Let $w\colon W\ra A$ be an annular [simple wrinkled fibration]{}. Then any fiber preserving diffeomorphism of ${\del_+}W$ extends to an auto-equivalence of $w$.
By Theorem \[T:annular SWFs <-> twisted SDs\] we can assume that $w$ is built from a twisted surface diagram ${\mathfrak{S}}=({\Sigma},{\Gamma},\mu)$ such that ${\del_+}W={\Sigma}(\mu)$. According to there are two types of automorphisms of ${\Sigma}(\mu)$, the *constant* ones coming from $C_{{\mathcal{M}}({\Sigma})}(\mu)$ and the *non-constant* ones originating from $\pi_1(\Diff({\Sigma}),\id)$. The statement that constant automorphisms of ${\del_+}W$ extend to auto-equivalences of $w$ is just a reformulation of Lemma \[T:diffeomorphis SDs -> equivalent SWFs\]. Thus it remains to treat the non-constant ones.
Recall that by Theorem \[T:Earle-Eells\] these only occur when ${\Sigma}$ has genus one. We can thus assume that ${\Sigma}=T^2$. A well known refinement of Theorem \[T:Earle-Eells\] states that the map $$\label{E:torus iso}
\pi_1\big(\Diff(T^2),\id\big)\ra \pi_1(T^2,x)$$ which sends an isotopy to the path traced out by a base point $x\in T^2$ during that isotopy is an isomorphism (see [@Earle-Eells]). Note that the fundamental group of $T^2$ is generated by the curves ${\gamma}_1$ and ${\gamma}_2$ (after choosing orientations, of course) if we take their unique intersection point as base point. Hence, we only have to extend the automorphisms coming from generators of $\pi_1\big(\Diff(T^2),\id\big)$ mapping to ${\gamma}_1$ and ${\gamma}_2$ in . If one parametrizes the torus by $S^1\times S^1\subset \C^2$ such that $S^1\times\set{1}$ maps to ${\gamma}_1$ and $\set{1}\times S^1$ maps to ${\gamma}_2$, then such generators are given by $$h^{{\gamma}_1}_t(\xi,\eta):=\big( e^{2\pi i\,t}\xi, \eta \big)
\quad\text{and}\quad
h^{{\gamma}_2}_t(\xi,\eta):=\big( \xi, e^{2\pi i\,t}\eta \big)
\quad\quad
(t\in[0,1])$$ and we denote the corresponding automorphisms of ${\Sigma}(\mu)$ by $$\varphi_i(x,t):=\big( h^{{\gamma}_i}_t(x),t \big).$$ In order to extend $\varphi_i$ to $Z_{\mathfrak{S}}$ we take one step back and homotope the path $h^{{\gamma}_i}$ to be constant outside the interval where the 2-handle corresponding to ${\gamma}_i$ is attached. These intervals (times \[0,1\]) are highlighted in Figure \[F:gluing\].
![The relevant regions for extending non-constant automorphisms.[]{data-label="F:gluing"}](gluing.pdf)
Outside the preimage of the regions shown in Figure \[F:gluing\] we can simply extend $\varphi_i$ as the identity. In these region, observe that $h^{{\gamma}_i}_t$ fixes ${\gamma}_i$ set wise at all times, it just rotates it more and more as $t$ increases. It is easy to see that these rotations can be extended across the 2-handles in a way that respects the fibration structure.
\[R:gluing without cusps\] The genus one case of Example \[eg:ADK sphere\] shows that this Lemma does not hold without in the absence of cusps. The above proof breaks down at the point where we need the vanishing cycles to generate the fundamental group.
[Simple wrinkled fibrations]{}over the disk and the sphere {#S:SWFs over disk and sphere}
==========================================================
We now leave the general theory behind and focus on untwisted surface diagrams, i.e. pairs $({\Sigma},{\Gamma})$ where ${\Gamma}$ is a closed circuit in ${\Sigma}$. By the correspondence established in the previous section such a surface diagram corresponds to an annular [simple wrinkled fibration]{}whose higher genus boundary component has trivial monodromy. We can thus fill this boundary component with ${\Sigma}\times D^2$ using some fiber preserving diffeomorphism of ${\Sigma}\times S^1$ to obtain a [simple wrinkled fibration]{}over the disk. (Note that the boundary of the disk is contained in the lower genus region; we will refer to such fibrations as *descending* [simple wrinkled fibrations]{}over the disk.) Furthermore, by Lemma \[T:boundary diffeomorphisms extend\] different choices of gluing diffeomorphisms produce equivalent [simple wrinkled fibrations]{}. Altogether, we have established the following.
\[T:SDs and descending SWFs\] There is bijective correspondence between (untwisted) surface diagrams up to equivalence and descending [simple wrinkled fibrations]{}over the disk up to equivalence.
As mentioned before, when we speak of surface diagrams, we will always mean untwisted surface diagrams. This will not lead to confusion since we will not encounter any twisted surface diagrams anymore.
For a surface diagram ${\mathfrak{S}}=({\Sigma},{\Gamma})$ we denote the corresponding [simple wrinkled fibration]{}by $w_{\mathfrak{S}}\colon Z_{\mathfrak{S}}\ra D^2$ or, by a slight abuse of notation, simply by $Z_{\mathfrak{S}}$ with the map to the disk implicitly understood. The boundary of $Z_{\mathfrak{S}}$ fibers over $S^1$ and if this boundary fibration is trivial, then we can *close off* to a [simple wrinkled fibration]{}over $S^2$. Recall that Theorem \[T:Williams existence\] tells us that we can obtain *all* smooth, closed, oriented 4-manifolds by this process. It is thus of great interest to understand which surface diagrams describe closed 4-manifolds. The following example indicates that this might be a hard problem.
\[eg:arbitrary monodromy\] Let ${\Sigma}$ be a closed, orientable surface together with a mapping class $\phi\in{\mathcal{M}}({\Sigma})$. Then any factorization of $\mu$ into positive Dehn twists yields a Lefschetz fibration over the disk whose boundary can be identified with the mapping torus ${\Sigma}(\phi)=({\Sigma}\times[0,1])/(x,1)\sim(\phi(x),0)$. As in Example \[eg:wrinkled Lefschetz fibrations\] we can turn this Lefschetz fibration into a descending [simple wrinkled fibration]{}without changing the boundary. Thus any surface bundle over the circle (with closed fibers) bounds some descending [simple wrinkled fibration]{}over the disk and any mapping class can be realized as the monodromy of a surface diagram.
In fact, the situation is very similar to the theory of Lefschetz fibrations. Any word in positive Dehn twists (or, equivalently, a finite sequence of [simple closed curves]{}) on a closed, oriented surface determines a Lefschetz fibration over the disk, the boundary fibers over the circle with monodromy given by the product of the Dehn twists and if this monodromy is trivial, then one can close off to a Lefschetz fibration over $S^2$. Just as an arbitrary product of Dehn twists will will not be isotopic to the identity, a surface diagram will not give rise to a [simple wrinkled fibration]{}over $S^2$. The advantage of the Lefschetz setting is the direct control over the boundary.
The monodromy of a surface diagram {#S:monodromy of SDs}
----------------------------------
In order to obtain a more intrinsic description of the boundary of $Z_{\mathfrak{S}}$ in terms of ${\mathfrak{S}}$ we need a little detour.
Let $a,b\subset {\Sigma}$ be a pair of [simple closed curves]{}in a surface ${\Sigma}$ that intersect transversely in a single point. We denote by ${\Sigma}_a$ and ${\Sigma}_b$ the surfaces obtained by surgery on the curves $a$ and $b$, respectively. To be concrete, we fix tubular $\nu a$ and $\nu b$ and picture ${\Sigma}_a$ (resp. ${\Sigma}_b$) as the result of filling in the two boundary components of ${\Sigma}\setminus\nu a$ (resp. ${\Sigma}\setminus\nu b$) with disks. By the assumption on intersections we can assume that $\nu(a\cup b):=\nu a\cup\nu b$ is diffeomorphic to a once punctured torus – for convenience we will also assume that it has a smooth boundary in ${\Sigma}$. Observe that ${\Sigma}\setminus \nu(a\cup b)$ has one boundary component and is contained in both ${\Sigma}_a$ and ${\Sigma}_b$ as a subsurface. Furthermore, the closure of $\nu b\setminus\nu a$ (resp. $\nu a\setminus\nu b$) is a disk in ${\Sigma}_a$ (resp. ${\Sigma}_b$). It follows that, up to isotopy, there is a unique diffeomorphism $$\kappa_{a,b}\colon{\Sigma}_a\ra {\Sigma}_b$$ which can be assumed to map $\nu b\setminus \nu a$ onto $\nu a\setminus\nu b$.
Now let ${\mathfrak{S}}=({\Sigma};{\gamma}_1,\dots,{\gamma}_l)$ be a surface diagram and consider the associated [simple wrinkled fibration]{}$w_{\mathfrak{S}}\colon Z_{\mathfrak{S}}\ra D^2$. Then each adjacent pair of curves ${\gamma}_i$ and ${\gamma}_{i+1}$ fits the above situation and we thus get a collection of diffeomorphisms $$\kappa_{{\gamma}_i,{\gamma}_{i+1}}\colon {\Sigma}_{{\gamma}_i}\ra {\Sigma}_{{\gamma}_{i+1}}.$$ Moreover, it follows from the definition of surface diagrams that the composition $$\mu_{\mathfrak{S}}:=\kappa_{{\gamma}_c,{\gamma}_1}\circ\kappa_{{\gamma}_{c-1},{\gamma}_c}\circ \dots\circ\kappa_{{\gamma}_1,{\gamma}_2}$$ maps ${\Sigma}_{{\gamma}_1}$ to itself and it is easy to see that its isotopy class does not depend on any of the implicit choices involved in its definition.
\[D:monodromy of SDs\] The mapping class $\mu_{\mathfrak{S}}\in{\mathcal{M}}({\Sigma}_{{\gamma}_1})$ represented by the diffeomorphism above is called the *monodromy* of ${\mathfrak{S}}$.
This name is justified by the following lemma.
\[T:boundary monodromy\] Let ${\mathfrak{S}}=({\Sigma},{\Gamma})$ be a surface diagram. Then the boundary fibration $(\del Z_{\mathfrak{S}},w_{\mathfrak{S}})$ can be identified with the mapping torus ${\Sigma}_{{\gamma}_1}(\mu_{\mathfrak{S}})$.
By the construction of $w_{\mathfrak{S}}$ its fiber over the origin is naturally identified with ${\Sigma}$. Furthermore, recall that the annular fibration associated to ${\mathfrak{S}}$ is equipped with a reference system whose reference paths we can naturally extend from the annulus to the disk by connecting them to the origin. The result is a collection of reference paths $R_1,\dots,R_c$ from the origin to the boundary of the disk and we denote its endpoints by $\theta_1\dots,\theta_c\in S^1$. Observe that such a reference path, $R_i$ say, gives rise to an identification of the fiber over $\theta_i$ with the surface ${\Sigma}_{{\gamma}_i}$ obtained from surgery on ${\gamma}_i$ where ${\gamma}_i$ is the vanishing cycle associated to $R_i$.
Now consider the region in the base bounded by two adjacent reference path $R_i$ and $R_{i+1}$. Using a suitable notion of parallel transport we see that the preimage of this region contains a trivial bundle with fiber ${\Sigma}\setminus{\nu({\gamma}_i\cup{\gamma}_{i+1})}$. In particular, the parallel transport along the boundary segment from $\theta_i$ to $\theta_{i+1}$ restricts to the identity on the complement of $\nu({\gamma}_i\cup{\gamma}_{i+1})$ and thus must be isotopic to $\kappa_{{\gamma}_i,{\gamma}_{i+1}}$ and the claim follows.
It is also possible to describe the monodromy in terms of the original surface ${\Sigma}$. This takes us on another small detour. Let $a\subset{\Sigma}$ be a non-separating [simple closed curve]{}in a surface ${\Sigma}$ and let ${\mathcal{M}}({\Sigma},a)$ denote the subgroup of ${\mathcal{M}}(S)$ consisting of all elements that fix $a$ up to isotopy. It is well known that there is a short exact sequence[^6] $$\label{E:cut sequence}
\xymatrix{
1 \ar[r] &
\scp{\tau_a} \ar[r] &
{\mathcal{M}}({\Sigma},a) \ar[r]^{\mathrm{cut}_a} &
{\mathcal{M}}({\Sigma}\setminus a) \ar[r] &
1
}$$ where ${\Sigma}\setminus a$ is viewed as a twice punctured surface. The complement ${\Sigma}\setminus a$ can be related to the surgered surface ${\Sigma}_a$ as follows. In ${\Sigma}_a$ there is an obvious pair of points, namely the centers of the surgery disks. If we denote by ${\Sigma}_a^*$ the surface obtained by marking these points, then ${\Sigma}\setminus a$ is canonically identified (at least up to isotopy) with ${\Sigma}_a^*$ and thus ${\mathcal{M}}({\Sigma}\setminus a)$ is canonically isomorphic to ${\mathcal{M}}({\Sigma}_a^*)$. Hence, we can define the *surgery homomorphism* $$\sigma_a\colon {\mathcal{M}}({\Sigma},a)\ra {\mathcal{M}}({\Sigma}_a)$$ as the composition $$\xymatrix{
{\mathcal{M}}({\Sigma},a) \ar@/^1pc/[rrr]^{\sigma_a} \ar[r]_{\mathrm{cut}_a} &
{\mathcal{M}}({\Sigma}\setminus a) \ar[r]_{\cong} &
{\mathcal{M}}({\Sigma}_a^*) \ar[r]_{\mathrm{forget}} &
{\mathcal{M}}({\Sigma}_a)
}$$ where the last map is induced by forgetting the marked points in ${\Sigma}_a^*$.
Applying this to surface diagram we obtain the following.
\[T:lifting the monodromy\] Let ${\mathfrak{S}}=({\Sigma};{\gamma}_1,\dots,{\gamma}_c)$ be a surface diagram. Then $$\tilde{\mu}_{\mathfrak{S}}:=\tau_{\tau_{{\gamma}_c}({\gamma}_1)} \circ
\tau_{\tau_{{\gamma}_{c-1}}({\gamma}_c)} \circ
\tau_{\tau_{{\gamma}_1}({\gamma}_2)} \in{\mathcal{M}}({\Sigma})$$ is contained in ${\mathcal{M}}({\Sigma},{\gamma}_1)$ and satisfies $\sigma_{{\gamma}_1}(\tilde{\mu}_{\mathfrak{S}})=\mu_{\mathfrak{S}}$.
We claim that this follows from the observation that $$\tau_{\tau_{{\gamma}_i}({\gamma}_{i+1})}({\gamma}_i)=\tau_{{\gamma}_{i}}\tau_{{\gamma}_{i+1}}\tau_{{\gamma}_{i}}\inv({\gamma}_i)={\gamma}_{i+1}.$$ Indeed, this obviously implies the first statement and the second follows from the fact that the diagrams $$\xymatrix{
{\Sigma}\ar[d]^{\tau_{\tau_{{\gamma}_i}({\gamma}_{i+1})}} &
{\Sigma}\setminus{\gamma}_{i} \ar[d] \ar[l] \ar[r] &
{\Sigma}_{{\gamma}_{i}}^* \ar[d]^{\kappa_{{\gamma}_i,{\gamma}_{i+1}}}
\\
{\Sigma}&
{\Sigma}\setminus{\gamma}_{i+1} \ar[l] \ar[r] &
{\Sigma}_{{\gamma}_{i+1}}^*
}$$ commute up to isotopy.
The above makes it interesting to study the map $\sigma_{{\gamma}_1}$ and its kernel.
\[T:generating MCG ficing a curve\] Let $a\subset{\Sigma}$ be a non-separating [simple closed curve]{}. Then the group ${\mathcal{M}}({\Sigma},a)$ is generated by elements of the form $\tau_c$ where $i(a,c)=0$ and $\Delta_{a,b}:=(\tau_a\tau_b)^3$ where $i(a,b)=1$.
We will refer to the mapping classes $\Delta_{a,b}$ as *$\Delta$-twists*.
It follows from the short exact sequence that we can obtain a generating set for ${\mathcal{M}}({\Sigma},a)$ by lifting a generating set for ${\mathcal{M}}({\Sigma}\setminus a)$ and adding the Dehn twist about $a$. As a generating set for ${\mathcal{M}}({\Sigma}\setminus a)$ we can take the collection Dehn twists and so called *half-twists* about simple arcs connecting the two punctures. Then the Dehn twists in ${\mathcal{M}}({\Sigma}\setminus a)$ have obvious lifts in ${\mathcal{M}}({\Sigma})$ and it is easy to see that each half-twist lifts to a $\Delta$-twist.
\[T:kernel of surgery homomorphism\] The kernel of the surgery homomorphism $\sigma_a\colon {\mathcal{M}}(S,a)\ra {\mathcal{M}}({\Sigma}_a)$ contains the Dehn twist about $a$ and all $\Delta$-twists involving $a$.
The expert will have noticed that the mapping class $\tilde{\mu}_{\mathfrak{S}}$ in Lemma \[T:lifting the monodromy\] is simply the monodromy of the boundary of the Lefschetz part of the simplified broken Lefschetz fibration obtained from $w_{\mathfrak{S}}$ by unsinking all the cusps. Of course, there are many different lifts of $\mu_{\mathfrak{S}}$ to ${\mathcal{M}}({\Sigma})$. For example, it follows from the braid relations for the pairs of adjacent curves that $$\begin{aligned}
\tilde{\mu}_{\mathfrak{S}}&= \tau_{{\gamma}_1}^{-c}
(\tau_{{\gamma}_c} \tau_{{\gamma}_1})
(\tau_{{\gamma}_{c-1}} \tau_{{\gamma}_c})
\dots
(\tau_{{\gamma}_1} \tau_{{\gamma}_2}) \\
&= \tau_{{\gamma}_1}^{-2c}
(\tau_{{\gamma}_c} \tau_{{\gamma}_1} \tau_{{\gamma}_c})
(\tau_{{\gamma}_{c-1}} \tau_{{\gamma}_c} \tau_{{\gamma}_{c-1}})
\dots
(\tau_{{\gamma}_1} \tau_{{\gamma}_2} \tau_{{\gamma}_1})
\end{aligned}$$ and since $\tau_{{\gamma}_1}$ is contained in the kernel of $\sigma_{{\gamma}_1}$ we obtain two other choices.
We illustrate these mapping class group techniques to produce many examples of surface diagrams with trivial monodromy.
\[eg:double monodromy\] Given an arbitrary circuit ${\Gamma}=({\gamma}_1,\dots,{\gamma}_l)$ in an oriented surface ${\Sigma}$ we can form a closed circuit $D{\Gamma}:=({\gamma}_1,\dots,{\gamma}_{l-1},{\gamma}_l,{\gamma}_{l-1},\dots,{\gamma}_2)$ which we call the *double* of ${\Gamma}$. We claim that the surface diagram $D{\mathfrak{S}}:=({\Sigma},D{\Gamma})$ has trivial monodromy. For convenience let us write $\tau_i=\tau_{{\gamma}_i}$. As explained above the monodromy of $D{\mathfrak{S}}$ can be lifted to ${\mathcal{M}}({\Sigma})$ as $$\begin{aligned}
\mu&=
(\tau_2 \tau_1 \tau_2)
\dots
(\tau_{l-2} \tau_{l-1} \tau_{l-2})
(\tau_{l-1} \tau_{l} \tau_{l-1})
(\tau_{l} \tau_{l-1} \tau_{l})
(\tau_{l-1} \tau_{l-2} \tau_{l-1})
\dots
(\tau_{1} \tau_{2} \tau_{1})\\
&= (\tau_2 \tau_1 \tau_2)
\dots
(\tau_{l-2} \tau_{l-1} \tau_{l-2})
\Delta_{{\gamma}_{l-1},{\gamma}_{l}}
(\tau_{l-1} \tau_{l-2} \tau_{l-1})
\dots
(\tau_{1} \tau_{2} \tau_{1}).
\end{aligned}$$ Our goal is to factor this expression into a sequence of $\Delta$-twists involving ${\gamma}_1$. The key observation is that $$\begin{aligned}
&(\tau_{l-2} \tau_{l-1} \tau_{l-2})
\Delta_{{\gamma}_{l-1},{\gamma}_{l}}
(\tau_{l-1} \tau_{l-2} \tau_{l-1}) \\
=&
(\tau_{l-2} \tau_{l-1} \tau_{l-2})
\Delta_{{\gamma}_{l-1},{\gamma}_{l}}
(\tau_{l-2} \tau_{l-1} \tau_{l-2}) \\
=&
(\tau_{l-2} \tau_{l-1} \tau_{l-2})
\Delta_{{\gamma}_{l-1},{\gamma}_{l}}
(\tau_{l-2} \tau_{l-1} \tau_{l-2})\inv
\Delta_{{\gamma}_{l-2},{\gamma}_{l-1}}
\\
=&
\Delta_{\tau_{l-2} \tau_{l-1} \tau_{l-2}({\gamma}_{l-1}),\tau_{l-2} \tau_{l-1} \tau_{l-2}({\gamma}_{l})}
\Delta_{{\gamma}_{l-2},{\gamma}_{l-1}} \\
=&
\Delta_{{\gamma}_{l-2},\tau_{l-2} \tau_{l-1} \tau_{l-2}({\gamma}_{l})}
\Delta_{{\gamma}_{l-2},{\gamma}_{l-1}}.
\end{aligned}$$ Applying this repeatedly we eventually obtain $$\mu=
\Delta_{{\gamma}_{1},\delta_{l}}
\Delta_{{\gamma}_{1},\delta_{l-1}}
\dots
\Delta_{{\gamma}_{1},\delta_{2}}$$ where $\delta_k:=(\tau_1 \tau_2 \tau_1)\dots(\tau_{k-2} \tau_{k-1} \tau_{k-2}) ({\gamma}_k)$. Hence, the monodromy of $D{\mathfrak{S}}$ is trivial by Corollary \[T:kernel of surgery homomorphism\].
If ${\Gamma}$ was a closed circuit to begin with so that ${\mathfrak{S}}=({\Sigma},{\Gamma})$ is a surface diagram, then one can show that $Z_{D{\mathfrak{S}}}$ closes off to $DZ_{\mathfrak{S}}=Z_{\mathfrak{S}}\cup_\del\overline{Z_{\mathfrak{S}}}$, the double of $Z_{\mathfrak{S}}$, whence the name.
Drawing Kirby diagrams {#S:Kirby diagrams}
----------------------
In this section we show how to translate surface diagrams into Kirby diagrams of the associated [simple wrinkled fibrations]{}. For the necessary background we refer the reader to [@GS]. Throughout, we use Akbulut’s *dotted circle notation* for 1-handles to avoid ambiguities for framing coefficients.
### Descending [simple wrinkled fibrations]{}
Let $w\colon Z\ra D^2$ be a descending [simple wrinkled fibration]{}of genus $g$ with surface diagram ${\mathfrak{S}}=({\Sigma}_g;{\gamma}_1,\dots,{\gamma}_c)$. Recall that the associated handle decomposition of $Z$ is obtained from (some handle decomposition of) ${\Sigma}_g\times D^2$ by attaching 2-handles along ${\gamma}_i\subset{\Sigma}_g\times\set{\theta_i}$ with respect to the fiber framing where $\theta_1,\dots,\theta_c\in S^1$ are ordered according to the orientation on $S^1$. So in order to draw a Kirby diagram for $Z$ we need to find a diagram for ${\Sigma}\times D^2$ in which the fibers of the boundary should be as clearly visible as possible.
A convenient choice is the diagram shown in Figure \[F:surface times disk 1\] which is induced from the obvious handle decomposition of ${\Sigma}_g$ with one 0-handle, $2g$ 1-handles and one 2-handle. One fiber of ${\Sigma}_g\times S^1$, which we identify with ${\Sigma}_g$, is clearly visible and the canonical generators $a_1,b_1,\dots,a_g,b_g$ for $H_1({\Sigma}_g)$ are also indicated. We have chosen the orientations such that $\scp{a_i,b_i}_{{\Sigma}_g}=1$. Another advantage of this picture is that the fiber framing agrees with the blackboard framing. One minor drawback is that the picture does not immediately show *all* fibers of ${\Sigma}_g\times S^1$ but only an interval worth of them (just thicken the surface a little). However, this is actually enough for our purposes since we only need the fibers over the interval $[\theta_1,\theta_c]\subset S^1$. To get the orientations right we require that the orientation of the fiber agrees with the standard orientation of the plane and, according to the “fiber first convention”, the positive $S^1$-direction points out of the plane.
![A diagram for ${\Sigma}_g\times D^2$ where fiber and blackboard framing agree. The red curves show a basis for $H_1({\Sigma}_g)$.[]{data-label="F:surface times disk 1"}](surface_times_disk_ff=bf.pdf)
With this understood, it is easy to locate the attaching curves of the fold handles in the diagram and it remains to determine their framing coefficients. More generally, we can describe the linking form of the link corresponding to the fold handles. It should be no surprise that the framing and linking information in the diagram depends on our choice of the handle decomposition for ${\Sigma}_g$.
Let ${\gamma}\subset{\Sigma}_g$ be a [simple closed curve]{}. After choosing an orientation its homology class $[{\gamma}]\in H_1({\Sigma})$ can be expressed as $$[{\gamma}]=\sum_{i=1}^g \big( n_{a_i}({\gamma})\,a_i + n_{b_i}({\gamma})\,b_i \big).$$ We identify ${\Sigma}_g$ with ${\Sigma}_g\times\set{0}$ and, by a slight abuse of notation, we continue to denote the canonical push-off of ${\gamma}$ to ${\Sigma}_g\times\set{z}$, $z\in D^2$, by ${\gamma}$.
\[T:framings and linking\] For a [simple closed curve]{}${\gamma}\subset{\Sigma}_g\times\set{\theta}$, $\theta\in [\theta_1,\theta_c]\subset S^1$, the framing coefficient of the fiber framing in Figure \[F:surface times disk 1\] is given by $$\label{E:fiber framing}
\mathrm{fr}({\gamma})= \sum_{i=1}^g n_{a_i}({\gamma}) n_{b_i}({\gamma}).$$ Furthermore, if ${\gamma}\subset{\Sigma}_g\times\set{\theta}$ and ${\gamma}'\subset{\Sigma}_g\times\set{\theta'}$, $\theta,\theta'\in[\theta_1,\theta_c]$, are two oriented [simple closed curves]{}, then their linking number in Figure \[F:surface times disk 1\] is $$\label{E:linking}
\begin{split}
\mathrm{lk}({\gamma},{\gamma}')
=&\frac{1}{2} \mathrm{sgn}(\theta-\theta') \big\langle {\gamma},{\gamma}' \big\rangle \\
&+\frac{1}{2} \sum_{i=1}^g \big[ n_{a_i}({\gamma})n_{b_i}({\gamma}')+n_{a_i}({\gamma}')n_{b_i}({\gamma}) \big]
\end{split}$$ where $\scp{{\gamma},{\gamma}'}$ is the algebraic intersection number of ${\gamma}$ and ${\gamma}'$ in ${\Sigma}_g$ and $\mathrm{sgn}$ denotes the sign of a real number[^7].
First observe that ${\gamma}\subset {\Sigma}_g\times\set{\theta}$ can be isotoped off the 2-handle to be completely visible in Figure \[F:surface times disk 1\] and, since the fiber framing and blackboard framing agree, its framing coefficient is given by its writhe in the diagram, i.e. the signed count of crossings with some chosen orientation. From the way the diagram is drawn it is clear that each crossing is caused by ${\gamma}$ running over $a_i$ *and* $b_i$ for some $i$ and that their signed sum is given by the right hand side of .
The statement about linking numbers follows from a similar count of crossings. Recall that the linking number of two oriented knots can be computed from any link diagram as half of the signed number of crossings. The second term on the right hand side of arises just as above. However, the first term deserves some explanation. Each (transverse) intersection point of ${\gamma}$ and ${\gamma}'$ in ${\Sigma}_g$ contributes a crossing in the diagram. Now, the sign of the crossing depends one two things: the sign of the intersection point and the information which strand is on top in the diagram. From Figure \[F:intersections and crossings\] we see that the contribution of each crossing is exactly as in .
![An intersection in a surface diagram and its crossing in the Kirby diagram.[]{data-label="F:intersections and crossings"}](crossing+linking.pdf)
\[R:intersection form\] Formula \[E:linking\] can be used to obtain a description of the intersection form of the 4-manifold $Z_{\mathfrak{S}}$ described by a surface diagram ${\mathfrak{S}}$ which only uses the data in ${\mathfrak{S}}$. Moreover, since \[E:linking\] only depends on the homology classes of the curves in ${\mathfrak{S}}$, so do the intersection form and, in particular, the signature of $Z_{\mathfrak{S}}$. We will return to this observation in a future publication.
The diagrams of [simple wrinkled fibrations]{}derived from Figure \[F:surface times disk 1\] are good for abstract reasoning, however, in practice it is convenient to start with a cleaner diagram for ${\Sigma}_g\times D^2$ such as the one shown in Figure \[F:surface times disk 2\]. In this picture, the fiber appears as the boundary sum of regular of the basis curves $\set{a'_i,b_i}_{i=1}^{g}$ which, in turn, appear as meridians to the dotted circles.
![A cleaner diagram of ${\Sigma}_g\times D^2$.[]{data-label="F:surface times disk 2"}](surface_times_disk_int2.pdf)
The framing coefficient of the fiber framing for [simple closed curves]{}on a fiber in Figure \[F:surface times disk 2\] can be computed as follows. It is not hard to see that Figure \[F:surface times disk 2\] is obtained from Figure \[F:surface times disk 1\] by a sequence of 1-handle slides and an isotopy of the 2-handle and vice versa. Note that these moves do not change the framing coefficients of any other 2-handles that might have been around. Moreover, during the moves, the $b$-curves remain fixed, while the $a$-curves undergo some changes. When pulling $a_i'$ in Figure \[F:surface times disk 2\] back to Figure \[F:surface times disk 1\] one obtains a representative for the element $$[a_1,b_1]\ast\dots\ast[a_{i-1},b_{i-1}]\ast a_i\in\pi_1({\Sigma}_g)$$ where $[x,y]=xyx\inv y\inv$. The important observation is that while this curve is not isotopic to $a_i$ it does represent the same homology class. As a consequence, formula can be used for Figure \[F:surface times disk 2\] with $a_i$ replaced by $a_i'$.
### Closing off and the last 2-handle {#S:the last 2-handle}
Recall that our motivation comes from Williams’ theorem that all closed, oriented 4-manifolds admit [simple wrinkled fibrations]{}over $S^2$. We have seen that these can be described (up to equivalence) by surface diagrams with trivial monodromy and we have already mentioned that it is in general not easy to check whether the monodromy of a given surface diagram is trivial. But the situation is even worse. Say that we know for some reason that a given surface diagram has trivial monodromy and let us also assume that the genus is at least three so that there are no gluing ambiguities. Even in this case it is not clear at all how the surface diagram encodes the information to complete the Kirby diagram.
To be more precise, let $w\colon X\ra S^2$ be a [simple wrinkled fibration]{}with surface diagram ${\mathfrak{S}}$. Let $\nu{\Sigma}_-$ be a of a lower genus fiber and let $Z:=X\setminus\nu{\Sigma}_-$. Then $w$ restricts to a descending [simple wrinkled fibration]{}on $Z$ and $\del Z$ can be identified with ${\Sigma}_-\times S^1$ so that ${\mathfrak{S}}$ must have trivial monodromy. We can draw a Kirby diagram for $Z$ as described in the previous section and to complete it to a diagram for $X$ we have to understand how to glue $\nu{\Sigma}_-$ back in.
We can choose a handle decomposition for $\nu{\Sigma}_-$ with one 0-handle, $2g({\Sigma}_-)$ 1-handles and one 2-handle. Turning this upside down results in a relative handle decomposition on $\del Z\cong {\Sigma}_-\times S^1$ with one 2-handle, $2g({\Sigma}_-)$ 3-handles and a 4-handle. The general theory tells us that the 3- and 4-handles attach in a standard way once we know how to attach the 2-handle. Unfortunately, it turns out to be rather difficult to locate this *last 2-handle* in the Kirby diagram for $Z$.
Our knowledge about the last 2-handle is a priori limited to the following observation. If we identify $\nu{\Sigma}_-$ with ${\Sigma}_-\times D^2$, then the attaching curve of the last 2-handle corresponds to $\set{p}\times \del D^2$ for some $p\in{\Sigma}_-$. In particular, we see that it must be attached along a section of the boundary fibration $(\del Z,w)$.
\[R:closing off by Kirby moves\] Given a surface diagram ${\mathfrak{S}}$ with trivial monodromy, there is a general method for finding possible last 2-handles for $Z_{\mathfrak{S}}$ which is not very conceptual but still useful in some situations.[^8] One considers a Kirby diagram for $Z_{\mathfrak{S}}$ as a surgery diagram for $\del Z_{\mathfrak{S}}$ and performs (3-dimensional) Kirby moves until the fibration structure is clearly visible as ${\Sigma}_-\times S^1$. In such a diagram it is easy to locate the possible attaching curves for last 2-handles. One can then pull back these curves to the original diagram by undoing the moves and dragging the curves along.
Just as in the Lefschetz case, the situation becomes easier if one knows that $Z_{\mathfrak{S}}$ can be closed off to a fibration over $S^2$ which admits a section. The proof of the following lemma is the same as in the Lefschetz case and we refer the reader to [@GS].
\[T:closing off with section\] Let $w\colon X\ra S^2$ be a [simple wrinkled fibration]{}with surface diagram ${\mathfrak{S}}$. If $w$ admits a section of self-intersection $k$, then the last two handle appears in the diagram for $Z_{\mathfrak{S}}$ as a $k$-framed meridian of the 2-handle corresponding to the fiber. Furthermore, if ${\mathfrak{S}}$ is a surface diagram and a meridian as above can be used to attach the last 2-handle, then the corresponding [simple wrinkled fibration]{}admits a section of self-intersection $k$.
In order to illustrate Remark \[R:closing off by Kirby moves\] and Lemma \[T:closing off with section\] as well as our method of drawing Kirby diagrams we give an example which is also a warm up for the next section.
\[eg:Kirby diagram example\] Let $a,b\subset{\Sigma}_g$ be a geometrically dual pair of [simple closed curves]{}. We claim that ${\mathfrak{S}}=({\Sigma}_g;a,\tau_b(a),b)$ is a surface diagram for ${\Sigma}_{g-1}\times S^2\#{\overline{\mathbb{C}P^2}}$.
![Manifolds with surface diagram $({\Sigma}_g;a,\tau_b(a),b)$[]{data-label="F:blow-up preview"}](blow-up_example.pdf)
We can assume that $a$ and $b$ are the standard generators $a_1$ and $b_1$ in Figure \[F:surface times disk 2\] and Figure \[F:blow-up preview\] shows the final Kirby diagram. In order to see how we got there let us first ignore all the blue components. What is left is just the Kirby diagram for $Z_{\mathfrak{S}}$. The framings on the fold handles can either be computed using Lemma \[T:framings and linking\] (together with Proposition \[T:Picard-Lefschetz formula\]) or by hand[^9]. We now perform the obvious handle moves: using the meridians to the two 1-handles on the left we first unlink the $-1$-framed fold handle (corresponding to $\tau_b(a)$) to obtain a $-1$-framed unknot isolated from the rest of the diagram, then we unlink the black 2-handle (corresponding to the fiber) and finally cancel the 1-handles and their meridians. Obviously, the thus obtained diagram shows ${\Sigma}_{g-1}\times D^2 \# {\overline{\mathbb{C}P^2}}$ and the boundary is clearly visible as ${\Sigma}_{g-1}\times S^1$. Moreover, it is easy to see that the last 2-handle can be attached along a 0-framed meridian to the fiber 2-handle and the resulting manifold is ${\Sigma}_{g-1}\times S^2 \# {\overline{\mathbb{C}P^2}}$ as claimed. Finally, since we attached the last 2-handle in a region that was not affected by the Kirby moves it will not change when we undo the moves again and we arrive at Figure \[F:blow-up preview\]. Lemma \[T:closing off with section\] then tells us that the corresponding [simple wrinkled fibration]{}will have a section of self-intersection zero.
Note that for $g\geq 3$ the way we have attached the last 2-handle is unique. In the lower genus cases there are more options. However, in any case one will end up with a blow-up of some surface bundle over $S^2$.
Relation to broken Lefschetz fibrations {#S:relation to BLFs}
---------------------------------------
Let $w\colon X\ra B$ be a [simple wrinkled fibration]{}. After trading all the cusps for Lefschetz singularities by applying Lekili’s unsinking modification we obtain a broken Lefschetz fibration $$\beta_w\colon X \ra B$$ with one round singularity, smoothly embedded in the base, and all its Lefschetz points on the higher genus side. If the base is the sphere or the disk, then $\beta_w$ is a *simplified broken Lefschetz fibration* in the sense of [@Baykur2] and thus induces another handle decomposition of $X$.
In order to relate these two handle decompositions, let us briefly review how a handle decomposition is obtained from a simplified broken Lefschetz fibration $\beta\colon X\ra B$. Much in the spirit of [simple wrinkled fibrations]{}one chooses a reference point in the higher genus region together with a collection of disjointly embedded arcs $L_1,\dots,L_k,R\subset B$, where $k$ is the number of Lefschetz singularities, emanating from the reference point such that each $L_i$ ends in a Lefschetz point and $R$ passes through the round singularity once. Such a system of arcs is known as a *Hurwitz system* for $\beta$. The arcs in a Hurwitz system then give rise to [simple closed curves]{}in the reference fiber ${\Sigma}$ to which we shall refer to as the *Lefschetz vanishing cycles* $\lambda_1,\dots,\lambda_k\subset{\Sigma}$ and the *round vanishing cycle* $\rho$. A handle decomposition of $X$ is then given as follows:
- Start with ${\Sigma}\times D^2$
- Going around $S^1$ attach a *Lefschetz handle* along the $\lambda_i$ pushed off into fibers over $S^1$, i.e. 2-handles with framing $-1$ the fiber framing
- Attach a *round 2-handle* along $\rho$
The round 2-handle decomposes into a 2-handle and a 3-handle such that the 3-handle goes over the 2-handle geometrically twice and the 2-handle is attached along $\rho$ the fiber framing. (For more details see [@Baykur2].)
Now let $w\colon X\ra B$ be a [simple wrinkled fibration]{}and let $\beta_w$ be the associated simplified broken Lefschetz fibration. Given a reference system ${\mathcal{R}}=\set{R_i}$ for $w$ with associated surface diagram $(\Sigma,{\Gamma})$ there is a canonical Hurwitz system for $\beta_w$. Since the unsinking homotopy is supported near the cusps we can assume that the nothing happens around the reference paths. Now observe that the arcs $R_i$ cut the higher genus region into triangles each containing a single Lefschetz singularity of $\beta_w$. Thus, up to isotopy, there is a unique arc $L_i$ in the triangle bounded by $R_i$ and $R_{i+1}$ going from the reference fiber to the Lefschetz singularity and for the round singularity we take the arc $R=R_1$. According to Lekili [@Lekili], the vanishing cycles of $\beta_w$ this Hurwitz system are given by $$\lambda_i=\tau_{{\gamma}_i}({\gamma}_{i+1}) \quad\text{and}\quad \rho={\gamma}_1.$$ We can go from the handle decomposition induced by $\beta_w$ to the one induced by $w$ using the following handlebody interpretation of the (un-)sinking deformation.
![A Lefschetz singularity (a) before and (b) after sinking.[]{data-label="F:SWF vs BLF"}](SWF_vs_BLF.pdf)
Assume that we have a Lefschetz singularity next to a fold arc that is *sinkable*, i.e. the Lefschetz and fold vanishing cycles intersect in one point. (In other words, it is the resulting of unsinking a cusp.) In terms of handle decompositions the situation before and after the sinking process is locally described in Figure \[F:SWF vs BLF\].[^10] (The Lefschetz 2-handle in (a) is the one that goes over both 1-handles. One readily checks that it is correctly framed.) Clearly, both pictures describe a 4-ball and they are related by an obvious 2-handle slide. Indeed, to go from (a) to (b) one has to slide the Lefschetz handle over the fold handle in such a way that it unlinks from the lower 1-handle. Note that his handle slide is compatible with the fibration structures in the sense that the attaching curves stay on the fibers. Moreover, it mysteriously adjusts the framings exactly as needed.
\[R:wrinkling via handles\] Although the handle slide described above seems to be a correct interpretation of Lekili’s (un-)sinking deformation it is a priori not obvious why this should be true. In fact, the deformation is a combination of wrinkling, merging and flipping (see [@Lekili], Figure 8) and does not seem very atomic. On the other hand, the handle slide is an atomic modification of the handlebodies. It would be interesting to see a 1-parameter family of Morse functions associated with the (un-)sinking deformation that would exhibit the handle slide.
This shows that, if we start we the handle decomposition of $\beta_w$, then sliding $\lambda_1$ over $\rho={\gamma}_1$ produces a fiber framed attaching curve $\lambda_1'$ which is isotopic to ${\gamma}_2$. Successively sliding $\lambda_{i}$ over $\lambda_{i-1}'\sim{\gamma}_i$ results in fiber framed attaching curves $\lambda_{i}'$ isotopic to ${\gamma}_{i+1}$. Altogether we end up with fiber framed curves $\lambda_1',\dots,\lambda_c',\rho$. The final observation is that $\lambda_c'$ is isotopic to $\rho={\gamma}_1$ and can be unlinked and isolated from the rest of the diagram to form a zero framed unknot which cancels the 3-handle coming from the round singularity. What we are left with is the decomposition associated to $w$.
Substitutions {#S:substitutions}
=============
Let ${\mathfrak{S}}=({\Sigma},{\Gamma})$ be a surface diagram and let $\Lambda$ be a subcircuit of ${\Gamma}$. If $\Lambda'$ is any circuit that starts and ends with the same curves as $\Lambda$, then we can build a new surface diagram $({\Sigma},{\Gamma}')$ where ${\Gamma}'$ is obtained by replacing $\Lambda$ with $\Lambda'$. We call this operation a *substitution of type $(\Lambda|\Lambda')$*[^11].
Passing to the associated [simple wrinkled fibrations]{}one can ask how such a substitution affects the total spaces. In the following we will treat two instances in which this question can be answered. Our main tool are the handle decompositions exhibited in the previous section.
Let $Z$ be a compact 4-manifold, possibly with nonempty boundary. Recall that the *blow-up* of $Z$ is the connected sum of $Z$ with either ${\overline{\mathbb{C}P^2}}$ or ${\mathbb{C}P^2}$ (taken in the interior of $Z$). Moreover, the *sum stabilization* of $Z$ usually means the connected sum with $S^2\times S^2$. We will be slightly more general and also allow connected sums with ${\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}$, the twisted $S^2$-bundle over $S^2$. For convenience, we let $$\mathbb{S}_k:=
\begin{cases}
S^2\times S^2, & \text{$k$ even}\\
{\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}, & \text{$k$ odd}
\end{cases}$$ and note that $\mathbb{S}_k$ is described by the $(0,k)$-framed Hopf link.
\[T:blow-up lemma\] Let ${\mathfrak{S}}=({\Sigma},{\Gamma})$ be a surface diagram and let ${\mathfrak{S}}'$ be obtained from ${\mathfrak{S}}$ by a substitution of type $$\label{E:blow-up configuration}
\big( a,b \,|\, a,\tau_b^{\pm1}(a),b \big).$$ Furthermore, let ${\mathfrak{S}}''$ be obtained by a substitution of type $$\label{E:stabilization configuration}
\big( a,b \,|\, a,b,\tau_b^{k}(a),b \big).$$ Then $Z_{{\mathfrak{S}}'}$ is diffeomorphic to the blow-up $Z_{\mathfrak{S}}\#\mp{\mathbb{C}P^2}$ and $Z_{{\mathfrak{S}}''}$ is diffeomorphic to the sum stabilization $Z_{\mathfrak{S}}\#\mathbb{S}_{-k}$.
Of course, any substitution is reversible so that whenever a surface diagram contains a configuration of the form $(a,\tau_b^{\pm1}(a),b)$ or $(a,b,\tau_b^{k}(a),b)$ the associated 4-manifold must be a blow-up or sum stabilization, respectively. We will call these *blow-up* (resp. *sum stabilization*) *configurations*.
By switching we can assume that ${\Gamma}=(\dots,a,b)$ and thus ${\Gamma}'(\dots,a,\tau_b^{\pm1}(a),b)$ and ${\Gamma}''=(\dots,a,b,\tau_b^{k}(a),b)$. Figure \[F:blow-up lemma\] shows the relevant parts of the handle decompositions of the associated 4-manifolds. The shaded ribbons indicate the regions that contain all the other fold handles. Note that the curves $a$ and $b$ appear as 0-framed meridians to the dotted circles.
![The relevant parts of the handle decompositions of $Z_{\mathfrak{S}}$, $Z_{{\mathfrak{S}}'}$ and $Z_{{\mathfrak{S}}''}$. All 2-handles without framing coefficient are $0$-framed.[]{data-label="F:blow-up lemma"}](b+s.pdf)
In the case of $Z_{{\mathfrak{S}}'}$ we can use the meridians to unlink the curve corresponding to $\tau_b^\pm(a)$ resulting in an unknot with framing $\mp 1$ which is isolated from the rest of the diagram. Furthermore, the rest of the diagram agrees with the diagram for $Z_{\mathfrak{S}}$ and the claim follows.
The argument for $Z_{{\mathfrak{S}}''}$ is almost the same. Again, by sliding over the meridians we can isolate the curves corresponding to $b$ and $\tau_b^k(a)$ from the rest of the diagram. This time we obtain a $(0,-k)$-framed Hopf link which represents a copy of $\mathbb{S}_{-k}$.
\[T:blow-up closed\] Let ${\mathfrak{S}}$, ${\mathfrak{S}}'$ and ${\mathfrak{S}}''$ be as in Lemma \[T:blow-up lemma\].
1. All three diagrams have the same monodromy.
2. If ${\mathfrak{S}}$ has trivial monodromy so that $Z_{\mathfrak{S}}$ closes off to a closed 4-manifold $X$, then $Z_{{\mathfrak{S}}'}$ (resp. $Z_{{\mathfrak{S}}'}$) closes off to $X\#\mp{\mathbb{C}P^2}$ (resp. $X\#\mathbb{S}_k$).
3. Any closed 4-manifold obtained from ${\mathfrak{S}}'$ (resp. ${\mathfrak{S}}''$) is a blow-up (resp. sum-stabilization) of a manifold obtained from ${\mathfrak{S}}$.
The first statement follows directly from Lemma \[T:blow-up lemma\] since connected sums with closed manifold (taken in the interior) do not change the boundary.
For the other statements, observe that if one knows how to apply the method from Remark \[R:closing off by Kirby moves\] for ${\mathfrak{S}}$, then one also knows it for ${\mathfrak{S}}'$ (resp. ${\mathfrak{S}}''$) and vice versa.
Another instance where a substitution corresponds to a well known cut-and-paste operation has been observed by Hayano ([@HayanoR2], Lemma 6.13). Assume that a surface diagram ${\mathfrak{S}}$ contains a curve $c\subset {\Sigma}$. If $d\subset{\Sigma}$ is geometrically dual to $c$, then one can perform a substitution of type $(c|c,d,c)$ and Hayano shows that if ${\mathfrak{S}}'$ denotes the resulting surface diagram, then $Z_{{\mathfrak{S}}'}$ is obtained from $Z_{\mathfrak{S}}$ by a surgery on the curve $\delta\subset{\Sigma}\subset Z_{\mathfrak{S}}$ with respect to its *fiber framing*, i.e. the framing induced by the its canonical framing in ${\Sigma}$ together with the framing of ${\Sigma}$ in $Z_{\mathfrak{S}}$ as a regular fiber of $w_{\mathfrak{S}}\colon Z_{\mathfrak{S}}\ra D^2$.
One immediately notices that our sum-stabilization substitution is a special case of this construction. However, it also leads the way to the following minor generalization of the surgery substitution which captures not only the fiber framed surgery but also the one with the opposite framing.
\[T:Hayano surgery\] Let ${\mathfrak{S}}$ and ${\mathfrak{S}}'$ be two surface diagram with the same underlying surface ${\Sigma}$ and let $c,d\subset{\Sigma}$ be a geometrically dual pair of [simple closed curves]{}. If ${\mathfrak{S}}'$ is obtained from ${\mathfrak{S}}$ by a substitution of type $(c|c,\tau_c^k(d),c)$, then $Z_{{\mathfrak{S}}'}$ is obtained from $Z_{\mathfrak{S}}$ by a surgery on $d\subset{\Sigma}\subset X$ the fiber framing when $k$ is even and the opposite framing when $k$ is odd.
As in Hayano’s proof, it is enough to work in a of $c\cup d$ which we can assume to be a punctured torus. Using our handle decomposition instead of the ones from broken Lefschetz fibrations, the effect of Hayano’s surgery substitution, i.e. the case when $k=0$, looks as in Figure \[F:Hayano surgery\] where $c$ (resp. $d$) appears as the meridian of the upper (resp. lower) 1-handle.
![Hayano’s surgery substitution: with (a) vanishing cycle $c$ and (b) vanishing cycles $(c,d,c)$. []{data-label="F:Hayano surgery"}](surgery_1.pdf)
To obtain the other even cases, observe that in Figure \[F:Hayano surgery\](b) we can slide the 2-handle corresponding to $d$ once over each 2-handle corresponding to $c$ in the same direction. Depending on the direction this changes the framing coefficient by $\pm2$ and one readily checks that the resulting curve diagram shows a with vanishing cycles $(c,\tau_c^{\mp2}(d),c)$. Repeating this trick one can obtain all configurations with even $k$ and they will all describe the fiber framed surgery on $d$.
As shown in [@GS Example 8.4.6] the surgery with the opposite framing can be realized by inserting a pair of a Lefschetz vanishing cycle and an achiral Lefschetz vanishing cycle which are both parallel to $d$. But Figure \[F:opposite surgery\] shows that the result is the same as a substitution of type $(c|c,\tau_c\inv(d),c)$ which corresponds to $k=-1$.
![Surgery with the opposite framing.[]{data-label="F:opposite surgery"}](surgery_2)
Moreover, the arguments for shifting the value of $k$ by multiples of $2$ works just as in the fiber framed case.
Using the above lemma the sum-stabilization can be interpreted as performing surgery on a null-homotopic curve with either of its framing. Indeed, as $d$ one takes one of the adjacent vanishing cycles of $c$ in ${\mathfrak{S}}$ which is clearly null-homotopic in $Z_{\mathfrak{S}}$.
It would be interesting to interpret other cut-and-paste operations on 4-manifolds as substitutions in surface diagrams. For example, it is reasonable to expect such an interpretation for certain rational blow downs which can be described in terms of Lefschetz fibrations (see [@Endo2]). However, we will settle for blow-ups and sum-stabilizations in this paper.
Manifolds with genus 1 [simple wrinkled fibrations]{} {#S:genus 1 classification}
=====================================================
In this section we prove Theorem \[T:genus 1 classification, intro\]. Our strategy is to use Proposition \[T:blow-up closed\] to construct some genus $1$ [simple wrinkled fibrations]{}and then show that this construction gives all such fibrations.
We begin with the construction of genus 1 [simple wrinkled fibrations]{}over $S^2$. As before, we denote by $\mathbb{S}_k$ the closed 4-manifolds described by the $(0,k)$-framed Hopf link and we define a family of manifolds $$\label{E:genus 1 manifold list}
X_{klmn}=\mathbb{S}_k \# l(S^2\times S^2) \# m{\mathbb{C}P^2}\# n{\overline{\mathbb{C}P^2}},
\quad k\in\set{0,1},\;\, l,m,n\geq0.$$ Note that these are precisely the manifolds in Theorem \[T:genus 1 classification, intro\]. Recall that $\mathbb{S}_k$ is an $S^2$-bundle over $S^2$. By performing a birth on a suitable bundle projection $\mathbb{S}_k\ra S^2$ we obtain a [simple wrinkled fibration]{}with two cusps. We can then use Lemma \[T:blow-up lemma\] to add the other summands at will. Thus, in order to prove Theorem \[T:genus 1 classification, intro\], it remains to show the following.
\[T:genus 1 classification\] Let $w\colon X\ra S^2$ be a [simple wrinkled fibration]{}of genus 1. Then $X$ is diffeomorphic to some $X_{klmn}$ described in .
\[R:reformulation of classification\] The reason for our small reformulation of Theorem \[T:genus 1 classification, intro\] is that, while the original formulation is cleaner, the new one is much more in tune with the structure of the proof.
The key to the proof of Proposition \[T:genus 1 classification\] is the simple nature of simple closed curves on the torus. Indeed, the two well known facts that two oriented [simple closed curves]{}on the torus are isotopic if and only if they are homologous and that the (absolute value of the) algebraic and geometric intersection numbers agree allow us to transfer the whole discussion of genus 1 surface diagrams into the homology group $H_1(T^2)\cong\Z\oplus\Z$ simply by choosing orientations on the curves. Building on this observation we obtain the following result about the structure of genus 1 surface diagrams.
\[T:genus 1 circuits\] Any closed circuit on the torus of length at least three contains blow-up or sum stabilization configurations (as described in Lemma \[T:blow-up lemma\]).
Let ${\Gamma}=({\gamma}_1,\dots,{\gamma}_c)$ be a (not necessarily closed) circuit on the torus of length $c\geq3$. As usual, we choose an arbitrary orientation on ${\gamma}_1$ and orient the remaining curves by requiring that $\scp{{\gamma}_i,{\gamma}_{i+1}}=1$ for $i<c$ so that we can consider each ${\gamma}_i$ as an element of $H_1(T^2)$.
We first observe that, since any two adjacent curves in ${\Gamma}$ algebraically dual, they form a basis of $H_1(T^2)$. In particular, for $i\geq3$ we can write $${\gamma}_i=k_i{\gamma}_{i-1}-{\gamma}_{i-2},\quad k_i\in\Z$$ where the coefficient of ${\gamma}_{i-2}$ determined by our convention that $\scp{{\gamma}_{i-1},{\gamma}_i}=1$. This shows that if we denote by $\sigma_i:=\scp{{\gamma}_1,{\gamma}_i}$ the algebraic intersection number between ${\gamma}_1$ and ${\gamma}_i$, then we have $\sigma_1=0$, $\sigma_2=1$ and the recursion formula $$\label{E:intersection recursion}
\sigma_{i}=k_i\sigma_{i-1}-\sigma_{i-2}$$ holds for $i\geq3$. At this point we note that ${\Gamma}$ is closed if and only if $|\sigma_c|=1$.
We claim that if $|k_i|\geq2$ for all $i\geq3$, then $|\sigma_{i+1}|>|\sigma_{i+1}|$ for all $i$. This follows inductively since $|\sigma_2|>|\sigma_1|$ and from we get $$\begin{aligned}
|\sigma_{i+1}|
&=|k_{i+1}\sigma_i-\sigma_{i-1}| \\
& \geq \big| |k_{i+1}||\sigma_i|-|\sigma_{i-1}| \big| \\
& = |k_{i+1}||\sigma_i|-|\sigma_{i-1}| > |\sigma_i|
\end{aligned}$$ where we have used the reverse triangle inequality, the induction hypothesis and the assumption that $|k_{i+1}|\geq2$. As a consequence, we see that if ${\Gamma}$ is closed, then we must have $|k_i|\leq1$ for some $i\geq3$.
Assume first that $k_i=\pm1$. For the sake of a cleaner notation we momentarily rename the relevant curves to $$\label{E:detecting a blow-up}
({\gamma}_{i-2},{\gamma}_{i-1},{\gamma}_i)=:(a,\xi,b).$$ By assumption, we have $b=\pm\xi-a$ and thus $\xi=\pm(a+b)$ and the orientation convention shows that $\scp{a,b}=\pm1$. By invoking the Picard-Lefschetz formula (Proposition \[T:Picard-Lefschetz formula\]) we obtain $$\begin{aligned}
\tau_a^{\pm1}(b)
&=b\pm\scp{a,b}a \\
&=a+b \\
&= \pm\xi
\end{aligned}$$ which, after forgetting the orientations again, reveals the excerpt of ${\Gamma}$ shown in as a blow-up configuration.
A similar argument exhibits a sum-stabilization configuration in the remaining case when $k_i=0$. The details are left to the reader.
The proof of Proposition \[T:genus 1 classification\], and thus of Theorem \[T:genus 1 classification, intro\] is now very easy.
Any genus one [simple wrinkled fibration]{}over $S^2$ can is obtained by closing off a manifold $Z_{\mathfrak{S}}$ associated to a surface diagram ${\mathfrak{S}}=(T^2,{\Gamma})$. Moreover, any such diagram ${\mathfrak{S}}$ can be closed off since the mapping class group of the lower genus fiber is trivial. By Lemma \[T:genus 1 circuits\] and Proposition \[T:blow-up closed\] (3) we can successively split off summands of the form $\pm{\mathbb{C}P^2}$ and $\mathbb{S}_k$ until the remaining surface diagram, say ${\mathfrak{S}}_0$ has a circuit of length two. It is easy to see that $Z_{{\mathfrak{S}}_0}$ is the trivial disk bundle $S^2\times D^2$. (Either by drawing a Kirby diagram or by observing that any [simple wrinkled fibration]{}with two cusps is homotopic to a bundle projection.) Thus there are exactly two ways to close off the fibration, producing a summand of the form $\mathbb{S}_0\cong S^2\times S^2$ or $\mathbb{S}_1\cong{\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}$.
Concluding remarks {#S:concluding remarks}
==================
The theory of simple wrinkled fibrations and surface diagrams is still in a very early stage and at this point it raises more questions then it provides answers. We would like to take the opportunity to point out some of the major problems in the subject as well as to indicate some further developments.
Closed 4-manifolds
------------------
The ultimate goal is to use surface diagrams to study *closed* 4-manifolds. Unfortunately, it turns out that most surface diagrams do *not* describe closed manifolds since they have non-trivial monodromy and it is usually a hard problem to determine whether a given surface diagram has trivial monodromy. The following is thus of great interest.
\[P:trivial monodromy conditions\] Find at least necessary conditions for a surface diagram to have trivial monodromy that are easier to check.
The next major problem was already mentioned in Section \[S:the last 2-handle\]. If a surface diagram of sufficiently high genus is known to have trivial monodromy, then it determines a unique closed 4-manifold together with a [simple wrinkled fibration]{}over $S^2$ by closing off the associated fibration over the disk. However, the way that the surface diagram encodes the closing off information is too implicit for practical purposes. For example, by simply looking at the surface diagram it not at all clear how to answer the following very reasonable questions about the corresponding [simple wrinkled fibration]{}over $S^2$:
- Does the fibration have a section?
- What can be said about the homology class of the fiber? (Is it trivial, primitive, torsion,... ?)
- What is the fundamental group, homology, etc. of the total space?
What is missing is one more piece of information which is roughly the (framed) attaching curve of the last 2-handle. One can reformulate this issue in terms of mapping class groups (see [@HayanoR2], for example)
\[P:closing off\] Find a practical method to determine the missing piece of information from a surface diagram with trivial monodromy.
Higher genus fibrations
-----------------------
The fact that any (achiral) Lefschetz fibration can be turned into a [simple wrinkled fibration]{}of one genus higher suggests the philosophy that [simple wrinkled fibrations]{}of a fixed genus might behave similarly as (achiral) Lefschetz fibrations of one genus lower.
This analogy works rather well for the lowest possible fiber genera. Indeed, our result about genus one [simple wrinkled fibrations]{}looks very similar to the (rather trivial) classification of genus zero (achiral) Lefschetz fibrations, the latter being blow-ups of either $S^2\times S^2$ or ${\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}$.
Following this train of thought one might hope to be able to say something useful about the classification of genus two [simple wrinkled fibrations]{}over $S^2$ but one should expect to be lost as soon as the genus is three or higher. However, it is nonetheless conceivable that part of the classification scheme that worked in the genus one case might carry over to higher genus fibrations, as we will now explain
Let ${\mathfrak{S}}=({\Sigma};{\gamma}_1,\dots,{\gamma}_l)$ be a surface diagram and assume that for some $2<k<l$ the curve ${\gamma}_k$ is geometrically dual to ${\gamma}_1$. Then there is an obvious way to decompose ${\mathfrak{S}}$ into the two smaller surface diagrams $({\Sigma};{\gamma}_1,\dots,{\gamma}_k)$ and $({\Sigma};{\gamma}_1,{\gamma}_k,\dots,{\gamma}_l)$. Repeating this process we eventually obtain a decomposition of ${\mathfrak{S}}$ into a collection of surface diagram with the property that no pair of non-adjacent curves has geometric intersection number one. Let us call such a surface diagram *irreducible*.
In terms of the [simple wrinkled fibration]{}associated to ${\mathfrak{S}}$ the above decomposition of ${\mathfrak{S}}$ should correspond to merging the fold arcs that induce ${\gamma}_1$ and ${\gamma}_k$. The result is a wrinkled fibration that naturally decomposes as a *boundary fiber sum* of the two [simple wrinkled fibrations]{}associated to the parts of the decomposition of ${\mathfrak{S}}$.
This suggests that any descending [simple wrinkled fibration]{}over the disk naturally decomposes into a boundary fiber sum of *irreducible* fibrations where we call a [simple wrinkled fibration]{}irreducible if its surface diagram is irreducible. Consequently, the classification of descending [simple wrinkled fibrations]{}splits into two parts: the classification of irreducible fibrations and understanding the effect of boundary fiber sums.
The genus one classification fits into this scheme as follows. Our arguments show that the only irreducible surface diagrams of genus one are given by the blow-up configurations $(a,\tau_a^{\pm1}(b),b)$ and the sum-stabilization configurations $(a,b,\tau_b^k(a),b)$ for $k\neq1$. Using the handle decompositions it is easy to identify the corresponding manifolds. (They are the connected sum of $S^2\times D^2$ with either $\pm{\mathbb{C}P^2}$, $S^2\times S^2$ or ${\mathbb{C}P^2}\#{\overline{\mathbb{C}P^2}}$.) Furthermore, the boundary fiber sums are performed along spheres and are thus easy to understand.
Making these arguments precise requires an understanding of the effect of merging folds and cusps on surface diagrams.
Uniqueness of surface diagrams
------------------------------
Given the fact that all closed 4-manifolds can be described by surface diagrams, it is natural to ask for a set of moves to relate different surface diagrams that describe the same manifold, similar to the situation of 3-manifolds and Heegaard diagrams.
A first step in this direction was taken by Williams [@Williams2] who relates the surface diagrams of homotopic [simple wrinkled fibrations]{}over $S^2$ of genus at least three. He shows that any two homotopic [simple wrinkled fibrations]{}can be connected by a special homotopy that is made up of four basic building blocks. These building blocks are simple enough to understand their effect on the initial surface diagram (see also the recent work of Hayano [@HayanoR2]).
So far this is completely analogous to the 3-dimensional context. A new phenomenon in the 4-dimensional context is that two [simple wrinkled fibrations]{}on a given 4-manifold are not necessarily homotopic. The structure of the set $\pi^2(X):=[X,S^2]$ of homotopy classes of maps from a closed 4-manifold to the 2-sphere – also known as the *second cohomotopy set* of $X$ – is described in [@cohomotopy] (see also the references therein). Our results show that an equivalence class of surface diagrams for $X$ determines an orbit of the action of the diffeomorphism group of $X$ on $\pi^2(X)$. This action is usually neither trivial[^12] nor transitive[^13]. Consequently, reparametrizing a surface diagram can change the homotopy class of its [simple wrinkled fibration]{}but one cannot expect to obtain all homotopy classes in this way.
A general method for relating broken fibrations in different homotopy classes is the *projection move* mentioned in [@Williams1] but it is not at all obvious how to interpret this procedure in terms of surface diagrams. Altogether, the problem of relating surface diagram with non-homotopic fibrations is still wide open.
[^1]: By now this can be considered as a special case of [@GK3] which appeared while we were writing this paper.
[^2]: A map $f\colon M^m\ra N^n$ is *locally modeled around $p\in M$* on $f_0\colon\colon\R^m\ra \R^n$ if there are local coordinates around $p$ and $f(p)$ mapping these points to the origin such that the coordinate representation of $f$ agrees with $f_0$.
[^3]: For both models there are orientation reversing diffeomorphisms which leave the map invariant
[^4]: Eliashberg’s classification of overtwisted contact structures and the Giroux correspondence between contact structures and open book decompositions
[^5]: The sign depends on how the fold and cusp models are embedded.
[^6]: For a proof that $\mathrm{cut}_a$ is well defined see [@Ivanov Section 7.5], the rest follows as in [@primer Chapter 3].
[^7]: To avoid any confusion, we use the convention that $\mathrm{sgn}(0)=0$.
[^8]: Compare Chapter 8.2 in [@GS] (p. 299f) for the Lefschetz case.
[^9]: The curve is simple enough to draw a parallel push-off in the fiber direction and compute the linking number
[^10]: These handle decompositions have already appeared in a disguised form in [@Lekili].
[^11]: Similar substitution techniques for Lefschetz fibrations are studied in [@Endo1; @Endo2].
[^12]: For example the diffeomorphism of $S^2\times S^2$ that interchanges the two factors also interchanges the projections onto the factors which are easily seen not to be homotopic.
[^13]: This follows from the fact that the diffeomorphism action on $H_2(X)$ preserves divisibility.
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---
abstract: 'Coclass theory can be used to define infinite families of finite $p$-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-$p$ cohomology rings. Here we prove that almost all groups in one of these infinite families have equivalent Quillen categories. We also show how the Quillen categories of the groups in an infinite family are connected to the Quillen category of their associated infinite pro-$p$-group of finite coclass.'
author:
- 'Bettina Eick and David J. Green'
date: 2 October 2013
title: |
The Quillen category of\
finite $p$-groups and coclass theory
---
Introduction
============
The coclass of a finite $p$-group of order $p^n$ and nilpotency class $c$ is defined as $n-c$. Leedham-Green and Newman [@LGN] proposed the use of coclass a primary invariant for the classification and investigation of finite $p$-groups. This suggestion was highly successful and has led to various new insights into the structure of finite $p$-groups. We refer to the book by Leedham-Green and McKay [@LGM] for details.
Eick and Leedham-Green [@ELG08] introduced the [*coclass families*]{}: these are certain infinite families of finite $p$-groups of fixed coclass; see Section \[coclass\] for the explicit definition. The groups in a coclass family have a similar structure and hence can be treated simultaneously in various structural investigations of finite $p$-groups. The significance of the coclass families is underlined by the result that the infinitely many finite $2$-groups of fixed coclass fall into finitely many coclass families and finitely many other groups.
The Quillen category ${\mathcal A}_p(G)$ of a finite group $G$ is the category whose objects are the elementary abelian $p$-subgroups of $G$ and whose morphisms are the injective group homomorphisms induced by conjugation with elements of $G$. The following is the main result of this paper, see Section \[equiv\] for a proof.
\[thm:main\] Let $(G_x \mid x \geq 0)$ be a coclass family of finite $p$-groups. Then there exists $x_0 \in {\mathbb{N}}$ so that the Quillen categories ${\mathcal A}_p(G_x)$ and ${\mathcal A}_p(G_y)$ are equivalent for all $x,y \geq x_0$.
For the proof of Theorem \[thm:main\] we consider $x \in {\mathbb{N}}$ large enough and define a special skeleton ${\overline}{{\mathcal A}}_p(G_x)$ for the Quillen category of $G_x$ together with an explicit functor $F : {\overline}{{\mathcal A}}_p(G_x) {\rightarrow}{\overline}{{\mathcal A}}_p(G_{x+1})$. We then show that this induces an isomorphism of categories. Hence the categories ${\mathcal A}_p(G_x)$ and ${\mathcal A}_p(G_{x+1})$ are equivalent.
As a by-product we also define a special skeleton ${\overline}{{\mathcal A}}_p(S)$ for the infinite pro-$p$-group $S$ associated to the coclass family $(G_x \mid x
\geq 0)$. For each large enough $x$ we then introduce a functor $F_S :
{\overline}{{\mathcal A}}_p(G_x) {\rightarrow}{\overline}{{\mathcal A}}_p(S)$. This functor $F_S$ is not necessarily injective or surjective on objects, but nonetheless it exhibits an interesting link between the Quillen category of an infinite pro-$p$-group of finite coclass and its associated coclass families.
Given the Quillen category ${\mathcal A}_p(G)$ of a finite $p$-group $G$ and given a field $k$ of characteristic $p$, we denote $${\overline}{H}^*(G,k) = \lim_{E \in {\mathcal A}_p(G)} H^*(E,k).$$ Restriction induces a natural homomorphism $$\phi_G : H^*(G,k) \rightarrow {\overline}{H}^*(G,k).$$ Quillen [@Quillen Th. 6.2] proved that $\phi_G$ is an *inseparable isogeny*. That is, every homogeneous element of the kernel of $\phi_G$ is nilpotent and for every element $s$ in the range of $\phi_G$ there exists $n \in {\mathbb{N}}$ so that $s^{p^n}$ is an element of the image of $\phi_G$. Thus $\phi_G$ induces a homeomorphism of the prime ideal spectra of $H^*(G, k)$ and ${\overline}{H}^*(G,k)$ and it shows how the Quillen category of a finite $p$-group influences its mod-$p$ cohomology ring. Theorem \[thm:main\] has the following immediate corollary.
\[cor:main\] Let $(G_i \mid i \geq 0)$ be a coclass family of finite $p$-groups. Then there exists $l \in {\mathbb{N}}$ so that ${\overline}{H}^*(G_i,k) \cong {\overline}{H}^*(G_j,k)$ for all $i,j \geq l$.
Carlson [@Car05] proved that the mod-$2$ cohomology rings of the finite $2$-groups of a fixed coclass fall into finitely many isomorphism types. The following conjecture expands this result to odd primes $p$ and it also strengthens it for prime $2$. Corollary \[cor:main\] can be considered as an indication that supports the following conjecture.
\[con:main\] Let $(G_i \mid i \geq 0)$ be a coclass family of finite $p$-groups. Then there exists $l \in {\mathbb{N}}$ so that $H^*(G_i,k) \cong H^*(G_j,k)$ holds for all $i,j \geq l$.
|
---
abstract: 'Three models were presented in *M.K. Khakim Habibi, Hamid Allaoui, Gilles Goncalves, Collaborative hub location problem under cost uncertainty, Computers & Industrial Engineering Volume 124, October 2018, Pages 393-410* as models for collaborative Capacitated Multiple Allocation Hub Location Problem. In this note, we point out a few flaws in modeling. In particular, we elaborate and explain that none of the those models incorporates any element of a collaborative activity.'
address: 'Département Réseaux et Télécommunications, Université d’Artois, F-62400 Béthune, France'
author:
- Shahin Gelareh
bibliography:
- 'sample.bib'
title: 'A note on ’Collaborative hub location problem under cost uncertainty’'
---
\[thm\][Lemma]{}
Introduction {#introduction .unnumbered}
============
In the relevant literature, the term ’*collaboration*’ is synonym for ’*the process of two or more people or organizations working together to complete a task or achieve a goal* (see, Marinez-Moyano, I. J. *Exploring the Dynamics of Collaboration in Interorganizational Settings*, Ch. 4, p. 83, in Schuman (Editor). Creating a Culture of Collaboration. Jossey-bass, 2006. ISBN 0-7879-8116-8.).\
Three models were presented in [@HABIBI2018393] as models for collaborative Capacitated Multiple Allocation Hub Location Problem. The base model of all is one of the very early and well-studied models of the classical hub location problem in [@EBERY2000614] with the same primitive assumptions which were set at the beginning of the century due to the computational limits of the CPUs in early 2000. Today, after two decades, the models of hub location have become much richer and those assumptions are overwritten by much more realistic features and constraints. In this structure (see [@EBERY2000614]) the hub-level structure is a complete sub-graph of the resulting network and a capacity is set on the volume of flow entering a hub node. No origin-destination path travels more than one hub edge while spoke nodes are allocated to as many hub node as they wish.
Literature review
=================
’*We are particularly interested in this problem considering the presence of uncertainty due to its applicability as well as the literature.*[@HABIBI2018393]’.\
It is often not very wise to talk about ’*lack of research*’ in some classical topic such as location, routing and scheduling: Interested readers are also referred to the recent HLPs literature review in , [@campbell2012twenty] and @Farahani:2013 as well as other contributions by and @KaraTaner:20111.\
The Hub-and-spoke structures is a very active research area and lots of development. Two of the most recent and very relevant work that subsume the ’collaborative’ modeling includes [@GROOTHEDDE2005567] for a work on collaborative, intermodal hub networks (a case study in the fast moving consumer goods market) and [@coopetitive] for a coopetitive hub location model that includes both competition collaboration/cooperation. One may conclude that we are far from a *research desert* when it comes to variants of hub location problem and hub-and-spoke structures.
Modeling and Problem Settings
=============================
In [@HABIBI2018393], the authors assume that ’*each supply chain is represented as a distribution network containing hub and spoke nodes*’. The authors do not specify whether every Supply Chain $SC$ has its own origin-destination product/service to fulfill demand nodes or the same demand can be fulfilled by any of the $SC$s.\
In [@HABIBI2018393], the original model in [@EBERY2000614] has been referred to as ’No Collaboration’ model wherein every supply chain operator operates its optimal network. Let $\chi, \alpha, \delta$ be the discount factors for collection, transfer and distribution. $W_{ij}$ represent the demand matrix and $\mathcal{N}$ stands as for the set of nodes. $C_{ij}$ and $F_k$ are the transportation cost per unit of flow and hub node setup costs, respectively. $\sigma_k^s$ is the so called supplementary cost for installing a shared hub node in scenario $s$. $(\tilde F)^s_k$ is the setup cost comprising supplementary cost $(\tilde F)^s_k=F_k+\sigma^s_k$ and $\Gamma_k$ is the capacity of hub node $k$. The variables are the following: $H_k$ is 1 if a hub is located at $k$, 0 otherwise. $Z_{ik}$ is the flow from origin $i$ to hub $k$; $Y_{kl}^i$ represents the flow from origin $i$ via hub link $k$ and $l$ and $X_{lj}^i$ represent the flow from origin $i$ to the destination $j$ via hub $l$.\
\[note:1\] It is unclear why in the ’No Collaboration’, the constraints in model are indexed for a network composed of all nodes in $\mathcal{N}$. One may conclude that all the SCs are operating on the same set of nodes, i.e. $\mathcal{N}$.\
The mathematical model in [@HABIBI2018393] follows: $$\begin{aligned}
{minmax}_{\forall s \in \mathcal{S}} ~~& L_s = \sum_{k\in\mathcal{N}}{F}_k H_k + \sum_{i\in \mathcal{N}}\left( \chi \sum_{k\in \mathcal{N}} C_{ik}Z_{ik} + \alpha \sum_{k\in \mathcal{N}}\sum_{l\in \mathcal{N}}C_{kl}Y_{kl}^i + \delta \sum_{l\in \mathcal{N}}\sum_{j\in \mathcal{N}}C_{ij}X_{lj}^i\right) \label{obj}\\
s.t.: &\nonumber\\
& \sum_{k\in \mathcal{N}} Z_{ik} = \sum_{j\in \mathcal{N}}W_{ij}, & \forall l\in \mathcal{N} \label{eq2}\\
& \sum_{l\in \mathcal{N}} X_{lj}^i = W_{ij}, & \forall i,j\in \mathcal{N} \label{eq3}\\
& \sum_{i\in \mathcal{N}} Z_{ik} \leq \Gamma_k H_k, & \forall k\in \mathcal{N} \label{eq4}\\
& \sum_{l\in \mathcal{N}} Y_{kl}^i + \sum_{j\in \mathcal{N}} X_{kj}^i = \sum_{l\in \mathcal{N}} Y_{lk}^i + Z_{ik} & \forall i,k\in \mathcal{N} \label{eq5}\\
& Z_{ik} \leq \sum_{j\in \mathcal{N}} W_{ij} H_k & \forall i,k\in \mathcal{N} \label{eq6}\\
& \sum_{i\in \mathcal{N}} X_{lj}^i \leq \sum_{i\in \mathcal{N}} W_{ij} H_l & \forall l,j\in \mathcal{N} \label{eq7}\\
& X_{lj}^i, Y_{kl}^i, Z_{ik}\geq 0 & \forall i,j,k,l\in \mathcal{N} \label{eq8}\\
& H_k\in \{0,1\} & \forall k\in \mathcal{N} \label{eq9}\end{aligned}$$
In the following we review the models and elaborate on the existing flaws in modeling collaboration:
Centralized Collaboration (CC)
==============================
Again, in the Centralized Collaboration model, the authors use the same set $\mathcal{N}$ to refer to the set of nodes for every $SC$. One may again, as in Note , conclude that both operators are operating on exactly the same set of nodes. The flaw in this model is the following.\
The objective function is the following: $$\begin{aligned}
{minmax}_{\forall s \in \mathcal{S}} & L_s = \sum_{k\in\mathcal{N}}\tilde{F}^s_k H_k + + \sum_{i\in \mathcal{N}}\left( \chi \sum_{k\in \mathcal{N}} C_{ik}Z_{ik} + \alpha \sum_{k\in \mathcal{N}}\sum_{l\in \mathcal{N}}C_{kl}Y_{kl}^i + \delta \sum_{l\in \mathcal{N}}\sum_{j\in \mathcal{N}}C_{ij}X_{lj}^i\right) \label{obj:CC}\\
s.t& \nonumber\\
& \eqref{eq2}-\eqref{eq9}\nonumber\end{aligned}$$ subject to the same set of constraints as in [@EBERY2000614], i.e. ’No Collaboration’ model. In this slightly modified model, the setup cost of hub node takes form of random variable with unknown distribution but discrete probability values.\
The authors claim that this is a model for ’ *forming a in order to achieve* ’.\
Insofar as the model , -, the network node set remains the same, the origin-destination (O-D) flows are the same as in NC and the amount of flow arriving to a node or leaving a node remains equal to the demand of every individual $SC$, which apparently operates on the entire network.\
The least one expected to convince oneself of having two network here would be that the flow originating from a node that is a member of more than one $SC$ be the sum of supply to all those nodes (which is not the case in this model), any sign, index or constraint differentiating between the variables or constraints of every one of those ’two’ networks.\
Nothing can be seen in this model that connect it to any other supply chain operator. This model is only in relation to itself. We are facing a single operator that is dealing with its own uncertainty in its setup costs. This operator has not even expanded its operation over a network of another operator as the set $\mathcal{N}$ remains unchanged. Therefore, this operator works on its own network and now has to deal with its own uncertainty in setup cost. This is a unilateral decision making and has nothing to do with any sort of collaboration.\
There are no two networks; there is no coalition and there is no share feature in anything of this mode.\
Therefore, unless proven otherwise, this model does not represent any form or shape of collaborative problem. As the literature is already aware of several example of similar work in CMAHLP with uncertainty in some parameters, this model by itself can hardly show any sign of novelty to be considered as a contribution.\
Centralized Collaboration with Uncertain Supplementary Cost (CCU)
=================================================================
In [@HABIBI2018393], when the minimax (CC) model is linearized in the form of a max-regret problem, the authors refer to it as a CCU model.\
The CCU model follows:
$$\begin{aligned}
Min & R \label{eq11}\\
&s.t& \nonumber\\
& \eqref{eq2}-\eqref{eq9} \nonumber\\
& R_s = \sum_{k\in\mathcal{N}}\tilde{F}^s_k H_k + + \sum_{i\in \mathcal{N}}\left( \chi \sum_{k\in \mathcal{N}} C_{ik}Z_{ik} + \alpha \sum_{k\in \mathcal{N}}\sum_{l\in \mathcal{N}}C_{kl}Y_{kl}^i + \delta \sum_{l\in \mathcal{N}}\sum_{j\in \mathcal{N}}C_{ij}X_{lj}^i\right) - L_s^* & \forall s\in\mathcal{S}\label{eq12}\\
& R\geq R_s, &\forall s\in S\label{eq13}\\
&R_s,R\in \mathbb{R}\forall s\in S\label{eq14}\end{aligned}$$
Again, beyond some algebra for getting to CCU from CC, there is no minimal link (even exchange of information on cost or any other thing) going beyond planning for anything more than one single entity (supply chain operator).\
In all these development a second (or any other) supply chain players is absolutely absent in any modeling. The O-D flow are the same, the $SC$ network is the same and we are not aware of any other $SC$ and its market share. Therefore, unless proven otherwise, this model does not represent any form or shape of collaborative problem (not even talking about location, routing or scheduling ...).
The network remains the same (i.e. $\mathcal{N}$) and this model has nothing to exchange or share with outside world. Therefore, there is no collaborative aspect in this modeling.\
Converting a minimax problem to a max-regret problem is closer to a textbook exercise. The literature of minmax regret model even in location problem is by far more advanced and this model by itself can hardly be considered as a contribution given the rich body of state-of-the-art research.
Optimized Collaboration with Uncertain Supplementary Cost (OCU)
===============================================================
Finally, we reach to a point where the authors talk about ’other’ and develop at *least some constraints* for distinct pairs of $SC$s. Two additional set of variables are introduced by the authors: $I_k=1$, if a hub located at $k$ is a *collaborative* one, 0 otherwise and $T_k=1$, if a hub located at $k$ is a *non-collaborative* one. The model becomes:
$$\begin{aligned}
Min & R \label{eq11}\\
&s.t& \nonumber\\
& \eqref{eq2}-\eqref{eq9}, \eqref{eq12}-\eqref{eq14} \nonumber\\
&H_k = I_k+T_k & \forall k\in\mathcal{N}\label{eq15}\\
&Z_{ik} \leq M(1-T_k) & \forall i\in SC_a, k\in SC_b, SC_a\neq SC_b:\forall SC_a,SC_b \in \mathcal{SC} \label{eq16}\\
&X_{lj}^i \leq M(1-T_l) & \forall i,j\in SC_a, l\in SC_b, SC_a\neq SC_b:\forall SC_a,SC_b \in \mathcal{SC} \label{eq17}\\
& Y_{kl}^i \leq M(1-T_l) & \forall i,k\in SC_a, l\in SC_b, SC_a\neq SC_b:\forall SC_a,SC_b \in \mathcal{SC} \label{eq18}\\
& Y_{kl}^i \leq M(1-T_k) & \forall i,l\in SC_a, k\in SC_b, SC_a\neq SC_b:\forall SC_a,SC_b \in \mathcal{SC} \label{eq19}\\
& Y_{kl}^i \leq M(1-T_k)(1-T_l) & \forall i\in SC_a, k,l\in SC_b, SC_a\neq SC_b:\forall SC_a,SC_b \in \mathcal{SC} \label{eq20}\\
& R_s = \sum_{k\in \mathcal{N}}\left( F_kT_k +\tilde{F}^s_k H_k \right. \nonumber\\
& \left.\sum_{k\in\mathcal{N}}(\chi \sum_{k\in \mathcal{N}} C_{ik}Z_{ik} + \alpha \sum_{k\in \mathcal{N}}\sum_{l\in \mathcal{N}}C_{kl}Y_{kl}^i + \delta \sum_{l\in \mathcal{N}}\sum_{j\in \mathcal{N}}C_{ij}X_{lj}^i)\right) - L_s^* & \forall s\in\mathcal{S}\label{eq21}\\
& I_k, T_k \in \{0,1\}, \forall k\ \in \mathcal{N}\label{eq22}\end{aligned}$$
According to the constraints $SC_a\subset \mathcal{N}$ and $SC_b\subset \mathcal{N}$, $SC_a \neq SC_b$ for every $SC_a, SCb \in SC$ and we assume it is implicitly said that $\bigcup_{a\in SC}SC_a=\mathcal{N}$ and $SC_a\cap SC_b, \forall ~~a,b$ may be non-empty. This contradicts in what has been concluded in Note because over there, every $SC$ for example $SC_a$ was solving the model on the entire $\mathcal{N}$ and not only on nodes of $SC_a$.\
But given the new notations let us suppose that the node set $\mathcal{N}$ is composed of union of node sets for every $SC_a \in SC$ nodes. This means that the authors basically have made up some subsets of $\mathcal{N}$ with possibly non-empty intersections (i.e. the sets are not necessarily mutually exclusive) and imposed additional constraints on the CCU model to achieve OCU.
There are a few flaws which are elaborated in the following:
The objective value of OCU
--------------------------
\[thm1\] The objective function of OCU can never be any better than the objective of CCU.
In simple words and without resorting to any complicated polyhedral theory and descriptions, one notices that OCU is no less constrained than model CCU and therefore the number of feasible solutions in OCU (in the space of $X,Y,Z$) is no more than those in CCU. In particular, given that $SC_a \neq SC_b$ for every $SC_a, SC_b \in SC$, one may say that OCU is in fact strictly more constrained than model CCU. As such, the set of feasible solutions in OCU is strictly contained in the set of feasible solutions of CCU (in the space of $X,Y,Z$) and no objective function delivers a better objective value on $\mathcal{P}(OCU)$ than on $\mathcal{P}(CCU)$, being polytopes of the corresponding problems. As such, any claim and computational results claiming otherwise is provably wrong.
Therefore, the so-called ’*collaboration*’, if any, is an absolute loss for the supply chain as a whole. The authors would not even need to conduct computational experiments to realize this fact. Moreover, any numerical outcome that contradicts this, is most probably a result of mistake and error, in an optimistic viewpoint.\
flaw in modeling:\#1
--------------------
According to the constraints -, no two subsets $SC_a$ and $SC_b$ are exactly the same. It is unclear whether the intersection is non-empty or not. Suppose $k$ is a node in the intersection of $SC_a$ and $SC_b$ and is a hub node. The first question arises here, is the following: who has paid the setup cost for such a hub node, was it $SC_a$ or $SC_b$? Which stakeholder in this problem description, if any, is in charge of setting up the hub nodes? How can one say that this hub belongs to $SC_a$ but not to $SC_b$, or the other way around?\
In addition, according to the constraints -, none of these two $SC$s are authorized to use this hub node unless $T$ variable becomes 0 (what authors call it a variable for determining ’non-collaborative’ nature of a hub node). First of all, for such a node in the intersection of node sets, we do not know to whom this hub belongs. Even more, if that node is also in the node set of any other $SC$, the later cannot use it either, unless pay extra to turn $I_k$ to 1.\
If we assume that the intersection of every pair of $SC_a$ and $SC_b$ is an empty set, the issue becomes even more serious. Suppose that $SC_a$ and $SC_b$ operating on disjoint node sets and they do not have any node in common. In this case, how can one say that there has been a flow from $i\in SC_a$ to $j\in SC_b$, i.e. $W_{ij}\neq 0$. How this flow even existed when every operator was operating on its own node set? Where does this ghost O-D flow come from.\
To that one can add the following: Given constraints -, this model does not deliver any optimal solution in, which for a hub node $k$, $T_k=1$.
In any case, it is clear that some unknown and absent stakeholder, as an autocracy, is present who forcefully wants all these nodes to be regrouped in one decision and one network, magically generates additional market share among the pairs that did not know each others before and solve one problem as a whole. This ghost stakeholder has only one goal, he does not want to see many players, just wants one single network.\
Given that, there is absolutely no collaboration going on here.
flaw in modeling:\#2
--------------------
An average reader understands that the OCU model is solved once and gives solution for the whole merger, otherwise it does not make sense that every $SC$ solve one separate OCU and pays the full cost of setup for a hub node that is in the intersection of two $SC$s and is already paid by one or more other $SC$. But if so, why we do not see any additional flow? The same $W$ matrix as before is used in OCU. The volume in this merger has not increased by summing up the volume of all $SC$s and the capacities remain the same even in this merger!. A single $SC$ given its volume of flow, set $\Gamma_k$ capacity on hub $k$ and when they all merge and volume must normally increase nothing happens to the volumes and structures.\
flaw in modeling:\#3
--------------------
The nonlinear constraints are totally irrelevant. They are already implied by the constrains -. Therefore, all trivial linearization algebra that follows, including the ’Proposition 1’ in [@HABIBI2018393] are irrelevant and over engineered.
flaw in modeling:\#4
--------------------
The definition of variables $T_{k}$ and $I_k$ is curious:
$$\begin{aligned}
I_k \left\{
\begin{array}{ccc}
1 & \mbox{if a hub located at $k$ is collaborative} \\
0 & \mbox{oterwise}. \\
\end{array}
\right.\hspace{2cm}
T_k \left\{
\begin{array}{ccc}
1 & \mbox{if a hub located at $k$ is non-collaborative} \\
0 & \mbox{oterwise}. \\
\end{array}
\right.\end{aligned}$$
Given constraints , i.e. $H_k=I_k+T_k$, if any of $I_k$ or $T_k$ takes 1, it means that $k$ is a hub node. Anyone of $I$ or $T$ would implies the other in the definition, so why should one declare two sets of variables and unnecessarily enlarge the polyhedral description in a higher dimension? In particular, just in the context of modeling (not approving the model itself) definition of variable $I_k$ apparently serves no purpose and $I_k$ is already implied by $T_k$.
flaw in modeling:\#5
--------------------
Cost sharing is not a post-processing phase. The model and solution must show that there is an added value in the collaboration. We know that OCU never generates any better feasible solution than any other model mentioned. The question is, when no positive sign is showing up, it means that the overall merger is loosing, while $SC$s are loosing at different levels.\
It is curious that when the overall cost increasing, no additional market is being generated as a result of the merger, no economy of scale is exploited by deploying more efficient (perhaps larger with higher capacity) transporter on any hub-level link, or anything else, this ’merger’ can make any sense for anyone or would mean anything similar to a ’*collaboration*’.\
The optimization must be made in presence of a motivating factor such as reduction of overall cost due to increase in market share, reduce in transportation cost due to the establishment of new hub-to-hub links or something that must become better through the so called ’*collaboration*’.
Conclusion
==========
None of the proposed models have much to do with any sort of ’collaboration’. It is very trivial, as shown above, that the heart of this article which is the OCU model does not deliver any better feasible solution. The proposed model can hardly have any real-life realisation. The proposed model OCU increases the overall cost of operation (perhaps gain for some but loose for the overall merger) and the postprocessing phase, rather than being a cost sharing mechanism is a disaster sharing technique. The model is basically promoting the notion of ’*loosing together and crying together*’.
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---
abstract: 'The analysis of networks affects the research of many real phenomena. The complex network structure can be viewed as a network’s state at the time of the analysis or as a result of the process through which the network arises. Research activities focus on both and, thanks to them, we know not only many measurable properties of networks but also the essence of some phenomena that occur during the evolution of networks. One typical research area is the analysis of co-authorship networks and their evolution. In our paper, the analysis of one real-world co-authorship network and inspiration from existing models form the basis of the hypothesis from which we derive new 3-lambda network model. This hypothesis works with the assumption that regular behavior of nodes revolves around an average. However, some anomalies may occur. The 3-lambda model is stochastic and uses the three parameters associated with the average behavior of the nodes. The growth of the network based on this model assumes that one step of the growth is an interaction in which both new and existing nodes are participating. In the paper we present the results of the analysis of a co-authorship network and formulate a hypothesis and a model based on this hypothesis. Later in the paper, we examine the outputs from the network generator based on the 3-lambda model and show that generated networks have characteristics known from the environment of real-world networks.'
author:
- |
Milos Kudelka\
\
\
\
Eliska Ochodkova\
\
\
\
Sarka Zehnalova\
\
\
\
bibliography:
- 'references.bib'
date: 19 October 2016
title: 'Around Average Behavior: 3-lambda Network Model'
---
<ccs2012> <concept> <concept\_id>10010147.10010341.10010346.10010348</concept\_id> <concept\_desc>Computing methodologies Network science</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
Network analysis is a phenomenon that affects research in many areas. One of the goals of network analysis is to describe the phenomena, properties, and principles that are universal and manifest in nature, society, and in the use of technology. As a network, we understand an ordered pair $G = (V, E)$ (undirected unweighted graph) of a set $V$ of nodes and a set $E$ of edges which are unordered pairs of nodes from $G$. The complex network structure can be viewed from the perspective of the network’s state at the time of the analysis. Networks can, therefore, be described by the properties known from the environment of real-world networks, including, in particular, the small-world, free-scale, high average clustering coefficient, assortativity [@newman2002assortative], community structure, shrinking diameter [@leskovec2005graphs], but also others such as core-periphery structure [@rombach2014core] and self-similarity [@song2005self]. Underlying processes that take place during the evolution of real-world networks are also examined. Some models are based on analyzing these processes, which allows using the formally described underlying process as a generative mechanism. Such a mechanism can generate networks possessing one or more known properties. Models that reveal key principles include those using the preferential attachment to generate network centers [@albert2002statistical] or triadic closure i.e. completing interconnections into triangles, capable of generating community structure [@bianconi2014triadic].
Models that provide key knowledge about networks are usually inherently simple. Most of them, however, focus on the question *“How to connect a new node into the network?”* Our question is, *“How does an existing node behave to its neighbors and other existing and new nodes during the network’s evolution?”*. The result of our focus on the behavior of *existing nodes* is a simple model *without memory* and with only *three parameters*. This new 3-lambda model is inspired by the evolution of the co-authorship network. For the analysis we used a network generated from a DBLP dataset and we worked with the assumption that in each publication is just one key author who picks out additional co-authors. In the analytically oriented experiment, we show that with such an assumption, the number of publications with a given number of co-authors corresponds approximately to a Poisson distribution. Based on the result of this experiment, we formulate a simple hypothesis and the resulting network growth model. This hypothesis assumes that one-step of the network growth is an interaction involving existing and new network nodes. In this respect our approach is similar to the model of collaborative networks published by Ramasco et al. [@ramasco2004self] and inspired by the analysis of co-authorship ego networks in research of Arnaboldi et al. [@Arnaboldi2016analysis]. 3-lambda is a stochastic model that estimates the number of nodes in the interaction using the Poisson distribution. In the experimental part of this paper, we describe the network generator based on our model and three experiments. The first experiment shows that generated networks have characteristics known from real-world networks, and how the selected properties change with a different setting of the generator. The second experiment shows how the properties of generated network change during its growth. The aim of the third experiment is to compare some characteristics of the DBLP network and large-scale generated networks.
The rest of the paper is organized as follows: Section \[sec:rel\] focuses on the related work. Section \[sec:dblp\] provides our findings and hypothesis on the real-world network extracted from the DBLP dataset. In Section \[sec:met\], we describe the 3-lambda model of collaborative network and network generator based on this model. Section \[sec:exp\] focuses on three experiments with generated networks. We conclude and briefly discuss open problems in Section \[sec:conc\].
Related work {#sec:rel}
============
In the last two decades, the analysis of real-world networks has received extraordinary attention. One of the sources of data is social networks, which are growing at an enormous rate. Notable and a long investigated source in this area are co-authorship and, in general, collaborative networks. A common feature of this type of network is that underlying processes proceed in cliques, which then become a fundamental building block of the network. Barabasi et al. [@barabasi2002evolution] presented and analyzed in detail a network model inspired by the evolution of co-authorship networks. The research presented by Ramasco et al.[@ramasco2004self] falls into the same area; it analyzed in detail the development of collaboration networks. Their model combines preferential edge attachment with the bipartite structure and depends on the act of collaboration. The rise of the giant connected component in the set of $k$-cliques of a classical random graph was described by Derenyi at al. [@derenyi2005clique] as well as a $k$-clique community, as a union of all $k$-cliques. A novel model of multi-layer network was proposed by Battiston et al. [@battiston2016emergence], their model captures a multi-faceted character of actors in collaborative networks.
The universally recognized principle is the so-called preferential attachment. At the moment of the connecting of new nodes to the network during its growth, there is a preference for selecting high degree nodes. The well-known Barabasi-Albert model [@albert2002statistical] is based on experimental work and analysis of large-scale data. Zuev et al. [@zuev2015emergence] described how preferential attachment together with latent network geometry explains the emergence of soft community structure in networks and non-uniform distribution of nodes.
One of the basic characteristics of some types of networks (social and biological), is their community structure. Understanding the principles upon which communities emerge is a key task. Growing network model using the triadic closure mechanism is able to display a nontrivial community structure, as was proposed by Bianconi at al. [@bianconi2014triadic]. The addition of links between existing nodes having a common neighbor as a local process leads to the emergence of preferential attachment as is stated by Shekatkar & Ambika [@shekatkar2015complex]. In another model for growing network proposed by Toivonen et al. [@toivonen2006model], communities arise from a mixture of random attachment and implicit preferential attachment.
A frequent feature of these approaches is that communities rise from a combination of links between existing nodes with their neighbors to new nodes. Some of these approaches do not use the preferential attachment for node selection because the scale-free property is the result of underlying processes.
Another well-known property of real-world networks is that communities have overlaps. A node may belong to more cliques simultaneously, and this property is the basis of the Clique Percolation Method presented by Palla et al. [@palla2005uncovering]. The clique graph, wherein cliques of a given order are represented as nodes in a weighted graph, is a conceptual tool to understand the $k$-clique percolation described by Evans [@evans2010clique]. Yang and Leskovec introduced the Community-Affiliation Graph [@yang2012community] based on observation, that community overlaps are denser than communities themselves.
Processes in the networks take place in time. Networks are from this perspective temporal, and each interaction is reflected in changes to the network structure. Application of the principles mentioned above in the course of network evolution allows us to examine how network structure changes over time. Holme & Saramaki [@holme2012temporal] presented a time-varying importance of nodes and edges together with a survey of existing approaches and the unification of terminology in the area of temporal networks research. Ramasco at al. [@ramasco2004self] studied social collaboration networks as dynamic networks growing in time by the continuous addition of new acts of collaboration and new actors. In real-world networks, particularly social ones, instances often have strong relations defined as interactions that are frequently repeated (nodes remember them), as well as weak relations representing the occasional interactions. Karsai et al. [@karsai2014time] explain, how is creating new relationships and strengthening existing links in networks important for network evolution.
There are also novel approaches focused on models which allow generating networks with predictable properties. For instance, Zhang et al. [@zheng2014simple] formulated a generative model as an optimization problem.
In our approach, we do not use preferential attachment in a straightforward manner. It is, however, a side effect of principles related to the nature of the formation of the community structure. In our model, we use a clique as a structural element of the network. A clique is the result of interaction among nodes and is the basis of the community structure. Our networks are generated as temporal because one interaction is the result of one step of the growth of the network.
Conclusion {#sec:conc}
==========
Evolution of real-world networks is influenced by many factors. The purpose of network models is to discover these factors and describe them in a simple way. Our research focused on analyzing behavioral patterns of nodes existing in a co-authorship network while participating in publishing activities. We described four roles of nodes involved in interactions. Based on the analysis of the DBLP dataset, we formulated the hypothesis, which assumes that the numbers of nodes involved in interactions revolve around an average, and they are independent Poisson variables. Based on this hypothesis, we defined the 3-lambda model of collaborative network. The model has three parameters and has no memory. In three experiments, based on three different settings corresponding to dyads, triads and larger groups behavior, we showed that networks generated by the 3-lambda model have the characteristics known from the environment of real-world social networks. Furthermore, we showed that the model can be understood as temporal. In one experiment we presented the development and stabilization of generated network properties in time. For future research there remain some open questions. They bear relation to recognizing other factors influencing the development of the network and to the detailed study of the dependence and predictability of properties of generated networks on the setting of three network parameters (lambdas).
Acknowledgments
===============
This work was supported by SGS, VSB-Technical University of Ostrava, under the grant “Parallel Processing of Big Data”, no. SP2016/97.
|
---
abstract: 'Black holes monopolize nowadays the center stage of fundamental physics. Yet, they are poorly understood objects. Notwithstanding, from their generic properties, one can infer important clues to what a fundamental theory, a theory that includes gravitation and quantum mechanics, should give. Here we review the classical properties of black holes and their associated event horizons, as well as the quantum and thermodynamic properties, such as the temperature, derived from the Hawking radiation, and the entropy. Then, using the black hole properties we discuss a universal bound on the entropy for any object, or for any given region of spacetime, and finally we present the arguments, first given by ’t Hooft, that, associating entropy with the number of quantum degrees of freedom, i.e., the logarithm of quantum states, via statistical physics, leads to the conclusion that the degrees of freedom of a given region are in the area $A$ of the region, rather than in its volume $V$ as naïvely could be thought. Surely, a fundamental theory has to take this in consideration.'
author:
- 'José P. S. Lemos'
title: 'Black hole entropy and the holographic principle [^1]'
---
Introduction
============
Black holes have been playing a fascinating role in the development of physics. They have entered into the physics domain through a combination of the disciplines of general relativity and astrophysics. Indeed, black holes arise naturally within the theory of general relativity, Einstein’s geometric theory of gravitation. Being exact vacuum solutions of the theory, they are thus, unequivocally, geometric objects. From the first solution in 1916, the Schwarzschild black hole, to the rotating Kerr black hole solution found in 1963, until they were accepted as the ultimate endpoints of the gravitational collapse of massive stars, as well as the gravitational collapse of huge amounts of matter (being it, clusters of stars, dark matter or any other matter form) in the center of galaxies, there has been a highly winding road (see, e.g., [@htww; @penrose65; @mtw] for the initial developments and references therein). The name black hole was coined in 1968 by Wheeler [@wheelerbh] (see also [@wheelerautobiog; @bhname]). Now there is no doubt that solar mass black holes abound in our Galaxy, and supermassive ones reign at the centers of galaxies playing their roles as energizers of their own neighborhoods, such as in quasars (see [@kormendy] for the status of supermassive black holes in galaxies). New theoretical developments show that black holes can form in various ways. They can be eternal being out there since the very initial universe, they can be pair created in a Schwinger type process, they can form from the collision of highly energetic particles as in accelerators, and in the more usually case, discussed in the original work that gave rise to the concept, they can form from the collapse of matter. These and other developments, which have put black holes at the center of studies in fundamental physics, were possible after Hawking discovered that they act as thermal quantum creators and radiators of particles [@hawking1975]. This result, albeit in a semiclassical regime, unites in one stroke, gravitation (represented by the universal constant of gravitation $G$, and the velocity of light $c$), quantum mechanics (represented by Planck’s constant $\hbar$) and statistical physics (represented by Boltzmann’s constant $k_{\rm B}$). Thus, black holes turn out to be in the forefront of physics, since by acting as unifying objects, through them one can test unifications ideas of gravity (and possibly other fields) and quantum mechanics.
Classical properties of black holes
===================================
The event horizon
-----------------
General relativity introduces the idea that gravitation is a manifestation of the geometry and curvature of spacetime. Its equations, Einstein’s equations, imply that objects, like test particles either massive or massless (like light), move as geodesics in the given underlying curved geometry which, in turn, is established by a certain concentration of matter and energy. Einstein’s equations imply in addition that a high concentration of matter and energy curve spacetime strongly. When the concentration of matter and energy is high enough, such as in a collapsing star, spacetime will be so curved that it tears itself, so to speak, and forms a black hole. And, like a burst of water in a river, that suddenly opens up a falls, once falling down the falls it is impossible to get back. Indeed, a black hole is a region where the spacetime curvature is so strong that the velocity required to escape from it is greater than the speed of light. The surface of the black hole is the limit of the region from where light cannot escape. Just outside this surface light and particles may escape and be detected at infinity. Inside the surface all particles fall through and never come out again. This boundary, defining what can be seen by observers outside the black hole, the boundary of the region of no return to the outside, is called the event horizon. Thus, in the case the black hole forms from a collapsing star, say, the event horizon is not to be identified with the surface of the star that formed the black hole. The matter that formed the black hole goes through its own horizon, and once inside the horizon, it will continue to collapse right down the falls it created, until a singularity forms, where the curvature of spacetime blows up, i.e., tidal forces disrupt spacetime itself. There is a horizon floating outside, whereas the surface of the star and indeed the whole of the star are now at the singularity. In a black hole, the singularity is hidden behind the horizon, in the hidden region, where nothing comes out. In brief, a black hole is not a solid body with a matter surface, it is a three surface in space and time bounded by a horizon. It is a pure gravitational object with an event horizon, from inside which there is no escape, and with a hidden singularity at its center. A singularity is an object nobody knows what it is. To know what lies inside a black hole, what is a singularity, it is certainly one of the most important problems to be solved in physics, but due to its complexity one sees it seldom discussed. Here we also do not discuss it. Rather, we are interested in the horizon and in the black hole properties, classical and quantum, exterior to it.
In the simplest case the horizon is a sphere, and the corresponding solution is called the Schwarzschild solution. In this case the horizon radius $R_{\rm bh}$ scales with the mass $M$ of the black hole, $R_{\rm bh}=2M\,\left(\frac{G}{c^2}\right)$, or in natural or Planck units ($G=1$, $c=1$, $\hbar=1$, $k_{\rm B}=1$, units which will be often used), one has $$R_{\rm bh}=2M\,,
\label{horizonradius}$$ so that the more massive the black hole the larger the horizon (see, e.g., [@mtw]). One can think of the horizon as a sphere of photons, or null geodesics, that are trying to get out radially, but due to the strong gravitational force, i.e., high curvature, stay fixed at $R_{\rm bh}=2M$. Photon spheres inside the horizon, in the hidden region, are dragged down to the singularity even if they are locally outgoing. On the other hand, photon spheres outside the horizon, moving radially, reach infinity, with those that originate in the horizon’s vicinity having to climb a huge gravitational field, or geometrical barrier, take a long time to do so.
Black holes have no hair
------------------------
In general relativity, time depends on the observer. Observers, particles, whatever that enters the black hole will go straight to the singularity, and will not come back, according to their own reckoning. On the other hand, observers that stay outside the horizon, see things differently. They cannot know what happens inside a black hole. Classically, nothing, no signal nor information, can emerge across the horizon and escape into the outside to be detected by an observer in the external world. However, at least one feature of the spherical black hole can be measured outside. One can know from the outside the mass of the black hole. Thus, if the Schwarzschild black hole was formed from the collapse of matter, the only property of the initial matter that can be known after it has collapsed is its mass, all the knowledge on the other properties of the initial matter has disappeared down the horizon. Moreover, additional matter falling into the black hole simply adds to the mass of the black hole, and disappears from sight, taking with itself its own properties. More generally, within general relativity, there are three parameters that can characterize a black hole. A spherically symmetric vacuum black hole, the Schwarzschild black hole, is characterized only by its mass $M$, with the horizon radius given by equation (\[horizonradius\]). A great deal of complexity is added if in addition to the mass $M$ the black hole possesses angular momentum $J$, with the horizon being now oblate instead of spherical. This is the important rotating Kerr black hole solution, an exact solution, that by introducing new dynamics, gave a totally new flare to classical black hole theory, which in turn was essential in the construction of the coupling of black holes to quantum mechanics. Adding electric charge $Q$ one has the Kerr-Newman black hole family, where for $J=0$ one calls the charged nonrotating black hole a Reissner-Nordström black hole. So, a generic Kerr-Newman black hole is characterized three parameters only, namely $(M,J,Q)$, this being the only knowledge one obtains out of a black hole (see, e.g., [@mtw]). Such a black hole can form from the collapse of an extremely complex rotating distribution of ions, electrons, radiation, all kinds of other matter, and myriads of other properties characterizing the matter itself. But once it has formed, for an external observer, the only parameters one can know from the outside are the the mass-energy $M$ of the matter that went in, its angular momentum $J$, and its electrical charge $Q$. One then says a black hole has no hair, since it has only three hairs, and someone with three hairs is effectively bald (see, e.g., [@mtw]). This property has measurable consequences on the spacetime outside the horizon. The black hole’s mass, angular momentum, and electric charge certainly change accordingly to the type of matter that is added onto it. Moreover, this change obeys strict rules, indeed, one can show that the laws of conservation of energy, angular momentum and charge, are still valid when a black hole is involved [@bardeenetal]. Thus, $M$, $J$, and $Q$ are observable properties that can be known through some form of external interaction with the black hole. On the other hand, all the other features that could possibly further characterize the black hole do not exist. Or if they exist they have vanished from sight. These other features certainly characterize usual matter, star matter say. But once the star has imploded into a black hole all the features, but three, disappear. Where are now those features? Is this property of hiding features, one that one can capitalize on, and discover new properties of the world? By a remarkable twist, quantum mechanics comes to the rescue.
Quantum properties of black holes
==================================
The questions raised lead us into the quantum realm and put black holes on a central scene to unify in one stroke, gravitation and quantum mechanics itself, within the framework of thermodynamics and through the concepts of black hole temperature and entropy. These results will also definitely impel into an ultimate bound on the entropy of a given region and to the the establishment of a new revolutionary holographic principle. Let us see each point one at a time, and then all altogether.
There are now many theories, of general relativity type, that have many different black hole solutions, with charges other than $M$, $J$, and $Q$. These theories, one way or the other have general relativity as a limit. So let us stick to general relativity, and moreover let us study the simplest case the Schwarzschild black hole, with its only one hair, the mass $M$ and the associated horizon radius $R$ given in equation (\[horizonradius\]). The Kerr black hole was very important to put nontrivial dynamics on a classical setting, it acted in this context as a catalyst, but after it induced forcefully the introduction of quantum phenomena in the whole scheme, one can use the simplest black hole, the Schwarzschild black hole, this is indeed sufficient to understand the profound ideas that lie underneath black hole physics.
Black hole thermodynamics
-------------------------
The first law of thermodynamics states that the total energy is conserved in an isolated system. It seems trivially obvious nowadays, but some time ago it was hard to understand the nature of heat as energy, an achievement that was accomplished after the work of Carnot first, and the further insights of Mayer, Joule, Kelvin, and Helmholtz in the 1840s [@lindley; @brush1; @brush2]. In its simple form it states that $dE=dQ$, where $Q$ is the amount of heat exchanged, and $dE$ is the variation in internal energy of the system. Another important idea in thermodynamics, and in physics in general, is the introduction of the concept of $dQ/T$, heat over temperature, a state variable. It was devised by Clausius in 1854, who also found an adequate name for it in 1865, entropy $S$, such that $dS=dQ/T$. In terms of state variables the first law can now be written as $dE=T\,dS$. This concept of entropy, also led Clausius to postulate a second law of thermodynamics by stating that equilibrium states have an entropy associated with them such that processes can occur only when the final entropy is larger than the initial entropy, i.e., in any closed system, entropy always increases or stays the same, $d
S\geq0$. As also worked out by him in a paper dealing with the “the nature of the motion which we call heat”, the entropy concept had an immediate impact in kinetic theory and statistical mechanics (see, e.g., [@brush2]). Both these advances are remarkable. Of course, Clausius could not know of its ramifications and problems that such a concept would introduce more than 100 years later, when applied to gravitating systems. A first hint of these problems appeared in studies on Newtonian gravitating systems, such as in clusters of stars [@lyndenbell1; @lyndenbell2], whereas one needs general relativity to apply thermodynamic and statistical mechanics concepts to fully general relativistic objects. Two such systems are the Universe itself and a black hole. That the entropy concept and its associated second law of entropy increase can have remarkable implications upon the Universe as a whole and on the arrow of time was first understood by Boltzmann within his statistical physics formulation (see, e.g., [@brush2; @eddington]), an issue that is still today under heavy discussion [@huw; @penrosebook]. But conundrums of a different caliber and with a more direct physical significance perhaps, involving physics at the most fundamental level, have arisen from the fact that a black hole has entropy, an entropy with a form never seen before. Indeed, black hole entropy is proportional to the area of the black hole, rather than the volume. Let us first see how the black hole entropy arises and then where the second law takes us to.
If one thinks, as before, of a black hole forming from the collapse of a matter star, one has an initial configuration, a star say, and a final configuration, a black hole. The star is specified by very many parameters and quantities, the black hole by the mass $M$ alone, in the spherical vacuum case. This led us to argue above that a black hole is a system specified by one macroscopic hair parameter only, the mass $M$ and hiding lots of other parameters perhaps located inside the black hole event horizon. Thus, the black hole acts like a black box. In physics there is another instance of this kind of black box situation, whereby a system is specified and usefully described by few parameters, but on a closer look there are many more other parameters that are not accounted for in the gross macroscopic description. This is the well know case of thermodynamics described above. For thermodynamical systems in equilibrium one gives the energy $E$, the volume $V$, and the number of particles $N$, say, and one can describe the system in a useful manner, obtaining from the laws of the thermodynamics its entropy and other important quantities. On doing this one does not worry that the system encloses a huge number of molecules and that the description hides its own microscopic features. Of course, one can then plunge into a deeper treatment and apply statistical mechanical methods to the particles constituting the thermodynamical system, using the distribution density function of Gibbs for classical particles or the density matrix for quantum ones in the appropriate ensemble, and then applying Boltzmann’s formula for the entropy $S=k_{\rm B}\,\ln\Omega$, where $\Omega$ is the number of states, or any other formula, like Gibbs’ formula, to make the connection to thermodynamics. Due to this black box analogy between a black hole and a thermodynamic system, one can ask first the question: Is thus a black hole a thermodynamic system? If yes, one should pursue and ask two further questions: Can one find the analogue of the constituent particles to allow for a statistical interpretation? To where can the black hole thermodynamic system lead us to, in terms of the ultimate fundamental theory?
The first, and then the subsequent questions, started to be answered through a combination of hints. From the Penrose and superradiance processes, deduced using Kerr black hole backgrounds, one could conclude that the area of a black hole would not decrease in such cases [@penroseprocess; @superradiancep1; @superradiancep2; @christodoulou], an idea that culminated with the underlying area law theorem, which states that in a broad class of circumstances, such as in black hole merger events, the area could never decrease, only increase or stay even in any process [@hawkingarealaw]. At about the same time, Wheeler raised the problem (see [@wheelerautobiog]), that when matter disappears into a black hole, its entropy is gone for good, and the second law seems to be transcended, i.e., in the vicinity of a black hole entropy can be dumped onto it, thus disappearing from the outside world, and grossly violating the second law of thermodynamics. Bekenstein, a Ph.D. student in Princeton at the time, solved part of the problem in one stroke. With the hint that the black hole area always increases, he postulated, entropy is area [@bekensteins-a]. Specifically, he postulated [@bekensteins-a], $S_{\rm bh}=\eta\,\frac{A_{\rm bh}}{A_{\rm
pl}}\;\,k_{\rm B}\,, \label{entropybekenstein}$ where $A_{\rm bh}$ is the black hole area, $\eta$ is a number of the order of unity or so, that could not be determined, $A_{\rm pl}$ is the Planck area, and $k_{\rm B}$ is the Boltzmann constant. Note that the Planck length $l_{\rm pl}\equiv\sqrt{\frac{G\hbar}{c^2}}$, of the order of $10^{-33}\,$cm, is the fundamental length scale related to gravity and quantum mechanics, and the Planck area is its square, $A_{\rm
pl}=l_{\rm pl}^2\sim 10^{-66}\,{\rm cm}^2$. Several physical arguments were invoked to why the entropy $S$ should go with $A_{\rm bh}$ and not with $\sqrt {A_{\rm bh}}$ or $A_{\rm bh}^2$. For instance, it cannot go with $\sqrt {A_{\rm bh}}$. This is because $A_{\rm bh}$ itself goes with $R_{\rm bh}^2\sim M^2$, for a Schwarzschild black hole, and when two black holes of masses $M_1$ and $M_2$ merge, the final mass $M$ obeys $M<M_1+M_2$ since there is emission of gravitational radiation. But if $S_{\rm bh}\propto\sqrt {A_{\rm bh}}
\propto M<M_1+M_2\propto S_{\rm bh1}+S_{\rm bh2}$ the entropy could decrease, so such a law is no good. The correct option turns out to be $S_{\rm
bh}\propto A_{\rm bh}$, the one that Bekenstein took. It seems thus, there is indeed a link between black holes and thermodynamics. In addition, it seems correct to understand that this phenomenum is a manifestation of an underlying fundamental theory of spacetime, a quantum theory of gravity, since the Planck area appears naturally in the formula, hinting that there must be a connection with some fundamental spacetime microscopic ingredient whose statistics connects to the thermodynamics.
Since there is a link between black holes and thermodynamics, black holes have entropy, one can then wonder whether black holes obey the first and second laws of thermodynamics (see, e.g., [@parker; @davies; @lemos1] for reviews on black hole particle creation and black hole thermodynamics). In relation to the first law, note that for a Schwarzschild black hole, the simplest case, one has that the area of the event horizon is given precisely by $A_{\rm
bh}=4\pi\,R_{\rm bh}^2$. Now, $R_{\rm bh}=2M$, so one has $A_{\rm bh}=16\pi M^2$ (in natural or Planck units). Then one finds $dM=1/(32\,\pi\,M)\, dA_{\rm
bh}$, which can be written as [@bardeenetal], $$dM=\frac{\kappa}{8\,\pi}\, dA_{\rm bh}\,,
\label{firstlawofbhs}$$ where $\kappa$ is the surface gravity of the black hole, a quantity that can be calculated independently and gives a measure of the acceleration of a particle at the event horizon. In the Schwarzschild case $\kappa=1/4\,M$. Equation (\[firstlawofbhs\]) is a simple dynamical equation for the black hole. When one compares it with the first law of thermodynamics, $
dE=TdS\,,
\label{firstlawofthermodynamics}
$ the similarity is striking, and since following Bekenstein $S_{\rm bh}
$ and $A_{\rm bh}$ are linked, and following Einstein $M$ and $E$ are linked, indeed they are the same quantity, one is tempted to associate $T$ and $\kappa$ [@waldbook]. But from thermodynamical arguments alone one cannot determine $\eta$ the dimensionless proportionality constant of order unity between entropy and area, and cannot also determine the constant of proportionality between $T$ and $\kappa$, related to $\eta$. Using quantum field theory methods in curved spacetime Hawking [@hawking1975], in a spectacular tour de force, showed that a Schwarzschild black hole radiates quantically as a black body at temperature $T_{\rm bh}=\frac{1}{8\pi
M}\left(\frac{\hbar\,c^3}{G\,k_{\rm B}}\right)$, uniting in one formula $\hbar$, $G$ and $c$, and $k_{\rm B}$. In natural Planck units, and returning to $\kappa$ this is, $$T_{\rm bh}=\frac{\kappa}{2\pi}\,,
\label{hawkingtemperature}$$ connecting definitely and physically the surface gravity with temperature, and closing the thermodynamic link. Moreover, from the first law of thermodynamics one obtains $\eta=1/4$, yielding finally $$S_{\rm bh}=\frac14\,A_{\rm bh}\,,
\label{entropyarealawgeometrical}$$ in natural units. Thus, Hawking radiation allows one to determine, on one hand, the relation between the temperature of the black hole and its surface gravity, and on the other hand, to fix once and for all the proportionality constant between black hole entropy and horizon area. The black hole entropy is one quarter of the event horizon’s area, when measured in Planck area units. For thermodynamic systems, this is a huge entropy, the entropy of a black hole one centimeter in radius is about $10^{66}$ in Planck units, of the order of the thermodynamic entropy of a cloud of water with $10^{-3}$ light years in radius. The Hawking radiation solved definitely the thermodynamic conundrum. The generalized first law is then given by a simple extension of equation (\[firstlawofbhs\]). $M$ is now the energy of the whole system, black hole plus matter, $T$ for the matter and $\kappa/2\pi$ for the black hole have the same values, for a system in equilibrium, and the entropy of the thermodynamic system is now $S=S_{\rm bh}+S_{\rm matter}$, a sum of the black hole entropy $S_{\rm
bh}$, and the usual entropy of the matter and radiation fields which we denote simply as $S_{\rm matter}$. It is advisable to separate the entropy into two terms, since one does not know for sure the meaning of black hole entropy.
What about the second law of thermodynamics, can it be embodied in a framework where black holes are present? The second law of thermodynamics mathematizes the evidence that many processes in nature are irreversible, hot coffee cools in the atmosphere, but cold coffee never gets hot spontaneously, and so on. The law states that the entropy of an isolated physical system never decreases, either remains constant, or it increases, usually. It holds in a world where gravitational physics is unimportant. What happens in gravitational systems in which there are black holes. Given that the black hole is a thermodynamic system, with entropy and temperature well defined, the second law of thermodynamics $d\,S\geq0$ should be obeyed. Indeed one can write the second law as $$dS_{\rm bh}+d S_{\rm matter}\geq0\,,
\label{gsl}$$ commonly called the generalized second law [@bekensteingeneralizedsecondlaw]. In words, the sum of the black hole entropy and the ordinary entropy outside the black hole cannot decrease. This generalized second law proved important in many developments, and its consequences are the main object of this review. The generalized second law has passed several tests. For instance, when a star collapses to form a black hole, one can show that the black hole has an entropy that far exceeds the initial entropy of the star. Also, when matter falls into an already existing black hole, the increase in black hole entropy always compensates for the lost entropy of the matter down the horizon. Another interesting example where the generalized second law holds involves Hawking radiation. Due to this radiation the black hole evanesces. Its mass decreases, and so the black hole area also decreases. This violates the area law theorem, but this is no problem, the theorem was proved classically. Then, the black hole entropy decreases indeed. However, one can show that the entropy in the emitted radiation exceeds by some amount the original entropy of the black hole, upholding the generalized second law [@sorkinwalzhang1981]. Using generic arguments hinged on a quantum definition of entropy, it is possible to argue, that due to lack of influence of the inside on the outside, the generalized second law is valid for processes involving black holes [@sorkin86]. We note that there are arguments that claim that one does not need the generalized second law, the ordinary second law alone is enough in itself, see, e.g., [@unruh82; @bekenstein83; @unruh83; @bekenstein94]. This controversy would merit a review in itself.
Black hole entropy
------------------
Before start discussing to where the generalized second law leads us, it is interesting to think about the consequences of black holes having entropy, as Bekenstein did almost immediately after his major discovery [@bekensteins-a]. Entropy is one of the most important concepts in everyday physics. Somehow, it is a recondite concept, and even more mysterious when black holes are involved. Let us see this.
Following Boltzmann, the entropy $S$ of a closed isolated system with fixed macroscopic parameters, is given by, $$S=k_{\rm B}\ln\Omega\,,
\label{boltzentropy0}$$ where again $k_{\rm B}$ is the Boltzmann constant, and $\Omega$ is the number of accessible microstates that the large system has. Each microstate $i$ has equal probability $p_i$ of occurring, so $p_i=1/\Omega$, and equation (\[boltzentropy0\]) can be written in the alternative form $S=-k_{\rm B}\ln\,p_i$. For open systems, that can exchange energy and other quantities, the entropy can be written in a more useful manner as $
%\begin{equation}
S=-k_{\rm B}\sum_ip_i\ln p_i\,,
\label{gibbs}
%\end{equation}
$ where $p_i$ is the probability of microstate $i$ occurring, which now due to the openness of the system is not anymore equal for each state, states with a given energy, the average energy, have a higher probability of occurring. This formula was given by Gibbs upon careful consideration of his ensemble theory and generalization of Boltzmann ideas (see, e.g., [@cowan] for the deduction of Gibbs entropy formula from equation (\[boltzentropy0\])). If the system is closed then $p_i=1/\Omega$ and Boltzmann equation (\[boltzentropy0\]) follows. In the Gibbs formulation, one works in a $6N$ dimensional classical phase space, and having to work with a continuum distribution probability density, the phase-space density $\rho$ (instead of $p_i$), one should write $S=-k_{\rm
B}\int\,d^{3N}q\,d^{3N}p\,\,\rho(q,p)\,\ln\rho(q,p)$, which is the continuum Gibbs entropy equation for a system of $N$ particles in three dimensional space with $6N$ classical degrees of freedom, $3N$ for the coordinates and $3N$ for the momenta. In this setting each point in the phase-space represents a state, a microstate, of the system. This was then generalized, in a natural way, although through a postulated basis, by Von Neumann to quantum systems. One postulates first that $\rho$ goes into the quantum operator $\hat\rho$ which gives the probability that the system is in some given microstate (essentially is $p_i$), and second that the entropy is $S=-k_{\rm
B}{\rm Tr}\,\hat\rho\ln\hat\rho$. This Von Neumann entropy should be calculated in some complete orthonormal basis of the appropriate state space, or Hilbert space. Since $\hat\rho$ can have non-diagonal terms, which can be suited for calculating quantities other than traces, the Von Neumann entropy is a generalization of the Gibbs entropy, although possibly not unique. All these formulas for the entropy can be useful, depending on the context one is working. Gibbs formula, for instance, has an interesting advantage sometimes. Indeed, the formula is the same as the one that emerged for the entropy in information theory, the Shannon entropy [@shannonbook]. The Shannon entropy first appeared in connection with a mathematical theory of communication, where it was perceived that the best measure of information is entropy. In fact, entropy in an informational context represents missing information. The Shannon entropy formula is given by $S=-k_{\rm S}\sum_i p_i\ln p_i\,$, where here, since the connection with temperature is unimportant, $k_{\rm B}$ is substituted by $k_{\rm
S}$, the Shannon constant, which generally is put equal to $1/\ln2$, so that the entropy is given in bits, a dimensionless quantity. Apart the constant used, $k_{\rm B}$ or $k_{\rm S}$, which is a matter of convenience, the two entropies are the same. However, Shannon entropy is applied to measure the information a given system (a computer for instance) has, basically how many bits the system has, whereas Gibbs entropy is applied to the thermodynamic system itself, essentially the number of molecules the system (a computer for instance) has. Both entropies can be given in Shannon units, of course. Gibbs entropy is usually much larger than Shannon entropy. The day bits are imprinted on molecules, rather than in chips, the two entropies will give the same number. The connection between information and entropy turns out to be very useful and important in black hole theory, see, e.g., [@bekensteininformation].
To try to understand the meaning of the black hole entropy given in equation (\[entropyarealawgeometrical\]), one can use the various formulas for the entropy presented above. But here, for our purposes, it is simpler if we explore Boltzmann’s formula (\[boltzentropy0\]). It chiefly claims that one way to think about entropy is that it is a measure, a logarithmic measure, of the number of accessible microstates that the isolated system has. Any system, including a black hole, should follow this rule. For black holes, there is a snag, we do not really know what those microstates are, so we cannot count them to take the entropy. There are several ideas. One idea is that the microstates could be associated to the singularity inside the event horizon, where the crushed matter and the demolished spacetime lie altogether. As in the ordinary matter case, one could think that rearranging these states, somehow lying on the singularity, do not affect the mass $M$ of the black hole (and $Q$, and $J$ for the other hairs, if there are those). There are problems with this interpretation for the entropy. The singularity is in principle spacelike, in addition it is certainly causally disconnected to the outside of the black hole, and therefore it is hard to imagine how it could influence any quantity exterior to itself, let alone to the exterior of event horizon. This interpretation is related to the interpretation that the degrees of freedom, are in some measure of the volume inside the horizon (see, e.g., [@bekensteinreviewhistory; @jacobson]). Moreover, such type of interpretations are very difficult to implement since no one knows really what goes on inside let alone in a a singularity, only with a fully developed theory of quantum gravity can one attempt to understand singularities. Another place where the microstates might be located is in the vicinity of the event horizon area as has been suggested many times (see, e.g., [@wheelerjourneybook] for a heuristic account, [@carlip99; @solodukhin; @diaslemos] for a particular implementation, and [@lemosreview] for a review). The idea beyond this suggestion is that for photons emitted near the horizon, only those with very high energies, indeed trans-Planckian energies, can arrive with some finite nonzero energy at infinity, and so, these photons probe near-horizon Planckian structures, i.e., probe quantum gravity. Indeed, light sent from the very vicinity of the horizon has to climb up the huge gravitational field, or if one prefers, the huge spacetime barrier set up by the black hole. In turn this means that the pulse of light an observer a distance away form the horizon receives has a much lower frequency (much higher wavelength) than the very high pulse frequency (very low wavelength) of the emitted pulse. This is the redshift effect. The nearer the horizon the pulse is emitted the higher the effect. Since in quantum mechanics frequency and energy are the same thing, $E=\hbar\omega$, the closer to the horizon the photon is emitted, the more energy it must get rid off as it travels towards the observer. In effect, there is an exponential gravitational redshift near the horizon so that the outgoing photons and other Hawking radiation particles originate from modes with extremely large, trans-Planckian, energies. But now this is very important, photons with very high energy, very low wavelengths, probe very small regions. So the Planckian and trans-Planckian photons, that arrive at the observer somehow come from regions of space and time that are themselves quantum gravity regions. There are various possibilities for these regions, such spacetime regions may be discrete, or may be fluctuating in a quantum foam structure, or whatever. Thus, if one can observe Hawking photons originating from very close to the horizon of a black hole, one is possibly seeing the quantum structure of the spacetime. In the context we are discussing, this means that the entropy should be a feature of the horizon region itself. Near the horizon quantum gravity and matter fields are being probed, and these, alone or together, can be the degrees of freedom one is looking for to generate the entropy of the black hole. This fact led thus to the proposal that the entropy is in the horizon area. This proposal is very interesting and may solve the degrees of freedom, or the entropy, problem. But this follow up from black hole thermodynamics is not our main concern here, see [@lemosreview] and references therein for more on that. We have commented on it solely to get a preliminary understanding of black hole entropy. Even without understanding where are those degrees of freedom that make up the black hole entropy one can derive some new consequences, such as the entropy bounds and the holographic principle.
An Entropy bound involving black holes
======================================
The generalized second law allows us to set bounds on the the entropy of a given system. Or, in terms of information, it sets bound on the information capacity any isolated physical system can have. Since this law involves gravitation, and gravitation together with quantum mechanics should provide a fundamental theory, the bound refers to the maximum entropy up to the ultimate level of description, a given region can have.
To obtain the bound let us think of the formation of a black hole from the collapse of some ordinary matter. It is interesting to consider thus an initial configuration, a star say, and a final configuration, a black hole. Consider then any approximately spherical isolated matter that is not itself a black hole, and that fits inside a closed surface of area $A$. If the mass can collapse to a black hole, the black hole will end up with a horizon area smaller than $A$, i.e., $A_{\rm bh}\leq A$. The black hole entropy, $S_{\rm bh}=A_{\rm
bh}/4$, is therefore smaller than $A/4$. According to the generalized second law, the entropy of the system cannot decrease. Therefore, the initial entropy of the matter system, $S_{\rm initial}^{\rm system}$, cannot be larger than $A_{\rm bh}/4$, and so not larger than $A/4$. It follows that the entropy of an isolated physical system with boundary area $A$ is necessary less than $A/4$, i.e., $S_{\rm initial}^{\rm
system}<A/4$. So, following ideas devised early by Bekenstein [@originalbekensteinbound], Susskind [@susskindsphericalbound] through such a simple argument developed this spherical bound. Putting $S_{\rm initial}^{\rm system}\equiv S$, to clarify the notation, Susskind’s bound reads $$S\leq\frac14\,A\,.
\label{entropybound}$$ One can now anticipate a result which will be further discussed in the next section: since $A$ is the number of Planck unit areas that tile the area $A$, the bound says that the number of quantum degrees of freedom, or the logarithm of the number of quantum states, of the system within an area $A$ is necessarily equal or less than one quarter of the number of Planck unit areas that fit in the area $A$. In brief, following [@susskindsphericalbound] the generalized second law implies the bound (\[entropybound\]), usually called the spherical holographic bound for reasons we will see below.
One example one can give that certainly satisfies the bound refers to two black holes in a box. Let us put two Schwarzschild black holes of masses $M_1$ and $M_2$ in a box. The entropy is $S=\frac14 \left(
A_{\rm bh1}+A_{\rm bh2}\right)$. Since $A_{\rm bh1} =4\pi
R_{\rm bh1}^2= 16\pi
M_1^2$ and $A_{\rm bh2} =4\pi R_{\rm bh2}^2= 16\pi M_2^2$, one has $S=4\pi
\left(M_1^2+M_2^2\right)$. Now, from a distance, the system should not be a large black hole of mass $M=M_1+M_2$, otherwise the argument is of no interest. So, there is a radius for the box $R$, with the associated area $A$, which obeys $A_{\rm bh1}+A_{\rm bh}<A$, i.e., $\frac14\left( A_{\rm bh1}+A_{\rm bh2}\right)<\frac14 A$. Finally, since $S=\frac14\left( A_{\rm bh1}+A_{\rm bh1}\right)$ one has $S<\frac14
A$. The bound is clearly satisfied, and it is saturated only when the box is a black hole. There are many other examples one can think of.
Now this bound suffers from some drawbacks, it only applies to systems which are initially nearly spherically symmetric, and not much strongly time dependent. In addition is not a covariant bound, and in general relativity, all statements should some way or another be put in a covariant form. These problems were cured by Bousso [@bousso1; @bousso2; @bousso3], who has managed to formulate a covariant entropy bound, (see also [@flanagan; @gaolemos1; @gaolemos2; @gaolemos3]). Susskind’s bound is a particular case of this Bousso’s covariant bound. In addition, the original Bekenstein bound [@originalbekensteinbound] can be derived through Susskind bound, and surely, through Bousso’s bound (see [@boussoderivationofbounds]). The covariant entropy bound is fascinating and has been proved correct in very many instances. However, due to its simplicity, it is useful to stick to the spherical bound given in equation (\[entropybound\]). This will take us more directly to the holographic principle.
Holographic principle
=====================
Definition
----------
What are the ultimate degrees of freedom, what are the degrees of freedom of quantum gravity, what are the fundamental constituents of spacetime? Portions of ordinary matter are made of molecules, which are made of atoms, which are made of electrons and nuclei, which nuclei are made of protons and neutrons, which are made of quarks, and so on including all known interactions and their associated particles, up to the quantum gravity level, the fundamental spacetime level. For ordinary matter and the corresponding cascade of constituents and interactions we know what and where the degrees of freedom are. For spacetime we do not, yet, unfortunately. However, even without knowing of what the spacetime is made of we can extract limits for the number of such degrees of freedom, and other relevant information from the entropy bounds discussed previously.
Indeed, based on his own ideas about entropy bounds and even before the spherical bound was advanced, ’t Hooft [@thooftonholoprinciple] proposed that the degrees of freedom of a region of space circumscribed within an area $A$ are in the area itself. This is counter to the results of everyday physics which give that the entropy is proportional to the volume of the region, and so the degrees of freedom of these usual systems are in this sense in the volume. A usual system has entropy, or information, inside it. For instance, in a book, the information is contained inside (in the volume), not in the cover (in the area). One knows that reading the title of a book is not enough at all to know what is inside. This also happens for all usual thermodynamic physical systems upon a statistical physics treatment. However, there are system that do not follow this rule. These are the black holes, which once more reveal themselves as the most fundamental objects to uncover the secrets of nature. For black holes, the entropy and information of what is inside is projected in the area. Since black holes are of fundamental importance, both in gravitation and quantum theory, they are the ones that dictate the ultimate rule that should be obeyed. Thus, the generalized second law, derived from black hole physics, together with the bounds above conjure to give the result that the degrees of freedom are in the area itself.
To be definitive, define first the number of degrees of freedom $N_{\rm f}$ of a quantum system as the logarithm of the number of quantum states $\Omega$ of the system, with $\Omega$ being the same as the dimension of the Hilbert space of the system, (parts of this exposition follows [@susskindsphericalbound; @bousso3; @thooftonholoprinciple]). So $N_{\rm f}=\ln \Omega$. This generalizes the idea of degrees of freedom of a classical system. To have an idea of what $N_{\rm f}$ means, take, for example, a spin system with 1000 spins, each spin being able to be up or down only. Such a system should have around 1000 degrees of freedom. In fact, from the definition above, since there are two states for each spin, one has that the number of states for the whole system is ${\cal N}=2^{1000}$. So its number of degrees of freedom is $N_{\rm f}=1000\ln 2$. In terms of information, following Shannon, this means that the system can store 1000 bits, or its Shannon entropy is 1000, as mentioned in Section III. This system is small, thermodynamics systems are huge in comparison. For a given isolated thermodynamic system, with entropy $S$, the number of independent quantum states is $\Omega={\rm e}^S$, in natural units, see equation (\[boltzentropy0\]). So, for a thermodynamic system, $N_{\rm f}$ is related to the entropy $S$, in fact, following the definition above, they are the same in natural units, $S=N_{\rm f}$. For instance, in order to see that such a definition is reasonable, recall that the number of states $\Omega$ of an ideal gas with fixed energy $E$, volume $V$, and number of particles $N$ can be written as $\Omega=\left[{\rm e}^{5/2}\,(V/N)(4\pi mE/3N)^{3/2}/h^3\right]^N$, so that since $V=L^3$, where $L$ is the dimension of the enclosure say, and $(2mE)^{1/2}=\bar p$, where $\bar p$ is a typical momentum of the particles, one finds $N_{\rm f}=6N\, \alpha$ where $\alpha$ is a number of the order one or so, proportional to a logarithm term. Thus, $N_{\rm f}$ as defined gives roughly the classical number $6N$ of degrees of freedom as expected. Of course, through this definition one has exactly $S=N_{\rm f}= 6N\left[{5/12}+(1/6)\ln\left((V/N)(4\pi
mE/3N)^{3/2}/h^3\right)\right]$, which is the Sackur-Tetrode formula (see, e.g., [@cowan]).
Let us suppose then that we are given a nearly spherical finite region of volume $V$ with boundary area $A$. Suppose again that, initially, gravitation is not strong enough, so that spacetime is not time dependent and all the relevant physical quantities are well defined. One can consider then that the nearly spherical region has some matter content. However, this content ultimately does not interest us, we can forget about the solid, liquid, gas, or vacuum that fills up the region. At the ultimate level one is only interested in the region itself, in the spacetime itself alone. One wants to know what are the states themselves of that region and what is their number, at the most fundamental level. So we want to know how many degrees of freedom are there for the fundamental system, or how much complexity there is at the fundamental level, or how much information one needs to specify the region. One way to start out and see where it leads to is to pick up a theory that has given fruitful results in ultra microscopic physics. This theory is quantum field theory. It works extremely well in flat spacetime, and with care it can be extrapolated to curved spacetime [@birrel]. A quantum field is described by harmonic oscillators at every point in spacetime. A quantum harmonic oscillator has an infinite number of states and so an infinite number of quantum degrees of freedom. So, there are infinite number of degrees of freedom at every spacetime point in a quantum field. Moreover, within a volume $V$ there are infinite number of points. Thus, a quantum field in a given spacetime background has, by this rationale, a huge infinite of infinite number of degrees of freedom. So it seems. However, one can easily argue, that the number is indeed huge, albeit finite. Indeed, gravity together with quantum theory show that there is a minimum length scale given by the Planck length $l_{\rm pl}$, and a maximum energy scale, the Planck mass $m_{\rm pl}$, beyond which any theory of distances and scales does not make sense. So, crudely, one might guess that there is one oscillator per $l_{\rm pl}$, each with maximum energy $m_{\rm pl}$ (more energy than this turns spacetime into a black hole). One can now think that each spacetime volume $V$ has $V/V_{\rm
pl}$ oscillators and each oscillator has a finite number of states $n$ say, which is large, (the highest energy state for each oscillator is given by the Planck energy). So, in Planck or natural units, the total number of states is $\Omega\simeq n^V$ and thus the number of degrees of freedom is $N_{\rm f}\simeq V\ln n$, i.e., $S\simeq V\ln n$ (see [@bousso3] for more details). So, if this conclusion is fully correct, a fundamental theory needs to account for an entropy proportional to the volume or bulk of each region being considered, i.e., the disorder of the region goes with the volume.
But this naïve reasoning fails when gravity is included. The fundamental theory has to include gravity, for sure, and when this is done one finds that a fundamental theory needs only to account for an entropy proportional to the surface area, and this is much less than entropy proportional to the volume. Let us see this in more detail. Entropy is a measure of the logarithm of the number of microstates of a given system macroscopically specified, so that, as seen, entropy is also a measure of the number of degrees of freedom of the system. Now we know that given an area $A$ there is a bound for the entropy $S\leq \frac14 A$ in Planck units (i.e., $\frac{S}{k_{\rm
B}}\leq \frac14 \frac{A}{A_{\rm pl}}$ restoring units) for any system. Any system, including the fundamental system, has to obey this bound. When we have an adequate quantum gravity it will give an entropy for the quantum system which is equal or lower than this bound. Now a black hole with this same area $A$ saturates the bound, so one can say there are systems that saturate the bound. Thus the number of degrees of freedom of a sphere of area $A$, and the related number of states are given by, respectively, $$N_{\rm f}=\frac14\,A\,,\quad {\rm and} \quad\Omega={\rm e}^{\frac14A}\,.
\label{numerofsatesfundamental}$$ That the number of states has to be given by (\[numerofsatesfundamental\]) can be argued more effectively using unitarity, which claims that an initial state evolves in a well defined manner to a final state, such that probability in quantum theory is conserved. Essentially, it says one can derive the final state from the initial and vice-versa. Given an initial object, or region, suppose that the number of states of the Hilbert space for it goes roughly with ${\rm e}^V$. Allow the object, or the region, to evolve into a black hole of the same size of the region. Then the new number of states is ${\rm
e}^{A/4}$, where $A$ is the area enclosing $V$. But this number is much less than the initial one, so one cannot recover the initial state from the final one, the states would not evolve unitarily. Thus, one should start with ${\rm e}^{A/4}$ as the initial number of states.
Now, the number given in (\[numerofsatesfundamental\]) is much smaller than the number $n^V$ guessed earlier, for lengths larger than about the Planck size. One can understand this much lower number than the one given by the naïve guess of quantum field theory, by invoking heuristic arguments coming from the inclusion of gravity (see again [@bousso3] for more details). It is true we have imposed, naïvely, that there is at most one Planck mass per Planck volume. So there is a high energy cut off, and modes with higher energy than that do not exist and do not contribute to the entropy. That is fine. But this cut off at large scales, scales larger than Planck scales, gives that, within a region of radius $R$ and assuming roughly a constant field density, the mass can scale as $M/M_{\rm Pl}\sim \left(R/R_{\rm
Pl}\right)^3$, i.e., $M\sim R^3$ in Planck units. This cannot be right, since we know that for sure $M/M_{\rm Pl}\buildrel<\over\sim
R/R_{\rm Pl}$, i.e., $M\buildrel<\over\sim R$ in Planck units. For $M\buildrel>\over\sim R$ one forms a black hole, the most massive object that can be localized in the sphere of radius $R$. Thus at face value it seems that one should rather assume that the field content (gravity and possibly other fields) density goes at most as $1/R^2$ rather than constant. And so there are many less states than naïvely one could guess. Due to gravity, a long range universal field, the energy of the field content is lower in large volumes than it could possibly be in small Planckian volumes. Thus crudely, the entropy, which in many ways is related to the energy, is also drastically reduced. The conclusion is that naïve field theory seems to yield more degrees of freedom than those that can be used for generating entropy, or to store information. So, there are at most $A/4$ degrees of freedom inside a region whose volume is surrounded by an area $A$. Most systems have less than $A/4$ degrees of freedom such as any system made of ordinary matter. One system that strictly matches the bound is a black hole, which has precisely $A/4$ degrees of freedom. If one has any system, and wants to excite more degrees of freedom than those given by the bound, then one forms a black hole. A black hole is an object that has the maximum entropy for an outside observer. Perhaps there can be other objects with such an entropy, e.g., quasi black holes (see [@weinberglue; @lemoszaslavskii]), but not object has larger entropy. Summing up what we have seen so far, we can say that Bekenstein’s and Hawking’s works, coupled to the Susskind bound, states that a fundamental theory, one in which gravity is included, has a number of degrees of freedom proportional to the area, which leads to fewer degrees of freedom, and so less entropy or less disorder, than the theory would have to have had the entropy of a region been proportional to the volume, rather than the area.
Then one can go a step further, as ’t Hooft did [@thooftonholoprinciple], actually before the spherical and the covariant bounds were discussed. If the maximum entropy, obtained from fundamental degrees of freedom, in a given region of space, is proportional to the area, rather than the volume, then the degrees of freedom should lie in the area of the region. This is the basis of the holographic principle. It states: a region with boundary area $A$ is totally described by at most $A/4$ degrees of freedom (in Planck or natural units), i.e., about one bit of information per Planck area. In a sense, the description of the processes that happen within the region’s volume, is projected into the surface of that region, in the same way as the visual perception of a three dimensional region can be encoded in a hologram, a two dimensional sheet. So, in principle, there are two possible descriptions, the volumetric or three dimensional, and the areal or two dimensional description, the latter one being certainly more economical. In order to grasp better this idea let us introduce the following allegory [@greene]. Imagine that a futuristic plane is surrounded by a hypothetical giant two dimensional spherical screen located in space. And that all the activities and happenings on such a planet, through illumination, are projected onto this screen. The image on the giant screen would be a blow down of the three dimensional world to two dimensions. If the projection is accurate enough, there are two sorts of people, one two dimensional, the other three dimensional. But in such an accurate case, people in both scenarios can think of themselves as equally alive, and the other as the mirage, each containing the same amount of information and each being described through equivalent mathematical theories, with no theory being more correct than the other.
Implementation
--------------
Concrete examples which satisfy the holographic principle have been found in anti-de Sitter spacetimes, i.e., spacetimes with a negative cosmological constant. A de Sitter universe is one with a positive cosmological constant that creates a universe uniformly accelerating, and present observational results indicate we leave in such a universe. On the other hand, a negative cosmological constant has the property of giving a uniform gravitational attraction over all of the spacetime. Such a spacetime has uniform negative curvature, and due to the relentless constant attraction this spacetime has a boundary, it is as if the spacetime is set up in a box with some definite length. For instance, in this spacetime, a massless particle can travel in a finite time from any point in the interior to spatial infinity and back again.
Anti-de Sitter spacetimes appear often in string theory, or its M-theory generalization, as well as in several corresponding low energy limits that yield supergravity theories. Now, when string theory is properly used in an anti-de Sitter spacetime one finds that it is equivalent to a quantum field theory on the boundary of that spacetime [@maldacenaadscft]. The first instance in which this holographic result was found is not directly applicable to our real universe, first because the cosmological constant of the universe is positive rather than negative, and second because in [@maldacenaadscft] it was found that the calculations simplify if one works in a five dimensional (four space and one time dimensions) anti-de Sitter spacetime, ${\rm AdS}_5$, rather than the more usual four dimensional one, ${\rm AdS}_4$. String theory is well formulated only in ten dimensions, so to be precise, one can also include compactified dimensions. In fact, the example is given in the context of ${\rm AdS}_5$ times the five sphere ${\rm S}_5$, so that the whole spacetime is ${\rm AdS}_5\times{\rm S}_5$. The equivalent boundary quantum field theory arises from the boundary of ${\rm
AdS}_5$. In this setting, one can argue that the physics experimented by an observer living in the bulk of the ${\rm AdS}_5$ spacetime can be completely described in terms of the physics taking place on the spacetime’s boundary. This initial result, valid for a five dimensional anti-de Sitter spacetime and its four dimensional boundary dual, was later exhibited in many other situations and other dimensions, including the more usual four dimensional anti-de Sitter spacetime, being in this case dual to a quantum field in three spacetime dimensions. Generically, one finds that the bulk and the boundary descriptions are equivalent, none of the descriptions is more complete or important than the other. In the bulk description gravitation operates and spacetime is d dimensional, say, whereas in the boundary description there is no gravity but a quantum field theory, conformal in nature, operating in a d$\,$-1 dimensional flat spacetime. It is as if there is a duality between this d dimensional spacetime and its d$\,$-1 dimensional boundary. It then means that two different theories, acting in spacetimes with different dimensions, are equivalent. This reinforces the idea that beings living in the d dimensional spacetime would be mathematically equivalent to beings living in the d$\,$-1 dimensional one, there is no way to distinguish between them. This is certainly an interesting implementation of the holographic principle of ’t Hooft [@thooftonholoprinciple]. Technically, it is also fruitful, since difficult calculations performed on the bulk spacetime can perhaps be easily done in the quantum field theory on the boundary and vice versa. For instance, one can show that a black hole in anti-de Sitter spacetime is equivalent to hot radiation in the boundary, and that the mysterious black hole entropy is equivalent to the radiation entropy [@wittenbhradiation]. In addition, it may give insights into the information problem in black hole physics. We have not yet explicitly mentioned it. This problem is related to the entropy interpretation problem of what and where are the degrees of freedom corresponding to the black hole entropy. We have argued that the black hole seems to hide many features inside the horizon. For instance it possibly hides all the information that the original star had before it collapsed into a black hole. But now, how can we get information to reobtain those features? If the inside of black holes are disconnected to the outside world, then classically this information seems to have disappeared. Is there an information loss, and with it a break of unitarity? It has not yet been shown that information is lost or not lost when one throws objects through a black hole, following Hawking’s original suggestion of information loss. But, in this string theory description, a black hole is now dual to a lower dimensional world in which it seems information is never lost. So there is hope in solving this problem. Another place where it can be of use is in quantum field theory itself. The reason is that anti-de Sitter spacetime yields relatively easy calculations, whereas calculations on quantum fields are technically hard. For instance, one cannot yet derive the proton and neutron properties from quantum chromodynamics, the theory of quarks, a well understood theory, but extremely hard to solve. One can now try to solve these properties using the above duality.
Conclusions
===========
We have trodden a long way from the first ideas in black holes with their associated event horizons all within the context of pure classical general relativity, passed through semiclassical calculations meeting the concepts of entropy and temperature for black holes, and then through the statistical physics connection of entropy and its associated second law, arriving at the maximum number of degrees of freedom a fundamental theory, one which includes quantum gravity, can have. Surprisingly, this number goes with the area $A$ of the region, rather than with the volume $V$. In turn this means first that the holographic principle should be valid, i.e., a region with boundary area $A$ is totally described by at most $A/4$ degrees of freedom (in natural units), and second we need a fundamental theory that incorporates this principle. As we have seen, local quantum field theory is certainly not such a theory. Taking this idea seriously, one can advance that the universe can indeed be described by a model with one less dimension, in the sense, that the formulation of the fundamental theory can be done in a lower dimension. Perhaps, as string theory suggests, the fundamental theory can be formulated in the dimensions we are used to (in our case three plus one spacetime dimensions), as well as in the holographic dimensions (in our case two plus one spacetime dimensions). In this case, one should also be able to find a dictionary, or a map, between both formulations. This idea that one can trade spacetime dimensions in a fundamental description, leads one to speculate that, conceivably, spacetime itself is not a so fundamental concept. Of course, if true, such considerations are destined to enter into the philosophy dominion and radically transform our notions of what space and time are. Is the universe a hologram? Is there a shadow universe in which our bodies exist in a compressed two dimensional form? The answer lies ahead.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank the Organizing Committee of the meeting [*Advances in Physical Sciences*]{}, held at the Universidade de Aveiro, September 2005, for the invitation to lecture on this topic, and in particular, I thank Luis Carlos for his patience with the manuscript. I thank António Luciano Videira, to whom this meeting is devoted, for initiating me in scientific matters, namely black hole thermodynamics back in 1980.
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|
---
abstract: 'Interactions of high momentum partons with Quark-Gluon Plasma created in relativistic heavy-ion collisions provide an excellent tomography tool for this new form of matter. Recent measurements for charged hadrons and unidentified jets at the LHC show an unexpected flattening of the suppression curves at high momentum, exhibited when either momentum or the collision centrality is changed. Furthermore, a limited data available for B probes indicate a qualitatively different pattern, as nearly the same flattening is exhibited for the curves corresponding to two opposite momentum ranges. We here show that the experimentally measured suppression curves are well reproduced by our theoretical predictions, and that the complex suppression patterns are due to an interplay of collisional, radiative energy loss and the dead-cone effect. Furthermore, for B mesons, we predict that the uniform flattening of the suppression indicated by the limited dataset is in fact valid across the entire span of the momentum ranges, which will be tested by the upcoming experiments. Overall, the study presented here, provides a rare opportunity for pQCD theory to qualitatively distinguish between the major energy loss mechanisms at the same (nonintuitive) dataset.'
author:
- Magdalena Djordjevic
title: 'Complex suppression patterns distinguish between major energy loss effects in Quark-Gluon Plasma'
---
Introduction
============
In the collisions of ultra-relativistic heavy ions at RHIC and LHC experiments, a new state of matter, called Quark-Gluon Plasma (QGP), is created. Rare high momentum probes transverse and interact with the medium, providing an excellent QGP tomography tool [@Bjorken]. Utilizing such tool requires comparing experimental data with theoretical predictions, where nonintuitive observations present a particular challenge for the theory. Such a challenge is provided by the recent measurements of suppression for charged hadrons [@ATLAS_CH], unidentified jets [@ATLAS_Jets] and B probes [@CMS_JPsi; @CMS_Bjets] at 2.76 TeV Pb+Pb collisions at the LHC. In particular, in Fig. \[Data\] (the left and the central panels) are shown ATLAS [@ATLAS_CH; @ATLAS_Jets] suppression ($R_{AA}$) data for different momentum ranges and as a function of both the number of participants ($N_{part}$), see the left panel, and momentum ($p_\perp$), see the central panel. In particular, ATLAS charged hadron ($h^\pm$) data [@ATLAS_CH] show that $R_{AA}$ [*vs.*]{} $N_{part}$ curves become increasingly flatter as one moves towards higher momentum ranges (compare purple, green and blue data points in the left panel). Furthermore, the central panel shows flattening (saturation) of $R_{AA}$ at high momentum, that can be observed for $R_{AA}$ [*vs.*]{} $p_\perp$ dependence corresponding to unidentified jets at ATLAS (red squares) [@ATLAS_Jets]. These observations are highly non-trivial: the left panel suggest that, while the lower momentum light flavor probes are very sensitive to the $N_{part}$ - and consequently to the system size and energy density- such sensitivity is significantly smaller for the high momentum probes. For the central panel, one observes an apparent plateau reached by $R_{AA}$ data at high $p_\perp$, leading to the question what energy loss mechanism is responsible for this effect. Moreover, a qualitatively different $R_{AA}$ [*vs.*]{} $N_{part}$ pattern is apparently observed for B mesons: $R_{AA}$ for non-prompt $J/\Psi$ at lower momentum [@CMS_JPsi] and B jets [@CMS_Bjets] at high momentum (the purple and the blue dots in the right panel of Fig. \[Data\], respectively), surprisingly show the same $R_{AA}$ [*vs.*]{} $N_{part}$ for these two opposite momentum ranges - both of them indicating small sensitivity to the increase in $N_{part}$; for observing the difference with $h^\pm$ data, compare the purple and the blue data points on the left and the right panels of Fig. \[Data\].
We will study the data patterns summarized above within our state-of-the-art dynamical energy loss formalism [@MD_PRC; @DH_PRL]. Briefly, the formalism takes into account that QGP consists of dynamical (moving) partons - which removes the widely used assumption of static scattering centers - and that the created medium has a finite size. Both collisional [@MD_Coll] and radiative [@MD_PRC; @DH_PRL] energy losses are calculated within the same theoretical framework, which is applicable to both light and heavy flavor, and includes finite magnetic mass [@MD_MagnMass] and running coupling [@MD_PLB]. This formalism is integrated in a numerical procedure which takes into account the up-to-date initial distributions [@Cacciari:2012; @Vitev0912], fragmentation functions [@DSS], path-length [@WHDG; @Dainese] and multi-gluon fluctuations [@GLV_MG]; importantly [*no free parameters*]{} are used in comparing the model predictions with the data. Our aim is not only showing that the dynamical energy loss formalism can well explain the complex $R_{AA}$ data patterns, but even more providing an intuitive explanation for the unexpected experimental observations, which may point to an anticipated example of a qualitative interplay between the major energy loss effects. Finally, we will provide predictions for the upcoming experimental measurements that will further test the mechanism proposed here. Note that predictions presented here are applicable to both 2.76 TeV and 5.02 TeV Pb+Pb collisions at the LHC, since in [@DBZ; @MD_5TeV], we predict that $R_{AA}$s at these two collision energies will be the same.
Results
=======
The predictions are generated by the dynamical energy loss formalism, where the computational procedure and the parameter set are described in detail in [@MD_PLB]. Briefly, we consider a QGP with $n_f{\,=\,}3$ and $\Lambda_{QCD}=0.2$ GeV. For the light quarks, we assume that their mass is dominated by the thermal mass $M{\,=\,}\mu_E/\sqrt{6}$, where the temperature dependent Debye mass $\mu_E (T)$ is obtained from [@Peshier], while the gluon mass is $m_g=\mu_E/\sqrt{2}$ [@DG_TM] and the charm (bottom) mass is $M{\,=\,}1.2$GeV ($M{\,=\,}4.75$GeV). Since various non-perturbative calculations [@Maezawa; @Nakamura; @Hart; @Bak] have shown that magnetic mass $\mu_M$ is different from zero in QCD matter created at the LHC and RHIC, the finite magnetic mass effect is also included in our framework. Moreover, from these non-perturbative QCD calculations it is extracted that magnetic to electric mass ratio is $0.4 < \mu_M/\mu_E < 0.6$, so the uncertainty in the predictions, presented in this section, will come from this range of screening masses ratio. Path-length distributions are taken from [@Dainese].
The temperatures for different centralities are calculated according to [@DDB_PLB]. As a starting point in this calculation we use the effective temperature ($T_{eff}$) of 304 MeV for 0-40$\%$ centrality Pb+Pb collisions at the LHC [@ALICE_T] experiments (as extracted by ALICE). As this temperature comes with $\pm 60$ MeV errorbar, we first ask how this uncertainty affects the calculated suppression. Conseqeuntly, in Fig. \[TDepLB\] we show how the variations (uncertainty in the average QGP temperature) influences the suppression results for different types of flavor (both light and heavy), and at different $p_\perp$ regions. We see that $R_{AA}$ dependence on the average temperature of QGP is almost linear. We also see that the change in the average temperature of the QGP does not significantly affect the suppression, i.e. maximal temperature uncertainly (of 60 MeV), leads to the change in $R_{AA}$ of less than 0.07. Furthermore, we see that dependence of $R_{AA}$ on $T_{eff}$ is almost the same for all parton energies and all types of flavor. We therefore conclude that this uncertainty in the effective temperature would basically lead to a systematic (constant value) shift in the predictions, so the results presented in this paper would not be affected by this uncertainty. Furthermore, extensive comparison [@MD_PLB; @MD_PRL; @DDB_PLB] of our theoretical predictions with experimental data (corresponding to different probes, experiments and centrality regions), shows a robust agreement when the experimentally measured average QGP temperature of $T_{eff}=304$ MeV is used, so we will further use this temperature as a starting point in the prediction calculations. Furthermore, note that these extensive comparisons use the same theoretical framework and the parameter set (corresponding to the standard literature values) as the predictions presented in this paper; consequently, the predictions presented here are well constrained, not only by the absence of the free parameters, but also by the agreement with an extensive set of other data.
In the upper left panel of Fig. \[DataVsTheory\], we provide predictions which agree very well with the ATLAS $h^\pm$ data [@ATLAS_CH] for three different momentum ranges ($7<p_\perp<9$, $20<p_\perp<23$, $65<p_\perp<90$ GeV). The predicted curves reproduce well the tendency observed in the data, i.e. as one moves to higher energy ranges, $R_{AA}$ [*vs.*]{} $N_{part}$ becomes increasingly flatter. Another tendency of $R_{AA}$ [*vs.*]{} $N_{part}$ is also apparent from the predictions, i.e. as one moves towards higher momentum ranges, the difference between the curves becomes increasingly smaller; we will call this the apparent saturation in $R_{AA}$ [*vs.*]{} $N_{part}$ curves.
In the lower left panel of Fig. \[DataVsTheory\], we see that our predictions agree well with the measured [@ATLAS_Jets] $R_{AA}$ [*vs.*]{} $p_\perp$ dependance. Moreover, we see that the “plateau” [@ATLAS_Jets], often referred as surprising, corresponds to the slow increase in the predicted curves, which we further call saturation in $R_{AA}$ [*vs.*]{} $p_{\perp}$ dependence. From the insert in this panel (where we use the logarithmic scale for $p_\perp$ and linear scale for $R_{AA}$), we see that this slow increase in $R_{AA}$ corresponds to the linear dependence on $\ln (p_\perp)$, as can be observed from both the experimental data and the theoretical predictions. Finally, in the upper right and the lower right panels of Fig. \[DataVsTheory\], we see that our predictions can also well reproduce the experimental data [@CMS_JPsi; @CMS_Bjets] for B probes, which indicate qualitatively substantially different pattern compared to $h^\pm$ data. Moreover, the calculated $R_{AA}$ [*vs.*]{} $N_{part}$ and $R_{AA}$ [*vs.*]{} $p_\perp$ are largely flat across the [*entire*]{} span of the momentum ranges, and the apparent saturation in $R_{AA}$ [*vs.*]{} $N_{part}$ curves - which is for light probes observed only at higher $p_\perp$ range - is for B probes predicted for the entire momentum span. As a digression, we here note that, while our predictions are generated for single particles, the corresponding experimental measurements for high $p_\perp$ single particles are not always available. Consequently, we here compare our predictions with both single particles and (in some cases) with jets. While single particles and jets are not equivalent observables, we think that such comparison is not unreasonable because both theoretical predictions and experimental data (when available) indicate an overlap (within errorbars) between single particle and jet data [@DBZ]. This gives us confidence that, when high $p_\perp$ single particle data become available at 5 TeV collision energy, these experimental data will likely largely overlap with the existing jet $R_{AA}$ data.
The predictions shown in Fig. \[DataVsTheory\] contain several features, similar to those indicated by the experimental data (see the Introduction). First, the flattening and the apparent saturation observed in $R_{AA}$ [*vs.*]{} $N_{part}$ curves for $h^\pm$ imply that, at high $p_\perp$, the predictions indicate significantly smaller sensitivity to the collision centrality, and consequently to the corresponding changes in the medium properties; this is in contrast to the lower $p_\perp$, where predictions exhibit a considerable sensitivity to the collision centrality. Furthermore, the saturation observed in $R_{AA}$ [*vs.*]{} $p_\perp$ predictions - consistent with the corresponding plateau in the experimental data - indicates an unexpectedly slow change of jet energy loss with the initial jet energy at high $p_\perp$. Finally, the $R_{AA}$ [*vs.*]{} $N_{part}$ pattern predicted for the B probes is also surprising: here, a qualitatively different pattern compared to the light probes is obtained, where B probes show small sensitivity to the medium properties across the entire momentum span - as opposed to the small $h^\pm$ sensitivity at only high $p_\perp$. As the available data for B probes indicated in the right panels are limited (and indirect, i.e. corresponding to the different observables), note that the calculated B meson results also correspond to novel predictions (expected to become available at the 5 TeV collision energies), whose comparison with the upcoming data will test how our formalism can explain qualitatively unexpected observations.
We therefore aim understanding the nonintuitive patterns in the predictions/data outlined above. For the light probes, this explanation is provided by the upper left panel of Fig. \[LightBottomPatterns\], which shows $R_{AA}$ [*vs.*]{} $p_{\perp}$ dependence for a family of curves corresponding to increasing collision centrality. Note two main properties of these curves: First, their shape, which leads to the saturation in $R_{AA}$ [*vs.*]{} $p_{\perp}$ exhibited in the central panel in Fig. \[DataVsTheory\]; this shape will be explained by the upper central and the right panels in Fig. \[LightBottomPatterns\]. Secondly, their density, which (non-uniformly) increases as one moves from lower to high $p_\perp$ - with that respect, it may be useful to observe $R_{AA}$ [*vs.*]{} $p_\perp$ curves as field flux lines. For visualizing how the relevant curve density changes, three vertical arrows are indicated in the upper left panel of Fig. \[LightBottomPatterns\] - these arrows relate to understanding $h^\pm$ $R_{AA}$ [*vs.*]{} $N_{part}$ predictions in the upper left panel of Fig. \[DataVsTheory\]. Specifically, the leftmost arrow, corresponding to lower ($\sim 10$ GeV) $p_\perp$, spans a much larger $R_{AA}$ range compared to the two right arrows, which correspond to higher $p_\perp$. This observation directly translates to the fact that $R_{AA}$ [*vs.*]{} $N_{part}$ curves are much steeper at lower, compared to high, $p_\perp$ ranges. Moreover, there is a much larger difference in $R_{AA}$ span between the leftmost and the central arrows, as compared to the central and the rightmost arrows; this being despite the fact that the three arrows are spaced equidistantly in momentum. A direct consequence of these differences in $R_{AA}$ span, is the apparent saturation in $R_{AA}$ [*vs.*]{} $N_{part}$ in the upper left panel of Fig. \[DataVsTheory\], i.e. the fact that there is an increasingly smaller difference between $R_{AA}$ [*vs.*]{} $N_{part}$ curves as one moves towards increasingly higher momentum ranges.
The shape of the total $R_{AA}$ [*vs.*]{} $p_\perp$ curves (leading to the saturation in the lower left panel of Fig. \[DataVsTheory\]) is a consequence of an interplay between the collisional and the radiative contributions to the suppression [^1]. As can be seen at the upper central panel of Fig. \[LightBottomPatterns\], the collisional contribution to the $R_{AA}$ is notable for smaller $p_\perp$, where it increases steeply with momentum, rapidly approaching 1 at high $p_\perp$, providing a small contribution to total $R_{AA}$ at high $p_\perp$ region. On the other hand, the radiative contribution decreases much slower with the momentum, and has a significant contribution to $R_{AA}$ even at higher $p_\perp$. Consequently, the steep increase in total $R_{AA}$ at lower $p_\perp$ is driven by the dominant collisional contribution to $R_{AA}$ in that momentum range, while the slow increase (apparent saturation) of $R_{AA}$ at high $p_\perp$ is due to the dominant radiative contribution. This interplay then explains the saturation (plateau) of $R_{AA}$ observed at high $p_\perp$. Moreover, such interplay between the collisional and the radiative contributions to $R_{AA}$, also determines the density of $R_{AA}$ [*vs.*]{} $p_\perp$ curves. As can be seen from the vertical arrows indicated in the upper central and the right panels in Fig. \[LightBottomPatterns\], the collisional contribution is responsible for the large span of total $R_{AA}$ with changing centrality at lower momentum. On the other hand, at higher momentum, the radiative contribution exhibits a significantly smaller and largely uniform $R_{AA}$ span, therefore resulting in the larger and more uniform total $R_{AA}$ curve density in that momentum range.
In the central row of Figure \[LightBottomPatterns\], we see that D mesons show the similar behavior as $h^\pm$. Therefore, for the purpose of analyzing different suppression patterns at the LHC, D mesons can be used as an alternative to $h^\pm$. While D mesons are experimentally harder to measure, from theoretical perspective they have a clear advantage over $h^\pm$. This is because $h^\pm$s are composed of both light quarks and gluons, so that $h^\pm$ presents an indirect probe of light flavor, which is significantly influenced by the fragmentation functions. On the other hand, D meson $R_{AA}$ is a clear probe of bare charm quark $R_{AA}$, i.e. D meson $R_{AA}$ is not influenced by fragmentation functions. Therefore, despite the obvious experimental difficulty in measuring the D meson (compared to $h^\pm$) $R_{AA}$ patterns, D meson has an obvious advantage over $h^\pm$ for analyzing the unintuitive interplays of collisional [*vs.*]{} radiative energy loss and dead-cone effect [@Kharzeev], discussed in this paper.
An intuitive explanation behind the different $R_{AA}$ [*vs.*]{} $N_{part}$ pattern observed for B probes (the right panel in Fig. 1) is provided by the lower panels of Fig. \[LightBottomPatterns\]. In distinction to $h^\pm$, in the lower left panel, we see that total $R_{AA}$ [*vs.*]{} $p_\perp$ curves have mostly uniform density, with a largely flat shape of the curves across the entire momentum range. The difference with respect to $h^\pm$ is clearly due to the radiative contribution to the total $R_{AA}$, as the curves corresponding to the B meson collisional contribution are largely equivalent to those for $h^\pm$ (compare the upper and lower central panels in Fig.\[LightBottomPatterns\]). In particular, note an unusual shape of radiative $R_{AA}$ [*vs.*]{} $p_\perp$ curves (the lower right panel), which is a consequence of a strong dead-cone effect [@Kharzeev] in bottom quark energy loss. This unusual shape leads to a large density of radiative $R_{AA}$ [*vs.*]{} $p_\perp$ curves at lower momentum, and to a largely uniform curve density for high momentum. As a consequence, for lower momentum, the effect of the relatively large $R_{AA}$ span for the collisional contribution is abolished by the small $R_{AA}$ span for the radiative contribution, this leading to the flat and uniform density for total $R_{AA}$ [*vs.*]{} $p_\perp$ curves observed in the lower left panel of Fig.\[LightBottomPatterns\]. Such curve shape then evidently leads to a largely uniform total $R_{AA}$ span across the whole range of momentum, as indicated by the three vertical arrows in the lower left panel of Fig.\[LightBottomPatterns\], which then leads to our predictions of the largely flat $R_{AA}$ [*vs.*]{} $p_\perp$ and almost overlapping $R_{AA}$ [*vs.*]{} $N_{part}$ curves for B mesons. Comparison of theoretical predictions with experimental data shown in the lower (left and right) panels of Fig. \[DataVsTheory\] show an indication that these predictions might be in accordance with experimental data. However, the data shown in Fig. \[DataVsTheory\] are indirect, very limited and correspond to different bottom observables, so more detailed experimental data at 5 TeV Pb+Pb collisions at the LHC are needed to confirm (or dispute) the predictions presented in this study.
Conclusion
==========
A starting point for this work is an observation of a plateau reached at high momentum for $R_{AA}$ [*vs.*]{} $p_\perp$ measurements. Starting from this observation, we here combined related experimental data, which reveal an unexpected pattern in the $R_{AA}$ data. We showed that these data patterns are well reproduced by the theoretical predictions for charged hadrons and unidentified jets, with no free parameters used. For B mesons, we predict that the tendency indicated by the limited available data will be exhibited across the entire span of the momentum ranges - this prediction will be tested by the upcoming experimental data expected from the 5 TeV Pb+Pb collisions at the LHC.
We showed that these complex data patterns have, in fact, a simple qualitative interpretation, where it is useful to observe $R_{AA}$ [*vs.*]{} $p_\perp$ curves as field flux lines whose density changes across different momentum. These curve properties - which lead to the unexpected dependence of $R_{AA}$ on $N_{part}$ and $p_\perp$, and to qualitatively different $R_{AA}$ patterns for the light and heavy (i.e. bottom) probes - are determined by an interplay of collisional, radiative energy loss and the dead-cone effect.
Consequently, the results presented here provide a rare opportunity to qualitatively assess how the theory can account for two crucial effects: First, different suppression patterns exhibited by different probe types (here B mesons vs. charged hadrons or D mesons), providing a clear test of the dead cone effect. Second, contributions of different energy loss mechanisms, providing a test of an interplay between the collisional and the radiative energy loss. This point is even more important having in mind extensive experimental efforts aimed at assessing contributions of various energy loss effects. Consequently, this study provides both an important test of explaining and predicting complex data patterns and a clear qualitative example for distinguishing between major energy loss mechanisms.
[*Acknowledgments:*]{} This work is supported by Marie Curie IRG within the $7^{th}$ EC Framework Programme (PIRG08-GA-2010-276913) and by the Ministry of Science of the Republic of Serbia under project numbers ON173052 and ON171004. I thank B. Blagojevic for help with numerics and Marko Djordjevic for useful discussions.
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[^1]: The total $R_{AA}$ is approximately (though not exactly) equal to the product of the radiative and collisional contributions
|
---
author:
- 'V. Ossenkopf, C. Trojan'
- 'J. Stutzki'
date: 'Received: 15 November 2000; accepted: 18 June 2001'
title: 'Massive core parameters from spatially unresolved multi-line observations'
---
Introduction
============
Whereas the average density in molecular clouds falls between about 50 and 1000 cm$^{-3}$, observations of high-dipole-moment molecules like CS or NH$_3$ reveal cores with densities up to $10^6$ cm$^{-3}$. Massive cores are typically somewhat warmer than their embedding molecular cloud and show sizes of about $1- 3$ pc and masses between some ten and some thousand solar masses. They appear as relatively bright objects in molecular line maps so that they are favoured objects from the viewpoint of the observations. The observed line widths of $2-15$ km/s are considerably larger than thermal, indicating turbulent motions, possibly induced by outflows from star-forming activity. Many of them are associated with young OB or T Tauri stars suggesting that massive cores are sites of massive and multiple star formation (c.f. [@Myers]). For a better understanding of star formation we have to know the physical parameters within these cores, i.e. the geometrical structure, the density, velocity, and temperature distribution.
Dense cores are best traced by molecules like CS, HC$_3$N, NH$_3$, and H$_2$CO characterised by high critical densities even for low transitions. Throughout this paper we will use the term “core” for the dense inner part of a cloud which is directly visible in CS, whereas the rest of the cloud may contribute to the radiative excitation but is mostly visible in CO rather than in CS. We call smaller substructures within the cores “clumps”. Meta-stable transitions of NH$_3$ provide reasonable estimates for the kinetic core temperature and the simultaneous observation of several lines from the same isotope provide combined information both on the density and the temperature structure. Here, we use observations of the 2–1, 5–4, and 7–6 transitions of CS and C$^{34}$S obtained with the FCRAO 14m and the KOSMA 3m telescopes to study 15 massive cores and complement our observations with CS data from the literature to derive the physical core parameters.
[@Plume] observed about 150 massive cores in CS and C$^{34}$S using the IRAM 30m telescope. With an escape probability approximation they derive relatively similar physical parameters for most cores. We compare their results to the parameters obtained for the cores from our sample and test to which extent these results reflect the constraints provided by the set of lines observed, the limitations of the data analysis or the real physical properties of the cores. To study the influence of the data analysis we compare the escape probability approximation with a self-consistent radiative transfer code computing the excitation conditions in an inhomogeneous core with internal turbulence.
In Sect. 2 we provide a short overview on the sample and the observations. Sect. 3 discusses the traditional way to derive the cloud parameters from the observations. Using the mathematical description of the radiative transfer problem in Appx. A we compute cloud parameters in an escape probability approximation. In Sect. 4 the fully self-consistent radiative transfer code from Appx. B is used to derive the cloud parameters. Sect. 5 compares the resulting data with parameters obtained from independent observations and discusses implications for the physics of massive cores.
Observations
============
The sample
----------
The selection of the sample of massive cores was determined by the need of bright “standard” sources for the SWAS satellite which has a spatial resolution of about 4 arcmin in the frequency range between 490 and 560 GHz. For a comparison to data obtained at similar angular resolution we used observations taken with the array receiver of the FCRAO 14m telescope at 98 GHz (about 1 arcmin resolution) and performed complementary observations with the 3m KOSMA telescope at 245 and 343 GHz where its beam size is approximately 2 arcmin.
The sources were selected from the SWAS source list (Goldsmith et al., [*priv. comm.*]{}) to be observable from the FCRAO and KOSMA and bright enough to be detectable in a reasonable integration time. Tab. \[tab\_sources\] lists the 15 sources selected with their central position.
-------------- -------------------------------- --------------------------------
W49A ${19}^{h}:{07}^{m}:{50.0}^{s}$ ${09}^{\circ}:{01'}:{32.0''}$
W33 ${18}^{h}:{11}^{m}:{19.8}^{s}$ ${-17}^{\circ}:{56'}:{21.0''}$
W51A ${19}^{h}:{21}^{m}:{25.4}^{s}$ ${14}^{\circ}:{24'}:{45.0''}$
W3(OH) ${2}^{h}:{23}^{m}:{15.4}^{s}$ ${61}^{\circ}:{38'}:{53.0''}$
W3 ${2}^{h}:{21}^{m}:{42.5}^{s}$ ${61}^{\circ}:{52'}:{22.0''}$
S255 ${6}^{h}:{09}^{m}:{58.4}^{s}$ ${18}^{\circ}:{00'}:{19.0''}$
S235B ${5}^{h}:{37}^{m}:{31.8}^{s}$ ${35}^{\circ}:{40'}:{18.0''}$
S106 ${20}^{h}:{25}^{m}:{39.8}^{s}$ ${37}^{\circ}:{12'}:{46.0''}$
Serpens ${18}^{h}:{27}^{m}:{25.0}^{s}$ ${01}^{\circ}:{12'}:{00.0''}$
DR21 ${20}^{h}:{37}^{m}:{14.0}^{s}$ ${42}^{\circ}:{09'}:{00.0''}$
Mon R2 ${6}^{h}:{05}^{m}:{14.0}^{s}$ ${-6}^{\circ}:{23'}:{00.0''}$
NGC2264 ${6}^{h}:{38}^{m}:{26.0}^{s}$ ${9}^{\circ}:{32'}:{00.0''}$
OMC-2 ${5}^{h}:{32}^{m}:{57.0}^{s}$ ${-5}^{\circ}:{12'}:{12.0''}$
$\rho$ Oph A ${16}^{h}:{23}^{m}:{17.9}^{s}$ ${-24}^{\circ}:{17'}:{18.0''}$
NGC2024 ${5}^{h}:{39}^{m}:{12.0}^{s}$ ${-1}^{\circ}:{56'}:{40.0''}$
-------------- -------------------------------- --------------------------------
: Selected source sample with central positions
\[tab\_sources\]
NGC2024 is not part of the SWAS sample but we have observed this region as a standard for comparison: it is relatively close ($\approx 450$ pc) and has been studied already by numerous authors using various techniques. An extended map of the cloud and its environment in CS 2–1 was provided by [@Lada91], and line profiles in four transitions of the main CS isotope are given by Lada et al. (1997). [@Mezger] have identified seven clumps in NGC2024 from dust observations using the IRAM 30m telescope whereas the FCRAO and KOSMA beams can only distinguish between the two bright clumps FIR3 and FIR5. The data analysis is performed for the position of the brightest clump FIR5. Here, the KOSMA beam also contains weak contributions from FIR4, FIR6, and FIR7.
-------------- ------- -------- ----------------- ------ ------- ----------------- -------- ------- ------------------
W49A$^{(a)}$ 3.77 35.17 8.75 $\pm$ 0.55 1.54 12.61 7.69 $\pm$ 0.19 1.23 12.15 9.28 $\pm$ 1.24
W49A$^{(b)}$ 4.34 32.76 7.10 $\pm$ 0.38 2.04 18.70 8.62 $\pm$ 0.15 1.06 9.52 8.43 $\pm$ 1.03
W33 9.74 67.24 6.49 $\pm$ 0.16 3.76 27.40 6.84 $\pm$ 0.05 2.94 17.44 5.57 $\pm$ 0.05
W51A 10.86 110.51 9.56 $\pm$ 0.09 4.99 67.89 12.78$\pm$ 0.09 3.38 40.63 11.28 $\pm$ 0.08
W3(OH) 5.50 26.48 4.53 $\pm$ 0.20 2.13 10.13 4.46 $\pm$ 0.08 1.25 5.81 4.40 $\pm$ 0.31
W3 7.67 38.97 4.77 $\pm$ 0.05 1.35 7.72 5.36 $\pm$ 0.28 1.56 8.44 5.08 $\pm$ 0.09
S255 8.02 21.55 2.53 $\pm$ 0.04 2.91 10.13 3.28 $\pm$ 0.03 1.67 5.29 2.99 $\pm$ 0.15
S235B 6.38 16.21 2.39 $\pm$ 0.05 0.85 2.67 2.97 $\pm$ 0.13 0.42 1.10 2.53 $\pm$ 0.19
S106 3.29 8.19 2.34 $\pm$ 0.10 0.83 2.11 2.40 $\pm$ 0.22 0.65 2.23 3.23 $\pm$ 0.78
Serpens 3.74 9.36 2.35 $\pm$ 0.13 0.67 2.52 3.59 $\pm$ 0.35 0.39 1.68 4.11 $\pm$ 0.65
DR21 6.48 25.00 3.62 $\pm$ 0.10 2.41 9.74 3.80 $\pm$ 0.18 2.92 10.92 3.51 $\pm$ 0.06
Mon R2 5.53 12.26 1.90 $\pm$ 0.16 2.15 6.52 2.38 $\pm$ 0.27 1.70 3.83 1.95 $\pm$ 0.19
NGC2264 5.64 21.90 3.66 $\pm$ 0.08 2.30 9.48 3.89 $\pm$ 0.08 1.25 5.42 3.91 $\pm$ 0.12
OMC-2 5.62 9.26 1.55 $\pm$ 0.05 1.96 5.93 2.85 $\pm$ 0.10 0.92 1.54 1.58 $\pm$ 0.25
$\rho$ Oph A 4.22 11.10 2.47 $\pm$ 0.25 2.07 2.30 1.04 $\pm$ 0.08 $<$0.2 - -
-------------- ------- -------- ----------------- ------ ------- ----------------- -------- ------- ------------------
\[tab\_lineparams\]
Observational details
---------------------
All sources except NGC2024 were observed in CS 2–1 by Howe ([*priv. comm.*]{}) using the FCRAO 14m telescope providing a resolution of 53$''$. and a main beam efficiency $\eta{_{\rm mb}}^{2-1}=0.58$. The cores were covered by 30-point maps with a sampling of 50$''$. The CS 2–1 spectra for the southern core in NGC2024 were taken from Lada et al. (1997).
The CS 5–4 and 7–6 observations used the dual channel KOSMA SIS receiver with noise temperatures of about 95 K in the 230 GHz branch and 120 K in the 345 GHz branch. The 3m telescope provides a spatial resolution of 110$''$ in CS 5–4 and 80$''$ in CS 7–6. The default observing mode for all cores were cross scans with a separation of 50$''$ between subsequent points. Only for NGC2024 complete 5$'\times$5$'$ maps were obtained. At the time of the observations the telescope surface provided main beam efficiencies $\eta{_{\rm mb}}^{5-4}=0.54$ and $\eta{_{\rm mb}}^{7-6}=0.48$ respectively for the two transitions. For all measurements we use the conservative estimate of about 10% uncertainty for the main beam efficiency, another 10% atmospheric calibration uncertainty and add another 5% for possible drifts etc. As systematic errors they might sum up linearly to a total calibration error of at most 25%.
The FCRAO spectrometer had a channel width of 19.5 kHz corresponding to a velocity spacing of 0.060 km/s. For the broad lines from the sample four velocity channels were binned. The resulting r.m.s. falls between 0.2 and 0.45 K. The KOSMA spectra were taken with the medium resolution spectrometer (MRS) and the low resolution spectrometer (LRS) providing channel widths of 167 kHz and 688 kHz, respectively. Depending on the different combinations of these backends with the receivers at 245 and 343 GHz we obtain velocity spacings between 0.15 and 0.84 km/s. The particular spacing is not important for the analysis performed here because none of the lines shows strong spectral substructure. All points were integrated up to a noise limit of 0.1 K per channel.
Results
-------
For most sources the CS 2–1 maps show an approximately elliptical intensity peak with a weak elongation at scales of a few times the resolution. In W49A, Serpens, DR21, Mon R2, OMC-2, $\rho$ Oph A, and NGC2024 we can distinguish a second intensity maximum apart from the central position. Tab. \[tab\_lineparams\] summarises the parameters of the line profiles at the central position for all cores. The majority of line profiles are approximately Gaussian as indicated by integrated line intensities close to the Gaussian value of $1.06\,T{_{\rm mb}}\Delta v$ in Tab. \[tab\_lineparams\]. Broad wings are only visible in S255, W33, and DR21.
W49 shows a double-peak structure which has been interpreted e.g. by [@Dickel] as the footprint of large scale collapse. They fitted HCO$^{+}$ line profiles by a spherical collapse model but concluded that additional components are needed to explain the observations. Using the radiative transfer code from Appx. B we have tested their infall model and found that, while reproducing the HCO$^{+}$ profiles, it completely fails to explain the CS observations. The enhanced blue emission characteristic for collapse is visible only in CS 7–6. In CS 2–1 and 5–4 we rather find an enhanced red emission. No spherically symmetric collapse model can explain these observations. Instead of constructing a more complex model we have simply decomposed the emission into two separate components with a relative velocity of 8.5 km/s in the line of sight denoted as W49A$^{\rm (a)}$ and W49A$^{\rm (b)}$. From the modelling in Sect. 4 it turns out that we cannot even distinguish whether the two components are moving towards each other or apart as long as they do not line up exactly along the line of sight. Hence, we will treat them separately in the following, ignoring any possible interaction. Further observations including other tracers should be included to better resolve the situation.
--------------- ------ ------- -----------------
C$^{32}$S 5–4 5.19 11.65 2.10 $\pm$ 0.02
C$^{32}$S 7–6 3.63 9.13 2.38 $\pm$ 0.08
C$^{34}$S 5–4 1.19 2.22 1.75 $\pm$ 0.09
C$^{34}$S 7–6 0.65 0.98 1.45 $\pm$ 0.13
--------------- ------ ------- -----------------
: Measured line parameters within NGC2024
\[tab\_ngc2024params\]
In NGC2024 we also mapped the less abundant isotope C$^{34}$S with KOSMA in addition to the main CS isotope observed in all cores. The observed line parameters are given in Tab. \[tab\_ngc2024params\]. Lada et al. (1997) provided detailed CS spectra for the southern core at the position of the FIR5. They obtained in the CS 2–1 transition $T{_{\rm mb}}=15.4$ K, $\Delta v=1.80$ km/s at 24$''$ resolution, in CS 5–4 $T{_{\rm mb}}=9.6$ K, $\Delta v=2.1$ km/s at 30$''$ resolution, in CS 7–6 $T{_{\rm mb}}=10.1$ K, $\Delta v=2.7$ km/s at 20$''$ resolution, and in CS 10–9 $T{_{\rm mb}}=10.6$ K, $\Delta v=2.1$ km/s at 14$''$ resolution.
Cloud parameters from the escape probability model {#sect_ep}
==================================================
To interpret the small amount of information contained in observations of at most five transitions showing mainly Gaussian profiles and essentially unresolved approximately circular symmetric intensity distributions, we need a simple cloud model that is both physically reasonable and characterised by few parameters. An obvious choice is a spherically symmetric model. This geometry reflects early phases and the large scale behaviour of several collapse simulations (e.g. [@Galli99]), whereas the inner parts of collapsing clouds are probably flattened structures (e.g. [@Li]).
Even in spherical geometry there is no simple way to solve the radiative transfer problem relating the cloud parameters to the emitted line intensities (see Appx. A.1). Thus we cannot compute the cloud properties directly from the observations.
Application of the escape probability approximation
---------------------------------------------------
A common approach is the escape probability approximation discussed in detail in Appx. A.2. Assuming that all cloud parameters, including the excitation temperatures, are constant within a spherical cloud volume one can derive a simple formalism relating the three parameters kinetic temperature $T{_{\rm kin}}$, gas density $n{_{\rm H_2}}$, and column density of radiating molecules on the scale of the global velocity variation $N{_{\rm mol}}/\Delta v$ to the line intensity at the cloud model surface. No assumption on molecular abundances is required.
Since a telescope does not provide a simple pencil beam we have to correct the model surface brightness temperature by the beam filling factor $\eta{_{\rm f}}$, given as the convolution integral of the normalised intensity distribution with the telescope beam pattern, to compute the observable beam temperature. Unfortunately, the brightness profile of the source is a non-analytic function where we can only give simple expressions for the central value observed in a beam much smaller than the source or for the integral value observed in a beam much larger than the source (Eqs. \[eq\_tmb\_lvg\] and \[eq\_tmb\_esc\]). For intermediate situations we approximate the beam temperature by starting from both limits and using a beam filling factor given by the convolution integral of two Gaussians. The difference between the two values provides an estimate of the error made in the beam convolution.
-------------- ----- ----- ----- ----- ----- -----
W49A 1.0 1.1 0.3 1.5 1.7 1.3
W33 1.7 1.7 1.4 1.7 1.3 1.0
W51A 1.9 1.7 1.7 2.1 2.2 1.1
W3(OH) 2.1 1.1 1.3 1.7 1.4 1.4
W3 1.0 0.6 0.8 1.0 0.6 0.3
S255 1.9 1.6 1.1 1.2 0.6 1.1
S235B 1.0 2.2 1.1 0.9 0.2 0.3
S106 3.0 1.6 2.5 2.2
Serpens 2.0 2.2 2.1 2.3
DR21 3.0 2.6 1.8 1.5 1.7 1.8
Mon R2 3.2 3.0 1.1 3.6 3.2 1.6
NGC2264 3.2 1.7 0.6 2.9 2.1
OMC-2 2.4 1.4 0.6 2.5 1.4 0.6
$\rho$ Oph A 1.3 3.0 2.6 1.9 2.9 2.5
NGC2024 1.1 1.1 1.3 1.3 2.5 2.3
-------------- ----- ----- ----- ----- ----- -----
: Source size corrected for beam convolution.
\[tab\_coresize\]
To compute the integral we fitted the observed brightness distributions by Gaussians. Most cores are well approximated by slightly elongated Gaussians. W49A, S235B, Serpens, and $\rho$ Oph A show asymmetric scans so that the size determination is somewhat uncertain. [[ ]{}The fit error is about 0.3$'$ for these three sources. For the rest of the cores we obtain typical values of less than 0.2$'$.]{} The true object size finally follows from the deconvolution of the measured intensity distribution with the telescope beam. The resulting source sizes in $\alpha$ and $\delta$ are given in Tab. \[tab\_coresize\]. As the geometric mean is sufficient to compute the beam filling factor we don’t expect any serious error from the fact that the cross-scans in $\alpha$ and $\delta$ do not necessarily trace the major axes of the brightness distribution. For sources which are considerably smaller than the beam widths of 53$''$, 107$''$, and 80$''$, respectively, only a rough size estimate is possible according to the nonlinearity of the deconvolution. This holds for W49A, W3, S235B, and partially S255. Most clouds, however, show an extent of the emission which is close to the beam size.
In general different values are obtained for the spatial FWHMs in the different lines. In Tab. \[tab\_coresize\] we find two classes of sources with respect to the variation of the source size depending on the transition observed. Most cores show a monotonic decrease of the visible size when going to higher transitions. This is expected from the picture that higher transitions are only excited in denser and smaller regions. Serpens, $\rho$ Oph A, and NGC2024, however show the smallest width of the fit in the CS 2–1 transition. This is explained by eye inspecting the 2–1 maps and corresponding high-resolution observations from the literature where we see that the three sources break up into several clumps which are only separated in the 53$''$ beam but unresolved in the KOSMA beams. In these cases, we have restricted the analysis to the major core seen in the CS 2–1 maps using its size to compute the beam filling, although this approach introduces a small error in the data analysis by assigning the whole flux measured in the higher transitions to this central core.
Applying the two limits for the beam size treatment (Eqs. \[eq\_tmb\_lvg\] and \[eq\_tmb\_ep\]) using the sizes from Tab. \[tab\_coresize\] we find that the resulting gas and column densities are the same within 20% except for W49A, W3, S235B, and NGC2264. The first three are small compared to the beams so that the results from Eq. (\[eq\_tmb\_lvg\]) have to be rejected and only Eq. (\[eq\_tmb\_ep\]) can be used. For NGC2264 we cannot provide a simple explanation for the difference so that we give a relatively large error bar covering the results from both approximations.
The size of the resulting parameter range in $T{_{\rm kin}}$, $n{_{\rm H_2}}$, and $N{_{\rm CS}}/\Delta v$ is determined by the accuracy of the observations. For two cores it was only possible to set a lower limit to the gas density. Moreover, we were not able to provide any good constraint to the cloud temperature for all sources. Values between about 30 K and 150 K are possible. Hence, an independent determination of the cloud temperatures is required. Several different methods based on optically thick CO, NH$_3$ or dust observations are discussed in the literature and we used the values from the references given in Tab. \[tab\_epparm\]. In addition to these values we also used 50 K as assumed by [@Plume] as “standard” temperature in the parameter determination for massive cores.
Resulting core parameters
-------------------------
----------------- --------------------- ------------------- ----- ----------------- -----
W49A$^{(a)}$ 20$^{\mathrm{a}}$ $\!\!>$ 7.310$^6$ 2.210$^{14}$ 1.3
50$^{\mathrm{a}}$ 1.310$^6$ 1.5 2.110$^{14}$ 1.2
W49A$^{(b)}$ 20$^{\mathrm{a}}$ 4.510$^6$ 1.6 2.810$^{14}$ 1.3
50$^{\mathrm{a}}$ 7.610$^5$ 1.5 2.710$^{14}$ 1.2
W33 40$^{\mathrm{b}}$ 1.710$^6$ 1.7 2.310$^{14}$ 1.2
50$^{\mathrm{c}}$ 1.210$^6$ 1.6 2.310$^{14}$ 1.2
W51A 20$^{\mathrm{a}}$ 2.110$^7$ 1.6 4.810$^{14}$ 1.7
50 1.110$^6$ 1.5 4.110$^{14}$ 1.4
57$^{\mathrm{a}}$ 8.910$^5$ 1.4 3.710$^{14}$ 1.4
W3(OH) 30$^{\mathrm{d}}$ 1.910$^6$ 1.5 1.310$^{14}$ 1.2
50 8.510$^5$ 1.4 1.310$^{14}$ 1.2
W3 30$^{\mathrm{e}}$ 7.110$^6$ 2.9 7.410$^{14}$ 1.5
50 9.810$^5$ 1.8 8.110$^{14}$ 1.4
55$^{\mathrm{f}}$ 7.410$^5$ 1.6 8.310$^{14}$ 1.3
S255 40$^{\mathrm{g}}$ 1.310$^6$ 1.6 1.310$^{14}$ 1.3
50 9.310$^5$ 1.5 1.210$^{14}$ 1.3
S235B 40$^{\mathrm{h}}$ 1.210$^6$ 1.5 3.710$^{13}$ 1.3
50 8.910$^5$ 1.5 3.610$^{13}$ 1.3
S106 10$^{\mathrm{i}}$ $\!\!>$6.310$^6$ 4.710$^{13}$ 1.4
25$^{\mathrm{i}}$ 7.110$^5$ 1.5 3.810$^{13}$ 1.4
50 2.510$^5$ 1.6 3.910$^{13}$ 1.4
Serpens 25$^{\mathrm{j}}$ 4.610$^5$ 1.2 2.810$^{13}$ 1.5
50 1.910$^5$ 1.6 2.810$^{13}$ 1.5
DR21 35$^{\mathrm{k}}$ 1.510$^6$ 1.7 9.310$^{13}$ 1.9
50 9.110$^5$ 1.6 8.710$^{13}$ 1.4
Mon R2 25$^{\mathrm{l}}$ 1.710$^6$ 1.8 4.510$^{13}$ 1.5
50$^{\mathrm{m}}$ 6.210$^5$ 1.7 3.710$^{13}$ 1.7
NGC2264 25$^{\mathrm{n}}$ 1.410$^6$ 2.4 7.910$^{13}$ 2.6
50 4.710$^5$ 2.3 7.810$^{13}$ 2.5
OMC-2 19$^{\mathrm{q}}$ 1.910$^6$ 1.6 6.310$^{13}$ 1.4
24$^{\mathrm{o,p}}$ 1.110$^6$ 1.5 6.210$^{13}$ 1.4
50 3.410$^5$ 1.7 6.210$^{13}$ 1.4
$\rho$ Oph A 25$^{\mathrm{r}}$ 9.510$^5$ 1.6 2.010$^{13}$ 1.4
50 3.710$^5$ 1.8 2.010$^{13}$ 1.5
NGC2024 25$^{\mathrm{s}}$ 9.110$^6$ 2.9 $>$4.510$^{14}$
40$^{\mathrm{t}}$ 2.110$^6$ 2.0 $>$3.510$^{14}$
50 1.610$^6$ 1.7 $>$3.510$^{14}$
\[tab\_epparm\]
----------------- --------------------- ------------------- ----- ----------------- -----
: Clump parameters derived from the escape probability model
\
$^{\mathrm{a}}$ [@Sievers] , $^{\mathrm{b}}$ [@Goldsmith],\
$^{\mathrm{c}}$ [@Haschick], $^{\mathrm{d}}$ [@Wilson],\
$^{\mathrm{e}}$ [@Tieftrunk98], $^{\mathrm{f}}$ [@Tieftrunk95],\
$^{\mathrm{g}}$ [@Jaffe], $^{\mathrm{h}}$ [@Nakano],\
$^{\mathrm{i}}$ [@Roberts97], $^{\mathrm{j}}$ [@McMullin],\
$^{\mathrm{k}}$ [@Garden], $^{\mathrm{l}}$ [@Montalban],\
$^{\mathrm{m}}$ [@Giannakopoulou], $^{\mathrm{n}}$ [@Kruegel],\
$^{\mathrm{o}}$ [@Castets], $^{\mathrm{p}}$ [@Batrla],\
$^{\mathrm{q}}$ [@Cesaroni], $^{\mathrm{r}}$ [@Liseau],\
$^{\mathrm{s}}$ [@Ho], $^{\mathrm{t}}$ [@Mezger]
Tab. \[tab\_epparm\] lists the parameters from the escape probability model for all cores. Whereas the column density is well constrained for most clouds, there is a considerable uncertainty in the gas density resulting from the unknown cloud temperature. At the temperature of 50 K we obtain average values and logarithmic standard deviation factors of $$\begin{aligned}
\langle n{_{\rm H_{2}}}\rangle =& 7.9\,10^5 \qquad &\times\!/\!\div 1.5 \\
\langle N{_{\rm CS}}/\Delta v\rangle =& 1.2\,10^{14} \qquad &\times\!/\!\div 2.8\nonumber\end{aligned}$$
From the 71 cores analysed by [@Plume] assuming this temperature they obtained $$\begin{aligned}
\langle n{_{\rm H_{2}}}\rangle =& 8.5\,10^5 \qquad &\times\!/\!\div 1.7 \\
\langle N{_{\rm CS}}/\Delta v\rangle =& 2.5\,10^{14} \qquad &\times\!/\!\div 3.1\nonumber\end{aligned}$$
This good agreement indicates first that both investigations study the same type of clouds visible in the CS transitions. Second, this shows that the static escape probability (Eq. \[eq\_tmb\_statep\]) used here and the LVG escape probability (Eq. \[eq\_tmb\_lvg\]) applied by [@Plume] differ only marginally as discussed already by [@Stutzki85]. Third, the observational data from both telescopes give almost equivalent results, i.e. the reliability of the parameters hardly profits from using the 10 times smaller beam of the IRAM telescope when the escape probability approximation is used.
For NGC2024 we are able to test the consistency of the results from the CS and the C$^{34}$S observations. The resulting hydrogen densities agree for both isotopes within 20% at all temperatures assumed. We obtain 910$^6$ cm$^{-3}$ at 25 K (Mezger et al. 1992) and 3.210$^6$ cm$^{-3}$ at 40 K (Ho et al. 1993). Unfortunately, this is a core where we can only give a lower limit to the column densities. The limits deviate by a factor 13, which is significantly different from the terrestrial isotopic ratio of 23 but close to the value 10 derived by [@Mundy] for the isotopic ratio in NGC2024.
The escape probability model provides a first estimate to the physical parameters but its limitations are obvious. It is definitely not justified to assume constant parameters within the whole cloud. Moreover, several observations are in contradiction to the parameters from the escape probability models. [@Lada97] detected the CS 10–9 transition in NGC2024 and [@Plume] observed the 10–9 and 14–13 transitions in S255 and W3(OH). The critical densities for these transitions are about 610$^7$ cm$^{-3}$ and 210$^8$ cm$^{-3}$ respectively. From the densities in Tab. \[tab\_epparm\] one would conclude that these transitions are not excited. Hence, a more sophisticated model has to be applied to obtain a physically reasonable explanation of the measurements. [@Plume] suggested a two-component model or continuous density gradients to resolve this contradiction. We will discuss a self-consistent radiative transfer model including a radial density profile in the following. \[sect\_epresults\]
Cloud parameters from the nonlocal model {#sect_nonlocal}
========================================
Line fitting by [*SimLine*]{}
-----------------------------
We performed nonlocal radiative transfer simulations using the line radiative transfer code [*SimLine*]{} introduced in detail in Appx. B. [*SimLine*]{} is a FORTRAN code to compute the profiles of molecular rotational lines in spherically symmetric clouds with arbitrary density, temperature and velocity distribution. It consists of two parts: the self-consistent solution of the balance equations for all level populations and energy densities at all radial points and the computation of the emergent line profiles observed by a telescope with finite beam width and arbitrary offset. The optical depths in the lines may vary from minus a few, corresponding to weak masing, to several thousand.
Already in the spherically symmetric description of a core we face a large number of parameters. For all quantities (hydrogen density, kinetic temperature, velocity dispersion, molecular abundances) a radial function has to be found. Regarding the limited amount of information available from the three transitions, this leaves many options open. We decided to assume simple power-law radial functions and a central region with constant parameters for all quantities in the core simulations. This reduces the number of parameters to two (central value and radial exponent) for each gas property, plus the outer and inner radius.
The parameter fit procedure used the multidimensional downhill simplex algorithm from Press et al. (1992). Although it is not the most efficient way in terms of convergence speed it turned out to be very robust in all situations considered. Because a downhill simplex code does not necessarily find the global minimum of the $\chi^2$ function we performed for each core several runs with randomly chosen initial simplex covering a large part of the physically reasonable parameter space. For all clouds we made at least 30 runs to get a rough idea of the topology of the $\chi^2$ function. For cores like W33 this turned out to be sufficient since only one large minimum showed up which was found in almost half of the runs. The other extreme is S106 where we needed almost 1000 runs to be sure that we found the global minimum. Here, the $\chi^2$ function was quite complex with numerous local minima. Future improvements of the fit procedure should include more sophisticated algorithms like simulated-annealing approaches.
The noise in the line profiles produces some graininess of the $\chi^2$ function when directly fitting the measured profiles. This results in a very slow convergence of the algorithm close to the minimum. A considerable acceleration can be obtained by not fitting the measured noisy data but a smooth approximation to them. The line profiles were represented by a superposition of a Gaussian and a Lorentzian profile. This allows a good characterisation of all measured profiles including the reproduction of asymmetric profiles, self-absorption dips and line wings.
Taking the three measured transitions and their spatial extent as the quantities to be reproduced by a $\chi^2$ fit we find that we are able to derive at most six parameters to a reasonable accuracy. Fits with seven or eight free parameters, although still slightly improving the numerical $\chi^2$ value, do not produce any significant changes above the noise limit. This is comparable to the results by [@Young] fitting the full position velocity map of a particular core in a single transition. There is however a number of additional parameters where the model can derive certain limits. Hence, we have to decide first which parameters should be fitted directly, which parameters may be constrained by preventing reasonable fits when outside a certain range, and which parameters can be guessed independently from a physical line of reasoning. This is discussed in detail for all quantities in the following subsections.
Unfortunately, it is impossible to give an comprehensive error estimate for the parameters derived from the simulations. This would need a description of the six-dimensional surface in the parameter space within which none of the observational error bars is exceeded. According to the complex topology of the $\chi^2$ function it is not possible to give an easy description for the six-dimensional valley around the global minimum or its boundaries.
As a simple alternative we performed only one-dimensional variations to get a rough estimate of the maximum error in the parameters that we must expect. After the convergence of the $\chi^2$ fit, each of the fit parameters was varied up and down until one of the computed lines deviated by the assumed maximum observational error of 25% from the measured lines. The central values of the different functions were varied independently. When changing the inner radius, the central values of the gas parameters were adjusted to keep the functions in the power-law region unchanged. When varying exponents the corresponding central values were corrected in such a way that the parameters at the density of 210$^{6}$cm$^{-3}$ remained constant. By the selection of this density as the fix point when changing the slope, we scan about the maximum possible range of the exponents. This provides a conservative estimate of the maximum error.
The turbulence description
--------------------------
In spite of the large number of free parameters in the cloud models it is impossible to fit the line profiles with a smooth density and velocity distribution. The typical self-absorbed profiles known for all microturbulent codes ([@White]) appear in this case. Thus we have to take into account the effects of internal clumping and turbulence leading to a more realistic picture and preventing strong self-absorption.
[*SimLine*]{} treats turbulence and clumping in a local statistical approximation following [@MSH] (see Appx. B.2). The cloud material is subdivided into small coherent units (clumps) with only thermal internal velocity dispersion. The relative motion of many units then provides the full velocity profile. [@MSH] showed that the effective optical depth of such an ensemble can be considerably reduced compared to the microturbulent approximation. The relative reduction of the total optical depth depends on the optical depth, i.e. of size the coherent units.
This description does not need any assumptions on the nature of the turbulence creating the internal cloud structure. The reduction occurs in the same way whether clumps are units of the same velocity in a medium of constant density, representing the behaviour of incompressible turbulence, or whether they are density enhancements in a thin inter-clump medium. The different nature of these two scenarios has to be taken into account, however, when computing the excitation. Both the average gas density, providing a measure for the column density and thus the line intensity, and the local density, providing the collisional coupling to the gas, enter the balance equations. In case of coherent units in velocity space both densities agree.
For density clumps we use the additional simplification to treat the cloud locally as a two-component medium neglecting the contribution of the inter-clump medium to the radiative transfer. Then the collisional excitation is provided by the density in the clumps and the column density is provided by the average density. The ratio between the two quantities reflects the filling factor of the volume occupied by dense clumps. From the mathematical point of view this is equivalent to the treatment of the cloud as a homogeneous medium where the clumping occurs only in velocity space and the abundance of the radiating molecules is reduced. Hence, it is impossible to separate the influence of the filling factor from that of the molecular abundance so that the line fit provides only a combined quantity which we will denote combined abundance in the following. Beside the modification of the molecular abundance the statistical turbulence description introduces the size of the coherent units as an additional parameter.
Fig. \[fig\_s255\_turb\] demonstrates the influence of the turbulence. It shows the central line profiles of the best fits to the S255 observations using different assumptions on the turbulent nature. We selected S255 here because we can exploit the advantage of additional data for the CS 10–9 transition measured by [@Plume] providing more constraints on the cloud model[^1]. In all models the core is optically thin in CS 7–6 and 10–9, so that the reduction of the optical depth by the turbulent clumping produces only minor changes in these lines. In CS 2–1 the differences are, however, most obvious. In the microturbulent description and for the incompressible turbulence we find self-absorbed line profiles. A reasonable fit to the observations is only possible using the turbulence model including clumps with enhanced density. The reduction of the effective optical depth of the cloud by increasing the optical depth of the coherent units produces narrower lines providing a better fit to the observed line profiles.
We find that the parameter fits do not provide accurate values for the clump size and the combined abundances reflecting the volume filling factor. They only constrain an interval of possible values. We obtained good fits to the observations for the full range of clump sizes between about 0.005 and 0.05 pc. This size scale is confirmed by independent determinations of clump sizes by high-resolution observations in some of our cores. For NGC2024 [@Mezger] determined a radius of dense condensations around 0.015 pc and the interferometric studies by [@Wiesemeyer] showed values between 0.005 and 0.01 pc. In W3 [@Tieftrunk98] observed compact clumps with a size of 0.02 pc and in OMC-2 [@Chini] found dust condensations with radii between 0.01 and 0.05 pc.
For a better comparison to turbulence theory we prefer to specify the cell size in terms of the turbulent correlation length $l{_{\rm corr}}$ which should be on the order of 0.1 pc (Miesch et al. 1994, Goodman et al. 1998). The size of the units which are coherent with respect to the line radiative transfer is smaller by the ratio of the thermal line width to the total velocity dispersion. Thus the size range found corresponds to correlation lengths between 0.04 and 0.4 pc. In the following computations we use the intermediate value of 0.1 pc as correlation length for all clouds.
Regarding the combined abundances we find two classes of objects. The majority of cores, including the example of S255, is well fitted by values between $10^{-11}$ and 10$^{-10}$, whereas a second class, consisting of W3, Serpens, and S106 needs values between $10^{-10}$ and 10$^{-9}$ for a good fit. Because we do not know the molecular abundances there is no way to translate these values directly into clump filling factors. Assuming the CS abundances of $1.3\,10^{-9}$ to $1.3\,10^{-8}$ obtained by [@Hatchell] for several star-forming cores, the first class corresponds to filling factors around 0.01, whereas the CS abundance of $4\,10^{-10}$ from [@Plume] corresponds to a filling factor of 0.1. The filling factors in the second group are ten times higher accordingly. In all following computations we have used a combined abundance factor of $3\, 10^{-11}$ for the first and $3\, 10^{-10}$ for the second group. In the translation to cloud masses we will use the intermediate CS abundance of $4\,10^{-9}$.
As an additional parameter quantifying turbulence in a one-dimensional cloud model we have to take a radial variation of the turbulent velocity distribution into account to explain the observed size-line width and size-density relations ([@Larson69]). Exponents of this radial dependence between about 0.1 and 0.7 are typically discussed (see e.g. [@Goodman]). We left the width and the exponent of the turbulent velocity distribution as free parameters and obtained exponents between 0.15 and 0.65. \[sect\_turbulence\]
The density structure
---------------------
Any physically reasonable cloud model should include a spatial dependence of the gas parameters. [[ ]{}Collapse simulations might provide reasonable model assumptions for the density structure.]{} [@Bodenheimer] and [@Shu] have shown that an isothermal sphere evolves into a power law density profile $n{_{\rm H_2}}
\propto r^{-\alpha_n}$ with $\alpha_n=2.0$. Homologous collapse simulations provide an exponent $\alpha_n=3.0$ (Dickel & Auer 1994) and the free-fall collapse discussed by [@Welch] results in a density exponent $\alpha_n=1.5$. The inside-out collapse model (Shu 1997) combined two regions of different exponents and recent more sophisticated collapse simulations (see e.g. [@Basu]) show more complex density structures with an average exponent $\alpha_n$ between 1.5 and 1.7. Dust observations of the density profile of protostellar cores show evidence both for cores with a typical $r^{-2}$ profile and for cores with flat density structure and a sharp outer edge ([@Andre]). [[ ]{} Thus we expect exponents between about 1.5 and 2.0 in our power-law density model which is bound by an outer cut-off and a central constant region.]{} Although this simple model may not reflect the whole complexity of the density profile we can hardly derive any more information from the limited observations available. [[ ]{}We left the central density and the density exponent as free parameters to be fitted.]{} The resulting exponents span the relatively wide range between 1.1 and 2.2.
In the fit of the radii confining the power-law density profile we face two problems. We cannot distinguish changes of the model parameters on the smallest size scales where even the highest observed transition is thermalised because of the high density. Thus we can only set an upper limit $R{_{\rm cent}}^{\rm max}$ to the radius of the central region where a transition from the power law behaviour in the envelope to constant parameters might occur. Only in four clouds – Serpens, Mon R2, $\rho$ Oph A, and the b-component of W49A – the strength of the CS 7–6 line sets an upper limit to the density so that we can derive the inner radius directly from the observations. In all parameter fits, we left the central radius as a free parameter. After the fit we increased the radius until the maximum deviation in one of the lines reached 5%. This provides a reasonable upper limit to the inner radius in all cases where the line profiles are independent of the density structure below this limit and gives only small modifications in the four cases where the inner radius was already well constrained by the fit.
The outer radius of the cloud is also quite uncertain. We can easily provide a value for the extent of gas at densities above about $3\,10^5$ cm$^{-3}$ based on the spatial extent of the CS 2–1 emission and the line profiles. But it is not possible to derive a reliable value for the extent of low density gas. We can only give a lower limit to the outer radius and thus the mass of the massive cores considered. As the density exponents are typically shallower than –2, a majority of mass could be present beyond this radial limit at low densities invisible in CS. We excluded the outer radius from the parameter fit using a sufficiently large value for all clouds and performed later a separate run reducing the radius to find the minimum outer radius $R{_{\rm cloud}}^{\rm min}$ in a way equivalent to the maximisation of the inner radius described above. \[sect\_radfunc\]
The temperature structure
-------------------------
------------ ----- ---- ------ ------ ----- ------ -----
$\alpha_T$
0.0 47 47 -1.5 3300 1.1 1200 1.8
-0.12 73 52 -1.3 2500 1.3 1500 2.2
-0.4 220 52 -1.0 1000 2.5 7000 3.6
------------ ----- ---- ------ ------ ----- ------ -----
: Resulting cloud parameters for S255 assuming different exponents for the radial temperature dependence and fitting only the central line profiles.
\
${}^{\mathrm a}$ CS 2–1 transition \[tab\_temp\]
The temperature distribution of massive cores is still a matter of debate (cf. [@Garay]). During early phases of cloud collapse the temperature should remain constant as long as the core remains optically thin. Deviations are to be expected, however, as protostellar sources are formed in most of our cores, leading to an internal heating of the cloud. Moreover in thin outer regions external heating can be important. Based on several observational results [@Scoville] set up a spherical cloud model with a warm inner region resulting in an temperature profile $T{_{\rm kin}} \propto r^{\alpha_T}$ with $\alpha_T\approx -0.4$.
Hence, we should also derive the core temperature and the temperature exponent from the radiative transfer model. In a first run we have investigated the influence of a temperature gradient in S255 when fitting only the central line profiles. We compared the best fit models to the S255 observations using either a constant cloud temperature, a temperature decaying with the exponent -0.4, or the temperature exponent as a free parameter. The latter case provided a best fitting exponent of -0.12. All three fits showed an excellent agreement in terms of the central line profiles falling almost exactly on the solid curves in Fig. \[fig\_s255\_turb\]. Thus, it is impossible to favour a certain exponent from the least squares fit of the line profiles only.
In Tab. \[tab\_temp\] we see the resulting values for the other core parameters. The average temperature given here is computed as the mass-weighted average up to $R{_{\rm cloud}}^{\rm min}$. As main difference we find a kind of compensation between temperature and density exponent. The sum of both exponents is kept approximately constant to fulfil the constraints given by the line ratios. Temperature and density gradient act in a similar manner, leading to higher excitation in regions which are either denser or warmer. The variation of the density gradient, however, changes the extent of the emission in the CS 2–1 transition (the change is much smaller in the higher transitions). Consequently, we can constrain the temperature exponent when taking the observed size of the source into account. The beam convolved FWHM of the CS 2–1 emission in S255 falls between 1.5$'$ and 2.1$'$ clearly excluding models with steep temperature gradients.
In the fit procedure applied to all cores we have thus included the fit of the spatial extent. In this way we can constrain the temperature exponent but we are not able to derive exact values. In the parameter fit we always start with an isothermal cloud and change the temperature gradient in steps of 0.1 until the model provides a simultaneous good fit to the central line profiles and the spatial FWHM of the emission. It turns out that the observations of all clouds except W33, W3 and Mon R2 can be fitted by an isothermal model. This indicates that a large part of the gas mass in the clouds is characterised by a uniform kinetic temperature. This isothermal behaviour, however, does not support the assumption of constant excitation temperatures like in escape probability model. The excitation temperatures show local variations which are steeper than the density gradient. In the derivation of the kinetic temperature structure we have to keep in mind, however, that we are not very sensitive to the exact value of the kinetic temperature gradient. Observations of higher transitions like CS 14–13 would be needed to obtain reliable values including full error estimates.
Other parameters
----------------
Beside the temperature gradient we can expect the formation of compact [H[II]{} ]{}regions in the centre of a star forming core. Using the parameters of the central [H[II]{} ]{}region derived by [@Dickel] for W49 ($R=0.2$ pc, $n{_{\rm e}}=2.6\,10^4$ cm$^{-3}$, $T{_{\rm e}}=10^4$ K) we have compared the resulting CS lines when either including or neglecting the [H[II]{} ]{}region in the radiative transfer computations (see Appx. \[sect\_hii\_region\]). We find that the influence of the [H[II]{} ]{}region is negligible in this example for the four CS transitions considered here.
[[ ]{}In general [H[II]{} ]{}regions have two dominant effects on the line profiles. The integrated brightness changes the molecular excitation throughout the cloud and is visible as continuum emission. Moreover, the molecular material in front of an [H[II]{} ]{}region appears partially in absorption. A large [H[II]{} ]{}region with electron densities being a factor 10 or more higher than in the example above, would thus result in distinct changes in the lines. Depending on the configuration and velocity structure, the CS lines may appear in absorption or with P Cygni profiles. Moreover, a strong continuum emission would be observed at all frequencies considered. However, bright [H[II]{} ]{}regions with high electron densities are unlikely to be that extended ([@Wood]). For ultracompact [H[II]{} ]{}regions the change of the line profiles by absorption is negligible due to the small angular size of the [H[II]{} ]{}region so that the molecular excitation is the remaining effect. From the lack of a bright continuum underlying the lines we can, however, exclude configurations with a bright [H[II]{} ]{}region here. Weaker compact or ultracompact [H[II]{} ]{}regions – although possible – were not included in the fit computations as they would only influence material in their close environment which cannot be resolved in this study.]{}
Regarding the chemical evolution of massive cores one should also expect a variation of the molecular abundances of CS and C$^{34}$S. However, [[ ]{}our present knowledge]{} is still insufficient to guess reliable values here (cf. [@Bergin]). As discussed in Sect. \[sect\_turbulence\], it is also not possible to fit the abundance independently from the clump filling factor, so that we adopted here a constant CS abundance of $4\,10^{-9}$ relative to H$_2$ ignoring any radial variation of the abundances.
An additional test is only possible for NGC2024 where C$^{34}$S is sufficiently bright so that we could include it in the fits. Fig. \[fig\_ngc2024\] shows the best fit model to all six available line profiles. Equivalent to the results [[ ]{}from the escape probability model]{} we get the best match for a relative molecular abundance $X($CS$)/X($C$^{34}$S$) \approx 13$. All six lines are simultaneously fitted.
None of the cores except W49A show clearly asymmetric profiles as a signature of systematic internal velocities. The weak wings seen [[ ]{}in a few of the other lines]{} at low resolution are insufficient to derive any collapse or outflow model. Thus we have fitted all cores with a static model.
-------------- ------- ----------------- ------ ------------ ----------------------- ----------------- ----- ------------- ------ ----------------- ------
W49A$^{(a)}$ 0.23 \[:0.50\] 3.6 \[2.2:\] 1.2\[1.0:1.4\]10$^7$ -1.8\[1.6:2.3\] 92 \[67:125\] 0.0 6.6\[5.1:9.1\] 0.15
W49A$^{(b)}$ 0.92 \[0.59:1.2\] 3.6 \[2.6:\] 9.9\[8.5:11.0\]10$^5$ -1.6\[1.4:1.7\] 109 \[85:138\] 0.0 7.1\[5.6:9.6\] 0.22
W33 0.27 \[:0.39\] 1.9 \[0.81:\] 9.1\[7.0:11.0\]10$^6$ -1.9\[1.1:2.6\] 36 \[30:42\] -0.2 4.0\[2.7:6.2\] 0.44
W51A 0.57 \[:0.88\] 3.9 \[2.1:\] 5.3\[4.3:6.2\]10$^6$ -1.6\[1.2:2.8\] 44 \[35:53\] 0.0 8.3\[6.2:12.0\] 0.23
W3(OH) 0.086 \[:0.16\] 2.2 \[0.54:\] 1.2\[0.99:1.4\]10$^7$ -1.7\[1.4:2.6\] 39 \[32:46\] 0.0 3.7\[2.8:5.3\] 0.08
W3 0.062 \[:0.10\] 1.4 \[0.30:\] 6.3\[4.8:7.8\]10$^6$ -2.0\[1.7:2.6\] 64 \[49:79\] -0.2 3.0\[2.1:5.2\] 0.30
S255 0.056 \[:0.077\] 1.1 \[0.66:\] 1.6\[1.4:1.8\]10$^7$ -1.5\[1.3:1.7\] 47 \[41:52\] 0.0 2.1\[1.6:2.9\] 0.14
S235B 0.098 \[:0.12\] 0.53 \[0.28:\] 8.3\[6.8:9.8\]10$^6$ -2.1\[1.5:2.7\] 20 \[17:22\] 0.0 1.7\[1.3:2.3\] 0.56
S106 0.024 \[:0.038\] 1.2 \[0.13:\] 8.8\[7.5:9.9\]10$^5$ -1.3\[1.2:1.4\] 137 \[103:174\] 0.0 1.2\[0.9:1.6\] 0.55
Serpens 0.031 \[0.024:0.036\] 0.68 \[0.072:\] 6.4\[5.6:7.2\]10$^5$ -1.6\[1.5:1.7\] 93 \[74:112\] 0.0 2.8\[2.2:3.6\] 0.47
DR21 0.11 \[:0.21\] 1.7 \[0.78:\] 6.5\[5.3:7.6\]10$^6$ -1.5\[1.3:1.9\] 83 \[61:108\] 0.0 1.8\[1.4:2.5\] 0.52
Mon R2 0.097 \[0.057:0.12\] 1.3 \[0.33:\] 1.9\[1.6:2.1\]10$^6$ -1.5\[1.3:1.8\] 79 \[61:98\] -0.2 1.7\[1.3:2.3\] 0.32
NGC2264 0.052 \[:0.085\] 1.2 \[0.31:\] 7.0\[5.9:8.0\]10$^6$ -1.4\[1.1:2.1\] 37 \[31:44\] 0.0 2.9\[2.2:4.0\] 0.20
OMC-2 0.019 \[:0.037\] 0.85 \[0.11:\] 1.4\[1.2:1.7\]10$^7$ -1.5\[1.1:2.0\] 27 \[24:31\] 0.0 1.2\[0.9:1.5\] 0.38
$\rho$ Oph A 0.079 \[0.060:0.10\] 0.68 \[0.16:\] 1.1\[0.9:1.3\]10$^6$ -1.1\[0.9:1.3\] 20 \[15:24\] 0.0 1.2\[0.9:1.7\] 0.65
NGC2024 0.019 \[:0.025\] 0.15 \[0.093:\] 8.0\[5.8:10.2\]10$^7$ -2.2\[1.9:2.5\] 36 \[31:40\] 0.0 1.2\[0.8:1.8\] 0.48
-------------- ------- ----------------- ------ ------------ ----------------------- ----------------- ----- ------------- ------ ----------------- ------
\[tab\_ltr\_results\]
Comparing the massive cores
---------------------------
Tab. \[tab\_ltr\_results\] lists the resulting best fit parameters for all cores with their error intervals. As discussed in Sect. \[sect\_radfunc\] the intervals for the radii are in most cases only limited at one end. The central density refers to the value at the radius $R{_{\rm cloud}}^{\rm min}$.
Comparing the different cores we find that the inner and outer radii are mainly determined by the selectional bias from the observability as one massive core. The most distant cores are only detectable with the KOSMA telescope if they are relatively large whereas at small distances only small cores are unresolved.
The minimum central density fitting the CS lines varies between $10^6$ cm$^{-3}$ for W49A$^{(b)}$ and $8\,10^7$ cm$^{-3}$ for NGC2024. The high value derived for NGC2024 is due to the availability of the CS 10–9 observations tracing higher densities. With the three CS lines measured for most cores, only the density range below about $10^7$ can be reliably traced, so that we have to take the central density for all clouds except W49A$^{(b)}$, Serpens, Mon R2, and $\rho$ Oph A as lower limits. The density exponent covers the range between –1.1 and –2.2 where the majority of clouds shows values around –1.6 corresponding to large-scale collapse models (Sect. \[sect\_radfunc\]).
Although we don’t have a sample where we can expect to set up statistically significant correlations we can interpret some general relations. It turns out that the parameters are not completely independent of each other. Whereas the majority of clouds shows an average temperature between 20 and 50K, the three clouds which needed a larger combined abundance factor corresponding to a higher clump filling factor in the turbulence description (W3, S106, Serpens) also tend to require relatively high temperatures. There is a clear correlation between the cloud temperature and the turbulent line width indicating that heating and turbulent driving might have a related cause. The cores with a significant temperature exponent also show a relatively steep density exponent whereas a steep density exponent itself does not necessarily require a temperature exponent. These internal relations should be explained from the physical nature of the clouds.
There are some peculiarities concerning four massive cores. The fit of the second component of W49A needs a large inner region with constant parameters or a very shallow decay of the density profile. Here, the main information that we get from the line profiles is the central density whereas we can hardly constrain the density exponent in the power-law range. Regarding the error bars of the parameters for Serpens we see that the the density structure of this relatively nearby cloud is well constrained by its observed size whereas the temperature structure is relatively poorly determined. In $\rho$ Oph it was not possible to fit the line profiles simultaneously with the size of the smallest core resolved in the observations. This can be explained by the clumpy structure of the whole region where the excitation of the core at the central position cannot be treated separately from the other clumps. Thus we have used for the core model a size which is six times larger than the smallest resolved clump and contains most of the strong emission observed. To fit the profiles and spatial extent of Serpens and S106 we have to assume a relatively large combined abundance (see Sect. \[sect\_turbulence\]) and a low hydrogen density in the clumps. This peculiarity could be removed when assuming some foreground CS that increases the apparent size of the massive core in CS 2–1 but does not contribute to the excitation in the core.
Discussion
==========
Restriction by the observed transitions
---------------------------------------
--------------- ------ ----- ----------- ------ ---- ------ ------
$n{_{\rm l}}$
4 0.06 1.1 1.510$^7$ -1.5 47 3600 1000
3 0.11 1.1 5.310$^6$ -1.4 49 3700 1000
--------------- ------ ----- ----------- ------ ---- ------ ------
: Resulting cloud parameters for S255 using either three or four transitions in the fit.
\[tab\_3lines\]
--------------- ----------- ----------- ------------------- -------------- ----------- ------------- ----------------
$n{_{\rm l}}$
4 \[:0.08\] \[0.66:\] \[1.6:2.2\]10$^7$ -\[1.3:1.7\] \[41:52\] \[1.6:2.9\] \[-0.01:0.42\]
3 \[:0.21\] \[0.68:\] \[4.4:6.1\]10$^6$ -\[1.2:1.8\] \[40:59\] \[1.7:3.2\] \[-0.07:0.70\]
--------------- ----------- ----------- ------------------- -------------- ----------- ------------- ----------------
\[tab\_3errors\]
The main constraint to the core parameters that we can derive is set by the molecule and the transitions observed. They are only sensitive to a relatively narrow density range. We can test the limitation of the fits provided by the restriction to three line profiles by comparing the results obtained for cores where we have additional CS 10-9 profiles. With the data from [@Plume] for S255 we investigate how much information is lost due to the lack of CS 10–9 data in most cores. Fig. \[fig\_3lines\] shows the resulting best fits to the central line profiles in S255 when either all four lines are fitted or only the information from the lower three transitions is used. Tab. \[tab\_3lines\] lists the corresponding model parameters from the fits. Isothermal models provided good fits to the data and the derived physical parameters are almost identical except for the inner radius. The additional information from the CS 10–9 transition can set a smaller limit here corresponding to the higher central densities. When predicting the 10–9 line data from the best fit of the three other lines the intensity is too low by only about 20% (Fig. \[fig\_3lines\]). Hence, we expect reliable results also for those cores where only three lines are observed but it would be favourable to add information from higher transitions for a better resolution of the densest inner region.
Moreover, a fourth line will reduce the error bars of the parameters. In Tab. \[tab\_3errors\] we compare the error obtained for the S255 observations using either the three- or the four-lines fit. As discussed above the possible range of the inner and outer radii is limited only in one direction. [[ ]{}Thus, the fourth line mainly reduces the uncertainty of the inner radius. It hardly influences the error of the outer radius and the density at the inner radius but it also reduces the error of the density exponent, the temperature and the velocity structure.]{} Thus the inclusion of additional lines in the model fits would also give a better constraint of the parameters derived.
Comparison of the methods
-------------------------
-------------- ----------- ------ ------ ----- ------- ------- -------
W49A$^{(a)}$ 1.410$^3$ 2.6 4.6 92 15000 22000 11000
W49A$^{(b)}$ 1.710$^3$ 0.80 4.8 109 19000 14000 11000
W33 4.610$^3$ 2.3 3.8 28 7400 6300 4000
W51A 3.910$^3$ 3.0 9.8 43 54000 30000 7500
W3(OH) 7.710$^2$ 1.0 1.5 39 1900 1900 2200
W3 2.710$^3$ 3.6 9.7 41 1700 1100 2200
S255 2.810$^3$ 1.1 0.78 47 870 530 2500
S235B 4.510$^3$ 0.66 0.22 20 180 210 1800
S106 7.210$^2$ 0.32 0.23 137 280 200 600
Serpens 7.310$^2$ 0.22 0.17 93 53 80 310
DR21 1.610$^3$ 0.82 0.79 83 1800 1800 3000
Mon R2 6.110$^2$ 0.22 0.18 54 310 260 950
NGC2264 1.110$^3$ 0.46 0.71 37 450 720 800
OMC-2 6.610$^2$ 0.34 0.24 27 95 52 400
$\rho$ Oph A 1.310$^3$ 0.12 0.12 20 96 34 160
NGC2024 2.210$^4$ 1.3 1.6 36 17 35 420
-------------- ----------- ------ ------ ----- ------- ------- -------
\[tab\_coreparams\]
Tab. \[tab\_coreparams\] shows quantities characterising the global properties of the clouds computed from the fit parameters in Tab. \[tab\_ltr\_results\]. The average density in the second column is given by the cloud mass within the outer radius. [[ ]{}We see the strong discrepancy between the average density (Tab. \[tab\_coreparams\]) and the central clump density (Tab. \[tab\_ltr\_results\]) reflecting a very inhomogeneous structure with low volume filling factor of dense clumps.]{}
In the column 3 and 4 of Tab. \[tab\_coreparams\] one can compare the average column density towards the centre in the nonlocal model with the column density computed from the escape probability model. [[ ]{}Here, we have used the molecular column densities from Tab. \[tab\_epparm\] assuming a CS abundance of $4\,10^{-9}$.]{} We find an agreement within a factor of about 1.5, despite the completely different analysis applied, except for W49A, W51, and W3 where the column density from the escape probability model is more than a factor of two higher and S235B where it is lower. S235B, W49A, and W3 are the smallest sources in our sample unresolved even in the CS 2–1 beam. Here, the beam filling factors used in the escape probability model are uncertain so that they may be responsible for the difference. For W51A, no simple explanation for the difference is obvious. It is however, by far the most massive core in our sample so that it might be somewhat peculiar from that point of view. In general, we find that the escape probability analysis provides a reasonable determination for the column density when we have a good estimate of the beam filling factor. It fails to derive correct densities or sizes.
Core masses
-----------
Columns 6 and 7 in Tab. \[tab\_coreparams\] shows the core masses computed in two different ways. Column 6 gives the integrated mass of the model cloud assuming the smallest fitting outer radius and the maximum possible central radius. The influence of the inner radius on the total mass is negligible, but the uncertainty from the lack of information on the outer radius has to be kept in mind. Increasing the amount of virtually invisible material around the core by increasing the outer cloud radius can easily increase the total mass by more than a factor 10. As the mass computation relies on the knowledge of the CS molecular abundance (Sect. \[sect\_turbulence\]), the resulting values are to be changed if the true abundances deviate from the assumed value of $4\,10^{-9}$.
Column 7 contains the core virial mass assuming equipartition of kinetic and gravitational energy in a homogeneous spherical cloud. We used the central CS 2–1 line profile and the size of the cloud visible in this transition to estimate the velocity dispersion in the line of sight and the radius. Following [@Lang] we obtain the virial mass by $$M{_{\rm vir}}= 0.0183 \Delta v^2 D \sqrt{\Delta \alpha \Delta\delta}$$ where $\Delta v$ is the FWHM of the line (Tab. \[tab\_lineparams\]), $D$ the distance of the source, and the $\Delta\alpha$ and $\Delta\delta$ the FWHM of the source in declination and rectascension in arcmin from Tab. \[tab\_coresize\].
For all clouds except $\rho$ Oph A the agreement between the two masses falls within a factor of two. The behaviour of $\rho$ Oph A is due to the radius of the excitation model which is larger than the smallest resolved core, as discussed above. Hence, it provides a larger mass than the virial estimate which uses this visible core size. The general good agreement is quite amazing regarding the uncertainty of the outer boundary of the models. The clouds seem to be virialised and the CS abundance estimate holds approximately for all clouds.
The agreement of the mass from the [*SimLine*]{} fits with the virial mass and independent estimates from the literature indicates that the cores are well confined and our minimum outer radius corresponds to a real, relatively sharp boundary for most cores in agreement with the results from continuum observations of several cores by André et al. (1999). Future investigations are, however, necessary to confirm this result because the nature of virialisation is still not understood and it is therefore not clear how much of the “invisible” low density mass would contribute to the virial mass.
Comparison with other observations
----------------------------------
To judge the reliability of the parameters derived here, we can compare them with core parameters obtained independently from observations in other tracers and with other telescopes. In general they provide only values for few of the cloud parameters but they may serve as an independent test of our results. We cannot include a complete discussion of the literature concerning the 15 massive cores considered here. Rather we restrict ourselves to a few selected observations showing the general power and limitations of the method.
-------------------------- ---------------- --------- --------- ------- ------- --
Mezger$^{\mathrm a}$ 0.007 4.8 22 19 20
Wiesemeyer$^{\mathrm b}$ $\approx$ 0.01 0.9-2.3 3.1-8.4 16-19 16-50
this paper$^{\mathrm c}$ 0.15 $>0.8$ 9.9 36 17
-------------------------- ---------------- --------- --------- ------- ------- --
: Comparison of the NGC2024 FIR5 parameters with results from [[ ]{}investigations based on other high resolution observations.]{}
\
${}^{\mathrm a}$ IRAM 30m 870 $\mu$m, SEST 1.3 mm continuum, [@Mezger]\
${}^{\mathrm b}$ IRAM PdB 3 mm, IRAM 30m 870 $\mu$m, VLA 1.3 cm continuum, [@Wiesemeyer]\
${}^{\mathrm c}$ KOSMA CS and C$^{34}$S, IRAM 30m CS as discussed in the text\
\[tab\_ngc2024\]
For NGC2024 we have compared our data with results from complementary high-resolution observations in Tab. \[tab\_ngc2024\]. [@Mezger] combined the results of SEST observations at 1.3 mm with IRAM 30m continuum maps at 870 $\mu$m to identify several clumps in NGC2024 and to deduce their physical properties from the continuum fluxes. The given values correspond to FIR5 falling at our central position. [@Wiesemeyer] used a spherically symmetric continuum transfer model to derive the physical parameters of FIR5 from 3 mm continuum observations taken with the IRAM Plateau de Bure interferometer combined with the data from [@Mezger] and VLA 1.3 cm observations of [@Gaume]. Depending on the assumed luminosity and dust properties they found a range of parameters fitting the observed continuum. The last line in Tab. \[tab\_ngc2024\] represents the results from [[ ]{}the [*SimLine*]{} fit to our data.]{} In contrast to Tab. \[tab\_coreparams\], the central column density given here is not averaged but computed towards a central clump to allow a better comparison with the dust observations which are able to resolve this clump.
We see that one cannot reveal the true radius of the core from our low resolution observations. Moreover, the CS observations cannot trace the same high densities as the dust observations so that they provide only a lower limit. It is, however, already close to the central density given by [@Wiesemeyer]. The mass and column density derived from our radiative transfer computations agree quite well with the values provided by the high-resolution observations. A possible explanation for the difference between the gas kinetic temperature and the dust temperatures was provided already by [@Schulz]. They performed NH$_3$ and CS observations of NGC2024 and obtained temperatures between 35 and 40 K at the position considered. Using a two-component dust model they demonstrated that these temperatures are also consistent with the observations by [@Mezger]. Altogether we are able to derive realistic values for the core parameters with a clumpy radiative transfer model even if we are not able to deduce the exact object size as we cannot resolve it.
W51 consists of three compact molecular cores located within about one arcmin. Interferometric observations by Young et al. (1998) and the combination of line and continuum measurements by Rudolph et al. (1990) seem to indicate collapse of the component W51e2 with a mass of about 40000 M$_{\sun}$. The mass determined by the [*SimLine*]{} fit is 54000 M$_{\sun}$. The FCRAO and KOSMA observations show no signatures of collapse as they are probably blurred by our low spatial resolution. Sievers et al. (1991) obtained temperatures between 20K and 57K and Zang & Ho (1997) derived 40-50 K for an inner region of about 0.2 pc and 25-30K for the outer cloud based on NH$_3$ observations. We were able to fit the observations with an isothermal cloud at 44 K, however cannot exclude such a temperature structure. Young et al. (1998) used an LTE code assuming spherical or spheroidal symmetry to simulate the inner 0.2 pc region of W51e2 fitting the observed ammonia data. They obtained density gradients of -1.8 to -2.2 somewhat steeper than in our fit, indicating that a dense central region might be surrounded by an envelope with a flatter density gradient. Their central densities between 1.5 and 2210$^6$ cm$^{-3}$ bracket our value of 510$^6$. The assumption of a smooth medium by Young et al. (1998) results in a central temperature estimate below 25K and a steep temperature gradient. i.e. significantly lower temperatures than in our clumpy turbulent model. We have tested this behaviour by trying to fit the data without clumping in our model and also got low temperatures below 20 K but quite bad $\chi^2$ values. Thus the correct treatment of the internal clumping is essential for a reliable temperature derivation.
W3(OH) was studied e.g. by Wilson et al. (1991) using VLA observations of methanol and OH and by Tieftrunk et al. (1998) and [@Helmich] with single dish observations of ammonia and HDO, respectively. The KOSMA beam covers several maser spots and a luminous mm continuum source – probably a class 0 object. Wilson et al. (1991) obtained a kinetic temperature of the molecular cores of 20K, whereas [@Tieftrunk98] derived 27K. Both agreed with our total mass estimate of about 2000 M$_{\sun}$. The radius of 1.3 pc determined by Tieftrunk is somewhat smaller than the value computed from our model, but clearly within the error bar. From the HDO excitation [@Helmich] conclude dust temperatures above 100K at densities between $10^5$ and $10^6$ cm$^{-3}$, with few embedded clumps at $10^7$ cm$^{-3}$. This clump density and the column density of about $2\,10^{23}$ cm$^{-2}$ agree approximately with the values from our spherical model. The temperature of 39 K determined from the CS observations falls into the range discussed but the difference to the methanol and ammonia values asks for an explanation. Around the massive core there is probably still an extended envelope of gas at low densities insufficient to excite the observed CS transitions (Tieftrunk et al. 1998).
W3 was studied in detail by Tieftrunk et al. (1995, 1997, 1998) using C$^{34}$S, C$^{18}$O, NH$_3$, and continuum observations. Our beam covers the two bright components W3 Main and W3 West. Moreover, the region contains some ultracompact [H[II]{} ]{}regions related to infrared sources. The molecular line emission peaks at a position close to W3 West. The line velocity of our CS observations at -42 km/s agrees with the velocity of W3 West indicating this component as the main originator of the observed CS emission. The combination of single dish and VLA observations by Tieftrunk et al. (1997, 1998) showed extended gas at a temperature of 25-45K, a density of 10$^4$ cm$^{-3}$, an size of about 1 pc, and a total mass of 1100-1400 M$_{\sun}$. This corresponds well to the parameters derived from the KOSMA observations. We have traced the emission to the somewhat larger radius of 1.4 pc, but within a radius of 1 pc we get about the same average density of $0.7\,10^4$ cm$^{-3}$. Our mass estimate of 1100 M$_{\sun}$ and the average temperature of 41 K also agree. [[ ]{}The VLA observations showed several very compact clumps with a size of 0.02 pc, densities of 10$^7$ cm$^{-3}$, and $T{_{\rm kin}}=250$ K. They are not resolvable from our data, but correspond to the clumps in the turbulence description and the core parameters derived from the radiative transfer model show a similar size and density.]{} In the smooth temperature distribution assumed in the radiative transfer model we are not able to resolve hot spots with 250 K but found the need for an increased temperature towards the centre.
From the group of low mass cores, Castets & Langer (1995) analysed CS observations of OMC-2 by means of an LVG analysis providing $T{_{\rm kin}}=24$ K, a density of $9\,10^5$ cm$^{-3}$ and a CS column density of $5.4\,10^{13}$ cm$^{-2}$ in agreement with our results. [[ ]{}They found already indications for substructure with clump radii of about 0.022 pc and higher densities in observations with higher resolution.]{} Our analysis shows densities of at least 1.410$^7$ cm$^{-3}$ at a scale of 0.019 pc. The virial mass of 71 $M_{\sun}$ computed by Castets & Langer is only somewhat smaller than our mass estimate of 95 M$_{\sun}$. Recent 1.3 mm observations by [@Chini] show at least 11 embedded condensations in OMC-2 with masses between 5 and 8 M$_{\sun}$ and temperatures between 20 and 33 K whereas 350 $\mu{}m$ continuum data by [@Lis98] reveal even 30 clumps but lower temperatures of 17 K supporting our approach of the clumpy cloud model.
The comparison shows that different tracers see different parts of a cloud corresponding to different physical conditions. Results from other authors based on CS observations agree in most cases quite well, whereas the results from other tracers may considerably differ. The relatively large uncertainty in the temperature structure that we cannot resolve within our analysis asks for additional observations in higher transitions or at better spatial resolution. For nearby clouds like NGC2024, OMC-2 or $\rho$ Oph A we get a good agreement with results from high-resolution or even interferometric observations, whereas for distant massive cores like W49A and W51A there are several open question, especially regarding the temperature structure. \[sect\_comparison\]
The physical nature of massive cores
------------------------------------
All massive cores seem to be approximately virialised independent of their internal structure with respect to the number, distribution and luminosity of young stars. Although one could expect that violent bipolar outflows observed in some cores will drastically change the energy balance in the core, the physics of the turbulence in the cores seems to be extremely stable guaranteeing a continuous state of virialisation.
The relatively sharp outer boundary suggested by the mass estimates can be interpreted in terms of collapse models. The collapse of an isothermal sphere would result in a self-similar density distribution without clear boundary whereas our results rather tend towards the scenario of a finite-size Bonnor-Ebert condensation ([@Bonnor]). The outer boundary is, however, not well determined but only set by the mass constraints because the radiative transfer model itself cannot exclude a continuation of the density structure to larger radii.
The density exponent of about –1.6 derived for most cores is consistent with several collapse models (see Sect. \[sect\_radfunc\]) but deviations from the exponent for particular clouds up to values around –2 have to be explained.
On the other hand we have seen that simple collapse models are not relevant for the massive cores considered here. Clumpiness is a main feature of all clouds and smooth microturbulent models are not able to explain the observed lines. In agreement with other high-resolution observations we find typical clump sizes of 0.01–0.02 pc at least for the nearby cores. In massive distant cores the situation might be more complex including a hierarchy of clump sizes resulting in a larger uncertainty of the temperature profile derived from our model.
Conclusions
===========
We have shown that the careful analysis of multi-line single dish observations with a relatively large beam can provide a set of information comparable to single-line interferometric observations. From a careful excitation analysis using a self-consistent radiative transfer computation it is possible to deduce some sub-resolution information. We can infer clump sizes, masses and densities at scales below a tenth of the beam size. However, interferometric observations are necessary to determine the exact core geometry including the location and number of clumps within a dense core.
The spherically symmetric radiative transfer code used here is able to take into account radial gradients in all quantities and internal clumpiness of the cloud. It enables a reliable deduction of the physical parameters from line profiles observed in sources with a size close to the spatial resolution limit. The method allows to analyse similar observations of objects like star-forming cores in distant galaxies unresolvable by all today’s means. For a better resolution of the internal temperature structure the approach should be combined with sophisticated models on the energy balance including the continuum radiative transfer in the future. Although the simple escape probability analysis gives a reasonable estimate for the column density, it fails regarding the density and temperature structure.
The line analysis shows two essential points:\
[*i)*]{} The main constraints on the structural quantities which can be deduced from the observations are set by the tracer. The range of densities and temperatures that one can determine from the radiative transfer calculations is restricted by the transitions observed. In case of the CS 2–1, 5–4, and 7–6 lines, the covered densities range from about $2\,10^5$ to $10^7$ cm$^{-3}$. With additional information from the CS 10–9 the upper limit can be extended by another factor 5. The information from the rarer C${}^{34}$S isotope cannot extend the density interval but reduces the error bars and provides better estimates for the clumpiness of the medium. The high resolution observations discussed in Sects. \[sect\_epresults\] and \[sect\_comparison\] show that different tracers provide access to different types of information whereas the parameters from our CS observations agree well with the CS results there.\
[*ii)*]{} Temperature and clumpiness are related quantities. When turbulent clumping in the cloud is neglected, the temperature determination will necessarily fail. On the other hand does accurate information on the clumpy structure of a massive core help to constrain the temperature structure. Additional observations in higher transitions or complementary estimates of the clumpiness will help to reduce the uncertainty of the temperatures. Thus spatial resolution is still essential. For nearby clouds we get a good agreement with results from other high-resolution observations, but for distant massive cores the temperature structure is still an open question.
All massive cores that we have analysed are characterised by turbulent clumpiness with typical clump sizes of 0.01–0.02 pc. The clouds are approximately virialised and show density gradients around –1.6 but with a scatter between –1.1 and –2.2. Large parts of the cores follow a constant temperature but we must admit a considerable uncertainty in the most inner and outer parts. The correlation between the cloud temperature and the turbulent line width indicates that related processes should be responsible for heating and turbulent driving.
Future observations of dense cores should focus on different tracers to gain access to additional information which cannot be deduced from a single tracer such as CS. As a drawback, the full uncertainty of today’s chemical models will enter and partially limit the interpretation of the observations.
We thank J. Howe for providing us with the CS 2–1 observational data. We are grateful to the anonymous referee for many detailed comments helping to improve the paper considerably. This project was supported by the *Deutsche Forschungsgemeinschaft* through the grant SFB 301C. The KOSMA 3m radio telescope at Gornergrat-Süd Observatory is operated by the University of Cologne and supported by the Land Nordrhein-Westfalen. The receiver development was partly funded by the Bundesminister für Forschung and Technologie. The Observatory is administered by the Internationale Stiftung Hochalpine Forschungsstationen Jungfraujoch und Gornergrat, Bern. The research has made use of NASA’s Astrophysics Data System Abstract Service.
The radiative transfer problem in the escape probability approximation
======================================================================
The general radiative transfer problem
--------------------------------------
The physical parameters of a cloud model and the emerging line profiles and intensities are linked by the radiative transfer problem. It relates the molecular emission and absorption coefficient at one point to the radiation field determined by the emission and transfer of radiation at other locations in the cloud. The quantity entering the balance equations for the level populations at a point $\vec{r}$ is the local radiative energy density $u$ within the frequency range for each transition: $$\begin{aligned}
u(\vec{r})&=&\int_{4\pi} \! d\Omega\; u(\vec{r},\vec{n}) \nonumber \\
\quad u(\vec{r},\vec{n})&=& {1 \over c} \int_{-\infty}^{\infty} d\nu\;
I_\nu(\vec{r},\vec{n}) \Phi_\nu (\vec{r},\vec{n})\;.
\label{ufirst}\end{aligned}$$ Here, $u(\vec{r},\vec{n})$ is the absorbable radiative energy coming from direction $\vec{n}$ and $I_\nu(\vec{r},\vec{n})$ is the intensity at a given frequency within this direction. It is determined by the radiative transfer equation $$\vec{n} \nabla I_\nu(\vec{r},\vec{n}) = - \kappa_\nu(\vec{r},\vec{n})
I_\nu(\vec{r},\vec{n}) + \epsilon_\nu(\vec{r},\vec{n})
\label{eq_transfer}$$ Assuming complete redistribution the profile for the absorption and the emission coefficients, $\kappa_\nu(\vec{n},s)$ and $\epsilon_\nu(\vec{n},s)$ is given by the same local line profile $\Phi_\nu$. For Maxwellian velocity distributions it is a Gaussian: $$\Phi_\nu(\vec{n},\vec{r})={1 \over \sqrt{\pi} \sigma}
\exp\left(-{(\nu-\vec{n}\vec{v}(\vec{r}))^2\over \sigma^2}\right)\;.$$ Here, $\vec{v}$ is the velocity of the local volume element written in units of the frequency $\nu$. The frequently used FWHM of the distribution is related to $\sigma$ by FWHM$=2\sqrt{\rm ln\,2}\,\sigma$.
For a molecular cloud this results in a huge system of integral equations interconnecting the level populations and intensities at all points within a cloud.
The escape probability model
----------------------------
A simple way to avoid the nonlinear equation system is the escape probability approximation that is widely applied to interpret molecular line data. It is based on the assumption that the excitation, and thus the absorption and emission coefficients, are constant within those parts of a cloud which are radiatively coupled. Then the radiative transfer equation (Eq. \[eq\_transfer\]) can be integrated analytically. We obtain for the integrated radiative energy density: $$u(\vec{n})={1 \over c} \left [\beta(\vec{n}) I{_{\rm bg}}(\vec{n})+\left(1-\beta(\vec{n})
\right) {\epsilon_l \over \kappa_l}\right]\;.
\label{eq_udir}$$ where the subscript $l$ denotes line integrated quantities. The term $\beta(\vec{n})$ is the probability that a photon can escape or penetrate along the line of sight $\vec{n}$ from the considered point to the boundary of the interaction region or vice versa. $$\begin{aligned}
\beta(\vec{n}) &= & \int_{-\infty}^{\infty} d\nu\; \Phi_\nu(\vec{r},\vec{n})
\times \exp (-\tau_\nu(\vec{r},\vec{n}))\\
{\rm with} && \tau_\nu(\vec{r},\vec{n})=\int_{-\infty}^{\vec{r}} ds_{\vec{n}}\;
\kappa_l(\vec{r}) \Phi_\nu(\vec{r},\vec{n})\end{aligned}$$ where the integration path $ds_{\vec{n}}$ follows the direction $\vec{n}$.
There are two main concepts to define the interaction region and thus to compute the escape probability. The first one is the large velocity gradient approximation introduced by [@Sobolev]. Here, the interaction region is determined by a velocity gradient in the cloud that displaces the line profiles along the line of sight so that distant regions are radiatively decoupled. When the resulting interaction region is sufficiently small it is justified to assume constant parameters. The escape probability then follows $$\begin{aligned}
\beta(\vec{n})
&=& \left. \left[1-\exp \left( -\tau{_{\rm LVG}}
\right)\right]\right/ \tau{_{\rm LVG}}\\
{\rm with} && \tau{_{\rm LVG}}={\kappa_l \over |(\vec{n}\nabla) (\vec{n}\vec{v})|}
\label{eq_lvg}\end{aligned}$$ (c.f. [@Ossenkopf97]). The radiative energy density in each direction is determined by two local quantities only: the source function $S=\epsilon_l/\kappa_l$ and the optical depth of the interaction region $\tau{_{\rm LVG}}$. The observable brightness temperature at the line centre is constant over all regions with the same velocity gradient $$T{_{\rm B}} \approx {c^2 \over 2 k \nu^2} \left( S - I{_{\rm bg}}\right)
\left[1-\exp(-2\tau{_{\rm LVG}})\right]
\label{eq_tmb_lvg}$$
In molecular clouds, the local velocity gradients are unknown, however. It is generally assumed that the total observed line width, which is composed from turbulent, thermal, and systematic contributions, can be used as the measure of the velocity gradient over the cloud size. This approach was applied by [@Plume] for the massive cores discussed in the text.
We used another method, the static escape probability model. It does not depend on the velocity structure but assumes a special geometry of the interaction region. [@Stutzki85] solved the problem for a homogeneous spherical cloud with constant excitation parameters. The resulting escape probability is taken to be constant $$\beta=\int_{-\infty}^{\infty} d\nu\; \Phi_\nu \times \exp (-\tau_\nu)\;.
\label{eq_beta_ep}$$ where $\tau_\nu$ is the optical depth at the cloud centre.
The surface brightness temperature towards the centre of the cloud is given by the same expression as Eq. (\[eq\_tmb\_lvg\]) when we use the line integrated optical depth at the cloud centre $\tau_l$ instead of $\tau{_{\rm LVG}}$. It decays with growing distance from the cloud centre. Averaged over the whole cloud, the brightness temperature at the line centre is given by $$\begin{aligned}
T{_{\rm B}} & \approx& {c^2 \over 2 k \nu^2} \left( S - I{_{\rm bg}}\right)
\left(1-e(-2\tau_l)\right) \label{eq_tmb_ep} \label{eq_tmb_esc}
\label{eq_tmb_statep}\\
{\rm with} & & e(x)={2 \over x^2} \left( 1 - \exp(-x)(1+x)\right) \nonumber \;.\end{aligned}$$ This value would be observed with a beam much larger than the cloud.
When the velocity gradient in the LVG approximation is computed from the total line width and the cloud size, it turns out that both methods agree when applied to observations with a small beam towards the cloud centre. Only for large-beam observations, they differ in the functions in Eq. (\[eq\_tmb\_lvg\]) and (\[eq\_tmb\_ep\]), which are either $\exp(-2\tau)$ or $e(-2\tau)$, but result in similar values.
By setting up a table of beam temperatures from Eqs. (\[eq\_tmb\_lvg\]) and (\[eq\_tmb\_ep\]) and comparing the observed line intensities with the tabulated values we can derive three parameters from the observations: the kinetic temperature $T{_{\rm kin}}$ and the gas density $n{_{\rm H_2}}$ providing mainly the source function, and the column density of the considered molecules relative to the line width $N/\Delta v$ providing the photon escape probability.
SimLine - A one-dimensional radiative transfer code
===================================================
The radiative transfer problem
------------------------------
[*SimLine*]{} solves the line radiative transfer problem discussed in Appx. A.1 in a spherically symmetric configuration by means of a $\lambda$-iteration. The code is similar to the concept described by [@Dickel] but it contains several extensions and achieves a higher accuracy from an adaptive discretisation of all independent quantities.
[*SimLine*]{} integrates the radiative transfer equation (\[eq\_transfer\]) for a number of rays numerically. In spherical symmetry it is sufficient to consider the propagation of radiation in one arbitrary direction which is taken as $z$ here. The integral is computed stepwise from $z_{i-1}$ to $z_{i}$ $$\begin{aligned}
I_\nu(p,z_i) &=& \exp \left( -\int_{z_{i-1}}^{z_i} \!\!\! dz\;
\kappa_\nu(p,z) \right) \Bigg[ I_z(\nu,p,z_{i-1}) \nonumber \\ &
+&\left. \int_{z_{i-1}}^{z_i} \!\!\! dz\;\epsilon_\nu(p,z) \exp \left( \int_{z_{i-1}}^z
\!\!\! dz' \kappa_\nu(p,z') \right) \right]
\label{transfer}\end{aligned}$$ where $p$ denotes the displacement variable perpendicular to the $z$ direction. To minimise the discretisation error the integral does not use the source function but only the emission and absorption coefficients which are linear in the level populations. They are assumed to change linearly between the grid points and the exact integration formula for a linear behaviour (which is not given here, but can be derived straight forward) is applied. This approach provides a reduction of the integration error to third order. The radial grid is dynamically adjusted to give a maximum variation of the level populations between two neighbouring points below a certain limit. In case of strong velocity gradients additional points are included on the $z$-scale for a sufficiently dense sampling of changes in the profile function. The incident radiation at the outer boundary of the cloud is assumed to follow a black body spectrum.
In spherical symmetry the spatial integration of the radiative energy density (Eq. \[ufirst\]) can be reduced to $$\begin{aligned}
u(r) &=& {2\pi\over r c} \int_{-r}^r dz' \int_{-\infty}^\infty d\nu\;
\Phi_\nu(p',z') I_\nu(p',z') \\
{\rm with}&& p'=\sqrt{r^2-z'^2} \nonumber
\label{uzint} \end{aligned}$$ The grid of rays tangential to the radial grid is refined by additional rays at intermediate $p'$ values to guarantee a sufficiently dense sampling on the $z'$ scale The integration uses a cubic spline interpolation.
With the values of the radiative energy density at each radial point and for each transition, the system of balance equations can be solved providing new level populations. Here, a LU decomposition algorithm with iterative improvement ([@Press]) is used. The new level populations are used in the next iteration as input for the radiative transfer equation. The whole $\lambda$-iteration scheme is solved using the convergence accelerator introduced by [@Auer87].
Depending on the physical situation the initial guess is either the optically thin limit, thermalisation or the solution of the radiative transfer equation using the LVG approximation (Eq. \[eq\_udir\]). The stability of the local radiation field is used as convergence criterion. The number of iterations required for convergence depends strongly on the optical depth of the model cloud. For the examples discussed in this paper only about a dozen iterations were necessary but other test cases with complex molecules like water, [[ ]{}non-monotonic]{} velocity gradients, and high optical depths require several hundred iterations.
The local turbulence approximation
----------------------------------
The turbulence description uses two additional parameters for each spatial point: the width of the velocity distribution $\sigma$ providing the local emission profile for optically thin lines and the correlation length of the macroturbulent density or velocity distribution $l{_{\rm corr}}$.
The width of the velocity distribution $\sigma$ is composed of a turbulent and a thermal contribution $$\sigma = {\nu_0 \over c} \,\sqrt{ {2kT{_{\rm kin}} \over m{_{\rm mol}}} + {2 \over 3}
\langle v{_{\rm turb}}^2 \rangle}$$ where a Maxwellian distribution of turbulent velocities is assumed. The relation between the FWHM and the variance of the turbulent velocity distribution is given by ${\rm FWHM}(v{_{\rm turb}})=\sqrt{8/3\times{\rm ln}2\;
\langle v{_{\rm turb}}^2 \rangle}$). The long range variation of the turbulence spectrum as described by means of a Kolmogorov or Larson exponent is simulated by a radially varying turbulent velocity dispersion $\sqrt{\langle v{_{\rm turb}}^2\rangle} \propto r^\gamma$. Exponents $\gamma$ between about 0.1 ([@Goodman]) and 0.7 (Fuller & Myers 1992) are observationally justified.
For the local treatment of coherent units in a turbulent medium the considered volume element is subdivided into numerous clumps with a thermal internal velocity dispersion. When each clump is characterized by a Gaussian density distribution of molecules with about the same velocity $ n(r)=n_0\times\exp(-r^2/r{_{\rm cl}}^2) $ the effective absorption coefficient at the considered velocity for the whole medium is $$\kappa{_{\rm eff}}=n{_{\rm cl}} \times \pi r{_{\rm cl}}^2 \int_0^{\tau{_{\rm cl}}}
{1-\exp(-\tau) \over \tau} d\tau$$ where $n{_{\rm cl}}$ is the number density of contributing cells and $\tau{_{\rm cl}}=\sqrt{\pi}\,\kappa r{_{\rm cl}}$ is their central opacity (Martin et al. 1984). As the clumps size $r{_{\rm cl}}$ is the length on which the abundance of molecules within the same thermal velocity profile is reduced by the factor $1/e$, we can compute it from the correlation length of the velocity or density structure by $ r{_{\rm cl}}=l{_{\rm corr}}\times \sigma{_{\rm th}} /\sigma$.
When the turbulent velocity dispersion $\sigma$ is at least three times as large as the thermal velocity dispersion $\sigma{_{\rm th}}$, we obtain an effective absorption coefficient $$\kappa{_{\rm eff,\nu}}=n{_{\rm ges}} \pi r{_{\rm cl}}^2 \times A(\tau{_{\rm cl}})
\times {\sigma{_{\rm th}} \over \sigma} \exp{}\left(-{(\nu-\nu_0)^2
\over \sigma^2}\right)
\label{eq_keff}$$ with $$A(\tau)={1\over \sqrt{\pi}}\int_{-\infty}^{\infty}dv
\int_0^{\tau \exp(-v^2)} {1-\exp(-\tau') \over \tau'} d\tau'$$ Here, $n{_{\rm ges}}$ is the total number density of clumps. In case of incompressible turbulence, i.e. clumping in velocity space, it is equal to the reciprocal cell volume. For small values of the clump opacity, $A(\tau{_{\rm cl}})$ is identical to $\tau{_{\rm cl}}$ and we reproduce the microturbulent limit. For large $\tau{_{\rm cl}}$, the function $A(\tau{_{\rm cl}})$ saturates and we obtain a significant reduction of the effective absorption coefficient. In case of density clumps, $\kappa{_{\rm eff}}(\nu)$ is further reduced by the filling factor entering $n{_{\rm ges}}$. In [*SimLine*]{}, this is simulated by a corresponding artificial reduction of the molecular abundance.
The central [H[II]{} ]{}region
------------------------------
To simulate the effect of a central continuum source in the cloud, it is possible to assume an [H[II]{} ]{}region in the cloud centre. The [H[II]{} ]{}region is characterised by two parameters, the electron density $n{_{\rm e}}$ and the kinetic electron temperature $T{_{\rm e}}$.
The absorption coefficient for electron-ion bremsstrahlung in the Rayleigh-Jeans approximation is given by: $$\kappa_\nu={8 \over 3\sqrt{2\pi}}\; {e^6 \over (4\pi\epsilon_0 m{_{\rm e}})^3 c}
\left(n{_{\rm e}} \over \nu\right)^2 \left(m{_{\rm e}}\over k T{_{\rm e}}\right)^{3/2}
{\rm ln}\Lambda
\label{eq_hii_kappa}$$ where it is assumed that the gas is singly ionised and $\Lambda$ is given by $$\Lambda=\left(2k T{_{\rm e}} \over \gamma m{_{\rm e}}\right)^{3/2}
{4\pi \epsilon_0 m{_{\rm e}} \over \pi\gamma e^2 \nu} \approx
4.96\,10^7 \left(T\over {\rm K}\right)^{3/2} {{\rm Hz} \over \nu}$$ for $T{_{\rm e}}<3.6\,10^5$K. The quantities $e$ and $m{_{\rm e}}$ denote the electron charge and mass, $c$ is the vacuum light velocity, and $\gamma=1.781$ (Lang 1980).
For a thermal plasma, the emission coefficient follows from the Planck function $$\epsilon_\nu=\kappa_\nu\times B_\nu(T{_{\rm e}})\;.
\label{eq_hii_ep}$$
In the radiative transfer computations the frequency dependence of these continuum coefficients is neglected within the molecular line width. Within the [H[II]{} ]{}region, we substitute the molecular coefficients in Eq. (\[transfer\]) by the quantities from Eqs. (\[eq\_hii\_kappa\]) and (\[eq\_hii\_ep\]) so that we locally switch to a continuum radiative transfer. \[sect\_hii\_region\]
Computation of beam temperatures
--------------------------------
When the level populations are known, the beam temperature relative to the background is computed by the convolution of the emergent intensity with the telescope beam. $$T{_{\rm mb}}={c^2 \over 2 k \nu_0^2} \;{\displaystyle \int_0^{2\pi} \!\!d\phi
\int_0^\infty \!\!dp \;p (I_\nu^S(p)-I{_{\rm bg}} )
f{_{\rm mb}}(p,\phi) \over \displaystyle
\int_0^{2\pi} \!\!d\phi \int_0^\infty \!\!dp\;
p f{_{\rm mb}}(p,\phi)}
\label{beam}$$ The intensity $I_\nu^S(p)$ is the value on the cloud surface $I_\nu^S(p)=I_\nu(p,\sqrt{R{_{\rm cloud}}^2-p^2})$. We assume a Gaussian profile for the telescope beam $$f{_{\rm mb}}(p,\phi)=
\exp{}\left(-(p-p{_{\rm offset}})^2(1+\phi^2)^2 \over \sigma{_{\rm mb}}^2\right)$$ The projected beam width is computed from the angular width by $\sigma{_{\rm mb}} = \pi/648000\, D\, \sigma{_{\rm mb}}['']
= 2.912\,10^{-6}\, D\, {\rm FWHM} ['']$ where $D$ is the distance of the cloud. The program computes a radial map with arbitrary spacings.
The general code design
-----------------------
The design of the code is directed towards a high accuracy of the computed line profiles. All errors in the different steps of the program are explicitly user controlled by setting thresholds. All discretisations necessary to treat the problem numerically are performed in an adaptive way, i.e. there is no predefined grid and all grid parameters will change during the iteration procedure. The system of balance equations is truncated whenever the excitation of all higher levels falls below a chosen accuracy limit.
Furthermore, the code was pushed towards a high flexibility, i.e. the ability to treat a very broad range of physical parameters with the same accuracy and without numerical limitations. The systematic velocities e.g. may range from 0 to several times the turbulent velocity and the optical depths may vary from negative values for weak masing to values of several thousands.
The program is not optimised towards a high speed. Other codes with lower inherentaccuracy may easily run a factor 10 faster and further improvements are possible. Nevertheless, the code is suitable for an interactive work even on a small PC with execution times of a few seconds for the models considered in this paper.
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[^1]: Unfortunately, [@Plume] listed only the peak intensity and FWHM so that we have to assume a Gaussian for the CS 10–9 profile here.
|
---
abstract: |
Events with isolated leptons play a prominent role in signatures of new physics phenomena at high energy collider physics facilities. In earlier publications, we examine the standard model contribution to isolated lepton production from bottom and charm mesons and baryons through their semileptonic decays $b, c \rightarrow l +
\mathrm{X}$, showing that this source can overwhelm the effects of other standard model processes in some kinematic domains. In this paper, we show that we obtain good agreement with recent Tevatron collider data, both validating our simulations and showing that we underestimate the magnitude of the heavy-flavor contribution to the isolated lepton yields. We also show that the isolation requirement acts as a narrow bandpass filter on the momentum of the isolated lepton, and we illustrate the effect of this filter on the background to Higgs boson observation in the dilepton mode. We introduce and justify a new rule of thumb: isolated electrons and muons from heavy-flavor decay are produced with roughly the same distributions as $b$ and $c$ quarks, but with 1/200 times the rates of $b$ and $c$ production, respectively.
author:
- Zack Sullivan
- 'Edmond L. Berger'
date: 'May 14, 2010'
title: 'Isolated leptons from heavy flavor decays: Theory and data'
---
Introduction {#sec:introduction}
============
[*Isolated*]{} leptons along with missing transverse energy ${{\slash\!\!\!\!E_T}}$ are typical signatures for new physics processes at collider energies. A much-anticipated example of charged dilepton production is Higgs boson decay, $H \rightarrow W^+ W^-$ followed by purely leptonic decays of the $W$ intermediate vector bosons. Charged trilepton production may arise from the associated production of a chargino $\tilde
{\chi}_1^{\pm}$ and a neutralino $\tilde{\chi}^0_2$ in supersymmetric (SUSY) models, followed by the leptonic decays of the chargino and neutralino.
There are many standard model (SM) sources of isolated leptons. The nature and magnitude of contributions from semileptonic decays of heavy flavors (bottom and charm quarks) are emphasized in two papers [@Sullivan:2006hb; @Sullivan:2008ki]. The role of heavy flavor backgrounds in $H \rightarrow W^+ W^-\rightarrow l^+ l^- +
{{\slash\!\!\!\!E_T}}$ at Fermilab Tevatron and CERN Large Hadron Collider (LHC) energies is presented in Ref. [@Sullivan:2006hb]. We simulate the contributions from processes with $b$ and $c$ quarks in the final state, including $b \bar{b} X$, $c \bar{c} X$, $W c$, $W b$, $W b
\bar{b}$. In Ref. [@Sullivan:2008ki], we study the signal and backgrounds for $\tilde {\chi}_1^{\pm} \tilde{\chi}^0_2 \rightarrow
l^+ l^- l^{\pm} + {{\slash\!\!\!\!E_T}}$. We include heavy-flavor contributions to the backgrounds from $b Z/\gamma^*$, $b \bar {b} Z/\gamma^*$, $c
Z/\gamma^*$, $c \bar {c} Z/\gamma^*$, $b \bar {b} W$, and $c \bar {c}
W$. We learn that isolation cuts do not generally remove leptons from heavy-flavor sources as backgrounds to multilepton searches. A sequence of complex physics cuts is needed, conditioned by the new physics one is searching for. Moreover, the heavy-flavor backgrounds cannot be easily extrapolated from more general samples. The interplay between isolation and various physics cuts tends to emphasize corners of phase space rather than the bulk characteristics. While the heavy-flavor backgrounds can be overwhelming, we propose specific new cuts that can help in dealing with them, and we suggest methods for [*in situ*]{} verification of the background estimates.
Our finding that the dominant backgrounds to low-momentum dilepton and trilepton signatures come from real $b$ and $c$ decays may be met with some skepticism. Since our publications, important Collider Detector at Fermilab (CDF) data [@Aaltonen:2008my] have appeared that allow us to make a quantitative comparison at Tevatron collider energies. We report the results of our comparison in this paper. Specifically, these data allow us to verify how well we model isolation of leptons for events classified as originating from the “Drell-Yan” processes, in which virtual photons $\gamma^*$ and intermediate vector bosons $W$ and $Z$ are produced and decay into leptons. In addition, since we predict the absolute rate of heavy-flavor production and the Drell-Yan processes, we can check whether the data agree with our prediction of the rates of isolated leptons from these sources.
In Sec. \[sec:compare\], we present our detailed comparison with the CDF data, using the same control regions defined in their study, and using the same detector simulations and event generation methods of our previous papers [@Sullivan:2006hb; @Sullivan:2008ki]. We obtain good agreement with the CDF data, both validating our simulations and showing that our estimates of the magnitude of the heavy-flavor contribution are conservative.
The added confidence in our understanding of the backgrounds from heavy-flavor sources motivates another look at one aspect of our study of $H \rightarrow W^+ W^-\rightarrow l^+ l^- + {{\slash\!\!\!\!E_T}}$. Our results show a sharp fall of the contribution from heavy-flavor decays at large values of the dilepton transverse mass distribution. This falloff is too steep to reflect only the drop with transverse momentum of the cross section for heavy-flavor production. In Sec. \[sec:isofilter\], we explain how isolation serves as a narrow bandpass filter on the momentum of the leptons, thus explaining the steep decrease at high mass.
A discussion of the implications of our results is found in Sec. \[sec:conclusions\]. We utilize the effect of the bandpass filter on $b$ and $c$ decays as a way to develop a simple rule-of-thumb that 1/200 of every produced bottom or charm quark is seen as an isolated muon, and another 1/200 is seen as an isolated electron, each with roughly the momentum of the heavy quark. We end this introduction with a general discussion of isolation and a summary of its effects.
Given a lepton track and a cone of size $\Delta R$ in rapidity and azimuthal angle space, the lepton is said to be [*isolated*]{} if the sum of the transverse energy of all other particles within the cone is less than a predetermined value (either a constant or a value that scales with the transverse momentum of the lepton). Our simulations based on the known semileptonic decays of bottom and charm hadrons show that leptons which satisfy isolation take a substantial fraction of the momentum of the parent heavy hadron. Moreover, isolation leaves $\sim 7.5 \times 10^{-3}$ muons per parent $b$ quark. The potential magnitude of the background from heavy-flavor decays may be appreciated from the fact that the inclusive $b \bar{b}$ cross section at LHC energies is about $5 \times 10^8$ pb. A suppression of $\sim
10^{-5}$ from isolation of two leptons still leaves a formidable rate of isolated dileptons. For the isolated leptons, our simulations show that roughly $1/2$ of the events satisfy isolation because the remnant is just outside whatever cone is used for the tracking and energy cuts, and another $1/2$ pass because the lepton took nearly all the energy, meaning there is nothing left to reject upon. The latter events are not candidates to reject with impact parameter cuts since they tend to point to the primary vertex. Although the decay leptons are “relatively” soft, we find that their associated backgrounds extend well into the region of new physics with relatively large mass scales, such as a Higgs boson with mass $\sim 160$ GeV.
Comparison to CDF {#sec:compare}
=================
In their analysis of isolated leptons from $b$ decay [@Aaltonen:2008my; @CDFweb], CDF defines several control regions in order to disentangle the effects of different backgrounds. In Fig.\[fig:regions\], we reproduce Fig. 5 of their paper [@Aaltonen:2008my] displaying the various regions. In order, control region Z is the $Z$-boson resonance, and corresponds to an opposite-sign dimuon invariant mass acceptance of $76 < M_{\mu^+\mu^-}
< 106$ GeV. Region A is a low missing transverse energy region, with ${{\slash\!\!\!\!E_T}}< 10$ GeV and $M_{\mu^+\mu^-} > 10.5$ GeV. The CDF region S is designed as a signal region for their trilepton study (but examines dimuons here) with ${{\slash\!\!\!\!E_T}}> 15$ GeV, $\le 1$ jet, $M_{\mu^+\mu^-} >
15$ GeV, and excludes region Z. Regions B, C, and D complement region S: region B is a subset of region Z, with ${{\slash\!\!\!\!E_T}}> 15$ GeV; region C is a subset of region A with $M_{\mu^+\mu^-} > 15$ GeV and excludes the overlap with region Z; and region D is the same as region S, but requires at least 2 jets.
{width="\columnwidth"}
Our goal is to compare directly with the CDF measurement of isolated leptons from $b$ decay [@Aaltonen:2008my; @CDFweb] in each of the control regions defined above using exactly the same detector simulations and methods as our previous papers [@Sullivan:2006hb; @Sullivan:2008ki]. We concentrate on the contributions from $b\bar b$ pair production, with the semileptonic decay $b\to\mu+X$, and the Drell-Yan process $p\bar p\to
\gamma^*/Z+X$. We also consider the contributions from $W$ bosons plus heavy flavors ($Wc$, $Wb$, $Wc+\mathrm{jet}$, $Wb+\mathrm{jet}$, $Wb\bar b$, and $Wc\bar c$), though they contribute significantly only to region S. We do not include muons from light-quark jets, such as $K$ or $\pi$ decays, which are usually classified as “fakes.” We also do not predict the rate of muons from $c\bar c$, as this was not separately identified by CDF.
We generate events with a customized version of MadEvent 3.0 [@Maltoni:2002qb] and run them through the PYTHIA 6.327 [@PYTHIA] showering Monte Carlo. Both programs use the CTEQ6L1 parton distribution functions [@Pumplin:2002vw] evaluated via an efficient evolution code [@Sullivan:2004aq]. The showered events are fed through a version of the PGS 3.2 [@Carena:2000yx] fast detector simulation, modified to match CDF geometries, efficiencies, and detailed reconstruction procedures [@Acosta:2005mu]. At the level of individual reconstructed leptons and jets we reproduce CDF full detector simulations and data acceptance to a few percent. In Fig. \[fig:mll\] we show the dilepton invariant mass distribution in region A. This figure compares well with Fig. 6c of the CDF analysis [@Aaltonen:2008my], indicating that our predictions of the shapes agree with the data.
![Dimuon invariant mass distribution from Drell-Yan and heavy-flavor production in region A. \[fig:mll\]](Fig2.ps){width="3in"}
An important point regarding our predictions is that they are absolutely normalized. We apply $K$-factors to the leading order rates of $1.4$ for Drell-Yan, and $1.4$ for $b\bar b$ production[^1]. These $K$-factors are calculated via the NLO program MCFM [@MCFM] including cuts that mimic the final sample in order to more accurately predict the result in the region of CDF acceptance.
In order to compare to the CDF data, we scale our results to those observed in region Z, the $Z$-peak, where we do not expect any signal from $b\bar b$ events. Theoretical errors due to parton distribution functions and Monte Carlo statistics are included, but they are smaller than the error propagated from the experimental measurement of region Z used to normalize the data. We see in Table \[tab:results\] that with this scaling we reproduce well the results for Drell-Yan production in control regions A and C — the regions with the most statistical significance.
------------------- ----------------- ---------------- ----------------- ------------------
\[-0.5ex\] Region DY $b\bar b$ DY $b\bar b$
Control Z $6419\pm 709$ — $6419\pm 752$ —
Control A $14820\pm 2242$ $9344\pm 1621$ $14222\pm 1615$ $5118\pm 584$
Control B $217 \pm 25$ — $58.9 \pm 24.9$ —
Control C $5770\pm 1043$ $2238\pm 384$ $4898\pm 584$ $924\pm 117$
Control D $7.8 \pm 1.5$ $9 \pm 4$ $9.8 \pm 9.9$ —$^*$
Control S $169 \pm 30$ $90 \pm 20$ $226 \pm 53.2$ $26 \pm 10^\dag$
------------------- ----------------- ---------------- ----------------- ------------------
: Comparison between the number of the Drell-Yan (DY) and $b\bar b$ events observed by CDF (from Table III of Ref. [@Aaltonen:2008my]) in each control region, and our predictions. Dashes indicate no events are expected or observed. ($^*$) We could not obtain enough statistics in Control D to compare. ($^\dag$) Includes $10\pm 3$ events from $W+\mathrm{heavy\ flavors}$ (mostly $Wc$). \[tab:results\]
Once we have normalized to the observed luminosity, we find that our predictions systematically *underestimate* the number of isolated leptons from $b$ decays in the overlapping control regions (A and C) by nearly a factor of two. This result suggests that our analysis has been conservative in estimating this generally ignored source of contamination. Part of the observed difference may be due to the small $K$-factor of $1.4$ we use to estimate the absolute normalization. However, even a $K$-factor of 2 would leave a systematic underestimate of $1.4$. The remainder could be due to contamination from $c\bar c$ decays that produce isolated muons on one side of the event, and resemble $b$’s on the other side. Regardless, the background is not only a large fraction of the entire data set, but it is more significant than the estimations of Refs.[@Sullivan:2006hb; @Sullivan:2008ki].
In our main analysis, we do not consider regions D and S of the CDF study significant for two reasons. First, the statistics are too small to make a definitive statement. Second, the choice of regions is sensitive to the reconstruction of missing transverse energy ${{\slash\!\!\!\!E_T}}$, which is difficult to model at the level of 10–15 GeV. One concern may be that events have “slipped” from the low ${{\slash\!\!\!\!E_T}}$ regions to the high ones. If this were due to a systematic shift in our reconstruction it should also appear in the Drell-Yan sample. We see a hint of this in our Drell-Yan estimate for region S. However, we see the opposite effect in isolated leptons from heavy-flavor decays, where our prediction of 16 events from $b\bar b$ and 10 events from $W+\mathrm{heavy\ flavors}$, underestimates the measured CDF region by a factor of 3.5.[^2] The underestimate of isolated leptons from heavy-flavor decays in region S is consistent with our reduced estimate of Drell-Yan in region B.
Given that region Z overlaps both regions C and S, a comparison that should reduce the sensitivity to ${{\slash\!\!\!\!E_T}}$ would be between the sums of these regions. In that case, our Drell-Yan result (C$+$S) of $5124
\pm 586$ events does agree a bit better with the CDF observation of $5939 \pm 1043$, but in both cases is within $1\sigma$. If we consider isolated leptons from heavy-flavor decays, we find regions (C$+$S) have $950 \pm 120$ events, which is still approximately a factor of two smaller than the combined CDF regions of $2328 \pm 385$. Hence, it appears unlikely that misestimations of ${{\slash\!\!\!\!E_T}}$ or $W+\mathrm{heavy\ flavor}$ decays, are responsible for our systematic underestimate of the heavy-flavor background.
Figure \[fig:mll\] demonstrates that the bulk of the CDF dimuon sample from heavy flavors is composed of muons with transverse momentum $p_T$ less than 10 GeV. One limitation of our study is that our original detector simulation was constructed and tuned for leptons with $p_T > 10$–20 GeV. It is somewhat surprising we are able to model the detector response as well as we do. Perhaps the $Z$ region, where we trust our detector simulation, is not representative enough of the low dilepton invariant-mass region. To this end, we also consider in Table \[tab:region\] normalizing our $b\bar b$ contribution in each region independently. We do this by extracting the $K$-factor necessary to exactly scale our Drell-Yan calculation to the measured Drell-Yan rate, and applying that to our prediction of the $b\bar b$ contamination. Under this method, our predicted background from $b\bar b$ increases by less than one standard deviation, and it is still about a factor of 2 smaller than CDF data.
Region CDF Our study
----------- ---------------- -----------------
Control A $9344\pm 1621$ $5333 \pm 833$
Control C $2238\pm 384$ $1089 \pm 213$
Control S $90 \pm 20$ $20 \pm 8^\dag$
: Comparison between the number of $b\bar b$ events observed in CDF control regions, and our predictions, where each region is normalized separately by the ratio of the CDF measurement of Drell-Yan production in that region divided by our prediction. ($^\dag$) Includes $8\pm 3$ events from $W+\mathrm{heavy\ flavors}$ (mostly $Wc$). \[tab:region\]
We conclude this section with the statement that our estimate of the contamination to isolated lepton sample from heavy-flavor decays appears conservative.
Lepton isolation acts as a momentum filter {#sec:isofilter}
==========================================
Having experimental verification that isolated leptons from heavy flavor decays play an important role, we clarify one aspect that experimental reconstruction has on the spectrum of these leptons. In Sec. II of Ref. [@Sullivan:2008ki], we explain in detail how leptons from $b$ and $c$ decays pass isolation cuts. Briefly, the probability to produce a lepton above some threshold, e.g., $p_{T\mu}
> 10$ GeV is convolved with the probability of missing the rest of the $b$ or $c$ decay remnant. For completeness, we reproduce the shape of this efficiency using an ATLAS-like detector simulation for muons as a function of the $p_{Tb}$ of the quark in Fig. \[fig:muvptb\].
![Normalized probability for a $b$ quark to produce an isolated muon with $p_{T\mu}>10$ GeV (solid) vs. the $b$ transverse momentum [@Sullivan:2008ki]. This curve is a multiplicative combination of the probability of producing a muon with $p_{T\mu}>10$ GeV (dotted) and the probability the muon will be isolated (dashed). The $b$ production spectrum is not included. \[fig:muvptb\]](Fig3.eps){width="3in"}
When the acceptance function is folded with a typical $b$ transverse momentum spectrum — whether from $b\bar b$ production, or any other spectrum that falls with $p_{T}$ — the peak of the resulting cross section of isolated muons comes from $b$’s whose $p_T$ is just above the muon $p_T$. This effect is clearly present in Fig.\[fig:muvbb\], where the peak for muons with $p_{T\mu} > 10$ GeV in $b\bar b$ production comes from 20 GeV $b$’s. Previously [@Sullivan:2008ki], we focused on the low-momentum end of this spectrum, noting that a significant fraction of isolated leptons arise from $b$’s just above threshold for production. Hence, to properly model this background, $b$ and $c$ must be modeled all the way down to threshold.
![Cross section for production of a muon of $P_{T\mu} > 10$ GeV from $b\bar b$ production and decay (solid), or an isolated muon (dashed). \[fig:muvbb\]](Fig4.ps){width="3in"}
In this section, we focus on the upper end of the isolated lepton spectrum as a function of $b$ (or $c$) transverse momentum. The long tail of acceptance in Fig. \[fig:muvptb\] is suppressed by the sharply falling $b$ transverse momentum spectrum. The net result is that the isolation acts as a *narrow bandpass filter on momentum*. This observation has a critical importance for higher-scale physics, such as Higgs boson production.
In the dilepton analysis of a Higgs boson decaying to $W^+W^-$ to dileptons we observe that the transverse-mass $M_T$ spectrum due to $b\bar b$/$c\bar c$ and $W+$heavy flavors drops sharply at large $M_T$. In Fig. \[fig:mthmm\](a) the background from heavy flavors falls through the middle of Higgs boson signals for masses in the range 140–200 GeV. Raising the $p_T$ threshold of the *second-highest $p_T$ lepton* from 10 GeV to 20 GeV pushes this leading edge to lower transverse mass, Fig. \[fig:mthmm\](b), thereby recovering our ability to extract a Higgs signal in the dilepton channel despite the heavy-flavor background.
Considering lepton isolation criteria as a *narrow bandpass filter*, we see that the sharp high-$M_T$ edge is due to the cutoff of large-momentum $b$’s by this filtering mechanism. Hence, even though we are modeling the tail of a steeply falling spectrum, the filtering effect of lepton isolation acts to safely suppress the region where we expect our $M_T$ shapes to be less-well defined. The net result of this filter is to provide a more robust determination of the shape of the background from isolated leptons.
Discussion {#sec:conclusions}
==========
In this paper we predict the rate of isolated muons from b decay in order to compare directly with data from the CDF Collaboration. The central conclusion to draw is that isolated leptons from $b$ decays are a large fraction of the low transverse momentum lepton sample. In the case of the CDF measurement we *under-predict* the measured rate of dimuons from $b$ decay using the same codes and procedures we use to estimate the background to dilepton and trilepton signatures at the Tevatron and LHC [@Sullivan:2006hb; @Sullivan:2008ki]. Hence, we are confident that these backgrounds will play an important role in the extraction of Higgs boson decays to $WW$, trilepton supersymmetry, and indeed all processes with any low transverse momentum electron or muon.
Given the broad nature of our conclusions, and the significant computing resources required to model these backgrounds properly, we introduce a new “rule-of-thumb” to determine whether these leptons may be problematic in any given analysis:
- Replace 1/200 of every produced $b$ or $c$ quark with a muon, and 1/200 with an electron having the same momentum as the $b$ or $c$.
- If the resulting background is more than 10% of the signal, it should be simulated more carefully, and eventually measured *in situ*.
This new rule-of-thumb works precisely because the lepton isolation criteria act as a bandpass filter selecting leptons from $b$ or $c$ quarks whose transverse momenta are only slightly above the momentum of the lepton. In general, this rule-of-thumb is valid for large transverse momentum leptons as well. Fortunately, the production rates for $b$ and $c$ quarks tend to fall rapidly (with the exception of $b$ decays from top quarks, which peak near 50–60 GeV). Overall, we strongly recommend that all analyses involving leptons consider the background from the decay of heavy-flavor hadrons, as many analyses are sensitive to regions of phase space in which this background is enhanced.
E. L. B. is supported by the U. S. Department of Energy under Contract No. DE-AC02-06CH11357. We gratefully acknowledge the use of JAZZ, a 350-node computer cluster operated by the Mathematics and Computer Science Division at Argonne as part of the Laboratory Computing Resource Center. We wish to thank John Strologas for discussions regarding details of the CDF data.
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[^1]: Note, this factor is smaller than the $K$ factor of $2.0$ typically assumed for $b\bar b$, and that would be obtained for a different choice of scale, and a more inclusive measurement.
[^2]: Using the lower estimate of 70 events from the CDF measurement and our upper estimate of 36 events leads to the same factor of 2 under-prediction observed in other regions.
|
---
abstract: 'In this paper we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.'
author:
- 'Marcello Carioni [^1]'
- 'Alessandra Pluda [^2]'
bibliography:
- 'cali\_c\_p.bib'
date:
-
-
title: Calibrations for minimal networks in a covering space setting
---
[**Keywords**: Minimal partitions, Steiner problem, covering spaces, calibrations]{}
[**Mathematics Subject Classification (2010):** 49Q20, 49Q05, 57M10]{}
Introduction
============
The classical Steiner problem, whose first modern formulation can be found in [@courant], can be stated as follows: given a collection $S$ of $m$ points of ${\mathbb{R}}^n$, find the connected set that contains $S$ with minimal length, namely $$\label{ste}
\inf \{{\mathcal{H}}^1(K) : K \subset {\mathbb{R}}^n, \mbox{ connected and such that } S \subset K\}\,$$ (we refer to [@Ivan] for a survey on the topic). In its highest generality the problem can be stated replacing the ambient space ${\mathbb{R}}^n$ with any metric space [@paopaostepanov]. From a computational point of view the Steiner problem is NP-hard, hence, it is interesting to attack it in new ways (for example by multiphase approximations à la Modica–Mortola [@belpaover; @orlandi; @sant2; @chambolle; @sant]) in order to improve the rate of convergence of the algorithms. From a theoretical perspective the solution of the Steiner problem by variational methods has received an increasing interest starting from several results of the 90’s, which establish an equivalence between the Steiner problem and the minimal partition problem [@ab1; @ab2; @morel; @tamanini], and ending to more recent approaches, such as currents or vector valued BV functions defined on a covering space [@cover; @brakke], currents with coefficients in a group [@annalisaandrea], rank-one tensor valued measures [@orlandi].
The first approach via covering space to the Steiner problem, and more in general to Plateau’s type problems, is due to Brakke [@brakke]. Having in mind a possible candidate minimizer (unoriented soap films, soap films with singularities, soap films touching only part of a knotted curve) for a certain Plateau’s type problem he introduces a double covering called *pair covering space* chosen compatibly to the soap film he wants to obtain as a minimizer. Then he minimizes the mass of integral currents defined on it, and only in some special cases, via a calibration argument, he proves that the minimizer coincides with his candidate. As a consequence of this *ad hoc* approach, he did not give an explicit proof of the equivalence with the Steiner problem; however his setting allows to describe a great variety of different objects, for istance soap films in higher dimension. In [@cover] the authors revive Brakke’s covering space approach, constructing an $m$–sheeted covering space $Y_\Sigma$ of $\mathbb{R}^2\setminus S$ and minimizing the total variation of vector valued $BV$ functions on $Y_\Sigma$ satisfying a certain constraint. In this setting the authors prove the equivalence between their minimization problem and the Steiner problem in the plane.
The first part of this paper is devoted to an improvement of the result in [@cover] reducing the vector valued problem in [@cover] to a scalar one: we minimize the *perimeter* in a family of finite perimeter sets in $Y_\Sigma$ instead of the total variation of constrained vector valued $BV$ functions. In particular we state the following minimization problem in $Y_\Sigma$: $$\label{minpro2}
\mathscr{A}_{constr}(S) = \inf \left\{P(E) : E\in\mathscr{P}_{constr}(Y_\Sigma)\right\}\,,$$ where $\mathscr{P}_{constr}(Y_\Sigma)$ is the space of sets of finite perimeter in $Y_\Sigma$ satisfying a suitable constraint. Once proved the existence of minimizers we show an equivalence between our minimization problem and the classical Steiner problem (we refer the reader to the beginning of Section \[equivalence\] for further explanations).
In the second part of the paper we introduce a notion of calibration suitable to our setting. In the context of minimal surfaces, a *calibration* for a $k$–dimensional oriented manifold in ${\mathbb{R}}^{n+1}$ is a closed $k$-form $\omega$ such that $\vert\omega\vert\leq 1$ (the so–called size condition) and $\langle \omega, \xi\rangle = 1$, where $\xi$ is the unit $k$-vector orienting the manifold. It is easy to see that the existence of a calibration for a certain manifold implies that the manifold is area minimizing in its homology class. In Definition \[caliconvering\] we adapt this notion to our setting taking advantage of the theory of Null-Lagrangians [@ab1; @ab2]: a calibration for $E \in \mathscr{P}_{constr}(Y_\Sigma)$ is a divergence-free vector field defined on the covering space $Y_\Sigma$ such that $\int_{Y_\Sigma} \Phi \cdot D\chi_E = P(E)$ and a suitable size condition for $\Phi$ is fulfilled (see Remark \[twoinsteadofone\]). As for minimal surfaces, we show that the existence of a calibration for a set $E\in \mathscr{P}_{constr}(Y_\Sigma)$ implies that it is a minimizer to . In order to show the usefulness of our theory we give the explicit examples of calibrations for the Steiner minimal configuration connecting two points, three points located at the vertices of any triangle and for the four vertices of a square.
A notion of calibration for the partition problem was firstly introduced by Morgan and Lawlor in [@lawmor]: their *paired calibration* technique allows to prove the minimality of soap films among all the competitors that split the domain in a fixed number of regions. In the context of the Steiner problem the limit of this approach is that it can be applied only when the points of $S$ belong to the boundary of a convex set. As mentioned previously, a notion of calibration, adapted to the covering space approach, is proposed also in [@brakke]. Finally, Marchese and Massaccesi in [@annalisaandrea] describe Steiner trees using currents with coefficient in groups and rephrase the Steiner problem as a minimum problem for the mass of currents. In this way they were able to introduce a related notion of calibration (see also [@coveringbis] for a comparison of the different notions). Both [@brakke] and [@annalisaandrea] have a companion paper devoted to numerical results (see [@brakke2] and [@MasOudVel16]). We underline that our approach, as the one introduced in [@brakke; @annalisaandrea], does not require that the points of $S$ lie on the boundary of a convex set.
The goal of the last part of the paper is to tackle the minimality of the Steiner minimal configurations for the vertices of a regular pentagon and of a regular hexagon using the theory of calibrations.
We remind that the explicit minimizers for the Steiner problem, if the points of $S$ are the vertices of regular $n$-gon, are well known. In particular for $n\geq 6$ the Steiner minimal network is the polygon without an edge. The first proof for the cases $n\leq 6$ and $n\geq 13$ is due to Jarnik and Kössler in 1934 [@jarnik]. Fifty years later, Du, Hwang and Weng proved the remaining cases [@Hwang].
The main purpose of searching for a calibration is to have an easy argument to show the minimality of a certain candidate. Unfortunately finding a calibration is in general not an easy task and only very few example are known even for the Steiner problem. As already anticipated, in this work we propose a calibration argument to prove in a strikingly easy way the minimality for the Steiner minimal configuration connecting the vertices of the regular pentagon and of the regular hexagon.
The interest of our technique goes beyond these specific results because it can be generalized to arbitrary configurations of points in ${\mathbb{R}}^2$ and suggests how to “decompose" the Steiner problem in several simpler convex problems that can be solved (and calibrated) separately. Moreover the authors believe that a similar idea could be applied to different variational problems. The idea is to divide the set of competitors in different families, denoted by $\mathcal{F}(\mathcal{J})$, and define an appropriate notion of calibration in each family with a weaker size condition. To be more precise all the competitors that belong to the same family share a property related to the projection of their essential boundary onto the base set $M$: for certain couples of indices $(i,j)$ in $\{1,\ldots,m\}\times\{1,\ldots,m\}$ the intersection of the projections onto $M$ of the boundary of the part of the set $E$ in the $i$–th sheet and in the $j$–th sheet is $\mathcal{H}^1$–negligible. As a consequence, the definition of calibration in a family does not require to verify the size condition for the pairs of sheets associated to these couples of indices $(i,j)$. Once identified a candidate minimizer in each family, we calibrate them and in conclusion we compare their energy to find the explicit global minimizers of Problem . Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon (the hexagon without one edge) and of the regular pentagon.
Finally we outline the structure of the paper: in the beginning of Section \[equivalence\] we summarize the setting introduced in [@cover] describing the construction of the covering space and we define finite perimeter sets on it. In Subsection \[problem\] we introduce the space $\mathscr{P}_{constr}(Y_\Sigma)$ and prove the existence of minimizers for Problem . Then, in Theorem \[regularity\], we present a regularity result for the essential boundary of minimizers proving a *local* equivalence with the problem of minimal partitions [@ab1; @ab2; @morel; @tamanini]. We conclude Section \[equivalence\] proving the equivalence between the classical Steiner problem and our minimization problem .
In Section \[seccal\], after giving the definition of calibration, we show that the existence of a calibration for $E \in \mathscr{P}_{constr}(Y_\Sigma)$ implies minimality of $E$ with respect to . We construct explicit examples of calibration for the Steiner minimal configuration connecting two points, three points and the four vertices of a square.
In Section \[famiglie\] we develop the notion of calibration in families and use this tool to prove the minimality of the Steiner minimal configurations spanning the vertices of a regular hexagon and of a regular pentagon.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors are warmly grateful to Giovanni Bellettini and Maurizio Paolini for several discussions and helpful conversations. Once this manuscript was already completed, the authors have been told by Frank Morgan that the idea about separating the competitors into classes for the hexagon problem was found independently by Gary Lawlor and presented at Lehigh University in 1998, but was not published even in preprint form. The authors would like to thank Frank Morgan and Gary Lawlor for their remarks about the history of the problem and for their kind mail correspondence.
Equivalence between Problem and the Steiner problem {#equivalence}
====================================================
In this section we consider the construction of the $m$–sheeted covering space $Y_\Sigma$ of $M:=\mathbb{R}^2\setminus S$ presented in [@cover] and we define sets of finite perimeter on the covering. In particular we define the space $\mathscr{P}_{constr}(Y_\Sigma)$ of the sets of finite perimeter $E$ in $Y_\Sigma$ satisfying a suitable boundary condition at infinity and such that for almost every $x$ in the base space there exists exactly one point $y$ of $E$ such that $p(y)=x$, where $p$ is the projection onto the base space (see Definition \[zerounoconstr\]). Notice that for every set $E\in \mathscr{P}_{constr}(Y_\Sigma)$ it is possible to show a formula (Proposition \[dueh\]) that relates its perimeter to the $\mathcal{H}^1$-measure of the projection onto the base space $M$ of its essential boundary. In Subsection \[problem\] we state the minimization problem and we prove existence and non–triviality of the minimizers.
The last part of the section is devoted to the equivalence between our minimization problem and the classical Steiner problem. On one side it is enough to show that given a minimizer for Problem , the network obtained as the closure of the projection onto $M$ of the essential boundary is a competitor for the Steiner problem. Roughly speaking given the $m$ points of $S$ in $\mathbb{R}^2$, the covering space of $M$ is constructed in such a way that if we consider a closed curve $\gamma$ (namely a loop) with index one with respect to at most $m-1$ points and with index zero with respect to at least one point of $S$, then $p^{-1}(\gamma)$ is connected. Combining this property of the covering with the constraint on the set it is possible to show that the set $S$ is contained in a connected component of the closure of the projection of the essential boundary. On the other hand we describe a procedure to construct a set in $\mathscr{P}_{constr}(Y_\Sigma)$ from a minimal Steiner graph.
Construction of the covering space {#setcov}
----------------------------------
For the rest of the paper we consider $S = \{p_1,\ldots,p_m\}$ a finite set of points in $\mathbb{R}^2$ and $M \coloneqq {\mathbb{R}}^2 \setminus S$. Moreover we fix an open, smooth and bounded set $\Omega \subset {\mathbb{R}}^2$ such that $\mbox{Conv}(S)\subset \Omega$, where $\mbox{Conv}(S)$ is the convex envelope of $S$.
\[admissible\] We denote by $\mbox{Cuts}(S)$ the set of all $\Sigma \coloneqq \bigcup_{i=1}^{m-1} \Sigma_i \subset \Omega$ such that:
- for $i=1,\ldots,m-1$, $\Sigma_i$ is a Lipschitz simple curve starting at $p_i$ and ending at $p_{i+1}$;
- if $m>2$ then $\Sigma_i \cap \Sigma_{i+1}=\{p_{i+1}\}$ for $i=1,\ldots,m-2$;
- $\Sigma_i \cap \Sigma_l = \emptyset$ for any $i,l=1,\ldots,m-1$ such that $|l - i|>1$.
Moreover we denote by $\mbox{\textbf{Cuts}}(S)$ the set of all pairs ${\bf{\Sigma}} \coloneqq (\Sigma,\Sigma')$ such that
- $\Sigma,\Sigma' \in \mbox{Cuts}(S)$ and $\Sigma\cap \Sigma' = S$;
- if $m>2$, for every $i=2,\ldots,m-1$ let $C_i(p_i,\varepsilon)$ be a circle centered at $p_i$ with radius $\varepsilon$ such that $C_i(p_i,\varepsilon)\cap \Sigma_{i-1}\neq\emptyset$ and $C_i(p_i,\varepsilon)\cap \Sigma_{i}\neq\emptyset$. Denote by $x_i$ (resp $y_i$) the intersection of $C_i$ with $\Sigma_{i-1}$ (resp. with $\Sigma_{i}$). Then there exists an arc of $C_i$ connecting $x_i$ and $y_i$ and not intersecting $\Sigma'$.
Fix ${\bf\Sigma}=(\Sigma,\Sigma') \in \mbox{\textbf{Cuts}}(S)$ and define $$D\coloneqq\mathbb{R}^2 \setminus \Sigma\,, \quad\quad D'\coloneqq\mathbb{R}^2 \setminus \Sigma'\,$$ and $$X = \bigcup_{j=1}^m(D,j)\;\, \cup \bigcup_{j'=m+1}^{2m} (D',j')$$ that is the space made of $m$ disjoint copies of $D$ and of $D'$.
at (0,0) [{width="60.00000%"}]{};
Let $I_i$ be the open, bounded set enclosed by $\Sigma_i$ and $\Sigma_i'$ and $O = \mathbb{R}^2 \setminus \bigcup_{i=1}^{m-1} \overline{I_i}$. Given $(x,j) \in (D,j)$ with $j\in\{1,\ldots,m\}$ and $(x',j') \in (D',j')$ with $j'\in\{m+1,\ldots,2m\}$, we define the equivalence relation $\sim$ in $X$ as $(x,j)\sim (x',j')$ if and only if one of the following conditions holds: $$\label{ide2}
\left\{\begin{array}{ll}
j\equiv j'\ (\mbox{mod}\ m),& x=x' \in O\,, \\
j\equiv j'-i\ (\mbox{mod}\ m), & x=x'\in I_i,\ i=1,\ldots , m-1\,.
\end{array}
\right.$$
We define $Y_\Sigma$ to be the topological quotient space induced by $\sim$, i.e. $$Y_\Sigma\coloneqq X \Big{/}\sim\,.$$
Finally we denote by $\tilde \pi: X \rightarrow Y_\Sigma$ the projection induced by the equivalence relation, by $\pi$ the projection from $X$ to the space $M$ and by $p: Y_\Sigma \rightarrow M$ the map that makes the following diagram commutative: $$\xymatrix{
X \ar[dr]_{\pi} \ar[r]^{\widetilde{\pi}} & Y_{\Sigma} \ar[d]^{p}\\
& M}$$
The map $p: Y_\Sigma \rightarrow M$ is well-defined and the pair $(Y_\Sigma, p)$ is a covering space of $M$.
The covering $(Y_\Sigma,p)$ of $M$ inherits the local structure of $M$, hence, in our case, it is a Riemannian manifold.
In order to be able to work on Euclidean spaces we define the following natural local parametri-\
zations of $Y_\Sigma$.
\[local\] For every $j=1,\ldots\,m$ we define the local parame-\
trizations $\psi_j:D\to\widetilde{\pi}\left((D,j )\right)$ as $$\psi_j(x)\coloneqq \widetilde{\pi}\left(\left(x,j\right)\right)\;\;\text{for every}\;x\in D\,,$$ or equivalently as $$\psi_j(x)\coloneqq \left({{ \left.\kern-\nulldelimiterspace p \vphantom{\big|} \right|_{\tilde \pi ((D,j))} }}\right)^{-1}(x)\;\;\text{for every}\;x\in D\,.$$ The local parametrizations $\psi_{j'}:D'\to\widetilde{\pi}\left((D',j' )\right)$, for $j'=m+1,\ldots\,2m$ are analogously defined.
Sets of finite perimeter on the covering space
----------------------------------------------
It is natural to endow the space $Y_\Sigma$ with a measure $\mu$ defined as the sum on every sheet of the covering space of the push-forward by the local parametrization introduced in Definition \[local\] of the Lebesgue measure $\mathscr{L}^2$. Given a Borel set $E\subset Y_\Sigma$ we define $$\mu(E) \coloneqq \sum_{j=1}^m \psi_{j\#}\mathscr{L}^2(E\cap \tilde \pi((D,j)))\,.$$
We define the pullback of a function $f$ by the local parametrizations $\psi_j$ and $\psi_{j'}$ in the following way:
\[pullback\] Consider a function $f:Y_{\Sigma}\to {\mathbb{R}}^n$. For $j=1,\ldots,m$ and $j'=m+1,\ldots\,2m$ we let $f^j:D\to \mathbb{R}^n$, $f^{j'}:D'\to {\mathbb{R}}^n$ be the maps defined by $$f^j\coloneqq f\circ \psi_j \qquad \text{and} \qquad
f^{j'}\coloneqq f\circ \psi_{j'}\,.$$
By construction, thanks to the identifications given by $\sim$, we have $$\label{ide}
\left\{\begin{array}{lll}
f^j=f^{j'}
&\text{in}\; O
&\text{if}\; j\equiv j'\ (\mbox{mod}\ m)\,, \\
f^j=f^{j'}
&\text{in}\; I_i
&\text{if}\; j\equiv j'-i\ (\mbox{mod}\ m),\ i=1,\ldots , m-1\,. \\
\end{array}
\right.$$
From now on by $\chi_E$ we mean the characteristic function of the set $E$.
Given a $\mu$–measurable set $E \subset Y_\Sigma$ we define $$E^j:=\{x\in D \,: \, \chi^j_E(x)=1\}=p(E\cap \tilde{\pi}((D,j)))$$ for $j\in\{1,\ldots,m\}$ and analogously $$E^{j'}:=\{x\in D' \,: \, \chi^{j'}_E(x)=1\}=p(E\cap\tilde{\pi}((D',j')))$$ for $j'\in\{m+1,\ldots,2m\}$.
\[presc\] It is useful sometimes to define functions $f$ (resp. sets $E$) on the covering space $Y_\Sigma$ prescribing first the parametrizations $f^j : D \rightarrow {\mathbb{R}}^n$ (resp. sets $E^j$) for every $j=1\ldots,m$ and then deducing $f^{j'}$ in $D' \setminus \Sigma$ (resp. sets $E^{j'}$) for every $j'=m+1,\ldots,2m$, according to .
We set $L^1(Y_\Sigma) \coloneqq L^1(Y_\Sigma;{\mathbb{R}};\mu)$ and analogously, $L^1_{loc}(Y_\Sigma) \coloneqq L^1_{loc}(Y_\Sigma;{\mathbb{R}};\mu)$. We also define the distributional gradient of a function $u\in L^1(Y_\Sigma)$ as the linear map $$Du(\psi) = -\int_{Y_\Sigma} u\, {{\rm div}\,}\psi \, d\mu$$ for $\psi \in C_c^1(Y_\Sigma, {\mathbb{R}}^2)$, where the space $C_c^1(Y_\Sigma, {\mathbb{R}}^2)$ is defined in the natural way by the local parametrizations.
Given $u\in L^1(Y_\Sigma)$ we say that $u \in BV(Y_\Sigma)$ if $Du$ is represented by a Radon measure with bounded total variation.
Given $E\subset Y_\Sigma$ a $\mu$-measurable set and $\Lambda \subset Y_\Sigma$ open, we define $$\label{perimeter}
P(E, \Lambda):=\vert D\chi_{E}\vert (\Lambda) = \sup \left\{\int_E {{\rm div}\,}\psi \, d\mu : \psi \in C_c^1(\Lambda, {\mathbb{R}}^2) ,
\|\psi\|_\infty\leq 1\right\}\,.$$ We say that $E$ is a set of finite perimeter in $\Lambda$ if $P(E,\Lambda) <\infty$. In the case $E$ is of finite perimeter in $Y_\Sigma$, the definition of $P$ can be extended to all Borel sets $\Lambda \subset Y_\Sigma$ and $\Lambda \rightarrow P(E, \Lambda)$ is a Borel measure in $Y_\Sigma$.
For every $t\in [0,1]$ define the set $E^t \subset Y_\Sigma$ as $$E^t = \left\{x\in Y_\Sigma : \lim_{r\rightarrow 0}\frac{\mu(E \cap B_r(x))}{\mu(B_r(x))}=t\right\} \,.$$ We denote by $\partial^{\ast}E$ the essential boundary of $E$ defined as $\partial^{\ast}E = Y_\Sigma \setminus (E^0 \cup E^1)$ (see [@afp page 158] for the definition of the essential boundary in the Euclidean setting).
\[represent\] Let $\Lambda$ be a Borel set in $Y_\Sigma$ and $E$ a set of finite perimeter in $Y_\Sigma$. Then defining $\Lambda^{j'}_\Sigma:= p(\Lambda \cap \tilde \pi((\Sigma \setminus S, j')))$ we have $$\label{perimetro}
P(E,\Lambda) = \sum_{j=1}^m P(E^j,\Lambda^j)
+ \sum_{j'=m+1}^{2m} P(E^{j'},\Lambda^{j'}_\Sigma)\,$$ and $$\label{derivata}
D\chi_E(\Lambda) = \sum_{j=1}^m D\chi_{E^j} (\Lambda^j)
+ \sum_{j'=m+1}^{2m} D\chi_{E^{j'}}(\Lambda^{j'}_\Sigma)\,.$$
We notice that a Borel set $\Lambda\subset Y_\Sigma$ can be decomposed in the union of the disjoint sets: $$\label{declambda}
\Lambda \cap \tilde \pi((D,j)),\ \ j=1,\ldots m\,,\qquad \Lambda \cap \tilde \pi((\Sigma \setminus S, j')),\ \ j'=m+1,\ldots,2m\,.$$ Hence it is enough to prove the statements for a $\Lambda \subset \tilde \pi ((D,j))$ for a fixed $j$ or a $\Lambda \subset \tilde \pi ((\Sigma\setminus S,j'))$ for a fixed $j'$. Let us assume without loss of generality that $\Lambda \subset \tilde \pi ((D,j))$ is open. Consider $\eta \in C^1(\Lambda)$ and compactly supported in $\Lambda$. Then noticing that as ${{ \left.\kern-\nulldelimiterspace p \vphantom{\big|} \right|_{\Lambda} }}$ is bijective we have $$\begin{aligned}
D\chi_E(\eta)=\int_{\Lambda} \chi_E\, {{\rm div}\,}\eta \, d\mu =
\int_{p(\Lambda \cap E)} \, {{\rm div}\,}(\eta\circ p^{-1}) \, d\mathscr{L}^2\,=D \chi_{p(E\cap \Lambda)}(\eta\circ p^{-1})\,.\end{aligned}$$ Moreover as ${{ \left.\kern-\nulldelimiterspace p \vphantom{\big|} \right|_{\Lambda} }}$ is an homeomorphism it is easy to verify that $\eta \in C_c^1(\Lambda)$ if and only if $\eta \circ p^{-1} \in C_c^1(p(\Lambda))$. Therefore taking the supremum on $\eta$ we have that $$P(E,\Lambda) = P(p(E\cap \Lambda)) = P(p(E\cap \Lambda\cap \tilde \pi((D,j)))) = P(E^j,\Lambda^j).$$ If $\Lambda \subset \tilde \pi ((\Sigma\setminus S,j'))$ the decomposition follows performing the same computation in $(D',j')$ and then using the outer regularity of the measure $\Lambda \rightarrow P(E,\Lambda)$. Formula can be proven similarly.
\[bvpar\] From the computations of the previous lemma, one can easily see that if $E$ is a set of finite perimeter in $Y_\Sigma$, then $E^j$ is a set of finite perimeter in $D$ for $j=1\,\dots\,m$ (and, respectively, in a similar way one can show that $E^{j'}$ is a set of finite perimeter in $D'$ for $j'=m+1\,\dots\,2m$).
The constrained minimum problem {#problem}
-------------------------------
We define our simplified version of the minimization problem introduced by Amato, Bellettini and Paolini in [@cover], where we use sets of finite perimeter instead of vector valued $BV$ functions.
\[zerounoconstr\] We denote by $\mathscr{P}_{constr}(Y_\Sigma)$ the space of the sets $E$ of finite perimeter in $Y_\Sigma$ such that
- $\sum_{p(y) = x} \chi_E(y) = 1\ \ $ for almost every $x\in M =\mathbb{R}^2\setminus S$ ,
- $\chi_{E^1}(x) = 1\ \ $ for every $x\in \mathbb{R}^2 \setminus \Omega$.
In other words a set $E$ of finite perimeter in $Y_\Sigma$ belongs to $\mathscr{P}_{constr}(Y_\Sigma)$ if for almost every $x$ in the base space there exists exactly one point $y$ of $E$ such that $p(y)=x$.
Notice that it is possible to produce different sets in $Y_\Sigma$ (by a permutation of the sheets) satisfying condition $i)$ in Definition \[zerounoconstr\] that have the same projection on to the base space. In order to avoid this unpleasant effect we decide to add the condition $ii)$ in Definition \[zerounoconstr\].
We state the constrained minimization problem as follows: $$\mathscr{A}_{constr}(S) = \inf \left\{P(E) : E\in\mathscr{P}_{constr}(Y_\Sigma)\right\}\,.$$
\[independence\] It can be proved as in [@cover] (see also [@cover3; @cover2]) that given ${\bf\Sigma}, \hat{{\bf\Sigma}} \in \mbox{\textbf{Cuts}}(S)$ and $E \in \mathscr{P}_{constr}(Y_\Sigma)$ there exists $\hat{E} \in \mathscr{P}_{constr}(Y_{\hat{\Sigma}})$ such that $p(\partial^\ast E) = p(\partial^\ast \hat E)$. This implies that the quantity $\mathscr{A}_{constr}(S)$ is independent on the choice of the cuts ${\bf\Sigma}$.
\[comp\] Let $(E_n)_{n\in {\mathbb{N}}}$ be a sequence of sets in $\mathscr{P}_{constr}(Y_\Sigma)$ such that $$\sup_{n\in {\mathbb{N}}} P(E_n) < + \infty\,.$$ Then there exists $E\in \mathscr{P}_{constr}(Y_\Sigma)$ and a subsequence $(E_{n})_{n \in {\mathbb{N}}}$ (not relabelled) converging to $E$ in $L^1(Y_\Sigma)$ as $n\rightarrow +\infty$.
Notice that, thanks to , there holds $P(E^j_n, D) \leq P(E_n)< c <\infty$. Therefore, up to subsequences, $E^j_n\to E^j$ in $L^1(D)$ for every $j=1,\ldots,m$ (without relabelling the subsequence). The set $E$ is determined according to Remark \[presc\]. As for every $x\in D$ we have that $p^{-1}(x) = \tilde \pi \circ \pi^{-1}(x) = \bigcup_{j=1}^m \psi_j(x)$, the property in Definition \[zerouno\] can be rephrased using the local parametrization in the following way: $$\sum_{j=1}^m \chi_{E^j_n}(x)=1\,.$$ for almost every $x\in M$. Hence letting $n \rightarrow +\infty$ we get that $\sum_{i=1}^m \chi_{E^j}(x)=1$ for almost every $x\in M$. Moreover, as $\chi_{E^1_n}= 1$ in $\mathbb{R}^2 \setminus \Omega$ for every $n$, we have $\chi_{E^1}= 1$ in $\mathbb{R}^2 \setminus \Omega$ and therefore $E \in \mathscr{P}_{constr}(Y_\Sigma)$.
There exists a minimizer for Problem .
The proof follows by the application of the direct method thanks to Lemma $\ref{comp}$ and the lower semicontinuity of the perimeter in $L^1_{loc}(Y_\Sigma;{\mathbb{R}};\mu)$, that holds because the perimeter defined in is supremum of continuous functional.
Some properties of the projection of the essential boundary {#properties}
-----------------------------------------------------------
\[dueh\] Given a set $E$ in $\mathscr{P}_{constr}(Y_\Sigma)$, we have $$P(E) = 2{\mathcal{H}}^1(p(\partial^\ast E))\,.$$
Consider a set $E$ in $\mathscr{P}_{constr}(Y_\Sigma)$, then thanks to Lemma \[represent\] we get $$\label{split}
P(E) = \sum_{j=1}^m P(E^j,D) + \sum_{j'=m+1}^{2m} P(E^{j'},\Sigma \setminus S)\,.$$ Notice firstly that $\partial^\ast E^j$ and $\partial^\ast E^{j'}$ are rectifiable sets in $D$ and $D'$ respectively, therefore they admit a generalized unit normal that we denote by $\nu_{j}$ and $\nu_{j'}$. Define, for $h,k = 1,\ldots,m$, the set $$A_{h,k} = \left\{x\in \partial^\ast E^h\cap \partial^\ast E^k : \nu_h(x),\nu_k(x)\; \text{exist and}\;
\nu_h(x)=-\nu_k(x) \right\}\,,$$ and, for $h',k'= m+1,\ldots,2m$ $$A_{h',k'} = \left\{x\in \partial^\ast E^{h'}\cap \partial^\ast E^{k'} :
\nu_{h'}(x),\nu_{k'}(x)\; \text{exist and}\; \nu_{h'}(x)=-\nu_{k'}(x) \right\}\,.$$ Suppose that $A_{h,k} \subset D$ for every $h,k = 1,\ldots,m$.\
As $E\in\mathscr{P}_{constr}(Y_\Sigma)$ the sets $A_{h,k}$ satisfy the following properties:
- the sets $\{A_{h,k} : k=1,\ldots,m\}$ are pairwise disjoint for every $h= 1,\ldots,m$;
- $\mathcal{H}^1\left(p(\partial^\ast E) \setminus \bigcup_{h<k}A_{h,k}\right) = 0$;
- ${\mathcal{H}}^1(A_{h,k} \cap \partial^\ast E^j) = 0$ for every $h,k\neq j$.
Hence using and the previous properties we have $$\begin{aligned}
P(E) & = & \sum_{h=1}^m P(E^h, \bigcup_{k=1}^m A_{h,k})
= \sum_{h,k=1}^m P(E^h,A_{h,k})\\
& = & \sum_{h<k} P(E^h,A_{h,k}) + \sum_{h>k} P(E^h,A_{h,k})\\
&=& 2{\mathcal{H}}^1(p(\partial^\ast E))\end{aligned}$$ as we wanted to prove.\
If $A_{h,k} \cap \Sigma \neq \emptyset$ for some $h,k$, then one can repeat the previous argument decomposing $p(\partial^\ast E)$ with the sets $A_{h,k}$ in $D$ and with the sets in $A_{h',k'}$ in $\Sigma \setminus S$ and then use $(\ref{split})$.
\[similar\] With a similar proof it is possible to prove that given a $\mu$–measurable set $A\subset \mathbb{R}^2$ and $E\in\mathscr{P}_{constr}(Y_\Sigma)$ we have $$P(E,p^{-1}(A)) =2 {\mathcal{H}}^1(A\cap p(\partial^\ast E))\,.$$
\[nonconst\] Let $A\subset \mathbb{R}^2$ be a nonempty, open set such that $p^{-1}(A\setminus S)$ is connected. Then, for every $E \in \mathscr{P}_{constr}(Y_\Sigma)$ $${\mathcal{H}}^1 (A \cap p(\partial^\ast E)) > 0\,.$$
By contradiction there exists $E \in \mathscr{P}_{constr}(Y_\Sigma)$ such that ${\mathcal{H}}^1 (A \cap p(\partial^\ast E)) = 0$. Then by Remark \[similar\] we obtain $
P(E, p^{-1}(A\setminus S)) = 2{\mathcal{H}}^1(A \cap p(\partial^\ast E)) =0$. By Lemma \[represent\] there holds $P(E^j, (p^{-1}(A\setminus S))^j) \leq P(E, p^{-1}(A\setminus S))=0$, for every $j\in\{1,\ldots, m\}$ and $P(E^{j'}, (p^{-1}(A\setminus S))^{j'}) \leq P(E, p^{-1}(A\setminus S))=0$, for every $j'\in\{m+1,\ldots, 2m\}$.
Hence $D\chi_{E^j}=0$ in $(p^{-1}(A\setminus S))^j$, that implies that the function $\chi_{E^j}$ is constant in $(p^{-1}(A\setminus S))^j$ for every $j\in\{1,\ldots, m\}$ (similarly $\chi_{E^{j'}}$ is constant in $(p^{-1}(A\setminus S))^{j'}$). Moreover writing the set $p^{-1}(A\setminus S)$ as $$p^{-1}(A\setminus S)=\bigcup_{j=1}^m\big[p^{-1}(A\setminus S)\cap \tilde{\pi}(D,j)\big]
\bigcup_{j'=m+1}^{2m}\big[p^{-1}(A\setminus S)\cap \tilde{\pi}(D',j')\big]$$ and applying $p$ to both sides we obtain that $$A\setminus S=\bigcup_{j=1}^m(p^{-1}(A\setminus S))^j
\bigcup_{j'=m+1}^{2m}(p^{-1}(A\setminus S))^{j'}.$$ Hence $A\setminus S$ is a connected set, union of open sets in which $\chi_{E^{j}}$ and $\chi_{E^{j'}}$ are constant. This implies that the value of all $\chi_{E^{j}}$ and $\chi_{E^{j'}}$ is the same for every $j,j'$ and, as a consequence, $\chi_E$ is constant in $p^{-1}(A\setminus S)$. This contradicts the validity of the constraint $i)$ of Definition \[zerounoconstr\].
The previous lemma implies that the projection via $p$ of the essential boundary of $E\in\mathscr{P}_{constr}(Y_\Sigma)$ touches $S$.
\[esse\] Let $E \in \mathscr{P}_{constr}(Y_\Sigma)$. Then $$S \subset \overline{p(\partial^\ast E)} \,.$$
Suppose by contradiction that there exists $p_i \in S\setminus \overline{p(\partial^\ast E)}$. Then there exists a ball $B$ with center in $p_i$ and such that $B\cap \overline{p(\partial^\ast E)} = \emptyset$. Notice that $p^{-1}(B\setminus \{p_i\})$ is path connected in $Y_\Sigma$. Indeed given two points $q_1$ and $q_2$ in $p^{-1}(B\setminus \{p_i\})$ it is possible to construct a connected path in $Y_\Sigma$ joining $q_1$ and $q_2$ following the identifications given by $\sim$ and crossing the cuts in the right order (see Definition \[admissible\] and the identifications defined in ); so we can apply Lemma $\ref{nonconst}$ to deduce that $${\mathcal{H}}^1(B\cap p(\partial^\ast E)) > 0\,,$$ that is a contradiction.
\[connected\] Consider $E\in \mathscr{P}_{constr}(Y_\Sigma)$ such that ${\mathcal{H}}^1(\overline{p(\partial^\ast E)})<+\infty$. If at least one point of $S$ is contained in a connected component $\mathcal{C}$ of $\overline{p(\partial^\ast E)}$, then the whole $S$ is contained in $\mathcal{C}$.
For any $p_j\in S$ let $\mathcal{C}_j$ be the connected component of $\overline{p(\partial^\ast E)}$ containing $p_j$. Suppose by contradiction that there exists $k_1\neq k_2$ such that $\mathcal{C}_{k_1} \neq \mathcal{C}_{k_2}$. As a consequence $\overline{p(\partial^\ast E)}$ is not connected. Hence there exist two non-empty disjoint sets $A,B \subset \overline{p(\partial^\ast E)}$, relatively closed in $\overline{p(\partial^\ast E)}$ with $A\cup B = \overline{p(\partial^\ast E)}$. Moreover it is possible to choose $A$ and $B$ satifying the properties above and such that there exists $p_A, p_B \in S$ with $p_A \in A$ and $p_B \in B$. Notice that, thanks to condition $ii)$ in Definition \[zerounoconstr\], the set $\overline{p(\partial^\ast E)}$ is bounded. Hence $A$ and $B$ are compact in ${\mathbb{R}}^2$. Let $\varepsilon > 0$ be such that $$A_\varepsilon \cap B = \emptyset\, ,$$ where we have denoted by $A_\varepsilon$ the open $\varepsilon$-neighbourhood of $A$. Notice that $\partial A_\varepsilon$ is a Lipschitz manifold for $\varepsilon$ small enough [@tubular]. Hence $\partial A_\varepsilon$ has a finite number of connected components and in particular it is a finite union of simple loops (see for example [@connectper Corollary 1]) that we denote by $\{\gamma_i\}_{i=1,\ldots,q}$. As $p_A \in A_\varepsilon$,
$$\label{indicescurve}
1 = \mbox{Ind}(\gamma_1 \circ \gamma_2 \ldots \circ \gamma_q ,p_A) = \sum_{i=1}^q \mbox{Ind}(\gamma_i, p_A)\,,$$
where we have denoted by $\mbox{Ind}(\gamma,p)$ the index of the loop $\gamma$ with respect to the point $p$ and by $\gamma \circ \sigma$ the concatenation of two curves $\gamma$ and $\sigma$. From we infer that there exists a loop $\gamma_Q$ such that $\mbox{Ind}(\gamma_Q, p_A) = 1$ and, as $p_B$ belongs to the unbounded connected component of ${\mathbb{R}}^2$ described by $\gamma_Q$, we have also that $\mbox{Ind}(\gamma_Q, p_B) = 0$.
Hence the loop $\gamma_Q$ is such that $\gamma_Q \cap A = \emptyset$, $\gamma_Q \cap B = \emptyset$ and it has index one with respect to at most $m-1$ points of $S$ and index zero with respect to at least one point of $S$. This implies that $\gamma_Q$ is crossing the cuts $\Sigma$ and $\Sigma'$ at least once and therefore $p^{-1}(\gamma_Q)$ is a closed loop in the covering space $Y_\Sigma$. Indeed tracking the path $p^{-1}(\gamma_Q)$ in $Y_\Sigma$ one can see that for every crossing of the cuts $\Sigma$ and $\Sigma'$ in the covering space the path is continuing to the next sheet (following the identification given in ). In doing so, $p^{-1}(\gamma_Q)$ is visiting all the sheets of $Y_\Sigma$ closing back at the starting point.
Additionally there exists an open $\varepsilon$-tubular neighborhood of $\gamma_Q$ such that $(\gamma_Q)_\varepsilon \cap A = \emptyset$ and $(\gamma_Q)_\varepsilon \cap B = \emptyset$. We infer that $p^{-1}((\gamma_Q)_\varepsilon)$ is path connected. Indeed given two points $q_1,q_2 \in p^{-1}((\gamma_Q)_\varepsilon)$ it is enough to connect them to $p^{-1}(\gamma_Q)$ with a path that do not intersect the cuts and use that $p^{-1}(\gamma_Q)$ is a closed loop in $Y_\Sigma$. This is a contradiction with Lemma \[nonconst\].
The next theorem is a regularity result for $p(\partial^\ast E)$ when $E \in \mathscr{P}_{constr}(Y_\Sigma)$ is a minimizer of Problem . To prove this theorem our strategy is to establish *locally* an equivalence between Problem and the partition problem [@ab1] and then use the known regularity results for it (see, for instance, [@morel; @tamanini]).
Given $A$ an open subset of $Y_\Sigma$, we say that $E\in \mathscr{P}_{constr}(Y_\Sigma)$ is a local minimizer for Problem in $A$ if for every $F\in \mathscr{P}_{constr}(Y_\Sigma)$ such that $E\Delta F \subset \subset A$, there holds $$P(E,A) \leq P(F,A)\,.$$
Consider the $m$–regular simplex in $\mathbb{R}^{m+1}$ centred in the origin and call $\{\alpha_1\,\ldots \,\alpha_m\}$ the vertices of the simplex and call $BV(A, \{\alpha_1\,\ldots \,\alpha_m\})$ the space of $BV$–functions with values in $\{\alpha_1\,\ldots \,\alpha_m\}$.
\[minpart\] Given $A$ an open bounded subset of ${\mathbb{R}}^2$, we say that $u\in BV(A, \{\alpha_1\,\ldots \,\alpha_m\})$ is a local minimizer for the partition problem in $A$ if for every $w\in BV(A, \{\alpha_1\,\ldots \,\alpha_m\})$ such that $\{u\neq w\} \subset \subset A$, there holds $$\vert Du \vert \left(A \right)\leq \vert Dw \vert \left(A \right)\,.$$
Given a set $E\in \mathscr{P}_{constr}(Y_\Sigma)$, consider $x\in \overline{p(\partial^\ast E)}$ such that $x\notin \Sigma \cup
\Sigma'$ and $r>0$ small enough such that $B_r(x) \cap (\Sigma \cup \Sigma') = \emptyset$. The associated vector valued function $u^\alpha$ in $BV(B_r(x), \{\alpha_1\,\ldots \,\alpha_m\})$ is canonically defined as $$u^{\alpha}(x) \coloneqq \alpha_j \quad\text{if}\; x\in E^j\cap B_r(x)\,,$$ for $j=1\,,\ldots\,,m$. Notice that by construction there holds $$\label{jp}
p(\partial^\ast E) \cap B_r(x) =J_{u^\alpha}\,,$$ where $J_{u^\alpha}$ is the jump set of $u^\alpha$.
\[equivalenzalocale\] Suppose that $E_{\min}\in \mathscr{P}_{constr}(Y_\Sigma)$ is a local minimizer for Problem in $p^{-1}(B_r(x))$ with $x\in \overline{p(\partial^\ast E)}$ and $r>0$ such that $B_r(x) \cap (\Sigma \cup \Sigma') = \emptyset$. Then the associated vector valued function $u_{\min}^\alpha$ in $BV(B_r(x), \{\alpha_1\,\ldots \,\alpha_m\})$ is a local minimizer for the partition problem in $p^{-1}(B_r(x))$.
Consider any $w\in BV(B_r(x),\{\alpha_1\,\ldots \,\alpha_m\})$ such that $\{u_{\min}^\alpha \neq w\} \subset \subset B_r(x)$. We associate to $w$ a set $F$ in $\mathscr{P}_{constr}(Y_\Sigma)$ defining its characteristic function as $$\chi_ {F^{j}}(x) \coloneqq\left\{
\begin{array}{ll}
1 &\text{if}\; w(x)=\alpha_j\,,\\
0 &\text{otherwise}\,
\end{array}
\right.$$ for $j=1\,,\ldots\,,m$ (see Remark \[presc\]). By construction we have that $J_{w}=p(\partial^\ast F) \cap B_r(x)$.
Then applying Remark \[similar\] and we obtain $$\begin{aligned}
&2\vert Du^\alpha_{\min}\vert
(B_r(x))=2\mathcal{H}^1\left(J_{u^\alpha_{\min}}\right)\\
&= 2\mathcal{H}^1\left(p(\partial^\ast E_{\min})\cap B_r(x)
\right)=P(E_{\min},p^{-1}(B_r(x)))\\
&\leq P(F,p^{-1}(B_r(x)))=2\mathcal{H}^1\left(p(\partial^\ast F) \cap
B_r(x)\right)\\
& =2\mathcal{H}^1\left( J_w \right)=2\vert Dw \vert(B_r(x))\,,\end{aligned}$$ hence $$\vert Du^\alpha_{\min}\vert (B_r(x))\leq \vert Dw \vert(B_r(x))\,.$$
We have exhibited a way to pass *locally* from our minimization problem in the covering space to a problem of minimal partition in $\mathbb{R}^2$ for . This is enough for our aim, that is obtaining the regularity of the minimizer $E$, but clearly it is possible, with a similar procedure, to show that given a local minimizer for the minimal partition problem, then the associated set $E\in \mathscr{P}_{constr}(Y_\Sigma)$ is a local minimizer for . We underline that in general the equivalence between the partition problem and Problem does not hold *globally*.
\[regularity\] Given $E_{min} \in \mathscr{P}_{constr}(Y_\Sigma)$ a minimizer of Problem , then there holds $$\label{reg1}
\mathcal{H}^1(\overline{p(\partial^\ast E_{\min})} \setminus
p(\partial^\ast E_{\min}))=0\,.$$ Moreover $\overline{p(\partial^\ast E_{\min})}$ is a finite union of segments meeting at triple junctions with angles of $120$ degrees.
Consider $x\in \overline{p(\partial^\ast E)}$ and suppose without loss of generality (thanks to Remark \[independence\]) that $x\notin \Sigma \cup \Sigma'$; then there exists $r>0$ such that $B_r(x)\cap (\Sigma \cup
\Sigma')=\emptyset$ and $E_{\min}\in \mathscr{P}_{constr}(Y_\Sigma)$ is a local minimizer for in $p^{-1}(B_r(x))$. Then by Lemma \[equivalenzalocale\] the associated function $u_{\min}^\alpha$ is a local minimizer for the partition problem. Thanks to the regularity results for the local partition problem (see [@tamanini Theorem $4.7$]) we have that $\mathcal{H}^1(\overline{p(\partial^\ast E_{\min})} \setminus
p(\partial^\ast E_{\min}))=0$. Moreover $\overline{p(\partial^\ast E_{\min})}$ inherits all the regularity properties of the minimum of the partition problem, namely the set $\overline{p(\partial^\ast E_{\min})}$ is finite union of segments meeting at triple junctions with angles of $120$ degree.
Proof of the equivalence
------------------------
In this section we prove that the minimization problem is equivalent to the Steiner problem in ${\mathbb{R}}^2$. First of all we need to prove that, given a solution of the Steiner problem for $S$, we can find a set $E\in \mathscr{P}_{constr}(Y_\Sigma)$ such that $\overline{p(\partial^\ast E)}$ is the Steiner network. We prove this statement for a smaller class of network, namely for the connected networks without loops. This result will be used again in Section \[famiglie\].
\[conetw\] A connected network is a finite union of $C^0$ injective curves that intersect each other only at their end points. We say that a connected network $\mathscr{S}$ connects the points of $S$ if $S\subset \mathscr{S}$ and the end points of the curves of $\mathscr{S}$ either have order one and are points of $\mathscr{S}$ or have order greater or equal than one. In the first case the end points are called leaves, in the latter case they are called multipoint. We call $L\subset \mathscr{S}$ the set of all leaves of $\mathscr{S}$
\[costruzione\] Consider $S=\{p_1, \ldots, p_m\}$ and $\mathscr{S}$ a connected network without loops that connects the $m$ points of $S$. Then, for an appropriate relabeling of the points of $S$ there exists an admissible pair of cuts ${\hat{\bf{{\Sigma}}}} \in \mbox{\textbf{Cuts}}(S)$ and a set $E_{\mathscr{S}} \in \mathscr{P}_{constr}(Y_{\hat{\Sigma}})$ such that $\overline{p(\partial^\ast E_{\mathscr{S}})}=\mathscr{S}$.
First of all we prove that, up to a permutation of the labelling of the point of $S$, there exists an admissible pair of cuts $\hat{{\bf{\Sigma}}} \in \mbox{\textbf{Cuts}}(S)$ such that $\hat{{\bf{\Sigma}}}\cap \mathscr{S}=S$.
We notice that in order to prove the previous claim it is sufficient to find $\hat{\Sigma} \in Cuts(S)$ such that $\hat{\Sigma} \cap \mathscr{S}=S$. Then by a continuous deformation it is immediate to construct $\hat{{\bf{\Sigma}}} \in \mbox{\textbf{Cuts}}(S)$ with $\hat{{\bf{\Sigma}}}\cap \mathscr{S}=S$. We build $\hat\Sigma$ in a constructive way. We remind that from Definition \[conetw\] follows that the set $S$ can be written as $L\cup \mathcal{M}:=\{\ell_1,\ldots,\ell_h\}\cup\{m_1,\ldots,m_k\}$ where $L$ is the set of leaves and $\mathcal{M}$ is a subset (possibly empty) of the set of all the multipoints of $\mathscr{S}$ and $m=h+k$. The first step of our construction is the following: fix $\ell_1\in L$ and follow the network $\mathscr{S}$ with the rule that at every multipoint we proceed along the closest curve with respect to the clockwise rotation. As the network is without loops, this procedure ends in a leaf $\ell_2\neq\ell_1$. We call $N_{\ell_1}$ the subnetwork described by the just defined procedure, $N_{\ell_1}$ contains $\ell_1,\ell_2$ and possibly some points of $\mathcal{M}$, let us say $m_i$ with $i\in\{1,\ldots,j\}, j\leq k$. Then there exists a Lipschitz curve $\Sigma_1$ that connects $\ell_1$ to $m_1$, for $i\in\{1,\ldots,j-1\}$ there exist Lipschitz curves $\Sigma_{i+1}$ that connects $m_i$ to $m_{i+1}$ and $\Sigma_{j+1}$ a Lipschitz curve from $m_j$ to $\ell_2$. Moreover we can choose all the Lipschitz curves in such a way that they do not intersect $\mathscr{S}\setminus S$ (for instance they can be obtained by continuous deformation of the subnetwork $N_{\ell_1}$). Step $2$ to step $h$ of the procedure are nothing else than an iteration of the procedure of the first step, starting from $\ell_i$ with $i\in\{2,\ldots,h-1\}$ with the extra requirement that at the $n-th$ step one ones does not connect with Lipschitz curves the points $m_i$, already visited in the steps $1$ to $n-1$ (see Figure \[puntiinterni\]). This will produce a family of Lipschitz curves $\{\Sigma_i\}_{i=1,\ldots,m-1}$ connecting the points of $S$ and not intersecting $\mathscr{S}\setminus S$. Then $\hat{\Sigma}=\cup_{i=1}^{m-1}\Sigma_i$ is the desired set of cuts.
Now we describe how to associate to $\mathscr{S}$ a set $E_{\mathscr{S}}$ in the covering space $Y_{\hat{\Sigma}}$. For $j=1,\ldots,m-1$ the set $E_{\mathscr{S}}^{m+1-j}$ is defined as the open set such that its boundary is composed by $\Sigma_j$ and the part of $\mathscr{S}$ connecting $\tilde{p}_j$ and $\tilde{p}_{j+1}$ and $E_{\mathscr{S}}^1=\mathbb{R}^2\setminus \cup_{j=1}^{m-1}E_{\mathscr{S}}^{m+1-j}$. Thanks to Remark \[presc\] the set $E_{\mathscr{S}}$ is well defined. By construction it is trivial that $E_{\mathscr{S}}$ satisfies the constraints of Definition \[zerounoconstr\] and that $\overline{p(\partial^\ast E_{\mathscr{S}})}=\mathscr{S}$.
The choice of $E^j$ in the previous construction is not arbitrary: if one chooses differently the sets $E^j$, one obtains a different set with perimeter greater than the perimeter of $E_{\mathscr{S}}$.
(1,1) node\[above\][$p_3=l_3$]{} (1,-1) node\[below\][$p_2=l_2$]{} (-1,-1) node\[below\][$p_1=l_1$]{} (-1,1) node\[above\][$p_4=l_4$]{}; (1,1) circle (1.7pt); (1,-1) circle (1.7pt); (-1,1) circle (1.7pt); (-1,-1) circle (1.7pt); (1,-1)–(0.42,0) (1,1)–(0.42,0) (0.42,0)–(-0.42,0) (-0.42,0)–(-1,-1) (-0.42,0)–(-1,1); (-1,-1)to\[out=-150,in=-60, looseness=1\] (1,-1) (-1,1)to\[out=160,in=-160, looseness=1\] (-1,-1) (1,1)to\[out=-20,in=20, looseness=1\] (1,-1); (1.05,-1)–(0.47,0) (1.05,1)–(0.47,0); (-0.47,0)–(-1.05,-1) (-0.47,0)–(-1.05,1); (-0.37,-0.1)–(0.37,-0.1) (-0.37,-0.1)–(-0.95,-1) (0.37,-0.1)–(0.95,-1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1);
(0,0.42)node\[left\][$p_3=m_1$]{} (1,1) node\[above\][$p_4=l_3$]{} (1,-1) node\[below\][$p_2=l_2$]{} (-1,-1) node\[below\][$p_1=l_1$]{} (-1,1) node\[above\][$p_5=l_4$]{}; (0,0.42)circle (1.7pt); (1,1) circle (1.7pt); (1,-1) circle (1.7pt); (-1,1) circle (1.7pt); (-1,-1) circle (1.7pt); (1,-1)–(0.42,0) (1,1)–(0.42,0) (0.42,0)–(-0.42,0) (-0.42,0)–(-1,-1) (-0.42,0)–(-1,1); (-1,-1)to\[out=-150,in=-120, looseness=1\] (0.42,0) (0.42,0)to\[out=-120,in=-100, looseness=1\] (1,-1) (-1,1)to\[out=160,in=-160, looseness=1\] (-1,-1) (1,1)to\[out=-20,in=20, looseness=1\] (1,-1); (1.05,-1)–(0.47,0) (1.05,1)–(0.47,0); (-0.47,0)–(-1.05,-1) (-0.47,0)–(-1.05,1); (-0.37,-0.1)–(0.37,-0.1) (-0.37,-0.1)–(-0.95,-1) (0.37,-0.1)–(0.95,-1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1);
(0,-0.42)node\[left\][$p_3=m_1$]{} (0,0.42)node\[left\][$p_4=m_2$]{} (1,1) node\[above\][$p_5=l_3$]{} (1,-1) node\[below\][$p_2=l_2$]{} (-1,-1) node\[below\][$p_1=l_1$]{} (-1,1) node\[above\][$p_6=l_4$]{}; (0,0.42)circle (1.7pt); (0,-0.42)circle (1.7pt); (1,1) circle (1.7pt); (1,-1) circle (1.7pt); (-1,1) circle (1.7pt); (-1,-1) circle (1.7pt); (1,-1)–(0.42,0) (1,1)–(0.42,0) (0.42,0)–(-0.42,0) (-0.42,0)–(-1,-1) (-0.42,0)–(-1,1); (-1,1)to\[out=160,in=-160, looseness=1\] (-0.42,0) (-0.42,0)to\[out=160,in=-160, looseness=1\] (-1,-1) (-1,-1)to\[out=-150,in=-120, looseness=1\] (0.42,0) (0.42,0)to\[out=-120,in=-100, looseness=1\] (1,-1) (1,1)to\[out=-20,in=20, looseness=1\] (1,-1); (1.05,-1)–(0.47,0) (1.05,1)–(0.47,0); (-0.47,0)–(-1.05,-1) (-0.47,0)–(-1.05,1); (-0.37,-0.1)–(0.37,-0.1) (-0.37,-0.1)–(-0.95,-1) (0.37,-0.1)–(0.95,-1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1);
\[samelength\] The Steiner problem and Problem are equivalent. In particular if $E\in\mathscr{P}_{constr}(Y_\Sigma)$ is a minimizer of Problem , then $\overline{p(\partial^\ast E})$ is a solution of the Steiner problem and if $\mathscr{S}$ is a minimizer for the Steiner problem, then its associated set $E_\mathscr{S}$ (constructed as in Proposition \[costruzione\]) is a minimizer for Problem .
Consider $E$ a minimizer for Problem in $Y_\Sigma$ and $\mathscr{S}$ a minimizer for the Steiner problem. Thanks to Proposition \[costruzione\] there exists an admissible pair of cuts ${\hat{\bf{{\Sigma}}}} \in \mbox{\textbf{Cuts}}(S)$ and a set $E_{\mathscr{S}} \in \mathscr{P}_{constr}(Y_{\hat{\Sigma}})$ such that $\overline{p(\partial^\ast E_{\mathscr{S}})}=\mathscr{S}$. Thanks to Remark \[independence\] Problem is independent on the choice of the cuts. Therefore $$\label{f}
P(E, Y_\Sigma) \leq P(E_\mathscr{S}, Y_{\hat{\Sigma}})\,.$$ Using Proposition \[dueh\] and inequality we have $$\label{d}
\mathcal{H}^1(p(\partial^\ast E))=\frac 12 P(E, Y_{\Sigma}) \leq \frac 12 P(E_\mathscr{S}, Y_{\hat{\Sigma}})=
\mathcal{H}^1(p(\partial^\ast E_\mathscr{S})) \leq \mathcal{H}^1(\mathscr{S})\,.$$ Thanks to Theorem \[regularity\] we know that $$\label{g}
\mathcal{H}^1(\overline{p(\partial^\ast E)})=\mathcal{H}^1(p(\partial^\ast E))\,.$$ Thanks to Corollary \[esse\] we have $S\subset \overline{p(\partial^\ast E)}$, and by Proposition \[connected\] there exists a connected component $\mathcal{C}_{E}$ of $\overline{p(\partial^\ast E)}$ that contains $S$. Hence $\mathcal{C}_{E}$ is a competitor for the Steiner problem for $S$ in $\mathbb{R}^2$, therefore from the minimality of $\mathscr{S}$ we get $$\label{e}
\mathcal{H}^1(\mathscr{S})\leq\mathcal{H}^1(\mathcal{C}_{E})
\leq\mathcal{H}^1(\overline{p(\partial^\ast E)})\,.$$ Combining , and we have $$\mathcal{H}^1(\overline{p(\partial^\ast E)})=\mathcal{H}^1(\mathscr{S})\,.$$ We obtain as well that $\mathcal{H}^1(\mathcal{C}_{E})=\mathcal{H}^1(\overline{p(\partial^\ast E)})$; therefore using again Theorem \[regularity\] we infer that $\overline{p(\partial^\ast E})$ is connected. The set $\overline{p(\partial^\ast E)}$ is a connected set that joints the point of $S$ and (by the minimality of $\mathscr{S}$) such that, for every connected set $\mathscr{T}\subset {\mathbb{R}}^2$ that connects the point of $S$ $$\mathcal{H}^1(\overline{p(\partial^\ast E)})= \mathcal{H}^1(\mathscr{S})\leq \mathcal{H}^1(\mathscr{T}) \,.$$ Hence $\overline{p(\partial^\ast E)}$ is a minimizer for the Steiner problem. On the other hand the constrained set $E_\mathscr{S}$ has the same perimeter of $E$, hence is a solution of Problem .
Calibrations {#seccal}
============
In this section we introduce the notion of calibration for the minimum problem and we show some explicit examples. In doing so, it is often convenient to consider vector fields that are not continuous and for which a divergence theorem still holds. For this reason we employ the notion of *approximately regular vector field* (in a slightly stronger version than in [@mumford]) and then we generalize it to the covering space setting.
Given $A\subset {\mathbb{R}}^{n}$, a Borel vector field $\Phi: A \rightarrow {\mathbb{R}}^{n}$ is approximately regular if it is bounded and for every Lipschitz hypersurface $\mathscr{M}$ in ${\mathbb{R}}^{n}$, $\Phi$ admits traces on $\mathscr{M}$ on the two sides of $\mathscr{M}$ (denoted by $\Phi^+$ and $\Phi^-$) and $$\label{app}
\Phi^+(x) \cdot \nu_{\mathscr{M}}(x) = \Phi^-(x) \cdot \nu_{\mathscr{M}}(x) = \Phi(x) \cdot \nu_\mathscr{M}(x),$$ for $\mathcal{H}^{n-1}$–a.e. $x \in \mathscr{M}\cap A$.
\[appreg\] Given $\Phi: Y_\Sigma\rightarrow {\mathbb{R}}^2$, we say that it is *approximately regular* in $Y_\Sigma$ if $\Phi^j$ and $\Phi^{j'}$ (see Definition \[pullback\]) are *approximately regular* for every $j=1,\ldots,m$ and $j'=m+1,\ldots,2m$.
\[caliconvering\] Given $E\in\mathscr{P}_{constr}(Y_\Sigma)$, a calibration for $E$ (with respect to the minimum problem ) is an *approximately regular* vector field $\Phi :Y_{\Sigma}\to\mathbb{R}^2$ such that:
1. ${{\rm div}\,}\Phi=0$ (in the sense of the distributions);
2. $\vert \Phi^i (x) - \Phi^j (x)\vert \leq 2$ for every $i,j = 1,\ldots m$ and for every $x\in D$;
3. $\int_{Y_\Sigma} \Phi \cdot D\chi_E=P(E)$.
\[twoinsteadofone\] At first sight the size condition (**2**) may sounds different in comparison with the classical notion of paired calibration. This difference is only apparent. Indeed we choose to minimize $P(E)$ that thanks to Proposition \[dueh\] is equal to the double of the length of the minimal network on to the base space.
\[divteo\] Consider $E,F\in \mathscr{P}_{constr}(Y_\Sigma)$ and let $\Phi : Y_\Sigma \to {\mathbb{R}}^2$ be an approximately regular vector field such that ${{\rm div}\,}\Phi = 0$ (in the sense of the distributions) in $Y_\Sigma$. Then $$\label{booh}
\int_{Y_\Sigma} \Phi \cdot D\chi_E = \int_{Y_\Sigma} \Phi \cdot D\chi_F.$$
See Appendix \[appe\].
In the following theorem we prove that our notion of calibration is indeed meaningful, in the sense that the existence of a calibration for a given $E\in \mathscr{P}_{constr}(Y_\Sigma)$ implies the minimality of $E$ for Problem .
\[impli\] If $\Phi :Y_{\Sigma}\to\mathbb{R}^{2}$ is a calibration for $E$, then $E$ is a minimizer of Problem .
Let $\Phi :Y_{\Sigma}\to\mathbb{R}^{2}$ be a calibration for $E\in \mathscr{P}_{constr}(Y_\Sigma)$ and let $F\in \mathscr{P}_{constr}(Y_\Sigma)$ a competitor. By Proposition \[divteo\] and (**1**) of Definition \[caliconvering\] we have $$\label{a}
\int_{Y_\Sigma} \Phi \cdot D\chi_E = \int_{Y_\Sigma} \Phi \cdot D\chi_F\,$$ and thanks to property (**3**) of Definition \[caliconvering\] $$\label{b}
P(E) = \int_{Y_\Sigma} \Phi \cdot D\chi_E\,.$$ Moreover, using we have
$$\begin{aligned}
\label{c}
\int_{Y_\Sigma} \Phi \cdot D\chi_F
&= \sum_{j=1}^m \int_{D} \Phi^j\cdot D\chi_{F^j}
+ \sum_{j'=m+1}^{2m} \int_{\Sigma} \Phi^{j'}\cdot D\chi_{F^{j'}}\\
&= \sum_{j=1}^m\int_{\partial^\ast F^j \cap D}\Phi^j \cdot \nu_{F^{j}} \, d{\mathcal{H}}^{1}
+ \sum_{j'=m+1}^{2m} \int_{\partial^\ast F^{j'} \cap \Sigma}\Phi^{j'} \cdot \nu_{F^{j'}} \, d{\mathcal{H}}^{1}\\
&= \int_{p(\partial^\ast F)\cap D}\sum_{j=1}^m \Phi^j \cdot \nu_{F^{j}}\chi_{\partial^*F^j} \, d{\mathcal{H}}^{1}
+ \int_{p(\partial^\ast F) \cap \Sigma} \sum_{j'=m+1}^{2m}\Phi^{j'} \cdot \nu_{F^{j'}} \chi_{\partial^*F^{j'}}\, d{\mathcal{H}}^{1}\\
&\leq \int_{p(\partial^\ast F)\cap D}
\Big|
\sum_{j=1}^m \Phi^j \cdot \nu_{F^{j}}\chi_{\partial^*F^j} \Big|\, d{\mathcal{H}}^{1}
+ \int_{p(\partial^\ast F) \cap \Sigma}\Big|
\sum_{j'=m+1}^{2m}\Phi^{j'} \cdot \nu_{F^{j'}}\chi_{\partial^*F^{j'}}\Big| \, d{\mathcal{H}}^{1}.\end{aligned}$$
As $F\in \mathscr{P}_{constr}(Y_\Sigma)$, for ${\mathcal{H}}^1$–a.e. $x\in p(\partial^\ast F) \cap D$ there exist exactly two distinct indices $j_1,j_2\in \{1,\ldots, m\}$ such that $x\in \partial^\ast F^{j_1}\cap \partial^\ast F^{j_2}$ and $\nu_{F^{j_1}}=-\nu_{F^{j_2}}$. Therefore using condition (**2**) of Definition \[caliconvering\] and the usual identifications given by $\sim$ we get that $$\label{c}
\int_{Y_\Sigma} \Phi \cdot D\chi_F \leq 2{\mathcal{H}}^1(p(\partial^\ast F)) = P(F),$$ where the last equality follows from Proposition \[dueh\].\
Combining Equations , and one obtains $$P(E) = \int_{Y_\Sigma} \Phi \cdot D\chi_E= \int_{Y_\Sigma} \Phi \cdot D\chi_F\leq P(F)\,.$$
\[const\] Given $\Phi: Y_\Sigma \rightarrow {\mathbb{R}}^2$ a calibration for $E\in \mathscr{P}_{constr}(Y_\Sigma)$, then for every $c\in {\mathbb{R}}^2$ we have that $\Phi + c$ is a calibration for $E$. Indeed if $\Phi$ is a calibration for $E\in \mathscr{P}_{constr}(Y_\Sigma)$ then it is easy to see that properties (**1**) and (**2**) hold for $\Phi + c$ as well. It remains to show that if $\int_{Y_\Sigma} \Phi \cdot D\chi_{E} = P(E)$, then $$\int_{Y_\Sigma} (\Phi + c) \cdot D\chi_{E} = P(E)\,,$$ that is that for every $c\in {\mathbb{R}}^2$ we have $\int_{Y_\Sigma} c \cdot D\chi_E = 0$. Following the computation in the proof of Theorem \[divteo\] we have that $$\int_{Y_\Sigma} c \cdot D\chi_E = \int_{\partial \Omega} c \cdot \nu_{\partial \Omega}\, d{\mathcal{H}}^1 = 0\,.$$
Examples of calibrations {#excal}
------------------------
We present here several examples of calibrations for Steiner configurations in the covering space setting. In the figures below the vector field is implicitly defined as the constant $\Phi=(0,0)$, where no arrows are drawn. Notice that, thanks to Definition \[appreg\], a calibration can admit discontinuities in the domain of definition provided that is fulfilled.
In order to introduce the reader to the calibration method in our setting we start with the trivial example of the minimality of the segment. In particular we show that the set $E\in\mathscr{P}_{constr}(Y_\Sigma)$ defined in such a way that the closure of its essential boundary is the segment connecting $p_1$ and $p_2$ is the minimizer of Problem . We recall that in this case the number of sheets of the covering is two. We define $E^1$ (resp. $E^2$) as the coloured subset of $(D,1)$ (resp. $(D,2)$) in Figure \[minduefig\] and the set $E$ is obtained as explained in Remark \[presc\].
at (0,0) [![The candidate minimizer $E$ and the vector field $\Phi$[]{data-label="minduefig"}](duepunti "fig:"){width="95.00000%"}]{};
Let us denote by $Q$ the dashed stripe in Figure \[minduefig\], by $A_1$ the set enclosed by $Q$ and the cut $\Sigma$ and by $A_2$ the complement of $A_1$ with respect to $Q$. We define a vector field $\Phi : Y_\Sigma \rightarrow {\mathbb{R}}$ prescribing its parametrization on the sheets $(D,1)$ and $(D,2)$: $$\Phi^1(x) = \left\{\begin{array}{ll}
(0,1) & x\in A_1\\
(0,-1) & x\in A_2\\
0 & \text{otherwise}
\end{array}
\right.
\quad
\Phi^2(x) = \left\{\begin{array}{ll}
(0,-1) & x\in A_1\\
(0,1) & x\in A_2\\
0 & \text{otherwise }.
\end{array}
\right.$$ We verify that the unit vector field $\Phi : Y_\Sigma \rightarrow {\mathbb{R}}$ defined as in Figure \[minduefig\] is a calibration of $E$. First notice that $\Phi$ is an approximately regular divergence free vector field in $Y_\Sigma$. Indeed, as a consequence of the identifications in the construction of the covering space, $\Phi$ is constant in $p^{-1}(Q)$. Since $\Phi$ is a piecewise constant vector field satisfying , its distributional divergence of is zero. Condition (**2**) in Definition \[caliconvering\] is trivially satisfied. Finally $$\begin{aligned}
\int_{Y_\Sigma} \Phi \cdot D\chi_E &=& \sum_{j=1}^2 \int_{D} \Phi^j\cdot D\chi_{E^j} \\
&=& \int_{p(\partial^\ast E)} \Phi^1 \cdot \nu_{E^1} d{\mathcal{H}}^1 + \int_{p(\partial^\ast E)} \Phi^2 \cdot \nu_{E^2} d{\mathcal{H}}^1 \\
&=& 2{\mathcal{H}}^1(p(\partial^\ast E)) \\
&=& P(E)\end{aligned}$$ where we have used Lemma \[represent\] and Proposition \[dueh\].
This shows, thanks to Theorem \[impli\], that $E$ is a minimizer for Problem .
at (0,0) [![Minimizer and calibration for three points, located at the vertices of an equilater triangle[]{data-label="min3punti"}](trepunti "fig:"){width="95.00000%"}]{};
Let us consider the case where $S$ consists of three points $p_1,p_2,p_3$. First of all we focus our attention on the case in which the three points are the vertices of an equilateral triangle. We set without loss of generality $p_1=(-\sqrt{3}/2,-1/2), \,p_2=(\sqrt{3}/2,-1/2)$ and $p_3=(0,1)$.
The set $E_{min} \in \mathscr{P}_{ constr}(Y_\Sigma)$ (constructed from the minimal triple junction connecting $p_1,p_2,p_3$ following the procedure of Proposition \[costruzione\]) is colored in Figure \[min3punti\]. We define $\Phi$ as in Figure \[min3punti\]. The vectors represented by the arrows are the following: $$\label{calitripuntoeq}
\Phi^1 = (-1, 1/\sqrt{3}) , \quad \Phi^2 = (1, 1/\sqrt{3}),\quad \Phi^3 = (0, -2/\sqrt{3}) \,.$$ It is easy to check that the conditions in Definition \[caliconvering\] are satisfied.
Notice that the calibration for three points $p_1,p_2$ and $p_3$ which are the vertices of a triangle with all angles of amplitude less or equal than $120$ degrees is the same (up to a rotation and minor modifications of the extension outside the cuts and the convex envelope of the points) of the calibration for the equilateral triangle that we have just explicitly shown. Indeed in this case the minimal Steiner network is again the union of three segments (possibly with different lenghts) meeting in a triple junction with angles of $120$ degrees.
Hence, it remains to consider the cases in which the three points of $S$ form a triangle with one angle greater or equal than $120$ degrees. For simplicity let $d(p_1,p_2)=d(p_2,p_3)$ and $\alpha$ be the angle between the segment $\overline{p_1p_2}$ (respectively $\overline{p_2p_3}$) and the horizontal line (as in Figure \[posizione3punti\]).
(0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1) (0,-0.3)to\[out= 90,in=-90, looseness=1\] (0,0.003); (0,0)to\[out= -45,in=135, looseness=1\] (0.1,-0.1) (0,0)to\[out= -135,in=45, looseness=1\] (-0.1,-0.1) (0,-0.3)to\[out= 90,in=-90, looseness=1\] (0,0.003); (0,-1.75)–(0,0.5) (-2,-1)–(0,-1) (2,-1)–(0,-1); (-1,-0.66)–(0,-1) (1,-0.66)–(0,-1); (0,-1) circle (1.7pt); (-1,-0.66) circle (1.7pt); (1,-0.66) circle (1.7pt); plot([2.\*sin(r)]{}, [2.\*cos(r)]{}) ; plot([2.\*sin(r)]{}, [2.\*cos(r)]{}) ; (-0.5,-0.925)node\[left\][$\alpha$]{} (0.5,-0.925)node\[right\][$\alpha$]{}; (-1,-0.66)node\[left\][$p_1$]{} (-0.2,-1)node\[below\][$p_2$]{} (1,-0.66)node\[right\][$p_3$]{};
It is well known that in this case the Steiner configuration that connects the points $p_1,p_2,p_3$ reduces to the two segments $\overline{p_1p_2}$ and $\overline{p_2p_3}$. Again we construct the set $E_{min} \in \mathscr{P}_{ constr}(Y_\Sigma)$ from the minimal Steiner configuration following the procedure of Proposition \[costruzione\].
at (0,0) [![Minimizer and calibration for three points[]{data-label="min3deg"}](min3deg "fig:"){width="48.50000%"}]{};
at (0,0) [![Minimizer and calibration for three points[]{data-label="min3deg"}](cali3deg "fig:"){width="48.50000%"}]{};
The calibration, depending on the fixed angle $\alpha \in (0,\pi/6)$, is the following (see Figure \[min3deg\]): $$\label{caliangologrande}
\Phi^1 = (0, 0) , \quad \Phi^2 = (2\sin\alpha, -2\cos\alpha),\quad \Phi^3 = (-2\sin\alpha, -2\cos\alpha) \,.$$
For $\alpha=\pi/6$ the calibration coincides, up to a rotation and a translation (see Remark \[const\]), to the one for the triangle .
at (0,0) [![Minimizers and calibration for the vertices of a square[]{data-label="cal4"}](fig4punti "fig:"){width="100.00000%"}]{};
Given $S=\{p_1,p_2,p_3,p_4\}$ located at the vertices of a square, the Steiner problem does not admit a unique solution. Therefore a calibration (if it exists) must calibrate all the minimizers.
The two candidates minimizers $E_{\min,1}, E_{\min,2}$ for Problem are shown in Figure \[cal4\] on the left (we draw only the sheets $(D,i)$ for $i=1,\ldots,4$). The calibration for $E_{min, 1}$ and $E_{min, 2}$ is defined as in Figure \[cal4\] on the right. Again, we draw the vector field only in the sheets $(D,i)$ for $i=1,\ldots,4$ and we employ the usual convention that where the vector field is not written it is equal to $(0,0)$. The reader can easily verify that the vector field defined in this way is a calibration on $Y_\Sigma$ for both the minimizers $E_{min, 1}$ and $E_{min, 2}$.
Calibrations in families {#famiglie}
========================
The aim of this section is to introduce a weaker definition of calibration. The idea is finding a way to divide the set of competitors in different families defining an appropriate notion of calibration in each family with a weaker condition (**2**). Then we calibrate the minimizers of each family separately and in conclusion we compare the energy of the minimizers to find the explicit solutions of Problem .
Let $\mathcal{J} \subset \{1,\ldots,m\}\times \{1,\ldots,m\}$ be a subset of the Cartesian product of the indices. Given $E \in \mathscr{P}_{constr}(Y_\Sigma)$ we define $$E^{i,j}:= \partial^\ast E^i\cap \partial^\ast E^j$$ and $$\label{famiglia}
\mathcal{F}(\mathcal{J}) := \{E \in \mathscr{P}_{constr}(Y_\Sigma):
{\mathcal{H}}^1(E^{i,j}) = 0 \mbox{ for every } (i,j) \in \mathcal{J}\}.$$
\[partcal\] Given $E \in \mathscr{P}_{constr}(Y_\Sigma)$, a calibration for $E$ in $\mathcal{F}(\mathcal{J})$ is an *approximately regular* vector field $\Phi :Y_{\Sigma}\to\mathbb{R}^2$ such that
1. ${{\rm div}\,}\Phi=0$ (in the sense of the distributions);
2. $|\Phi^i(x) - \Phi^j(x)| \leq 2$ for every $i,j = 1,\ldots m$ such that $(i,j) \notin \mathcal{J}$ and for every $x\in D$;
3. $\displaystyle \int_{Y_\Sigma} \Phi \cdot D\chi_{E}= P(E)$.
\[implipar\] Given $\mathcal{J}$ as above and $E\in\mathcal{F}(\mathcal{J})$, if $\Phi :Y_{\Sigma}\to\mathbb{R}^2$ is a calibration for $E$ in the family $\mathcal{F}(\mathcal{J})$, then $$P(E)\leq P(F)$$ for every $F\in \mathcal{F}(\mathcal{J})$. In particular $E$ minimizes the perimeter in the class $\mathcal{F}(\mathcal{J})$.
The proof is similar to that of Theorem \[impli\], and it is omitted.
We want to use Definition \[partcal\] to validate the minimality of a candidate minimizer for the Steiner problem. To this aim we need to assign any competitor in $\mathscr{P}_{constr}(Y_\Sigma)$ to at least one family. Notice that there exist sets $E\in \mathscr{P}_{constr}(Y_\Sigma)$ such that $ {\mathcal{H}}^1(E^{i,j})>0$ for every couple of indices $(i,j)\in \{1,\ldots,m\}\times \{1,\ldots,m\}$, hence it is not possible to cover $\mathscr{P}_{constr}(Y_\Sigma)$ with non-trivial families $\mathcal{F}(\mathcal{J}_i)$. For this reason we restrict to $\mathscr{P}^T_{constr}(Y_\Sigma)$.
We call $T$ the set of all connected networks without loops (see Definition \[conetw\]). Moreover we call $\mathscr{P}_{constr}^T(Y_\Sigma)$ the set of all $E\in\mathscr{P}_{constr}(Y_\Sigma)$ such that $\overline{p(\partial^\ast E)}$ is an element of $T$.
Consider the following:
\[stp\] Given $S$ a finite sets of points in $\mathbb{R}^2$ we look for a network in $T$ with minimal length that connects the points of $S$.
It is well known that Problem \[stp\] is equivalent to the Steiner problem defined in (see, for instance [@paopaostepanov]). Therefore if we define $\mathscr{A}^T_{constr}(S) = \inf \left\{P(E) : E\in\mathscr{P}^T_{constr}(Y_\Sigma)\right\}$ we infer, thanks to Theorem \[samelength\], that $\mathscr{A}^T_{constr}(S) = \mathscr{A}_{constr}(S)
$.
\[mincalifam\] Suppose that there exists $\mathcal{J}_1, \ldots, \mathcal{J}_N \subset \{1,\ldots,m\}\times \{1,\ldots,m\}$ such that $$\label{parti}
\mathscr{P}^T_{constr}(Y_\Sigma) \subseteq \bigcup_{i = 1}^N \mathcal{F}(\mathcal{J}_i).$$ If for every $i=1,\ldots,N$ there exists a calibration $\Phi_i$ for $E_i$ in $\mathcal{F}(\mathcal{J}_i)$, then $$\label{mineq}
\mathscr{A}_{\rm{constr}}(S)= \min\{P(E_i): i= 1,\ldots,N\}.$$ In other words the set $E_i$ with less perimeter is the absolute minimizer of Problem .
Fix $F\in \mathscr{P}^T_{constr}(Y_\Sigma)$. Thanks to there exists at least one $i\in \{1,\ldots,N\}$ such that $F\in \mathcal{F}(\mathcal{J}_i)$. Proposition \[implipar\] implies that $P(E_i)\leq P(F)$. Thus $$\min\{P(E_i): i= 1,\ldots,N\} \leq P(F)\,,$$ that is .
In Subsection \[aaaah\] we will see that it is relevant *how coarse* is the decomposition of $\mathscr{P}^T_{constr}(Y_\Sigma)$ in families $\mathcal{F}(\mathcal{J}_i)$. Indeed if the cardinality of $\mathcal{J}_i$ is small, then one needs less families to cover $\mathscr{P}^T_{constr}(Y_\Sigma)$, but the task of finding an explicit calibration in $\mathcal{F}(\mathcal{J}_i)$ results more challenging.
Examples {#aaaah}
--------
Suppose that the finite set $S=\{p_1,\ldots,p_m\}$ consists of $m$ points located on the boundary of a smooth, open, convex set $A\subset \mathbb{R}^2$. For simplicity we label the points on the boundary of $A$ in a anticlockwise order and from now on we consider the indices $i=1,\ldots, m$ cyclically identified modulus $m$.
It is not restrictive to suppose that each competitor $\Gamma\in T$ is contained in $A$, so that $T$ induces a partition of $A\setminus \Gamma$ in $m$ connected sets $\{A_\Gamma^1,\ldots,A_\Gamma^m\}$ labelled in such a way that $\{p_i,p_{i+1}\}\subset\partial A^{m+1-i}$. Calling $A_\Gamma^{i,j} := \partial A_\Gamma^{i} \cap \partial A_\Gamma^{j}$ for every $i,j= 1,\ldots,m$, the Steiner problem can be rephrased as $$\label{ste2}
\min\left\{\sum_{i<j} {\mathcal{H}}^1(A_\Gamma^{i,j}): \Gamma\in T\right\}\,.$$
We now suggest a general and explicit way to cover $\mathscr{P}^T_{constr}(Y_\Sigma)$ with families $\mathcal{F}(\mathcal{J}_i)$ in order to use the notion of calibration in $\mathcal{F}(\mathcal{J}_i)$ and Proposition \[mincalifam\] to show explicit solution of Problem .
\[1split\] Consider $\Gamma \in T$ inducing the partition $\{A_\Gamma^1,\ldots,A_\Gamma^m\}$ and suppose that there exists $i\neq j$ such that $A^{i,j}_\Gamma \neq \emptyset$. Then for every $0\leq k_1<i<k_2<j \leq m$ (or $0\leq j<k_1<i<k_2 \leq m$) we have that $${\mathcal{H}}^1(A_\Gamma^{k_1,k_2})=0\,.$$
By contradiction it is enough to notice that if ${\mathcal{H}}^1(A_\Gamma^{k_1,k_2})>0$, then the interior of $\overline{A_\Gamma^{k_1} \cup A_\Gamma^{k_2}}$ is an open connected set that separates $A_\Gamma^i$ and $A_\Gamma^j$. Hence we infer $A_{\Gamma}^{i,j} = \emptyset$.
We construct the covering space $Y_\Sigma$ choosing an admissible pair of cuts in the following way: the cut $\Sigma'$ coincides with $\partial A$ and the cut $\Sigma$ lies outside $A$. Then, thanks to Proposition \[costruzione\], it is possible to associate to the network $\Gamma$, and hence to the partition $\{A_\Gamma^1,\ldots,A_\Gamma^m\}$, a set $E_{\Gamma}$ in the covering space $Y_\Sigma$ (simply setting $E_\Gamma^j=A_\Gamma^j$) such that $$\label{poi}
p(\overline{\partial^\ast E_\Gamma}) =\Gamma= \bigcup_{i,j}A_\Gamma^{i,j} \quad \mbox{and}
\quad E^{i,j}_\Gamma = A_\Gamma^{i,j}.$$
Thanks to Lemma \[1split\] is trivially true replacing $A_\Gamma^{i,j}$ with $E_\Gamma^{i,j}$.
We define $$\mathcal{F}_{i,j} = \{\Gamma \in T: A^{i,j}_\Gamma \neq \emptyset\}\,.$$ It is easy to see that $T$ can be covered in the following way: $$\label{spliiiiit}
T = \bigcup_{k=2}^{\lfloor\frac{m}{2}\rfloor} \bigcup_{ |i-j| = k} \mathcal{F}_{i,j}\,.$$ This covering induces automatically a covering of $\mathscr{P}_{constr}(Y_\Sigma)$. Consider for instance a family $\mathcal{F}_{i,i+\lfloor\frac{m}{2}\rfloor}$ for a fixed $i$ in $\{1,\ldots,m\}$. Thanks to Lemma \[1split\] we have that $${\mathcal{H}}^1(E^{k,l}_{\Gamma}) = 0$$ for all $(k,l) \in\{i+1,\ldots,i+\lfloor\frac{m}{2}\rfloor-1\}\times\{i+\lfloor\frac{m}{2}\rfloor+1,i-1\}$. This property defines a family $\mathcal{F}(\mathcal{J})$ according to with $\mathcal{J}=\{i+1,\ldots,i+\lfloor\frac{m}{2}\rfloor-1\}\times\{i+\lfloor\frac{m}{2}\rfloor+1,i-1\}$.
\[penta\]
Consider $S=\{p_1,p_2,p_3,p_4,p_5\}$ located at the vertices of a regular pentagon. First we divide the elements of $\mathscr{P}^T_{constr}(Y_\Sigma)$ in families. According to , we cover $T$ with five families $\mathcal{F}_{i,j}$ as follows: $$T= \bigcup_{|i-j| = 2} \mathcal{F}_{i,j}\,.$$ Then we split again each family $\mathcal{F}_{i,j}$ in two subfamilies $\mathcal{F}^1_{i,j}$ and $\mathcal{F}^2_{i,j}$ defined as $$\mathcal{F}^1_{i,j} = \mathcal{F}_{i,j} \cap \mathcal{F}_{i,j+1} \quad \mbox{and} \quad \mathcal{F}^2_{i,j} =
\mathcal{F}_{i,j} \cap \mathcal{F}_{i-1,j}\,.$$ This produces in principle $10$ families, but it is easy to see that $\mathcal{F}^k_{i,j}=\mathcal{F}^{k'}_{i',j'}$ for some $i,j,i',j'\in\{1,2,3,4,5\}$ and $k,k'\in\{1,2\}$. Hence, we obtain $$\mathscr{P}^T_{constr}(Y_\Sigma)=\bigcup_{i=1}^5\mathcal{F}(\mathcal{J}_i)\,,$$ with $$\begin{aligned}
\mathcal{J}_1=\{(1,3),(1&,4),(2,4)\},\quad\mathcal{J}_2=\{(1,3),(1,4),(3,5)\},
\quad\mathcal{J}_3=\{(1,3),(2,5),(3,5)\}, \\
&\mathcal{J}_4=\{(1,4),(2,4),(2,5)\}, \quad\mathcal{J}_5=\{(2,4),(2,5),(3,5)\}\,.\end{aligned}$$
It is known that the Steiner problem for $S$ has $5$ minimizers $\mathscr{S}_i$ for $i = 1,\ldots,5$ (obtained by rotation one from the other). Denoted by $E_{min,i}\in \mathscr{P}^T_{constr}(Y_\Sigma)$ for $i = 1,\ldots,5$ the sets associated with the minimizers of the Steiner problem, it is easy to see that that $E_{min,i}\in\mathcal{F}(\mathcal{J}_i)$ for $i = 1,\ldots,5$. Our aim is to prove that $E_{min,i}$ is a minimizer in $\mathcal{F}(\mathcal{J}_i)$ constructing a vector field $\Phi_i$ that is a calibration for $E_{min,i}$ in $\mathcal{F}(\mathcal{J}_i)$.
On the left of Figure \[cali5\] is shown the set $E_{min,5}\in \mathscr{P}^T_{constr}(Y_\Sigma)$, on the right a calibration for $E_{min,5}$ in $\mathcal{F}(\mathcal{J}_{5})$. The vector field represented by the arrows is the following: $$\begin{aligned}
\Phi^1=(0,0),\quad \Phi^2 =(2,0), \quad \Phi^3=(1,-\sqrt{3}),\quad
\Phi^4(-1,-\sqrt{3}),\quad \Phi^5=(-2,0)\,\end{aligned}$$ and it is easy to verify that it is indeed a calibration for $E_{min,5}$ in $\mathcal{F}(\mathcal{J}_{5})$.
As the minimizers $\mathscr{S}_i$ with $i = 1,\ldots,5$ for the Steiner problem are obtained by rotation one from the other, it is easy to construct for $E_{min,i}$ (with $i=1,2,3,4$) a calibration in $\mathcal{F}(\mathcal{J}_i)$ similar to the one for $E_{min,5}$ in $\mathcal{F}(\mathcal{J}_5)$.
To summarize we have split the set $\mathscr{P}_{constr}^T(Y_\Sigma)$ and we have exhibited a calibration in each family for the corresponding $E_{min,i}$. As $\mathcal{H}^1(\mathscr{S}_i)=\mathcal{H}^1(\mathscr{S}_j)$ for $i,j\in\{1,\ldots,5\}$, thanks to Proposition \[dueh\] we have also that $P(E_{min,i}) = P(E_{min,j})$ for every $i,j\in \{1,\ldots,5\}$. Thus applying Proposition \[mincalifam\] we infer that $E_{min,i}$ are minimizers of Problem for every $i=1\ldots,5$, as we wanted to prove.
at (0,0) [![The minimizer $u_{min,5}$ for five points at the vertices of a regular pentagon and a calibration for the family $\mathcal{F}(\mathcal{J}_5)$[]{data-label="cali5"}](min5punti "fig:"){width="44.00000%"}]{};
at (0,0) [![The minimizer $u_{min,5}$ for five points at the vertices of a regular pentagon and a calibration for the family $\mathcal{F}(\mathcal{J}_5)$[]{data-label="cali5"}](cali5punti "fig:"){width="44.00000%"}]{};
In [@coveringbis] we prove that if $\Phi:Y\to\mathbb{R}^2$ is a calibration for $E\in\mathcal{P}_{constr}$, then $E$ is a minimizer not only among all (constrained) finite perimeter sets, but also in the larger class of finite linear combinations of characteristic functions of (constrained) finite perimeter sets. Then if there exists an element of this larger class with strictly less energy of the minimizer of Problem , a calibration for such a minimizer cannot exist. This is the case when $S=\{p_1,\ldots,p_5\}$ with $p_i$ the vertices of a regular pentagon. This counterexample can be constructed adapting the example by Bonafini [@bonafini] in the framework of rank one tensor valued measures to our setting. Hence the tool of the calibration in families is necessary, in this case, to prove the minimality of the candidate by a calibration argument.
We fix the points of $S$ as the vertices of a regular hexagon in the following way: $p_1=(-1/2,\sqrt{3}/2)$, $p_2=(-1,0)$, $p_3=(-1/2,-\sqrt{3}/2)$, $p_4=(1/2,-\sqrt{3}/2)$, $p_5=(1,0)$, $p_6=(1/2,\sqrt{3}/2)$.
As in Example \[penta\] we start covering $\mathscr{P}^T_{constr}(Y_\Sigma)$ with explicit families $\mathcal{F}(\mathcal{J})$ of competitors. From we get $$T = \left( \bigcup_{ |i-j| = 3} \mathcal{F}_{i,j} \right) \cup
\left( \bigcup_{ |i-j| = 2} \mathcal{F}_{i,j}\right)\,.$$
For given $i,j\in \{1,\ldots,6\}$ such that $|i-j|=3$ we further split $\mathcal{F}_{i,j}$ in four classes $(\mathcal{F}^k_{i,j})_{k=1,\ldots,4}$ as follows: $$\mathcal{F}^1_{i,j} = \mathcal{F}_{i,j} \cap (\mathcal{F}_{i,j-1} \cup \mathcal{F}_{i,j+1}), \qquad \mathcal{F}^2_{i,j} = \mathcal{F}_{i,j} \cap (\mathcal{F}_{i+1,j} \cup \mathcal{F}_{i-1,j})\, ,$$ $$\mathcal{F}^3_{i,j} = \mathcal{F}_{i,j} \cap (\mathcal{F}_{i,j-1} \cup \mathcal{F}_{i+1,j}), \qquad \mathcal{F}^4_{i,j} = \mathcal{F}_{i,j} \cap (\mathcal{F}_{i-1,j} \cup \mathcal{F}_{i,j+1})\, .$$
As in the previous example, thanks to Lemma \[1split\] applied to the families $ \mathcal{F}^k_{i,j}$ with $i,j\in \{1,\ldots,6\}$ and $k\in\{1,\ldots,4\}$, we can associate the respectively families $\mathcal{F}(\mathcal{J}_i)$ with $J_i$ defined as follows: $$\begin{aligned}
&\mathcal{J}_1=\{(2,4),(2,5),(2,6),(3,5),(3,6),(4,6)\},
&\mathcal{J}_2=\{(1,3),(1,4),(1,5),(3,5),(3,6),(4,6)\},\\
&\mathcal{J}_3=\{(1,4),(1,5),(2,4),(2,5),(2,6), (4,6)\},
&\mathcal{J}_4=\{(1,3),(1,5),(2,5),(2,6),(3,5),(3,6)\},\\
&\mathcal{J}_5=\{(1,3),(1,4),(2,4),(2,6),(3,6),(4,6)\},
&\mathcal{J}_6=\{(1,3),(1,4),(1,5),(2,4),(2,5),(3,5)\},\\
&\mathcal{J}_7=\{(1,5),(2,4),(2,5),(2,6),(3,5),(3,6)\},
&\mathcal{J}_8=\{(1,3),(1,4),(2,6),(3,5),(3,6),(4,6)\},\\
&\mathcal{J}_9=\{(1,3),(1,4),(1,5),(2,4),(2,5),(4,6)\},
&\mathcal{J}_{10}=\{(1,3),(2,5),(2,6),(3,5),(3,6), (4,6)\},\\
&\mathcal{J}_{11}=\{(1,3),(1,4),(1,5),(2,4),(3,6),(4,6)\},
&\mathcal{J}_{12}=\{(1,4),(1,5),(2,4),(2,5),(2,6),(3,5)\}.\\\end{aligned}$$ We notice that the families $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$ can be obtained from the first one by a cyclic permutation of the indices. Also the families $\mathcal{F}(\mathcal{J}_8)$ and $\mathcal{F}(\mathcal{J}_9)$ can be obtained by a cyclic permutation of the indices from $\mathcal{F}(\mathcal{J}_7)$, and the same holds for the families $\mathcal{F}(\mathcal{J}_{10})$, $\mathcal{F}(\mathcal{J}_{11})$, and $\mathcal{F}(\mathcal{J}_{12})$ (see Figure \[familieshex\]).
Consider now the case in which $|i-j|=2$ for given $i,j\in \{1,\ldots,6\}$. Here the situation is easier, as we find two families $\mathcal{F}_{i,j}$ and it is not necessary to consider a further refinement of the classes. In terms of $\mathcal{F}(\mathcal{J}_i)$ we get $$\mathcal{J}_{13}=\{(1,4),(2,4),(2,5),(2,6),(3,6),(4,6)\}\, ,\;
\mathcal{J}_{14}=\{(1,3),(1,4),(1,5),(2,5),(3,5), (3,6)\}\, ,$$ where again $\mathcal{F}(\mathcal{J}_{14})$ is obtained by a cyclic permutation of the indices from $\mathcal{F}(\mathcal{J}_{13})$. In conclusion the subdivision in families is as follows: $$\mathscr{P}^T_{constr}(Y_\Sigma)=\bigcup_{i=1}^{14}\mathcal{F}(\mathcal{J}_i)\,.$$
at (0,0) [![On the left we represent the projection onto the base set of the essential boundary of an element of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, then of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{7,8,9\}$ and $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{10,11,12\}$, on the right $\mathcal{F}(\mathcal{J}_{i})$ with $i\in\{13,14\}$. []{data-label="familieshex"}](fam1 "fig:"){width="23.10000%"}]{};
at (0,0) [![On the left we represent the projection onto the base set of the essential boundary of an element of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, then of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{7,8,9\}$ and $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{10,11,12\}$, on the right $\mathcal{F}(\mathcal{J}_{i})$ with $i\in\{13,14\}$. []{data-label="familieshex"}](fam2 "fig:"){width="23.10000%"}]{};
at (0,0) [![On the left we represent the projection onto the base set of the essential boundary of an element of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, then of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{7,8,9\}$ and $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{10,11,12\}$, on the right $\mathcal{F}(\mathcal{J}_{i})$ with $i\in\{13,14\}$. []{data-label="familieshex"}](fam3 "fig:"){width="23.10000%"}]{};
at (0,0) [![On the left we represent the projection onto the base set of the essential boundary of an element of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, then of the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{7,8,9\}$ and $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{10,11,12\}$, on the right $\mathcal{F}(\mathcal{J}_{i})$ with $i\in\{13,14\}$. []{data-label="familieshex"}](fam4 "fig:"){width="23.10000%"}]{};
It is well know that if the points of $S$ lies at the vertices of a regular hexagon, then there are six minimizers $\mathscr{S}_i$ for the Steiner problem. Calling $E_{\min,i}$ with $i\in\{1,\ldots,6\}$ the sets in $\mathscr{P}_{constr}(Y_\Sigma)$ associated to $\mathscr{S}_i$, we have that $E_{\min,i}\in \mathcal{F}(\mathcal{J}_i)$ for $i\in \{1,\ldots,6\}$.
In order to use Proposition \[mincalifam\] we have to find a calibration $\Phi^i$ for an explicit set $E_{i}\in \mathcal{F}(\mathcal{J}_i)$ for every $i\in\{1,\ldots,14\}$. The global minimizers $E_{\min,i}$ are clearly minimizers in their families, so they are the natural candidate minimizers for the families $ \mathcal{F}(\mathcal{J}_i)$ for $i\in\{1,\ldots,6\}$; it is more challenging to propose a minimizer for the other families. Our candidate minimizers are shown in Figure \[candidati\].
at (0,0) [![From left to right: the projection onto the base set of the essential boundary of candidate minimizers in the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, $i\in\{7,8,9\}$ $i\in\{10,11,12\}$ and $i\in\{13,14\}$.[]{data-label="candidati"}](minimizer1 "fig:"){width="23.10000%"}]{};
at (0,0) [![From left to right: the projection onto the base set of the essential boundary of candidate minimizers in the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, $i\in\{7,8,9\}$ $i\in\{10,11,12\}$ and $i\in\{13,14\}$.[]{data-label="candidati"}](minimizer2 "fig:"){width="23.10000%"}]{};
at (0,0) [![From left to right: the projection onto the base set of the essential boundary of candidate minimizers in the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, $i\in\{7,8,9\}$ $i\in\{10,11,12\}$ and $i\in\{13,14\}$.[]{data-label="candidati"}](minimizer3 "fig:"){width="23.10000%"}]{};
at (0,0) [![From left to right: the projection onto the base set of the essential boundary of candidate minimizers in the family $\mathcal{F}(\mathcal{J}_i)$ with $i\in\{1,\ldots,6\}$, $i\in\{7,8,9\}$ $i\in\{10,11,12\}$ and $i\in\{13,14\}$.[]{data-label="candidati"}](minimizer4 "fig:"){width="23.10000%"}]{};
We write explicitly the calibration $\Phi_i$ (written, as usual, by means of the pullbacks on every sheet $\Phi_i^j$) for $E_{\min,i}\in \mathcal{F}(\mathcal{J}_i)$ with $i=1,10,13$. The other calibrations are easy variants of the previous ones. Moreover we give the expression of the vector field only inside $p^{-1}(\mbox{Conv}(S))$, as the divergence free extension in $Y_\Sigma$ can be easily achieved. We get $$\begin{array}{lll}
\Phi^{1}_1=(0,0)\,, &\Phi^{2}_1 =(\sqrt{3},1)\,, &\Phi^{3}_1=(\sqrt{3},-1)\,,\\
\Phi^{4}_1=(0,-2)\,,&\Phi^{5}_1=(-\sqrt{3},-1)\,,& \Phi^{6}_1=(-\sqrt{3},1)\,,\\ \\
\Phi^{1}_{10}=(0,0)\,,&\Phi^{2}_{10} =\left( \frac{3\sqrt{3}}{\sqrt{7}},\frac{1}{\sqrt{7}}\right)\,,
&\Phi^{3}_{10}=\left( \frac{2\sqrt{3}}{\sqrt{7}}\,,-\frac{4}{\sqrt{7}}\right)\,,\\
\Phi^{4}_{10}=\left( -\frac{\sqrt{3}}{\sqrt{7}}\,,-\frac{5}{\sqrt{7}}\right) \,,
&\Phi^{5}_{10}=\left( -\frac{4\sqrt{3}}{\sqrt{7}},-\frac{6}{\sqrt{7}}\right) \,,
&\Phi^{6}_{10}=\left( -\frac{3\sqrt{3}}{\sqrt{7}},-\frac{1}{\sqrt{7}}\right) \,,\\ \\
\Phi^{1}_{13}=(0,0)\,, &\Phi^{2}_{13} =(2,0)\,, &\Phi^{3}_{13}=(1,-\sqrt{3})\,,\\
\Phi^{4}_{13}=(0,-2\sqrt{3})\,,
&\Phi^{5}_{13}=(-1,-\sqrt{3})\,, &\Phi^{6}_{13}=(-2,0)\,.
\end{array}$$
The vector field $\Phi_{10}$ is a calibration for the set $E\in \mathscr{P}^T_{constr}(Y_\Sigma)$ associated to the third network represented in Figure \[candidati\] for $i=1$. It can be computed easily as follows: firstly we rotate the points of $S$ and the network by a rotation of angle $\alpha = -\arctan{(\frac{1}{3\sqrt{3}})}$ with the following rotation matrix: $$R(\alpha) = \begin{bmatrix}
\frac{3\sqrt{3}}{2\sqrt{7}} & \frac{1}{2\sqrt{7}}\\
-\frac{1}{2\sqrt{7}} &\frac{3\sqrt{3}}{2\sqrt{7}}\, \\
\end{bmatrix}
\,.$$ The network obtained in this way can be calibrated with the following vector field: $$\begin{array}{lll}
\widetilde{\Phi}^{1}_{10}=(0,0)\,, &\widetilde{\Phi}^{2}_{10} =(2,0)\,, &\widetilde{\Phi}^{3}_{10}=(1,-\sqrt{3})\,,\\
\widetilde{\Phi}^{4}_{10}=(-1,-\sqrt{3})\,,
&\widetilde{\Phi}^{5}_{10}=(-3,-\sqrt{3})\,, &\widetilde{\Phi}^{6}_{10}=(-2,0)\,\,.
\end{array}$$ Finally we rotate back with the matrix $R(-\alpha)$ to get $\Phi_{10}$.
An easy computation shows that $P(E_{min,i}) < P(E_{min,j})$ for $i\in \{1,\ldots,6\}$ and $j\in \{7,\ldots,14\}$. Thus applying Proposition \[mincalifam\] we infer that $E_{min,i}$ are minimizers of $\mathscr{A}_{constr}(S)$ for every $i=1,\ldots,6$, as we wanted to prove.
The above division in families is the finest possible one. It has the advantage that we obtain constant calibrations. However from the numerical point of view this choice is not convenient. Indeed the complexity is simply shifted from solving the non–convex original problem to finding all the families. It would be better to consider a more coarse division in families in which one can in any case find a calibration.
Appendix: Divergence theorem on $Y_\Sigma$ {#appe .unnumbered}
==========================================
Consider $E,F\in \mathscr{P}_{constr}(Y_\Sigma)$ and let $\Phi : Y_\Sigma \to {\mathbb{R}}^2$ be an approximately regular vector field such that ${{\rm div}\,}\Phi = 0$ (in the sense of distributions) in $Y_\Sigma$. Then $$\int_{Y_\Sigma} \Phi \cdot D\chi_E = \int_{Y_\Sigma} \Phi \cdot D\chi_F.$$
From , one gets: $$D\chi_E = \sum_{j=1}^m D\chi_{E^j} {\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt
\vrule height .5pt width 6pt depth 0pt}}\nolimits}D
+ \sum_{j'=m+1}^{2m} D\chi_{E^{j'}}{\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt
\vrule height .5pt width 6pt depth 0pt}}\nolimits}(\Sigma\setminus S)\,.$$ Fix $\varepsilon>0$ and call $\Omega_\varepsilon$ the $\varepsilon$–tubular neighbourhood of $\Omega$. Using the divergence theorem for approximately regular vector fields in $D\subset {\mathbb{R}}^2$ (see [@mumford]) and the definition of $D\chi_{E^{j'}}$ we have $$\begin{aligned}
& \int_{Y_\Sigma} \Phi \cdot D\chi_E
= \sum_{j=1}^m \int_{D} \Phi^j\cdot D\chi_{E^j}
+ \sum_{j'=m+1}^{2m} \int_{\Sigma} \Phi^{j'}\cdot D\chi_{E^{j'}}\nonumber\\
=&-\sum_{j=1}^m \int_{\Sigma} \big[(\Phi^j)^+ \chi_{E^j}^+ - (\Phi^j)^-\chi_{E^j}^-\big] \cdot \nu_\Sigma\, d{\mathcal{H}}^{1}
- \sum_{j=1}^m \int_{\partial \Omega_\varepsilon} \chi_{E^{j}}\Phi^j
\cdot \nu_{\partial \Omega_\varepsilon}\, d{\mathcal{H}}^1
\nonumber\\
+& \sum_{j'=m+1}^{2m} \int_{\Sigma} \big[(\chi_{E^{j'}})^+
- (\chi_{E^{j'}})^-\big]\Phi^{j'} \cdot \nu_\Sigma \, d{\mathcal{H}}^1\, ,\label{for}\end{aligned}$$ where $ \nu_{\partial \Omega_\varepsilon}$ is the inner unit normal to $\partial \Omega_\varepsilon$. From now on we will call $$a_{h}^+:=(\Phi^h)^+ \cdot \nu_{\Sigma},
\quad a_{h}^-:=(\Phi^h)^- \cdot \nu_{\Sigma} \quad \mbox{ for } h=1,\ldots,2m\, .$$ As $\Phi$ is approximately regular (see Definition \[app\] and Definition \[appreg\]) one has that $$\label{approx}
a_{j'}^+(x) = a^-_{j'}(x) = (\Phi^{j'} \cdot \nu_\Sigma)(x) \quad \mbox{ for }{\mathcal{H}}^1- a.e\ x\in \Sigma$$ for every $j'=m+1, \ldots, 2m$. Moreover as $E\subset Y_\Sigma$, one can verify that $$\label{t1}
\sum_{j=1}^m a^+_j(x) (\chi_{E^j})^+(x) = \sum_{j'=m+1}^{2m} a^+_{j'}(x) (\chi_{E^{j'}})^+(x)$$ and $$\label{t2}
\sum_{j=1}^m a^-_j(x) (\chi_{E^j})^-(x) = \sum_{j'=m+1}^{2m} a^-_{j'}(x) (\chi_{E^{j'}})^-(x)$$ for ${\mathcal{H}}^1$- a.e $x\in \Sigma$.\
Using , , on Formula it is easy to see that $$\int_{Y_\Sigma} \Phi \cdot D\chi_E
=- \sum_{j=1}^m \int_{\partial \Omega_\varepsilon} \Phi^j \chi_{E^{j}}
\cdot \nu_{\partial \Omega_\varepsilon}\, d{\mathcal{H}}^1 \,$$ and analogously $$\int_{Y_\Sigma} \Phi \cdot D\chi_F
=- \sum_{j=1}^m \int_{\partial \Omega_\varepsilon} \Phi^j \chi_{F^{j}}
\cdot \nu_{\partial \Omega_\varepsilon}\, d{\mathcal{H}}^1\, .$$ As $E, F \in \mathscr{P}_{constr}(Y_\Sigma)$, the functions $\chi_{E^j}$ and $\chi_{F^j}$ have same boundary conditions on $\partial \Omega_\varepsilon$ for every $j$. This gives equation as we wanted to prove.
[^1]: Institut für Mathematik, Karl–Franzens–Universität, Heinrichstrasse 36, 8010, Graz, Austria
[^2]: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy
|
---
author:
- |
Stephen William Semmes\
Rice University\
Houston, Texas
title: 'Potpourri, 10'
---
Let $(M, d(x, y))$ be a metric space. Thus $M$ is a nonempty set and $d(x, y)$ is a real-valued function defined for $x, y \in M$ such that $d(x, y) \ge 0$ for all $x, y \in M$, $d(x, y) = 0$ if and only if $x = y$, $d(x, y) = d(y, x)$ for all $x, y \in M$, and $$d(x, z) \le d(x, y) + d(y, z)$$ for all $x, y, z \in M$. If we have the stronger condition that $$d(x, z) \le \max(d(x, y), d(y, z))$$ for all $x, y, z \in M$, then we say that $d(x, y)$ is an *ultrametric* on $M$.
Of course the real line ${\bf R}$ with the standard metric $|x
- y|$ is a metric space. We shall discuss some more examples soon.
Let $(M, d(x, y))$ be a metric space, and let $E$ be a subset of $M$. We say that $E$ is bounded if the collection of real numbers $d(x, y)$, $x, y \in E$ are bounded, and in this event we define the diameter of $E$ by $$\diam E = \sup \{d(x, y) : x, y \in E\}.$$ The diameter of the empty set is defined to be $0$, and we can also define the diameter of an unbounded subset of $M$ to be $+\infty$.
For each $p \in M$ and $r > 0$ define the open and closed balls in $M$ with center $p$ and radius $r$ by $$B(p, r) = \{x \in M : d(x, p) < r\}$$ and $$\overline{B}(p, r) = \{x \in M : d(x, p) \le r\},$$ respectively. Thus $$B(p, r) \subseteq \overline{B}(p, r)$$ and $$\diam \overline{B}(p, r) \le 2 \, r$$ for all $p \in M$ and $r > 0$.
If $E$ is a subset of $M$ and $\mathcal{A}$ is a family of subsets of $M$, then we say that $\mathcal{A}$ is an *admissible covering* of $E$ if $$E \subseteq \bigcup_{A \in \mathcal{A}} A$$ and $\mathcal{A}$ has at most countably many elements. In some situations one might wish to restrict one’s attention to admissible coverings by bounded subsets of $M$. One might also be interested in coverings which contain only finitely many subsets of $M$.
For example, if $E$ is any subset of $M$, then we can simply cover $E$ by $E$ itself, or by $M$. If $p$ is any element of $M$, then the family of balls $B(p, l)$, with $l$ a positive integer, is an admissible covering of any subset of $M$ by bounded subsets of $M$.
Suppose that $h(t)$ is a monotone increasing continuous real-valued function defined for nonnegative real numbers $t$ such that $h(0) = 0$, $h(t) > 0$ when $t > 0$. Put $$h(+\infty) = \sup \{h(t) : 0 \le t < \infty\},$$ which is equal to $+\infty$ if $h(t)$ is unbounded.
If $E$ is any subset of $M$, then we define $\mu_h(E)$, the Hausdorff content of $E$ relative to the function $h$, to be the infimum of $$\sum_{A \in \mathcal{A}} h(\diam A)$$ over all admissible coverings $\mathcal{A}$ of $E$. As a special case, if $\alpha$ is a positive real number, then we can take $h(t) =
t^\alpha$. For this we write $\mu^\alpha(E)$ instead of $\mu_h(E)$, and $\mu^\alpha(E)$ is called the $\alpha$-dimensional Hausdorff content of $E$.
Of course $\mu_h(E) \ge 0$ for all subsets $E$ of $M$, and $\mu_h(E) = 0$ if $E$ is the empty set, or if $E$ contains only one element. Using the trivial covering of $E$ by $E$ we get that $$\label{mu_h(E) le h(diam E)}
\mu_h(E) \le h(\diam E).$$
In general $\mu_h(E)$ may be equal to $+\infty$ when $E$ is unbounded. If $h$ is unbounded and we restrict ourselves to admissible coverings with only finitely many elements, then the analogue of $\mu_h(E)$ is equal to $+\infty$ when $E$ is unbounded.
If $E$, $\widetilde{E}$ are subsets of $M$ with $$\widetilde{E} \subseteq E,$$ then $$\mu_h(\widetilde{E}) \le \mu_h(E).$$ If $E_1$, $E_2$ are subsets of $M$, then $$\mu_h(E_1 \cup E_2) \le \mu_h(E_1) + \mu_h(E_2).$$ This follows easily by combining arbitrary admissible coverings of $E_1$, $E_2$ to get admissible coverings of $e_1 \cup E_2$, and it would also work for the analogue of $\mu_h(E)$ defined using finite coverings of $E$. If $E_1, E_2, E_3, \ldots$ is a sequence of subsets of $M$, then $$\mu_h\bigg(\bigcup_{j=1}^\infty E_j\bigg)
\le \sum_{j=1}^\infty \mu_h(E_j),$$ because arbitrary admissible coverings of the $E_j$’s can be combined to give an admissible covering of the union. This uses the fact that a countable union of countable sets is a countable set.
For the analogue of $\mu_h$ defined using finite coverings we have in particular that $\mu_h(E) = 0$ when $E$ is a finite set. If we use countable coverings as above then $\mu_h(E) = 0$ when $E$ is at most countable.
Notice that the closure of a bounded subset of a metric space $M$ is also bounded and has the same diameter. Using this one can check that one gets the same result for $\mu_h(E)$ if one restricts one’s attention to admissible coverings of $E$ by closed subsets of $M$.
The same remark applies to the analogue of $\mu_h(E)$ based on finite coverings of $E$. For this it follows that one gets the same answer for the closure of $E$ as for $E$. This does not work in general for $\mu_h(E)$ based on coverings which are at most countable.
If $A$ is any subset of $M$ and $r$ is a positive real number, put $$A(r) = \{x \in M : \hbox{ there is an } a \in A
\hbox{ such that } d(x, a) < r\}.$$ Thus $A \subseteq A(r)$, and one can check that $A(r)$ is always an open subset of $M$.
If $A$ is a bounded subset of $M$, then $A(r)$ is bounded too, and one can check that $$\diam A(r) \le \diam A + 2 \, r.$$ In the case of an ultrametric we have that the diameter of $A(r)$ is less than or equal to the maximum of the diameter of $A$ and $r$. Using this one can show that we can restrict ourselves to coverings by open sets and get the same answers for Hausdorff content.
If $E$ is a compact subset of $M$, then for each open covering of $E$ there is a finite subcovering of $E$. Thus the versions of $\mu_h(E)$ based on coverings which are finite or at most countable give the same result when $E$ is compact.
Let $\epsilon > 0$ be given, and let $E$ be a subset of $M$. An admissible covering $\mathcal{A}$ of $E$ in $M$ is called an $\epsilon$-covering if $$\diam A < \epsilon$$ for all $A \in \mathcal{A}$. If $M$ is a separable metric space, meaning that there is a dense subset of $M$ which is at most countable, then there is an $\epsilon$-covering of $M$, and hence of any subset of $M$, for all $\epsilon > 0$.
As a variant of the Hausdorff content associated to $h(t)$, define $\mathcal{H}_{h, \epsilon}(E)$ to be the infimum of $\sum_{A
\in \mathcal{A}} h(\diam A)$ over all $\epsilon$-coverings $\mathcal{A}$ of $E$, if there are any, and otherwise put $\mathcal{H}_{h, \epsilon}(E) = + \infty$. If $\epsilon_1$, $\epsilon_2$ are positive real numbers with $\epsilon_1 \le
\epsilon_2$, then $$\mathcal{H}_{h, \epsilon_2}(E) \le \mathcal{H}_{h, \epsilon_1}(E).$$ For all $\epsilon > 0$ we have that $$\mu_h(E) \le \mathcal{H}_{h, \epsilon}(E).$$
Just as for Hausdorff content, we can restrict ourselves to $\epsilon$-coverings by open or closed subsets of $M$ and get the same result for $\mathcal{H}_{h, \epsilon}(E)$. If $E$ is a compact subset of $M$, then $\mathcal{H}_{h, \epsilon}(E)$ is equal to its analogue for finite coverings.
The Hausdorff measure of a subset $E$ of $M$ associated to the function $h(t)$ is defined by $$\mathcal{H}_h(E)
= \sup_{\epsilon > 0} \mathcal{H}_{h, \epsilon}(E).$$ Thus $$\mu_h(E) \le \mathcal{H}_{h, \epsilon}(E) \le \mathcal{H}_h(E)$$ for all $\epsilon > 0$. If $\alpha$ is a positive real number and $h(t) = t^\alpha$, then we write $\mathcal{H}^\alpha_\epsilon(E)$, $\mathcal{H}^\alpha(E)$ in place of $\mathcal{H}_{h, \epsilon}(E)$, $\mathcal{H}_h(E)$.
Notice that $\mathcal{H}_h(E) = 0$ when $E$ is the empty set or $E$ consists of just one element. If $E$, $\widetilde{E}$ are subsets of $M$ and $\widetilde{E} \subseteq E$, then $\mathcal{H}_h(\widetilde{E}) \le \mathcal{H}_h(E)$.
If $E_1, E_2, \ldots$ are subsets of $M$, then $$\mathcal{H}_{h, \epsilon} \bigg(\sum_{j=1}^\infty E_j \bigg)
\le \sum_{j=1}^\infty \mathcal{H}_{h, \epsilon}(E_j)$$ for all $\epsilon > 0$, and therefore $$\mathcal{H}_h\bigg(\sum_{j=1}^\infty E_j \bigg)
\le \sum_{j=1}^\infty \mathcal{H}_h(E_j).$$ For the analogue of $\mathcal{H}_h$ based on finite coverings we still have finite subadditivity.
Suppose that $E_1$, $E_2$ are subsets of $M$ and that there is an $\eta > 0$ such that $d(x, y) \ge \eta$ for all $x \in E_1$ and all $y \in E_2$. In this event we have that $$\mathcal{H}_{h, \epsilon}(E_1 \cup E_2)
\ge \mathcal{H}_{h, \epsilon}(E_1) + \mathcal{H}_{h, \epsilon}(E_2)$$ for all $\epsilon \in (0, \eta]$. Hence $$\mathcal{H}_h(E_1 \cup E_2)
\ge \mathcal{H}_h(E_1) + \mathcal{H}_h(E_2).$$
Notice that $\mu_h(E) = 0$ for a subset $E$ of $M$ if and only if for each $\eta > 0$ there is an admissible covering $\mathcal{A}$ of $E$ such that $$\sum_{A \in \mathcal{A}} h(A) < \eta.$$ This implies that $\mathcal{H}_h(E) = 0$, since the diameters of the covering sets $A$ have to be small anyway.
Let us consider the special case where $M$ is the real line with the standard metric. For this we can restrict ourselves to coverings by intervals in the definitions of Hausdorff measure and Hausdorff content, since every subset of the real line is contained in an interval with the same diameter. We can allow unbounded intervals such as the real line itself to accommodate unbounded subsets of the real line.
If $a$, $b$ are real numbers with $a \le b$, let $[a, b]$ be the usual closed interval with endpoints $a$, $b$, consisting of the real numbers $x$ such that $a \le x \le b$. Recall that this is a compact subset of the real line.
Clearly $$\mu^1([a, b]) \le b - a.$$ For each $\epsilon > 0$ we can cover $[a, b]$ by finitely many closed intervals, each of length less than $\epsilon$, and so that the sum of their lengths is equal to $b - a$, which implies that $$\mathcal{H}^1_\epsilon ([a, b]) \le b - a$$ and therefore $$\mathcal{H}^1([a, b]) \le b - a.$$
Conversely, $$\mu^1([a, b]) \ge b - a.$$ This reduces to the observation that if $I_1, \ldots, I_l$ are intervals in the real line such that $$[a, b] \subseteq I_1 \cup \cdots \cup I_l,$$ then $b - a$ is less than or equal to the sum of the lengths of the $I_j$’s. In summary we have that $$\mathcal{H}^1([a, b]) = \mu^1([a, b]) = b - a.$$
Let $X_1, X_2, \ldots$ be a sequence of finite sets, where each $X_j$ has $n_g \ge 2$ elements, and let $X$ denote the space of sequences $x = \{x_j\}_{j=1}^\infty$ with $x_j \in X_j$ for each $j$. For each nonnegative integer $l$ and each $x \in X$ let $N_l(x)$ be the set of $y \in X$ such that the $j$th terms of $x$ and $y$ are equal when $j \le l$. These “neighborhoods” of points in $X$ generate a topology for $X$ in the usual way, in which a subset $U$ of $X$ is open if for each $x \in U$ there is an $l \ge 0$ such that $N_l(x) \subseteq U$. If $x, y \in X$ and $y \in N_l(x)$, then $N_l(y) = N_l(x)$, as one can easily see, and thus $N_l(x)$ is an open subset of $X$.
Let us call a subset of $X$ of the form $N_l(x)$ for some $x
\in X$ and $l \ge 0$ a cell. Thus cells are open subsets of $X$, and one can check that they are also closed, because the complement of a cell can be expressed as a union of cells. With this topology $X$ becomes a compact Hausdorff space which is totally disconnected, which is to say that $X$ does not contain any connected subsets with more than one element. If $\mathcal{C}_1$, $\mathcal{C}_2$ are cells in $X$, then either $\mathcal{C}_1 \subseteq \mathcal{C}_2$, or $\mathcal{C}_2 \subseteq \mathcal{C}_1$, or $\mathcal{C}_1 \cap
\mathcal{C}_2 = \emptyset$. We can be more precise and say that if $x, y \in X$ and $l$, $p$ are nonnegative integers with $l \le p$, then either $N_p(y) \subseteq N_l(x)$ or $N_l(x) \cap N_p(y) =
\emptyset$.
Suppose that $\{r_l\}_{l=0}^\infty$ is a strictly decreasing sequence of positive real numbers such that $\lim_{l \to \infty} r_l =
0$. Define a distance function on $X$ by saying that the distance from $x$ to $y$, $x, y \in X$ is equal to $0$ when $x = y$ and to $r_l$ when the $j$th terms of $x$, $y$ are equal for $j \le l$ and different for $j = l+1$. With respect to this distance function, $N_l(x)$ is the same as the set of $y \in X$ whose distance to $x$ is less than or equal to $r_l$. This defines an ultrametric on $X$ compatible with the topology generated by these standard neighborhoods in $X$.
If $A$ is a subset of $X$, then there is a cell in $X$ which contains $A$ and which has the same diameter as $A$, and therefore one may restrict one’s attention to admissible coverings by cells in $X$ in the definition of the Hausdorff content or Hausdorff measure of a subset of $X$. Of course $$\mu_h(\mathcal{C}) \le h(r_l)$$ for any cell $\mathcal{C}$ in $X$ of diameter $r_l$. If $p > l$, then $\mathcal{C}$ is the union of $$\prod_{i = l + 1}^p n_i$$ cells of diameter $r_p$. Hence $$\mathcal{H}_{h, \epsilon}(\mathcal{C})
\le \bigg(\prod_{i=l+1}^p n_i \bigg) \, h(r_p)$$ when $r_p < \epsilon$.
A particularly nice situation occurs when the function $h(t)$ satisfies $$h(r_l) = n_l \, h(r_{l+1})$$ for all $l \ge 0$. In this event we get that $$\mathcal{H}_h(\mathcal{C}) \le h(r_l)$$ for any cell $\mathcal{C}$ in $X$ with diameter $r_l$. Moreover, $$\mu_h(\mathcal{C}) \ge h(r_l)$$ under these conditions. This is because any finite covering of $\mathcal{C}$ can be refined to a covering of cells of the same diameter, and any covering of $\mathcal{C}$ by cells of diameter $r_p$, $p \ge l$, has to include all of the cells contained in $\mathcal{C}$ of diameter $r_p$. If the $n_j$’s are all equal to some positive integer $n$ and $r_l = r^l$ for some $r \in (0, 1)$, then we can take $$h(t) = t^\alpha$$ where $$n \, r^\alpha = 1.$$
More generally, suppose that $(M, d(x, y))$ is a metric space with $d(x, y)$ an ultrametric. Let $p_1$, $p_1$ be elements of $M$, and let $r_1 \le r_2$ be positive real numbers. If $$B(p_1, r_1) \cap B(p_2, r_2) \ne \emptyset,$$ then $$B(p_1, r_1) \subseteq B(p_2, r_2).$$ Similarly, if $$\overline{B}(p_1, r_1) \cap \overline{B}(p_2, r_2) \ne \emptyset,$$ then $$\overline{B}(p_1, r_1) \subseteq \overline{B}(p_2, r_2).$$ The diameter of a ball of radius $r$ in $M$ is less than or equal to $r$.
One can check that every open ball in $M$ is a closed subset of $M$. Also, every closed ball in $M$ is an open subset of $M$. In particular, $M$ is totally disconnected.
In our space $X$ as before, it is natural to say that a cell of the form $N_l(x)$ should have measure equal to $1$ when $l = 0$, and to $$\frac{1}{\prod_{i=1}^l n_i}$$ when $l > 0$. In this way the cells of the same diameter have the same measure.
Let $l$ be a nonnegative integer, and let $E_l$ be a subset of $X$ such that $E_l$ contains exactly one element from each of the cells in $X$ of diameter $r_l$. If $f$ is a continuous real-valued function on $X$, then we can associate to $f$ the Riemann sum $$\frac{1}{\prod_{i=1}^l n_i} \sum_{x \in E_l} f(x).$$ The integral of $f$ can be defined as a limit of Riemann sums as $l
\to \infty$. Recall that because $X$ is compact, $f$ is automatically uniformly continuous on $X$. This implies that the Riemann sums converge and are independent of the choices of the $E_l$’s.
If $\mathcal{C}$ is a cell in $X$, then the function on $X$ equal to $1$ on $\mathcal{C}$ and to $0$ on the complement of $\mathcal{C}$ in $X$ is a continuous function on $X$. The integral of this function is equal to the measure of the cell as defined before. Basically we simply have a probability distribution on $X$ which is uniformly distributed at each step in the obvious way.
If $(M, d(x, y))$ and $(N, \rho(u, v))$ are metric spaces, then a mapping $f : M \to N$ is said to be $C$-Lipschitz for some $C
\ge 0$ if $$\rho(f(x), f(y)) \le C \, d(x, y)$$ for all $x, y \in M$. This is equivalent to saying that for each bounded subset $A$ of $M$, $f(A)$ is a bounded subset of $N$ with diameter less than or equal to $C$ times the diameter of $A$ in $M$. In this case, if $h(t)$ is a function on $[0, \infty)$ as before and $\widetilde{h}(t) = h(C^{-1} \, t)$, then for each subset $E$ of $M$ we have that the Hausdorff content of $f(E)$ in $N$ with respect to $\widetilde{h}$ is less than or equal to the Hausdorff content of $E$ in $M$ with respect to $h$. Similarly, $\mathcal{H}_{\widetilde{h}, C
\, \epsilon}(f(E))$ in $N$ is less than or equal to $\mathcal{H}_{h,
\epsilon}(E)$ in $M$ for all $\epsilon > 0$. As a consequence, $\mathcal{H}_{\widetilde{h}}(f(E))$ in $N$ is less than or equal to $\mathcal{H}_h(E)$ in $M$.
A real-valued function $f(x)$ on $M$ is $C$-Lipschitz if and only if $$f(x) \le f(y) + C \, d(x, y)$$ for all $x, y \in M$. For each $p \in M$ we have that $f_p(x) = d(x,
p)$ is $1$-Lipschitz. More generally, if $A$ is a nonempty subset of $M$, then $$\dist(x, A) = \inf \{d(x, y) : y \in A\}$$ is $1$-Lipschitz. If the $1$-dimensional Hausdorff content of $M$ is equal to $0$, then the same must be true of the image of any real-valued Lipschitz function on $M$. In particular, $M$ must be totally disconnected.
Let $\phi(t)$ be a monotone increasing continuous real-valued function defined for nonnegative real numbers $t$ such that $\phi(0) =
0$ and $\phi(t) > 0$ when $t > 0$. If $(M, d(x, y))$ is a metric space and $d(x, y)$ is actually an ultrametric on $M$, then $\phi(d(x,
y))$ is also an ultrametric on $M$ which defines the same topology as $d(x, y)$. In particular, this works when $\phi(t) = t^a$ for some $a
> 0$.
Now assume further that $\phi(t)$ is subadditive, so that $$\phi(x + y) \le \phi(x) + \phi(y)$$ for all $x, y \ge 0$. This implies that $\phi(t)$ is uniformly continuous under these conditions. Namely, $$\phi(x) \le \phi(x + r) \le \phi(x) + \phi(r),$$ so that uniform continuity follows from continuity at $0$. If $(M,
d(x, y))$ is a metric space, then $\phi(d(x, y))$ is a metric on $M$ which defines the same topology as $d(x, y)$ does. If $0 < a \le 1$, then $\phi(t) = t^a$ satisfies these conditions.
In either of these two situations, if $A$ is a bounded subset of $M$ with respect to $d(x, y)$, then $A$ is bounded with respect to $\phi(d(x, y))$. The diameter of $A$ with respect to $\phi(d(x, y))$ is equal to $\phi$ of the diameter of $A$ with respect to $d(x, y)$.
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abstract: 'Exotic multi-quark states are examined in the quark model and in QCD. Current status of theoretical studies of the pentaquark $\Theta^+$ is reported. We show recent analyses of multi-quark components of baryons. A novel method to extract a compact excited state in the lattice QCD is applied. We also discuss how to determine mixings of different Fock components in the hadron in the QCD sum rule approach.'
address: |
Department of Physics, H-27, Tokyo Institute of Technology\
Meguro, Tokyo 152-8551 Japan
author:
- 'Makoto Oka[^1]'
title: 'Dynamics of Multiquark Systems: Mass, Width and Exotics'
---
Introduction: Quark Model
=========================
The quark model describes mesons and baryons in terms of constituent quarks. Mesons are $q\bar q$ states and baryons are $qqq$, while all the other combinations of quarks, such as $qq\bar q\bar q$, $qqqq\bar q$, are called exotic. The constituent quarks must be effective degrees of freedom which are valid only in low energy regime. They must have the same conserved charges as the QCD quarks: baryon number 1/3, spin 1/2, color 3 and flavor 3. They acquire dynamical masses of order $\sim 300-500$ MeV induced by chiral symmetry breaking of the QCD vacuum.
The ground-state mesons and baryons have been classified based on $SU(3)\times SU(2) \to SU(6)$ symmetry, i.e., the pseudoscalar and vector meson nonets, and the octet and decuplet baryons. Their mass spectra, in particular, the pattern of SU(3) breaking, and the electro-magnetic properties are well reproduced by minimum dynamics of quarks, quark confinement and color-magnetic type interactions, except for pseudoscalar mesons, i.e., the octet pseudoscalars, $\pi, K, \eta$, are very light according to chiral symmetry breaking, while the $\eta'$ mass is large due to the $U_A(1)$ anomaly.
The quark model based on $SU(6) \times O(3)$ symmetry, however, encounters some difficulties when it is applied to excited hadron states. An example is the scalar meson nonet; ($\sigma(600), a_0(980), f_0(980), K_0^*(800)$). ($K_0^*$ has been indicated in $K\pi$ final states in $J/\psi$ and $D$ meson decays, but not yet established[@kappa].) Their mass ordering as $q\bar q$ states is expected to be like $m(\sigma) \sim m(a_0) < m(f_0)$, assuming the ideal mixing, i.e., $$\begin{aligned}
&& \sigma\sim \frac{u\bar u + d\bar d}{\sqrt{2}} , \qquad
a_0\sim \frac{u\bar u - d\bar d}{\sqrt{2}} , \qquad
f_0 \sim s\bar s \nonumber\end{aligned}$$ Furthermore, while they are classified as $^3P_0$ states, their spin-orbit partners $J= 1$ and 2 states are not observed in their vicinity.
A possible solution of the difficulty is to consider four-quark exotic states for the scalar mesons[@4q]. Suppose that diquarks with flavor 3, color 3 and spin 0, i.e., $$\begin{aligned}
&& U=(\bar d\bar s)_{S=0,C=3,f=3}\qquad D=(\bar s\bar u)_{S=0,C=3,f=3}
\qquad S=(\bar u\bar d)_{S=0,C=3,f=3} .
\label{diquark}\end{aligned}$$ are building blocks of the scalar mesons. Then the scalar nonets in the ideal mixing may appear as $$\begin{aligned}
&& \sigma\sim S\bar S \sim (ud)(\bar u\bar d) \nonumber\\
&& a_0\sim \frac{1}{\sqrt{2}} (U\bar U - D\bar D) \sim \frac{1}{\sqrt{2}} ((ds)(\bar d \bar s)-(su)(\bar s\bar u))
\nonumber\\
&& f_0 \sim \frac{1}{\sqrt{2}} (U\bar U + D\bar D) \sim \frac{1}{\sqrt{2}} ((ds)(\bar d \bar s)+(su)(\bar s\bar u))
\nonumber\end{aligned}$$ Then one sees that the strange quark counting predicts the observed mass pattern, $m(\sigma) < m(a_0) \sim m(f_0)$. It also explains that the $J=0$ state is isolated without $J=1, 2$ partners.
Thus one sees that multi-quark components may help to explain anomalies in the scalar meson nonets. There are other hadrons which are suspected to contain exotic multi-quark components, such as $D_s^*$, $X(3872)$ and $\Lambda(1405)$, mainly because they do not fit well in the spectra of the ordinary mesons or baryons.
The next question is whether the QCD dynamics allows such states? We actually have a simple reason, for instance, why $\Lambda (1405)$ is possible to be a 5-quark state. $\Lambda(1405)$ is a $J^{\pi} = 1/2^-$ flavor singlet baryon. The three quark ($uds$) configuration has to contain orbital excitation $L=1$ with spin 1/2, leading to $J=1/2^-$ and $3/2^-$ states. Then the candidate of the $3/2^-$ partner is $\Lambda(1520)$, but the spin-orbit splitting is unusually large compared to the nonstrange baryons with $L=1$. On the other hand, the 5-quark content, $udsu\bar u+udsd\bar d$ may be realized in $L=0$ without orbital excitation, and thus has advantage over the $L=1$ excited states. In terms of the diquark language, one may assign the $L=0$ and $S=1/2$ configuration. This gives an isolated $J=1/2$ state and has no difficulty of large LS splitting. Therefore it is quite interesting and important to answer whether realistic dynamical calculation indeed gives multi-quark states a lower energy.
Pentaquark $\Theta^+$
=====================
We first look at the status of the pentaquark $\Theta^+$ in the quark picture. The pentaquark $\Theta^+$ is a baryon with strangeness $+1$, whose minimal quark content is $uudd\bar s$. It may be affiliated to a member of flavor antidecuplet[@Theta-quark]. How does the known dynamics of the quark model work for $\Theta^+$? It has been clarified that the “standard” quark model, which explains the ground state mesons and baryons very well, does not easily yield the small mass and the tiny width of $\Theta^+$ simultaneously. Recently variational techniques for solving multi-body bound states have been applied to the 5-quark systems. The results[@Theta-variational] show that the typical masses of spin $1/2^{\pm}$ and $3/2^{\pm}$ pentaquarks are more than 500 MeV above the threshold of $NK$, and thus the masses are 1.9 GeV or higher. Furthermore, if they have such large masses, various decay channels are open and their widths must be very large. It was pointed out that even at the mass 1.54 GeV, the decay widths of $1/2^-$ 5-quark states[@Hosaka-width], which decay into S wave $NK$ states, must be a few hundred MeV or larger.
In order to bring such high-mass resonances down to the observed mass[@Nakano], one may require a very strong correlation. Diquark correlation is a possibility[@Theta-diquark]. Most dynamical models of quarks, such as one-gluon exchange, instanton induced interaction, and so on, give attraction to color $\bar 3$, flavor $\bar 3$, spin 0 diquarks (Eq.(\[diquark\])). It is, however, noted that the same interaction causes (often stronger) attraction to $q \bar q$ color singlet, flavor 8, spin 0 system (i.e., pseudoscalar mesons). Then the ground state of the pentaquark may well be a state of a baryon and a meson far separated. Thus, the di-quark scenario may not work unless the spin (or other quantum numbers) is chosen so that it hinders a pseudoscalar subsystem. One such case is that the spin of $\Theta^+$ is 3/2[@Hosaka-width; @spin3_2].
Even if the observed $\Theta^+$ might not be what we expected originally, the techniques developed in the study of the pentaquark $\Theta^+$ are useful in studying other exotic multiquark resonances as well as multiquark components of ordinary hadrons. In particular, the variational method is powerful and useful to judge whether a certain quark model allows multiquark hadrons as a ground state.
Pentaquarks in Lattice QCD
==========================
As the model calculations have various ambiguity in treating multiquark systems with color singlet sub-systems, direct applications of QCD to exotics are desperately needed. In fact, QCD does not a priori exclude exotic multi-quark bound (resonance) states as far as they are color-singlet. We here show some of the results from the lattice QCD simulations and the approach using QCD sum rules.
Lattice QCD is powerful in understanding non-perturbative physics of QCD: vacuum structure, phase diagrams, mass spectra of ground state mesons and baryons, interactions of quarks, . . . However, at this moment the lattice QCD has two rather severe restrictions: (1) Light quarks are too expensive. The simulations are made at a large quark mass and require (often drastic) extrapolation to physical quark masses for $u$ and $d$ quarks. (2) No direct access to resonance poles is possible. It is hard to distinguish resonances from hadron scattering states. Real number simulations can not access complex poles. Thus applications to exotic hadrons are yet limited.
We have developed a new method, called hybrid boundary condition (HBC) method, to extract the 5-quark resonances out of the meson-baryon background[@Ishii]. The basic idea of HBC is to apply anti-periodic boundary conditions to certain quarks, which effectively raise the energy of the lowest-energy meson-baryon scattering state. Comparing the results with the standard boundary condition with the hybrid one, one can determine whether a state seen in the lattice simulation is a compact resonance state or a hadronic continuum state. This technique has been applied to the pentaquark $\Theta^+$ first and later to $\Lambda(1405)$ and the other possible exotics.
The results for $\Theta^+$ pentaquark are summarized as follows.\
(1) The negative parity $1/2^-$ state appears at $m\sim 1.75$ GeV, which seems consistent with $NK$ ($L=0$) scattering state on the lattice. The HBC analysis confirms that the observed state is not a compact 5-quark state.\
(2) The positive parity $1/2^+$ state appears at 2.25 GeV, which is too heavy for $\Theta^+$.\
(3) The $J =3/2$ states are generated by three different operators: di-quark type, $NK^*$, and color-twisted $NK^* $ operators. The results show that the mass of $3/2^-$ state is around 2.11 GeV, that is consistent with $NK^*$ ($L=0$) threshold, while the positive parity $3/2^+$ is around 2.42 GeV, consistent with $NK^*$ ($L=1$) threshold.
Thus no candidate for compact 5-quark state is found. Most lattice QCD parties agree with these results with a few exceptions. The QCD sum rules also give the consistent results.
A similar method has been applied to study the penta-quark nature of the flavor singlet $\Lambda (1/2^-)$ state[@LQCD-Lambda]. In this case, one may twist the spatial boundary condition only of quarks in the quenched approximation, while antiquarks satisfy the standard boundary condition. This “HBC” will enable us to isolate $\Lambda^*$ resonance from $N\bar K$ and $\Sigma\pi$ scattering states. The results are satisfactory, showing that the 5-quark operator gives $m_{5Q}$ = 1.63(7) GeV, which is around the $m(N)+m(K)$ threshold on the current choice of the lattice parameters. It is further interesting that the HBC analysis indicates that this 5-quark state seems a compact resonance and thus it is a strong candidate for $\Lambda(1405)$. In contrast, the 3-quark operator gives a higher mass, $m_{3Q}$ = 1.79(8) GeV.
Exotic Multi-quark States in QCD Sum Rule
=========================================
In nature, it is expected that exotic multi-quark components mix with the standard Fock state in hadrons. In such mixed states, one may ask how large the mixing probability is of the exotic multi-quark components.
It turns out that such a mixing is not easily quantified. It would be natural to consider the strengths of the couplings of the 3-quark and 5-quark operators to the physical state and then evaluate the mixing angle. However, such a procedure is largely dependent on the definition and normalization of the local operators. Indeed, a numerical factor can be easily hidden in the local operators and thus the magnitudes of the coupling strengths are ambiguous.
This problem happens to be more fundamental than just the definition of the operators. In the field theory one may not be able to “measure” the number of quarks without ambiguity because no conserved charge corresponding the number of quarks, $N(q)+N(\bar q)$, is available. Namely, the Fock space separation may not be unique. Thus we have to consider the concept of the “number of quarks” in the context of the quantum mechanical interpretation of the field theoretical state.
In recent study[@SINNO], we propose two ways to “define” the ratio of the Fock space probability. In the first approach, we define local operators in the context of a 5-quark operator $J_5$. As the 5-quark operator contains $qqq$ component, one can write the operator into $J_5= \tilde J_5 + \tilde J_3$. Then the mixing parameter may be defined as the ratio of the couplings to $\tilde J_5$ and $\tilde J_3$. This mixing angle can be evaluated from the correlation functions with an assumption that the poles are at the same position. The result is model independent, but it depends on the choice of the operators. Therefore it does not necessarily have a direct relation to the mixing parameters employed in the quark models.
In order to define a mixing angle more appropriate to the quark models, one must determine the normalization of the operators using a quark model wave function. One may use the MIT bag model wave functions for the normalization, assuming that the bag model states (with define number of quarks) are normalized properly.
The above two methods have been applied to the $a_0$ scalar meson and we have found that the physical state for $a_0$ is indeed given by the mixings of $q\bar q$ and $qq\bar q\bar q$ states. It turns out that the 4-quark mixing probability is about 90%, which does not strongly depend on the choice of the definition.
In contrast, for the flavor singlet $\Lambda^* (1/2^-)$ state, we find that the 3-quark operator gives a higher mass and thus the lowest energy state is predominantly 5-quark state. In such a case, the mixing is not properly defined in our method.
Conclusion
==========
A technique using the hybrid boundary conditions seems to work in lattice QCD to distinguish compact states from scattering states. The quenched lattice QCD suggests that $\Lambda(1405)$ is predominantly a 5-quark state. The mixing of the multi-quark components is not quantified from the field theoretical viewpoint. However, one can define a set of useful mixing amplitudes using the well-defined matrix elements of local operators. Whether this definition of the mixing is relevant in the quark model is yet an open problem. The QCD sum rule indicates a large 4-quark components in scalar mesons.
The contents of this paper come from various collaborations. I acknowledge Drs. N. Ishii, H. Suganuma, T. Doi, Y. Nemoto, H. Iida, F. Okiharu, S. Takeuchi, A. Hosaka, T. Shinozaki, T. Nishikawa, J. Sugiyama, and T. Nakamura for productive collaborations and discussions.
[9]{} Particle Data Group, Jour. of Phys. [**G33**]{} (2006) 1. R.L. Jaffe, Phys. Rev. [**D15**]{} (1977) 267. M. Oka, Prog. Theor. Phys. [**112**]{} (2004) 1; R.L. Jaffe, Phys. Rept. [**409**]{} (2005)1. S. Takeuchi, K. Shimizu, Phys. Rev. [**C71**]{} (2005) 062202; E. Hiyama et al., Phys. Lett. [**B633**]{} (2006) 237. A. Hosaka, M. Oka, T. Shinozaki, Phys. Rev. [**D71**]{} (2005) 074021. T. Nakano et al., LEPS collaboration, Phys. Rev. Lett. [**91**]{} (2003) 012002. R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. [**91**]{} (2003) 232003. S. Takeuchi, K. Shimizu, Phys. Rev. [**C71**]{} (2005) 062202. N. Ishii et al., Phys. Rev. [**D71**]{} (2005) 034001; N. Ishii et al., Phys. Rev. [**D72**]{} (2005) 074503. Y. Nemoto et al., Phys. Rev. [**D68**]{} (2003) 094505; N. Ishii et al., in preparation. J. Sugiyama et al., in preparation.
[^1]: email: [email protected]
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[**Reciprocity in parity violating non-Hermitian systems** ]{}
[Ananya Ghatak [^1] Brijesh Kumar Mourya[^2] Raka Dona Ray Mandal [^3] and Bhabani Prasad Mandal [^4]]{}
[ Department of Physics, Banaras Hindu University, Varanasi-221005, INDIA.\
]{}
[**Abstract**]{}
Reciprocity is shown so far only when the scattering potential is either real or parity symmetric complex. We extend this result for parity violating complex potential by considering several explicit examples: (i) we show reciprocity for a PT symmetric (hence parity violating) complex potential which admits penetrating state solutions analytically for all possible values of incidence energy and (ii) reciprocity is shown to hold at certain discrete energies for two other parity violating complex potentials.
Introduction
============
Over a decade and half fully consistent quantum theory for non-Hermitian systems [@ben4]-[@benr] have been developed extensively with its application in different branches of physics [@opt1]-[@cal]. Recently scattering due to complex potentials has attracted huge attention due to its rich properties and wide applicabilities and usefulness in the study of different optical systems [@opt1]-[@eqv1]. Exceptional points (EPs) [@ep0]-[@ep2], spectral singularity (SS) [@ss1]-[@ss3], invisibility [@aop]-[@inv1], reciprocity [@aop]-[@resc], critical coupling (CC) [@cc0]-[@cc4] and coherent perfect absorption (CPA) [@cpa00]-[@cpa4] are among the interesting features of complex scattering. In particular, the CPA which is time reversed counter part of lasing effect has become the center of all such studies in optics due to the discovery of anti-laser [@cpa00]-[@cpa011] which has a number of technological implications. Other exciting feature is reciprocity which is the topic of discussion in the present work. In the case of scattering due to real potential, transmission coefficient ($R_l$) for left incident particle is always equal to that ($R_r$) of for the right incident particle, and the system is called reciprocal. Reciprocity holds good even for complex potentials which respect parity [@reci1; @reci2]. However reciprocity is known to be violated in the case of complex potentials which are not parity invariant. In fact PT symmetric non-Hermitian systems are always parity violating and hence non-reciprocal. Reciprocity has not been investigated in details for the PT symmetric non-Hermitian systems and to the best of our knowledge not a single example of parity violating PT-symmetric reciprocal system is known till date. Therefore it is worth investigating reciprocity in parity violating non-Hermitian systems. In this work we show that reciprocity can be restored even for the parity violating non-Hermitian systems. We consider three specific examples in support of our claim. In the first example we consider complexified Morse type potential [@aop] which admits bound, reflecting, penetrating and free state solutions depending on the energy of the incident particle. We show that reciprocity is restored for the entire range of incident energy even though the potential is parity violating and complex, admitting penetrating state solutions. In the case of reflecting states of the same model and in the case of parity violating (PT symmetric or non PT-symmetric) non-Hermitian double delta potential [@ram], we show that reciprocity is valid only at some specific value of incident energy. These results are shown graphically for the last two models. The most interesting part of the non-Hermitian double delta potential is that the reciprocity is restored only at extreme situations where spectral singularity or reflectionlessness occurs.
The plan of the paper is as follows. We systematically analyse the conditions of reciprocity in the case of both parity symmetric as well as parity violating real potential in section 2. In section 3 we use the result of section 2 to show the reciprocity for three parity violating complex potentials. Section 4 is kept for summary and discussion.
Reciprocity : Real Potential
=============================
In this section we obtain the general condition for reciprocity in usual quantum mechanics. The arguments are valid for both real and complex scattering and will be used in the later section. To demonstrate this we consider the general solution of Schroedinger equation for the scattering state as the superposition of the two independent solutions as,
(x)=a U(x)+ b V(x) \[sc\] where $U(x)$ and $V(x)$ are two independent scattering state solutions and $a, b$ are any complex numbers. Now in the case of scattering, $U(x)$ and $V(x)$ are written in general asymptotic form, so that the wave function for $x\rightarrow +\infty$ is written as, \^+(x) &=& a (u\_1\^+(k) e\^[ikx]{}+u\_2\^+(k) e\^[-ikx]{})+ b (v\_1\^+(k) e\^[ikx]{}+v\_2\^+(k) e\^[-ikx]{})\
&=&( au\_1\^+(k)+b v\_1\^+(k)) e\^[ikx]{}+( au\_2\^+(k)+b v\_2\^+(k)) e\^[-ikx]{} \[+\] Similarly for $x\rightarrow -\infty$ the wave function is written as, \^-(x) = ( au\_1\^-(k)+b v\_1\^-(k)) e\^[ikx]{}+( au\_2\^-(k)+b v\_2\^-(k)) e\^[-ikx]{} \[-\] From the Eqs. (\[+\],\[-\]) we can calculate the scattering amplitudes.
For left incidence: $au_2^+(k)+b v_2^+(k)=0$, and hence $r_l$ is evaluated as, r\_l= \[rl\]
For right incidence: $au_1^-(k)+b v_1^-(k)=0$, so r\_r= \[rr\] From Eqs. (\[rl\],\[rr\]) the condition for reciprocity ($R_l=R_r$)is written as, u\_2\^+ v\_2\^-(k)-v\_2\^+u\_2\^-(k)&=& v\_1\^-u\_1\^+(k)-u\_1\^- v\_1\^+(k)\
u\_2\^+ v\_2\^-(k)-v\_2\^+u\_2\^-(k)&=&v\_1\^-u\_1\^+(k)-u\_1\^- v\_1\^+(k)\[ler\] Now we discuss parity symmetric and parity violating cases separately using these general results.
Parity symmetric potential
---------------------------
For a parity symmetric potential (real or complex) the general wave functions are written in terms of the odd and even parity solutions. We consider $U(x)=U(-x)$ and $V(x)=-V(-x)$. This implies U(x) U\^+(x)U\^+(-x) U(x- )=U\^-(x) ; \[Ps\] and V(x) V\^+(x)V\^+(-x) -V(x- )=-V\^-(x) ; \[Ps\] From the above equation Eq. (\[Ps\]) we see that $U^+(-x)=U^-(x)$ and hence we obtain, u\_1\^+(k) e\^[-ikx]{}+u\_2\^+(k) e\^[ikx]{}&=&u\_1\^-(k) e\^[ikx]{}+u\_2\^-(k) e\^[-ikx]{},\
u\_1\^+(k)=u\_2\^-(k) &;& u\_2\^+(k)=u\_1\^-(k) \[u1u2\] Similarly due to $V^+(-x)=-V^-(x)$ we get, v\_1\^+(k)=-v\_2\^-(k) ; v\_2\^+(k)=-v\_1\^-(k) \[v1v2\] Due to these relations in Eq. (\[u1u2\]) and Eq. (\[v1v2\]), the condition of reciprocity in Eq. (\[ler\])is satisfied and the reflection coefficients as well as the amplitudes are equal (i.e. $r_l=r_r$ and $R_l=R_r$) for arbitrary parity symmetric potential.\
Parity violating real potential
-------------------------------
For a real potential, the wave function is always chosen as real in the form, $\psi(x)=1/2\left\{\phi(x) +\phi^*(x) \right\}$ where $\phi^*(x)$ and $\phi(x)$ are two independent solution of Schroedinger equation. From the asymptotic forms of $\psi$ in Eq. (\[+\]) and (\[-\]) we have the following equations for ${\psi^-}^*=\psi^-$ and by considering left incidence case we obtain && a\^\*[u\_1\^-]{}\^\*(k)+b\^\* [v\_1\^-]{}\^\*(k)= au\_2\^-(k)+b v\_2\^-(k)\
&&=\
&& N(r\_l)= \^\* , r\_l= \[rl1\] where $N(r_l)$ and $D(r_l)$ denote the numerator and denominator respectively in the expression of $(r_l)$ in equation (4). N(r\_r)= \^\* , r\_r= \[rr1\] It is clear from equations (11) and (12) that the magnitudes of $r_l$ and $r_r$ are same, they only differ by a phase. Thus in this parity violating case $R_l= R_r$ even though $r_l\not =r_r$. This proves the reciprocity for parity invariant real scattering case.
Parity violating complex potential
==================================
In this section we consider three different cases of parity violating complex potential to study the reciprocity. We complexify Morse type potential, which admits bound, reflecting, penetrating and free state solutions in two different way. We show that the scattering state for this PT symmetric complex potential is reciprocal only at some specific incident energy, whereas the penetrating states are always reciprocal, even though the potential violate parity symmetry. In another case we consider PT-symmetric as well as non PT-symmetric but Parity violating complex double delta potential to show that reciprocity is restored only at the energy values where SS and reflectionlessness occur.
Morse-type potential with scattering states
-------------------------------------------
The one dimensional real Morse-type potential [@aop] is written as, V\^[I]{}(x)=V\_0 \^2’ {+(’)}\^2 \[realm\] This potential admits bound, reflecting, penetrating and free state solutions depending on the energy of the incident particle. We complexified this potential as $\mu'\rightarrow i
\mu $ so that the corresponding Hamiltonian becomes PT-symmetric non-Hermitian. The Schroedinger equation for this case is written as, d\^2\_m /dz\^2+\[-v (2 i)-v (2 i )z+v \^2(i) [ [sech]{}]{}\^2 z\] \_m=0 \[compm\] where, $z=(x-i\mu d)/d$, $v=(2md^2/\hbar^2)V_0$, $\epsilon=(2md^2/\hbar^2)E$. Here $z^{PT}=-z$ and $z^P\not=\pm z$, so the above potential is invariant only under PT-transformation. The scattering states solutions of Schroedinger equation for this non-Hermitian potential are, U\_[m]{} (z) &=& Ne\^[i(k\_+-k\_-)z]{}(e\^z+e\^[-z]{})\^[i(k\_++k\_-)z]{}\
&.& F(-ik\_+-ik\_-+-,-ik\_+-ik\_- ++;1-ik\_+;)\
V\_[m]{} (z) &=& Ne\^[i(k\_+-k\_-)z]{}( e\^z+e\^[-z]{})\^[i(k\_++k\_-)z]{}. [()]{}\^[ik\_+]{}\
&.& F(ik\_+-ik\_-+-,ik\_+-ik\_- ++;1+ik\_+;) .\
where = ; k\_+= ; k\_-= \[k+-\] The general scattering wave function for this Morse-like potential is written in the superposition form of the two independent solutions as, \_m (z)=AU\_[m]{}(z)+BV\_[m]{}(z) \[m1\] with $A$ and $B$ as the arbitrary constants using the standard properties of hyper-geometric functions the asymptotic forms of $\psi_1$ and $\psi_2$ are written in equivalent notations of Eqs. (\[+\]) and (\[-\]) as, u\_[m1]{}\^+&=&N ; u\_[m2]{}\^+=0 ; v\_[m1]{}\^+=0 ; v\_[m2]{}\^+=N ; \
u\_[m1]{}\^-&=&N G2(k) ; u\_[m2]{}\^-=N G1(k) ; v\_[m1]{}\^-=N G4(k) ; v\_[m2]{}\^-=N G3(k) ; \[aa\] G1&=& ;\
G2&=& ;\
G3&=& ;\
G4&=& .\
\[gg\] The total wave function behaves asymptotically as, \_[m]{}\^+ (z+)&=&A Ne\^[ik\_+z]{}+B Ne\^[-ik\_+z]{} ;\
\_[m]{}\^- (z-)&=&N ; \[m1+-\] The left and right handed reflection amplitudes which can be constructed by using the Eqs. ( \[rl\]), (\[rr\]) and (\[aa\]) (or also can be calculated by (\[m1+-\])) are written as, r\_l= ; r\_r=- . The condition for reciprocity becomes G1(k)=G4(k)\[m1r\] This can always be seen directly from equation (6).
Eq. (\[m1r\]) is only satisfied for discrete values of ’$k$’ i.e. only for discrete values of incident particle energy (Fig. 1). Thus for this PT-symmetric non-Hermitian potential one gets reciprocity for certain particle energies without obtaining unitarity. Fig. 1 is showing the discrete incidence energies for which the scattering is reciprocal even for a PT symmetric non-Hermitian system.\
 (a)  (b)\
[**Fig.1:**]{} [*Shows different points where reciprocity is restored even for PT-symmetric non-Hermitian system. In 1(a) we have two energy points (in atomic unit) where $R_l=R_r$ for $v=1.9$ and $\mu=\pi/5$. 1 (b) shows three such energy points (with $\mu=\pi/10$) where reciprocity is restored.* ]{}\
Parity violating non-Hermitian double-delta potential
-----------------------------------------------------
Let us consider the following double-delta potential , $$V^{II}(x)= \lambda [\delta(x- \frac {a}{2})-\delta(x+ \frac {a}{2})]$$ where $\lambda$ and $a$ are constant parameters. This potential becomes (i) PT-symmetric non-Hermitian for imaginary $\lambda $ and (ii) non PT-symmetric non-Hermitian for complex $\lambda $. In either cases the non-Hermitian potential is parity violating. The reflection amplitudes for left and right incident particle with energy $E$ are written as, r\_l&=& ;\
r\_r&=&- ; k= . \[drr\] It is easy to see from these equations of reflection amplitudes that scattering due to this non-Hermitian double delta potential is non-reciprocal independent of the fact that PT-symmetry is broken or not. Further we show that reciprocity is achieved at some special situations where spectral singularity or reflectionlessness occurs. From Eq. \[drr\] it is easily seen that at the energies $E_*$ for which $\left [\cos (2k_*a)+i\sin(2k_*a)\right ]+(\frac{k_*}
{\lambda })^2=1$ both $r_l$ and $r_r$ are infinite. This is nothing but the condition of spectral singularity for the energy of the particle incident from either side. On the other situation reciprocity is restored due to the case of bidirectional reflectionlessness ($r_l=r_r=0$) at discrete incidence energies $E_{**}=\frac{n^2\pi^2}{2a^2m}$. Fig. 2 explains both the conditions where the left and right handed reflection amplitudes behave in exactly similar way.\
 (a)  (b)\
[*Fig. 2: (a, b) shows the restoration of reciprocity for parity asymmetric double delta non-Hermitian potential when $a=1, \lambda =20 i$ and $a=-1, \lambda =2.01 i - 6.1$ respectively at different discrete energies. Reciprocity and SS or reflectionlessness occurs at the same energy values.*]{}\
Penetrating state of a complex potential and reciprocity
--------------------------------------------------------
The potential in Eq. \[realm\] is complexified in PT-symmetric manner by taking the parameter $d$ imaginary (i.e. $d\rightarrow id$) and keeping the other parameters real. For this case the time independent Schroedinger equation (TISE) takes the form, +V\^I(z)=E\[schr\] where z is taken as $z=-iX/d-\mu$, with $X=x+i\zeta $. Note that the differential term in this equation comes with wrong sign due to the presence of $d^2$ term. However this equation can be interpreted as TISE for a upside down potential of the original one with energy eigenvalues (-E). It behaves like a potential barrier with a maxima at x=0. We have penetrating state solutions if the particle has negative energy with magnitude less than $ve^{-2\mu}$. On the other hand it accepts free states when the energy of the particle is more than the barrier height.
The penetrating state solutions are given by, \_1 (z) &=& Ne\^[-az]{}(e\^z+e\^[-z]{})\^[-bz]{} F(b+-,.\
&& . b++ ;a+b+1;)\
\_2 (z) &=& Ne\^[bz]{}(e\^z+e\^[-z]{})\^[az]{} F(-a+-,.\
&&. -a++ ;1-a-b;)\
\[penn\] with a\^2+b\^2=--v2; 2ab=-v2; and $k_+=\sqrt{\epsilon +ve^{2\mu}}$, $k_-=\sqrt{\epsilon +ve^{-2\mu}}$ so that a=-1/2(ik\_++ik\_-); b=1/2(-ik\_++ik\_-) . The wave function for this penetrating state is written in a general form as, (z)=A\_1(z)+B\_2(z) \[ppsi\] We calculate the left and right handed penetrating amplitudes from the asymptotic behavior of Eq. (\[ppsi\]) as, r\_l= r\_r=-, \[rtlp\] where P1, P2, P3, P4 are expressed in terms of Gamma function written as, P1&=&\
P2&=&\
P3&=&\
P4&=&\
where $\gamma'=\sqrt{-v\cosh^2 (\mu)+1/4} $. Here we note $r_l\not=r_r$ but $P1^*=P4; \ \
P2^*=P3$ (due to real $k_+$ and $k_-$ ). Thus the second condition of Eq. (\[ler\]) is being satisfied so that $$\ R_l\equiv \mid r_l\mid^2=\mid r_r\mid^2\equiv R_r \ .$$ This implies that we have reciprocity for this non-Hermitian PT-symmetric system for all values of energy. However as expected unitarity is violated, i.e $R+T\not=1$ (both for left and right handed cases) for this model. We further observe no SS is present for penetrating states and the potential never becomes reflectionless.
Conclusion
==========
Scattering due to real potential is always reciprocal. Further it has been shown that even complex scattering obeys reciprocity if the potential is parity symmetric. This implies that scattering due to PT symmetric non-Hermitian systems are supposed to be non-reciprocal as PT symmetric non-Hermitian system is essentially parity violating. In this present work we show that reciprocity is valid even for certain PT symmetric non-Hermitian systems. In support of our claim we consider, three explicit examples. The scattering states of PT-symmetric complex Morse potential is reciprocal only at certain specific incident energies. In another example we consider PT-symmetric as well as non PT-symmetric complex double delta potential to show restoration of reciprocity only at SS and reflectionless points. Most interesting result is for penetrating states of PT symmetric non-Hermitian Morse potential which is shown to be reciprocal at all incident energy. Unitarity of scattering S-matrix and reciprocity are two important characteristics of scattering theory. Their validity/non validity in the case of complex scattering is extremely important. Our results are one step forward in the investigation of reciprocity of complex scattering. No example of PT-symmetric non-Hermitian potentials is known so far when both unitarity and reciprocity hold good. It will be worth finding more and more parity violating non-Hermitian systems which are reciprocal as well as unitary.\
[**Acknowledgments**]{} BKM and BPM acknowledge the financial support from the Department of Science and Technology (DST), Gov. of India under SERC project sanction grant No. SR/S2/HEP-0009/2012. AG acknowledges the Council of Scientific & Industrial Research (CSIR), India for Senior Research Fellowship.
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[^1]: e-mail address: [email protected]
[^2]: e-mail address: [email protected]
[^3]: e-mail address: [email protected]
[^4]: e-mail address: [email protected], [email protected]
|
---
abstract: 'In this work we present an experimental setup to show the suitability of ROS 2.0 for real-time robotic applications. We disclose an evaluation of ROS 2.0 communications in a robotic inter-component (hardware) communication case on top of Linux. We benchmark and study the worst case latencies and missed deadlines to characterize ROS 2.0 communications for real-time applications. We demonstrate experimentally how computation and network congestion impacts the communication latencies and ultimately, propose a setup that, under certain conditions, mitigates these delays and obtains bounded traffic.'
author:
- |
**Carlos San Vicente Gutiérrez**, **Lander Usategui San Juan**,\
**Irati Zamalloa Ugarte**, **Víctor Mayoral Vilches**\
Erle Robotics S.L.\
Vitoria-Gasteiz,\
Álava, Spain\
bibliography:
- 'references.bib'
title: '**Towards a distributed and real-time framework for robots: Evaluation of ROS 2.0 communications for real-time robotic applications** '
---
Introduction
============
In robotic systems, tasks often need to be executed with strict timing requirements. For example, in the case of a mobile robot, if the controller has to be responsive to external events, an excessive delay may result in non-desirable consequences. Moreover, if the robot was moving with a certain speed and needs to avoid an obstacle, it must detect this and stop or correct its trajectory in a certain amount of time. Otherwise, it would likely collide and disrupt the execution of the required task. These kind of situations are rather common in robotics and must be performed within well defined timing constraints that usually require real-time capabilities[^1]. Such systems often have timing requirements to execute tasks or exchange data over the internal network of the robot, as it is common in the case of distributed systems. This is, for example, the case of the Robot Operating System (ROS)[@quigley2009ros]. Not meeting the timing requirements implies that, either the system’s behavior will degrade, or the system will lead to failure.
Real-time systems can be classified depending on how critical to meet the corresponding timing constraints. For hard real-time systems, missing a deadline is considered a system failure. Examples of real-time systems are anti-lock brakes or aircraft control systems. On the other hand, firm real-time systems are more relaxed. An information or computation delivered after a missing a deadline is considered invalid, but it does not necessarily lead to system failure. In this case, missing deadlines could degrade the performance of the system. In other words, the system can tolerate a certain amount of missed deadlines before failing. Examples of firm real-time systems include most professional and industrial robot control systems such as the control loops of collaborative robot arms, aerial robot autopilots or most mobile robots, including self-driving vehicles.
Finally, in the case of soft real-time, missed deadlines -even if delivered late- remain useful. This implies that soft real-time systems do not necessarily fail due to missed deadlines, instead, they produce a degradation in the usefulness of the real-time task in execution. Examples of soft-real time systems are telepresence robots of any kind (audio, video, etc.).\
As ROS became the standard software infrastructure for the development of robotic applications, there was an increasing demand in the ROS community to include real-time capabilities in the framework. As a response, ROS 2.0 was created to be able to deliver real-time performance, however, as covered in previous work [@DBLP:journals/corr/abs-1804-07643] and [@2018arXiv180810821G], the ROS 2.0 itself needs to be surrounded with the appropriate elements to deliver a complete distributed and real-time solution for robots.\
For distributed real-time systems, communications need to provide Quality of Services (QoS) capabilities in order to guarantee deterministic end-to-end communications. ROS 2 communications use Data Distribution Service (DDS) as its communication middleware. DDS contains configurable QoS parameters which can be tuned for real-time applications. Commonly, DDS distributions use the Real Time Publish Subscribe protocol (RTPS) as a transport protocol which encapsulates the well known User Datagram Protocol (UDP). In Linux based systems, DDS implementations typically use the Linux Networking Stack (LNS) for communications over Ethernet.\
In previous work [@DBLP:journals/corr/abs-1804-07643], we analyzed the use of layer 2 *Quality of Service (QoS)* techniques such as package prioritization and Time Sensitive Networking (TSN) scheduling mechanisms to bound end-to-end latencies in Ethernet switched networks. In [@2018arXiv180810821G], we analyzed the real-time performance of the LNS in a Linux PREEMPT-RT kernel and observed some of the current limitations for deterministic communications over the LNS in mixed-critical traffic scenarios. The next logical step was to analyze the real-time performance of ROS 2.0 communications in a PREEEMPT-RT kernel over Ethernet. Previous work [@7743223] which investigated the performance of ROS 2.0 communication showed promising results and discussed future improvements. However, the mentioned study does not explore the suitability of ROS 2.0 for real-time applications and the evaluation was not performed on an embedded platform.\
In this work, we focus on the evaluation of ROS 2.0 communications in a robotic inter-component communication use-case. For this purpose, we are going to present a setup and a set of benchmarks where we will measure the end-to-end latencies of two ROS 2.0 nodes running in different static load conditions. We will focus our attention on worst case latencies and missed deadlines to observe the suitability of ROS 2.0 communications for real-time applications. We will also try to show the impact of different stressing conditions in ROS 2.0 traffic latencies. Ultimately, we attempt to find a suitable configuration to improve determinism of ROS 2.0 and establish the limits for such setup in an embedded platform.\
The content below is structured as follows: section \[background\] presents some background of ROS 2.0 and how its underlying communication middleware is structured. Section \[setup\_results\] shows the experimental results obtained while using four different approaches. Finally, Section \[conclusions\] provides a discussion of the results.\
Background
==========
![Overview of ROS 2 stack for machine to machine communications over Ethernet[]{data-label="ros2_stack"}](images/layer.jpg){width="40.00000%"}
ROS is a framework for the development of robot applications. A toolbox filled with utilities for robots, such as a communication infrastructure including standard message definitions, drivers for a variety of software and hardware components, libraries for diagnostics, navigation, manipulation and many more. Altogether, ROS simplifies the task of creating complex and robust robot behavior across a wide variety of robotic platforms. ROS 2.0 is the new version of ROS which extends the initial concept (originally meant for purely research purposes) and aims to provide a distributed and modular solution for situations involving teams of robots, real-time systems or production environments, amidst others.
Among the technical novelties introduced in ROS 2.0, Open Robotics explored several options for the ROS 2.0 communication system. They decided to use the DDS middleware due to its characteristics and benefits compared to other solutions. As documented in [@ros2design], the benefit of using an end-to-end middleware, such as DDS, is that there is less code to maintain. DDS is used as a communications middleware in ROS 2.0 and it typically runs as userspace code. Even though DDS has specified standards, third parties can review audit, and implement the middleware with varying degrees of interoperability.\
As pointed out in the technical report [@osrf_rt], to have real-time performance, both a deterministic user code and an real-time operating system are needed. In our case, we will use a PREEMPT-RT patched Linux kernel as the core of our operating system for the experiments. Following the programming guidelines of the PREEMPT-RT and with a suitable kernel configuration, other authors[@Cerqueira_acomparison] demonstrated that it is possible to achieve system latency responses between 10 and 100 microseconds.\
Normally, by default, DDS implementations use the Linux Network Stack (LNS) as transport and network layer. This makes the LNS a critical part for ROS 2.0 performance. However, the network stack is not optimized for bounded latencies but instead, for throughput at a given moment. In other words, there will be some limitations due to the current status of the networking stack. Nevertheless, LNS provides QoS mechanisms and thread tuning which allows to improve the determinism of critical traffic at the kernel level.\
An important part of how the packets are processed in the Linux kernel relates actually to how hardware interrupts are handled. In a normal Linux kernel, hardware interrupts are served in two phases. In the first, an Interrupt Service Routine (ISR) is invoked when an interrupt fires, then, the hardware interrupt gets acknowledged and the work is postponed to be executed later. In a second phase, the soft interrupt, or “bottom half” is executed later to process the data coming from the hardware device. In PREEMPT-RT kernels, most ISRs are forced to run in threads specifically created for the interrupt. These threads are called IRQ threads [@lwn_irq_threads]. By handling IRQs as kernel threads, PREEMPT-RT kernels allow to schedule IRQs as user tasks, setting the priority and CPU affinity to be managed individually. IRQ handlers running in threads can themselves be interrupted so the latency due to interrupts is mitigated. For our particular interests, since our application needs to send critical traffic, it is possible to set the priority of the Ethernet interrupt threads higher than other IRQ threads to improve the network determinism.\
Another important difference between a normal and a PREEMPT-RT kernel is within the context where the softirq are executed. Starting from kernel version 3.6.1-rt1 on, the soft IRQ handlers are executed in the context of the thread that raised that Soft IRQ [@lwn_softirq]. Consequently, the NET\_RX soft IRQ, which is the softirq for receiving network packets, will normally be executed in the context of the network device IRQ thread. This allows a fine control of the networking processing context. However, if the network IRQ thread is preempted or it exhausts its NAPI[^2] weight time slice, it is executed in the ksoftirqd/n (where n is the logical number of the CPU).\
Processing packets in ksoftirqd/n context is troublesome for real-time because this thread is used by different processes for deferred work and can add latency. Also, as the ksoftirqd thread runs with SCHED\_OTHER policy, it can be easily preempted. In practice, the soft IRQs are normally executed in the context of the Ethernet IRQ threads and in the ksoftirqd/n thread, for high network loads and under heavy stress (CPU, memory, I/O, etc.). The conclusion here is that, in normal conditions, we can expect reasonable deterministic behavior, but if the network and the system are loaded, the latencies can increase greatly.\
Experimental setup and results {#setup_results}
==============================
This section presents the setup used to evaluate the real-time performance of ROS 2.0 communications over Ethernet in a PREEMPT-RT patched kernel. Specifically, we measure the end-to-end latencies between two ROS 2.0 nodes in different machines. For the experimental setup, we used a GNU/Linux PC and an embedded device which could represent a robot controller (RC) and a robot component (C) respectively.\
The PC used for the tests has the following characteristics:
- Processor: Intel(R) Core(TM) i7-8700K CPU @ 3.70GHz (6 cores).
- OS: Ubuntu 16.04 (Xenial).
- ROS 2.0 version: ardent.
- Kernel version: 4.9.30.
- PREEMPT-RT patch: rt21.
- Link capacity: 100/1000 Mbps, Full-Duplex.
- NIC: Intel i210.
In the other hand, the main characteristics of the embedded device are:
- Processor: ARMv7 Processor (2 cores).
- ROS 2 version: ardent
- Kernel version: 4.9.30.
- PREEMPT-RT patch: rt21.
- Link capacity: 100/1000 Mbps, Full-Duplex.
The RC and the C are connected point to point using a CAT6e Ethernet wire as shown in figure \[scenario\].
Round-trip test setup {#round-trip}
---------------------
The communications between the robot controller and the robot component are evaluated with a round-trip time (RTT) test, also called ping-pong test. We use a ROS 2.0 node as the client in RC and a ROS 2.0 node as the server in C. The round-trip latency is measured as the time it takes for a message to travel from the client to the server, and from the server back to the client. The message latency is measured as the difference between the time-stamp taken before sending the message (T1) in the client and the time-stamp taken just after the reception of the message in the callback of client (T2), as shown in figure \[ping-pong\].\
![Graphical presentation of the measured DDS round-trip latency. T1 is the time-stamp when data is send from the DDS publisher and T 2 is the time-stamp when data is received at the DDS publisher. Round-trip latency is defined as T2 - T1.[]{data-label="ping-pong"}](images/dds.jpg){width="40.00000%"}
The client creates a publisher that is used to send a message through a topic named ‘ping’. The client also creates a subscriber, which waits for a reply in a topic named ‘pong’. The server creates a subscriber which waits for the ‘ping’ topic and a publisher that replies the same received message through the ‘pong’ topic. The ‘ping’ and ‘pong’ topics are described by a custom message with an UINT64 used to write the message sequence number and a U8\[\] array to create a variable payload.\
For the tests performed in this work, we used a publishing rate of 10 milliseconds. The sending time is configured by waking the publisher thread with an absolute time. If the reply arrives later than 10 milliseconds and the current sending time has expired, the message is published in the next available cycle. If there is no reply in 500 milliseconds, we consider that the message was lost. Also, if the measured round-trip latency is higher than 10 milliseconds, we consider it a missed deadline, as shown in figure \[overrun\].\
![Time plot example with a 10 millisecond deadline and missed deadlines. (The image corresponds to Test2.D with DDS2)[]{data-label="overrun"}](images/time_iperf80_stress_rt_opsl.png){width="50.00000%"}
For all the experiments we have used the same ROS 2.0 QoS profile. We followed the guidelines described in the inverted pendulum control of the ROS 2.0 demos [@pendulum_demo]: best-effort reliability, KEEP\_LAST history and a history depth of 1. This configuration is not optimized for reliability but for low latencies. One of the motivations behind this configuration is that the reliable mode can potentially block, which is an unwanted behavior in real-time threads.\
All the experiments are run using three different DDS implementations.\
Experimental results {#results}
--------------------
### Test 1. System under load {#exp1}
In this test, we want to explore how communications get affected when the system is under heavy load. Additionally, we aim to show how a proper real-time configuration can mitigate these effects.\
For the ‘ping’ and ‘pong’ topics messages, we use a payload of 500 Bytes [^3]. To generate load in the system we use the tool ‘*stress*’ generating CPU stress, with 8 CPU workers, 8 VM workers, 8 I/O workers, and 8 HDD workers in the PC and 2 workers per stress type in the embedded device.\
We run the following tests:\
- Test1.A: System idle with no real-time settings.
- Test1.B: System under load with no real-time settings.
- Test1.C: System idle with real-time settings.
- Test1.D: System under load without real-time settings.
In the experiment Test1.A we run the round-trip test in normal conditions (Idle), that is, no other processes apart from the system default are running during the test. Figure \[Test1.A\] shows stable latency values with a reasonable low jitter (table \[tab:idle\]). We get latencies higher than 4 milliseconds for an specific DDS, which means that we probably can expect higher worst case latencies with a longer test duration.\
In the experiment Test1.B, we run the round-trip test with the system under load. Figure \[Test1.B\] shows how, in this case, latencies are severely affected resulting in a high number of missed deadlines (table \[tab:stress\]). This happens mainly because all the program threads are running with SCHED\_OTHER scheduling and are contending with the stress program for CPU time. Also, some latencies might also be caused due to memory page faults causing the memory not to be locked. The results show how sensitive non-real time processes are to system loads. As we wait until the arrival of the messages to send the next and we are fixing a 10 minutes duration for the test, the number of messages sent during each test might be different if deadlines are missed.\
In the experiment Test1.C we repeat the round-trip test configuring the application with real-time properties. The configuration used includes memory locking, the use of the TSLF allocators for the ROS 2.0 publishers and subscriptors, as well as running the round-trip program threads with SCHED\_FIFO scheduling and real-time priorities. The configuration of the DDS threads has been done in a different manner, depending on each DDS implementation. Some DDS implementations allowed to configure their thread priorities using the Quality of Service (QoS) profiles (through XML files), while others did not. In the cases where it was feasible, it allowed a finer real-time tuning since we could set higher priorities for the most relevant threads, as well as configuring other relevant features such memory locking and buffer pre-allocation. For the DDS implementations in which it was not possible to set the DDS thread priorities, we set a real-time priority for the main thread, so that all the threads created did inherit this priority.\
Figure \[Test1.D\] shows a clear improvement for all the DDS implementations when comparing to case Test1.B (table \[tab:stress\_rt\]), where we did not use any real-time settings. Henceforth, all the tests in the following sections are run using the real-time configurations.
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### Test 2. Concurrent traffic {#exp2}
In this second test, we want to observe the impact of non-critical traffic in the round-trip test measurements. We generate certain amount of traffic from RC to C and from C to RC using the tool *iperf*. [^4]\
In this experiment, we do not prioritize the traffic, which means the concurrent traffic will use the same Qdisc and interrupt threads than the critical traffic. In other words, we mix both non-critical and critical traffic together and analyze the impact of simply doing so.\
We run the test generating 1 Mbps, 40 Mbps and 80 Mbps concurrent traffic with and without stressing the system:
- Test2.A: System idle with 1 Mbps concurrent traffic.
- Test2.B: System under load with 1 Mbps concurrent traffic.
- Test2.C: System idle with 40 Mbps concurrent traffic.
- Test2.D: System under load with 40 Mbps concurrent traffic.
- Test2.E: System idle with 80 Mbps concurrent traffic.
- Test2.F: System under load with 80 Mbps concurrent traffic.
For the non stressed cases Test2.A, Test2.C and Test2.E, we observe that the concurrent traffic does not affect the test latencies significantly (figures \[Test2.A\], \[Test2.C\], \[Test2.E\]). When we stress the system, we can see how for 1 Mbps and 40 Mbps the latencies are still under 10 milliseconds. However, for 80 Mbps, the communications are highly affected resulting in a high number of lost messages and missed deadlines.\
As we explained in \[background\], depending on the network load, packets might be processed in the *ksoftirqd* threads. Because these threads run with no real-time priority, they would be highly affected by the system load. For this reason, we can expect higher latencies when both, concurrent traffic and system loads, are combined. For high network loads, the context switch would occur more frequently or even permanently. For medium network loads, the context switch happens intermittently with a frequency that correlates directly with the exact network load circumstances at each given time.\
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### Test 3. Increasing message payload size {#exp3}
In this third experiment, we increase the ‘Ping’ topic message payload to observe the determinism for higher critical bandwidth traffic. By increasing the payload, messages get fragmented. This implies that higher bandwidth is used and, as it happened in the previous test\[exp2\], depending on the traffic load, the packets may be processed in the *ksoftirqd* threads or not. Once again, we expect this action to have negative consequences to bound latencies.\
For this test we use two different payload sizes: 32 Kbytes and 128 Kbytes.
- Test3.A: System idle with 32 Kbytes payload.
- Test3.B: System under load with 32 Kbytes payload.
- Test3.C: System under load with 128 Kbytes payload.
For Test3.A, 32 Kbytes and the system stressed, we can start observing some high latencies (figure \[Test3.B\]), however there are no missed deadlines and message losses during a 10 minute duration test. On the other hand, for 128 Kbytes, we start observing missed deadlines and even packet loss in a 10 minutes test window. (\[tab:128k\_stress\_rt\]).\
In the previous experiment \[exp2\], non-critical traffic was the cause of high latencies. As the context switch from the Ethernet IRQ thread to the *ksoftirqd* threads occurs for a certain amount of consecutive frames, in this case the critical traffic is causing by its own the context switch. This must be taken into account when sending critical messages with high payloads.\
It is worth noting, that for this experiment, the DDS QoS were not optimized for throughput. Also, the publishing rates could not be maintained in a 10 millisecond window, thereby the missed deadlines statistics have been omitted in the plots.
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### Test 4. Tuning the kernel threads priorities {#exp4}
As suspected, the main problems demonstrated previously in \[exp2\] and \[exp3\] were caused by the packet processing (switching) in the ksoftirqd threads. In this experiment, we decided to tune the kernel threads to mitigate the problem.\
We configured the threads with the following priorities:
- Ethernet IRQ threads: priority 90
- ROS 2 executor threads: priority 80
- DDS threads: priority 70-80 [^5]
- ksoftirqd/n threads: 60
We repeated Test2.F (System under load with 80 Mbps concurrent traffic) using the new configuration.
- Test4.A: System under load with 80 Mbps concurrent traffic.
Comparing to Test2.F (Figures \[Test2.F\] and \[Test4.A\]) we observe a clear improvement with the new configuration. We observe almost no missed deadlines, nor message loss for a 10 minutes duration test. This suggests that the high latencies observed in \[exp2\] and \[exp3\] were caused because of the lack of real-time priority of *ksoftirqd* when there is a high bandwidth traffic in the networking stack.\
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### Test 5. Long term test {#exp5}
All the previous experiments were run within a 10 minute duration window. While this duration can be enough to evaluate the performance and identify some problems, it is usually not enough to observe the worst cases for a given configuration. Because some delays may appear just when specific events happen, not observing missed deadlines does not guarantee that it cannot happen in the long term. Therefore, we decided to repeat the test from experiment Test2.C for a 12 hour window.\
- Test5.A: System under load with 1 Mbps concurrent traffic. Duration 12 hours.
- Test5.B: System under load with 40 Mbps concurrent traffic. Duration 12 hours.
For 40 Mbps we observed some message lost that we did not observe in a 10 minute long test. However, the number of messages lost is reasonably low and depending on how critical is the application, it can be considered acceptable. For 1 Mbps we did not observe any message loss, nor a relevant amount of missed deadlines. We did observe a very low jitter when compared to the 40 Mbps case.
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Conclusion and future work {#conclusions}
==========================
In this work, we presented an experimental setup to show the suitability of ROS 2.0 for real-time robotic applications. We have measured the end-to-end latencies of ROS 2.0 communications using different DDS middleware implementations in different stress conditions. The results showed that a proper real-time configuration of the ROS 2.0 framework and DDS threads reduces greatly the jitter and worst case latencies.\
We also observed the limitations when there is non-critical traffic in the Linux Network Stack. Some of these problems can be avoided or minimized by configuring the network interrupt threads and using Linux traffic control QoS methods. Based on our results, we conclude that it seems possible to achieve firm and soft real-time Ethernet communications with mixed-critical traffic by using the Linux Network Stack but not hard real-time due to the observed limitations. There is ongoing work in the Linux kernel which may eventually improve the determinism in the Linux Network Stack [@xdp], [@AF_XDP] [@ETF].\
For the moment, the best strategies to optimize ROS 2.0 communications for real-time applications are to a) configure the application with real-time settings, b) configure the kernel threads accordingly to the application settings and c) limit the network and CPU usage of each one of the machines involved in the communication.\
Regarding the DDS middleware implementations evaluated, we observed differences in the performance and on the average latencies. These differences may be caused by a variety of reasons such as the DDS implementation itself, but also because of the ROS 2.0 RMW layer or even by the configurations used for the experiments. Regardless of the performance, we observed a similar behavior in terms of missed deadlines and loss messages which confirms the interest of DDS for real-time scenarios. In future work, we will evaluate several methods to limit the network and CPU usage. One way to achieve this is using the Linux control groups (cgroups) to isolate the application in exclusive CPUs. Using cgroups, it is possible to set the priority of the application traffic using ‘net\_prio’. This would help to isolate critical traffic from non-critical traffic. Also, we will evaluate the impact of non-critical traffic from another ROS 2.0 node in the same process or from the same node. For that purpose, we will focus on how ROS 2.0 executors and the DDS deal with mixed-critical topics.
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 827 & 981 & 2007 & 0/60001 & 0/60001\
DDS 2 & 1059 & 1237 & 4216 & 0/60001 & 0/60001\
DDS 3 & 1105 & 1335 & 3101 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 1079 & 1329 & 2141 & 0/60001 & 0/60001\
DDS 2 & 1301 & 1749 & 2161 & 0/60001 & 0/60001\
DDS 3 & 1394 & 1809 & 2735 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 810 & 2524 & 29481 & 233/56297 & 0/56297\
DDS 2 & 1082 & 2150 & 26764 & 236/58905 & 0/58905\
DDS 3 & 1214 & 2750 & 22492 & 133/58822 & 0/58822\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 769 & 1197 & 1823 & 0/60001 & 0/60001\
DDS 2 & 1008 & 1378 & 2774 & 0/60001 & 0/60001\
DDS 3 & 1082 & 1568 & 2092 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 976 & 1165 & 1930 & 0/60001 & 0/60001\
DDS 2 & 1340 & 1628 & 2153 & 0/60001 & 0/60001\
DDS 3 & 1343 & 1582 & 2151 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 1037 & 1283 & 3947 & 0/60001 & 0/60001\
DDS 2 & 1358 & 1744 & 4825 & 0/60001 & 0/60001\
DDS 3 & 1458 & 1835 & 4836 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 939 & 1157 & 1553 & 0/60001 & 0/60001\
DDS 2 & 1333 & 1603 & 2122 & 0/60001 & 0/60001\
DDS 3 & 1320 & 1607 & 2198 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 1011 & 1383 & 59762 & 240/36315 & 1230/36315\
DDS 2 & 1375 & 1805 & 57138 & 224/29613 & 1575/29613\
DDS 3 & 1388 & 1803 & 47058 & 119/51569 & 451/51569\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 991 & 1273 & 3509 & 0/60000 & 0/60000\
DDS 2 & 1365 & 1769 & 6405 & 0/60000 & 2/60000\
DDS 3 & 1382 & 1759 & 6274 & 0/60000 & 0/60000\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 2808 & 3226 & 3839 & 0/60001 & 0/60001\
DDS 2 & 4218 & 5505 & 6012 & 0/30000 & 0/30000\
DDS 3 & 10006 & 11655 & 12722 & 0/30001 & 0/30001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 2913 & 3292 & 4961 & 0/60001 & 0/60001\
DDS 2 & 3991 & 4664 & 7175 & 0/30000 & 0/30000\
DDS 3 & 9828 & 10809 & 35285 & 3/30000 & 0/30000\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 9000 & 9369 & 179909 & 2/30000 & 75/30000\
DDS 2 & 11511 & 14151 & 176074 & 1991/18241 & 1192/18241\
DDS 3 & 35832 & 37238 & 48398 & 0/14997 & 0/14997\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 985 & 1283 & 4942 & 0/4319751 & 3/4319751\
DDS 2 & 1319 & 1808 & 6075 & 0/4320001 & 0/4320001\
DDS 3 & 1398 & 1803 & 6906 & 0/4320000 & 5/4320000\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 1048 & 1353 & 1587 & 0/60001 & 3/60001\
DDS 2 & 1418 & 1821 & 2274 & 0/60001 & 0/60001\
DDS 3 & 1336 & 1781 & 2089 & 0/60001 & 0/60001\
[|c|c|c|c|c|c|]{}\
& Min($\mu$s) & Avg($\mu$s) & Max($\mu$s) & Missed deadlines & Message loss\
DDS 1 & 1015 & 1310 & 1670 & 0/60001 & 0/60001\
DDS 2 & 1409 & 1726 & 2411 & 0/60001 & 0/60001\
DDS 3 & 1443 & 1788 & 2603 & 0/60001 & 0/60001\
[^1]: Note that there is a relevant difference between having *well defined deadlines* and having the necessity to *meet such deadlines* in a strict manner, which is what real-time systems deliver.
[^2]: ‘New API’ or NAPI for short is an extension to the device driver packet processing framework, which is designed to improve the performance of high-speed networking.
[^3]: This makes a total packet size of 630 Bytes, summing the sequence number sub-message and UDP and RTPS headers.
[^4]: When the system was stressed the embedded devices were not capable of generating more than 20 Mbps steadily.
[^5]: Some DDS threads were configured with specific priorities using the QoS vendor profile XML
|
---
author:
- |
Luca Viano\
EPFL\
`[email protected]`\
Yu-Ting Huang\
EPFL\
`[email protected]`\
Parameswaran Kamalaruban [^1]\
LIONS, EPFL\
`[email protected]`\
Volkan Cevher\
LIONS, EPFL\
`[email protected]`\
bibliography:
- 'neurips\_2020.bib'
title: |
Robust Inverse Reinforcement Learning under\
Transition Dynamics Mismatch
---
Broader Impact {#sec.impact .unnumbered}
==============
We address the transition dynamics mismatch issue in the inverse reinforcement with a theoretically grounded algorithm and demonstrate its efficacy via extensive experiments. Our results apply to the finite MDP setting. Once we extend our work to the high dimensional and continuous control setting, it would be widely useful to the control/RL community. Nevertheless, we want to highlight the importance of robustness in IRL, especially when it comes to topics related to safety, such as autonomous driving and healthcare. Even though our adversarial training method improves robustness, being overly conservative might result in lower performance. Thus one should carefully tune the robustness related hyperparameters, which is $\alpha$ in our case.
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement n 725594 - time-data), and the Swiss National Science Foundation (SNSF) under grant number 407540\_167319.
[^1]: Corresponding author.
|
---
abstract: 'An explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces in $\mathbb{R}^{n}$. Necessary and sufficient conditions are given for such a mapping to be an isomorphism. Results apply, for example, to elliptic, parabolic, and generalized p-parabolic operators.'
author:
- |
G. N. Hile\
University of Hawaii, Honolulu, HI 96822, USA\
Email : [email protected]\
Tel : 808-956-6533\
Fax : 808-956-9139
title: Fundamental Solutions and Mapping Properties of Semielliptic Operators
---
**Keywords:** Semielliptic Operator, Fundamental Solution, Isomorphism
Introduction
============
Let $\mathcal{L}$ denote a linear *semielliptic* partial differential operator, acting on suitable real or complex $m\times1$ vector functions $u=u\left( x\right) $, $x\in\mathbb{R}^{n}$, according to$$\mathcal{L}u=\sum_{\alpha\cdot\gamma=\ell}A_{\alpha}\partial^{\alpha}u\quad.\label{op}$$ The coefficients $\left\{ A_{\alpha}\right\} $ are constant $m\times m$ matrices with real or complex entries, indexed by multi-indices $\alpha
=\left( \alpha_{1},\cdots,\alpha_{n}\right) $ in $\mathbb{R}^{n}$. The positive integer $\ell$ is the *order* of $\mathcal{L}$, $$\ell=\max\left\{ \left\vert \alpha\right\vert :A_{\alpha}\neq0\right\}
\quad,$$ and $\gamma$ is a fixed vector of rational numbers, $$\gamma=\left( \gamma_{1},\gamma_{2},\cdots,\gamma_{n}\right) =\left(
\frac{\ell}{\ell_{1}},\frac{\ell}{\ell_{2}},\ldots,\frac{\ell}{\ell_{n}}\right) \quad,\label{gamma}$$ with each $\ell_{k}$ a positive integer. The *semiellipticity condition* on $\mathcal{L}$ requires that its *symbol*, the matrix of polynomial functions$$L\left( x\right) =\sum_{\alpha\cdot\gamma=\ell}A_{\alpha}\left( ix\right)
^{\alpha}\quad,\label{ell}$$ be invertible for all nonzero $x$ in $\mathbb{R}^{n}$. (Alternative adjectives to *semielliptic*, all used by various authors, are *quasielliptic*, *semi-elliptic*, and *quasi-elliptic*). As explained for example in [@HMZ], §2, a consequence of the semiellipticity condition is that $\max_{k}\ell_{k}=\ell$, so that $\gamma
_{k}\geq1$ for each $k$ and $\gamma_{k}=1$ for at least one $k$. A further consequence is that, for each $k$, $1\leq k\leq n$, the term$$A_{\ell_{k}e_{k}}\partial^{\ell_{k}}/\partial x_{k}^{\ell_{k}}\qquad,$$ corresponding to $\alpha=\ell_{k}e_{k}$ where $e_{k}$ is the $k$th unit coordinate vector in $\mathbb{R}^{n}$, appears in $\mathcal{L}$ with $A_{\ell_{k}e_{k}}$ an invertible matrix. This term is the only unmixed derivative with respect to $x_{k}$ appearing in $\mathcal{L}$, and $\ell_{k} $ is the highest order of differentiation with respect to $x_{k}$ appearing in $\mathcal{L}$.
We consider the subclass of such operators satisfying the additional condition$$\left\Vert \gamma\right\Vert :=\sum_{k=1}^{n}\gamma_{k}>\ell\quad
.\label{gamcond}$$ We show that for such operators $\mathcal{L}$ a fundamental solution is given explicitly by the iterated integral$$F\left( x\right) =\left( 2\pi\right) ^{-n}\int_{0}^{\infty}\int
_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma\left( z\right) e^{-t\sigma\left(
z\right) }L\left( z\right) ^{-1}\ dz\ dt\qquad\left( x\neq0\right)
\quad,\label{eff}$$ where $\sigma$ is the function$$\sigma\left( x\right) =\sum_{k=1}^{n}x_{k}{}^{2\ell_{k}}\quad.\label{sigma}$$ In particular, $F$ has the properties
- $F\in C^{\infty}\left( \mathbb{R}^{n}\backslash\left\{ 0\right\}
\right) $,
- $\mathcal{L}F\left( x\right) =0$ for $x\neq0$,
- for all complex $m\times1$ vector functions $\varphi$ in $C_{0}^{\infty
}\left( \mathbb{R}^{n}\right) $, $F\ast\varphi\in C^{\infty}\left(
\mathbb{R}^{n}\right) $ and $\mathcal{L}\left( F\ast\varphi\right)
=\varphi$.
It must be pointed out that the order of integration is important in (\[eff\]), as Fubini’s Theorem does not apply, and interchanging orders of integration will likely destroy convergence of the integrals.
Of course other methods are known for constructing fundamental solutions for partial differential equations with constant coefficients. Unfortunately these methods do not always produce representations highly useful for investigation of solutions of the corresponding equations.
We use our fundamental solution to investigate properties of the operator $\mathcal{L}$ as a mapping between weighted function spaces. We introduce the vector of integers$$\underline{\ell}=\left( \ell_{1},\ell_{2},\ldots,\ell_{n}\right)
\quad,\label{ellvec}$$ and a corresponding weight function$$\rho\left( x\right) =\sigma\left( x\right) ^{1/\left( 2\ell\right)
}=\left( \sum_{k=1}^{n}x_{k}{}^{2\ell_{k}}\right) ^{1/\left( 2\ell\right)
}\quad.\label{rho}$$ We define function spaces $W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, consisting of complex $m\times1$ vector functions $u$ on $\mathbb{R}^{n}$ with finite norm$$\left\Vert u\right\Vert _{r,p,s;\underline{\ell}}=\sum_{\alpha\cdot\gamma\leq
r}\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha
}u\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{norm}$$ We demonstrate (Theorem \[main\]) that, if $1<p<\infty$ and $\left\Vert
\gamma\right\Vert >\ell$, then the mapping$$\mathcal{L}:W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline
{\ell}\right) \longrightarrow W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \label{mapit}$$ is an isomorphism if and only if $$-\left\Vert \gamma\right\Vert /p<s<\left\Vert \gamma\right\Vert -\ell
-\left\Vert \gamma\right\Vert /p\quad.\label{conit}$$
Investigations of mappings between weighted Sobolev spaces, analogous to (\[mapit\]), began with Cantor [@CA], who considered the scalar operator $\mathcal{L}=\Delta$, the Laplace operator in $\mathbb{R}^{n}$. Then $m=1$, $\ell=2$, $\gamma=\left( 1,\ldots,1\right) $, and the weight function $\rho$ is equivalent to the Euclidean norm. Cantor showed that in this special case (\[mapit\]) is an isomorphism provided that $n>2$, $n/\left( n-2\right)
<p<\infty$, and $-n/p<s<n-2-n/p$. Cantor was building on the work of Walker and Nirenberg [@WA1; @WA2; @NW], who showed that certain elliptic differential operators have finite dimensional null spaces as mappings between (unweighted) Sobolev spaces $W^{\ell,p}\left( \mathbb{R}^{n}\right) $.
McOwen [@MC1] removed Cantor’s restriction $p>n/\left( n-2\right) $ for the Laplace operator, and further listed various conditions, similar to (\[conit\]), that guarantee a power $\Delta^{k}$ of the Laplacian is a Fredholm map having certain properties. In particular, for $\Delta^{k}$ we have $\underline{\ell}=\left( 2k,\ldots,2k\right) $, and if $n>2k$ and $-n/p<s<n-2k-n/p$, the map$$\Delta^{k}:W_{s}^{2k,p}\left( \mathbb{R}^{n},\mathbb{C},\underline{\ell
}\right) \longrightarrow W_{s+2k}^{0,p}\left( \mathbb{R}^{n},\mathbb{C},\underline{\ell}\right)$$ is an isomorphism. Lockhart [@LOC] extended this result to scalar elliptic operators of order $\ell$, with constant coefficients and only highest order terms; for these operators (\[mapit\]) is an isomorphism provided that $-n/p<s<n-\ell-n/p$. Lockhart and McOwen [@LOC; @LM; @MC2] consider also elliptic operators with variable coefficients continuous at infinity; for such operators conditions are given under which the mapping (\[mapit\]) is Fredholm.
In [@HM], the author and C. Mawata considered the mapping (\[mapit\]) for the case of the heat operator in $\mathbb{R}^{n}$, $\mathcal{L}=\partial_{t}-\Delta u$. They gave conditions under which the mapping is Fredholm, and in particular showed that $\mathcal{L}$ is an isomorphism provided that $-\left( n+2\right) /p<s<n-\left( n+2\right) /p$.
In the last section of the paper are examples demonstrating how the main isomorphism theorem extends known results on elliptic operators, and produces new results for parabolic and generalized r-parabolic operators.
Of special relevance to this work is that of G. V. Demidenko [@DEM2; @DEM3] who, working in weighted Sobolev spaces somewhat different from ours, studied mapping properties of the semielliptic opertor (\[op\]). When translated to the notation of this paper, his result on isomorphic properties asserts that the mapping$$\mathcal{L}:W_{-\ell}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \longrightarrow L^{p}\left( \mathbb{R}^{n},\mathbb{C}^{m}\right)$$ is an isomorphism provided that $1<p<\left\Vert \gamma\right\Vert /\ell$. This statement is a special case of our isomorphism result, obtained by taking $s=-\ell$ in (\[mapit\]) and (\[conit\]). Demidenko did not use fundamental solutions in his investigations, but rather used integral representations to present what he called approximate solutions of equations $\mathcal{L}u=f$, converging in $L^{p}$ to actual solutions. A modification of Demidenko’s representations led to our discovery of formula (\[eff\]) for a fundamental solution.
Demidenko [@DEM4; @DEM5; @DEM6] has recently extended his isomorphism results to operators of the form$$\mathcal{L}=\left[
\begin{tabular}
[c]{cc}$\mathcal{L}_{1}$ & $\mathcal{L}_{2}$\\
$\mathcal{L}_{3}$ & $0$\end{tabular}
\ \ \right] \qquad,$$ where $\mathcal{L}_{1}$ is a square matrix semielliptic operator, and $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ are rectangular matrix differential operators having certain properties related to semiellipticity.
Notation and Preliminaries
==========================
In formula (\[op\]) we use the conventional notation$$\alpha\cdot\gamma=\alpha_{1}\gamma_{1}+\alpha_{2}\gamma_{2}+\cdots+\alpha
_{n}\gamma_{n}\qquad,\qquad\partial^{\alpha}=\frac{\partial^{\left\vert
\alpha\right\vert }}{\partial x_{1}^{\alpha_{1}}\partial x_{2}^{\alpha_{2}}\cdots\partial x_{n}^{\alpha_{n}}}\quad,$$ with $\left\vert \alpha\right\vert =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ representing the *length* of the multi-index $\alpha$. Some authors write (\[op\]) in the equivalent formulation$$\mathcal{L}u=\sum_{\alpha/\underline{\ell}=1}A_{\alpha}\partial^{\alpha}u\quad,$$ where $\alpha/\underline{\ell}$ is defined as the sum $\alpha_{1}/\ell
_{1}+\cdots+\alpha_{n}/\ell_{n}$. However we prefer (\[op\]), as the vector $\gamma$ proves useful in some of our representations.
For a vector $x\in\mathbb{R}^{n}$ and for a complex matrix $M=\left(
m_{ij}\right) $, we employ the usual norms$$\left\vert x\right\vert =\left( x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)
^{1/2}\qquad,\qquad\left\vert M\right\vert =\left( \sum_{i,j}\left\vert
m_{ij}\right\vert ^{2}\right) ^{1/2}\quad.$$ We also at times use an alternate norm for vectors, as specified by$$\left\Vert x\right\Vert :=\left\vert x_{1}\right\vert +\left\vert
x_{2}\right\vert +\cdots+\left\vert x_{n}\right\vert \quad,$$ and as demonstrated already in (\[gamcond\]). (The length $\left\vert
\alpha\right\vert $ of a multi-index $\alpha$ is not the Euclidean length, but rather the same as $\left\Vert \alpha\right\Vert $; however we conform to custom and use $\left\vert \alpha\right\vert $, with the expectation that the correct interpretation will be clear from the context.)
The function $\rho$ of (\[rho\]) serves as an *anisotropic length* of vectors $x$ in $\mathbb{R}^{n}$. In [@HMZ], §2 and §5, one finds verifications of the inequalities$$\rho(x+y)\leq\rho(x)+\rho(y)\qquad,\qquad\left\vert x^{\alpha}\right\vert
\leq\rho(x)^{\alpha\cdot\gamma}\quad.\label{tri}$$ For positive real numbers $t$ and for $x\in\mathbb{R}^{n}$ we denote$$t^{\gamma}x:=\left( t^{\gamma_{1}}x_{1},t^{\gamma_{2}}x_{2},\cdots
,t^{\gamma_{n}}x_{n}\right) \quad.$$ Straightforward calculations confirm that$$\left( t^{\gamma}x\right) ^{\alpha}=t^{\alpha\cdot\gamma}x^{\alpha}\qquad,\qquad\rho\left( t^{\gamma}x\right) =t\rho(x)\qquad,\qquad L\left(
t^{\gamma}x\right) =t^{\ell}L\left( x\right) \quad.\label{hom}$$ If we fix $x$ in the last two of these inequalities and choose $t=1/\rho
\left( x\right) $, so that $\rho\left( t^{\gamma}x\right) =1$, we deduce that $$c_{1}\left( \mathcal{L}\right) \rho(x)^{\ell}\leq\left\vert L\left(
x\right) \right\vert \leq c_{2}\left( \mathcal{L}\right) \rho(x)^{\ell
}\quad,\label{rhoineq1}$$$$c_{3}\left( \mathcal{L}\right) \rho(x)^{-\ell}\leq\left\vert L\left(
x\right) ^{-1}\right\vert \leq c_{4}\left( \mathcal{L}\right)
\rho(x)^{-\ell}\qquad\left( x\neq0\right) \quad,\label{rhoineq2}$$ where $\left\vert \cdots\right\vert $ here is the matrix norm, and $$\begin{aligned}
c_{1}\left( \mathcal{L}\right) & =\min_{\rho(y)=1}\ \left\vert
L(y)\right\vert \qquad,\qquad c_{2}\left( \mathcal{L}\right) =\max
_{\rho(y)=1}\ \left\vert L(y)\right\vert \quad,\\
c_{3}\left( \mathcal{L}\right) & =\min_{\rho(y)=1}\ \left\vert
L(y)^{-1}\right\vert \qquad,\qquad c_{4}\left( \mathcal{L}\right)
=\max_{\rho(y)=1}\ \left\vert L(y)^{-1}\right\vert \quad.\end{aligned}$$ It is further demonstrated in [@HMZ], §2, that, for $x$ in $\mathbb{R}^{n}$ and multi-indices $\alpha$, there are nonnegative constants $c_{5}\left( \mathcal{L}\right) $ and $c_{6}\left( \mathcal{L},\alpha\right) $ such that$$\left\vert \partial^{\alpha}L(x)\right\vert \leq\left\{
\begin{tabular}
[c]{cc}$c_{5}\left( \mathcal{L}\right) \rho\left( x\right) ^{\ell-\alpha
\cdot\gamma}$ & if $\alpha\cdot\gamma\leq\ell,$\\
$0$ & otherwise,
\end{tabular}
\right. \label{rhoineq3}$$$$\left\vert \partial^{\alpha}L\left( x\right) ^{-1}\right\vert \leq
c_{6}\left( \mathcal{L},\alpha\right) \rho\left( x\right) ^{-\ell
-\alpha\cdot\gamma}\qquad\left( x\neq0\right) \quad.\label{rhoineq4}$$
A Fundamental Solution
======================
We demonstrate that formula (\[eff\]) does indeed prescribe a fundamental solution for the differential operator $\mathcal{L}$. First we investigage in some detail the inner integral of (\[eff\]). For multi-indices $\beta$ and points $x$ in $\mathbb{R}^{n}$, and for $t>0$, we define $m\times m$ matrix valued functions $$\begin{aligned}
J\left( x,t\right) & =\int_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma
(z)e^{-t\sigma(z)}L\left( z\right) ^{-1}\ dz\quad,\label{ja}\\
J_{\beta}\left( x,t\right) & =\int_{\mathbb{R}^{n}}e^{ix\cdot z}\left(
iz\right) ^{\beta}\sigma(z)e^{-t\sigma(z)}L\left( z\right) ^{-1}\ dz\quad.\label{jay}$$ Observe that $J_{\beta}=J$ when $\beta=\left( 0,\ldots,0\right) $. Also, $J_{\beta}$ is the formal derivative $\partial_{x}^{\beta}J$, arising by differentiation of $J$ under the integral; we will discuss validity of this action, as well as convergence of these integrals and other properties.
We require a lemma concerning convergence of more elementary integrals.
\[rhoint\]Let $s$ be any real constant, and assume $t>0$.
(a) The integral$$\int_{\rho\left( x\right) \geq1}\rho\left( x\right) ^{s}\ dx$$ is finite if and only if $s<-\left\Vert \gamma\right\Vert $, in which case$$\int_{\rho\left( x\right) \geq t}\rho\left( x\right) ^{s}\ dx=t^{s+\left\Vert \gamma\right\Vert }\int_{\rho\left( x\right) \geq1}\rho\left( x\right) ^{s}\ dx\quad.\label{ja2}$$
(b) The integral$$\int_{\rho\left( x\right) \leq1}\rho\left( x\right) ^{s}\ dx$$ is finite if and only if $s>-\left\Vert \gamma\right\Vert $, in which case$$\int_{\rho\left( x\right) \leq t}\rho\left( x\right) ^{s}\ dx=t^{s+\left\Vert \gamma\right\Vert }\int_{\rho\left( x\right) \leq1}\rho\left( x\right) ^{s}\ dx\quad.\label{ja1}$$
Given $0\leq r<R\leq\infty$ and $s\in\mathbb{R}$, let $I\left( r,R,s\right)
$ be the integral$$I\left( r,R,s\right) =\int_{r\leq\rho\left( x\right) \leq R}\rho\left(
x\right) ^{s}\ dx\quad.\label{ei}$$ In this integral we make the change of integration parameter $z=t^{\gamma}x$, where $t$ is a positive constant; then $dz=t^{\left\Vert \gamma\right\Vert
}\ dx$, $\rho\left( z\right) =t\rho\left( x\right) $, to derive$$I\left( tr,tR,s\right) =t^{s+\left\Vert \gamma\right\Vert }I\left(
r,R,s\right) \quad.\label{ja3}$$ Setting $r=1$, $R=2$, and $t=2^{m}$ gives$$I\left( 2^{m},2^{m+1},s\right) =2^{m\left( s+\left\Vert \gamma\right\Vert
\right) }I\left( 1,2,s\right) \quad.$$ From the geometric summation$$I\left( 1,\infty,s\right) =\sum_{m=0}^{\infty}I\left( 2^{m},2^{m+1},s\right) =I\left( 1,2,s\right) \sum_{m=0}^{\infty}2^{m\left( s+\left\Vert
\gamma\right\Vert \right) }$$ it follows that $I\left( 1,\infty,s\right) <\infty$ if and only if $s+\left\Vert \gamma\right\Vert <0$. Then we obtain (\[ja2\]) by setting $r=1$, $R=\infty$ in (\[ja3\]). Likewise, from$$I\left( 0,1,s\right) =\sum_{m=-1}^{-\infty}I\left( 2^{m},2^{m+1},s\right)
=I\left( 1,2,s\right) \sum_{m=-1}^{-\infty}2^{m\left( s+\left\Vert
\gamma\right\Vert \right) }$$ it follows that $I\left( 0,1,s\right) <\infty$ if and only if $s+\left\Vert
\gamma\right\Vert >0$, in which case (\[ja1\]) is obtained by setting $r=0
$, $R=1$ in (\[ja3\]).
The next lemma is proved in [@HMZ], §5.
\[hmzlemma\]Given real numbers R and S with $0\leq R<S$, there exists a real valued function $\psi$ in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right)
$, with support in the region where $\rho\left( x\right) <S$, such that $0\leq\psi\leq1$, $\psi(x)=1$ if $\rho(x)\leq R$, and for any multi-index $\alpha$ and $x$ in $\mathbb{R}^{n}$,$$\left\vert \partial^{\alpha}\psi(x)\right\vert \leq C(\ell,\alpha)\left(
S-R\right) ^{-\alpha\cdot\gamma}\quad.$$
The following lemma, somewhat technical in nature, gathers pertinent information regarding the integrals $J_{\beta}$.
\[jaylemma\]Each integral $J_{\beta}\left( x,t\right) $ converges absolutely for $x\in\mathbb{R}^{n}$ and $t>0$, with$$\left\vert J_{\beta}\left( x,t\right) \right\vert \leq C\left(
\mathcal{L},\beta\right) \frac{t^{-1/2}}{\left[ t^{1/\left( 2\ell\right)
}+\rho\left( x\right) \right] ^{\beta\cdot\gamma+\left\Vert \gamma
\right\Vert }}\quad.\label{jaybound}$$ Moreover, $J\in C^{\infty}\left[ \mathbb{R}^{n}\times\left( 0,\infty\right)
\right] $, with differentiation of $J$ under the integral of all orders allowed. In particular, for $x\in\mathbb{R}^{n}$ and $t>0$,$$\begin{aligned}
\frac{\partial^{\beta}}{\partial x^{\beta}}J\left( x,t\right) & =J_{\beta
}\left( x,t\right) \quad,\label{jayy}\\
\frac{\partial^{k}}{\partial t^{k}}\frac{\partial^{\beta}}{\partial x^{\beta}}J\left( x,t\right) & =\left( -1\right) ^{k}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\left( iz\right) ^{\beta}\sigma\left( z\right)
^{k+1}e^{-t\sigma\left( z\right) }L\left( z\right) ^{-1}\ dz\quad
,\label{jayyy}$$ with (\[jayyy\]) likewise converging absolutely. If also $s>0$, then$$J_{\beta}\left( x,t\right) =s^{\beta\cdot\gamma+\left\Vert \gamma\right\Vert
+\ell}J_{\beta}\left( s^{\gamma}x,s^{2\ell}t\right) \quad.\label{jayz}$$
Using (\[rho\]), (\[tri\]), and (\[rhoineq2\]), we derive for the integrand of (\[jayyy\]) the bound$$\left\vert e^{ix\cdot z}\left( iz\right) ^{\beta}\sigma(z)^{k+1}e^{-t\sigma(z)}L\left( z\right) ^{-1}\right\vert \leq c_{4}\left(
\mathcal{L}\right) \rho\left( z\right) ^{\beta\cdot\gamma+2k\ell+\ell
}e^{-t\sigma\left( z\right) }\quad.\label{jayb}$$ This exponential decay confirms the absolute convergence of (\[jayyy\]), and (when $k=0$) of $J_{\beta}\left( x,t\right) $.
Next we write$$J_{\beta}\left( s^{\gamma}x,s^{2\ell}t\right) =\int_{\mathbb{R}^{n}}e^{is^{\gamma}x\cdot z}\left( iz\right) ^{\beta}\sigma(z)e^{-s^{2\ell
}t\sigma(z)}L\left( z\right) ^{-1}\ dz\quad,$$ and in the integral make the change of variable $y=s^{\gamma}z$, with$$y^{\beta}=s^{\beta\cdot\gamma}z^{\beta}\quad,\quad\sigma\left( y\right)
=s^{2\ell}\sigma\left( z\right) \quad,\quad L\left( y\right) =s^{\ell
}L\left( z\right) \quad,\quad dy=s^{\left\Vert \gamma\right\Vert }dz\quad,$$ to obtain (\[jayz\]).
We consider differentiating the integral on the right of (\[jayyy\]), which we will refer to as $Q\left( x,t\right) $, with respect to $x_{k}$. Recalling that $\left\vert e^{ir}-1\right\vert \leq\left\vert r\right\vert $ for $r\in\mathbb{R}$, we obtain for $h\neq0$ the estimate$$\left\vert \frac{1}{h}\left[ Q\left( x+he_{k},t\right) -Q\left(
x,t\right) \right] \right\vert \leq\int_{\mathbb{R}^{n}}\left\vert
z_{k}\right\vert \left\vert e^{ixz}\left( iz\right) ^{\beta}\sigma
(z)^{k+1}e^{-t\sigma(z)}L\left( z\right) ^{-1}\right\vert \ dz,$$ and note that the latter integral converges in view of (\[jayb\]). By the dominated convergence theorem, differentiation of $Q\left( x,t\right) $ with respect to $x_{k}$ under the integral is valid. In particular,$$\frac{\partial}{\partial x_{k}}J_{\beta}\left( x,t\right) =J_{\beta+e_{k}}\left( x,t\right) \quad,$$ and an induction argument confirms (\[jayy\]). In a similar way, differentiation of $Q\left( x,t\right) $ under the integral with respect to $t$ can be justified with use of the inequality $\left\vert e^{z}-1\right\vert
\leq\left\vert z\right\vert e^{\left\vert z\right\vert }$; then (\[jayyy\]) follows by induction.
It remains only to establish the bound (\[jaybound\]). First we bound the integral$$J_{\beta}\left( x,1\right) =\int_{\mathbb{R}^{n}}e^{ix\cdot z}\left(
iz\right) ^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\ dz\quad.\label{jay1}$$
By Lemma \[hmzlemma\], there exists a real valued function $\psi$ in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $, with $0\leq\psi\leq1$, $\psi\left( z\right) =0$ if $\rho\left( z\right) \geq2$, $\psi\left(
z\right) =1$ if $\rho\left( z\right) \leq1$, and $\left\vert \partial
^{\alpha}\psi\left( z\right) \right\vert \leq C\left( \ell,\alpha\right) $ for any multi-index $\alpha$ in $\mathbb{R}^{n}$. Given $\varepsilon>0$, we set $$\varphi_{\varepsilon}\left( z\right) =1-\psi\left[ \left( \varepsilon
^{-1}\right) ^{\gamma}z\right] =1-\psi\left( \frac{z_{1}}{\varepsilon
^{\gamma_{1}}},\frac{z_{2}}{\varepsilon^{\gamma_{2}}},\cdots,\frac{z_{n}}{\varepsilon^{\gamma_{n}}}\right) \quad,$$ so that $\varphi_{\varepsilon}\in C^{\infty}\left( \mathbb{R}^{n}\right) $, $0\leq\varphi_{\varepsilon}\leq1$, and$$\varphi_{\varepsilon}\left( z\right) =\left\{
\begin{array}
[c]{cc}0\quad, & \text{if }\rho\left( z\right) \leq\varepsilon\\
1\quad, & \text{if }\rho\left( z\right) \geq2\varepsilon
\end{array}
\right. \quad,\quad\left\vert \partial^{\alpha}\varphi_{\varepsilon}\left(
z\right) \right\vert \leq C\left( \ell,\alpha\right) \varepsilon
^{-\alpha\cdot\gamma}\quad.\label{phi}$$ But if $\alpha\neq0$, $\partial^{\alpha}\varphi_{\varepsilon}(z)$ vanishes except where $\varepsilon\leq\rho(z)\leq2\varepsilon$; thus the second inequality of (\[phi\]) implies also$$\left\vert \partial^{\alpha}\varphi_{\varepsilon}\left( z\right) \right\vert
\leq C\left( \ell,\alpha\right) \rho\left( z\right) ^{-\alpha\cdot\gamma
}\quad.\label{phib}$$ (For $\alpha=0$ inequality (\[phib\]) is trivial.)
As (\[jay1\]) converges absolutely, for any multi-index $\alpha$ we may write$$\begin{aligned}
\left( ix\right) ^{\alpha}J_{\beta}\left( x,1\right) & =\left(
ix\right) ^{\alpha}\ \lim_{\varepsilon\rightarrow0}\ \int_{\mathbb{R}^{n}}\varphi_{\varepsilon}\left( z\right) e^{ix\cdot z}\left( iz\right)
^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\ dz\\
& =\lim_{\varepsilon\rightarrow0}\ \int_{\rho\left( z\right) \geq
\varepsilon}\left( \partial_{z}^{\alpha}e^{ix\cdot z}\right) \varphi
_{\varepsilon}\left( z\right) \left( iz\right) ^{\beta}\sigma
(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\ dz\quad.\end{aligned}$$ We integrate by parts, taking into account the exponential decay of the integrand at infinity (which we discuss in more detail later), as well as the vanishing of $\varphi_{\varepsilon}\left( z\right) $ and all its derivatives on the surface $\rho\left( z\right) =\varepsilon$, to obtain$$\begin{aligned}
& \left( ix\right) ^{\alpha}J_{\beta}\left( x,1\right) \nonumber\\
& =\left( -1\right) ^{\left\vert \alpha\right\vert }\lim_{\varepsilon
\rightarrow0}\int_{\rho\left( z\right) \geq\varepsilon}e^{ix\cdot z}\partial^{\alpha}\left[ \varphi_{\varepsilon}\left( z\right) \left(
iz\right) ^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\right]
\ dz\ .\label{jbeta}$$
Now, the derivative $$\partial^{\alpha}\left[ \varphi_{\varepsilon}\left( z\right) \left(
iz\right) ^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\right]$$ is a finite linear combination of products of the form$$\partial^{\eta}\varphi_{\varepsilon}\left( z\right) \ \partial^{\mu}z^{\beta}\ \partial^{\nu}\left[ \sigma\left( z\right) e^{-\sigma\left(
z\right) }\right] \ \partial^{\tau}L\left( z\right) ^{-1}\quad,$$ where $\eta$, $\mu$, $\nu$, and $\tau$ are multi-indices in $\mathbb{R}^{n}$ with $\eta+\mu+\nu+\tau=\alpha$. From (\[phib\]) and (\[rhoineq4\]),$$\left\vert \partial^{\eta}\varphi_{\varepsilon}\left( z\right) \right\vert
\leq C\left( \ell,\eta\right) \rho\left( z\right) ^{-\eta\cdot\gamma}\quad,\quad\left\vert \partial^{\tau}L\left( z\right) ^{-1}\right\vert \leq
C\left( \mathcal{L},\tau\right) \rho\left( z\right) ^{-\ell-\tau
\cdot\gamma}\ .\label{bound1}$$ In view of the formula$$\partial^{\mu}z^{\beta}=\left\{
\begin{array}
[c]{cc}\frac{\beta!}{\left( \beta-\mu\right) !}z^{\beta-\mu} & ,\quad\text{if }\mu\leq\beta\quad\text{,}\\
0 & ,\quad\text{otherwise\quad,}\end{array}
\right.$$ we may use the second inequality of (\[tri\]) to obtain $$\left\vert \partial^{\mu}z^{\beta}\right\vert \leq C\left( \beta\right)
\rho\left( z\right) ^{\left( \beta-\mu\right) \cdot\gamma}\quad
.\label{bound2}$$
From (\[rho\]) and (\[hom\]), for $t>0$ and any multi-index $\omega$ we infer that$$\sigma\left( z\right) =t^{-2\ell}\sigma\left( t^{\gamma}z\right)
\qquad,\qquad\partial^{\omega}\sigma\left( z\right) =t^{\omega\cdot
\gamma-2\ell}\partial^{\omega}\sigma\left( t^{\gamma}z\right) \quad.$$ If $z\neq0$ we may choose $t=\sigma\left( z\right) ^{-1/(2\ell)}=\rho\left(
z\right) ^{-1}$, so that $\rho\left( t^{\gamma}z\right) =1$; then we obtain$$\left\vert \partial^{\omega}\sigma\left( z\right) \right\vert \leq C\left(
\omega,\underline{\ell}\right) \rho\left( z\right) ^{2\ell-\omega
\cdot\gamma}\quad,\label{bound3}$$ where $C\left( \omega,\underline{\ell}\right) =\sup_{\rho\left( z\right)
=1}\left\vert \partial^{\omega}\sigma\left( z\right) \right\vert $.
Any derivative $\partial^{\nu}\left[ \sigma\left( z\right) e^{-\sigma
\left( z\right) }\right] $ is a finite linear combination of terms of the form$$e^{-\sigma\left( z\right) }\left[ \prod_{k=1}^{K}\partial^{\nu^{k}}\sigma\left( z\right) \right] \quad,$$ where $1\leq K\leq1+\left\vert \nu\right\vert \leq1+\left\vert \alpha
\right\vert $ and $\left\{ \nu^{k}\right\} $ are multi-indices with $\nu
^{1}+\nu^{2}+\cdots+\nu^{K}=\nu$. As everything in this linear combination depends upon $\nu$ and $\underline{\ell}$, we obtain with use of (\[bound3\]) the estimate$$\begin{aligned}
\left\vert e^{-\sigma\left( z\right) }\left[ \prod_{k=1}^{K}\partial
^{\nu^{k}}\sigma\left( z\right) \right] \right\vert & \leq e^{-\sigma
\left( z\right) }\prod_{k=1}^{K}C\left( \nu^{k},\underline{\ell}\right)
\rho\left( z\right) ^{2\ell-\nu^{k}\cdot\gamma}\\
& \leq C\left( \nu,\underline{\ell}\right) \rho\left( z\right)
^{2K\ell-\nu\cdot\gamma}e^{-\sigma\left( z\right) }\quad,\end{aligned}$$$$\left\vert \partial^{\nu}\left[ \sigma\left( z\right) e^{-\sigma\left(
z\right) }\right] \right\vert \leq C\left( \nu,\underline{\ell}\right)
e^{-\sigma\left( z\right) }\cdot\left\{
\begin{array}
[c]{cc}\rho\left( z\right) ^{2\left( 1+\left\vert \alpha\right\vert \right)
\ell-\nu\cdot\gamma} & ,\quad\text{if }\rho\left( z\right) \geq1\ .\\
\rho\left( z\right) ^{2\ell-\nu\cdot\gamma} & ,\quad\text{if }\rho\left(
z\right) \leq1\ .
\end{array}
\right.$$
Upon combining this bound with (\[bound1\]) and (\[bound2\]), while noting that all multi-indices and linear combinations are determined ultimately by $\alpha$, $\beta$, and $\mathcal{L}$, we deduce that$$\begin{aligned}
& \left\vert \partial^{\alpha}\left[ \varphi_{\varepsilon}\left( z\right)
\left( iz\right) ^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right)
^{-1}\right] \right\vert \nonumber\\
& \leq C\left( \mathcal{L},\alpha,\beta\right) e^{-\sigma\left( z\right)
}\cdot\left\{
\begin{array}
[c]{cc}\rho\left( z\right) ^{2\left\vert \alpha\right\vert \ell+\ell+\beta
\cdot\gamma-\alpha\cdot\gamma} & ,\quad\text{if }\rho\left( z\right)
\geq1\ ,\\
\rho\left( z\right) ^{\ell+\beta\cdot\gamma-\alpha\cdot\gamma} &
,\quad\text{if }0<\rho\left( z\right) \leq1\ .
\end{array}
\right. \label{intbound}$$ Note that the displayed exponential decay at infinity justifies our previous integrations by parts. By Lemma \[rhoint\], for integrability of this last expression near $\rho\left( z\right) =0$ we require that $\ell+\beta
\cdot\gamma-\alpha\cdot\gamma>-\left\Vert \gamma\right\Vert $. If this condition holds we may let $\varepsilon\rightarrow0$ inside the integral in (\[jbeta\]), as the bounds (\[intbound\]) are independent of $\varepsilon
$. We have the pointwise limits $\varphi_{\varepsilon}\left( z\right)
\rightarrow1$ and $\partial^{\nu}\varphi_{\varepsilon}\left( z\right)
\rightarrow0$ if $\nu\neq0$; thus (\[jbeta\]) results in$$x^{\alpha}J_{\beta}\left( x,1\right) =i^{\left\vert \alpha\right\vert }\int_{\mathbb{R}^{n}}e^{ix\cdot z}\partial^{\alpha}\left[ \left( iz\right)
^{\beta}\sigma(z)e^{-\sigma(z)}L\left( z\right) ^{-1}\right] \ dz\quad
,\label{jbetaparts}$$ provided that $\alpha\cdot\gamma<\ell+\beta\cdot\gamma+\left\Vert
\gamma\right\Vert $. Our argument thus far ensures that this integral converges absolutely, and indeed (\[intbound\]) and (\[jbetaparts\]) imply the bound$$\left\vert x^{\alpha}J_{\beta}\left( x,1\right) \right\vert \leq C\left(
\mathcal{L},\alpha,\beta\right) \qquad\text{( if }\alpha\cdot\gamma
<\ell+\beta\cdot\gamma+\left\Vert \gamma\right\Vert \text{ )}\quad
.\label{jay1est}$$
Now in (\[jay1est\]) we choose $\alpha=N\ell_{k}e_{k}$, where $N$ is a nonnegative integer and $e_{k}$ is the unit multi-index in the $k$th coordinate direction. The condition $\alpha\cdot\gamma=N\ell_{k}\gamma
_{k}=N\ell<\ell+\beta\cdot\gamma+\left\Vert \gamma\right\Vert $ leads to the requirement$$0\leq N<1+\frac{\beta\cdot\gamma+\left\Vert \gamma\right\Vert }{\ell}\quad.\label{enn}$$ We have $x^{\alpha}=x_{k}{}^{N\ell_{k}}$, and hence (\[jay1est\]) gives$$\left\vert x_{k}{}^{N\ell_{k}}J_{\beta}\left( x,1\right) \right\vert \leq
C\left( \mathcal{L},N,\beta\right) \quad.\label{jab}$$ If $\delta_{1}$, $\delta_{2}$, $\ldots$, $\delta_{n}$ each have either of the values $+1$ or $-1$, then application of (\[jab\]), and (\[jay1est\]) with $\alpha=0$, gives$$\left\vert \left( 1+\delta_{1}x_{1}{}^{N\ell_{1}}+\delta_{2}x_{2}{}^{N\ell_{2}}+\cdots+\delta_{n}x_{n}{}^{N\ell_{n}}\right) J_{\beta}\left(
x,1\right) \right\vert \leq C\left( \mathcal{L},N,\beta\right) \quad.$$ Now, given any $x$ in $\mathbb{R}^{n}$, we may choose the values $\left\{
\delta_{k}\right\} $ so that this inequality becomes$$\left( 1+\left\vert x_{1}\right\vert ^{N\ell_{1}}+\left\vert x_{2}\right\vert
^{N\ell_{2}}+\cdots+\left\vert x_{n}\right\vert ^{N\ell_{n}}\right)
\left\vert J_{\beta}\left( x,1\right) \right\vert \leq C\left(
\mathcal{L},N,\beta\right) \quad.$$ But $$\left[ 1+\rho\left( x\right) \right] ^{N\ell}\leq C\left( N,\ell
,n\right) \left( 1+\left\vert x_{1}\right\vert ^{N\ell_{1}}+\left\vert
x_{2}\right\vert ^{N\ell_{2}}+\cdots+\left\vert x_{n}\right\vert ^{N\ell_{n}}\right) \quad,$$ and thus, provided that (\[enn\]) holds, $$\left\vert J_{\beta}\left( x,1\right) \right\vert \leq\frac{C\left(
\mathcal{L},N,\beta\right) }{\left[ 1+\rho\left( x\right) \right]
^{N\ell}}\quad\text{.}$$ We may choose an integer $N$ satisfying (\[enn\]) so that $N\geq\left(
\beta\cdot\gamma+\left\Vert \gamma\right\Vert \right) /\ell$; then we obtain$$\left\vert J_{\beta}\left( x,1\right) \right\vert \leq\frac{C\left(
\mathcal{L},\beta\right) }{\left[ 1+\rho\left( x\right) \right]
^{\beta\cdot\gamma+\left\Vert \gamma\right\Vert }}\quad.\label{jb1}$$
Finally, we set $s=t^{-1/\left( 2\ell\right) }$ in (\[jayz\]) to obtain$$J_{\beta}\left( x,t\right) =t^{-\left( \beta\cdot\gamma+\left\Vert
\gamma\right\Vert +\ell\right) /\left( 2\ell\right) }J_{\beta}\left(
t^{-\gamma/\left( 2\ell\right) }x,1\right) \quad.\label{jb2}$$ Applying then (\[jb1\]), noting that $\rho\left( t^{-\gamma/\left(
2\ell\right) }x\right) =t^{-1/\left( 2\ell\right) }\rho\left( x\right)
$, yields (\[jaybound\]).
Our proposed fundamental solution (\[eff\]) for $\mathcal{L}$ can be written in terms of $J$ as the $m\times m$ matrix valued function$$F\left( x\right) =\left( 2\pi\right) ^{-n}\int_{0}^{\infty}J\left(
x,t\right) \ dt\qquad\left( x\neq0\right) \quad.\label{effb}$$ In view of (\[jayy\]), the formal derivative $\partial^{\beta}F$ of (\[effb\]) is$$F_{\beta}\left( x\right) :=\left( 2\pi\right) ^{-n}\int_{0}^{\infty
}J_{\beta}\left( x,t\right) \ dt\qquad\left( x\neq0\right) \quad
.\label{effbeta}$$ We will show that, under the added restriction $\left\Vert \gamma\right\Vert
>\ell$, these integrals converge absolutely if $x\neq0$. When $x=0$, (\[effbeta\]) and (\[jb2\]) give$$F_{\beta}\left( 0\right) =\left( 2\pi\right) ^{-n}J_{\beta}\left(
0,1\right) \int_{0}^{\infty}t^{-\left( \beta\cdot\gamma+\left\Vert
\gamma\right\Vert +\ell\right) /\left( 2\ell\right) }\ dt\quad.$$ As the integral on the right is infinite for any value of $\beta$, formulas (\[effb\]) and (\[effbeta\]) are undefined at $x=0$.
\[effthm\]Suppose $\left\Vert \gamma\right\Vert >\ell$. Then
(a) for $x\neq0$ each integral (\[effbeta\]) for $F_{\beta}\left(
x\right) $ converges absolutely, and $$\left\vert F_{\beta}\left( x\right) \right\vert \leq C\left( \mathcal{L},\beta\right) \rho\left( x\right) ^{\ell-\beta\cdot\gamma-\left\Vert
\gamma\right\Vert }\qquad\left( x\neq0\right) \ ,\label{effest}$$
(b) $F\in C^{\infty}\left( \mathbb{R}^{n}\backslash\left\{ 0\right\}
\right) $, and $\partial^{\beta}F\left( x\right) =F_{\beta}\left(
x\right) $ for all multi-indices $\beta$ and all nonzero $x$ in $\mathbb{R}^{n}$,
\(c) for $s>0$ and $x\in\mathbb{R}^{n}$,$$F_{\beta}\left( s^{\gamma}x\right) =s^{\ell-\beta\cdot\gamma-\left\Vert
\gamma\right\Vert }F_{\beta}\left( x\right) \quad,\label{effhom}$$
(d) $\mathcal{L}F=0$ in the region $\mathbb{R}^{n}\backslash\left\{
0\right\} $.
By (\[jaybound\]),$$\int_{0}^{\infty}\left\vert J_{\beta}\left( x,t\right) \right\vert \ dt\leq
C\left( \mathcal{\mathcal{L}},\beta\right) \int_{0}^{\infty}\frac{t^{-1/2}}{\left[ t^{1/\left( 2\ell\right) }+\rho\left( x\right) \right]
^{\beta\cdot\gamma+\left\Vert \gamma\right\Vert }}\ dt\ .$$ For $x\neq0$ we make the change of integration parameter $t=\rho\left(
x\right) ^{2\ell}s^{2\ell}$, to obtain$$\int_{0}^{\infty}\left\vert J_{\beta}\left( x,t\right) \right\vert \ dt\leq
C\left( \mathcal{\mathcal{L}},\beta\right) \rho\left( x\right)
^{\ell-\beta\cdot\gamma-\left\Vert \gamma\right\Vert }2\ell\int_{0}^{\infty
}\frac{s^{\ell-1}}{\left( s+1\right) ^{\beta\cdot\gamma+\left\Vert
\gamma\right\Vert }}\ ds\quad.$$ The latter integral converges at zero since $\ell\geq1$, and at infinity since $\beta\cdot\gamma+\left\Vert \gamma\right\Vert \geq\left\Vert \gamma
\right\Vert >\ell$; thus$$\int_{0}^{\infty}\left\vert J_{\beta}\left( x,t\right) \right\vert \ dt\leq
C\left( \mathcal{\mathcal{L}},\beta\right) \rho\left( x\right)
^{\ell-\beta\cdot\gamma-\left\Vert \gamma\right\Vert }\qquad\left(
x\neq0\right) \quad.\label{intjay}$$ Hence (a) follows, with this inequality and (\[effbeta\]) implying (\[effest\]).
To verify the assertions of (b), for $x\neq0$ we examine a difference quotient$$\frac{F_{\beta}\left( x+se_{k}\right) -F_{\beta}\left( x\right) }{s}=\left( 2\pi\right) ^{-n}\int_{0}^{\infty}\frac{J_{\beta}\left(
x+se_{k},t\right) -J_{\beta}\left( x,t\right) }{s}\ dt\quad.\label{dq}$$ If $x\neq0$ and $\left\vert s\right\vert <\left\vert x\right\vert /2$, then the line connecting $x$ to $x+se_{k}$ misses the origin, and there is a number $r$ between $0$ and $s$ so that$$\frac{J_{\beta}\left( x+se_{k},t\right) -J_{\beta}\left( x,t\right) }{s}=\frac{\partial}{\partial x_{k}}J_{\beta}\left( x+re_{k},t\right)
=J_{\beta+e_{k}}\left( x+re_{k},t\right) \quad;$$ then by (\[jaybound\]),$$\left\vert \frac{J_{\beta}\left( x+se_{k},t\right) -J_{\beta}\left(
x,t\right) }{s}\right\vert \leq C\left( \mathcal{L},\beta\right)
\frac{t^{-1/2}}{\left[ t^{1/\left( 2\ell\right) }+\rho\left(
x+re_{k}\right) \right] ^{\beta\cdot\gamma+\gamma_{k}+\left\Vert
\gamma\right\Vert }}\ \ .$$ By the triangle inequality of (\[tri\]), we have $\rho\left( x+re_{k}\right) \geq\rho\left( x\right) -\rho\left( re_{k}\right) \geq\rho\left(
x\right) /2$ if $s$ (and thus $r$) is sufficiently small, and we obtain$$\left\vert \frac{J_{\beta}\left( x+se_{k},t\right) -J_{\beta}\left(
x,t\right) }{s}\right\vert \leq C\left( \mathcal{L},\beta\right)
\frac{t^{-1/2}}{\left[ t^{1/\left( 2\ell\right) }+\rho\left( x\right)
/2\right] ^{\beta\cdot\gamma+\gamma_{k}+\left\Vert \gamma\right\Vert }}\ \ .$$ The condition $\left\Vert \gamma\right\Vert >\ell$ ensures that the right side of this inequality is an integrable function of $t$ on $\left( 0,\infty
\right) $; as it is also independent of $s$ we may let $s\rightarrow0$ in (\[dq\]) and conclude that$$\frac{\partial}{\partial x_{k}}F_{\beta}\left( x\right) =\left(
2\pi\right) ^{-n}\int_{0}^{\infty}J_{\beta+e_{k}}\left( x,t\right)
\ dt=F_{\beta+e_{k}}\left( x\right) \quad.$$ An induction arguement now confirms (b).
For $s>0$ and $x\in\mathbb{R}^{n}$, use of (\[effbeta\]) and (\[jayz\]) gives$$F_{\beta}\left( x\right) =\left( 2\pi\right) ^{-n}s^{\beta\cdot
\gamma+\left\Vert \gamma\right\Vert +\ell}\int_{0}^{\infty}J_{\beta}\left(
s^{\gamma}x,s^{2\ell}t\right) \ dt\quad.$$ In the last integral we make the change of integration parameter $r=s^{2\ell
}t$ to obtain (\[effhom\]).
To verify statement (d), we use (\[op\]), the formula $\partial^{\alpha
}F=F_{\alpha}$, (\[effbeta\]), (\[jay\]), and (\[ell\]) to write, for $x\neq0$,$$\begin{aligned}
\mathcal{L}F\left( x\right) & =\left( 2\pi\right) ^{-n}\int_{0}^{\infty
}\int_{\mathbb{R}^{n}}e^{ix\cdot z}L\left( z\right) \sigma\left( z\right)
e^{-t\sigma\left( z\right) }L\left( z\right) ^{-1}\ dz\ dt\nonumber\\
& =I\left( 2\pi\right) ^{-n}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma\left( z\right) e^{-t\sigma\left( z\right)
}\ dz\ dt\quad,\label{funda}$$ where $I$ is the $m\times m$ identity matrix. It is straightforward to verify that$$\int_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma\left( z\right) e^{-t\sigma\left(
z\right) }\ dz=-\frac{d}{dt}\int_{\mathbb{R}^{n}}e^{ix\cdot z}e^{-t\sigma
\left( z\right) }\ dz\quad,$$ and so we obtain$$\mathcal{L}F\left( x\right) =I\left( 2\pi\right) ^{-n}\left[
\int_{\mathbb{R}^{n}}e^{ix\cdot z}e^{-t\sigma\left( z\right) }\ dz\right]
_{t=\infty}^{t=0^{+}}\quad,\label{eval}$$ provided that the evaluations at $t=0^{+}$ and $t=\infty$ exist. To address this question we consider a scalar valued function$$g_{k}\left( s,t\right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{isr}e^{-tr^{2k}}\ dr\qquad\left( s\in\mathbb{R},\text{\ }t>0\right) \quad,$$ where $k$ is a positive integer. According to the discussion in chapter 9, section 2, of the book of Friedman [@FR], $g_{k}\left( s,t\right) $ is a fundamental solution of the parabolic differential equation$$\frac{\partial u\left( s,t\right) }{\partial t}=\left( -1\right)
^{k+1}\frac{\partial^{2k}u\left( s,t\right) }{\partial s^{2k}}\quad,$$ and for $s\in\mathbb{R}$ and $t>0$ satisfies an inequality$$\left\vert g_{k}\left( s,t\right) \right\vert \leq C_{1}\left( k\right)
t^{-1/\left( 2k\right) }\exp\left[ -C_{2}\left( k\right) \left(
\frac{s^{2k}}{t}\right) ^{1/\left( 2k-1\right) }\right] \quad,\label{gee}$$ where $C_{1}\left( k\right) $ and $C_{2}\left( k\right) $ are positive constants. (See Theorem 1 in chapter 9 of [@FR], or §2 of [@HAM] for a more detailed treatment.) In particular, $g_{k}\left( s,t\right) $ vanishes at $t=\infty$, and at $t=0^{+}$ provided that $s\neq0$. We now write$$\left( 2\pi\right) ^{-n}\int_{\mathbb{R}^{n}}e^{ix\cdot z}e^{-t\sigma\left(
z\right) }\ dz=\left( 2\pi\right) ^{-n}\int_{\mathbb{R}^{n}}\prod_{j=1}^{n}e^{ix_{j}z_{j}}e^{-tz_{j}^{2\ell_{j}}}\ dz=\prod_{j=1}^{n}g_{\ell_{j}}\left( x_{j},t\right) \ ,$$$$\left\vert \left( 2\pi\right) ^{-n}\int_{\mathbb{R}^{n}}e^{ix\cdot
z}e^{-t\sigma\left( z\right) }\ dz\right\vert \leq\prod_{j=1}^{n}\left\vert
g_{\ell_{j}}\left( x_{j},t\right) \right\vert \quad.$$ From the bound (\[gee\]) on $g_{k}$ we deduce that the product on the right vanishes at $t=\infty$, and at $t=0^{+}$ provided that $x_{j}\neq0$ for some $j$; thus (\[eval\]) gives $\mathcal{L}F\left( x\right) =0$ if $x\neq0$.
We define an integral operator $\mathcal{S}$, prescribed on suitable $m\times1$ complex vector functions $f$ on $\mathbb{R}^{n}$ according to$$\mathcal{S}f\left( x\right) =F\ast f\left( x\right) =\int_{\mathbb{R}^{n}}F\left( x-y\right) \ f\left( y\right) \ dy\quad.\label{ess}$$
\[phithm\]Assume $\left\Vert \gamma\right\Vert >\ell$, and let $f$ be an $m\times1$ complex vector function in the space $C_{0}^{\infty}\left(
\mathbb{R}^{n}\right) $. Then the integral (\[ess\]) converges absolutely for all $x$ in $\mathbb{R}^{n}$, and $\mathcal{S}f\in C^{\infty}\left(
\mathbb{R}^{n}\right) $ with$$\mathcal{L}\left( \mathcal{S}f\right) =f\quad.$$
From (\[ess\]), and (\[effest\]) with $\beta=0$, we find that$$\left\vert \mathcal{S}f\left( x\right) \right\vert \leq C\left(
\mathcal{L}\right) \int_{\mathbb{R}^{n}}\rho\left( x-y\right)
^{\ell-\left\Vert \gamma\right\Vert }\ \left\vert f\left( y\right)
\right\vert \ dy\quad.$$ Since $f$ has compact support the integral on the right converges at infinity, and by Lemma \[rhoint\] it converges near $y=x$ because $\ell-\left\Vert
\gamma\right\Vert >-\left\Vert \gamma\right\Vert $. Thus (\[ess\]) converges absolutely.
From (\[ess\]) and (\[effb\]),$$\mathcal{S}f\left( x\right) =\left( 2\pi\right) ^{-n}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}J\left( x-y,t\right) \ f\left( y\right) \ dt\ dy\quad
,\label{essb}$$ and then from (\[intjay\]) with $\beta=0$,$$\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\left\vert J\left( x-y,t\right)
\ f\left( y\right) \right\vert \ dt\ dy\leq C\left( \mathcal{\mathcal{L}}\right) \int_{\mathbb{R}^{n}}\rho\left( x-y\right) ^{\ell-\left\Vert
\gamma\right\Vert }\ \left\vert f\left( y\right) \right\vert \ \ dy\quad.$$ As again this integral is finite, we may interchange orders of integration in (\[essb\]) and substitute (\[ja\]) to obtain$$\mathcal{S}f\left( x\right) =\left( 2\pi\right) ^{-n}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\left( x-y\right) \cdot
z}\sigma(z)e^{-t\sigma(z)}L\left( z\right) ^{-1}f\left( y\right)
\ dz\ dy\ dt.\label{essc}$$ Looking at the inner two integrals in (\[essc\]), we use (\[rho\]) and (\[rhoineq2\]) to estimate$$\begin{aligned}
& \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\left\vert e^{i\left(
x-y\right) \cdot z}\sigma(z)e^{-t\sigma(z)}L\left( z\right) ^{-1}f\left(
y\right) \right\vert \ dz\ dy\\
& \leq C\left( \mathcal{L}\right) \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\rho(z)^{\ell}\ e^{-t\sigma(z)}\ \left\vert f\left( y\right)
\right\vert \ dz\ dy\\
& \leq C\left( \mathcal{L}\right) \left\Vert f\right\Vert _{1,\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\rho(z)^{\ell}\ e^{-t\sigma(z)}\ dz<\infty\quad.\end{aligned}$$ Thus we may interchange orders of integration in these two integrals to write$$\mathcal{S}f\left( x\right) =\left( 2\pi\right) ^{-n/2}\int_{0}^{\infty
}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma\left( z\right) e^{-t\sigma\left(
z\right) }L\left( z\right) ^{-1}\widehat{f}\left( z\right) \ dz\ dt\quad
,\label{essd}$$ where $\widehat{f}$ is the $n$-dimensional Fourier transform of $f$,$$\widehat{f}\left( z\right) =\left( 2\pi\right) ^{-n/2}\int_{\mathbb{R}^{n}}e^{-iy\cdot z}\ f\left( y\right) \ dy\quad.$$
Next we use (\[rhoineq2\]) once more to estimate$$\begin{aligned}
& \int_{0}^{\infty}\int_{\mathbb{R}^{n}}\left\vert e^{ix\cdot z}\sigma\left(
z\right) e^{-t\sigma\left( z\right) }L\left( z\right) ^{-1}\widehat
{f}\left( z\right) \right\vert \ dz\ dt\\
& \leq C\left( \mathcal{L}\right) \int_{\mathbb{R}^{n}}\rho\left(
z\right) ^{-\ell}\left\vert \widehat{f}\left( z\right) \right\vert \int
_{0}^{\infty}\sigma\left( z\right) e^{-t\sigma\left( z\right) }\ dt\ dz\\
& =C\left( \mathcal{L}\right) \int_{\mathbb{R}^{n}}\rho\left( z\right)
^{-\ell}\left\vert \widehat{f}\left( z\right) \right\vert \ dz\quad.\end{aligned}$$ As is well known, if $f\in C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $ then $\left\vert \widehat{f}\left( z\right) \right\vert $ decreases at infinity faster than any power of $\left\vert z\right\vert $. Thus the last integral converges at infinity, and by Lemma \[rhoint\] also at zero as we assume $\ell<\left\Vert \gamma\right\Vert $. Hence we may once more interchange orders of integration in (\[essd\]) to arrive at$$\begin{aligned}
\mathcal{S}f\left( x\right) & =\left( 2\pi\right) ^{-n/2}\int
_{\mathbb{R}^{n}}e^{ix\cdot z}L\left( z\right) ^{-1}\widehat{f}\left(
z\right) \ \int_{0}^{\infty}\sigma\left( z\right) e^{-t\sigma\left(
z\right) }\ dt\ dz\nonumber\\
& =\ \left( 2\pi\right) ^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot z}L\left(
z\right) ^{-1}\widehat{f}\left( z\right) dz\quad.\label{esse}$$ It is an easy manner to check that we may differentiate (\[esse\]) under the integral to obtain, for any multi-index $\alpha$,$$\partial^{\alpha}\mathcal{S}f\left( x\right) =\left( 2\pi\right)
^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\left( iz\right) ^{\alpha}L\left(
z\right) ^{-1}\widehat{f}\left( z\right) \ dz\quad.\label{essf}$$ Indeed, in view of (\[tri\]) and (\[rhoineq2\]), absolute convergence of these integrals is confirmed by$$\int_{\mathbb{R}^{n}}\left\vert e^{ix\cdot z}\left( iz\right) ^{\alpha
}L\left( z\right) ^{-1}\widehat{f}\left( z\right) \right\vert \ dz\leq
C\left( \mathcal{L}\right) \int_{\mathbb{R}^{n}}\rho\left( z\right)
^{\alpha\cdot\gamma-\ell}\left\vert \widehat{f}\left( z\right) \right\vert
\ dz<\infty\quad.$$
Finally, from (\[op\]) and (\[essf\]) it follows that$$\begin{aligned}
\mathcal{L}\left( \mathcal{S}f\right) \left( x\right) & =\left(
2\pi\right) ^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot z}L\left( z\right)
L\left( z\right) ^{-1}\widehat{f}\left( z\right) \ dz\\
& =\left( 2\pi\right) ^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\widehat
{f}\left( z\right) \ dz=f\left( x\right) \quad,\end{aligned}$$ with the last equality the Fourier inversion theorem for functions in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $.
We mention here related integral representations of Demidenko [@DEM2; @DEM3], who introduced integral operators $\left\{ P_{h}\right\} $ defined by$$\begin{aligned}
& \left( 2\pi\right) ^{n}P_{h}f\left( x\right) \\
& =\int_{h}^{h^{-1}}t^{-\left\Vert \gamma\right\Vert /\ell}\int
_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\left( x-y\right) \cdot\left(
t^{-\gamma/\ell}z\right) }2\kappa\sigma\left( z\right) ^{\kappa}e^{-\sigma\left( z\right) ^{\kappa}}L\left( z\right) ^{-1}f\left(
y\right) \ dz\ dy\ dt\ ,\end{aligned}$$ where $\kappa$ is a suitable positive integer. Using formulas of Uspenskiĭ [@USP] regarding certain averagings of functions, Demidenko showed that, as $h\rightarrow0$ and under suitable regularity conditions on $f$, the functions $\left\{ P_{h}f\right\} $ converge in a weighted Sobolev norm on $\mathbb{R}^{n}$ to a solution $u$ of $\mathcal{L}u=f$. A modification of this development leads to the formula for the fundamental solution $F$ and to the integral operator $\mathcal{S}$ (which, when written as a triple integral, closely resembles $P_{h}$ after some changes in integration parameters).
Function Spaces
===============
We introduce function spaces useful in working with semielliptic operators.
Given $0\leq r<\infty$, $1\leq p\leq\infty$, and a domain $\Omega$ in $\mathbb{R}^{n}$, we say a complex $m\times1$ vector function $u$ is in the space $W^{r,p}\left( \Omega,\mathbb{C}^{m},\underline{\ell}\right) $ provided that $u$ and its weak derivatives $\partial^{\alpha}u$, $0\leq
\alpha\cdot\gamma\leq r$, are in $L^{p}\left( \Omega\right) $; the norm of $u$ in this space is$$\left\Vert u\right\Vert _{r,p;\Omega,\underline{\ell}}=\sum_{\alpha\cdot
\gamma\leq r}\left\Vert \partial^{\alpha}u\right\Vert _{p,\Omega}\quad.\label{norm1}$$ (We assume always that $\gamma$, $\ell$, and $\underline{\ell}$ are related by (\[gamma\]) and (\[ellvec\]), with $\ell=\max_{k}\ \ell_{k}$.) We say that $u\in W_{loc}^{r,p}\left( \Omega,\mathbb{C}^{m},\underline{\ell}\right) $ whenever $u\in W^{r,p}\left( \Omega_{0},\mathbb{C}^{m},\underline{\ell
}\right) $ for all bounded open sets $\Omega_{0}$ with closure in $\Omega$.
If $u$ is defined in all of $\mathbb{R}^{n}$ and $s$ is a real number, we say $u$ is in the weighted Sobolev space $W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ provided that the weak derivatives $\partial^{\alpha}u$, $0\leq\alpha\cdot\gamma\leq r$, are in $L_{loc}^{p}\left( \mathbb{R}^{n}\right) $ and $u$ has finite norm$$\left\Vert u\right\Vert _{r,p,s;\underline{\ell}}=\sum_{\alpha\cdot\gamma\leq
r}\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha
}u\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{norm2}$$ Obviously the spaces $W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ are decreasing with respect to $s$; that is$$s_{1}\leq s_{2}\Longrightarrow W_{s_{2}}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \subset W_{s_{1}}^{r,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \quad.$$ We are concerned in this paper mainly with the cases $r=\ell$ and $r=0$. Note that in the space $W_{s}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ the norm (\[norm2\]) simplifies to$$\left\Vert u\right\Vert _{0,p,s;\underline{\ell}}=\left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}\quad.$$
For $0<R<S\leq\infty$ we define in $\mathbb{R}^{n}$ the bounded open sets$$\Omega\left( R\right) =\left\{ x:\rho\left( x\right) <R\right\}
\qquad,\qquad\Omega\left( R,S\right) =\left\{ x:R<\rho\left( x\right)
<S\right\} \quad.\label{omega}$$
Following is a density theorem for the spaces $W_{s}^{r,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $.
\[densthm\]If $0\leq r<\infty$, $s\in\mathbb{R}$, and $1\leq
p<\infty$, then $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $ is dense in the space $W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $; that is, given a complex $m\times1$ vector function $u$ in $W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $\varepsilon>0$, there exists a complex $m\times1$ vector function $\varphi$ in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $ such that$$\left\Vert u-\varphi\right\Vert _{r,p,s;\underline{\ell}}=\sum_{\alpha
\cdot\gamma\leq r}\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma
}\partial^{\alpha}\left( u-\varphi\right) \right\Vert _{p,\mathbb{R}^{n}}<\varepsilon\quad.\label{eps0}$$
Let $u$ be as described, and suppose $\varepsilon>0$. Let $R$ be a real constant, $R\geq1$. By Lemma \[hmzlemma\] there exists a real valued function $\psi$ in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $, with support in the region where $\rho\left( x\right) <2R$, such that $0\leq
\psi\leq1$, $\psi\equiv1$ where $\rho\left( x\right) \leq R$, and for any multi-index $\alpha$ and $x$ in $\mathbb{R}^{n}$,$$\left\vert \partial^{\alpha}\psi(x)\right\vert \leq C\left( \ell
,\alpha\right) R^{-\alpha\cdot\gamma}\quad.\label{psineq}$$ We set $v=\psi u$, so that $v\in W_{s}^{r,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, $v\equiv u$ where $\rho\leq R$, and $v\equiv0 $ where $\rho\geq2R$. Then for any multi-index $\alpha$ with $\alpha\cdot\gamma\leq r$,$$\begin{gathered}
\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha}(v-u)\right\Vert
_{p,\mathbb{R}^{n}}=\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha
}(v-u)\right\Vert _{p,\Omega(R,\infty)}\nonumber\\
\leq\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha}v\right\Vert
_{p,\Omega(R,2R)}+\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha
}u\right\Vert _{p,\Omega(R,\infty)}\quad.\label{dens1}$$ Use of the product rule for differentiation, along with (\[psineq\]), gives$$\begin{gathered}
\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha}v\right\Vert
_{p,\Omega(R,2R)}=\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\sum_{\beta
\leq\alpha}\binom{\alpha}{\beta}\partial^{\beta}u\,\partial^{\alpha-\beta}\psi\right\Vert _{p,\Omega(R,2R)}\\
\leq\sum_{\beta\leq\alpha}\left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\binom{\alpha}{\beta}\partial^{\beta}u\,C\left( \ell,\alpha-\beta\right)
R^{-\left( \alpha-\beta\right) \cdot\gamma}\right\Vert _{p,\Omega
(R,2R)}\quad.\end{gathered}$$ Given the requirement $\alpha\cdot\gamma\leq r$ and $\beta\leq\alpha$, there are only a finite number of possible values of $\alpha$ and $\beta$ in these manipulations, depending on $\underline{\ell}$ and $r$. Also, $R\leq1+\rho
\leq3R$ in $\Omega(R,2R)$. It follows that$$\begin{aligned}
& \left\Vert (1+\rho)^{s+\alpha\cdot\gamma}\partial^{\alpha}v\right\Vert
_{p,\Omega(R,2R)}\\
& \leq C(\underline{\ell},r)\sum_{\beta\leq\alpha}\left\Vert (1+\rho
)^{s+\alpha\cdot\gamma}\partial^{\beta}u\,\left( 1+\rho\right) ^{-\left(
\alpha-\beta\right) \cdot\gamma}\right\Vert _{p,\Omega(R,2R)}\\
& \leq C(\underline{\ell},r)\sum_{\beta\leq\alpha}\left\Vert (1+\rho
)^{s+\beta\cdot\gamma}\,\partial^{\beta}u\right\Vert _{p,\Omega(R,\infty
)}\quad.\end{aligned}$$ Then from this inequality and (\[dens1\]),$$\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha
}(v-u)\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \underline{\ell},r\right)
\sum_{\beta\leq\alpha}\left\Vert (1+\rho)^{s+\beta\cdot\gamma}\,\partial
^{\beta}u\right\Vert _{p,\Omega(R,\infty)}\ .$$ Since the norm (\[norm2\]) is assumed finite, the right side of this last inequality tends to $0$ as $R\rightarrow\infty$; thus we may choose $R$ large enough that$$\left\Vert v-u\right\Vert _{r,p,s;\underline{\ell}}=\sum_{\alpha\cdot
\gamma\leq r}\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha}\left( v-u\right) \right\Vert _{p,\mathbb{R}^{n}}<\varepsilon/2\quad.\label{eps1}$$
Now we use a standard argument involving mollifiers to verify there is a complex $m\times1$ vector function $\varphi$ in $C_{0}^{\infty}\left(
\mathbb{R}^{n}\right) $ such that $$\left\Vert \varphi-v\right\Vert _{r,p,s;\underline{\ell}}<\varepsilon
/2\quad,\label{eps2}$$ which when combined with (\[eps1\]) yields (\[eps0\]). Let $\eta$ be a nonnegative function in $C_{0}^{\infty}\left( R^{n}\right) $ vanishing outside the unit ball $\left\vert x\right\vert \leq1$, with $\int\eta\;dx=1$. For $t>0$ set $\eta_{t}(x)=t^{-n}\eta(x/t)$, and let $v_{t}$ be the convolution $v_{t}=\eta_{t}\ast v$. The support of $v$ lies in some ball of radius $S/2$ centered at $0$, and we may assume $S\geq1$. It follows that $v_{t}\in C_{0}^{\infty}\left( R^{n}\right) $ with support in the ball of radius $S$ about $0$ if $t<S/2$. For $\alpha\cdot\gamma\leq r$ we have $\partial^{\alpha}(v_{t})=\left( \partial^{\alpha}v\right) _{t}$ and $\left\Vert \left( \partial^{\alpha}v\right) _{t}-\partial^{\alpha
}v\right\Vert _{p,\mathbb{R}^{n}}\rightarrow0$ as $t\rightarrow0$. For $\left\vert x\right\vert \leq S$ with $S\geq1$, crude estimates yield$$1\leq1+\rho(x)\leq1+nS\quad.$$ Therefore, for any $\alpha$ with $\alpha\cdot\gamma\leq r$, as $t\rightarrow0
$ we have$$\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha
}\left( v_{t}-v\right) \right\Vert _{p,\mathbb{R}^{n}}\leq C\left(
s,r,S,n\right) \left\Vert \partial^{\alpha}\left( v_{t}-v\right)
\right\Vert _{p,\mathbb{R}^{n}}\longrightarrow0\;\;\;.$$ Thus, if we let $\varphi=v_{t}$ we have (\[eps2\]) if $t$ is sufficiently small.
Demidenko [@DEM1] has also introduced special weighted function spaces for use with semielliptic operators. He defined a space $W_{p,\tau}^{\underline
{\ell}}\left( \mathbb{R}^{n}\right) $, with norm$$\left\Vert u,W_{p,\tau}^{\underline{\ell}}\left( \mathbb{R}^{n}\right)
\right\Vert =\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\left\langle
x\right\rangle \right) ^{-\tau\left( 1-\alpha\cdot\gamma/\ell\right)
}\partial^{\alpha}u\right\Vert _{p,\mathbb{R}^{n}}\quad,$$ where $\left\langle x\right\rangle $ is defined by$$\left\langle x\right\rangle ^{2}=\sum_{k=1}^{n}x_{k}{}^{2\ell_{k}}=\rho\left(
x\right) ^{2\ell}\quad.$$ In terms of $\rho$ this norm is equivalent to$$\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right)
^{-\tau\left( \ell-\alpha\cdot\gamma\right) }\partial^{\alpha}u\right\Vert
_{p,\mathbb{R}^{n}}\quad.$$ When $\alpha\cdot\gamma=\ell$ the weight reduces to $1$, regardless of $\tau$; thus this norm appears fundamentally different from (\[norm2\]). However, in the case $\tau=1$, Demidenko’s norm corresponds to our norm $\left\Vert
u\right\Vert _{\ell,p,-\ell;\underline{\ell}}$, and the space $W_{p,1}^{\underline{\ell}}\left( \mathbb{R}^{n}\right) $ is equivalent to $W_{-\ell}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{n},\underline{\ell
}\right) $. Demidenko has shown [@DEM1] that $C_{0}^{\infty}\left(
\mathbb{R}^{n}\right) $ is dense in $W_{p,\tau}^{\underline{\ell}}\left(
\mathbb{R}^{n}\right) $ whenever $0\leq\tau\leq1$.
Apriori Bound
=============
The next result, taken here as a lemma, is a special case of Theorem 2 of [@HMZ].
\[apr\]Let $\Omega$ be an open subset in $\mathbb{R}^{n}$, and let $\Omega_{0}$ be a bounded open set whose closure lies in $\Omega$. If $1<p<\infty$ and $\alpha$ is a multi-index with $\alpha\cdot\gamma\leq\ell$, then for all complex $m\times1$ functions $u$ in the space $W^{\ell,p}\left(
\Omega,\mathbb{C}^{m},\underline{\ell}\right) $,$$\left\Vert \partial^{\alpha}u\right\Vert _{p,\Omega_{0}}\leq C\left(
\mathcal{L},p,\Omega,\Omega_{0}\right) \left[ \left\Vert u\right\Vert
_{p,\Omega}+\left\Vert \mathcal{L}u\right\Vert _{p,\Omega}\right] \quad.$$
Following is our fundamental apriori bound regarding the operator $\mathcal{L}$ of (\[op\]).
\[apbnd\]Let $u$ be a complex $m\times1$ function in the space $W_{loc}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $. If $1<p<\infty$ and $s\in\mathbb{R}$, then$$\begin{aligned}
\left\Vert u\right\Vert _{\ell,p,s;\underline{\ell}} & =\sum_{\alpha
\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma
}\partial^{\alpha}u\right\Vert _{p,\mathbb{R}^{n}}\nonumber\\
& \leq C\left( \mathcal{L},s,p\right) \left[ \left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}+\left\Vert \left(
1+\rho\right) ^{s+\ell}\mathcal{L}u\right\Vert _{p,\mathbb{R}^{n}}\right]
\quad.\label{apriori}$$
Let $u$, $p$, and $s$ be as described. We assume the right side of (\[apriori\]) is finite, as otherwise the inequality is trivial. Let $\alpha$ be a multi-index such that $\alpha\cdot\gamma\leq\ell$. We use the notation (\[omega\]).
First, in the region $\Omega(4)$, where $\rho(x)<4$, Lemma \[apr\] implies $$\left\Vert \partial^{\alpha}u\right\Vert _{p,\Omega(2)}\leq C\left(
\mathcal{L},p\right) \left[ \left\Vert u\right\Vert _{p,\Omega
(4)}+\left\Vert \mathcal{L}u\right\Vert _{p,\Omega(4)}\right] \quad.$$ As $1\leq1+\rho(x)\leq5$ in $\Omega(4)$, this inequality implies$$\begin{aligned}
& \int_{\Omega(2)}\left( 1+\rho\right) ^{\left( s+\alpha\cdot
\gamma\right) p}\left\vert \partial^{\alpha}u\right\vert ^{p}\ dx\label{add1}\\
& \leq C\left( \mathcal{L},s,p\right) \left[ \int_{\Omega(4)}\left(
1+\rho\right) ^{sp}\left\vert u\right\vert ^{p}\ dx+\int_{\Omega(4)}\left(
1+\rho\right) ^{\left( s+\ell\right) p}\left\vert \mathcal{L}u\right\vert
^{p}\ dx\right] \quad.\nonumber\end{aligned}$$
Next let $t$ be a real constant, $t\geq1$, and define a function $v$ by$$v(x)=u\left( t^{\gamma}x\right) \quad.$$ Calculations show that$$\partial^{\alpha}v(x)=t^{\alpha\cdot\gamma}\left( \partial^{\alpha}u\right)
\left( t^{\gamma}x\right) \qquad,\qquad\mathcal{L}v(x)=t^{\ell}\left(
\mathcal{L}u\right) \left( t^{\gamma}x\right) \quad.$$ Again by Lemma \[apr\], $$\int_{\Omega(2,4)}\left\vert \partial^{\alpha}v(x)\right\vert ^{p}\ dx\leq
C\left( \mathcal{L},p\right) \left[ \int_{\Omega(1,8)}\left\vert
v(x)\right\vert ^{p}\ dx+\int_{\Omega(1,8)}\left\vert \mathcal{L}v(x)\right\vert ^{p}\ dx\right] \quad,$$ or in terms of $u$,$$\begin{aligned}
& \int_{\Omega(2,4)}\left\vert t^{\alpha\cdot\gamma}\left( \partial^{\alpha
}u\right) \left( t^{\gamma}x\right) \right\vert ^{p}\ dx\\
& \leq C\left( \mathcal{L},p\right) \left[ \int_{\Omega(1,8)}\left\vert
u\left( t^{\gamma}x\right) \right\vert ^{p}\ dx+\int_{\Omega(1,8)}\left\vert
t^{\ell}\left( \mathcal{L}u\right) \left( t^{\gamma}x\right) \right\vert
^{p}\ dx\right] \quad.\end{aligned}$$ In these last integrals we make the change of integration parameter $y=t^{\gamma}x$, with $\rho(y)=t\rho(x)$, $dy=t^{\left\Vert \gamma\right\Vert
}\ dx$, and obtain$$\begin{aligned}
& t^{(\alpha\cdot\gamma)p}\int_{\Omega(2t,4t)}\left\vert \left(
\partial^{\alpha}u\right) \left( y\right) \right\vert ^{p}\ dy\\
& \leq C\left( \mathcal{L},p\right) \left[ \int_{\Omega(t,8t)}\left\vert
u\left( y\right) \right\vert ^{p}\ dy+t^{\ell p}\int_{\Omega(t,8t)}\left\vert \mathcal{L}u(y)\right\vert ^{p}\ dy\right] \quad.\end{aligned}$$ But in $\Omega(t,8t)$ with $t\geq1$, we have $t\leq1+\rho(y)\leq9t$, and so we may multiply this inequality by $t^{sp}$ to obtain$$\begin{aligned}
& \int_{\Omega(2t,4t)}\left( 1+\rho\right) ^{\left( s+\alpha\cdot
\gamma\right) p}\left\vert \partial^{\alpha}u\right\vert ^{p}\ dy\\
& \leq C\left( \mathcal{L},s,p\right) \left[ \int_{\Omega(t,8t)}\left(
1+\rho\right) ^{sp}\left\vert u\right\vert ^{p}\ dy+\int_{\Omega
(t,8t)}\left( 1+\rho\right) ^{\left( s+\ell\right) p}\left\vert
\mathcal{L}u\right\vert ^{p}\ dy\right] \quad.\end{aligned}$$ Now in this inequality we take $t=2^{m}$ for $m=0,1,2,\ldots$, and add all the resulting inequalities to (\[add1\]) to arrive at$$\begin{aligned}
& \int_{\mathbb{R}^{n}}\left( 1+\rho\right) ^{\left( s+\alpha\cdot
\gamma\right) p}\left\vert \partial^{\alpha}u\right\vert ^{p}\ dy\\
& \leq C\left( \mathcal{L},s,p\right) \left[ \int_{\mathbb{R}^{n}}\left(
1+\rho\right) ^{sp}\left\vert u\right\vert ^{p}\ dy+\int_{\mathbb{R}^{n}}\left( 1+\rho\right) ^{\left( s+\ell\right) p}\left\vert \mathcal{L}u\right\vert ^{p}\ dy\right] \quad,\end{aligned}$$ which leads to$$\left\Vert \left( 1+\rho\right) ^{s+\alpha\cdot\gamma}\partial^{\alpha
}u\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},s,p\right) \left[
\left\Vert \left( 1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}+\left\Vert \left( 1+\rho\right) ^{s+\ell}\mathcal{L}u\right\Vert
_{p,\mathbb{R}^{n}}\right] \ .$$ Finally, we sum over all $\alpha$ such that $\alpha\cdot\gamma\leq\ell$ to obtain (\[apriori\]).
The Operator $\mathcal{S}$
==========================
We investigate properties of the operator $\mathcal{S}$ as a mapping between certain function spaces. We require another technical lemma.
\[klemma\]For $x$ in $\mathbb{R}^{n}$, and for real numbers $\xi$ and $\eta$, let$$K\left( x,\xi,\eta\right) =\int_{\mathbb{R}^{n}}\rho\left( x-y\right)
^{\xi}\left[ 1+\rho\left( y\right) \right] ^{\eta}\ dy\quad.$$ If$$\xi+\left\Vert \gamma\right\Vert >0\qquad,\qquad\eta+\left\Vert \gamma
\right\Vert >0\qquad,\qquad\xi+\eta+\left\Vert \gamma\right\Vert
<0\quad,\label{conds}$$ then$$K\left( x,\xi,\eta\right) \leq C\left( \xi,\eta,\underline{\ell}\right)
\left[ 1+\rho\left( x\right) \right] ^{\xi+\eta+\left\Vert \gamma
\right\Vert }\quad.\label{kay}$$
Note that conditions (\[conds\]) imply that $\xi,\eta<0$.
Fixing $x$ in $\mathbb{R}^{n}$, we partition $\mathbb{R}^{n}$ into three disjoint regions,$$\begin{aligned}
R_{1} & =\left\{ y:\rho\left( x-y\right) \leq\frac{1+\rho(x)}{2}\right\}
\quad,\\
R_{2} & =\left\{ y:\frac{1+\rho(x)}{2}<\rho\left( x-y\right) <2\left[
1+\rho(x)\right] \right\} \quad,\\
R_{3} & =\left\{ y:2\left[ 1+\rho(x)\right] \leq\rho\left( x-y\right)
\right\} \quad,\end{aligned}$$ and write$$K\left( x,\xi,\eta\right) =K_{1}\left( x,\xi,\eta\right) +K_{2}\left(
x,\xi,\eta\right) +K_{3}\left( x,\xi,\eta\right) \quad,$$ where$$K_{i}\left( x,\xi,\eta\right) =\int_{R_{i}}\rho\left( x-y\right) ^{\xi}
\left[ 1+\rho\left( y\right) \right] ^{\eta}\ dy\qquad,\qquad
i=1,2,3\quad.$$
As $\rho$ satisfies the triangle inequality, in the region $R_{1}$ we have$$\begin{gathered}
1+\rho(x)\leq1+\rho(y)+\rho(x-y)\leq1+\rho(y)+\frac{1+\rho(x)}{2}\quad,\\
1+\rho(x)\leq2\left[ 1+\rho(y)\right] \quad.\end{gathered}$$ We use the fact that $\eta<0$, along with Lemma \[rhoint\](b) and $\xi>-\left\Vert \gamma\right\Vert $, to derive$$\begin{aligned}
K_{1}(x,\xi,\eta) & \leq\left[ \frac{1+\rho\left( x\right) }{2}\right]
^{\eta}\int_{\rho\left( x-y\right) \leq\left[ 1+\rho(x)\right] /2}\rho\left( x-y\right) ^{\xi}\ dy\\
& \leq2^{-\eta}\left[ 1+\rho\left( x\right) \right] ^{\eta}\int
_{\rho\left( z\right) \leq1+\rho(x)}\rho\left( z\right) ^{\xi}\ dz\\
& =2^{-\eta}\left[ 1+\rho\left( x\right) \right] ^{\eta}\left[
1+\rho\left( x\right) \right] ^{\xi+\left\Vert \gamma\right\Vert }\int_{\rho\left( z\right) \leq1}\rho\left( z\right) ^{\xi}\ dz\\
& =C\left( \xi,\eta,\underline{\ell}\right) \left[ 1+\rho(x)\right]
^{\xi+\eta+\left\Vert \gamma\right\Vert }\quad.\end{aligned}$$
In the region $R_{2}$,$$\rho(y)\leq\rho(x-y)+\rho(x)\leq2\left[ 1+\rho(x)\right] +\rho
(x)\leq3\left[ 1+\rho(x)\right] \quad.$$ We use $\xi<0$ and $-\left\Vert \gamma\right\Vert <\eta<0$, along with Lemma \[rhoint\](b), to derive$$\begin{aligned}
K_{2}\left( x,\xi,\eta\right) & \leq\int_{\left[ 1+\rho(x)\right]
/2<\rho\left( x-y\right) <2\left[ 1+\rho(x)\right] }\left[ \frac
{1+\rho(x)}{2}\right] ^{\xi}\left[ 1+\rho(y)\right] ^{\eta}\ dy\\
& \leq2^{-\xi}\left[ 1+\rho(x)\right] ^{\xi}\int_{\rho\left( y\right)
\leq3\left[ 1+\rho(x)\right] }\rho\left( y\right) ^{\eta}\ dy\\
& =C\left( \xi,\eta,\underline{\ell}\right) \left[ 1+\rho(x)\right]
^{\xi+\eta+\left\Vert \gamma\right\Vert }\quad.\end{aligned}$$
In the region $R_{3}$, $$\begin{aligned}
\rho(x-y) & \leq\rho(x)+\rho(y)\leq\frac{\rho(x-y)}{2}+\rho(y)\quad,\\
\rho(x-y) & \leq2\rho(y)\leq2\left[ 1+\rho(y)\right] \quad.\end{aligned}$$ We use the fact that $\eta<0$, along with Lemma \[rhoint\](a) and $\xi
+\eta<-\left\Vert \gamma\right\Vert $, to derive$$\begin{aligned}
K_{3}\left( x,\xi,\eta\right) & \leq\int_{2\left[ 1+\rho(x)\right]
\leq\rho\left( x-y\right) }\rho\left( x-y\right) ^{\xi}\left[ \frac
{\rho\left( x-y\right) }{2}\right] ^{\eta}\ dy\\
& \leq2^{-\eta}\int_{1+\rho(x)\leq\rho(z)}\rho(z)^{\xi+\eta}\ dz\leq C\left(
\xi,\eta,\underline{\ell}\right) \left[ 1+\rho(x)\right] ^{\xi
+\eta+\left\Vert \gamma\right\Vert }\quad.\end{aligned}$$
Combining finally our estimates for $K_{1}$, $K_{2}$, and $K_{3}$ gives (\[kay\]).
\[esslemma\]Suppose $\left\Vert \gamma\right\Vert >\ell$, $1\leq
p\leq\infty$, let $s$ be a real number in the range$$\ell-\left\Vert \gamma\right\Vert /p<s<\left\Vert \gamma\right\Vert
-\left\Vert \gamma\right\Vert /p\quad,\label{hyp}$$ and let $f$ be a complex $m\times1$ vector function such that $\left\Vert
\left( 1+\rho\right) ^{s}f\right\Vert _{p,\mathbb{R}^{n}}<\infty$. Then the integral$$\mathcal{S}f\left( x\right) =F\ast f\left( x\right) =\int_{\mathbb{R}^{n}}F\left( x-y\right) \ f\left( y\right) \ dy\quad.\label{essit}$$ converges absolutely for almost all $x$ in $\mathbb{R}^{n}$, and$$\left\Vert \left( 1+\rho\right) ^{s-\ell}\mathcal{S}f\right\Vert
_{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p,s\right) \left\Vert \left(
1+\rho\right) ^{s}f\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{essbnd}$$ If moreover $p>\left\Vert \gamma\right\Vert /\ell$, then in fact (\[essit\]) converges absolutely for all $x$ in $\mathbb{R}^{n}$, and$$\left\vert \mathcal{S}f\left( x\right) \right\vert \leq C\left(
\mathcal{L},p,s\right) \left[ 1+\rho\left( x\right) \right]
^{\ell-s-\left\Vert \gamma\right\Vert /p}\left\Vert \left( 1+\rho\right)
^{s}f\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{essbndb}$$
From (\[essit\]), and (\[effest\]) with $\beta=0$, $$\begin{aligned}
\left\vert \mathcal{S}f\left( x\right) \right\vert & \leq\int
_{\mathbb{R}^{n}}\left\vert F\left( x-y\right) \right\vert \ \left\vert
f(y)\right\vert \ dy\nonumber\\
& \leq C\left( \mathcal{L}\right) \int_{\mathbb{R}^{n}}\rho\left(
x-y\right) ^{\ell-\left\Vert \gamma\right\Vert }\left\vert f\left( y\right)
\right\vert \ dy,\label{essbd}$$
First consider the case $1<p<\infty$. Let $q$ be defined by the usual relation $1/p+1/q=1$. In general, two finite and nonempty open intervals $\left(
a,b\right) $ and $\left( c,d\right) $ intersect if and only if $a<d$ and $c<b$. Condition (\[hyp\]) implies$$\frac{\ell}{q}<s+\frac{\left\Vert \gamma\right\Vert -\ell}{p}\qquad,\qquad
s<\frac{\left\Vert \gamma\right\Vert }{q}\quad;$$ thus there is a real number $r$ such that$$\frac{\ell}{q}<r<\frac{\left\Vert \gamma\right\Vert }{q}\qquad,\qquad
s<r<s+\frac{\left\Vert \gamma\right\Vert -\ell}{p}\quad.\label{intcond}$$ By (\[essbd\]) and Hölder’s inequality,$$\begin{aligned}
\left\vert \mathcal{S}f\left( x\right) \right\vert & \leq C\left(
\mathcal{L}\right) \left( \int_{\mathbb{R}^{n}}\rho\left( x-y\right)
^{\ell-\left\Vert \gamma\right\Vert }\left[ 1+\rho(y)\right] ^{-rq}\ dy\right) ^{1/q}\\
& \cdot\left( \int_{\mathbb{R}^{n}}\rho\left( x-y\right) ^{\ell-\left\Vert
\gamma\right\Vert }\left[ 1+\rho(y)\right] ^{rp}\left\vert f(y)\right\vert
^{p}\ dy\right) ^{1/p}\quad.\end{aligned}$$ As (\[intcond\]) and $\ell\geq1$ imply that $\xi=\ell-\left\Vert
\gamma\right\Vert $ and $\eta=-rq$ satisfy the hypotheses of Lemma \[klemma\], we may apply that result and conclude that$$\begin{aligned}
\left\vert \mathcal{S}f\left( x\right) \right\vert ^{p} & \leq C\left(
\mathcal{L},p,s\right) \left[ 1+\rho(x)\right] ^{-rp+\ell p/q}\\
& \cdot\int_{\mathbb{R}^{n}}\rho\left( x-y\right) ^{\ell-\left\Vert
\gamma\right\Vert }\left[ 1+\rho(y)\right] ^{rp}\left\vert f(y)\right\vert
^{p}\ dy\quad.\end{aligned}$$ Therefore,$$\begin{aligned}
\left[ \left\Vert \left( 1+\rho\right) ^{s-\ell}\mathcal{S}f\right\Vert
_{p,\mathbb{R}^{n}}\right] ^{p} & =\int_{\mathbb{R}^{n}}\left[
1+\rho\left( x\right) \right] ^{\left( s-\ell\right) p}\left\vert
\mathcal{S}f\left( x\right) \right\vert ^{p}\ dx\\
& \leq C\left( \mathcal{L},p,s\right) \int_{\mathbb{R}^{n}}\left[
1+\rho(y)\right] ^{rp}\left\vert f(y)\right\vert ^{p}\\
& \cdot\int_{\mathbb{R}^{n}}\rho\left( x-y\right) ^{\ell-\left\Vert
\gamma\right\Vert }\left[ 1+\rho(x)\right] ^{sp-rp-\ell}\ dx\ dy\ .\end{aligned}$$ Again, (\[intcond\]) implies that $\xi=\ell-\left\Vert \gamma\right\Vert $ and $\eta=sp-rp-\ell$ satisfy the hypotheses of Lemma \[klemma\]; applying that lemma (with the roles of $x$ and $y$ reversed), we obtain$$\left[ \left\Vert \left( 1+\rho\right) ^{s-\ell}\mathcal{S}f\right\Vert
_{p,\mathbb{R}^{n}}\right] ^{p}\leq C\left( \mathcal{L},p,s\right)
\int_{\mathbb{R}^{n}}\left[ 1+\rho(y)\right] ^{sp}\left\vert f(y)\right\vert
^{p}\ dy\quad,$$ and thereby (\[essbnd\]). Note that we have verified that the integral on the right of (\[essbd\]) defines a function of $x$ in the space $L_{loc}^{p}\left( \mathbb{R}^{n}\right) $. In particular, for almost all $x$ this integral must be finite, and consequently the integral defining $\mathcal{S}f\left( x\right) $ absolutely convergent.
The case $p=1$, when (\[hyp\]) reduces to $\ell-\left\Vert \gamma\right\Vert
<s<0$, is simpler. We multiply (\[essbd\]) by $\left[ 1+\rho\left(
x\right) \right] ^{s-\ell}$, integrate over $\mathbb{R}^{n}$ with respect to $x$, and apply Lemma \[klemma\] as above to obtain (\[essbnd\]) with $p=1$.
For the case $p=\infty$, when (\[hyp\]) reduces to $\ell<s<\left\Vert
\gamma\right\Vert $, we apply first (\[essbd\]) and then Lemma \[klemma\] to derive$$\begin{aligned}
\left\vert \mathcal{S}f\left( x\right) \right\vert & \leq C\left(
\mathcal{L}\right) \left\Vert \left( 1+\rho\right) ^{s}f\right\Vert
_{\infty}\int_{\mathbb{R}^{n}}\rho\left( x-y\right) ^{\ell-\left\Vert
\gamma\right\Vert }\left\vert 1+\rho\left( y\right) \right\vert ^{-s}\ dy\\
& \leq C\left( \mathcal{L},s\right) \left\Vert \left( 1+\rho\right)
^{s}f\right\Vert _{\infty}\left[ 1+\rho\left( x\right) \right] ^{\ell
-s}\quad.\end{aligned}$$ This inequality implies (\[essbnd\]), as well as (\[essbndb\]), for the case $p=\infty$.
Next assume $\left\Vert \gamma\right\Vert /\ell<p<\infty$. Application of Hölder’s inequality to (\[essbd\]) gives$$\left\vert \mathcal{S}f\left( x\right) \right\vert \leq C\left(
\mathcal{L}\right) \left( \int_{\mathbb{R}^{n}}\rho\left( x-y\right)
^{\left( \ell-\left\Vert \gamma\right\Vert \right) q}\left[ 1+\rho
(y)\right] ^{-sq}\ dy\right) ^{1/q}\left\Vert \left( 1+\rho\right)
^{s}f\right\Vert _{p,\mathbb{R}^{n}}\ .$$ Conditions (\[hyp\]), and $p>\left\Vert \gamma\right\Vert /\ell$, ensure that Lemma \[klemma\] applies with $\xi=\left( \ell-\left\Vert
\gamma\right\Vert \right) q$ and $\eta=-sq$, resulting in (\[essbndb\]).
\[essthm\]Suppose $\left\Vert \gamma\right\Vert >\ell$, $1<p<\infty$, and let $s$ be a real number in the range$$-\left\Vert \gamma\right\Vert /p<s<\left\Vert \gamma\right\Vert -\ell
-\left\Vert \gamma\right\Vert /p\quad.\label{hypb}$$ Let $f$ be a complex $m\times1$ vector function in the space $W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $; i.e, such that $\left\Vert \left( 1+\rho\right) ^{s+\ell}f\right\Vert
_{p,\mathbb{R}^{n}}<\infty$. Then the integral$$\mathcal{S}f\left( x\right) =\int_{\mathbb{R}^{n}}F\left( x-y\right)
\ f\left( y\right) \ dy\quad.$$ converges absolutely for almost all $x$ in $\mathbb{R}^{n}$, and $\mathcal{S}f\in W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ with $\mathcal{L}\left( \mathcal{S}f\right)
=f$ and$$\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right)
^{s+\alpha\cdot\gamma}\partial^{\alpha}\left( \mathcal{S}f\right)
\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p,s\right)
\left\Vert \left( 1+\rho\right) ^{s+\ell}f\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{essbndc}$$
We replace $s$ by $s+\ell$ in Lemma \[esslemma\], and deduce that the integral $\mathcal{S}f\left( x\right) $ converges absolutely for almost all $x$ in $\mathbb{R}^{n}$, with$$\left\Vert \left( 1+\rho\right) ^{s}\mathcal{S}f\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p,s\right) \left\Vert \left( 1+\rho\right)
^{s+\ell}f\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{essbnd0}$$
By Theorem \[densthm\] there exists a sequence $\left\{ \varphi
_{k}\right\} $ of complex $m\times1$ vector functions in $C_{0}^{\infty
}\left( \mathbb{R}^{n}\right) $ converging to $f$ in $W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, so that $$\left\Vert \left( 1+\rho\right) ^{s+\ell}\left( \varphi_{k}-f\right)
\right\Vert _{p,\mathbb{R}^{n}}\rightarrow0\quad.\label{phieff}$$ Theorem \[phithm\] implies $\mathcal{S}\varphi_{k}\in C^{\infty}\left(
\mathbb{R}^{n}\right) $ for each $k$, with $$\mathcal{L}\left( \mathcal{S}\varphi_{k}\right) =\varphi_{k}\quad
.\label{ellphi}$$ As (\[essbnd0\]) must apply also to each $\varphi_{k}$ and to $\varphi
_{k}-f$, we have$$\left\Vert \left( 1+\rho\right) ^{s}\mathcal{S}\varphi_{k}\right\Vert
_{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p,s\right) \left\Vert \left(
1+\rho\right) ^{s+\ell}\varphi_{k}\right\Vert _{p,\mathbb{R}^{n}}\quad,\label{phikay}$$$$\left\Vert \left( 1+\rho\right) ^{s}\mathcal{S}\left( \varphi_{k}-f\right)
\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p,s\right)
\left\Vert \left( 1+\rho\right) ^{s+\ell}\left( \varphi_{k}-f\right)
\right\Vert _{p,\mathbb{R}^{n}}\quad.\label{phikaf}$$ We apply Theorem \[apbnd\] to each function $\mathcal{S}\varphi_{k}$ and obtain$$\begin{aligned}
\left\Vert \mathcal{S}\varphi_{k}\right\Vert _{\ell,p,s;\underline{\ell}} &
=\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right)
^{s+\alpha\cdot\gamma}\partial^{\alpha}\left( \mathcal{S}\varphi_{k}\right)
\right\Vert _{p,\mathbb{R}^{n}}\\
& \leq C\left( \mathcal{L},s,p\right) \left[ \left\Vert \left(
1+\rho\right) ^{s}\left( \mathcal{S}\varphi_{k}\right) \right\Vert
_{p,\mathbb{R}^{n}}+\left\Vert \left( 1+\rho\right) ^{s+\ell}\varphi
_{k}\right\Vert _{p,\mathbb{R}^{n}}\right] \quad,\end{aligned}$$ which when combined with (\[phikay\]) yields$$\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right)
^{s+\alpha\cdot\gamma}\partial^{\alpha}\left( \mathcal{S}\varphi_{k}\right)
\right\Vert _{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},s,p\right)
\left\Vert \left( 1+\rho\right) ^{s+\ell}\varphi_{k}\right\Vert
_{p,\mathbb{R}^{n}}\quad.\label{phikab}$$ But (\[phikab\]) must apply also to each difference $\varphi_{k}-\varphi
_{j}$, and in view of (\[phieff\]) we conclude that the sequence $\left\{
\mathcal{S}\varphi_{k}\right\} $ is Cauchy in the space $W_{s}^{\ell
,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. Hence $\left\{ \mathcal{S}\varphi_{k}\right\} $ converges in that space to some function, which must be $\mathcal{S}f$ because of (\[phikaf\]) and (\[phieff\]). In particular, $\mathcal{S}f\in W_{s}^{\ell,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. Letting $k\rightarrow\infty$ in (\[phikab\]) and (\[ellphi\]) gives (\[essbndc\]) as well as $\mathcal{L}\left( \mathcal{S}f\right) =f$.
Mapping Properties
==================
We combine our results thus far to draw conclusions about mapping properties of the partial differential operator $$\mathcal{L}=\sum_{\alpha\cdot\gamma=\ell}A_{\alpha}\partial^{\alpha}\quad,\label{op1}$$ as described in the introduction.
For complex $m\times1$ vector functions $u$ and $v$ on $\mathbb{R}^{n}$, we define the inner product (when it exists)$$\left\langle u,v\right\rangle =\int_{\mathbb{R}^{n}}u\cdot v\ dx=\int
_{\mathbb{R}^{n}}v^{\ast}u\ dx\quad,\label{ip}$$ where $^{\ast}$ denotes the conjugate transpose operation.
Recall that functions $u\in W_{s}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $v\in W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ have the respective norms$$\left\Vert u\right\Vert _{0,p,s;\underline{\ell}}=\left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}\qquad,\qquad\left\Vert
v\right\Vert _{0,q,-s;\underline{\ell}}=\left\Vert \left( 1+\rho\right)
^{-s}v\right\Vert _{q,\mathbb{R}^{n}}\quad.$$ If $1/p+1/q=1$, then (\[ip\]) is defined for such $u$ and $v$, and according to Hölder’s inequality,$$\left\vert \left\langle u,v\right\rangle \right\vert \leq\left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}\left\Vert \left(
1+\rho\right) ^{-s}v\right\Vert _{q,\mathbb{R}^{n}}=\left\Vert u\right\Vert
_{0,p,s;\underline{\ell}}\left\Vert v\right\Vert _{0,q,-s;\underline{\ell}}\quad.\label{hold}$$ Indeed, a standard argument confirms that, if $1<p,q<\infty$ and $1/p+1/q=1$, then the spaces $W_{s}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ are duals of one another.
If $u\in W_{loc}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline
{\ell}\right) $ and $\varphi$ is a complex $m\times1$ vector function in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $, we have by the usual integration by parts that$$\left\langle \mathcal{L}u,\varphi\right\rangle =\int_{\mathbb{R}^{n}}\mathcal{L}u\cdot\varphi\ dx=\int_{\mathbb{R}^{n}}u\cdot\mathcal{L}^{\ast
}\varphi\ dx=\left\langle u,\mathcal{L}^{\ast}\varphi\right\rangle
\quad,\label{parts1}$$ where $\mathcal{L}^{\ast}$ is the *adjoint operator* to $\mathcal{L}$,$$\mathcal{L}^{\ast}=\sum_{\alpha\cdot\gamma=\ell}\left( -1\right)
^{\left\vert \alpha\right\vert }A_{\alpha}{}^{\ast}\,\partial^{\alpha}\quad.$$ Also, as $\left( \mathcal{L}^{\ast}\right) ^{\ast}=\mathcal{L}$, we have$$\left\langle \mathcal{L}^{\ast}u,\varphi\right\rangle =\left\langle
u,\mathcal{L}\varphi\right\rangle \quad.\label{parts2}$$ If we let $L^{\ast}\left( x\right) $ denote the symbol (\[ell\]) for $\mathcal{L}^{\ast}$, then by a brief calculation,$$L^{\ast}\left( x\right) =L\left( x\right) ^{\ast}\quad.$$ Consequently, $L^{\ast}\left( x\right) $ is invertible whenever $L\left(
x\right) $ is invertible, and semiellipticity of $\mathcal{L}$ implies the same for $\mathcal{L}^{\ast}$. Moreover, all results proved thus far for $\mathcal{L}$ are equally valid for $\mathcal{L}^{\ast}$. We let $F^{\ast}$ denote the fundamental solution for the adjoint operator $\mathcal{L}^{\ast}
$, $$F^{\ast}\left( x\right) =\left( 2\pi\right) ^{-n}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{ix\cdot z}\sigma\left( z\right) e^{-t\sigma\left(
z\right) }L^{\ast}\left( z\right) ^{-1}\ dz\ dt\qquad\left( x\neq0\right)
\quad,$$ and $\mathcal{S}^{\ast}$ the corresponding convolution operator,$$\mathcal{S}^{\ast}f\left( x\right) =F^{\ast}\ast f\left( x\right)
=\int_{\mathbb{R}^{n}}F^{\ast}\left( x-y\right) f\left( y\right)
\ dy\quad.$$ Obviously, Theorems \[effthm\], \[phithm\], and \[essthm\] apply as well to $\mathcal{L}^{\ast}$, $F^{\ast}$, and $\mathcal{S}^{\ast}$.
In accordance with (\[parts1\]) and (\[parts2\]), for complex $m\times1$ vector functions $u$ and $f$ in $L_{loc}^{1}\left( \mathbb{R}^{n}\right) $ we say that $u$ is a *distributional solution* in $\mathbb{R}^{n}$ of the equation (a)$\ \mathcal{L}u=f$, or (b)$\ \mathcal{L}^{\ast}u=f$, provided that, respectively,$$\text{(a)\ \ }\left\langle u,\mathcal{L}^{\ast}\varphi\right\rangle
=\left\langle f,\varphi\right\rangle \text{\qquad,\qquad(b)\ \ }\left\langle
u,\mathcal{L}\varphi\right\rangle =\left\langle f,\varphi\right\rangle \quad,$$ for all complex $m\times1$ vector functions $\varphi$ in $C_{0}^{\infty
}\left( \mathbb{R}^{n}\right) $.
Elementary estimates confirm that, for suitable positive constants $K_{1}\left( \underline{\ell}\right) $ and $K_{2}\left( \underline{\ell
}\right) $ and for $x\in\mathbb{R}^{n}$,$$K_{1}\left( \underline{\ell}\right) \left( 1+\left\vert x\right\vert
\right) ^{1/\ell}\leq1+\rho\left( x\right) \leq K_{2}\left( \underline
{\ell}\right) \left( 1+\left\vert x\right\vert \right) \quad.\label{rhoabs}$$ For $z\in\mathbb{C}^{n}$ and $x\in\mathbb{R}^{n}$ we define$$\left\Vert z\right\Vert _{x}:=\left[ 1+\rho\left( x\right) \right]
^{-\ell}\left\vert z\right\vert \quad,\label{temp}$$ and conclude that, for another positive constant $K_{3}\left( \underline
{\ell}\right) $,$$K_{3}\left( \underline{\ell}\right) \left( 1+\left\vert x\right\vert
\right) ^{-\ell}\left\vert z\right\vert \leq\left\Vert z\right\Vert _{x}\leq\left( 1+\left\vert x\right\vert \right) ^{\ell}\left\vert z\right\vert
\quad.$$ This inequality demonstrates, according to the criterion of Hörmander ([@HO], § 22.1), that (\[temp\]) defines a *temperate norm* on $\mathbb{C}^{n}$, parametrized by $x\in\mathbb{R}^{n}$. For nonzero $x\in\mathbb{R}^{n}$ and for $z\in\mathbb{C}^{n}$, use of (\[rhoineq2\]) leads to$$\left\vert z\right\vert =\left\vert L\left( x\right) ^{-1}L\left( x\right)
z\right\vert \leq\left\vert L\left( x\right) ^{-1}\right\vert \left\vert
L\left( x\right) z\right\vert \leq c_{4}\left( \mathcal{L}\right)
\rho\left( x\right) ^{-\ell}\left\vert L\left( x\right) z\right\vert
\quad.$$ But if $\rho\left( x\right) \geq1$ then $\rho\left( x\right) ^{-\ell}\leq2^{\ell}\left( 1+\rho\left( x\right) \right) ^{-\ell}$, and we obtain$$\left\vert z\right\vert \leq C\left( \mathcal{L}\right) \left\Vert L\left(
x\right) z\right\Vert _{x}\qquad\text{,\qquad if }\rho\left( x\right)
\geq1\quad.\label{crit1}$$ From (\[temp\]) and (\[rhoineq3\]) we deduce that$$\begin{aligned}
\left\Vert \partial^{\alpha}L\left( x\right) z\right\Vert _{x} &
\leq\left[ 1+\rho\left( x\right) \right] ^{-\ell}\left\vert z\right\vert
\cdot\left\{
\begin{tabular}
[c]{cc}$c_{5}\left( \mathcal{L}\right) \rho\left( x\right) ^{\ell-\alpha
\cdot\gamma}$ & ,\ if\ $\alpha\cdot\gamma\leq\ell$,\\
$0$ & ,\ otherwise.
\end{tabular}
\ \ \ \ \right. \\
& \leq c_{5}\left( \mathcal{L}\right) \left[ 1+\rho\left( x\right)
\right] ^{-\alpha\cdot\gamma}\left\vert z\right\vert \quad.\end{aligned}$$ Then with use of (\[rhoabs\]) and the inequality $$\alpha\cdot\gamma=\sum_{k=1}^{n}\alpha_{k}\frac{\ell}{\ell_{k}}\geq\sum
_{k=1}^{n}\alpha_{k}=\left\vert \alpha\right\vert \quad,$$ we find that there is a constant $C\left( \mathcal{L},\alpha\right) $ such that $$\left\Vert \partial^{\alpha}L\left( x\right) z\right\Vert _{x}\leq C\left(
\mathcal{L},\alpha\right) \left( 1+\left\vert x\right\vert \right)
^{-\left\vert \alpha\right\vert /\ell}\left\vert z\right\vert \quad
,\label{crit2}$$ where obviously $0<1/\ell\leq1$. Inequalities (\[crit1\]) and (\[crit2\]) demonstrate that $\mathcal{L}$ is a matrix *hypoelliptic operator*, as defined by Hörmander ([@HO], § 22.1). As a consequence (see [@HO]), if $f$ is of class $C^{\infty}$ in an open set in $\mathbb{R}^{n}
$, then any distributional solution of $\mathcal{L}u=f$ in that open set likewise is of class $C^{\infty}$.
The preceding observations yield the following regularity result.
\[apdit\]Suppose $\left\Vert \gamma\right\Vert >\ell$, $1<p<\infty$, $s\in\mathbb{R},$ $f\in W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, and $u\in W_{s}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. If $u$ is a distributional solution in $\mathbb{R}^{n}$ of $\mathcal{L}u=f$, then $u\in W_{s}^{\ell
,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, and$$\left\Vert u\right\Vert _{\ell,p,s;\underline{\ell}}\leq C\left(
\mathcal{L},s,p\right) \left[ \left\Vert \left( 1+\rho\right)
^{s}u\right\Vert _{p,\mathbb{R}^{n}}+\left\Vert \left( 1+\rho\right)
^{s+\ell}f\right\Vert _{p,\mathbb{R}^{n}}\right] \quad.\label{apbd}$$
Let $B$ be any open ball in $\mathbb{R}^{n}$. As the function $f\chi_{B}$ is in the space $W_{s^{\prime}+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ for all $s^{\prime}\in\mathbb{R}$, Theorem \[essthm\] asserts that the function $\mathcal{S}\left( f\chi_{B}\right) $ is in $W_{s^{\prime}}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ for some $s^{\prime}$, with $\mathcal{L}\left(
\mathcal{S}\left( f\chi_{B}\right) \right) =f\chi_{B}$. Thus $w=u-\mathcal{S}\left( f\chi_{B}\right) $, a distributional solution in $B$ of $\mathcal{L}w=0$, is in $C^{\infty}\left( B\right) $. As $B$ is arbitrary in $\mathbb{R}^{n}$, $u\in W_{loc}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. By Theorem \[apbnd\], $u\in
W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right)
$, and (\[apbd\]) holds.
\[dual\]Assume $\left\Vert \gamma\right\Vert >\ell,$ $1<p<\infty$, $1/p+1/q=1$, and $s\in\mathbb{R}.$
a)If $u\in W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $v\in W_{-s-\ell}^{\ell,q}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, then$$\left\langle \mathcal{L}u,v\right\rangle =\left\langle u,\mathcal{L}^{\ast
}v\right\rangle \qquad,\qquad\left\langle \mathcal{L}^{\ast}u,v\right\rangle
=\left\langle u,\mathcal{L}v\right\rangle \quad.\label{vec1}$$
b)If $f\in W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $$-\frac{\left\Vert \gamma\right\Vert }{p}<s<\frac{\left\Vert \gamma\right\Vert
}{q}-\ell\quad,\label{conb}$$ then $\mathcal{S}f$ and $\mathcal{S}^{\ast}f$ are in $W_{-s-\ell}^{\ell
,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, with $\mathcal{L}\left( \mathcal{S}f\right) =\mathcal{L}^{\ast}\left(
\mathcal{S}^{\ast}f\right) =f$; moreover, for $u\in W_{s}^{\ell,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $,$$\left\langle \mathcal{L}u,\mathcal{S}^{\ast}f\right\rangle =\left\langle
\mathcal{L}^{\ast}u,\mathcal{S}f\right\rangle =\left\langle u,f\right\rangle
\quad.\label{vec2}$$
a)By Theorem \[densthm\], there exists a sequence $\left\{
\varphi_{k}\right\} $ of complex $m\times1$ vector functions in $C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $ converging to $v$ in the space $W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $; that is, with$$\sum_{\alpha\cdot\gamma\leq\ell}\left\Vert \left( 1+\rho\right)
^{-s-\ell+\alpha\cdot\gamma}\partial^{\alpha}\left( v-\varphi_{k}\right)
\right\Vert _{q,\mathbb{R}^{n}}\longrightarrow0\quad.$$ Since $u\in W_{loc}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, for each $\varphi_{k}$ we have$$\left\langle \mathcal{L}u,\varphi_{k}\right\rangle =\left\langle
u,\mathcal{L}^{\ast}\varphi_{k}\right\rangle \quad.$$ Since also $\left\Vert \left( 1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}<\infty$ and $\left\Vert \left( 1+\rho\right) ^{s+\ell}\mathcal{L}u\right\Vert _{p,\mathbb{R}^{n}}<\infty$, in view of (\[hold\]) we may let $k\rightarrow\infty$ in this equation to obtain the left equation of (\[vec1\]). The right equation of (\[vec1\]) follows similarly, or by replacing $\mathcal{L}$ with $\mathcal{L}^{\ast}$.
b)We apply Theorem \[essthm\] but with $p$ replaced by $q$ and $s$ replaced by $-s-\ell$. The hypothesis (\[hypb\]) is replaced by $$-\left\Vert \gamma\right\Vert /q<-s-\ell<\left\Vert \gamma\right\Vert
-\ell-\left\Vert \gamma\right\Vert /q\quad,$$ which follows from (\[conb\]). The theorem concludes that $\mathcal{S}f$ and $\mathcal{S}^{\ast}f$ are in $W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, with $\mathcal{L}\left(
\mathcal{S}f\right) =\mathcal{L}^{\ast}\left( \mathcal{S}^{\ast}f\right)
=f$. Application of (\[vec1\]) to $v=\mathcal{S}^{\ast}f$ and $v=\mathcal{S}f$ yields (\[vec2\]).
\[main\]Assume $\left\Vert \gamma\right\Vert >\ell$, $1<p<\infty$, $1/p+1/q=1$, $s\in\mathbb{R}$, and consider the mapping$$\mathcal{L}:W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline
{\ell}\right) \longrightarrow W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \quad.\label{map}$$
a)If $-\left\Vert \gamma\right\Vert /p<s$, then the mapping is one-to-one.
b)If $-\left\Vert \gamma\right\Vert /p<s<\left\Vert \gamma\right\Vert
/q-\ell$, the mapping is onto,
c)If $s<-\left\Vert \gamma\right\Vert /p$, the mapping is not one-to-one.
d)If $s\geq\left\Vert \gamma\right\Vert /q-\ell$ the mapping is not onto.
e)If $s=-\left\Vert \gamma\right\Vert /p$, the mapping is not bounded below.
Consequently, (\[map\]) is an isomorphism if and only if $$-\left\Vert \gamma\right\Vert /p<s<\left\Vert \gamma\right\Vert /q-\ell
\quad.\label{iso}$$
a)Under the given assumptions, assume $u\in W_{s}^{\ell,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $\mathcal{L}u=0$. We may choose $s$ smaller, if necessary, so that (\[conb\]) holds. Then by Lemma \[dual\](b), for all $f$ in $W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ we have$$\left\langle u,f\right\rangle =\left\langle \mathcal{L}u,\mathcal{S}^{\ast
}f\right\rangle =\left\langle 0,\mathcal{S}^{\ast}f\right\rangle =0\quad.$$ As $u\in W_{s}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $ and $W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ is the dual space, we infer that $u=0$. Thus (\[map\]) is one-to-one.
b)Under the given assumptions, let $f\in W_{s+\ell}^{0,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. By Theorem \[essthm\], $\mathcal{S}f\in W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and $\mathcal{L}\left(
\mathcal{S}f\right) =f$. Thus $\mathcal{L}$ is onto.
c)Since $\mathcal{L}$ has no zero order term, any constant vector $u$ solves $\mathcal{L}u=0$. But for such $u$, (\[norm2\]) gives$$\left\Vert u\right\Vert _{\ell,p,s;\underline{\ell}}=\left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}=\left\vert u\right\vert
\left( \int_{\mathbb{R}^{n}}\left( 1+\rho\right) ^{sp}\ dx\right)
^{1/p}\quad,$$ which according to Lemma \[rhoint\] is finite whenever $sp<-\left\Vert
\gamma\right\Vert $. Thus, if $s<-\left\Vert \gamma\right\Vert /p$, then $u\in
W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ and (\[map\]) is not one-to-one.
d)We consider the case $s>\left\Vert \gamma\right\Vert /q-\ell$, but delay the case $s=\left\Vert \gamma\right\Vert /q-\ell$ until after the proof of (e). Let $v$ be any nonzero $m\times1$ constant function, and let $f $ be the function$$f\left( x\right) =\left[ 1+\rho\left( x\right) \right] ^{-\left(
s+\ell+\left\Vert \gamma\right\Vert \right) }v\quad.$$ By Lemma \[rhoint\],$$\left\Vert v\right\Vert _{\ell,q,-s-\ell;\underline{\ell}}=\left\vert
v\right\vert \left( \int_{\mathbb{R}^{n}}\left( 1+\rho\right) ^{-\left(
s+\ell\right) q}\ dx\right) ^{1/q}<\infty\quad,$$ since $-\left( s+\ell\right) q<-\left\Vert \gamma\right\Vert $; also,$$\begin{aligned}
\left\Vert f\right\Vert _{0,p,s+\ell;\underline{\ell}} & =\left(
\int_{\mathbb{R}^{n}}\left( 1+\rho\right) ^{\left( s+\ell\right)
p}\left\vert f\right\vert ^{p}\ dx\right) ^{1/p}\\
& =\left\vert v\right\vert \left( \int_{\mathbb{R}^{n}}\left(
1+\rho\right) ^{-\left\Vert \gamma\right\Vert p}\ dx\right) ^{1/p}<\infty\quad,\end{aligned}$$ as $-\left\Vert \gamma\right\Vert p<-\left\Vert \gamma\right\Vert $. Thus $v\in W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline
{\ell}\right) $ and $f\in W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. If $f=\mathcal{L}u$ for some $u$ in $W_{s}^{\ell,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $, then Lemma \[dual\](a) requires that $$\begin{aligned}
0 & =\left\langle u,0\right\rangle =\left\langle u,\mathcal{L}^{\ast
}v\right\rangle =\left\langle \mathcal{L}u,v\right\rangle =\left\langle
f,v\right\rangle \\
& =\int_{\mathbb{R}^{n}}f\cdot v\ dx=\left\vert v\right\vert ^{2}\int_{\mathbb{R}^{n}}\left[ 1+\rho\left( x\right) \right] ^{-\left(
s+\ell+\left\Vert \gamma\right\Vert \right) }\ dx>0\quad,\end{aligned}$$ a contradiction. Thus (\[map\]) is not onto.
e)Let $s=-\left\Vert \gamma\right\Vert /p$. Given $R\geq1$, let $\psi$ be the function described in the proof of Theorem \[densthm\], satisfying $\psi\in C_{0}^{\infty}\left( \mathbb{R}^{n}\right) $, $0\leq\psi\leq1$, $\psi\left( x\right) =1$ for $\rho\left( x\right) \leq R$, $\psi\left(
x\right) =0$ for $\rho\left( x\right) \geq2R$, and $$\left\vert \partial^{\alpha}\psi\left( x\right) \right\vert \leq C\left(
\ell,\alpha\right) R^{-\alpha\cdot\gamma}\quad.$$ Let $v$ be any nonzero constant $m\times1$ vector, and let $u$ be the function $u\left( x\right) =\psi\left( x\right) v$. Then$$\left\vert \mathcal{L}u\left( x\right) \right\vert =\left\vert \sum
_{\alpha\cdot\gamma=\ell}A_{\alpha}\partial^{\alpha}\psi\left( x\right)
v\right\vert \leq\left\{
\begin{tabular}
[c]{ll}$C\left( \mathcal{L}\right) \left\vert v\right\vert R^{-\ell}$ & , if
$R\leq\rho\left( x\right) \leq2R\,,$\\
$0$ & , otherwise\thinspace.
\end{tabular}
\ \ \ \right.$$ As $1\leq R\leq\rho\leq2R$ implies $\rho\leq1+\rho\leq2\rho$, we have$$\left\Vert \left( 1+\rho\right) ^{s+\ell}\mathcal{L}u\right\Vert
_{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L}\right) \left\vert v\right\vert
R^{-\ell}\left( \int_{R\leq\rho\left( x\right) \leq2R}\rho\left( x\right)
^{\left( s+\ell\right) p}\ dx\right) ^{1/p}\quad.$$ In the last integral we set $x=R^{\gamma}y$, $\rho\left( x\right)
=R\rho\left( y\right) $, $dx=R^{\left\Vert \gamma\right\Vert }dy$ and obtain$$\begin{aligned}
\int_{R\leq\rho\left( x\right) \leq2R}\rho\left( x\right) ^{\left(
s+\ell\right) p}\ dx & =R^{\left( s+\ell\right) p+\left\Vert
\gamma\right\Vert }\int_{1\leq\rho\left( y\right) \leq2}\rho\left(
y\right) ^{\left( s+\ell\right) p}\ dy\\
& =C\left( \mathcal{L},p\right) R^{\left( s+\ell\right) p+\left\Vert
\gamma\right\Vert }\quad,\end{aligned}$$ and thereby, upon setting $s=-\left\Vert \gamma\right\Vert /p$,$$\left\Vert \left( 1+\rho\right) ^{s+\ell}\mathcal{L}u\right\Vert
_{p,\mathbb{R}^{n}}\leq C\left( \mathcal{L},p\right) \left\vert v\right\vert
\quad.\label{ellu}$$ On the other hand, Lemma \[rhoint\] ensures that, as $R\rightarrow\infty$,$$\begin{aligned}
\left\Vert \left( 1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}} &
\geq\left( \int_{\rho\left( x\right) \leq R}\left[ 1+\rho\left( x\right)
\right] ^{sp}\left\vert u\left( x\right) \right\vert ^{p}\ dx\right)
^{1/p}\\
& =\left\vert v\right\vert \left( \int_{\rho\left( x\right) \leq R}\left[
1+\rho\left( x\right) \right] ^{-\left\Vert \gamma\right\Vert }\ dx\right)
^{1/p}\longrightarrow\infty\quad.\end{aligned}$$ Combining this result with (\[ellu\]), we conclude that, as $R\rightarrow
\infty$,$$\frac{\left\Vert \mathcal{L}u\right\Vert _{0,p,s+\ell;\underline{\ell}}}{\left\Vert u\right\Vert _{\ell,p,s;\underline{\ell}}}\leq\frac{C\left(
\mathcal{L},p\right) \left\vert v\right\vert }{\left\Vert \left(
1+\rho\right) ^{s}u\right\Vert _{p,\mathbb{R}^{n}}}\longrightarrow0\quad.$$ By choosing $R$ large enough we can make the left side of this expression as small as desired; thus the mapping (\[map\]) is not bounded below in the case $s=-\left\Vert \gamma\right\Vert /p$.
We are left with only the case $s=\left\Vert \gamma\right\Vert /q-\ell$. We assume (\[map\]) is onto, and we will produce a contradiction. The condition $\left\Vert \gamma\right\Vert >\ell$ implies $s>-\left\Vert \gamma\right\Vert
/p$, and the result of (a) confirms the mapping is one-to-one. Consequently, as a bounded, onto, one-to-one.mapping from one Banach space to another, $\mathcal{L}$ has a bounded inverse mapping $\mathcal{L}^{-1}$. In particular, $$\mathcal{L}^{-1}:W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \longrightarrow W_{s}^{\ell,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right)$$ and there is a positive constant $M$ such that$$\left\Vert \mathcal{L}^{-1}v\right\Vert _{\ell,p,s;\underline{\ell}}\leq
M\left\Vert v\right\Vert _{0,p,s+\ell;\underline{\ell}}\quad.$$
Given any function $f$ in $W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, we may define a mapping $T$ from $W_{s+\ell
}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $ into $\mathbb{C}$, according to$$T\left( v\right) =\left\langle \mathcal{L}^{-1}v,f\right\rangle \quad.$$ Obviously $T$ is linear, and it is also bounded, as verified by$$\begin{aligned}
\left\vert T\left( v\right) \right\vert & =\left\vert \left\langle
\mathcal{L}^{-1}v,f\right\rangle \right\vert \leq\left\Vert \mathcal{L}^{-1}v\right\Vert _{0,p,s;\underline{\ell}}\left\Vert f\right\Vert
_{0,q,-s;\underline{\ell}}\\
& \leq\left\Vert \mathcal{L}^{-1}v\right\Vert _{\ell,p,s;\underline{\ell}}\left\Vert f\right\Vert _{0,q,-s;\underline{\ell}}\leq M\left\Vert
f\right\Vert _{0,q,-s;\underline{\ell}}\left\Vert v\right\Vert _{0,p,s+\ell
;\underline{\ell}}\quad.\end{aligned}$$ Since $W_{-s-\ell}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $ is the dual of $W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, there exists a function $u$ in $W_{-s-\ell}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $ such that$$T\left( v\right) =\left\langle \mathcal{L}^{-1}v,f\right\rangle
=\left\langle v,u\right\rangle \qquad,\qquad\forall v\in W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \quad.$$ Giving any complex $m\times1$ vector function $\varphi$ in $C_{0}^{\infty
}\left( \mathbb{R}^{n}\right) $ we may choose $v=\mathcal{L}\varphi$ to obtain$$\left\langle \varphi,f\right\rangle =\left\langle \mathcal{L}\varphi
,u\right\rangle \qquad,\qquad\left\langle f,\varphi\right\rangle =\left\langle
u,\mathcal{L}\varphi\right\rangle \quad.$$ This relation implies that $u$ is a distributional solution in $\mathbb{R}^{n}$ of the equation $\mathcal{L}^{\ast}u=f$. By Proposition \[apdit\] applied to $\mathcal{L}^{\ast}$, and with $p$ replaced by $q$ and $s$ by $-s-\ell$, we have $u\in W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $. As $f$ is arbitrary in $W_{-s}^{0,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, this argument shows that the mapping $$\mathcal{L}^{\ast}:W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \longrightarrow W_{-s}^{0,q}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right)$$ is onto. But the result of (e), with $\mathcal{L}$ replaced by $\mathcal{L}^{\ast}$, $p$ by $q$, and $s$ by $-s-\ell$, applies to this mapping, as $-s-\ell=-\left\Vert \gamma\right\Vert /q$. Since $\mathcal{L}^{\ast}$ is onto but not bounded below, it cannot be one-to-one. Hence there is a function $w$ in $W_{-s-\ell}^{\ell,q}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell
}\right) $ such that $w\neq0$ and $\mathcal{L}^{\ast}w=0$. By Lemma \[dual\](a) we then have, for all $u$ in $W_{s}^{\ell,p}\left(
\mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $,$$\left\langle \mathcal{L}u,w\right\rangle =\left\langle u,\mathcal{L}^{\ast
}w\right\rangle =\left\langle u,0\right\rangle =0\quad.$$ But we assume (\[map\]) is onto, so we have $\left\langle f,w\right\rangle
=0$ for all $f$ in $W_{s+\ell}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) $, a contradiction if $w\neq0$.
As a special case of Theorem \[main\], we take $s=-\ell$ and find that the mapping$$\mathcal{L}:W_{-\ell}^{\ell,p}\left( \mathbb{R}^{n}\mathbb{C}^{m},\underline{\ell}\right) \longrightarrow W_{0}^{0,p}\left( \mathbb{R}^{n},\mathbb{C}^{m},\underline{\ell}\right) \label{dimi}$$ is an isomorphism provided that $\ell<\left\Vert \gamma\right\Vert /p$. This result has already been obtained by Demidenko [@DEM2; @DEM3], who wrote (\[dimi\]) with the notation$$\mathcal{L}:W_{p,1}^{\underline{\ell}}\left( \mathbb{R}^{n}\right)
\longrightarrow L_{p}\left( \mathbb{R}^{n}\right) \quad.$$
Examples
========
We give a few examples to which results of the paper apply.
Consider a parabolic opertor in $\mathbb{R}^{n+1}=\mathbb{R}^{n}\times\mathbb{R}$,$$\mathcal{L}u=\sum_{\left\vert \alpha\right\vert =\ell}A_{\alpha}\partial
_{x}^{\alpha}u-I\partial_{t}u\quad,\label{par}$$ where each $A_{\alpha}$ is a complex constant $m\times m$ matrix, $I$ is the $m\times m$ identity, and $u=u\left( x_{1},\ldots,x_{n},t\right) $. The usual parabolicity condition (see [@FR]) requires that each eigenvalue $\lambda\left( x\right) $ of the matrix$$P\left( x\right) =\sum_{\left\vert \alpha\right\vert =\ell}A_{\alpha}\left(
ix\right) ^{\alpha}$$ satisfy an inequality$$\operatorname{Re}\ \lambda\left( x\right) \leq-\delta\left\vert x\right\vert
^{\ell}\qquad\left( \delta>0\right) \quad.\label{par1}$$ For (\[par\]), formula (\[ell\]) gives$$L\left( x,t\right) =\sum_{\left\vert \alpha\right\vert =\ell}A_{\alpha
}\left( ix\right) ^{\alpha}-itI\quad.$$ Thus $L\left( x,t\right) $ is invertible if $\left( x,t\right) \neq0$, as (\[par1\]) shows that $P\left( x\right) $ has no purely imaginary eigenvalue when $x\neq0$. Also for (\[par\]), whose order is $\ell$, we determine that$$\underline{\ell}=\left( \ell,\ldots,\ell,1\right) \qquad,\qquad
\gamma=\left( 1,\ldots,1,\ell\right) \qquad,\qquad\left\Vert \gamma
\right\Vert =n+\ell>\ell\quad,$$$$\rho\left( x,t\right) =\left( t^{2}+\sum_{k=1}^{n}x_{k}{}^{2\ell}\right)
^{1/2\ell}\quad.$$ Thus Theorem \[main\] applies, and we conclude that the mapping (\[map\]) is an isomorphism if and only if$$-\frac{n+\ell}{p}<s<n-\frac{n+\ell}{p}\quad.$$ For the special case of the heat equation,$$\mathcal{L}u=\Delta u-\partial u/\partial t\quad,$$ this result was proved in [@HM].
Consider in $\mathbb{R}^{n}$ the operator$$\mathcal{L}u=\sum_{\left\vert \alpha\right\vert =\ell}A_{\alpha}\partial
_{x}^{\alpha}u\quad,$$ where again each $A_{\alpha}$ is a complex $m\times m$ matrix. The symbol is$$L\left( x\right) =\sum_{\left\vert \alpha\right\vert =\ell}A_{\alpha}\left(
ix\right) ^{\alpha}\quad,$$ and the semiellipticity requirement that $L\left( x\right) $ be invertible if $x\neq0$ reduces to the usual requirement for ellipticity of the operator. We have$$\underline{\ell}=\left( \ell,\ell,\ldots,\ell\right) \qquad,\qquad
\gamma=\left( 1,1,\ldots,1\right) \qquad,\qquad\left\Vert \gamma\right\Vert
=n\quad,$$$$\rho\left( x\right) =\left( \sum_{k=1}^{n}x_{k}{}^{2\ell}\right)
^{1/2\ell}\quad.$$ Obviously $\rho\left( x\right) $ is equivalent to the simpler weight function $\left\vert x\right\vert $. Theorem \[main\] applies only if $\left\Vert \gamma\right\Vert =n>\ell$, in which case the mapping (\[map\]) is an isomorphism if and only if$$-\frac{n}{p}<s<n-\ell-\frac{n}{p}\quad.$$
Let $k$ and $r$ be positive integers, and let $\mathcal{L}$ be the operator$$\mathcal{L}u=\sum_{j=0}^{k}\ \sum_{\left\vert \beta\right\vert =jr}A_{\beta,k-j}\partial_{x}^{\beta}\partial_{t}^{k-j}u\quad,\label{pparab}$$ where again $u=u\left( x_{1},\ldots,x_{n},t\right) $, and each $A_{\beta}$ is a complex constant $m\times m$ matrix. The semiellipticity condition is that the matrix$$\sum_{j=0}^{k}\ \sum_{\left\vert \beta\right\vert =jr}A_{\beta,k-j}\left(
ix\right) ^{\beta}\left( it\right) ^{k-j}\label{ppar}$$ be invertible when $\left( x,t\right) \neq0$. The order of this operator is $\ell=kr$, while$$\underline{\ell}=\left( kr,\ldots,kr,k\right) \qquad,\qquad\gamma=\left(
1,\ldots,1,r\right) \qquad,\qquad\left\Vert \gamma\right\Vert =n+r\quad,$$$$\rho\left( x,t\right) =\left( t^{2k}+\sum_{j=1}^{n}x_{k}{}^{2kr}\right)
^{1/2kr}\quad.$$ Our condition $\left\Vert \gamma\right\Vert >\ell$ reduces to $n>\left(
k-1\right) r$, and the isomorphism condition (\[iso\]) becomes$$-\frac{n+r}{p}<s<n-\left( k-1\right) r-\frac{n+r}{p}\quad.$$
In the scalar case $m=1$, and with $A_{0,k}=i^{-k}$, (\[ppar\]) becomes$$f\left( x,t\right) :=t^{k}+\sum_{j=1}^{k}\ \sum_{\left\vert \beta\right\vert
=jr}A_{\beta,k-j}\left( ix\right) ^{\beta}\left( it\right) ^{k-j}\quad.$$ The operator $\mathcal{L}$ in this case is said to be r-parabolic" *[@IT; @MIZ] under the additional assumption that* $\operatorname{Im}\ t\geq\delta>0$ for each root $t$ of $f\left( x,t\right)
=0$, as $x $ ranges over $\mathbb{R}^{n}$ with $\left\vert x\right\vert =1$.
[99]{} M. Cantor, Spaces of functions with asymptotic conditions on $R^{n}$, Indiana Univ. Math. J. 24 (1975), 897-902.
G. V. Demidenko, On weighted Sobolev spaces and integral operators determined by quasi-elliptic equations, Russian Acad. Sci. Dokl. Math. 49 (1994), 113-118.
G. V. Demidenko, On quasielliptic operators in $R^{n}$, Siberian Math. J. 39 (1998), 884-893.
G. V. Demidenko, Isomorphic properties of quasi-elliptic operators, Doklady Mathematics 59 (1999), 102-106.
G. V. Demidenko, Isomorphic properties of one class of differential operators and their applications, Siberian Math. J. 42 (2001), 865-883.
G. V. Demidenko, On properties of a class of matrix differential operators in $R^{n}$, Selçuk J. Appl. Math. 3 (2002), 23-36.
G. V. Demidenko, Isomorphic properties of one class of matrix differential operators, Doklady Mathematics 68 (2003), 4-7.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
D. T. Haimo and C. Markett, A representation theory for solutions of a higher order heat equation, I, J. Math. Anal. Appl. 168 (1992), 89-107.
G. N.Hile and C. P. Mawata, The behaviour of the heat operator on weighted Sobolev spaces, Trans. Amer. Math. Soc. 350, no. 4 (1998), 1407-1428.
G. N. Hile, C. P. Mawata, and C. Zhou, Apriori bounds for semielliptic operators, J. Differential Equations, vol. 176 (2001), 29-64.
L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985.
M. Itano, Some remarks on the Cauchy problem for p-parabolic equations, Hiroshima Math. J. 4 (1974), 211-228.
R. B. Lockhart, Fredholm properties of a class of elliptic operators on non-compact manifolds, Duke Math. J. 48 (1981), 289-312.
R. B. Lockhart and R. C. McOwen, On elliptic systems in $R^{n}$, Acta. Math. 150 (1983), 125-135.
R. C. McOwen, The behavior of the Laplacian on weighted Sobolev Spaces, Comm. Pure Appl. Math. 32 (1979), 783-795.
R. C. McOwen, On elliptic operators in $R^{n}$, Comm. Partial Differential Equations 5 (1980), 913-933.
S. Mizohata, Le problème de Cauchy pour les équations paraboliques, J. Math. Soc. Japan 8 (1956), 269-299.
L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators in $R^{n}$, J. Math. Anal. Appl. 42 (1973), 271-301.
S. V. Uspenskiĭ, On the representation of functions defined by a class of hypoelliptic operators, Trudy Mat. Inst. Steklov Akad. Nauk 117 (1972), 292-299; English transl. in Proc. Steklov Inst. Math. 117 (1972), 343-352.
H. F. Walker, On the null-spaces of first-order elliptic partial differential operators in $R^{n}$, Proc. Amer. Math. Soc. 30 (1971), 278-286.
H. F. Walker, On the null-spaces of elliptic partial differential operators in $R^{n}$, Trans. Amer. Math. Soc. 173 (1972), 263-275.
|
---
abstract: 'The analytic structure of the S-matrix of singular quantum mechanics is examined within a multichannel framework, with primary focus on its dependence with respect to a parameter ($\Omega$) that determines the boundary conditions. Specifically, a characterization is given in terms of salient mathematical and physical properties governing its behavior. These properties involve unitarity and associated current-conserving Wronskian relations, time-reversal invariance, and Blaschke factorization. The approach leads to an interpretation of effective nonunitary solutions in singular quantum mechanics and their determination from the unitary family.'
author:
- 'Horacio E. Camblong,$^{1}$ Luis N. Epele,$^2$ Huner Fanchiotti,$^2$ and Carlos A. García Canal$^2$'
title: |
Analytic Structure of the S-Matrix\
for Singular Quantum Mechanics
---
Introduction {#sec:introduction}
============
Singular potentials are known to pose notoriously subtle difficulties that call for an extension of the rules of ordinary quantum mechanics. Some of the outstanding problematic features of singular quantum mechanics (SQM) were addressed in the pioneering work of Ref. [@cas:50] and in the early literature reviewed in Refs. [@spector_RMP] and [@Newton]. These features can be ultimately traced to an indeterminacy in the boundary conditions [@landau:77]. More recent general frameworks were proposed in Ref. [@Esposito] using polydromy properties [@Stroffolini], and in Ref. [@Smatrix-singularQM] within a general multichannel framework. In addition, significant progress has been made for specific problems related to the renormalization of the inverse square potential (ISP) [@Gupta-1993; @HEC-LP1; @HEC-LP2; @HEC:CQM-renormalization] and contact interactions [@Jackiw-1991-Beg], their associated conformal symmetry [@Alfaro-Fubini-Furlan-1976; @Jackiw_SO(21)_1; @Jackiw_SO(21)_2] and anomaly [@HEC-CQManomaly1; @HEC-CQManomaly2], and other cases of SQM renormalization [@Beane:2000wh].
In this paper, we reexamine the generic form of the S-matrix for SQM and shed light on the existence of an intriguing connection between the unitary and nonunitary solutions. A particular form of this connection can be found in the pioneering work of Ref. [@radin:75], where it was pointed out that, while the basic results of Ref. [@cas:50] focus on the unitary but nonunique solutions of SQM, the path-integral treatment of Ref. [@nel:64] for the inverse square potential (ISP) generates a distinct and unique nonunitary solution with perfect absorption. Thus, in Ref. [@radin:75], an average expression at the level of Green’s functions is used to relate particular cases of unitary and nonunitary solutions. Subsequently, it was pointed out [@bawin_coon_wire] that a similar result can be obtained for the S-matrix of the two-dimensional ISP: an average of unitary solutions coincides with the solution for perfect absorption.
In essence, we display the existence of a network of relations supported on current-conservation equations or concomitant unitarity of an appropriately defined S-matrix. We implement our proposal within our recently developed general multichannel framework [@Smatrix-singularQM] that not only includes the ISP in any number of dimensions but also applies to more singular potentials. Thus, we address broad analytic properties of the S-matrix that can be derived from general principles—determining, among other things, the functional Blaschke-type form of the S-matrix [@Blaschke1; @Blaschke2; @Garnett:1981], in addition to other related mathematical properties.
The remainder of this paper is organized as follows. In Sec \[sec:SQM-multichannel-setup\] we outline the basic definitions and setup of the multichannel SQM framework. Section \[sec:S-matrix-structure\] is the core of the paper, with an examination of the analytic properties of the S-matrix vis-à-vis its boundary value indeterminacy. Section \[sec:unitary-nonunitary\_connection\] explores the unitary-nonunitary connection arising from the S-matrix structure. The conclusions are summarized in Sec \[sec:conclusions\].
Singular Quantum Mechanics (SQM): Multichannel Framework and Setup of the Problem {#sec:SQM-multichannel-setup}
=================================================================================
Following Ref. [@Smatrix-singularQM], we consider a quantum-mechanical problem or equivalent system described by a Schrödinger-type equation, i.e., the normal invariant form of the generic linear second-order differential equation that can be obtained via a Liouville transformation [@Forsyth]: $$\left[
\frac{d^{2}}{dr^{2}}
\,
+ J(r)
\right]
u (r) = 0
\; .
\label{eq:singular-QM_radial}$$ In Eq. (\[eq:singular-QM\_radial\]), for a broad range of physical applications, one can rewrite the normal invariant [@Forsyth] as $J(r)
=
k^{2}
-
%V({\bf r})
V( r )
-
\left[\left( l + \nu \right)^{2} - 1/4 \right]/r^{2}
$, with parameters $l$ and $\nu$ (usually associated with angular momentum and spatial dimensionality) to be adjusted for each physical or mathematical application (details can be found in Refs. [@Smatrix-singularQM] and [@HEC:CQM-renormalization], where a class of anisotropic potentials are also subsumed in this formulation).
In addition to an irregular singularity at infinity, we assume the existence of a singular point at a finite location that we take to be $r=0$, with behavior $$J( r )
% J({\bf r})
\sim
-
V( r )
% V({\bf r})
\sim
\lambda
\,
r^{-p}
\label{eq:potential-near-singularity}$$ for $r \sim 0$, where
- $p=2$, with a regular singularity. This marginally singular case is known as the ISP, defining long-range conformal quantum mechanics (CQM).
- $p>2$, with an irregular singularity. This properly singular case yields the family of strongly singular power-law potentials.
Thus, dominant contribution (\[eq:potential-near-singularity\]) in the neighborhood of the origin defines the all-important power-law class of potentials in SQM. Moreover, with minor adjustments, our problem could be generalized to include additional singular points, as well as logarithmic behavior near the singularities.
Let us consider the set of solutions ${\mathcal B}_{\rm sing}
=
\biggl\{
u_{+}(r) , u_{-}(r)
\biggr\}
\;
\label{eq:singularity-basis}
$ with [*outgoing/ingoing wave-like behavior near the singularity at $r=0$*]{}. This solution set serves as a fundamental basis for the two-dimensional solution space. Thus, it conveys information regarding the singular point as a generalized form of the solutions proposed in Ref. [@Vogt-Wannier] for the inverse quartic potential. Some explicit expressions are shown below.
In addition, the singular point at infinity involves another set of solutions with [*outgoing/ingoing wave-like behavior as $r \sim \infty$*]{}: $
{\mathcal B}_{\rm asymp}
=
\biggl\{
u_{1}(r) , u_{2}(r)
\biggr\}
%\; \label{eq:asymptotic-basis}
%\end{equation}
$, and which are motivated by the determination of physical observables via the [*asymptotic S-matrix*]{} $ S_{\rm asymp} $ (see below). Explicitly, $$u_{1,2} (r)
\stackrel{( r \rightarrow \infty )}{\sim}
\frac{1}{ \sqrt{k} }
\,
e^{\mp i \pi/4}
\,
e^{\pm i kr}
\; ,
\label{eq:asymptotic-wave-normalization}$$ where: (i) the chosen phases are adopted for comparison with asymptotic expansions of Hankel functions; (ii) the normalization is enforced with the WKB amplitude factor $k^{-1/2}$, which is re-evaluated in Sec. \[sec:S-matrix-structure\] from Wronskian properties. Correspondingly, the linear relation between the two bases, with the associated resolutions of $u(r)$, $$u (r)
=
C^{ \mbox{\tiny $ (+)$} }
u_{+}(r)
+
C^{ \mbox{\tiny $ (-)$} }
u_{-} (r)
=
C^{ \mbox{\tiny $ (1)$} }
u_{1}(r)
+
C^{ \mbox{\tiny $ (2)$} }
u_{2} (r)
\; ,
\label{eq:wave-function_bases}$$ provide a formal solution to the most general problem of SQM with one [*finite singular point*]{}, as sketched in Fig. 1. In summary, the connection between the bases ${\mathcal B}_{\rm sing}$ and ${\mathcal B}_{\rm asymp}$ constitutes a two-channel framework, where each “channel” is associated with a singular point (with one finite singular point and the second singular point located at infinity)—generalizations to multiple singularities are possible as a multichannel setup [@Smatrix-singularQM], as will be further discussed elsewhere.
In particular, it is convenient to rewrite the distinct sides of Eq. (\[eq:wave-function\_bases\]) as $$u (r) \propto
\Omega
\, u_{+}(r) + u_{-} (r)
\;
\label{eq:wave-function_near-origin-basis}$$ and $$u (r) \propto
\hat{S}_{\rm asymp}
\,
u_{1} (r) + u_{2} (r)
\; .
\label{eq:wave-function_asymptotic-basis}$$ Here, with the use of the proportionality symbol, the ratio $$\Omega =
\frac{
C^{ \mbox{\tiny $ (+)$} }
}{
C^{ \mbox{\tiny $ (-)$} }
}
\;
\label{eq:singularity-parameter}$$ provides a “singularity parameter” related to the indeterminacy of the boundary conditions at the finite singular point. Thus, $\Omega$ specifies an auxiliary “boundary condition,” i.e., it gauges the relative probability amplitudes of outgoing (emission) to ingoing (absorption) waves in the neighborhood of $r=0$. Similarly, we fix the normalization for the point at infinity with $$S_{\rm asymp}
=
e^{i \pi \left( l+\nu \right)}
\,
\hat{S}_{\rm asymp}
\; ,
\label{eq:S-matrix_and_reduced-S-matrix}$$ in terms of the [*reduced matrix elements*]{} $\hat{S}_{\rm asymp}$ and an $l$- and $d$-dependent phase factor. Equations (\[eq:wave-function\_asymptotic-basis\]) and (\[eq:S-matrix\_and\_reduced-S-matrix\]) yield the S-matrix from which the physical observables are extracted. In addition, by the way they are constructed as outgoing/ingoing waves, the basis functions satisfy $$\begin{aligned}
[u_{1} (r)]^{*} & = u_{2} (r)
\label{eq:conjugate-basis-12-relation}
\; ,
\\
[u_{+} (r)]^{*} & = u_{-} (r)
\label{eq:conjugate-basis-pm-relation}
\; ,\end{aligned}$$ for $r \in \mathbb{R}$.
For the sake of completeness, we provide explicit expressions for the dominant behavior of the “singularity basis” [@Smatrix-singularQM],
[ u\_ (r) ]{} & , \[eq:Wannier\_waves\_p>2\]\
[ ]{} = ( r )\^[ i ]{} & . \[eq:Wannier\_waves\_p=2\]
Here, the normalization is also enforced with the WKB amplitude factor $(k_{\rm WKB})^{-1/2}$, see again further details in Sec. \[sec:S-matrix-structure\]. Moreover, the arbitrary floating inverse length $\mu$ for the regular singular case $p=2$, which is mandatory by its asymptotic conformal invariance, arises from the integration constant in the WKB solution or via dimensional homogeneity in the associated Cauchy-Euler differential equation. However, for the irregular singular case $p>2$, the integration constant in the exponent only appears at a higher order in the asymptotic expansion with respect to $1/r$; specifically, the Bessel-function solutions of Eq. (\[eq:singular-QM\_radial\]) as $r \rightarrow 0$, $u / \sqrt{r} \propto H^{(1,2)}_{-1/n} \left( - 2 \sqrt{\lambda} \, r^{-n/2}/n \right)$, where $n= p-2>0$, combined with the asymptotics of Hankel functions, provide the correct outgoing/incoming behavior of Eq. (\[eq:Wannier\_waves\_p>2\]). In addition, the conformal case $p=2$ includes the Langer correction [@Langer] corresponding to the critical coupling $\lambda =1/4$, with shifted square-root coupling constant $
\Theta^{2}
\equiv
\lambda - 1/4
$; in this case, for particular instances of nonrelativistic quantum mechanics, the angular momentum is merged with the marginally singular $p=2$ term at the same order, leading to an effective interaction coupling—but this does not occur for quantum fields in black hole backgrounds and other relativistic applications [@Smatrix-singularQM; @HEC-CQManomaly1; @HEC-CQManomaly2; @HEC:CQM-renormalization]. Parenthetically, when the potential has a long-range tail for $r \sim \infty$ given by $V ( r ) \sim - \lambda r^{-\delta}$, the asymptotic behavior involves an extra phase in the form $\sqrt{k} \, u_{1,2} (r)
\stackrel{( r \rightarrow \infty)}{\sim}
\,
e^{\mp i \pi/4}
\,
e^{\pm i kr}
\,
e^{\pm i \lambda r^{1-\delta}/2k (1 -\delta)}
%\; ,
%\label{eq:modified_asymptotic-wave-normalization}
%\end{eqnarray}
$ which can be similarly derived by WKB integration for the irregular singular point at infinity [@Smatrix-singularQM].
Analytic Structure of the S-matrix {#sec:S-matrix-structure}
==================================
For the remainder of this paper, we will write the reduced asymptotic S-matrix, defined by Eq. (\[eq:wave-function\_asymptotic-basis\]), as $\hat{S} \equiv \hat{S}_{\rm asymp} $. The S-matrix of physical interest follows from Eq. (\[eq:S-matrix\_and\_reduced-S-matrix\]). From Sec. \[sec:SQM-multichannel-setup\], the existence of a complex function $$\hat{S}
=
\hat{S} (\Omega)$$ of the variable $\Omega$ relies on the singular nature of the potential via Eqs. (\[eq:wave-function\_near-origin-basis\]) and (\[eq:wave-function\_asymptotic-basis\]). Specifically, the boundary condition indeterminacy at the finite singular point (e.g., the origin) is described by the arbitrary parameter $\Omega$ that represents the ratio of the amplitude coefficients for the required outgoing and ingoing waves at the singularity. By the nature of the solutions, $\Omega$ is generically a complex number.
In addition, $ \hat{S} (\Omega)$ is a [*meromorphic function*]{}, as follows constructively from the definition of the “parameters” $\hat{S}$ and $\Omega$ in the scattering process. In effect, the linear relationship between the coefficients $(C^{ \mbox{\tiny $ (+)$} } , C^{ \mbox{\tiny $ (-)$}}) $ and $(C^{ \mbox{\tiny $ (1)$} } , C^{ \mbox{\tiny $ (2)$}}) $, or between the corresponding bases, as displayed by Eq. (\[eq:wave-function\_bases\]), implies a fractional linear transformation for the relationship between $\hat{S}$ and $\Omega$. Thus, properties of Möbius functions can be used to further understand this formalism [@Smatrix-singularQM]. By contrast, in this paper, we reverse the logic and focus on the basic principles that ultimately generate this remarkable analytic structure of the S-matrix for singular potentials.
The central results we address below are based on the analytic structure of the S-matrix that relies on Blaschke factorization. For our current purposes, some language, properties, and theorems of Blaschke products are in order. Let $\mathbb{D} = \left\{ \Omega \in \mathbb{C} : \; \; |\Omega| < 1 \right\}$ and $\overline{\mathbb{D}}= \left\{ \Omega \in \mathbb{C} : \; \; |\Omega| \leq 1 \right\}$ be the open and closed unit disks in the complex plane, respectively; and the boundary $\mathbb{T} = \left\{ \Omega \in \mathbb{C} : \; \; |\Omega| = 1 \right\} = \partial
\overline{\mathbb{D}}
$ be the unit circle. Then [@Blaschke1; @Blaschke2],
> Let $F(z)$ be a holomorphic function on $\mathbb{D}$ that can be extended to a continuous function on $\overline{\mathbb{D}}$. If $F$ is a mapping of the unit disk $\overline{\mathbb{D}}$ to itself that preserves the disk boundary $\mathbb{T}$ (i.e., $|F| = 1$ if $|z| = 1 $), then $F$ admits a finite Blaschke product factorization, $$F(z) =
> \zeta \;
> \prod_{j=1}^{n}
> \left(
> \frac{z-z_{j}}{ 1 - z_{j}^{*} \, z}
> \right)
> \; ,
> \label{eq:Blaschke_factorization}$$ where $\zeta$ is a phase factor ($|\zeta| =1$) independent of $z$ and $z_{j}$ are the zeros of $F(z)$ in $\mathbb{D}$.
For the case when a bounded analytic function satisfies the conditions above on the open disk $\mathbb{D}$, Carathéodory’s theorem yields a possibly infinite factorization with an appropriate behavior of the given sequence of zeros [@Garnett:1981]. In the finite case above, the number $n$ is the degree, ${\rm deg} \, F$, of the mapping.
Specifically, for our generic treatment of SQM, the mapping $ \hat{S} (\Omega)$ is indeed restricted to the closed unit disk, $\overline{\mathbb{D}}$, i.e., $|S| \leq 1$ iff $|\Omega| \leq 1 $. This property can be established directly from “conserved currents” (with the usual probabilistic interpretation in the particular case of quantum mechanics proper applications) as described by Wronskian properties of pairs of solutions of the governing differential equation. In effect, from the Wronskian $W[\psi_{1},\psi_{2}]$ of any two solutions of Eq. (\[eq:singular-QM\_radial\]) (and/or their complex conjugates, which are also solutions), and through the definition $J[u]=W[u^{*},u]/i$ of conserved currents, it follows that $J[u_{\pm}] = \pm 2$ and similarly $J[u_{1,2}]= \pm 2$ (guaranteed by the WKB normalization). More generally, the form $J[u,v]=W[u^{*}, v]/i$ is conjugate-symmetric sesquilinear, $J[u,\lambda_{1} v_{1} + \lambda_{2} v_{2}]
=
\lambda_{1} J[u, v_{1}]
+
\lambda_{2} J[u, v_{2}]
$ and $\left( J[u,v] \right)^{*}= J[v,u]$, and further satisfies $\left( J[u,v] \right)^{*}= -J[u^{*},v^{*}]$. Then, from these definitions, one can derive all the relevant current-conservation relations, including explicit identities for reflection and transmission coefficients. Moreover, the Hermitian quadratic form $J[u]\equiv J[u,u]$ is an SU(1,1) inner product in the specific sense that $J[u^{*}]=- J[u]$, leading to $J[\lambda_{1} u + \lambda_{2} u^{*}] = \left( |\lambda_{1}|^{2} - |\lambda_{2}|^{2} \right) \, J[u]$; thus, for wave function (\[eq:wave-function\_bases\]) that is the general solution to Eq. (\[eq:singular-QM\_radial\]), $$\frac{1}{2} \,
J [u ] =
|C^{ (1)} |^{2}
-
|C^{ (2)} |^{2}
=
|C^{ (+)} |^{2}
-
|C^{ (-)} |^{2}
\; .$$ This implies that $
\bigl| \hat{S} \bigr|^{2} -1
=
\left[
\left| \Omega \right|^{2} -1
\right]
\,
|C^{(-)}/C^{(2)}|^{2}
$, whence $${\rm sgn}
\left( J \right)
=
{\rm sgn}
\left[
\bigl| \hat{S} \bigr|^{2} -1
\right]
=
{\rm sgn}
\left[
\left| \Omega \right|^{2} -1
\right]
\; .
\label{eq:S-matrix_beta_sign_connection}$$ Therefore, $\bigl| \hat{S} \bigr| \leq 1$ iff $\left| \Omega \right| \leq 1$ (with one-to-one correspondence of the equal signs), so that from the ensuing map $ \hat{S} (\Omega)$ one concludes that $ \hat{S} (\Omega)$ admits the Blaschke factorization (\[eq:Blaschke\_factorization\]), with $z \equiv \Omega$ and $F \equiv \hat{S}$, up to a global phase.
The characterization of the form of the asymptotic S-matrix concludes with the restriction to a single Blaschke factor, which is due to the existence of a unique zero $\Omega_{1} = {\mathcal R}^{*}$ for the S-matrix (except for $|\mathcal{R}|=1$; see below). This property can be established from general arguments, as follows. First, from the generic framework of Sec. \[sec:SQM-multichannel-setup\], one can depict the zeros and poles of the S-matrix using Fig. 1. In effect, the zero of $\hat{S}$ occurs when the building block $u_{1}$ is suppressed. Second, define the solutions $\check{u}_{1}$ and $\check{u}_{-}$, with modified normalizations adapted to the standard 1D scattering problem, $$\begin{aligned}
\check{u}_{1}
& = &
\left\{
\begin{array}{ll}
u_{+} + {\mathcal R} u_{-}
\; & \; \; {\rm for} \; r \sim 0
\\
{\mathcal T} u_{1}
\; & \; \; {\rm for} \; r \sim \infty
\end{array}
\right.
\label{eq:tilde-u_1}
\; ,
\\
\check{u}_{-}
& = &
\left\{
\begin{array}{ll}
u_{2} + {\mathcal R'} u_{1}
\; & \; \; {\rm for} \; r \sim \infty
\\
{\mathcal T'} u_{-}
\; & \; \; {\rm for} \; r \sim 0
\;
\end{array}
\label{eq:tilde-u_-}
\; ,
\right.\end{aligned}$$ where $( {\mathcal R} , {\mathcal T}) $ and $( {\mathcal R}' , {\mathcal T}') $ are, respectively, the right-moving and left-moving reflection and transmission amplitude coefficients. Third, by time-reversal invariance (given by the complex conjugate and reversal of the arrows in Fig. 1), we can see that this amounts to $
\check{u}_{1}^{*}
= \left( {\mathcal T} u_{1} \right)^{*}
= {\mathcal T}^{*} u_{2}$, where Eq. (\[eq:conjugate-basis-12-relation\]) was used—this effectively suppresses $u_{1}$ and selects the zero of $\hat{S}$; thus, $\check{u}_{1}^{*}
%\stackrel{ (r \rightarrow 0)}{\sim}
=
[ u_{+} ]^{*} + {\mathcal R}^{*} [ u_{-} ]^{*}
=
u_{-} + {\mathcal R}^{*} u_{+} $, leading to $$\Omega_{1} = \left. \Omega \right|_{\rm zero \; of \; \hat{S} } = {\mathcal R}^{*}
\;$$ \[by comparison against Eq. (\[eq:wave-function\_near-origin-basis\])\]. Fourth, from the physical bound $|{\mathcal R}| \leq 1 $, which is established through $|{\mathcal R}|^2 + |{\mathcal T}|^2 = 1$ via Wronskian relations (see below), it follows that $$%\noindent
\text{\em
for $|{\mathcal R}| < 1 $,
there is a unique zero $\Omega_{1} = {\mathcal R}^{*}$ of the S-matrix,
with $\Omega_{1} \in
{\mathbb{D}} $.}
% \overline{\mathbb{D}} $.}
\label{eq:unique-zero}$$ So far, we have only established that $\Omega_{1} \in
\overline{\mathbb{D}} $, but a zero on the unit circle $\mathbb{T} = \partial \overline{\mathbb{D}} $ is to be rejected, as shown below. Fifth, by comparison against Eq. (\[eq:Blaschke\_factorization\]), one can explicitly write the reduced S-matrix as a single Blaschke factor times a global phase, $$\hat{S}
=
\Delta
\,
\frac{
\Omega
-
{\mathcal R}^{*}
}{
{\mathcal R}
\,
\, \Omega
-
1
}
\label{eq:asymptotic-S-matrix_from-MC-S-matrix}
\; .$$ Sixth, regarding the general Blaschke-factor of Eq. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\]), when $|\mathcal{R}| =1$, the S-matrix is trivial with respect to $\Omega$, in the sense that it is restricted to the unit circle, $|\hat{S}| =1$ and $\Omega$-independent, as can be easily verified (this is a general property of the Blaschke factors); thus, in that case, being a constant, $\hat{S}$ has neither zeros nor poles—incidentally, there is no contradiction herein because $\mathcal{T}=0$ when $|\mathcal{R}| =1$, and the asymptotic form of Eq. (\[eq:tilde-u\_1\]) becomes degenerate, failing to produce an actual zero of $\hat{S}$. Finally, in Eq. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\]), it is also possible to identify the global phase as $$\Delta =
- \frac{ {\mathcal T} }{ {\mathcal T}^{*}}
=
\frac{ {\mathcal R'} }{ {\mathcal R}^{*}}
\label{eq:S-matrix_Mobius-phase}
\; ,$$ from the following general arguments: (i) $\Delta ={\mathcal R'}/ {\mathcal R}^{*}$ from $\hat{S} (\Omega =0) = {\mathcal R'}$ (i.e., the corresponding S-matrix is the left-moving reflection coefficient); (ii) by means of a Wronskian relation, the Stokes’ reciprocity relation $ {\mathcal T} {\mathcal R}^{*}
+
{\mathcal T}^{*}
{\mathcal R'}
= 0
$ is established. Parenthetically, a network of Wronskian relations $W \bigl[ \check{u}_{1}^{*}, \check{u}_{1} \bigr]$, $W \bigl[ \check{u}_{-}^{*}, \check{u}_{-} \bigr]$, $W \bigl[ \check{u}_{1}^{*}, \check{u}_{-} \bigr]$, and $W \bigl[ \check{u}_{1}, \check{u}_{-} \bigr]$, implies the following four conditions: $|{\mathcal R}|^{2} + |{\mathcal T}|^{2} = 1
$, $|{\mathcal R'}|^{2} + |{\mathcal T'}|^{2} = 1$, ${\mathcal R^{*} } {\mathcal T'} + {\mathcal T^{*} } {\mathcal R'} = 0$, and ${\mathcal T'} = {\mathcal T}$ (with the last two being used in the above argument).
It should be noticed that various particular cases of Eq. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\]) have appeared in the literature in Refs. [@perelomov:70; @Alliluev_ISP; @aud:99] (for $\Omega =0$) and in Ref. [@Voronin] (for $\Omega =0$ and $\Omega =\infty$), while the general form has been derived by other techniques in Refs. [@Esposito; @Stroffolini; @Smatrix-singularQM].
As a corollary of relations (\[eq:unique-zero\])–(\[eq:S-matrix\_Mobius-phase\]),
> [*the function $\hat{S} (\Omega)$ is [*analytic*]{} in the closed unit circle $\overline{\mathbb{D}} $, i.e., it is a meromorphic function with no poles for $| \Omega | \leq 1$.*]{}
This is simply due to the fact that the unique purported pole is located at $\Omega_{2} = 1/{\mathcal R}$, as shown by the Blaschke factor—but this location of $\Omega_{2}$ could also be established independently by a similar subset of arguments, via the suppression of the building block $u_{2}$ combined with Eq. (\[eq:tilde-u\_1\]) leading to $\check{u}_{1}
=
u_{+} + {\mathcal R} u_{-} $ (i.e, $\Omega$ given by the explicit ratio $ \Omega_{2} = 1/\mathcal{R} $). Thus, by a reversal of the reflection-coefficient bound discussed above, $|\Omega_{2}| = |1/{\mathcal R}| = 1/|\Omega_{1}| \geq 1$; in addition, the trivial case $|{\mathcal R}| =1 $ can be dealt with separately \[as established in the argument following Eq. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\])\], thus leading to $\hat{S}$ being a constant ($\Omega$-independent) of modulus one, so that the stricter condition $|\Omega_{2}| > 1$ is enforced.
Incidentally, the S-matrix of Eq. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\]), being a single Blaschke factor, defines a well-known Möbius transformation $\hat{S} (\Omega)$ that is an automorphism of the unit disk [@Needham_Complex].
Determination of Nonunitary Solutions from the Whole Family of Unitary Solutions {#sec:unitary-nonunitary_connection}
================================================================================
The general properties discussed in Sec. \[sec:S-matrix-structure\] for the two-channel case (with one singularity other than asymptotic infinity) yield a holomorphic function $ \hat{S} (\Omega)$ with a unique zero at $\Omega_{1} = \mathcal{R}^{*}$. Furthermore, the scattering bound $|{\mathcal R}| \leq 1$ (familiar restriction on the reflection coefficients) leads to a location of the corresponding pole outside the unit disk $\mathbb{D}$, so that the function $ \hat{S} (\Omega)$ is guaranteed to be analytic in $ \overline{\mathbb{D}} $ (i.e., $|\Omega | \leq 1$).
As a result, Cauchy’s integral formula [@Needham_Complex] implies the following [*“average characterization” of nonunitary solutions for all the cases with absorption due to a singular potential*]{}, $$\hat{S}(\Omega)
=
\frac{1}{2 \pi i }
\oint\limits_{{\mathbb{T}} }
d \Omega' \, \frac{ \hat{S} (\Omega') }{\Omega'-\Omega }
\; ,
\label{eq:Cauchy-theorem}$$ where $\mathbb{T} = \partial \overline{\mathbb{D}} $ is the unit circle consisting of the whole family of unitary S-matrix values, i.e., the whole family of solutions to Eq. (\[eq:singular-QM\_radial\]) with “elastic” or self-adjoint boundary conditions. This amounts to $\Omega' = e^{i\chi} $, where $\chi \in \mathbb{R}$ is a phase specifying the generic self-adjoint condition. The one-to-one correspondence of the conditions $|\Omega' | = 1$ and $|S (\Omega' )| = |\hat{S} (\Omega' ) | = 1 $ can be seen from Eq. (\[eq:S-matrix\_beta\_sign\_connection\]). With this notation, and going back to the primitive form of asymptotic S-matrix (\[eq:S-matrix\_and\_reduced-S-matrix\]) by reintroducing the required phase factors (whenever appropriate), Eq. (\[eq:Cauchy-theorem\]) can be rewritten in the suggestive form $${S}(\Omega)
=
\int_{0}^{2\pi}
\frac{d \chi}{2 \pi }
\;
\frac{ {S}(\Omega' = e^{i\chi}) }{ 1 -\Omega \, e^{-i\chi} }
\; .
\label{eq:Cauchy-average}$$ Equation (\[eq:Cauchy-average\]) is an ensemble average with a nonuniform (shifted) weight of the whole family of unitary S-matrices. Therefore, we have established the general identity relating unitary and nonunitary solutions for $|\Omega |< 1$, i.e., all the cases that exhibit [*net absorption*]{}.
The particular case of “perfect absorption,” $\Omega = 0$, involves a symmetric ensemble average with uniform weight, i.e., $${S} (\Omega=0)
=
\int_{0}^{2\pi}
\frac{d \chi}{2 \pi }
\;
{S} (\Omega' = e^{i\chi})
\; ,$$ which corresponds to the particular case considered in Refs. [@radin:75] and [@bawin_coon_wire]. It is also noteworthy that this is equal to the left-moving reflection coefficient: $\hat{S} (\Omega=0) = {\mathcal R}'$ \[see Eqs. (\[eq:asymptotic-S-matrix\_from-MC-S-matrix\]) and (\[eq:S-matrix\_Mobius-phase\])\]. Our present work shows that this result is not accidental but the logical consequence of general physical principles applied to the S-matrix—following a similar methodology to the S-matrix approach in field theory. Remarkably, this is a generic property of SQM (either conformal or an irregular singularity): while singular potentials support this analytic “structure,” the distinct case (not tackled herein) of absorption by complex (optical) potentials generally does not.
Conclusions {#sec:conclusions}
===========
We have shown that, given the existence of a singular problem (in the sense of SQM), a basic set of general principles: linearity, unitarity of the multichannel S-matrix with the related network of current-conservation statements (encoded by reflection and transmission coefficients), and time-reversal symmetry allow the complete determination of the analytic structure of the asymptotic S-matrix of SQM with respect to the parameter $\Omega$ that specifies boundary conditions. This analytic structure involves Möbius transformations of the Blaschke-factor type associated with the unit disk. Due to its generality, this approach also provides the rationale to predict and explain the intriguing average relationship between unitary and nonunitary solutions at the level of the asymptotic S-matrix.
This work was partially supported by the University of San Francisco Faculty Development Fund (H.E.C.); and ANPCyT, Argentina (L.N.E., H.F., and C.A.G.C.).
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|
---
abstract: 'In strong gravitational lens systems, the light bending is usually dominated by one main galaxy, but may be affected by other mass along the line of sight (LOS). Shear and convergence can be used to approximate the contributions from less significant perturbers (e.g. those that are projected far from the lens or have a small mass), but higher order effects need to be included for objects that are closer or more massive. We develop a framework for multiplane lensing that can handle an arbitrary combination of tidal planes treated with shear and convergence and planes treated exactly (i.e., including higher order terms). This framework addresses all of the traditional lensing observables including image positions, fluxes, and time delays to facilitate lens modelling that includes the non-linear effects due to mass along the LOS. It balances accuracy (accounting for higher-order terms when necessary) with efficiency (compressing all other LOS effects into a set of matrices that can be calculated up front and cached for lens modelling). We identify a generalized multiplane mass sheet degeneracy, in which the effective shear and convergence are sums over the lensing planes with specific, redshift-dependent weighting factors.'
bibliography:
- 'multiplane-full.bib'
title: A new hybrid framework to efficiently model lines of sight to gravitational lenses
---
gravitational lensing: strong – gravitational lensing: weak.
Introduction {#S:intro}
============
In galaxy-scale strong gravitational lens systems, there is often a single galaxy that dominates the lens potential. A few systems are compound lenses having two or three lens galaxies within the Einstein radius [e.g. @Koopmans1608; @Rusin1359; @Winn0134], and many more have significant contributions from a group or cluster environment [e.g. @Young81; @Kundic97; @Fischer98; @Tonry98; @Tonry99; @Keeton00; @Kneib00; @Fassnacht06; @Iva]. In all of these cases, the light bending effectively occurs in a single lens plane. If there are any massive objects along the line of sight (LOS; e.g. individual galaxies, galaxy groups or clusters, or cosmic filaments), the additional lens planes may affect the light rays in ways that cannot be ignored.
A dramatic example occurs when two galaxies at different redshifts lie close enough in projection (roughly speaking, their Einstein radii need to overlap) that both act as strong lenses. This ‘two-screen lensing’ can produce new lensing phenomena that have been studied in detail theoretically [@KA88; @Erdl93; @PW95; @Moller01; @Werner08; @Rhie09]. The effect is rare because it requires close alignment; it has been identified in two of the few hundred known galaxy-scale lens systems [@Chae2114; @Gavazzi0946; @Sonnenfeld12].
It is more common to have many objects projected outside the Einstein radius [e.g. @Tonry00], which produce an accumulation of small perturbations that couple to the main lens. To study this scenario, one common approach is to assume that each object contributes only tidal effects—shear and convergence—to the lensing potential. Neglecting higher order effects, similar to what is used for cosmic shear studies [e.g. @Munshi08], makes it possible to use the statistical distribution of galaxies and large-scale structure to predict lensing perturbations [e.g. @Seljak94; @Bar-Kana96; @KKS97]. In strong lens modelling, the amplitude and direction of the shear are often treated as free parameters to be optimized in individual lens systems [e.g. @KKS97].
This widely used approach, which only includes tidal effects, has three possible limitations. First, it may not be appropriate to omit higher order effects beyond shear when a perturber is massive and/or close to the lens. Secondly, the shear is assumed to originate in the main lens plane, neglecting non-linear effects that arise from having mass in multiple planes [see @Jaroszynski12]. Thirdly, lens models themselves cannot constrain any external convergence because of the mass sheet degeneracy [@Falco85]. To avoid biases in derived and cosmological parameters, lens model results must be adjusted after the fact to account for external convergence. It is customary to use independent data such as weak lensing [@Nakajima09; @Fadely10] or the number density of galaxies near the lens [@Suyu10; @Suyu13; @Collett13], although @Schneider13 have questioned the efficacy of this approach as it probes the density field on scales of arcminutes, much larger than the arcsecond scales relevant for strong lensing. Even when galaxies near the lens are modelled explicitly [e.g. @Morgan2033; @CSK0435; @Vuissoz2033; @Fadely12] and/or external convergence is included, the mass is typically assumed to be in the main lens plane, neglecting redshift effects.
One approach to study the redshift effects due to mass along the LOS is to examine mathematical aspects of strong lensing with multiple lens planes [@Levine93; @Kayser93; @Petters95a; @Petters95b; @PLW]. Such studies yield rigorous results but are typically limited to general issues such as bounds on the number of images, counting rules for different types of lensed images, and classifications of caustic geometry. They do not help us account for specific, observed LOS structures in models of real lens systems.
Yet another approach is to write down the multiplane lens equation [e.g. @BN86; @Kovner87; @SEF; @PLW] and then perform ray-tracing calculations through appropriate three-dimensional mass distributions [e.g. @Refsdal70; @SW88a; @SW88b; @Jaroszynski89; @Jaroszynski91; @Jaroszynski92; @Rauch91; @Lee97; @Premadi98; @Wambsganss98; @Wambsganss05; @Hilbert07; @Hilbert09; @Collett13; @Petkova13]. The full multiplane lens equation properly captures the redshift dependences and the couplings between redshift planes, but it can be computationally impractical. There may be hundreds of objects projected close enough to a lens to affect the light rays [e.g. @Iva; @Kurtis; @Wong], making it too expensive to evaluate the enormous number of times required in careful lens modelling.
In this paper, we present a framework for multiplane lensing that consolidates the various approaches to provide an efficient, general way to quantify LOS effects for observed lens systems (see also @Wong; McCully et al., in preparation). Our approach balances the accuracy of the full multiplane lens equation with the efficiency of the tidal approximation. Specifically, our framework can handle an arbitrary collection of “main” planes (strong lenses) that are treated exactly and tidal planes that are approximated with shear and convergence (weak lenses), at any location along the LOS. After reviewing the setup ([Section \[S:setup\]]{}), we analyse the lens equation and magnification tensor ([Section \[S:leqn\]]{}) and time delays ([Section \[S:tdel\]]{}) in the multiplane context. We then examine a multiplane version of the gauge symmetry known as the mass sheet degeneracy ([Section \[S:masssheet\]]{}).
Setup {#S:setup}
=====
Our discussion of multiplane gravitational lensing follows Chapter 9 of the book by @SEF [hereafter SEF] and Section 6.4 of the book by @PLW, which in turn draw on papers by @BN86 and @Kovner87. In particular, our analysis of LOS shear in [Section \[S:leqn-1main\]]{} is equivalent to the discussion of the generalized quadrupole lens in Section 9.3 of SEF.
Definitions {#S:defn}
-----------
Consider $N$ galaxies with redshifts $z_i$, indexed by increasing redshift so $z_1 \le z_2 \le \ldots \le z_N < z_s$. (It is fine to have more than one galaxy at a given redshift.) The source is in plane $N+1$, which is labelled with the index $s$. Let $D_i$ and $D_{is}$ be the angular diameter distances from the observer to galaxy $i$ and from galaxy $i$ to the source (respectively). For $i<j$ let $D_{ij}$ be the angular diameter distance from galaxy $i$ to galaxy $j$.
Let galaxy $i$ have lensing potential $\phi_i({\mbox{\boldmath $x$}}_i)$ and surface mass density $\Sigma_i({\mbox{\boldmath $x$}}_i)$. The lensing effects are functions of the angular position ${\mbox{\boldmath $x$}}_i$ of a light ray as it passes through plane $i$, which in general is not the same as the observed position on the sky. The position ${\mbox{\boldmath $x$}}_i$ depends on how the light is bent by other planes, as characterized by the lens equation (\[eq:multi-x\]). The lensing potential and surface mass density are related by the Poisson equation $$\nabla^2\phi_i({\mbox{\boldmath $x$}}_i) = 2 \frac{\Sigma_i({\mbox{\boldmath $x$}}_i)}{{\Sigma_{{\rm cr},i}}}\ ,$$ where the critical surface density for lensing for plane $i$ is $${\Sigma_{{\rm cr},i}}= \frac{c^2}{4\uppi G}\ \frac{D_s}{D_i D_{is}}\ .$$ The deflection angle from galaxy $i$ is then $${\mbox{\boldmath $\alpha$}}_i({\mbox{\boldmath $x$}}_i) = \nabla\phi_i({\mbox{\boldmath $x$}}_i)\,.$$ It is useful to introduce the matrix of second derivatives, or the ‘tidal tensor’: $${{\bf\Gamma}}_i = \frac{\partial{\mbox{\boldmath $\alpha$}}_i}{\partial{\mbox{\boldmath $x$}}_i}
= \left[\begin{array}{cc}
\kappa_i + \gamma_{\mathrm{c},i} & \gamma_{\mathrm{s},i} \\
\gamma_{\mathrm{s},i} & \kappa_i - \gamma_{\mathrm{c},i}
\end{array}\right] ,$$ where we define the convergence ($\kappa$) and shear ($\gamma$) components from galaxy $i$: $$\begin{aligned}
\kappa_i &=& \frac{1}{2} \left( \frac{\partial^2\phi_i}{\partial x_i^2} +
\frac{\partial^2\phi_i}{\partial y_i^2} \right) , \\
\gamma_{\mathrm{c},i} &=& \frac{1}{2} \left( \frac{\partial^2\phi_i}{\partial x_i^2} -
\frac{\partial^2\phi_i}{\partial y_i^2} \right) , \\
\gamma_{\mathrm{s},i} &=& \frac{\partial^2\phi_i}{\partial x_i \partial y_i}\ .\end{aligned}$$ Note that the convergence can be obtained from the trace of ${{\bf\Gamma}}$, while the shear components are given by the traceless, symmetric part of ${{\bf\Gamma}}$.
We can Taylor expand the lens potential for a perturbing galaxy about the centre of the main lens galaxy as $$\phi({\mbox{\boldmath $x$}}) = \phi(0) + \alpha^a(0) x^a + \frac{1}{2}\Gamma^{ab} x^a x^b+ \frac{1}{6} {{\mathcal F}}^{abc} x^a x^b x^c + \cdots
\label{eq:phiser}$$ where $a,b,c$ are vector or tensor component indices and we have adopted the Einstein notation of summing over repeated indices. ${{\mathcal F}}$ is the flexion tensor of third derivatives defined by $${{\mathcal F}}^{abc} \equiv \left. \frac{\partial^3 \phi}{\partial x^a \partial x^b \partial x^c} \right|_{x=0}.$$ In [equation (\[eq:phiser\])]{}, the $\phi(0)$ term is the zeropoint of the potential, which is unobservable. The ${\mbox{\boldmath $\alpha$}}(0)$ term corresponds to a uniform deflection that is degenerate with a translation of the source plane coordinates. Thus, the first significant term is the second-order one. If we can neglect higher order terms and truncate the expansion at second order, we have $$\begin{aligned}
\label{eq:shearapprox}
\phi_i ({\mbox{\boldmath $x$}}_i) &\approx& \frac{1}{2} {\mbox{\boldmath $x$}}_i \cdot {{\bf\Gamma}}_i(0) {\mbox{\boldmath $x$}}_i\,, \\
{\mbox{\boldmath $\alpha$}}_i ({\mbox{\boldmath $x$}}_i) &\approx& {{\bf\Gamma}}_i(0) {\mbox{\boldmath $x$}}_i\,, \\
{{\bf\Gamma}}_i({\mbox{\boldmath $x$}}_i) &\approx& {{\bf\Gamma}}_i(0)\,.\end{aligned}$$ This defines the tidal approximation, which we employ for all planes in which the higher-order terms beyond shear are sufficiently small. (We quantify the accuracy of the tidal approximation in a forthcoming paper; McCully et al. in preparation) In the remainder of the paper we drop $(0)$ for simplicity. We refer to planes that employ the tidal approximation as ‘tidal planes,’ and planes that are treated exactly as ‘main planes.’
For illustration, the lensing potential of a point mass is given by $$\phi({\mbox{\boldmath $x$}}) = R_{\mathrm{E}}^2 \ln \left| {\mbox{\boldmath $x$}}- {\mbox{\boldmath $r$}}_{\mathrm{p}} \right|$$ where ${\mbox{\boldmath $r$}}_{\mathrm{p}}$ and $R_{\mathrm{E}}$ are the position and Einstein radius of the perturber, respectively. If we let $|{\mbox{\boldmath $x$}}| = x$, $|{\mbox{\boldmath $r$}}_{\mathrm{p}}| = r_\mathrm{p}$, and $\theta$ be the angle between the perturber and the image position as measured from the origin, then we can rewrite the potential using the law of cosines as $$\phi(x,\theta) = \frac{1}{2}R_{\mathrm{E}}^2 \ln\left(r_{\mathrm{p}}^2 + x^2 - x r_{\mathrm{p}} \cos\theta\right).$$ If we assume the projected offset of the perturber is large compared to the image positions ($r_{\mathrm{p}} \gg x$), then we can expand the logarithm as $$\begin{gathered}
\phi(x,\theta) \approx R_{\mathrm{E}}^2 \left[ \ln(r_{\mathrm{p}}) - \cos(\theta) \frac{x}{r_{\mathrm{p}}} - \frac{1}{2} \cos(2\theta) \frac{x^2}{r_{\mathrm{p}}^2} \right. \\
\left. - \frac{1}{3}\cos(3\theta)\frac{x^3}{r_{\mathrm{p}}^3} + \cdots\right].\end{gathered}$$ We see that a point mass has ${{\bf\Gamma}}\propto R_E^2/r_p^2$ and ${{\mathcal F}}\propto R_E^2/r_p^3$.
Multiplane lensing {#S:multiplane}
------------------
The lens equation is constructed by working “backwards” from the observer, through the lens planes one by one, until we reach the source. If ${\mbox{\boldmath $x$}}_j$ is the position in plane $j$, we have (see equation 9.7a of SEF, and equation 6.29 of @PLW) $${\mbox{\boldmath $x$}}_{j} = {\mbox{\boldmath $x$}}_1 - \sum_{i=1}^{j-1} \beta_{ij} {\mbox{\boldmath $\alpha$}}_i({\mbox{\boldmath $x$}}_i)\,,
\label{eq:multi-x}$$ where $$\label{eq:beta}
\beta_{ij} = \frac{D_{ij} D_s}{D_j D_{is}}\ .$$ Note that the lens equation for plane $j$ depends on all planes in front of $j$ ($i < j$), so this amounts to a recursion relation that we can use to start with angular coordinates on the observer’s sky (${\mbox{\boldmath $x$}}_1$) and work our way up in redshift until we reach the source plane (${\mbox{\boldmath $x$}}_s = {\mbox{\boldmath $x$}}_{N+1}$). Some authors [e.g. @Seitz94; @Hilbert09] write the recursion relation in a different form, but we find [equation (\[eq:multi-x\])]{} to be useful.
The Jacobian matrix for the mapping between the coordinates on the sky and the coordinates in plane $j$ is $$\label{eq:multi-A}
{{\textbf{\textsf{A}}}}_{j} \ =\ \frac{\partial{\mbox{\boldmath $x$}}_{j}}{\partial{\mbox{\boldmath $x$}}_1}
\ =\ {{\textbf{\textsf{I}}}}- \sum_{i=1}^{j-1} \beta_{ij} \frac{\partial{\mbox{\boldmath $\alpha$}}_i}{\partial{\mbox{\boldmath $x$}}_i}
\frac{\partial{\mbox{\boldmath $x$}}_i}{\partial{\mbox{\boldmath $x$}}_1}
\ =\ {{\textbf{\textsf{I}}}}- \sum_{i=1}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {{\textbf{\textsf{A}}}}_i\,,$$ where ${{\textbf{\textsf{I}}}}$ is the $2\times2$ identity matrix. The lensing magnification tensor is the inverse of the Jacobian matrix for the source plane: ${{\bf \mu}}= {{\textbf{\textsf{A}}}}_s^{-1}$.
The general form for the multiplane time delay is (see equation 6.22 of @PLW) $$T = \sum_{i=1}^{s-1} \tau_{i\,i+1} \left [ \frac{1}{2} |{\mbox{\boldmath $x$}}_{i+1} - {\mbox{\boldmath $x$}}_i|^2 - \beta_{i\,i+1} \phi_i({\mbox{\boldmath $x$}}_i)\right ],
\label{eq:full_t}$$ where $$\tau_{ij} =\frac{1+z_i}{c} \frac{D_i D_j}{D_{ij}}$$ is a distance combination with dimensions of time. We can omit the redshift dependence if we measure $D_i$, $D_j$, and $D_ij$ as comoving rather than angular diameter distances.
Throughout the derivation we use the following identities from the definitions of $\beta_{ij}$ and $\tau_{ij}$ (see Section 6.4.1 in @PLW): $$\begin{aligned}
\beta_{is} &=& 1 \qquad(\forall i)\,, \label{eq:b_1}\\
\tau_{is} &=& \beta_{ij} \tau_{ij} \qquad(\forall ij)\,, \label{eq:t_b} \\
\frac{1}{\tau_{ik}} &=& \frac{1}{\tau_{ij}} + \frac{1}{\tau_{jk}} \qquad (i<j<k)\, \label{eq:t_ik}.\end{aligned}$$ Also, to simplify the notation we define versions of $\beta$ and $\tau$ with a single subscript as $$\beta_i \equiv \beta_{i\,i+1}\,,
\qquad
\tau_i \equiv \tau_{i\,i+1}\,.$$
Lens Equation and Magnification Tensor {#S:leqn}
======================================
In this section we work with the multiplane lens equation and magnification tensor. We start by using the tidal approximation for all planes other than the plane containing the main lens galaxy. We then generalize to arbitrary combinations of tidal and main planes.
One ‘main’ plane {#S:leqn-1main}
----------------
{width="100.00000%"}
Suppose there is a single ‘main’ lens plane ($i = \ell$) and all other galaxies can be treated with the tidal approximation as illustrated in Fig. \[fig:1plane\]. (This case has been studied previously by @Kovner87 and SEF.) Using [equation (\[eq:shearapprox\])]{}, we can write the recursion relations for the position and Jacobian matrix as $$\begin{aligned}
{\mbox{\boldmath $x$}}_j &=& {\mbox{\boldmath $x$}}_1 - \sum_{i=1,i\ne\ell}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {\mbox{\boldmath $x$}}_i
- \beta_{\ell j} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell)\,, \\
\label{eq:vxj}
{{\textbf{\textsf{A}}}}_j &=& {{\textbf{\textsf{I}}}}- \sum_{i=1,i\ne\ell}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {{\textbf{\textsf{A}}}}_i
- \beta_{\ell j} {{\bf\Gamma}}_\ell({\mbox{\boldmath $x$}}_\ell) {{\textbf{\textsf{A}}}}_\ell\,.
\label{eq:Amatj}\end{aligned}$$ We separate the terms with $i = \ell$ and write ${\mbox{\boldmath $\alpha$}}_\ell$ and ${{\bf\Gamma}}_\ell$ explicitly because we do *not* use the tidal approximation for the main plane.
It is interesting to consider the position ${\mbox{\boldmath $x$}}'_j$ and Jacobian matrix ${{\textbf{\textsf{B}}}}_j$ that we would get if we were to omit the main plane. These quantities must be used with care because they do not include contributions from the main plane (which will be added back in later), but they will prove to be valuable. These modified quantities have the form $$\begin{aligned}
{\mbox{\boldmath $x$}}'_j &=& {\mbox{\boldmath $x$}}_1 - \sum_{i=1,i\ne\ell}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {\mbox{\boldmath $x$}}'_i \,,
\label{eq:vx'j} \\
{{\textbf{\textsf{B}}}}_j &=& {{\textbf{\textsf{I}}}}- \sum_{i=1,i\ne\ell}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {{\textbf{\textsf{B}}}}_i \,.
\label{eq:Bmatdef}\end{aligned}$$ In the foreground of the main lens plane ($j \le \ell$), we clearly have ${\mbox{\boldmath $x$}}'_j = {\mbox{\boldmath $x$}}_j$ and ${{\textbf{\textsf{B}}}}_j = {{\textbf{\textsf{A}}}}_j$ because the trajectory has not yet been affected by the main plane. (Recall that we trace a light ray backwards from the observer.) The situation is different; however, in the background of the main lens plane ($j > \ell$). Taking the difference between [equations (\[eq:Amatj\]) and (\[eq:Bmatdef\])]{}, we have $${{\textbf{\textsf{A}}}}_j-{{\textbf{\textsf{B}}}}_j = - \beta_{\ell j} {{\bf\Gamma}}_\ell {{\textbf{\textsf{A}}}}_\ell
- \sum_{i=\ell+1}^{j-1} \beta_{ij} {{\bf\Gamma}}_i ({{\textbf{\textsf{A}}}}_i-{{\textbf{\textsf{B}}}}_i) \,.$$ Note that the sum now includes only terms with $i > \ell$, because ${{\textbf{\textsf{A}}}}_i-{{\textbf{\textsf{B}}}}_i=0$ for $i \le \ell$. Now if we multiply through by $(- {{\bf\Gamma}}_\ell {{\textbf{\textsf{B}}}}_\ell)^{-1}$ from the right and use the fact that ${{\textbf{\textsf{A}}}}_\ell = {{\textbf{\textsf{B}}}}_\ell$, we obtain $$\begin{aligned}
{{\textbf{\textsf{C}}}}_{\ell j} &\equiv & ({{\textbf{\textsf{A}}}}_j-{{\textbf{\textsf{B}}}}_j) (- {{\bf\Gamma}}_\ell {{\textbf{\textsf{B}}}}_\ell)^{-1} \\
&= & \beta_{\ell j} {{\textbf{\textsf{I}}}}- \sum_{i=\ell+1}^{j-1} \beta_{ij} {{\bf\Gamma}}_i {{\textbf{\textsf{C}}}}_{\ell i} .
\label{eq:Cmatdef}\end{aligned}$$ Equation (\[eq:Cmatdef\]) is a recursion relation for ${{\textbf{\textsf{C}}}}_{\ell j}$ that involves only LOS effects, specifically only planes in between the main plane and plane $j$. In other words, ${{\textbf{\textsf{C}}}}_{\ell j}$ is independent of the main lens. There is, of course, a dependence on the main lens in converting between ${{\textbf{\textsf{C}}}}_{\ell j}$ and ${{\textbf{\textsf{A}}}}_j$ with $${{\textbf{\textsf{A}}}}_j = {{\textbf{\textsf{B}}}}_j - {{\textbf{\textsf{C}}}}_{\ell j} {{\bf\Gamma}}_\ell {{\textbf{\textsf{B}}}}_\ell\,.$$
The matrices ${{\textbf{\textsf{B}}}}_j$ and ${{\textbf{\textsf{C}}}}_{\ell j}$ turn out to have an additional use when we consider the positions. Returning to [equations (\[eq:vxj\]) and (\[eq:vx’j\])]{} and writing out terms, we find that in the tidal approximation we have the simple relation $${\mbox{\boldmath $x$}}'_j = {{\textbf{\textsf{B}}}}_j {\mbox{\boldmath $x$}}_1 \,,$$ for all $j$. In the foreground ($j \le \ell$) we of course have ${\mbox{\boldmath $x$}}_j = {\mbox{\boldmath $x$}}'_j$. In the background ($j > \ell$), the positions ${\mbox{\boldmath $x$}}_j$ and ${\mbox{\boldmath $x$}}'_j$ are different, and in fact we have $${\mbox{\boldmath $x$}}_j \ =\ {\mbox{\boldmath $x$}}'_j - {{\textbf{\textsf{C}}}}_{\ell j} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell)
\ =\ {{\textbf{\textsf{B}}}}_j {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{C}}}}_{\ell j} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) \,.
\label{eq:singlelenseq}$$ Note that the deflection depends on the position in the main lens plane ${\mbox{\boldmath $x$}}_\ell$, not the observed sky plane ${\mbox{\boldmath $x$}}_1$. Therefore tidal effects from foreground planes couple to the deflection from the main lens plane and cannot be mimicked by a standard, linear shear term in the lens plane.[^1] The resulting non-linear effects are important for the multiplane mass sheet degeneracy and for lens modelling (see Sections \[S:masssheet\] and \[S:conclusions\]).
To summarize, in the case of a single main plane plus a collection of planes that can be treated with the tidal approximation, we can separate the full multiplane lensing analysis into pieces that depend only on the LOS (${{\textbf{\textsf{B}}}}_\ell$, ${{\textbf{\textsf{B}}}}_s$, and ${{\textbf{\textsf{C}}}}_{\ell s}$) and pieces that depend on the main lens plane (${\mbox{\boldmath $\alpha$}}_\ell$ and ${{\bf\Gamma}}_\ell$, both of which are evaluated at the position ${\mbox{\boldmath $x$}}_\ell = {{\textbf{\textsf{B}}}}_\ell {\mbox{\boldmath $x$}}_1$). We can combine the pieces into the lens equation and Jacobian matrix as follows: $$\begin{aligned}
{\mbox{\boldmath $x$}}_s &=& {{\textbf{\textsf{B}}}}_s {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{C}}}}_{\ell s} {\mbox{\boldmath $\alpha$}}_\ell({{\textbf{\textsf{B}}}}_\ell {\mbox{\boldmath $x$}}_1) \,,
\label{eq:xfin1} \\
{{\textbf{\textsf{A}}}}_s &=& {{\textbf{\textsf{B}}}}_s - {{\textbf{\textsf{C}}}}_{\ell s} {{\bf\Gamma}}_\ell {{\textbf{\textsf{B}}}}_\ell \,.
\label{eq:Afin1}\end{aligned}$$ This represents a *complete* description of the multiplane lensing in this scenario; there are no approximations involved in the treatment of multiplane lensing itself. The only approximation used here is the tidal approximation for the perturbing galaxies.
The multiplane lens equation (\[eq:xfin1\]) is identical to the quadrupole lens equation in SEF and equivalent to the results from @Kovner87 and @Bar-Kana96. From a formal standpoint, the equation can be made to look like the standard single-plane lens equation through a suitable change of variables [@Bar-Kana96; @Schneider97; @CRKanal]. From a practical standpoint, however, the change of variables is of limited use because lens modelling needs to use *observed* coordinates. (Observed and scaled coordinates can be related to one another only if the transformation matrices are known, in which case one might as well use equation \[eq:xfin1\].)
Small-shear limit {#S:leqn-small}
-----------------
It is instructive to consider the preceding analysis in the limit where all the LOS shears are small. If we make Taylor series expansions and work to linear order in the LOS shears, we obtain $$\begin{aligned}
{{\textbf{\textsf{B}}}}_s &\approx& {{\textbf{\textsf{I}}}}- {{\bf\Gamma}}_{\rm tot}\,, \\
{{\textbf{\textsf{B}}}}_\ell &\approx& {{\textbf{\textsf{I}}}}- {{\tilde{{\bf\Gamma}}}}_{\rm f} \,, \\
{{\textbf{\textsf{C}}}}_{\ell s} &\approx& {{\textbf{\textsf{I}}}}- {{\tilde{{\bf\Gamma}}}}_{\rm b} ,\end{aligned}$$ where $${{\bf\Gamma}}_{\rm tot} = \sum_{i=1,i\neq\ell}^{N} {{\bf\Gamma}}_i
\label{eq:fgbg1}$$ are simple sums of the foreground and background tidal tensors (with uniform weighting), while $$\begin{aligned}
\label{eq:fgbg2}
{{\tilde{{\bf\Gamma}}}}_{\rm f} = \sum_{i=1}^{\ell-1} \beta_{i \ell} {{\bf\Gamma}}_i
\quad\mbox{and}\quad
{{\tilde{{\bf\Gamma}}}}_{\rm b} = \sum_{i=\ell+1}^{N} \beta_{\ell i} {{\bf\Gamma}}_i\end{aligned}$$ are sums where the different planes have different weighting factors $\beta_{i \ell} \ne 1$ and $\beta_{\ell i} \ne 1$. The different weighting factors between ${{\bf\Gamma}}$ and ${{\tilde{{\bf\Gamma}}}}$ will be important for the discussion of the mass sheet degeneracy (Section \[S:masssheet\]). Note that @Wong used ${{\bf\Gamma}}_{\rm tot}$ to characterize environmental effects for observed lenses. The sums above are discretized versions of the integrals used in cosmic shear calculations [e.g. @Munshi08].
Multiple ‘main’ planes {#S:leqn-Nmain}
----------------------
We now extend the framework to allow arbitrary combinations of main planes (which are given full treatment) and tidal planes, illustrated in Fig. \[fig:2plane\]. We do not make any particular assumptions about how the planes are distributed in redshift; there may be 0, 1, or many tidal planes in between any two main planes. As noted above, more than one galaxy may be at a given redshift. Our notation is as follows: Roman letters $(i,j)$ are used to sequentially index all planes (both main and shear). Greek letters $(\mu,\nu)$ are used to sequentially index main planes only. Also, $\ell_\mu$ denotes the Roman index of the main plane $\mu$; in other words, $\{\ell_1,\ell_2,\ldots,\ell_\mu,\ldots\}$ are the indices of the main planes. The source plane counts as a main plane, but with index $s = N+1$.
{width="100.00000%"}
To set the stage, let us re-examine the multiplane lens equations for the case in which all planes are main (equations \[eq:multi-x\] and \[eq:multi-A\]) and the case with a single main plane (equations \[eq:xfin1\] and \[eq:Afin1\]). The first term in each case represents what would happen if the main planes were not present: in [equation (\[eq:multi-x\])]{} this is characterized by the identity matrix because if we remove all planes we are left with no lensing; while in [equation (\[eq:xfin1\])]{} there is distortion from all the tidal planes, which is characterized by the matrix ${{\textbf{\textsf{B}}}}_s$. The terms in the sums represent the combined contributions from the main plane(s) in the foreground of the plane being evaluated. In [equation (\[eq:multi-x\])]{} the light ray experiences no distortions in between planes, so the connecting factor is just a scalar ($\beta_{ij}$) that encodes the relative distances between planes $i$ and $j$. In [equation (\[eq:xfin1\])]{}, by contrast, the light ray may be sheared in between main planes, so the connecting factor becomes a matrix (${{\textbf{\textsf{C}}}}_{\ell s}$) that includes not only the distance factors but also the shears in between the main planes.
We can now understand the form of the lens equations for a general combination of main and tidal planes: $$\begin{aligned}
{\mbox{\boldmath $x$}}_{i} &=& {{\textbf{\textsf{B}}}}_i {\mbox{\boldmath $x$}}_1 - \sum_{\ell \in \{\ell_\mu<i\}} {{\textbf{\textsf{C}}}}_{\ell i} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) \,,
\label{eq:xfinN} \\
{{\textbf{\textsf{A}}}}_{i} &=& {{\textbf{\textsf{B}}}}_i - \sum_{\ell \in \{\ell_\mu<i\}} {{\textbf{\textsf{C}}}}_{\ell i} {{\bf\Gamma}}_\ell {{\textbf{\textsf{A}}}}_\ell \,.
\label{eq:AfinN}\end{aligned}$$ Again note that the deflections depend on the positions in the main planes ${\mbox{\boldmath $x$}}_\ell$. Also, these sums only include main planes. At each step in the recursion, ${\mbox{\boldmath $\alpha$}}_\ell$ and ${{\bf\Gamma}}_\ell$ are to be evaluated at the position ${\mbox{\boldmath $x$}}_\ell$. The matrix ${{\textbf{\textsf{B}}}}_\ell$ represents the net effects of the tidal planes in between the observer and the main plane with index $\ell$, which can be found recursively as follows: $${{\textbf{\textsf{B}}}}_j = {{\textbf{\textsf{I}}}}- \sum\limits_{i=1,i\not \in \{\ell_\mu\}}^{j-1}\beta_{ij}{{\bf\Gamma}}_i {{\textbf{\textsf{B}}}}_i$$ where this sum does not include any of the main planes (even if they happen to lie between the observer and plane $j$). The matrix ${{\textbf{\textsf{C}}}}_{\ell j}$ represents the net effects of the tidal planes in between the main plane $\ell$ and plane $j$ whose recursion relation is $${{\textbf{\textsf{C}}}}_{\ell j} = \beta_{\ell j} {{\textbf{\textsf{I}}}}- \sum\limits_{i=\ell+1, i\not\in\{\ell_\mu\}}^{j-1} \beta_{i j}{{\bf\Gamma}}_i{{\textbf{\textsf{C}}}}_{\ell i},$$ where again this sum only includes tidal planes. Note that ${{\textbf{\textsf{B}}}}_j$ and ${{\textbf{\textsf{C}}}}_{\ell j}$ are defined for arbitrary $j$, but [equations (\[eq:xfinN\]) and (\[eq:AfinN\])]{} show that only the matrices associated with main planes need to be stored for later use. The benefit of this approach for lens modelling is that the bulk of the computational effort goes into determining ${{\textbf{\textsf{B}}}}_{\ell_\mu}$ and ${{\textbf{\textsf{C}}}}_{\ell_\mu \ell_\nu}$, but that step needs to be done only once. Once those matrices are stored, the mass model in the main plane(s) can be varied without having to recompute the full LOS.
Time Delay {#S:tdel}
==========
We now turn to time delays. As before, we start with a single main plane plus a collection of tidal planes, and then generalize to an arbitrary combination of tidal and main planes. To set the context, it is useful to recall the classic expression for the time delay in single-plane lensing. The single-plane time delay can be written in several different forms, the most familiar of which is $$T \propto \frac{1}{2} |{\mbox{\boldmath $x$}}-{\mbox{\boldmath $x$}}_s|^2 - \phi({\mbox{\boldmath $x$}})\,.$$ We can expand the quadratic term as $$T \propto \frac{1}{2} ( {\mbox{\boldmath $x$}}^2 - {\mbox{\boldmath $x$}}\cdot {\mbox{\boldmath $x$}}_s - {\mbox{\boldmath $x$}}_s \cdot {\mbox{\boldmath $x$}}+ {\mbox{\boldmath $x$}}^2_s) - \phi({\mbox{\boldmath $x$}})\,. \label{eq:t_quad}$$ In terms of the deflection angle ${\mbox{\boldmath $\alpha$}}$, we can rewrite this as $$T \propto \frac{1}{2} | {\mbox{\boldmath $\alpha$}}|^2 - \phi({\mbox{\boldmath $x$}})\,.$$ We can even mix these two forms giving $$T \propto \frac{1}{2} ({\mbox{\boldmath $x$}}-{\mbox{\boldmath $x$}}_s)\cdot {\mbox{\boldmath $\alpha$}}- \phi({\mbox{\boldmath $x$}})\,.
\label{eq:t_alpha}$$ While these forms may look rather distinct, they are all equivalent. We will see below how the different forms are useful.
Single ‘main’ plane {#S:tdel-1main}
-------------------
The general expression for the multiplane time delay depends explicitly on *all* of the ${\mbox{\boldmath $x$}}_j$ and $\phi_j$. Our goal is to write the time delay in terms of $({\mbox{\boldmath $x$}}, {\mbox{\boldmath $x$}}_s, \phi_\ell)$ or equivalently $({\mbox{\boldmath $x$}}, {\mbox{\boldmath $\alpha$}}_\ell,\phi_\ell)$. To that end, we substitute for the position coordinates, explicitly separate out the main plane lens potential, and implement the tidal approximation for all other planes ($\phi_j\approx \frac{1}{2} {\mbox{\boldmath $x$}}_j \cdot{{\bf\Gamma}}_j{\mbox{\boldmath $x$}}_j$). This yields $$T =\sum\limits_{i=1}^{s-1} \frac{1}{2} \tau_{i} \left[ {\mbox{\boldmath $x$}}_{i+1} - {\mbox{\boldmath $x$}}_i \right]^2 -\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell}) - \sum\limits_{i=1,i\neq\ell}^{s-1} \frac{1}{2} \tau_{i} \beta_{i} {\mbox{\boldmath $x$}}_i\cdot{{\bf\Gamma}}_i{\mbox{\boldmath $x$}}_i .
\label{eq:t1}$$ We would like to eliminate ${{\bf\Gamma}}_i$, so it is necessary to digress to derive a few useful identities. We start by examining $${{\textbf{\textsf{B}}}}_{j+1}-{{\textbf{\textsf{B}}}}_j = - \beta_{j} {{\bf\Gamma}}_j{{\textbf{\textsf{B}}}}_j - \sum\limits_{i\neq\ell}^{j-1} (\beta_{i\,j+1}-\beta_{ij}){{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i,$$ and $${{\textbf{\textsf{B}}}}_{j}-{{\textbf{\textsf{B}}}}_{j-1} = - \beta_{j-1} {{\bf\Gamma}}_{j-1}{{\textbf{\textsf{B}}}}_{j-1} - \sum\limits_{i\neq\ell}^{j-2} (\beta_{ij}-\beta_{i\,j-1}){{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i.
\label{eq:a-}$$ Combining these and using [equation (\[eq:t\_b\])]{}, we can cancel the sum to obtain (see also @Seitz94) $${{\textbf{\textsf{B}}}}_{j+1} = \left[\left(1 +\frac{\tau_{j-1}}{\tau_{j}}\right){{\textbf{\textsf{I}}}}-\beta_{j} {{\bf\Gamma}}_j \right] {{\textbf{\textsf{B}}}}_j - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{B}}}}_{j-1}.
\label{eq:Bjidentity}$$ Rearranging, we can solve for ${{\bf\Gamma}}_j$: $$\beta_{j} {{\bf\Gamma}}_j = \left(1 +\frac{\tau_{j-1}}{\tau_{j}}\right){{\textbf{\textsf{I}}}}- {{\textbf{\textsf{B}}}}_{j+1} {{\textbf{\textsf{B}}}}^{-1}_{j} - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{B}}}}_{j-1}{{\textbf{\textsf{B}}}}^{-1}_{j}.
\label{eq:b_g}$$ Following the same procedure yields a similar result for ${{\textbf{\textsf{C}}}}_{\ell j}$: $$\beta_{j} {{\bf\Gamma}}_j = \left(1 +\frac{\tau_{j-1}}{\tau_{j}}\right){{\textbf{\textsf{I}}}}- {{\textbf{\textsf{C}}}}_{\ell \,j+1} {{\textbf{\textsf{C}}}}^{-1}_{\ell j} - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{C}}}}_{\ell \,j-1}{{\textbf{\textsf{C}}}}^{-1}_{\ell j}.
\label{eq:c_g}$$ We now multiply the lens equation (\[eq:singlelenseq\]) by $\beta_{j} {{\bf\Gamma}}_j$ from the left, and use [equation (\[eq:b\_g\])]{} when ${{\bf\Gamma}}_i$ multiplies ${{\textbf{\textsf{B}}}}_i$ and [equation (\[eq:c\_g\])]{} when ${{\bf\Gamma}}$ multiplies ${{\textbf{\textsf{C}}}}_{\ell j}$ yields $$\begin{gathered}
\beta_j {{\bf\Gamma}}_j{\mbox{\boldmath $x$}}_j = \left[\left(1 + \frac{\tau_{j-1}}{\tau_j}\right) {{\textbf{\textsf{B}}}}_j - {{\textbf{\textsf{B}}}}_{j+1} - \frac{\tau_{j-1}}{\tau_j}{{\textbf{\textsf{B}}}}_{j-1} \right] {\mbox{\boldmath $x$}}_1 \\
- \left[\left(1 + \frac{\tau_{j-1}}{\tau_j}\right) {{\textbf{\textsf{C}}}}_{\ell j} - {{\textbf{\textsf{C}}}}_{\ell\,j+1} - \frac{\tau_{j-1}}{\tau_j}{{\textbf{\textsf{C}}}}_{\ell\,j-1} \right]{\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell).\end{gathered}$$ Collecting terms and again using [equation (\[eq:singlelenseq\])]{} yields $$\beta_j{{\bf\Gamma}}_j {\mbox{\boldmath $x$}}_j = \left(1+\frac{\tau_{j-1}}{\tau_{j}}\right){\mbox{\boldmath $x$}}_j - {\mbox{\boldmath $x$}}_{j+1} - \frac{\tau_{j-1}}{\tau_{j}} {\mbox{\boldmath $x$}}_{j-1}.
\label{eq:x_g}$$ Using this relation in [equation (\[eq:t1\])]{} gives $$\begin{gathered}
T = \sum\limits_{i=1}^{s-1} \frac{1}{2} \tau_{i} \left[ {\mbox{\boldmath $x$}}_{i+1}^2 - 2 {\mbox{\boldmath $x$}}_{i+1} \cdot {\mbox{\boldmath $x$}}_{i} + {\mbox{\boldmath $x$}}_i^2 \right] -\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell})
\\
- \sum\limits_{i=1,i\neq\ell}^{s-1} \frac{1}{2} \tau_{i} {\mbox{\boldmath $x$}}_i \cdot\left[\left(1+\frac{\tau_{i-1}}{\tau_{i}}\right){\mbox{\boldmath $x$}}_i - {\mbox{\boldmath $x$}}_{i+1} - \frac{\tau_{i-1}}{\tau_{i}} {\mbox{\boldmath $x$}}_{i -1} \right] .
\label{eq:t_sum}\end{gathered}$$ The identity term in the second sum is identical to the second quadratic term in the first sum but with opposite sign. Also, the first $i,i+1$ cross term in the first sum matches the $i,i+1$ cross term in the second sum. These terms cancel except for the main plane term $j=\ell$ that we explicitly removed from the second sum.
The other terms in the first sum have the same form as the remaining terms in the second sum, but with indices decremented by 1. We therefore reindex the remaining terms in the second sum with $i \rightarrow i+1$. These terms become $$\sum\limits^{s-1}_{i=1,i\neq\ell} \tau_{i-1} {\mbox{\boldmath $x$}}_{i}^2 \rightarrow \sum\limits^{s-2}_{i=0,i\neq \ell-1} \tau_{i} {\mbox{\boldmath $x$}}_{i+1}^2$$ and $$\sum\limits^{s-1}_{i=1,i\neq\ell} \tau_{i-1} {\mbox{\boldmath $x$}}_{i} \cdot {\mbox{\boldmath $x$}}_{i-1}. \rightarrow \sum\limits^{s-2}_{i=0,i\neq \ell-1} \tau_{i} {\mbox{\boldmath $x$}}_{i+1} \cdot {\mbox{\boldmath $x$}}_i.$$ These match the terms in the first sum but have opposite sign and therefore all of the sums cancel. The only surviving terms are $s-1$ and $\ell -1$ terms from removing the main plane and reindexing. There is also an $i=0$ term from the second reindexed sum. This term would have $\tau_{0,1}$ as a coefficient. Taking the zero plane to be the observer, we have $D_0=0$ and therefore $\tau_{0,1}=0$. This leaves us with $$\begin{gathered}
T = \frac{1}{2} \left[ \tau_{\ell} {\mbox{\boldmath $x$}}_\ell \cdot \left({\mbox{\boldmath $x$}}_\ell - {\mbox{\boldmath $x$}}_{\ell +1} \right) + \tau_{\ell-1} {\mbox{\boldmath $x$}}_\ell\cdot \left({\mbox{\boldmath $x$}}_\ell - {\mbox{\boldmath $x$}}_{\ell-1} \right) \right.\\
\left.+ \tau_{s-1} {\mbox{\boldmath $x$}}_s \cdot \left({\mbox{\boldmath $x$}}_s - {\mbox{\boldmath $x$}}_{s-1} \right) \right] -\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell}).
\label{eq:Txdiff}\end{gathered}$$
The terms in [equation (\[eq:Txdiff\])]{} are of the form ${\mbox{\boldmath $x$}}_j - {\mbox{\boldmath $x$}}_{j-1}$. To handle these terms, we need some additional technical results. Consider the difference ${{\textbf{\textsf{B}}}}_j-{{\textbf{\textsf{B}}}}_{j-1}$. Multiplying [equation (\[eq:a-\])]{} through by $\tau_{j-1}$ gives $$\begin{aligned}
\tau_{j-1} {{\textbf{\textsf{B}}}}_{j}-\tau_{j-1} {{\textbf{\textsf{B}}}}_{j-1}&=-\tau_{j-1\,s}{{\bf\Gamma}}_{j-1}{{\textbf{\textsf{B}}}}_{j-1} - \sum\limits_{i\neq\ell}^{j-2} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i \\
&=- \sum\limits_{i\neq\ell}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i.
\end{aligned}$$ It is therefore convenient to define a new set of matrices: $${{\textbf{\textsf{F}}}}_j\equiv \tau_{j-1} {{\textbf{\textsf{B}}}}_{j}-\tau_{j-1} {{\textbf{\textsf{B}}}}_{j-1} = - \sum\limits_{i\neq\ell}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i
\label{eq:Fmatdef}$$ and similarly for ${{\textbf{\textsf{C}}}}_{\ell j}$, $${{\textbf{\textsf{G}}}}_{\ell j}\equiv \tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell j}-\tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell\,j-1} = \tau_{\ell s}{{\textbf{\textsf{I}}}}- \sum\limits_{i=\ell+1}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{C}}}}_{\ell i}.
\label{eq:Gmatdef}$$ Both the ${{\textbf{\textsf{F}}}}_j$ and ${{\textbf{\textsf{G}}}}_{\ell j}$ matrices have dimensions of time. Therefore terms in the time delay that include these matrices will not include an explicit $\tau_{ij}$ as a coefficient.
Using these relations along with the lens equation \[[equation (\[eq:singlelenseq\])]{}\], we find $$\tau_{j-1} \left({\mbox{\boldmath $x$}}_j - {\mbox{\boldmath $x$}}_{j-1}\right) = {{\textbf{\textsf{F}}}}_j {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{G}}}}_{\ell j} {{\textbf{\textsf{C}}}}_{\ell s}^{-1} \left({{\textbf{\textsf{B}}}}_s {\mbox{\boldmath $x$}}_1 - {\mbox{\boldmath $x$}}_s \right).
\label{eq:diffx}$$ Substituting this into [equation (\[eq:t\_sum\])]{}, we have $$\begin{gathered}
T=\frac{1}{2} {\mbox{\boldmath $x$}}_\ell \cdot \left [ ({{\textbf{\textsf{F}}}}_\ell - {{\textbf{\textsf{F}}}}_{\ell+1}) {\mbox{\boldmath $x$}}_1 - ({{\textbf{\textsf{G}}}}_{\ell \ell} - {{\textbf{\textsf{G}}}}_{\ell \,\ell+1}) {{\textbf{\textsf{C}}}}^{-1}_{\ell s} \left({{\textbf{\textsf{B}}}}_s{\mbox{\boldmath $x$}}_1-{\mbox{\boldmath $x$}}_s \right) \right ] \\
+ \frac{1}{2}{\mbox{\boldmath $x$}}_s \cdot \left[{{\textbf{\textsf{F}}}}_s {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{G}}}}_{\ell s} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} ({{\textbf{\textsf{B}}}}_s {\mbox{\boldmath $x$}}_1- {\mbox{\boldmath $x$}}_s)\right] -\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell}).\end{gathered}$$ Note that the ${{\textbf{\textsf{F}}}}_{j}$ do not include the main plane, so ${{\textbf{\textsf{F}}}}_\ell = {{\textbf{\textsf{F}}}}_{\ell+1}$. Also, as the ${{\textbf{\textsf{G}}}}_{\ell j}$ only include the background planes, ${{\textbf{\textsf{G}}}}_{\ell \ell}=0$ and ${{\textbf{\textsf{G}}}}_{\ell\,\ell+1} = \tau_{\ell s}$. With these simplifications, we have our final expression: $$\begin{gathered}
T=\frac{1}{2}\tau_{\ell s}{{\textbf{\textsf{B}}}}_{\ell} {\mbox{\boldmath $x$}}_1 \cdot {{\textbf{\textsf{C}}}}^{-1}_{\ell s} ({{\textbf{\textsf{B}}}}_s {\mbox{\boldmath $x$}}_1 - {\mbox{\boldmath $x$}}_s) -\tau_{\ell s} \phi_\ell({{\textbf{\textsf{B}}}}_\ell{\mbox{\boldmath $x$}}_1)\\
+ \frac{1}{2}{\mbox{\boldmath $x$}}_s\cdot ( {{\textbf{\textsf{F}}}}_s {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{G}}}}_{\ell s} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s{\mbox{\boldmath $x$}}_1 + {{\textbf{\textsf{G}}}}_{\ell s} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {\mbox{\boldmath $x$}}_s) .
\label{eq:single_sch_compare}\end{gathered}$$ This form is most like [equation (\[eq:t\_quad\])]{} and will be useful to compare to previous calculations with a single main plane. We can rewrite the result in an equivalent form that more closely resembles [equation (\[eq:t\_alpha\])]{} by reordering terms and substituting for ${\mbox{\boldmath $\alpha$}}_\ell$ and ${\mbox{\boldmath $x$}}_\ell$: $$T=\frac{1}{2}\bigl[ {\mbox{\boldmath $x$}}_s \cdot {{\textbf{\textsf{F}}}}_s {\mbox{\boldmath $x$}}_1 + \tau_{\ell s}{\mbox{\boldmath $x$}}_\ell \cdot {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) - {\mbox{\boldmath $x$}}_s \cdot {{\textbf{\textsf{G}}}}_{\ell s} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) \bigr] - \tau_{\ell s}\phi_\ell({\mbox{\boldmath $x$}}_\ell).
\label{eq:single_final}$$ This is the form that we will compare to our final results for multiple main planes (below).
We note that SEF previously derived an expression for the time delay in the case of a single main plane with multiple tidal planes. The analyses are complementary, because our approach is algebraic (we start with the general expression for the multiplane time delay and manipulate the expression to look for simplifications), whereas the approach in SEF is based on solving a partial differential equation. By Fermat’s principle, setting the derivative of the time delay equal to zero should give the lens equation. In the case of a single main plane, the only independent variable is the position in that plane (the positions in all of the tidal planes can be written in terms of ${\mbox{\boldmath $x$}}_\ell$). Therefore, the time delay must have $$\frac{\partial T}{\partial {\mbox{\boldmath $x$}}_\ell} \propto {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s{{\textbf{\textsf{B}}}}^{-1}_{\ell} {\mbox{\boldmath $x$}}_\ell - {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {\mbox{\boldmath $x$}}_s - \frac{\partial \phi_\ell}{\partial {\mbox{\boldmath $x$}}_\ell}$$ The right-hand side is linear in ${\mbox{\boldmath $x$}}_\ell$, so $T$ must be quadratic, and SEF find (in our notation) $$\begin{gathered}
T \propto \frac{1}{2} {{\textbf{\textsf{B}}}}_{\ell} {\mbox{\boldmath $x$}}_1 \cdot {{\textbf{\textsf{C}}}}^{-1}_{\ell s} ({{\textbf{\textsf{B}}}}_s {\mbox{\boldmath $x$}}_1 - {\mbox{\boldmath $x$}}_s) - \frac{1}{2} {\mbox{\boldmath $x$}}_{s} \cdot ({{\textbf{\textsf{C}}}}^{-1}_{\ell s})^{{{\mathrm{T}}}} ({{\textbf{\textsf{B}}}}_\ell {\mbox{\boldmath $x$}}_1 - {{\textbf{\textsf{B}}}}_\ell{{\textbf{\textsf{B}}}}^{-1}_s {\mbox{\boldmath $x$}}_s)\\
- \phi_\ell({\mbox{\boldmath $x$}}_\ell) + \mbox{const}.
\label{eq:sch}\end{gathered}$$ Applying Fermat’s principle does not specify the proportionality factor or an additive ‘constant’ (really, any term that is independent of ${\mbox{\boldmath $x$}}_\ell$). Comparing [equations (\[eq:single\_sch\_compare\]) and (\[eq:sch\])]{}, we see that the proportionality constant is $\tau_{\ell s}$ [which is not surprising; also see @Schneider97]. The terms in the first set of parentheses are identical to our solution. In the second set of parentheses, the first term in [equation (\[eq:sch\])]{} is equivalent to the corresponding term in our expression if the following identity holds: $$\tau_{\ell s} ({{\textbf{\textsf{C}}}}^{-1}_{\ell s})^{{{\mathrm{T}}}} {{\textbf{\textsf{B}}}}_\ell = {{\textbf{\textsf{G}}}}_{\ell s} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s - {{\textbf{\textsf{F}}}}_s.
\label{eq:identity}$$ The proof of this identity is given in Appendix \[appendix\]. We note that the second term in the second set of parentheses in [equation (\[eq:sch\])]{} is not equivalent to the corresponding term in our expression, but the difference term is quadratic in ${\mbox{\boldmath $x$}}_s$ and independent of ${\mbox{\boldmath $x$}}_\ell$ so it is part of the ‘const’ term in [equation (\[eq:sch\])]{}. Such a term does not affect *differential* time delays, which are the observables of interest.
Thus, we conclude that our expression is equivalent to that given by SEF, at least for differential time delays. We acknowledge that our algebraic approach is more complicated than the Fermat principle argument used by SEF, at least for the case of a single main plane. However, for multiple main planes, the Fermat principle approach would require solving an arbitrarily large system of coupled partial differential equations, while our algebraic approach is easily generalized, as we are about to see. Also, our algebraic approach can pin down terms that are quadratic in ${\mbox{\boldmath $x$}}_s$, which may be of formal interest even if they are unimportant for observable time delays.
Multiple ‘main’ planes {#S:tdel-Nmain}
----------------------
We now extend this analysis to an arbitrary combination of main planes and tidal planes. We again do not make any assumptions about the redshift distributions of the planes or how the planes are ordered. As above, we denote the index of main planes as $\ell \in \{\ell_1,\ell_2,\ldots,\ell_\mu,\ldots\}$. We begin with the lens equation (\[eq:xfinN\]) for multiple main planes. We substitute this expression into the full time delay expression, [equation (\[eq:full\_t\])]{}, and separate the main plane indices: $$\begin{gathered}
T = \sum\limits_{i=1}^{s-1} \frac{1}{2} \tau_{i} \left[{\mbox{\boldmath $x$}}_{i+1} -{\mbox{\boldmath $x$}}_i \right]^2
- \sum\limits_{i=1,i\not\in\{\ell_\mu\}}^{s-1} \frac{1}{2} \tau_{i} \beta_{i} {\mbox{\boldmath $x$}}_i \cdot {{\bf\Gamma}}_i {\mbox{\boldmath $x$}}_{i}\\ -\sum\limits_{\ell\in\{\ell_\mu\}}\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell}).\end{gathered}$$ As ${{\textbf{\textsf{B}}}}_j$ and ${{\textbf{\textsf{C}}}}_{\ell j}$ only depend on the tidal planes, the relationships between these matrices and ${{\bf\Gamma}}_j$, [equations (\[eq:b\_g\]) and (\[eq:c\_g\])]{}, still hold in the case of multiple main planes. It is useful to point out that [equation (\[eq:c\_g\])]{} generalizes to each $\ell \in \{\ell_\mu\}$. Using these relations and expanding the quadratic terms, analogous to [equation (\[eq:t\_sum\])]{}, we can rewrite the time delay as $$\begin{gathered}
T = \sum\limits_{i=1}^{s-1} \frac{1}{2} \tau_{i} \left[ {\mbox{\boldmath $x$}}_{i+1}^2 - 2 {\mbox{\boldmath $x$}}_{i+1}\cdot {\mbox{\boldmath $x$}}_i + {\mbox{\boldmath $x$}}_i^2 \right] - \sum\limits_{\ell\in\{\ell_\mu\}}\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell})\\- \sum\limits_{i=1,i\not\in\{\ell_\mu\}}^{s-1} \frac{1}{2} \tau_{i} {\mbox{\boldmath $x$}}_i \cdot \left[ \left(1-\frac{\tau_{i-1}}{\tau_{i}} \right ) {\mbox{\boldmath $x$}}_i - {\mbox{\boldmath $x$}}_{i+1} - \frac{\tau_{i-1}}{\tau_{i}}{\mbox{\boldmath $x$}}_{i-1} \right].\end{gathered}$$ As in the single-plane case, the identity term in the second term matches the second quadratic term in the first sum. These cancel, leaving only the main planes from the first sum. Again, the $i,i+1$ cross term in the second sum cancels one of the cross terms in the first sum, leaving only the main plane terms. As in the single-plane case, we see that the remaining terms are identical but that the indices in the second sum are decremented by 1. We reindex the sums with $i\rightarrow i+1$: $$\sum\limits_{i=1,i\not\in\{\ell_\mu\}}^{s-1} \tau_{i-1} {\mbox{\boldmath $x$}}_{i}^2 \rightarrow \sum\limits_{i=0,i+1\not\in\{\ell_\mu\}}^{s-2} \tau_{i} {\mbox{\boldmath $x$}}_{i+1}^2$$ and $$\sum\limits_{i=1,i\not\in\{\ell_\mu\}}^{s-1} \tau_{i-1}{\mbox{\boldmath $x$}}_{i}\cdot{\mbox{\boldmath $x$}}_{i-1} \rightarrow \sum\limits_{i=0,i+1\not\in\{\ell_\mu\}}^{s-2} \tau_{i}{\mbox{\boldmath $x$}}_{i+1}\cdot{\mbox{\boldmath $x$}}_i.$$ These terms now cancel in the sums leaving only the $s-1$ and the set of $\{\ell_\mu -1\}$ terms.
We are left with $$\begin{gathered}
T = \tau_{s-1} {\mbox{\boldmath $x$}}_s \cdot ({\mbox{\boldmath $x$}}_s - {\mbox{\boldmath $x$}}_{s-1}) + -\sum\limits_{\ell\in\{\ell_\mu\}}\tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell})\\
+ \frac{1}{2} \sum\limits_{\ell \in \{\ell_\mu\}} \tau_{\ell}{\mbox{\boldmath $x$}}_\ell\cdot({\mbox{\boldmath $x$}}_\ell - {\mbox{\boldmath $x$}}_{\ell+1}) + \tau_{\ell-1}{\mbox{\boldmath $x$}}_\ell\cdot({\mbox{\boldmath $x$}}_\ell - {\mbox{\boldmath $x$}}_{\ell-1}).
\label{eq:TNtmp}\end{gathered}$$ The analysis proceeds as it did after [equation (\[eq:Txdiff\])]{}. Our expressions for ${{\textbf{\textsf{F}}}}_j$ and ${{\textbf{\textsf{G}}}}_{\ell j}$ remain basically unchanged except that $\ell$ becomes a free index that runs over the main planes: $${{\textbf{\textsf{F}}}}_j\equiv \tau_{j-1} {{\textbf{\textsf{B}}}}_{j}-\tau_{j-1} {{\textbf{\textsf{B}}}}_{j-1} = - \sum\limits_{i=1,i\not \in \{\ell_\mu\}}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i
\label{eq:Fdef}$$ and $${{\textbf{\textsf{G}}}}_{\ell j}\equiv \tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell j}-\tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell \, j-1} = \tau_{\ell s}{{\textbf{\textsf{I}}}}- \sum\limits_{i=\ell+1,i \not \in \{\ell_\mu\}}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{C}}}}_{\ell i}.
\label{eq:Gdef}$$ The identity in [equation (\[eq:diffx\])]{} generalizes to $$\tau_{j-1} ({\mbox{\boldmath $x$}}_j - {\mbox{\boldmath $x$}}_{j-1}) = {{\textbf{\textsf{F}}}}_j {\mbox{\boldmath $x$}}_1 - \sum\limits_{\ell \in \{ \ell_\mu < j\}} {{\textbf{\textsf{G}}}}_{\ell j} {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell).$$ Substituting this into [equation (\[eq:TNtmp\])]{} yields $$\begin{gathered}
T = \frac{1}{2} {\mbox{\boldmath $x$}}_s \cdot\left[{{\textbf{\textsf{F}}}}_s {\mbox{\boldmath $x$}}_1 - \sum\limits_{\ell \in \{\ell_\mu\}} {{\textbf{\textsf{G}}}}_{\ell s} {\mbox{\boldmath $\alpha$}}_\ell\right] \\
+ \sum\limits_{\ell \in \{\ell_\mu\}}\left\{ \frac{1}{2} {\mbox{\boldmath $x$}}_\ell \cdot \left [ {{\textbf{\textsf{F}}}}_\ell {\mbox{\boldmath $x$}}_1 - \sum\limits^{\ell' < \ell}_{\ell' \in \{\ell_\mu\}} {{\textbf{\textsf{G}}}}_{\ell' \ell} {\mbox{\boldmath $\alpha$}}_{\ell'} \right.\right.\\
\left.\left.+ {{\textbf{\textsf{F}}}}_{\ell+1} {\mbox{\boldmath $x$}}_1 - \sum\limits^{\ell' < \ell+1}_{\ell' \in \{\ell_\mu\}} {{\textbf{\textsf{G}}}}_{\ell' \, \ell+1} {\mbox{\boldmath $\alpha$}}_{\ell'}\right] - \tau_{\ell s} \phi_\ell({\mbox{\boldmath $x$}}_{\ell}) \right\}.\end{gathered}$$ Recall that ${{\textbf{\textsf{F}}}}_j$ and ${{\textbf{\textsf{G}}}}_{\ell j}$ are both independent of main planes so ${{\textbf{\textsf{F}}}}_\ell = {{\textbf{\textsf{F}}}}_{\ell+1}$ and ${{\textbf{\textsf{G}}}}_{\nu \ell} = {{\textbf{\textsf{G}}}}_{\nu\,\ell+1}$. Therefore, as before, the ${{\textbf{\textsf{F}}}}_j$ and ${{\textbf{\textsf{G}}}}_{\ell j}$ terms cancel. There is an important subtlety here, however. The second sum with the ${{\textbf{\textsf{G}}}}_{\nu\,\ell+1}$ includes one more main plane than the previous corresponding sum, namely ${{\textbf{\textsf{G}}}}_{\ell\,\ell+1}{\mbox{\boldmath $\alpha$}}_\ell = \tau_{\ell s} {\mbox{\boldmath $\alpha$}}_\ell$. We also substitute ${\mbox{\boldmath $x$}}_\ell$ and ${\mbox{\boldmath $x$}}_s$ from the lens [equation (\[eq:xfinN\])]{}, finally giving us $$\begin{gathered}
T= \frac{1}{2} {\mbox{\boldmath $x$}}_s \cdot {{\textbf{\textsf{F}}}}_s {\mbox{\boldmath $x$}}_1 + \sum\limits_{\ell \in \{\ell_\mu\}}\left[\frac{1}{2}\tau_{\ell s} {\mbox{\boldmath $x$}}_\ell \cdot {\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) \right. \\
\left.-\frac{1}{2}{\mbox{\boldmath $x$}}_s \cdot {{\textbf{\textsf{G}}}}_{\ell s}{\mbox{\boldmath $\alpha$}}_\ell({\mbox{\boldmath $x$}}_\ell) -\tau_{\ell s}\phi_\ell({\mbox{\boldmath $x$}}_\ell) \right].
\label{eq:TfinN}\end{gathered}$$ This result immediately becomes the single main plane time delay, [equation (\[eq:single\_final\])]{}, by dropping the sum over main planes.
In practice, we can tabulate all of the LOS effects by calculating all of the ${{\textbf{\textsf{B}}}}_j$, ${{\textbf{\textsf{C}}}}_{\ell j}$, ${{\textbf{\textsf{F}}}}_j$, and ${{\textbf{\textsf{G}}}}_{\ell j}$ matrices. The benefit of this approach is that all of the LOS calculations can be done up front and performed only once. We can save these matrices and then vary the main plane potentials without ever having to recalculate the full LOS.
Mass Sheet Degeneracy {#S:masssheet}
=====================
For traditional, single-plane lensing, @Falco85 showed that certain transformations of the lens potential leave the image positions and flux ratios unchanged. One notable transformation is the ‘mass sheet degeneracy.’ In the single-plane case, the lens equation has the form $${\mbox{\boldmath $x$}}_s = {\mbox{\boldmath $x$}}- \nabla\phi({\mbox{\boldmath $x$}}).
\label{eq:leqn1}$$ If we apply the transformation $$\phi({\mbox{\boldmath $x$}}) \rightarrow (1 - \kappa) \phi({\mbox{\boldmath $x$}}) + \frac{\kappa}{2} {\mbox{\boldmath $x$}}^2
\label{eq:phitran}$$ the entire right-hand side of [equation (\[eq:leqn1\])]{} gets multiplied by $(1-\kappa)$. Because the source position is unobservable, we can define a rescaled source coordinate $(1-\kappa){\mbox{\boldmath $y$}}={\mbox{\boldmath $x$}}_s$ and then write the transformed lens equation as $$(1- \kappa) {\mbox{\boldmath $y$}}= (1- \kappa) {\mbox{\boldmath $x$}}- (1- \kappa)\nabla\phi({\mbox{\boldmath $x$}}),$$ The $(1-\kappa)$ factors cancel, so the transformed equation is formally equivalent to the original. A similar cancellation occurs for the fluxes if we rescale the source flux, which is permitted if the intrinsic flux of the source is unknown and constraints come from flux ratios rather than absolute fluxes.[^2] Time delays are different, however. The transformation (\[eq:phitran\]) causes differential time delays to be rescaled by $$\Delta T' = (1-\kappa) \Delta T,$$ which is important when using time delays to constrain the Hubble constant [e.g. @Fadely10; @Suyu10; @Suyu13]. Overall, the mass sheet degeneracy can be viewed as a type of gauge invariance analogous to what is seen with potentials in electricity and magnetism.
Before proceeding to the multiplane case, it is useful to examine a case with external convergence and shear in the lens plane. We can write the potential as $$\phi({\mbox{\boldmath $x$}}) = \phi_{\mathrm{g}}({\mbox{\boldmath $x$}}) + \frac{1}{2} {\mbox{\boldmath $x$}}\cdot {{\bf\Gamma}}{\mbox{\boldmath $x$}}$$ where $\phi_{\mathrm{g}}({\mbox{\boldmath $x$}})$ is the potential due to the main galaxy. The mass sheet degeneracy still applies to this situation, but the transformation is slightly different: $$\phi_{\mathrm{g}}({\mbox{\boldmath $x$}}) \rightarrow (1 - \kappa) \phi({\mbox{\boldmath $x$}}) + \frac{\kappa}{2} {\mbox{\boldmath $x$}}\cdot \left({{\textbf{\textsf{I}}}}- {{\bf\Gamma}}\right) {\mbox{\boldmath $x$}}.
\label{eq:sheartrans}$$ This form of the mass sheet degeneracy produces the same rescaling of observables as before.
We have found a similar gauge symmetry for the case of a single main plane with an arbitrary collection of tidal planes along the LOS. If we start with the lens equation (\[eq:xfin1\]) and make the transformation $$\phi({\mbox{\boldmath $x$}}_\ell) \rightarrow (1-\kappa)\phi({\mbox{\boldmath $x$}}_\ell) +\frac{\kappa}{2}{\mbox{\boldmath $x$}}_\ell \cdot {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s {{\textbf{\textsf{B}}}}_\ell^{-1} {\mbox{\boldmath $x$}}_\ell
\label{eq:phi_singlemain}$$ we find that the observables scale in the same way as the original mass sheet degeneracy. The form of this transformation is reminiscent of [equation (\[eq:sheartrans\])]{}, so we define an ‘effective’ tidal tensor by $${{\bf\Gamma}}_{\rm{eff}} \equiv {{\textbf{\textsf{I}}}}- {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s {{\textbf{\textsf{B}}}}_\ell^{-1}.
\label{eq:Geff}$$ To build some intuition about this quantity, it is useful to examine the small-shear limit. Substituting expressions from [Section \[S:leqn-small\]]{} yields $${{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s {{\textbf{\textsf{B}}}}_\ell^{-1} \approx ( {{\textbf{\textsf{I}}}}- {{\tilde{{\bf\Gamma}}}}_{\rm b} )^{-1} ( {{\textbf{\textsf{I}}}}- {{\bf\Gamma}}_{\rm tot}) ({{\textbf{\textsf{I}}}}- {{\tilde{{\bf\Gamma}}}}_{\rm f})^{-1}.$$ If we make the additional, stronger assumption that the sums over tidal planes are also small, we can further simplify this expression. Using a Taylor series expansion of the inverses and keeping only the first-order terms in ${{\bf\Gamma}}$’s, we obtain $${{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s {{\textbf{\textsf{B}}}}_\ell^{-1} \approx ( {{\textbf{\textsf{I}}}}+ {{\tilde{{\bf\Gamma}}}}_{\rm b} ) ( {{\textbf{\textsf{I}}}}- {{\bf\Gamma}}_{\rm tot}) ({{\textbf{\textsf{I}}}}+ {{\tilde{{\bf\Gamma}}}}_{\rm f}).$$ Multiplying this out and keeping only linear terms in ${{\bf\Gamma}}$’s, we find $${{\bf\Gamma}}_{\rm{eff}} \approx \sum\limits_{i = 1, i\neq\ell}^{N} \left(1 - \beta \right) {{\bf\Gamma}}_i,
\label{eq:Geff-approx}$$ where $\beta$ is $\beta_{i \ell}$ in the foreground and $\beta_{\ell i}$ in the background. In other words, ${{\bf\Gamma}}_{\rm eff}$ is approximately the sum of all of the tidal planes weighted by the redshift factor $(1-\beta)$. This has the same form as the effective shear that was found by @Iva. We will comment further on the use of ${{\bf\Gamma}}_{\rm{eff}}$ in [Section \[S:conclusions\]]{}.
The mass sheet degeneracy is more subtle for multiple main planes. As an example, consider the lens equation for two main planes: $$\begin{aligned}
{\mbox{\boldmath $x$}}_s &= {\mbox{\boldmath $x$}}_1 - {\mbox{\boldmath $\alpha$}}_1({\mbox{\boldmath $x$}}_1) - {\mbox{\boldmath $\alpha$}}_2({\mbox{\boldmath $x$}}_2) \\
&= {\mbox{\boldmath $x$}}_1 - {\mbox{\boldmath $\alpha$}}_1({\mbox{\boldmath $x$}}_1) - {\mbox{\boldmath $\alpha$}}_2({\mbox{\boldmath $x$}}_1 - \beta_{1 2} {\mbox{\boldmath $\alpha$}}_1({\mbox{\boldmath $x$}}_1)).
\end{aligned}$$ Any transformation that involves a rescaling of ${\mbox{\boldmath $\alpha$}}_1$ (like that in equation \[eq:phitran\]) would create a rescaling that appears inside the argument of ${\mbox{\boldmath $\alpha$}}_2$. In order for the transformation to create an overall rescaling similar to that for external convergence, the composition of the deflection functions ${\mbox{\boldmath $\alpha$}}_1$ and ${\mbox{\boldmath $\alpha$}}_2$ would have to be proportional to ${\mbox{\boldmath $\alpha$}}_2$, which appears to be a restrictive constraint. Therefore, it remains to be seen how the multiple-main-plane mass sheet degeneracy applies in practice.
Conclusions {#S:conclusions}
===========
To avoid possible biases in strong lensing studies, it is important to account for LOS effects. We have presented a lensing framework that fills the gap between using the full multiplane lens equation (which can be computationally expensive) and treating everything in the tidal approximation (which omits higher order effects that can be significant for objects that are projected near the lens and/or are massive). The framework can properly account for the non-linear effects from any mixture of ‘main’ planes (strong lenses) that are given full treatment and ‘tidal’ planes (weak lenses) that are treated using the tidal approximation. Our framework can be used to calculate all of the standard lensing observables. The general expressions for the lens equation, magnification tensor, and time delay are as follows (from equations \[eq:xfinN\], \[eq:AfinN\], and \[eq:TfinN\]):
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(We emphasize that ${\mbox{\boldmath $\alpha$}}_\ell$, ${{\bf\Gamma}}_\ell$, and $\phi_\ell$ all need to be evaluated at ${\mbox{\boldmath $x$}}_\ell$, which is important for reasons discussed below.) These expressions are more accurate than what we have termed the single main plane case, because they allow for higher order effects in planes other than the main lens plane. Yet they are more efficient than the full multiplane lens equation because the recursive sums only include main planes. All of the tidal planes—which may number in the hundreds for realistic lines of sight—can be compressed into the following matrices (from equations \[eq:Bmatdef\], \[eq:Cmatdef\], \[eq:Fmatdef\], and \[eq:Gmatdef\]): $$\begin{aligned}
{{\textbf{\textsf{B}}}}_j &= {{\textbf{\textsf{I}}}}- \sum\limits_{i=1,i\not \in \{\ell_\mu\}}^{j-1}\beta_{ij}{{\bf\Gamma}}_i {{\textbf{\textsf{B}}}}_i, \\
{{\textbf{\textsf{C}}}}_{\ell j} &= \beta_{\ell j} {{\textbf{\textsf{I}}}}- \sum\limits_{i=\ell+1, i\not\in\{\ell_\mu\}}^{j-1} \beta_{i j}{{\bf\Gamma}}_i{{\textbf{\textsf{C}}}}_{\ell i}, \\
{{\textbf{\textsf{F}}}}_j &\equiv \tau_{j-1} {{\textbf{\textsf{B}}}}_{j}-\tau_{j-1} {{\textbf{\textsf{B}}}}_{j-1}
\ =\ - \sum\limits_{i=1,i\not \in \{\ell_\mu\}}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{B}}}}_i, \\
{{\textbf{\textsf{G}}}}_{\ell j}&\equiv \tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell j}-\tau_{j-1} {{\textbf{\textsf{C}}}}_{\ell \,j-1}
\ =\ \tau_{\ell s}{{\textbf{\textsf{I}}}}- \sum\limits_{i=\ell+1,i \not \in \{\ell_\mu\}}^{j-1} \tau_{is}{{\bf\Gamma}}_i{{\textbf{\textsf{C}}}}_{\ell i}.\end{aligned}$$ These matrices can be computed once at the start of any lensing analysis and stored for repeated use.
To date, a common modelling approach has been to incorporate the main lens galaxy and any strong perturbers (assumed to lie in the same plane as the lens) into ${\mbox{\boldmath $\alpha$}}_\ell$, to fit for an external shear in the main plane, and then to correct for remaining LOS effects through an external convergence (e.g. @Hilbert09 [@Suyu10; @Collett13; @Suyu13], but see @Schneider13). Our analysis leads to two remarks. First, these LOS convergence corrections are been calibrated by ray tracing through cosmological simulations to compute the total convergence from a direct sum of all the mass along the LOS. We find, however, that the key quantities are the effective convergence and shear, which are given by (from equations \[eq:Geff\] and \[eq:Geff-approx\]) $$\begin{aligned}
{{\bf\Gamma}}_{\rm{eff}} \ \equiv\ {{\textbf{\textsf{I}}}}- {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s {{\textbf{\textsf{B}}}}_\ell^{-1}
\ \approx\ \sum\limits_{i = 1, i\neq\ell}^{N} (1 - \beta) {{\bf\Gamma}}_i ,\end{aligned}$$ where $\beta$ is $\beta_{i \ell}$ in the foreground of the main lens plane, and $\beta_{\ell i}$ in the background. The $\beta$ weighting factors depend on the redshift of the main lens galaxy as well as the redshifts of the source and the plane in question, so the effective shear and convergence cannot be tabulated in a general way that is independent of particular lens systems.
Secondly, most existing lens models have been fit to the positions of the images on the sky (which we have denoted by ${\mbox{\boldmath $x$}}_1$). Each main plane actually needs to be evaluated using the position ${\mbox{\boldmath $x$}}_\ell$ of the light ray in that plane. This distinction gives rise to non-linearities that cannot be mimicked by a simple shear and can lead to systematic uncertainties in lens models if not handled properly (McCully et al., in preparation). In principle, the ‘corrective’ approach to lens modelling could account for the non-linear effects by using ${\mbox{\boldmath $x$}}_\ell = {{\textbf{\textsf{B}}}}_\ell {\mbox{\boldmath $x$}}_1$, where the matrix ${{\textbf{\textsf{B}}}}_\ell$ can be calibrated by ray tracing.
A different approach to lens modelling is to directly incorporate LOS effects by building full three-dimensional mass models like those used by @Wong. Then all of the non-linear effects are automatically included, and the convergence and shear are computed self-consistently from an underlying mass distribution. In order to employ our hybrid framework effectively, we need to understand when it is acceptable to use the tidal approximation and when we need to treat a plane exactly. In a forthcoming paper (McCully et al., in preparation), we use realistic beams like those in @Wong to test the tidal approximation. We also quantify bias and scatter in lens models associated with different ways of handling the LOS. The framework presented here serves as the foundation for such detailed treatments of LOS effects in strong lensing.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee, Peter Schneider, for very detailed and helpful comments. We thank Phil Marshall, Roger Blandford, Sherry Suyu, and Stefan Hilbert for helpful conversations. CM and CRK acknowledge funding from NSF grants AST-0747311 and AST-1211385. KCW is supported by an EACOA Fellowship awarded by the East Asia Core Observatories Association, which consists of the Academia Sinica Institute of Astronomy and Astrophysics, the National Astronomical Observatory of Japan, the National Astronomical Observatory of China, and the Korea Astronomy and Space Science Institute. AIZ acknowledges funding from NSF grants AST-0908280 and AST-1211874, as well as NASA grants ADP-NNX10AD476 and ADP-NNX10AE88G. She also thanks the John Simon Guggenheim Memorial Foundation and the Center for Cosmology and Particle Physics at NYU for their support. Image credits: Centaurus A—Jean-Charles Cuillandre, Giovanni Anselmi, Hawaiian Starlight; Leo I—Oliver Stein.
Matrix Identities {#appendix}
=================
In Section \[S:tdel-1main\], we found that our expression for the time delay is equivalent to an alternative expression found by SEF only if the following identity holds (see equation \[eq:identity\]):
$$\tau_{\ell s} ({{\textbf{\textsf{C}}}}^{-1}_{\ell s})^{{\mathrm{T}}}{{\textbf{\textsf{B}}}}_\ell = {{\textbf{\textsf{G}}}}_{\ell s} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s - {{\textbf{\textsf{F}}}}_s.$$
Moving ${{\textbf{\textsf{C}}}}^{-1}_{\ell s}$ to the right-hand side and using the definitions of ${{\textbf{\textsf{F}}}}_s$ and ${{\textbf{\textsf{G}}}}_{\ell s}$ from [equations (\[eq:Fdef\]) and (\[eq:Gdef\])]{} yields $$\begin{aligned}
\tau_{\ell s} {{\textbf{\textsf{B}}}}_\ell
&= \tau_{s-1} {{\textbf{\textsf{C}}}}_{\ell s}^{{\mathrm{T}}}\left[ \left({{\textbf{\textsf{C}}}}_{\ell s} - {{\textbf{\textsf{C}}}}_{\ell \, s-1} \right ) {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s - \left({{\textbf{\textsf{B}}}}_s - {{\textbf{\textsf{B}}}}_{s-1}\right)\right] \\
&= \tau_{s-1} {{\textbf{\textsf{C}}}}_{\ell s}^{{\mathrm{T}}}\left( {{\textbf{\textsf{B}}}}_{s-1} - {{\textbf{\textsf{C}}}}_{\ell \, s -1} {{\textbf{\textsf{C}}}}^{-1}_{\ell s} {{\textbf{\textsf{B}}}}_s \right).
\end{aligned}
\label{eq:identity2}$$ This is the form of the identity we seek to prove.
We find it helpful to begin with two special cases. First, if all tidal planes are in the foreground then $\ell = s-1$ so $\tau_{s-1} = \tau_{\ell s}$ and ${{\textbf{\textsf{B}}}}_{s-1} = {{\textbf{\textsf{B}}}}_{\ell}$. Also, the background matrices are ${{\textbf{\textsf{C}}}}_{\ell s} = {{\textbf{\textsf{I}}}}$ and ${{\textbf{\textsf{C}}}}_{\ell \, s-1} = {{\textbf{\textsf{0}}}}$. Therefore the right-hand side of [equation (\[eq:identity2\])]{} reduces to $\tau_{\ell s} {{\textbf{\textsf{B}}}}_{\ell}$, which proves the identity for this case. Second, consider a single tidal plane that lies in the background and is characterized by the tidal matrix ${{\bf\Gamma}}$. In this case, the matrices are as follows: $$\begin{gathered}
{{\textbf{\textsf{B}}}}_\ell = {{\textbf{\textsf{I}}}},
\qquad
{{\textbf{\textsf{B}}}}_{s-1} = {{\textbf{\textsf{I}}}},
\qquad
{{\textbf{\textsf{B}}}}_s = {{\textbf{\textsf{I}}}}- {{\bf\Gamma}},\nonumber
\\
{{\textbf{\textsf{C}}}}_{\ell \, s-1} = \beta {{\textbf{\textsf{I}}}},
\qquad
\rm{and} \qquad {{\textbf{\textsf{C}}}}_{\ell s} = {{\textbf{\textsf{I}}}}- \beta {{\bf\Gamma}}.\end{gathered}$$ Therefore the left-hand side of [equation (\[eq:identity2\])]{} is $\tau_{\ell s} {{\textbf{\textsf{I}}}}$, while the right-hand side is $$\begin{aligned}
\mbox{RHS}
&=& \tau_{s-1} ({{\textbf{\textsf{I}}}}-\beta{{\bf\Gamma}}) \left[ {{\textbf{\textsf{I}}}}- \beta ({{\textbf{\textsf{I}}}}-\beta{{\bf\Gamma}})^{-1} ({{\textbf{\textsf{I}}}}-{{\bf\Gamma}}) \right] \nonumber \\
&=& \tau_{s-1} \left[ ({{\textbf{\textsf{I}}}}-\beta{{\bf\Gamma}}) - \beta ({{\textbf{\textsf{I}}}}-{{\bf\Gamma}}) \right] \nonumber \\
&=& \tau_{s-1} (1-\beta) {{\textbf{\textsf{I}}}}.\end{aligned}$$ Using [equations (\[eq:b\_1\])–(\[eq:t\_ik\])]{} we find $\tau_{s-1} (1-\beta) = \tau_{\ell s}$, which proves the identity for this case.
To prove the identity (\[eq:identity2\]) in general, we need to review some ancillary results and establish some new ones. From [equation (\[eq:Bjidentity\])]{} and the ensuing discussion, we have recursion relations for ${{\textbf{\textsf{B}}}}_j$ and ${{\textbf{\textsf{C}}}}_{\ell j}$: $$\begin{aligned}
{{\textbf{\textsf{B}}}}_{j+1} &=& {{\textbf{\textsf{M}}}}_j {{\textbf{\textsf{B}}}}_j - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{B}}}}_{j-1} , \label{eq:Brecur}\\
{{\textbf{\textsf{C}}}}_{\ell \, j+1} &=& {{\textbf{\textsf{M}}}}_j {{\textbf{\textsf{C}}}}_{\ell \, j} - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{C}}}}_{\ell \, j-1} , \label{eq:Crecur}\end{aligned}$$ where we define $${{\textbf{\textsf{M}}}}_j \equiv \left(1+\frac{\tau_{j-1}}{\tau_{j}}\right) {{\textbf{\textsf{I}}}}- \beta_j {{\bf\Gamma}}_j .
\label{eq:Mmatdef}$$ Note that ${{\bf\Gamma}}_j$ is symmetric and so ${{\textbf{\textsf{M}}}}_j$ is symmetric as well. We define a new matrix with the structure that we are looking for on the right-hand side of [equation (\[eq:identity2\])]{}: $${{\textbf{\textsf{W}}}}_j \equiv {{\textbf{\textsf{B}}}}_j - {{\textbf{\textsf{C}}}}_{\ell j} {{\textbf{\textsf{C}}}}_{\ell s}^{-1} {{\textbf{\textsf{B}}}}_s .
\label{eq:Wmatdef}$$ We can combine [equations (\[eq:Brecur\]) and (\[eq:Crecur\])]{} to write a recursion relation for ${{\textbf{\textsf{W}}}}_j$: $${{\textbf{\textsf{W}}}}_{j+1} = {{\textbf{\textsf{M}}}}_j {{\textbf{\textsf{W}}}}_j - \frac{\tau_{j-1}}{\tau_{j}} {{\textbf{\textsf{W}}}}_{j-1} .
\label{eq:Wrecur}$$
It is convenient to define a generalized version of the ${{\textbf{\textsf{C}}}}$ matrices (compare equation \[eq:Cmatdef\]), $${{\textbf{\textsf{C}}}}_{ik} \equiv \beta_{ik} {{\textbf{\textsf{I}}}}- \sum_{j=i+1}^{k-1} \beta_{jk} {{\bf\Gamma}}_j {{\textbf{\textsf{C}}}}_{ij} .$$ Note that the sum runs over the second index of ${{\textbf{\textsf{C}}}}$. Because ${{\textbf{\textsf{C}}}}_{ij}$ contains ${{\bf\Gamma}}$ matrices between planes $i$ and $j$, the products of ${{\bf\Gamma}}$ matrices have indices that decrease to the right. If we restrict attention to matrices ${{\textbf{\textsf{C}}}}_{is}$ in which the second index is $s$, we can write an alternative form of the sum as $$\begin{aligned}
{{\textbf{\textsf{C}}}}_{is} \ &=\ {{\textbf{\textsf{I}}}}- \sum_{j=i+1}^{s-1} \beta_{ij} {{\textbf{\textsf{C}}}}_{js} {{\bf\Gamma}}_j\\
&=\ {{\textbf{\textsf{I}}}}+ \sum_{j=i+1}^{s-1} \frac{\beta_{ij}}{\beta_{j}} {{\textbf{\textsf{C}}}}_{js} \left[ {{\textbf{\textsf{M}}}}_j - \left(1+\frac{\tau_{j-1}}{\tau_{j}}\right) {{\textbf{\textsf{I}}}}\right] .
\end{aligned}$$ Here, the sum runs over the first index of ${{\textbf{\textsf{C}}}}$, and because ${{\textbf{\textsf{C}}}}_{js}$ contains ${{\bf\Gamma}}$ matrices between planes $j$ and $s$, the factor of ${{\bf\Gamma}}_j$ needs to be on the right in order to have the matrix product arranged with indices that decrease to the right. In the second step, we use [equation (\[eq:Mmatdef\])]{} to replace ${{\bf\Gamma}}_j$ with ${{\textbf{\textsf{M}}}}_j$. Note that $$\begin{gathered}
{{\textbf{\textsf{C}}}}_{i-1 \, s} - {{\textbf{\textsf{C}}}}_{is}
= \frac{\beta_{i-1}}{\beta_{i}} {{\textbf{\textsf{C}}}}_{is} \left[ {{\textbf{\textsf{M}}}}_{i} - \left(1+\frac{\tau_{i-1}}{\tau_{i}}\right) {{\textbf{\textsf{I}}}}\right]\\
+ (\tau_{is} - \tau_{i-1 \, s}) \sum_{j=i+1}^{s-1} \frac{1}{\beta_{j} \tau_{js}} {{\textbf{\textsf{C}}}}_{js} \left[ {{\textbf{\textsf{M}}}}_{j} - \left(1+\frac{\tau_{j-1}}{\tau_{j}}\right) {{\textbf{\textsf{I}}}}\right] ,
\label{eq:Ctmp}\end{gathered}$$ where we make use of [equations (\[eq:b\_1\])–(\[eq:t\_ik\])]{}. We can write a similar expression for ${{\textbf{\textsf{C}}}}_{is} - {{\textbf{\textsf{C}}}}_{i+1 \, s}$ and combine it with [equation (\[eq:Ctmp\])]{} to eliminate the sum. Again using [equations (\[eq:b\_1\])–(\[eq:t\_ik\])]{} to simplify yields $${{\textbf{\textsf{C}}}}_{i-1 \, s} = \frac{\beta_{i-1}}{\beta_{i}} \left( {{\textbf{\textsf{C}}}}_{is} {{\textbf{\textsf{M}}}}_i - \frac{\tau_{is}}{\tau_{i+1 \, s}} {{\textbf{\textsf{C}}}}_{i+1 \, s} \right).
\label{eq:Crecur2}$$ We can simplify one step further by introducing a scaled version of the matrices: $${{\widetilde{{\textbf{\textsf{C}}}}}}_i \equiv \frac{1}{\beta_i} {{\textbf{\textsf{C}}}}_{is} .
\label{eq:Chatdef}$$ With this definition, [equation (\[eq:Crecur2\])]{} becomes $${{\widetilde{{\textbf{\textsf{C}}}}}}_{i-1} = {{\widetilde{{\textbf{\textsf{C}}}}}}_i {{\textbf{\textsf{M}}}}_i - \frac{\tau_{i}}{\tau_{i+1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{i+1},
\label{eq:Crecur3}$$ where we use [equation (\[eq:t\_b\])]{} to put $\tau_{is}/\beta_i = \tau_i$ (and similar for index $i+1$). If we start from index $s$ and work our way down, the first few matrices are $$\begin{aligned}
{{\widetilde{{\textbf{\textsf{C}}}}}}_{s-1} &= {{\textbf{\textsf{I}}}}, \label{eq:Cs-1}\\
{{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2} &= {{\textbf{\textsf{M}}}}_{s-1}, \label{eq:Cs-2}\\
{{\widetilde{{\textbf{\textsf{C}}}}}}_{s-3} &={{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{I}}}}, \label{eq:Cs-3}\\
{{\widetilde{{\textbf{\textsf{C}}}}}}_{s-4} &= {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-3} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{M}}}}_{s-3} - \frac{\tau_{s-3}}{\tau_{s-2}} {{\textbf{\textsf{M}}}}_{s-1}. \label{eq:Cs-4}\end{aligned}$$
There is one more useful technical result: $${{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} = {{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1}^{{{\mathrm{T}}}}.
\label{eq:Csymm}$$ We prove this by induction (see @Seitz94 for a similar argument). The relation is trivial for $j=s-2$ because ${{\widetilde{{\textbf{\textsf{C}}}}}}_{s-1} = {{\textbf{\textsf{I}}}}$. It is manifestly true for $j=s-3$ because we can evaluate the left- and right-hand sides explicitly using [equations (\[eq:Cs-2\]) and (\[eq:Cs-3\])]{}: $$\begin{aligned}
\mbox{LHS} &= {{\textbf{\textsf{M}}}}_{s-1} \left( {{\textbf{\textsf{M}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-1} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{I}}}}\right) \nonumber\\
&=\ {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-1} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{M}}}}_{s-1}, \\
\mbox{RHS} &= \left( {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{I}}}}\right) {{\textbf{\textsf{M}}}}_{s-1}\nonumber\\
&= {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-1} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{M}}}}_{s-1}.\end{aligned}$$ (Note that because ${{\textbf{\textsf{M}}}}_j$ is symmetric, $({{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2})^{{{\mathrm{T}}}} = {{\textbf{\textsf{M}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-1}$.) Now, if we postulate that [equation (\[eq:Csymm\])]{} is true for index $j$, we can ask what it implies for index $j-1$: $$\begin{aligned}
{{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1}^{{{\mathrm{T}}}}
&=& {{\widetilde{{\textbf{\textsf{C}}}}}}_j \left( {{\textbf{\textsf{M}}}}_j {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} - \frac{\tau_{j}}{\tau_{j+1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1}^{{{\mathrm{T}}}} \right) \nonumber\\
&=& \left( {{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\textbf{\textsf{M}}}}_j - \frac{\tau_{j}}{\tau_{j+1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1} \right) {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} \nonumber \\
&=& {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}}.\end{aligned}$$ In the first line we use [equation (\[eq:Crecur3\])]{} for ${{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1}^{{{\mathrm{T}}}}$, and in the second line we use [equation (\[eq:Csymm\])]{} to replace ${{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1}^{{{\mathrm{T}}}}$ with ${{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}}$ according to our postulate. We see that if [equation (\[eq:Csymm\])]{} is true for index $j$ then it is also true for index $j-1$, which completes the proof by induction.
Now we have all the pieces needed to prove [equation (\[eq:identity2\])]{}. We start with a trivial relation from [equation (\[eq:Wmatdef\])]{} with $j=s$: $${{\textbf{\textsf{0}}}}= {{\textbf{\textsf{W}}}}_s .$$ We use [equation (\[eq:Wrecur\])]{} on the right-hand side: $${{\textbf{\textsf{0}}}}= {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{W}}}}_{s-1} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{W}}}}_{s-2} .
\label{eq:Wtmp1}$$ We can solve this to find $${{\textbf{\textsf{W}}}}_{s-2} = \frac{\tau_{s-1}}{\tau_{s-2}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} ,
\label{eq:Ws-2}$$ where we use [equation (\[eq:Cs-2\])]{} to write this in a form involving ${{\widetilde{{\textbf{\textsf{C}}}}}}$, for reasons that will become clear. Now, we return to [equation (\[eq:Wtmp1\])]{}, again apply [equation (\[eq:Wrecur\])]{} to the first term, and rearrange to find $$\begin{aligned}
{{\textbf{\textsf{0}}}}&=\ \left( {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{M}}}}_{s-2} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\textbf{\textsf{I}}}}\right) {{\textbf{\textsf{W}}}}_{s-2} - \frac{\tau_{s-3}}{\tau_{s-2}} {{\textbf{\textsf{M}}}}_{s-1} {{\textbf{\textsf{W}}}}_{s-3}\\
&=\ {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-3} {{\textbf{\textsf{W}}}}_{s-2} - \frac{\tau_{s-3}}{\tau_{s-2}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2} {{\textbf{\textsf{W}}}}_{s-3} ,
\end{aligned}$$ where we use [equations (\[eq:Cs-2\]) and (\[eq:Cs-3\])]{}. We solve for ${{\textbf{\textsf{W}}}}_{s-3}$ and use [equation (\[eq:Ws-2\])]{}: $$\begin{aligned}
{{\textbf{\textsf{W}}}}_{s-3} &= \frac{\tau_{s-1}}{\tau_{s-3}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{-1} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-3} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} \nonumber \\
&= \frac{\tau_{s-1}}{\tau_{s-3}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{-1} \left( {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2} {{\textbf{\textsf{M}}}}_{s-2} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-1} \right) {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} \nonumber \\
&= \frac{\tau_{s-1}}{\tau_{s-3}} \left( {{\textbf{\textsf{M}}}}_{s-2} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-2}^{{{\mathrm{T}}}} - \frac{\tau_{s-2}}{\tau_{s-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-1}^{{{\mathrm{T}}}} \right) {{\textbf{\textsf{W}}}}_{s-1} \nonumber\\
&= \frac{\tau_{s-1}}{\tau_{s-3}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{s-3}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1}.
\label{eq:Ws-3}\end{aligned}$$ We use [equation (\[eq:Crecur3\])]{} in the second step, [equation (\[eq:Csymm\])]{} in the third step, and [equation (\[eq:Crecur3\])]{} again in the fourth step. Repeating the analysis reveals the pattern that ${{\textbf{\textsf{W}}}}_j$ can be written as $${{\textbf{\textsf{W}}}}_j = \frac{\tau_{s-1}}{\tau_{j}} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} .
\label{eq:Wj}$$ We can prove this result by induction. First, repeatedly applying [equation (\[eq:Crecur3\])]{} to [equation (\[eq:Wtmp1\])]{} yields $${{\textbf{\textsf{0}}}}= {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1} {{\textbf{\textsf{W}}}}_j - \frac{\tau_{j-1}}{\tau_{j}} {{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\textbf{\textsf{W}}}}_{j-1} ,$$ which we can solve to find $${{\textbf{\textsf{W}}}}_{j-1} = \frac{\tau_{j}}{\tau_{j-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{-1} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1} {{\textbf{\textsf{W}}}}_j .$$ If we postulate that [equation (\[eq:Wj\])]{} holds for index $j$, we can write $$\begin{aligned}
{{\textbf{\textsf{W}}}}_{j-1} &=& \frac{\tau_{s-1}}{\tau_{j-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{-1} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} \nonumber\\
&=& \frac{\tau_{s-1}}{\tau_{j-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{-1} \left( {{\widetilde{{\textbf{\textsf{C}}}}}}_j {{\textbf{\textsf{M}}}}_j - \frac{\tau_{j}}{\tau_{j+1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1} \right) {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} \nonumber\\
&=& \frac{\tau_{s-1}}{\tau_{j-1}} \left( {{\textbf{\textsf{M}}}}_j {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} - \frac{\tau_{j}}{\tau_{j+1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j+1}^{{{\mathrm{T}}}} \right) {{\textbf{\textsf{W}}}}_{s-1} \nonumber \\
&=& \frac{\tau_{s-1}}{\tau_{j-1}} {{\widetilde{{\textbf{\textsf{C}}}}}}_{j-1}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1}.\end{aligned}$$ Therefore if [equation (\[eq:Wj\])]{} holds for index $j$ then it also holds for index $j-1$, which completes the proof by induction.
To finish the full proof, we use [equation (\[eq:Chatdef\])]{} to write $$\frac{1}{\tau_j} {{\widetilde{{\textbf{\textsf{C}}}}}}_j^{{{\mathrm{T}}}} = \frac{1}{\beta_j \tau_j} {{\textbf{\textsf{C}}}}_{js}^{{{\mathrm{T}}}} = \frac{1}{\tau_{js}} {{\textbf{\textsf{C}}}}_{js}^{{{\mathrm{T}}}} ,$$ using [equation (\[eq:t\_b\])]{}. Then [equation (\[eq:Wj\])]{} becomes $$\tau_{js} {{\textbf{\textsf{W}}}}_j = \tau_{s-1} {{\textbf{\textsf{C}}}}_{js}^{{{\mathrm{T}}}} {{\textbf{\textsf{W}}}}_{s-1} .$$ We evaluate this at $j = \ell$ and use ${{\textbf{\textsf{W}}}}_\ell = {{\textbf{\textsf{B}}}}_\ell$, which holds because ${{\textbf{\textsf{C}}}}_{\ell\ell} = {{\textbf{\textsf{0}}}}$. This yields our final result $$\tau_{\ell s} {{\textbf{\textsf{B}}}}_\ell = \tau_{s-1} {{\textbf{\textsf{C}}}}_{\ell s}^{{{\mathrm{T}}}} ({{\textbf{\textsf{B}}}}_{s-1} - {{\textbf{\textsf{C}}}}_{\ell \, s-1} {{\textbf{\textsf{C}}}}_{\ell s}^{-1} {{\textbf{\textsf{B}}}}_s ) ,$$ which is the identity we sought to prove.
[^1]: Equation (\[eq:singlelenseq\]) can be made formally equivalent to the standard single-plane lens equation with a suitable transformation of the lens potential [@Schneider97]. In that case, however, the effective mass model differs from the true mass distribution of the lens. The fact that the lens potential must be distorted further emphasizes that tidal contributions from foreground planes create important non-linear effects.
[^2]: Type Ia supernovae can be used to break the mass sheet degeneracy because their intrinsic luminosity can be inferred from their light-curve shapes [e.g. @Kolatt98].
|
---
abstract: 'Fast secure random number generation is essential for high-speed encrypted communication, and is the backbone of information security. Generation of truly random numbers depends on the intrinsic randomness of the process used and is usually limited by electronic bandwidth and signal processing data rates. Here we use a multiplexing scheme to create a fast quantum random number generator structurally tailored to encryption for distributed computing, and high bit-rate data transfer. We use vacuum fluctuations measured by seven homodyne detectors as quantum randomness sources, multiplexed using a single integrated optical device. We obtain a real-time random number generation rate of 3.08 Gbit/s, from only 27.5 MHz of sampled detector bandwidth. Furthermore, we take advantage of the multiplexed nature of our system to demonstrate an unseeded strong extractor with a generation rate of 26 Mbit/s.'
author:
- Ben Haylock
- Daniel Peace
- Francesco Lenzini
- Christian Weedbrook
- Mirko Lobino
title: Multiplexed Quantum Random Number Generation
---
[^1]
[^2]
Introduction
============
Information security[@Ware:67] is a foundation of modern infrastructure with quantum optics set to play a prevalent role in the next generation of cryptographic hardware[@Zhang:14]. Randomness is a core resource for cryptography and considerable effort has gone into making systems suitable for supplying high bit rate streams of random bits. The randomness properties of the source have a profound effect on the security of the encryption, with several examples of compromised security from an attack on the random number generator [@Debian; @Dorrendorf:09; @Nohl:08]. In this area, quantum optics has provided advantages over previous methods, enabling random number generation with high speeds and enhanced security [@H-C:17; @Ma:16; @Hart:17].
The gold standard for security in random number generators comes from device independent quantum random number generators (QRNGs) [@Nie:16], where the output is certified as random regardless of the level of trust in the generator. These generators require an experimental violation of a Bell-type inequality, an extremely difficult task, limiting generation rates to well below practical requirements ($<$kbit/s) [@Pironio:10; @Bierhorst]. Other approaches based on the Kochen-Specker theorem to prove value indefiniteness of the measurement, have demonstrated faster but not yet usable generation rates (25kbit/s) [@Kulikov:17; @Abbott:12]. Recent advances[@Liu:18] have built upon the notion of Bell inequality violations using device independent quantum random number generators. Remarkably, these allow the closure of any security loopholes related to the how the device is made, i.e., independent of the device implementations. Currently, high-speed (Mbit/s-Tbit/s) quantum random number generation relies on trusted or semi-trusted generators, where the independence of the randomness from classical noise is experimentally tested [@Gabriel:10; @Li:14; @Lunghi:15; @Mitchell:15; @Marangon:2017; @Virte:14; @Wayne:09; @Xu:12]. While these systems have no quantum physical guarantee of their randomness, they are usually denoted as QRNGs due to the quantum mechanical origin of the randomness. This trade-off between speed and security is mostly caused by the experimental complexity of fully secure implementations.
Entropy sources sufficient for randomness generation rates up to 1.2Tb/s have been demonstrated[@Sakuraba:15], However, these systems are not capable of real time random number generation at full speed due to processing bandwidth limitations, instead performing off-line processing on captured data to generate randomness. Thus these schemes are not suited for providing high-speed random numbers for cryptography. Regardless of generation procedure, implementations of randomness extraction are limited by electronic logic speeds or detection bandwidths. Therefore, for high speed real time random number generation, the most feasible solution is to create many parallel sources with lower entropy rates.
Multiplexed quantum random number generation has previously been theoretically proposed as a solution to post-processing bottlenecks in the real-time rate of QRNGs [@Hart:17; @Ma:16]. Previous demonstrations of a parallel nature remain focused on increasing single-channel data rates. Gräfe et al. extend single photon path-encoding based QRNGs to the multi-mode case increasing the bitrate for a fixed measured photon flux[@Grafe:14]. Haw et al. sample two separate frequency slices of their homodyne detector bandwidth in a vacuum fluctuation QRNG, enabling them to generate randomness at twice their digitization rate[@Haw:15]. Both demonstrations remain ultimately rate-limited by generation[@Grafe:14] or detection[@Haw:15] rates. The goal of multiplexing should be to increase the data rate regardless of single channel bandwidth limitations as shown by IDQuantique by multiplexing together four separate devices on a single interface[@IDQuantique].
A multiplexing architecture is more versatile than increasing the rate of a single data stream, allowing the use of more complex extraction techniques and randomness distribution at rates faster than a single channel capacity. Randomness extractors are algorithms which, given a bit string from a weakly random physical source, produce a shorter sequence of truly random bits[@Shaltiel:11]. Previous works have largely used seeded extractors, which require a uniform random seed to convert the input to random bits. Such extraction is often described as randomness expansion, as it cannot extract true randomness without already having some at the input[@H-C:17]. Seeded extractors rely on the uniformity of the seed and independence of the output from this seed for security. We can relax both of these requirements with a multi-source extractor.
A single random output is produced from two weakly random inputs in a multi-source extractor. The main advantage of this approach is that random numbers can be generated without any initial random seed. Many examples of multi-source extractors exist that allow for unseeded extraction with low entropy loss, including constructions which are strong extractors in the presence of quantum side information[@Kasher:10].
We experimentally demonstrate a high-speed parallel quantum random generator whose total rate is not limited by generation or detection rates but rather by the number of parallel channels used. While Gbps real-time generation rates have been previously been demonstrated[@Ugajin:17; @Zhang:16; @Marangon:18], our scheme provides a simple method scheme to overcome electronic and processing bandwidth limitations.
Random Number Generation Scheme
===============================
A schematic of the multiplexed QRNG scheme is shown in Figure \[fig1\]. The noise source of each channel of our multiplexed design comes from homodyne measurements of vacuum state [@Gabriel:10; @Shen:10] (see inset in Figure \[fig2\](a)). A laser is sent onto a 50-50 beamsplitter while vacuum enters the other port, subsequently the two outputs of the beamsplitter are detected on two photodiodes and the difference between the two photocurrents is amplified. The homodyne current is proportional to a measurement of the quadrature operator of the vacuum state, and its value is independent and unpredictable within a Gaussian distribution with zero mean.
![Scheme for multiplexed quantum random number generator based on quadrature measurements of the vacuum state. A low noise, Koheras Boostik laser at 1550nm is coupled in and out of a lithium niobate waveguide network through butt-coupled fiber arrays. Light from the outputs is sent into seven homodyne detectors. The detector signals are sent to the ADC and FPGA for digitization, processing and randomness extraction. This schematic shows four channels, the experimental implementation used up to seven channels.[]{data-label="fig1"}](Fig1.png){width="49.00000%"}
The homodyne detectors (HDs) used in this demonstration follow the design of Kumar et al.[@Kumar:12] and have an electronic bandwidth of 100MHz. The quantum efficiencies of the photodiodes used (PD20, oemarket) are $>$67% (70% typical). The quantum signal to classical noise ratio (QCNR) is defined as QCNR$=10\log_{10}(\sigma_Q^2/\sigma_E^2)$ where $\sigma_Q^2$ is the quantum noise variance, and $\sigma_E^2$ is the classical noise variance. Using the equation $\sigma_Q^2=\sigma_M^2-\sigma_E^2$, QCNR can be calculated from experimental measurement of the variance of the output of the homodyne detector with the local oscillator off ($\sigma_E^2$), and with the local oscillator on ($\sigma_M^2$). Figure \[fig2\](a) shows the results for all seven homodyne detectors. We see that for all channels of our design this ratio exceeds 10dB across 30 MHz, with a measured common mode rejection ratio of $>$27dB across all seven detectors. To confirm that our detectors were measuring vacuum fluctuations, we determine the linearity of the noise as a function of the laser power (see Figure \[fig2\](b)). As the QCNR is the ratio between this response and the value at zero power (blue trace), it also scales linearly with input power. The independence of the outcomes of each of the channels was verified from cross-correlation measurements shown in Figure \[fig2\](c). The cross-correlation for ideal uniform data is $1/\sqrt{n}=3.16\times 10^{-4}$. All 21 channel pairing cross-correlations lie between $2.83\times 10^{-4}$ and $3.51\times 10^{-4}$.
Integrated optics provides a compact and stable way to implement the set of beamsplitters needed to feed many homodyne detectors. We fabricate a 1:32 multiplexer using annealed proton exchanged waveguides in lithium niobate with a device footprint of 60mm x 5mm[@Lenzini:15]. The device has insertion losses of $\approx$7 dB ($\approx$22dB total loss per channel) and we choose balanced outputs to send to the seven homodyne detectors. Total input power to the array is approximately 160mW ($\approx$1mW per channel at output), which will ultimately limit the maximum number of channels. This limitation may be overcome by using several multiplexed lasers as the input. Our choice of lithium niobate is designed for a semi-integrated system, where a compact butterfly diode laser, the waveguide device, a butt coupled linear photodiode array, and all the electronics could be housed on a single circuit board.
Several data processing steps are implemented in order to transform the analog signals from the homodyne detectors into a stream of random bits (see Fig. 1). First, the analog output of each detector is digitized into 12 bits per sample using an analog to digital converter (ADC, Texas Instruments ADS5295EVM). In our demonstration no anti-aliasing filter is used and as such frequencies higher than the detection bandwidth contribute to the signal. The digitized results from each outcome are sent in parallel into a field programmable gate array (FPGA, Altera Arria II GX Development Kit) for the remainder of the randomness extraction protocols. If the outputs are to be multiplexed back together rather than used in parallel, multiplexing occurs after the randomness extractor. Multiplexing is done by interleaving channel by channel and the output of the extractors is written to the on-chip memory of the FPGA and may be transferred to a computer via a USB cable for randomness testing. Randomness verification tests are performed off-line by transferring experimental data to a PC. In any application the memory connected to the FPGA is equally suited to storing and sending the randomness as the memory of a PC.
Three different extraction methods are demonstrated that convert the unpredictable measurement outcomes of the homodyne detectors into random bit streams. In the first extractor (A), which we call ‘raw bit extraction’, we take the eight least significant bits (LSBs) from the ADC and discard the remaining 4 bits per sample. This extractor follows the design of the ’environmental immunity’ procedure of [@Symul:11]. This extractor is designed to minimise the influence of the classical noise on the output signal. Demonstration of the security of this protocol is through the high entropy of the experimental output, rather than from any theoretical proof.
The second extractor (B) is based on the second draft of NIST Special Publication 800-90B[@Sonmez:16]. The authors list a set of vetted randomness extractors, one of which is the keyed algorithm CMAC (Cipher-based Message Authentication Code)[@Dworkin:05] with the AES (Advanced Encryption Standard)[@NIST:01] block cipher. For an input with k bits of min-entropy i.e., one where the maximum probability of any outcome is bounded by $2^k$ , when $\leq k/2$ bits are taken from the 128 bit output of the extractor, full-entropy output bits are produced [@Sonmez:16]. The remaining $\lfloor128-k/2\rfloor$ bits are used to refresh the seed. We take eight LSBs from sixteen consecutive digitization samples to form the 128-bit input to each run of the extractor. The AES hash is implemented on the FPGA using the TinyAES core[@Hsing].
![(a) Quantum to classical noise ratio (QCNR) of all seven homodyne detectors used, with inset showing a schematic description of a homodyne detector, orange hemispheres represent photodiodes. Power used is 1mW and all powers are measured at the output of the waveguide device. (b) Linearity of homodyne detector response with increasing local oscillator power, shown using a representative (channel 4). Blue - 0$\mu$W, orange -200$\mu$W, yellow - 400$\mu$W, purple - 600$\mu$W, green - 800$\mu$W. Inset, a plot of the linear response at 5MHz. (c) The largest positive (blue) and negative (red) correlations between any pairwise combinations of eight-bit encoded homodyne measurements from the seven channels. The green line represents the ideal value for the size of the data set (10 million samples).[]{data-label="fig2"}](qrng_fig2_qj.png){width="49.00000%"}
The third extractor(C) takes advantage of the fact that we have many independent sources, and as such can use a multi-source extractor. Examples of both weak (seed-dependent, non-reusable seed) and strong (seed-independent, reusable seed) extractors have been shown for QRNGs. The security of these extractors relies on the quality of the previously created random seed. Multi-source extractors discard the necessity for a truly random seed. Instead, they take two or more partially random bit-strings from weak randomness sources and produce a truly random output. Given sufficient randomness of the inputs, a strong multi source extractor outputs bits that are uncorrelated with any of the inputs, providing randomness even with full knowledge of all but one of the inputs. We implement a single bit two-source extractor, as described in [@Bouda:12], which produces a uniform output providing at least one of the inputs has more min-entropy than the output number of bits. Each extractor takes two 36-bit strings from two different homodyne detectors, each consisting of three 12-bit samples. As such we need an even number of input channels for this extractor, and using six of the detectors we create three of these extractors and multiplex the outputs together.
Entropy Source Evaluation
=========================
We first evaluate the worst-case conditional min-entropy of the 12-bit output of the ADC for each channel to find the amount of entropy sourced from the measurement of the vacuum state. Using the procedure described by Haw et al. [@Haw:15] for the worst-case conditional min-entropy ($H_{min}$) we can find a lower bound for the maximum extractable randomness given the discretized measured distribution $M_{dis}$, conditioned on the classical side information $E$ [@Haw:15; @Renner:08]: $$H_{min}(M_{dis}|E)=-\log_2 \left[\max_{e\in \mathbb{R}} \max_{m_i\in M_{dis}} P_{M_{dis}|E}(m_i|e)\right]
\label{minent}$$ where $m_i$ and $e$ are the samples from their respective distributions. Taking a maximum classical noise spread of $e_{max}=5\sigma_E$, and using a representative sample of $1\times 10^6$ samples per channel we numerically evaluate Equation \[minent\] and find a worse case min entropy of $H_{min}(M_{dis}|E)\geq 9.201$ across all seven channels, for a near optimal digitization range. This value describes the entropy component immune to classical noise sources; however, it does not describe the component immune to quantum side information, as the extractors we use are not secure against such attacks.
Raw 8 Bit AES Two Source
-------------------------- ----------- ------------------- -----------------
Extractor Channels 7 7 3
Sampling Rate (MSPS) 55 50 52
Bits per Sample 8 8 $\times$ 63/128 12 $\times$1/72
Generation Rate (Gbps) 3.08 1.37 0.026
Min-Entropy per 8 bits 7.897 7.902 7.890
IID Test[@Sonmez:16] Pass Pass Pass
Randomness Test[@Rukhin] Pass Pass Pass
: Summary of results for our three constructions. The bitrate can be found by multiplication of sampling rate, number of extractors, and extracted bits per sample.The min-entropy per 8 bits for extractor C can be obtained from the reported min-entropy per bit = 0.986 multiplied by eight.
\[table\]
If each sample in a test set from a noise source is mutually independent and have the same probability distribution, that noise source is considered to be independent and identically distributed (IID). The NIST SP800-90B entropy assessment package [@github] uses a range of statistical tests to attempt to prove that a sample is not IID. If none of the tests fail, the noise source is assumed IID. The output entropy of the raw bit extraction (A) is tested using the entropy estimate procedure from NIST SP800-90B, and find the sample passes the IID test with an entropy of 7.897 bits. This extractor is designed to minimise the effect of the classical noise on the output signal. The total bit rate of this construction is given by the product of the sample rate (55MSPS), extracted bits per sample (8), and number of channels (7), and is 3.08 Gbit/s. We sample at least 27.5 MHz of the homodyne detector bandwidth as allowed by the Shannon-Hartley limit[@Hart:17]. The sampling rate is limited by the interface between the ADC and FPGA. Thus, we generate 112 Mbit/s per MHz of single channel detector bandwidth. We note that previous implementations have sampled more than an order-of-magnitude more detector bandwidth with superior detectors and digitization [@Haw:15], which will enable parallel QRNG from vacuum to reach much faster rates than in this demonstration.
Using this entropy estimate of the 8-bit raw data we construct a vetted CMAC keyed extractor (B), taking 63 out of 128 bits of the output to ensure the number of bits we use is less than half the input entropy. Entropy tests of the output give a min-entropy of 7.902 bits and the sample passes the IID test. Finally, we measure the output entropy of our two-source extractor(C) to be 0.986 bits as it is a single bit extractor, and it also passes the IID test. Whilst the output bit rate is much lower because it requires two 36 bit inputs (72 raw bits total) to produce one output bit, it removes the necessity for an externally generated seed, prevalent in most QRNG demonstrations. The results for all three extractors are summarized in Table \[table\]. The NIST statistical test suite \[37\] is also used to identify any statistical correlations that may make the data non-random. We run the test suite on each of our constructions over a minimum sample size of $7\times 10^8$ bits and find extraction methods A, B, and C pass all tests.
![Correlation between the output of a single two-source extractor and its inputs 1(a) and 2(b). Both positive (squares) and negative (triangles) correlations are plotted, with the y-axis shared between plots. The black dotted line represents the correlation of perfectly random data for the sample size, $4.8\times 10^{-4}$.[]{data-label="fig3"}](Input_Output_Correlation5.png){width="49.00000%"}
The two-source extractor we implement is strong given perfectly independent inputs i.e., the output is uncorrelated to either of the inputs. We quantify the effect of experimental imperfections in the input independence by calculating the cross-correlation between each input and the output as a measure of extractor strength, shown in Fig. 3 for $4.3\times 10^6$ samples. The theoretical correlation of perfectly random data for the sample size is $4.8\times 10^{-4}$.
Conclusion
==========
In summary, we have demonstrated a high-speed parallel/ multiplexed quantum random number generator, a configuration ideally suited to a range of platforms, as well as capable of enhancing real time QRNG rates. Parallelisation of random number generation is an effective way to increase the real-time bit rate of QRNG’s, and to supply quantum random numbers to distributed or cluster based computation and parallel communication systems. Furthermore, the parallel architecture allows us to demonstrate a high-speed un-keyed strong extraction(C) to create random numbers without the need for an external provider of uniform random seeds. True randomness sources that do not need a random seed have practical security by relying only on the validity of the partially random sources, and not requiring an external source of true randomness.
The highly specialized nature of the optical and electronic components makes a system on a single circuit board the realistic short-term integration option for GHz rate QRNG. To continue the scaling of this system to hundreds of channels full integration of a high power laser, waveguides, photodiodes, and processing electronics on a single chip will be necessary, and silicon offers a suitable platform for both electronic and optical components [@Silverstone:16]. The simplicity of the entropy source makes quantum vacuum fluctuations an excellent choice for parallelisation with Gbps single channel bitrates reported.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Stefan Morley and Xingxing Xing for electronics support and Zachary Vernon for comments on the manuscript. BH is supported by the Australian Government Research Training Program Scholarship. This research is financially supported by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (CE170100012) and the Griffith University Research Infrastructure Programme.This work was performed in part at the Queensland node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and microfabrication facilities for Australia’s researchers.
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[^1]: This author contributed equally to this work
[^2]: This author contributed equally to this work
|
---
abstract: 'In this paper, we consider a nonlinear second order equation modelling rocket motion in the gravitational field obstructed by the drag force. The proofs of the main results are based on topological fixed point approach.'
address: |
Departament of Mathematics and Computer Science\
University of Łódź\
S. Banacha 22\
90-238 Łódź\
Instystut Matematyczny\
Uniwersytet Wrocławski\
Pl. Grunwaldzki 2/4\
50-368 Wrocław
author:
- Dorota Bors and Robert Stańczy
title: Some nonlinear second order equation modelling rocket motion
---
[^1]
Introduction
============
In this paper we consider the following second order differential equation $$\ddot{z}\left( t\right) =\alpha\left( t\right) \left( \dot{z}\left(
t\right) +a\right) ^{2}\exp\left( -\frac{z\left( t\right) }{H}\right)
+\beta\left( t\right) \label{BVP}$$ with suitably chosen functions $\alpha, \beta$ to be defined below by (\[albe\]) and some constants $a, H$ to be determined later and suitable initial or boundary conditions derived from some model describing the motion of the rocket, cf. [@Mes]. Specifically, it can be derived from the one-dimensional first oder differential equation $$m\left( t\right) \dot{v}\left( t\right) =\left( w\left( t\right)
-v\left( t\right) \right) \dot{m}\left( t\right) +f^{ext}\left(
t\right). \label{R1}$$ This equation can be used to model a rocket motion with decreasing mass $m=m\left(
t\right) $ and under external force $f^{ext}\left(
t\right)$. The rocket should be equipped with the engine that emits mass $m_{1}=m_{1}\left( t\right) $ with velocities $w=w\left( t\right) ,$ $t\in\left[ t_{0},t_{1}\right] .\,$Therefore, the difference between the mass of the rocket $m\left( t\right) $ and the mass of the gas emitted at time $t$ is constant and equal $m_{0}$ which is some positive value, i.e. $m(t)=m_{1}\left( t\right) +m_{0}$. In $m_{0}$ we include all components of the rocket infrastructure like the engine, empty tank, guidance equipment, payload and the like. The function $f^{ext}=f^{ext}\left( t\right) $ denotes all external forces that stimulate the rocket motion for example the gravity force or the drag force and in fact may depend also on the position and the velocity of the rocket, cf. (\[fex\])
The expression $m\dot{v}(t)$ is the force acting on the rocket body which is given by the amount of mass expelled and the velocity of the mass relative to the rocket body, i.e. $ w(t)-v(t) \dot{m}_{1}(t)$. The rocket moves forward since the rocket loses mass as it accelerates, so $\dot{m}_{1}(t)$ must be negative for any rocket. The term $w-v \frac{dm_{1}}{dt}$ is called the thrust of the rocket and can be interpreted as an additional force on the rocket due to the gas expulsion. The detailed derivation of the rocket equation one can find, among others, in [@Kos], [@Mes] and [@Par].The equation $\left( \ref{R1}\right) $ is so called the Meščerskii rocket equation, for details see [@Mes], [@Kos], where alos the case of more that just one emitted mass was considered. If we assume, the external force to be zero and a constant relative velocity of the emitted mass, i.e. $f^{ext}=0$ and $w\left(
t\right) -v\left( t\right) =:c<0$ for $t\in\left[ t_{1},t_{2}\right] $, then the equation $\left( \ref{R1}\right) $ has the form of the well-known Tsiolkovskii rocket equation. The corresponding solution to the Tsiolkovskii equation with the initial conditions $m\left( t_{1}\right)
=\tilde{m}$ and $v\left( t_{1}\right) =\tilde{v}$ can be easily found after integration of $\left( \ref{R1}\right) $ and reads as follows$$v\left( t\right) =\tilde{v}+c\ln\frac{m\left( t\right) }{\tilde{m}}.$$ The Tsiolkovskii rocket equation in the gravitational field, i.e. $f^{ext}=-mg$ with the initial conditions $m\left( t_{1}\right) =\tilde{m}$ and $v\left( t_{1}\right) =\tilde{v}\ $has a solution of the form$$\label{Tso}
v\left( t\right) =\tilde{v}-g\left( t-t_{1}\right) +c\ln\frac{m\left(
t\right) }{\tilde{m}}.$$ Assume that $f^{ext}=-mg-D$ with $D=\frac{1}{2}\rho v^{2}AC_{D}$ where $C_{D}$ is a drag coefficient, $A$ is the cross sectional area of the rocket, $\rho$ is the air density that changes with altitude $x,$ and can be approximated by $\rho=\rho_{0}\exp\left( -\frac{x}{H}\right) $ where $H\approx8000m$ is the so called “scale height” of the atmosphere, and finally $\rho_{0}$ is the air density at sea level. In that case it is impossible to solve the equation explicitly. However, it is interesting to note that the effect of drag loss is usually quite small and is often reasonable to ignore it in the first approach. In practice, the drag force is approximately only about 2% of the gravity force.
Except for the rocket equation, the Tsiolkovskii equation can also be applied to the one photon decay of the excited nucleus. Moreover, it is possible to identify the Meščerskii equation with the bremsstrahlung equation where the electron is in the permanent decay. Many other decays can be investigated from the point of view of the Meščerskii equation, as one can see in [@Par].
In the paper we will consider the equation $\left( \ref{R1}\right) ,$ with a given relative velocity of emitted masses $c_{i}\left( t\right)
=v_{i}\left( t\right) -v\left( t\right) $ for any $i=1,...,N$ and $t\in\left[ t_{1},t_{2}\right] .$ Chemical rockets produce exhaust jets at velocities $\left\vert c_{i}\right\vert $ from $2500$ $m/s$ to $4500$ $m/s.$
If we assume that the relative velocity is a given function $c\left(
t\right) =w\left( t\right) -v\left( t\right) ,$ we obtain the following equation$$m\left( t\right) \dot{v}\left( t\right) =c\left( t\right) \dot{m}\left(
t\right) +f^{ext}\left( t\right) .$$ Let us put $v\left( t\right) =\dot{x}\left( t\right) $ where $x\left(
t\right) $ denotes rocket position at time $t$. Furthermore, we assume that the external force depends not only on time $t$ but also on the rocket position $x\left( t\right) $ and its velocity $v\left( t\right) $ according to gravitational and drag forces, i.e. $$\label{fex}
f^{ext}\left( t\right) =f\left( t,x\left( t\right) ,v\left( t\right)
\right) =-m\left( t\right) g-\frac{AC_{D}\rho_{0}}{2}v^{2}\left( t\right)
\exp\left( -\frac{x\left( t\right) }{H}\right)$$ such that $f:\left[ t_{0},t_{1}\right] \times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$.Therefore, we get the equation $$m\left( t\right) \ddot{x}\left( t\right) =c\left( t\right) \dot
{m}\left( t\right) -m\left( t\right) g-\frac{AC_{D}\rho_{0}}{2}\dot{x}^{2}\left( t\right) \exp\left( -\frac{x\left( t\right) }{H}\right)
\label{R2}$$ for any $t\in\left[ t_{0},t_{1}\right] .$We provide our equation $\left( \ref{R2}\right) $ with the boundary conditions$$x\left( t_{0}\right) =x_{0},\text{ }x\left( t_{1}\right) =x_{1},
\label{R3}$$ which guarantee that the rocket attains the given point in the required period of time.
After a shift $z\left( t\right) =x\left( t\right) -y\left( t\right) $ by a linear function$$y\left( t\right) =a\left( t-t_{0}\right) +x_{0}. \label{T}$$ where $a=\frac{x_{1}-x_{0}}{t_{1}-t_{0}}$ and equation $\left( \ref{R2}\right) $ reads $$\ddot{z}\left( t\right) =c\left( t\right) \frac{\dot{m}\left( t\right)
}{m\left( t\right) }-g-\frac{AC_{D}\rho_{0}}{2}\frac{\left( \dot{z}\left(
t\right) +a\right) ^{2}}{m\left( t\right) }\exp\left( -\frac{z\left(
t\right) +y\left( t\right) }{H}\right) \label{R4}$$ for any $t\in\left[ t_{0},t_{1}\right] $ and $z$ satisfies the homogenous Dirichlet boundary conditions$$z\left( t_{0}\right) =z\left( t_{1}\right) =0. \label{R4b}$$ Consequently, one can reduce considered equation to a aforementioned form$$\ddot{z}\left( t\right) =\alpha\left( t\right) \left( \dot{z}\left(
t\right) +a\right) ^{2}\exp\left( -\frac{z\left( t\right) }{H}\right)
+\beta\left( t\right) \label{BVP}$$ where$$\begin{aligned}
\alpha\left( t\right) & =-\frac{AC_{D}\rho_{0}}{2m\left( t\right) }\exp\left( -\frac{y\left( t\right) }{H}\right) ,\label{albe}\\
\beta\left( t\right) & =c\left( t\right) \frac{\dot{m}\left( t\right)
}{m\left( t\right) }-g.\nonumber\end{aligned}$$
Integral formulation\[Sec1\]
============================
It should be noted that due to the presence of the derivative $\dot{z}$ the equation $\left( \ref{R4}\right) $ is not in a variational form. Therefore our approach differs from the one employed in [@BorWal] (for extension to elliptic problems see [@Bor]) and makes use of integral formulation approach. Specifically the boundary value problem $\left( \ref{BVP}\right) $ with $\left( \ref{R4b}\right) $ can be reformulated in the following way$$z\left( t\right) ={\displaystyle\int\limits_{t_{0}}^{t_{1}}}
G\left( t,s\right) F\left( s,z\left( s\right) ,\dot{z}\left( s\right)
\right) ds+b\left( t\right) \label{Int}$$ where the nonlinear term $F$ is defined as $$F\left( s,z,\dot{z}\right) =\alpha\left( s\right) \left( \dot
{z}+a\right) ^{2}\exp\left( -\frac{z}{H}\right)$$ with the abuse of notation, namely $\dot{z}$ denotes in the above formula the third independent variable and the function $b$ is defined by$$b\left( t\right) ={\displaystyle\int\limits_{t_{0}}^{t_{1}}}
G\left( t,s\right) \beta\left( s\right) ds.$$ Recall that $\alpha\left( t\right) $ and $\beta\left( t\right) $ are defined by formulas $\left( \ref{albe}\right) .$ The function $G$ is the Green function, i.e. the continuous symmetric function satisfying the following conditions.
1. the homogeneous equation, i.e. for any $t\neq s$$$\frac{\partial^{2}G}{\partial t^{2}}\left( t,s\right) =0\,,$$
2. the boundary conditions, i.e. for any $s\in\left[ t_{0},t_{1}\right]$ $$G\left( t_{0},s\right) =G\left( t_{1},s\right) =0\,,$$
3. the jump condition, i.e. for any $t\in(t_{0},t_{1})$ $$\lim_{s\rightarrow t^{+}}\frac{\partial G}{\partial t}\left( t,s\right)
-\lim_{s\rightarrow t^{-}}\frac{\partial G}{\partial t}\left( t,s\right) =1\,.$$
This function can be obtained from $\left( \ref{BVP}\right) $ by double integration and use of the the boundary conditions $\left( \ref{R4b}\right)
$ and it reads $$G\left( t,s\right) =\left\{
\begin{array}
[c]{c}\frac{1}{t_{1}-t_{0}}\left( t-t_{0}\right) \left( t_{1}-s\right) \text{,
}t<s,\\
\frac{1}{t_{1}-t_{0}}\left( s-t_{0}\right) \left( t_{1}-t\right) \text{,
}t>s.
\end{array}
\right. \label{Green}$$ with the derivative$$\frac{\partial G}{\partial t}\left( t,s\right) =\left\{
\begin{array}
[c]{c}\frac{1}{t_{1}-t_{0}}\left( t_{1}-s\right) \text{, }t<s,\\
\frac{1}{t_{1}-t_{0}}\left( t_{0}-s\right) \text{, }t>s.
\end{array}
\right. \label{Greender}$$
Main results
============
We shall work in the space $C^{1}=C^{1}\left( \left[ t_{0},t_{1}\right]
,\mathbb{R}\right) $ of continuously differentiable functions equipped with the natural norm$$\left\Vert z\right\Vert =\max\left\{ \left\vert z\right\vert ,\left\vert
\dot{z}\right\vert \right\}$$ where $\left\vert z\right\vert =\sup_{t\in\left[ t_{0},t_{1}\right]
}\left\vert z\left( t\right) \right\vert $ and $\left\vert \dot
{z}\right\vert =\sup_{t\in\left[ t_{0},t_{1}\right] }\left\vert \dot
{z}\left( t\right) \right\vert $ for any $z\in C^{1}.$ Next we can denote the right hand side of $\left( \ref{Int}\right) $ as the value of the integral operator $Sz,$ i.e.$$Sz(t)={\displaystyle\int\limits_{t_{0}}^{t_{1}}}
G\left( t,s\right) F\left( s,z\left( s\right) ,\dot{z}\left( s\right)
\right) ds+b\left( t\right) .\label{Sz}$$ Thus the existence of solution to $\left( \ref{Int}\right) $ is reduced to finding a fixed point of the operator $S,$ i.e. such a $z\in C^{1}$ that $$Sz=z.$$ First of all, note that the operator $S$ is well-defined, continuous and compact. Indeed, for any $z\in C^{1}$ the function $Sz$ is continuous since the functions appearing under the integral sign $\left( \ref{Sz}\right) $ are uniformly continuous and $\left( Sz\right) ^{\prime}$ is also continuous by the integrability of $\frac{\partial G}{\partial t},$ cf. $\left(
\ref{Greender}\right) .$ The continuity of the operator $S$ follows from the smoothness of $F$ with respect both to $z$ and $\dot{z}$ and boundness of the functions $G$ and $\frac{\partial G}{\partial t}.$ The compactness of the operator $S$ requires the straightforward application of the Ascoli-Arzèla theorem.
Sign insensitive case
---------------------
In this subsection we provide crude but immediate estimates yielding by the Schauder fixed point theorem the existence of a fixed point for the operator $S.$ To this end we start with simple estimates of the following sup norms $|\cdot|$$$\begin{aligned}
\left\vert Sz\right\vert & \leq G_{0}\left\vert \alpha\right\vert \left(
\left\vert \dot{z}\right\vert ^{2}+2a\left\vert \dot{z}\right\vert +a\right)
+\left\vert b\right\vert \label{estimates}\\
\left\vert \left( Sz\right) ^{\prime}\right\vert & \leq G_{1}\left\vert
\alpha\right\vert \left( \left\vert \dot{z}\right\vert ^{2}+2a\left\vert
\dot{z}\right\vert +a\right) +\left\vert b\right\vert \nonumber\end{aligned}$$ where $$\begin{aligned}
G_{0} & =\sup_{t\in\left[ t_{0},t_{1}\right] }{\displaystyle\int\limits_{t_{0}}^{t_{1}}}
G\left( t,s\right) ds,\\
G_{1} & =\sup_{t\in\left[ t_{0},t_{1}\right] }{\displaystyle\int\limits_{t_{0}}^{t_{1}}}
\left|\frac{\partial G}{\partial t}\left( t,s\right) \right| ds.\end{aligned}$$ Using formula $\left( \ref{Green}\right) $ one can easily derive the value of $$G_{0}=\frac{3}{8}\left( t_{1}-t_{0}\right) ^{2}$$ and $$G_{1}=\left( t_{1}-t_{0}\right) ^{2}.$$
The boundary value problem $\left( \ref{R4b}\right) -\left( \ref{BVP}\right) $ admits at least one solution provided $\left( \ref{aest}\right) $ and $\left( \ref{best}\right) $ are satisfied.
As announced we will make use of the Schauder fixed point theorem for the operator $S$.defined by $\left( \ref{Sz}\right) .$ We should prove that it maps some ball $B\left( 0,R\right) $ in $C^{1}$ into itself. Using the preliminary estimates $\left( \ref{estimates}\right) $ and defining $$G_{2}=\left\vert \alpha\right\vert \max\left\{ G_{0},G_{1}\right\}=\left\vert \alpha\right\vert\left( t_{1}-t_{0}\right) ^{2}$$ the invariance of the ball $B\left( 0,R\right) $ under the action of $S$ can be reduced to the following condition$$G_{2}\left( R^{2}+2aR+a\right) +\left\vert b\right\vert \leq R$$ which can be satisfied by some $R>0$ if $$\Delta=\left( 2aG_{2}-1\right) ^{2}-4G_{2}\left( G_{2}a+\left\vert
b\right\vert \right) >0,$$ i.e. $$\left\vert b\right\vert <\frac{4a\left( a-1\right) G_{2}^{2}-4aG_{2}+1}{4G_{2}}\label{best}$$ and the value in the nominator is positive which is guaranteed provided $$G_{2}\in\left( 0,\frac{a-\sqrt{a}}{2a\left( a-1\right) }\right)
\cup\left( \frac{a+\sqrt{a}}{2a\left( a-1\right) },\infty\right) \text{,
}a>1.\label{aest}$$
The conditions $\left( \ref{aest}\right) $ and $\left( \ref{best}\right) $ are satisfied if $\beta$ is sufficiently small.
If $A=0,$ i.e. we neglect the drag force, accounting for negligable, in real world model, part, our assumptions are naturally satisfied. In such a context it is natural that one can attain the destination point in the desired period of time. In fact since the equation is linear one can derive the explicit formula for the solution.
Sign sensitive case
-------------------
In this section we provide more subtle a priori estimates taking into regard the negative sign of some terms appearing in $\left( \ref{BVP}\right) $ and instead of using Schauder Theorem, we apply more subtle Schafer Theorem. First, we neglect negative term in (\[BVP\]) and integrate inequality $$\ddot{z}(t)\le \beta (t)$$ using boundary conditions. Thus we get by (\[Tso\]) the bound for $v$ and hence for $z$. Next we multiply by $\dot{z}$ both sides of (\[BVP\]) and integrate after dividing by $\left( \dot{z}+a\right) ^{2}$... to get better a priori bounds.
[9]{} Bors D., Walczak S., Multidimensional second order systems with controls, Asian Journal of Control [**12**]{} (2010), pp. 159–167
Bors D., Superlinear elliptic systems with distributed and boundary controls, Control and Cybernetics [**34**]{} (2005), pp. 987–1004
Kosmodemianskii A.A. (1966), The course of theoretical mechanics II, Moscow (in Russian)
Meščerskii I.V. (1962) Works on mechanics of the variable mass, Moscow (in Russian)
Pardy M., The rocket equation for decays of elementary particles, Preprint arXiv:hep-ph/0608161v1 (2008)
Peraire J., Variable mass systems: the rocket equation, MIT OpenSourceWare, Massachusetts Institute of Technology, Available online (2004)
[^1]: This work has been partially supported by the Polish Ministry of Science project N N201 418839
|
---
abstract: 'This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more than 50 years. We illustrate our approach by showing that: (1) there are precisely 78[,]{}892 $(3,3,3;13)$ Ramsey colorings; and (2) if there exists a $(4,3,3;30)$ Ramsey coloring then it is (13,8,8) regular. Specifically each node has 13 edges in the first color, 8 in the second, and 8 in the third. We conjecture that these two results will help provide a proof that no $(4,3,3;30)$ Ramsey coloring exists implying that $R(4,3,3)=30$.'
author:
- Michael Codish
- Michael Frank
- Avraham Itzhakov
- Alice Miller
title: |
Solving Graph Coloring Problems with\
Abstraction and Symmetry[^1]
---
Introduction {#sec:intro}
============
This paper introduces a general methodology that applies to solve graph edge-coloring problems and demonstrates its application in the search for Ramsey numbers. These are notoriously hard graph coloring problems that involve assigning $k$ colors to the edges of a complete graph. In particular, $R(4,3,3)$ is the smallest number $n$ such that any coloring of the edges of the complete graph $K_n$ in three colors will either contain a $K_4$ sub-graph in the first color, a $K_3$ sub-graph in the second color, or a $K_3$ sub-graph in the third color. The precise value of this number has been sought for more than 50 years. Kalbfleisch [@kalb66] proved in 1966 that $R(4,3,3)\geq
30$, Piwakowski [@Piwakowski97] proved in 1997 that $R(4,3,3)\leq
32$, and one year later Piwakowski and Radziszowski [@PR98] proved that $R(4,3,3)\leq 31$. We demonstrate how our methodology applies to provide further evidence that $R(4,3,3)=30$.
Solving hard search problems on graphs, and graph coloring problems in particular, relies heavily on breaking symmetries in the search space. When searching for a graph, the names of the vertices do not matter, and restricting the search modulo graph isomorphism is highly beneficial. When searching for a graph coloring, on top of graph isomorphism, solutions are typically closed under permutations of the colors: the names of the colors do not matter and the term often used is “weak isomorphism” [@PR98] (the equivalence relation is weaker because both node names and edge colors do not matter). When the problem is to compute the set of all solutions modulo (weak) isomorphism the task is even more challenging. Often one first attempts to compute all of solutions of the coloring problem, and to then apply one of the available graph isomorphism tools, such as `nauty` [@nauty] to select representatives of their equivalence classes modulo (weak) isomorphism. However, typically the number of solutions is so large that this approach is doomed to fail even though the number of equivalence classes itself is much smaller. The problem is that tools such as `nauty` apply after, and not during, search. To this end, we first observe that the technique described in [@DBLP:conf/ijcai/CodishMPS13] for graph isomorphism applies also to weak isomorphism, facilitating symmetry breaks during the search for solutions to graph coloring problems. This form of symmetry breaking is an important component in our methodology but on its own cannot provide solutions to hard graph coloring problems.
When confronted with hard computational problems, a common strategy is to consider approximations which focus on “abstract” solutions which characterize properties of the actual “concrete” solutions. To this end, given a graph coloring problem with $k$ colors on $n$ nodes, we introduce the notion of an $n\times k$ *degree matrix* in which each of $n$ rows describes the degrees of a coresponding node in the $k$ colors. In case the graph coloring problem is too hard to solve directly, we seek, possibly an over approximation of, all of the degree matrices of its solutions. This enables a subsequent independent search of solutions “per degree matrix” facilitating so called “embarrassingly parallel” search.
After laying the ground for a methodology based on symmetry breaking and abstraction we apply it to the problem of computing the Ramsey number $R(4,3,3)$ which reduces to determining if there exists a $(4,3,3)$ coloring of the complete graph $K_{30}$. We first characterize the degrees of the nodes in each of the three colors in any such coloring, if one exists. To this end, we show that if there is such a graph coloring then, up to swapping the colors two and three, all of its vertices have degrees in the three colors corresponding to the following triples: $(13, 8, 8)$, $(14, 8, 7)$, $(15, 7, 7)$, $(15, 8, 6)$, $(16, 7, 6)$, $(16, 8, 5)$. Then, we demonstrate that any potential $(4,3,3;30)$ coloring with a node with degrees $(d_1,d_2,d_3)$ in the corresponding colors must have three corresponding embedded graphs $G_1, G_2, G_3$ which are $(3,3,3;d_1)$, $(4,2,3;d_2)$, and $(4,3,2;d_3)$ colorings. For all of the cases except when the degrees are $(13,8,8)$ these sets of colorings are known and easy to compute. Based on this, we show using a SAT solver that there can be no nodes with degrees $(14, 8, 7)$, $(15, 7, 7)$, $(15, 8, 6)$, $(16, 7, 6)$ or $(16, 8, 5)$ in any $(4,3,3;30)$ coloring. Thus, we prove that any such coloring would have to be $(13,8,8)$ regular, meaning that all nodes are of degree 13 in the first color and of degree 8 in the second and third color.
In order to apply the same proof technique for the case where the graph is $(13,8,8)$ regular we need to first compute the set of all $(3,3,3;13)$ colorings, modulo weak isomorphism. This set of graphs does not appear in previously published work. So, we address the problem of computing the set of all $(3,3,3;13)$ Ramsey colorings, modulo weak isomorphism. This results in a set of 78[,]{}892 graphs. The set of $(3,3,3;13)$ Ramsey colorings has recently been independently computed by at least three other researchers: Richard Kramer, Ivan Livinsky, and Stanislaw Radziszowski [@stas:personalcommunication].
Finally, we describe the ongoing computational effort to prove that there is no $(13,8,8)$ regular $(4,3,3)$ Ramsey coloring. Using the embedding approach, and given the 78[,]{}892 (3,3,3;13) colorings there are 710[,]{}028 instances to consider. Over the period of 3 months we have verified using a SAT solver that 315[,]{}000 of these are not satisfiable. When this effort completes we will know if the value of $R(4,3,3)$ is 30 or 31.
Throughout the paper we express graph coloring problems in terms of constraints via a “mathematical language”. Our implementation uses the , finite-domain constraint compiler [@jair2013], which solves constraints by encoding them to CNF and applying an underlying SAT solver. The solver can be applied to find a single (first) solution to a constraint, or to find all solutions for a constraint modulo a specified set of (integer and/or Boolean) variables. Of course, correctness of our results assumes a lack of bugs in the tools we have used including the constraint solver and the underlying SAT solver. To this end we have performed our computations using four different underlying SAT solvers: MiniSAT [@minisat; @EenS03], CryptoMiniSAT [@Crypto], Glucose [@Glucose; @AudemardS09], and Lingeling [@Lingeling; @Biere14]. configures directly with MiniSAT, CryptoMiniSAT, and Glucose. For the experiments with Lingeling we first apply to generate a CNF (dimacs) file and subsequently invoke the SAT solver. Lingeling, together with Druplig [@BiereLing], provides a proof certificate for unsat instances (and we have taken advantage of this option). All computations were performed on a cluster with a total of $228$ Intel E8400 cores clocked at 2 GHz each, able to run a total of $456$ parallel threads. Each of the cores in the cluster has computational power comparable to a core on a standard desktop computer. Each SAT instance is run on a single thread.
The notion of a “degree matrix” arises in the literature with several different meanings. Degree matrices with the same meaning as we use in in this paper are considered in [@CarrollIsaak2009]. Gent and Smith [@GentS00], building on the work of Puget [@Puget93], study symmetries in graph coloring problems and recognize the importance of breaking symmetries during search. Meseguer and Torras [@MeseguerT01] present a framework for exploiting symmetries to heuristically guide a depth first search, and show promising results for $(3,3,3;n)$ Ramsey colorings with $14\leq n\leq 17$. Al-Jaam [@Jaam07] proposes a hybrid meta-heuristic algorithm for Ramsey coloring problems, combining tabu search and simulated annealing. While all of these approaches report promising results, to the best of our knowledge, none of them have been successfully applied to solve open instances or improve the known bounds on classical Ramsey numbers. Our approach focuses on symmetries due to weak-isomorphism for graph coloring and models symmetry breaking in terms of constraints introduced as part of the problem formulation. This idea, advocated by Crawford [[*et al.*]{}]{} [@crawford96], has previously been explored in [@DBLP:conf/ijcai/CodishMPS13] (for graph isomorphism), and in [@Puget93] (for graph coloring).
Graph coloring has many applications in computer science and mathematics, such as scheduling, register allocation and synchronization, path coloring and sensor networks. Specifically, many finite domain CSP problems have a natural representation as graph coloring problems. Our main contribution is a general methodology that applies to solve graph edge coloring problems. The application to potentially compute an unknown Ramsey number is attractive, but the importance here is in that it shows the utility of the methodology.
Preliminaries
=============
An $(r_1,\ldots,r_k;n)$ Ramsey coloring is an assignment of one of $k$ colors to each edge in the complete graph $K_n$ such that it does not contain a monochromatic complete sub-graph $K_{r_i}$ in color $i$ for $1\leq i\leq k$. The set of all such colorings is denoted ${{\cal R}}(r_1,\ldots,r_k;n)$. The Ramsey number $R(r_1,\ldots,r_k)$ is the least $n>0$ such that no $(r_1,\ldots,r_k;n)$ coloring exists. In the multicolor case ($k>2$), the only known value of a nontrivial Ramsey number is $R(3,3,3)=17$. The value of $R(4,3,3)$ is known to be equal either to 30 or to 31. The numbers of $(3,3,3;n)$ colorings are known for $14\leq n\leq 16$ but prior to this paper the number of colorings for $n=13$ was unpublished. Recently, the set of all $(3,3,3;13)$ colorings has also been computed by other researchers [@stas:personalcommunication], and they number 78[,]{}892 as reported also in this paper. More information on recent results concerning Ramsey numbers can be found in the electronic dynamic survey by Radziszowski [@Rad].
In this paper, graphs are always simple, i.e. undirected and with no self loops. Colors are associated with graph edges. The set of neighbors of a node $x$ is denoted $N(x)$ and the set of neighbors by edges colored $c$, by $N_c(x)$. For a natural number $n$ denote $[n]=\{1,2,\ldots,n\}$. A graph coloring, in $k$ colors, is a pair $(G,\kappa)$ consisting of a simple graph $G=([n],E)$ and a mapping $\kappa\colon E\to[k]$. When $\kappa$ is clear from the context we refer to $G$ as the graph coloring. The sub-graph of $G$ induced by the color $c\in[k]$ is the graph $G^c=([n],{\left\{~e\in E \left|
\begin{array}{l}\kappa(e)=c\end{array}
\right. \right\}})$. The sub-graph of $G$ on the $c$ colored neighbors of a node $x$ is the projection of the labeled edges in $G$ to $N_c(x)\times N_c(x)$ and denoted $G^c_x$. We typically represent $G$ as an $n\times n$ adjacency matrix, $A$, defined such that $$A_{i,j}= \begin{cases}\kappa(i,j) & \mbox{if } (i,j) \in E\\
0 & \mbox{otherwise}
\end{cases}$$ If $A$ is the adjacency matrix representing the graph $G$, then we denote the Boolean adjacency matrix corresponding to $G^c$ as $A[c]$. We denote the $i^{th}$ row of a matrix $A$ by $A_i$. The color-$c$ degree of a node $x$ in $G$ is denoted $deg_{G^c}(x)$ and is equal to the degree of $x$ in the induced sub-graph $G^c$. When clear from the context we write $deg_{c}(x)$. Let $G=([n],E)$ and $\pi$ be a permutation on $[n]$. Then $\pi(G) =
(V,{\left\{~ (\pi(x),\pi(y)) \left|
\begin{array}{l} (x,y) \in E\end{array}
\right. \right\}})$. Permutations act on adjacency matrices in the natural way: If $A$ is the adjacency matrix of a graph $G$, then $\pi(A)$ is the adjacency matrix of $\pi(G)$ obtained by simultaneously permuting with $\pi$ both rows and columns of $A$.
$$\begin{aligned}
\varphi_{adj}^{n,k}(A) &=& \hspace{-2mm}\bigwedge_{1\leq q<r\leq n}
\left(\begin{array}{l}
1\leq A_{q,r}\leq k ~~\land~~ A_{q,r} = A_{r,q} ~~\land ~~ A_{q,q} = 0
\end{array}\right)
\label{constraint:simple}
\\
\varphi_{K_3}^{n,c}(A) &=& \hspace{-2mm}\bigwedge_{1\leq q<r<s\leq n}\hspace{-3mm}
\neg~ \bigg(A_{q,r} = A_{q,s} = A_{r,s} = c\bigg)
\label{constraint:nok3}
\\
\varphi_{K_4}^{n,c}(A) &=& \hspace{-2mm}\bigwedge_{1\leq q<r<s<t\leq n}\hspace{-4mm}
\neg\left(\begin{array}{l}
A_{q,r} = A_{q,s} = A_{q,t} = A_{r,s} = A_{r,t} = A_{s,t} = c
\end{array}\right)
\label{constraint:nok4}\end{aligned}$$
$$\begin{aligned}
\label{constraint:r333}
\varphi_{(3,3,3;n)}(A) & = & \varphi_{adj}^{n,3}(A) \land \hspace{-2mm}
\bigwedge_{1\leq c\leq 3} \hspace{-1mm}
\varphi_{K_3}^{n,c}(A) \\
\label{constraint:r334}
\varphi_{(4,3,3;n)}(A) & = & \varphi_{adj}^{n,3}(A) \land \hspace{-2mm}
\bigwedge_{1\leq c\leq 2} \hspace{-1mm}
\varphi_{K_3}^{n,c}(A) \land\varphi_{K_4}^{n,3}(A)~~\end{aligned}$$
A graph coloring problem is a formula $\varphi(A)$ where $A$ is an $n\times n$ adjacency matrix of integer variables together with a set (conjunction) of constraints $\varphi$ on these variables. A solution is an assignment of integer values to the variables in $A$ which satisfies $\varphi$ and determine both the graph edges and their colors. We often refer to a solution as an integer adjacency matrix and denote the set of solutions as $sol(\varphi(A))$. Figure \[fig:gcp\] illustrates the two graph coloring problems we focus on in this paper: $(3,3,3;n)$ and $(4,3,3;n)$ Ramsey colorings. In Constraint (\[constraint:simple\]), $\varphi_{adj}^{n,k}(A)$, states that the graph $A$ has $n$ vertices, is $k$ colored, and is simple (symmetric, and with no self loops). In Constraints (\[constraint:nok3\]) and (\[constraint:nok4\]), $\varphi_{K_3}^{n,c}(A)$ and $\varphi_{K_4}^{n,c}(A)$ state that the $n$ vertex graph $A$ has no embedded sub-graph $K_3$, and respectively $K_4$, in color $c$. In Constraints (\[constraint:r333\]) and (\[constraint:r334\]), the formulas state that a graph $A$ is a $(3,3,3;n)$ and respectively a $(4,3,3;n)$ Ramsey coloring.
For graph coloring problems, solutions are typically closed under permutations of vertices and of colors. Restricting the search space for a solution modulo such permutations is crucial when trying to solve hard graph coloring problems. It is standard practice to formalize this in terms of graph (coloring) isomorphism.
\[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorings with $G=([n],E_1)$ and $H=([n],E_2)$. We say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are weakly isomorphic, denoted $(G,{\kappa_1})\approx(H,{\kappa_2})$ if there exist permutations $\pi \colon [n] \to [n]$ and $\sigma \colon [k] \to [k]$ such that $(u,v) \in E_1 \iff
(\pi(u),\pi(v)) \in E_2$ and $\kappa_1(u,v) = \sigma(\kappa_2(\pi(u),
\pi(v)))$. When $\sigma$ is the identity permutation, (i.e. $\kappa_1(u,v) =
\kappa_2(\pi(u),\pi(v))$) we say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are isomorphic. We denote such a weak isomorphism thus: $(G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$.
The following lemma emphasizes the importance of weak graph isomorphism as it relates to Ramsey numbers. Many classic coloring problems exhibit the same property.
Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be graph colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\sigma}
(H,\kappa_2)$. Then, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n)$
$\iff$ $(H,\kappa_2) \in
{{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n).$$
Assume that $(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n)$ and in contradiction that $(H,\kappa_2) \notin
{{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$. Let $R$ denote a monochromatic clique of size $r_s$ in $H$ and $R^{-1}$ the inverse of $R$ in $G$. From Definition \[def:weak\_iso\], $(u,v) \in R \iff (\pi^{-1}(u), \pi^{-1}(v))\in R^{-1}$ and $\kappa_2(u,v) = \sigma^{-1}(\kappa_1(u,v))$. Consequently $R^{-1}$ is a monochromatic clique of size $r_s$ in $(G,\kappa_1)$ in contradiction to $(G,\kappa_1)$ $\in$ ${{\cal R}}(r_1,r_2,\ldots,r_k;n)$.
Codish [[*et al.*]{}]{} introduce in [@DBLP:conf/ijcai/CodishMPS13] an approach to break symmetries due to graph isomorphism (without colors) during the search for a solution to general graph problems. Their approach involves adding a symmetry breaking predicate ${\textsf{sb}}^*_\ell(A)$, as advocated by Crawford [[*et al.*]{}]{} [@crawford96], on the variables of the adjacency matrix, $A$, when solving graph problems. In [@CodishMPS14] the authors show that the symmetry breaking approach of [@DBLP:conf/ijcai/CodishMPS13] holds also for graph coloring problems where the adjacency matrix consists of integer variables (the proofs for the integer case are similar to those for the Boolean case).
**[@DBLP:conf/ijcai/CodishMPS13].** \[def:SBlexStar\] Let $A$ be an $n\times n$ adjacency matrix. Then, viewing the rows of $A$ as strings, $${\textsf{sb}}^*_\ell(A) = \bigwedge_{i<j}
A_{i}\preceq_{\{i,j\}}A_{j}$$ where $s\preceq_{\{i,j\}}s'$ is the lexicographic order on strings $s$ and $s'$ after simultaneously omitting the elements at positions $i$ and $j$.
Table \[tab:333n1\] illustrates the impact of the symmetry breaking technique introduced by Codish [[*et al.*]{}]{} in [@CodishMPS14] on the search for $(3,3,3;n)$ Ramsey colorings. The column headed by “\#${\setminus}_{\approx}$” specifies the known number of colorings modulo weak isomorphism [@Rad]. The columns headed by “\#vars” and “\#clauses” indicate, respectively, the number of variables and clauses in the corresponding CNF encodings of the coloring problems with and without the symmetry breaking constraint. The columns headed by “time” indicate the time (in seconds, on a single thread of the cluster) to find all colorings iterating with a SAT solver. The timeout assumed here is 24 hours. The column headed by “\#” specifies the number of colorings found when solving with the symmetry break. These include colorings which are weakly isomorphic, but far fewer than the hundreds of thousands generated without the symmetry break (until the timeout). The results in this table were obtained using the CryptoMiniSAT solver [@Crypto].
$\left[
\begin{smallmatrix}
0 &1 &1 &1 &1 &1 &2 &2 &2 &2 &2 &3 &3 &3 &3 &3 \\
1 &0 &2 &2 &3 &3 &1 &1 &2 &2 &3 &1 &1 &2 &3 &3 \\
1 &2 &0 &3 &2 &3 &1 &2 &1 &3 &2 &2 &3 &1 &1 &3 \\
1 &2 &3 &0 &3 &2 &2 &1 &3 &1 &2 &3 &2 &1 &3 &1 \\
1 &3 &2 &3 &0 &2 &2 &3 &1 &2 &1 &1 &3 &3 &2 &1 \\
1 &3 &3 &2 &2 &0 &3 &2 &2 &1 &1 &3 &1 &3 &1 &2 \\
2 &1 &1 &2 &2 &3 &0 &3 &3 &1 &1 &2 &3 &2 &3 &1 \\
2 &1 &2 &1 &3 &2 &3 &0 &1 &3 &1 &3 &2 &2 &1 &3 \\
2 &2 &1 &3 &1 &2 &3 &1 &0 &1 &3 &2 &1 &3 &2 &3 \\
2 &2 &3 &1 &2 &1 &1 &3 &1 &0 &3 &1 &2 &3 &3 &2 \\
2 &3 &2 &2 &1 &1 &1 &1 &3 &3 &0 &3 &3 &1 &2 &2 \\
3 &1 &2 &3 &1 &3 &2 &3 &2 &1 &3 &0 &2 &1 &1 &2 \\
3 &1 &3 &2 &3 &1 &3 &2 &1 &2 &3 &2 &0 &1 &2 &1 \\
3 &2 &1 &1 &3 &3 &2 &2 &3 &3 &1 &1 &1 &0 &2 &2 \\
3 &3 &1 &3 &2 &1 &3 &1 &2 &3 &2 &1 &2 &2 &0 &1 \\
3 &3 &3 &1 &1 &2 &1 &3 &3 &2 &2 &2 &1 &2 &1 &0
\end{smallmatrix}\right]$ $\left[
\begin{smallmatrix}
0 &1 &1 &1 &1 &1 &2 &2 &2 &2 &2 &3 &3 &3 &3 &3 \\
1 &0 &2 &2 &3 &3 &1 &1 &2 &2 &3 &1 &1 &2 &3 &3 \\
1 &2 &0 &3 &2 &3 &2 &3 &1 &1 &2 &1 &2 &3 &1 &3 \\
1 &2 &3 &0 &3 &2 &1 &2 &1 &3 &2 &2 &3 &1 &3 &1 \\
1 &3 &2 &3 &0 &2 &3 &2 &2 &1 &1 &3 &1 &3 &2 &1 \\
1 &3 &3 &2 &2 &0 &2 &1 &3 &2 &1 &3 &3 &1 &1 &2 \\
2 &1 &2 &1 &3 &2 &0 &3 &3 &1 &1 &2 &3 &2 &1 &3 \\
2 &1 &3 &2 &2 &1 &3 &0 &1 &1 &3 &3 &2 &2 &3 &1 \\
2 &2 &1 &1 &2 &3 &3 &1 &0 &3 &1 &2 &1 &3 &3 &2 \\
2 &2 &1 &3 &1 &2 &1 &1 &3 &0 &3 &3 &2 &1 &2 &3 \\
2 &3 &2 &2 &1 &1 &1 &3 &1 &3 &0 &1 &3 &3 &2 &2 \\
3 &1 &1 &2 &3 &3 &2 &3 &2 &3 &1 &0 &2 &1 &2 &1 \\
3 &1 &2 &3 &1 &3 &3 &2 &1 &2 &3 &2 &0 &1 &1 &2 \\
3 &2 &3 &1 &3 &1 &2 &2 &3 &1 &3 &1 &1 &0 &2 &2 \\
3 &3 &1 &3 &2 &1 &1 &3 &3 &2 &2 &2 &1 &2 &0 &1 \\
3 &3 &3 &1 &1 &2 &3 &1 &2 &3 &2 &1 &2 &2 &1 &0
\end{smallmatrix} \right]$ $\left[
\begin{smallmatrix}
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5 \\
5 & 5 & 5
\end{smallmatrix} \right]$
\[333\_16\]
Figure \[333\_16\] depicts, on the left and in the middle, the two non-isomorphic colorings $(3,3,3;16)$ represented as adjacency graphs in the form found using the approach of Codish [[*et al.*]{}]{} [@CodishMPS14]. Note the lexicographic order on the rows in both matrices. These graphs are isomorphic to the two colorings reported in 1968 by Kalbfleish and Stanton [@KalbfleischStanton68] where it is also proven that there are no others (modulo weak isomorphism). The $16\times 3$ degree matrix (right) describes the degrees of each node in each color as defined below in Definition \[def:dm\]. The results reported in Table \[tab:333n1\] also illustrate that the approach of Codish [[*et al.*]{}]{} is not sufficiently powerful to compute the number of $(3,3,3;13)$ colorings. Likewise, it does not facilitate the computation of $R(4,3,3)$.
In the following we make use of the following results from [@PR98].
\[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $R(4,3,3)=31$ if and only if there exists a $(4,3,3;30)$ coloring $\kappa$ of $K_{30}$ such that: (1) For every vertex $v$ and $i\in\{2,3\}$, $5\leq deg_{i}(v)\leq
8$, and $13\leq deg_{1}(v)\leq 16$. (2) Every edge in the third color has at least one endpoint $v$ with $deg_{3}(v)=13$. (3) There are at least 25 vertices $v$ for which $deg_{1}(v)=13$, $deg_{2}(v)=deg_{3}(v)=8$.
The following is a direct consequence of Theorem \[thm:433\].
\[cor:degrees\] If $G$ is a $(4,3,3;30)$ coloring, and assuming without loss of generality that the degree of color two is greater equal to the degree of color three, then every vertex in $G$ has degrees in the corresponding colors corresponding to one of the triplets $(13, 8,
8)$, $(14, 8, 7)$, $(15, 7, 7)$, $(15, 8, 6)$, $(16, 7, 6)$, $(16,
8, 5)$.
Consider a vertex $v$ in a $(4,3,3;n)$ coloring and focus on the three subgraphs induced by the neighbors of $v$ in each of the three colors. The following states that these must be corresponding Ramsey colorings.
\[cor:embed\] Let $G$ is a $(4,3,3;n)$ coloring and $v$ be any vertex with degrees $(d_1,d_2,d_3)$ in the corresponding colors. Then, $d_1+d_2+d_3=n-1$ and $G^1_v$, $G^2_v$, and $G^3_v$ are respectively $(3,3,3;d_1)$, $(4,2,3;d_2)$, and $(4,3,2;d_3)$ colorings.
Note that by definition a $(4,2,3;n)$ coloring is a $(4,3;n)$ coloring in colors 1 and 3 and likewise a $(4,3,2;n)$ coloring is a $(4,3;n)$ coloring in colors 1 and 2. For $n\in\{14,15,16\}$, the set of all $(3,3,3;n)$ colorings modulo (weak) isomorphism are known and consist respectively of 2, 2 and 15 colorings. Similary, for $n\in\{5,6,7,8\}$ the set of all $(4,3;n)$ colorings modulo (weak) isomorphism are known and consist respectively of 9, 15, 9, and 3 Ramsey colorings.
Searching for Ramsey Colorings with Embeddings
==============================================
In this section we apply a general approach where, when seeking a $(r_1,\ldots,r_k;n)$ Ramsey coloring one selects a “preferred” vertex, call it $v_1$, and based on its degrees in each of the $k$ colors, embeds $k$ subgraphs which are corresponding smaller colorings. Using this approach, we apply Corollaries \[cor:degrees\] and \[cor:embed\] to establish that a $(4,3,3;30)$ coloring, if one exists, must be $(13,8,8)$ regular. Specifically, all vertices have 13 neighbors by way of edges in the first color and 8 neighbors each, by way of edges in the second and third colors.
\[thm:regular\] Any $(4,3,3;30)$ coloring, if one exists, is $(13,8,8)$ regular.
By computation as described in the rest of this section.
We seek a $(4,3,3;30)$ coloring of $K_{30}$, represented as a $30\times 30$ adjacency matrix $A$. We focus on the degrees, $(d_1,d_2,d_3)$ in each of the three colors, of the vertex $v_1$, corresponding to the first row in $A$, as prescribed by Corollary \[cor:degrees\]. For each such degree triplet, except for the case $(13,8,8)$, we take each of the known corresponding colorings for the subgraphs $G^1_{v_1}$, $G^2_{v_1}$, and $G^3_{v_1}$ and embed them in $A$. We then apply a SAT solver, to complete the remaining cells in $A$ to satisfy Constraint \[constraint:r334\] of Figure \[fig:gcp\]. If the SAT solver fails, then no such completion exists.
To illustrate the approach, consider the case where $v_1$ has degrees $(14,8,7)$ in the three colors. Figure \[embed\_14\_8\_7\] details one of the embeddings corresponding to this case. The first row of $A$ specifes the colors of the edges of the 29 neighbors of $v_1$. The symbol “$\_$” indicates an integer variable that takes a value between 1 and 3. The neighbors of $v_1$ in color 1 form a submatrix of $A$ embedded in rows (and columns) 2–15 of the matrix in the Figure. By Corollary \[cor:embed\] these are a $(3,3,3;14)$ Ramsey coloring and there are 115 possible such colorings modulo weak isonmorphism. The Figure details one of them. Similarly, there are 3 possible subgraphs for the neighbors of $v_1$ in color 2, (the $3$ $(4,2,3;8)$ colorings). In Figure \[embed\_14\_8\_7\], rows (and columns) 16–23 detail one such coloring. Finally, there are 9 possible subgraphs for the neighbors of $v_1$ in color 3, (the $9$ $(4,3,2;7)$ colorings). In Figure \[embed\_14\_8\_7\], rows (and columns) 24–30 detail one such coloring.
To summarize, Figure \[embed\_14\_8\_7\] is a partial instantiated adjacency matrix in which the first row determines the degrees of $v_1$ in each of the three colors, and where 3 corresponding subgraphs are embedded. The uninstantiated values in the matrix must be completed to obtain a solution that satisfies Constraint \[constraint:r334\] of Figure \[fig:gcp\]. This can be determined using a SAT solver. For the specific example in Figure \[embed\_14\_8\_7\], the CNF generated using our tool set consists of 33[,]{}959 clauses, involves 5[,]{}318 Boolean variables, and is shown to be unsatisfiable in 52 seconds of computation time. For the case where $v_1$ has degrees $(14,8,7)$ in the three colors this is one of $115\times 3\times 9 = 3105$ instances that need to be checked.
Table \[table:regular\] summarizes the experiment which proves Theorem \[thm:regular\]. For each of the possible degrees of vertex 1 in a $(4,3,3;30)$ coloring as prescribed by Corollary \[cor:degrees\], except $(13,8,8)$, and for each possible choice of colorings for the derived subgraphs $G^1_{v_1}$, $G^2_{v_1}$, and $G^3_{v_1}$, we apply a SAT solver to show that Constraint \[constraint:r334\] of Figure \[fig:gcp\] cannot be satisfied. The table details for each degree triple, the number of instances, their average size (number of clauses and Boolean variables), and the average and total times to show that the constraint is not satisfieable.
$v_1$ degrees \# instances \# clauses (avg.) \# vars (avg.) unsat (avg).) unsat (total)
--------------- ------------------ ------------------- ---------------- --------------- ---------------
(16,8,5) 54 (2\*3\*9) 32432 5279 51 sec. 0.77 hrs.
(16,7,6) 270 (2\*9\*15) 32460 5233 420 sec. 31.50 hrs.
(15,8,6) 90 (2\*3\*15) 33607 5450 93 sec. 2.32 hrs.
(15,7,7) 162 (2\*9\*9) 33340 5326 1554 sec. 69.94 hrs.
(14,8,7) 3105 (115\*3\*9) 34069 5324 294 sec. 253.40 hrs.
: Proving that any $(4,3,3;30)$ Ramsey coloring is $(13,8,8)$ regular (summary).[]{data-label="table:regular"}
To gain confidence in our implementation, we illustrate its application to find a $(4,3,3;29)$ coloring which is known to exist. This experiment involves some reverse engineering. In 1966 Kalbfleisch [@kalb66] reported the existence of a circulant $(3,4,4;29)$ coloring. Encoding Constraint \[constraint:r334\] with $n=29$, together with a constraint that states that the adjacency matrix $A$ is circulant, results in a CNF with 146[,]{}506 clauses and 8[,]{}394 variables. Using a SAT solver, we obtain a corresponding $(4,3,3;29)$ coloring in less than two seconds of computation time. The solution is $(12,8,8)$ regular and permuting its first row to be of the form $01111111111112222222233333333$ we extract from it three corresponding subgraphs: $G^1_{v_1}$, $G^2_{v_1}$ and $G^3_{v_1}$ which are respectively $(3,3,3;12)$, $(4,2,3;8)$ and $(4,3,2;8)$ Ramsey colorings. An embedding of these three in a $29\times 29$ adjacency matrix is depicted as Figure \[embed\_12\_8\_8\].
Applying a SAT solver to complete this embedding to a $(4,3,3;29)$ coloring that satsifies Constraint \[constraint:r334\] involves a CNF with 30[,]{}944 clauses and 4[,]{}736 variables and requires under two minutes of computation time.
#### Proving that $R(4,3,3)=30$.
To apply the embedding approach described in this section to prove that there is no $(4,3,3;30)$ Ramsey coloring which is $(13,8,8)$ regular would require considering all $(3,3,3;13)$ colorings modulo weak isomorphism. Doing this would constitute a proof that $R(4,3,3)=30$. We defer this discussion until after Section \[sec:dm2\] where we describe how we compute the set of all 78[,]{}892 $(3,3,3;13)$ Ramsey colorings modulo weak isomorphism.
Degree Matrices for Graph Colorings
===================================
We introduce an abstraction on graph colorings defined in terms of *degree matrices* and an equivalence relation on degree matrices. Our motivation is to solve graph coloring problems by first focusing on an over approximation of their degree matrices. The equivalence relation on degree matrices enables us to break symmetries during search when solving graph coloring problems. Intuitively, degree matrices are to graph edge-colorings as degree sequences are to graphs.
\[def:dm\] Let $A$ be a graph coloring on $n$ vertices with $k$ colors. The *degree matrix* of $A$, denoted $\alpha(A)$ is an $n\times k$ matrix, $M$ such that $M_{i,j} = deg_j(i)$ is the degree of vertex $i$ in color $j$. For a set ${{\cal A}}$ of graph colorings we denote $\alpha({{\cal A}}) = {\left\{~\alpha(A) \left|
\begin{array}{l}A\in{{\cal A}}\end{array}
\right. \right\}}$.
A degree matrix, $M$, is said to [*represent*]{} the set of graphs weakly-isomorphic to a graph with degrees as in $M$. We say that two degree matrices are equivalent if they represent the same sets of graph colorings.
\[def:conc\] Let $M$ and $N$ be $n\times k$ degree matrices. Then, $\gamma(M) =
{\left\{~A \left|
\begin{array}{l}A\approx A',~\alpha(A')=M\end{array}
\right. \right\}}$ is the set of graph colorings represented by $M$ and we say that $M\equiv N \Leftrightarrow
\gamma(M)=\gamma(N)$. For a set ${{\cal M}}$ of degree matrices we denote $\gamma({{\cal M}}) = \cup{\left\{~\gamma(M) \left|
\begin{array}{l}M\in{{\cal M}}\end{array}
\right. \right\}}$.
Due to properties of weak-isomorphism (vertices as well as colors can be reordered) we can exchange both rows and columns of a degree matrix without changing the set of graphs it represents. In our construction we assume that the rows and columns of a degree matrix are sorted lexicographically. Observe also that the columns of a degree matrix each form a graphic sequence (when sorted).
For an $n\times k$ degree matrix $M$ we denote by $lex(M)$ the smallest matrix with rows and columns in the lexicographic order (non-increasing) obtained by permuting rows and columns of $M$.
The following implies that for degree matrices we can assume without loss of generality that rows and columns are lexicographically ordered.
\[thm:dme\] If $M$, $N$ are degree matrices then $M\equiv N$ if and only if there exists permutations $\pi \colon [n] \to [n]$ and $\sigma
\colon [k] \to [k]$ such that, for $1\leq i \leq n$, $1\leq j\leq
k$, $M_{i,j} = N_{\pi(i),\sigma(j)}$.
Let $M$ and $N$ be degree matrices. Then, $$\begin{array}{l}
M\equiv N \xLeftrightarrow{\mbox{\tiny{~Defn.~\ref{def:conc}~}}}
\gamma(M)=\gamma(N) \xLeftrightarrow{\mbox{\tiny ~Defn.~\ref{def:conc}~}}
\forall_{G\approx H}.G\in\gamma(M)\leftrightarrow H\in\gamma(N)
\xLeftrightarrow{\mbox{\tiny ~Defn.~\ref{def:weak_iso}~}}\\
\exists_{\pi,\sigma}.\alpha(G)_{i,j}=\alpha(H)_{\pi(i),\sigma(j)}
\xLeftrightarrow{\mbox{\tiny ~Defn.~\ref{def:dm}~}}
M_{i,j} = N_{\pi(i),\sigma(j)}
\end{array}$$
\[cor:lexM\] $M \equiv lex(M)$.
The result follows from Theorem \[thm:dme\] because $M$ and $lex(M)$ are related by permutations of rows and columns.
The degree matrix on the right of Figure \[333\_16\] describes both of the graphs in the figure.
Solving Graph Coloring Problems with Degree Matrices {#sec:adm}
====================================================
Let $\varphi(A)$ be a graph coloring problem in $k$ colors on an $n\times n$ adjacency matrix, $A$. Assuming that ${{\cal A}}=sol(\varphi(A))$ is too hard to compute, either because the number of solutions is too large or because finding even a single solution is too hard, our strategy is to first compute an over-approximation ${{\cal M}}$ of degree matrices such that $\gamma({{\cal M}})\supseteq{{\cal A}}$ and to then use ${{\cal M}}$ to guide the computation of ${{\cal A}}$. We denote the set of solutions of the graph coloring problem, $\varphi(A)$, which have a given degree matrix, $M$, by $sol_M(\varphi(A))$ and we have $$\label{eq:solM}
sol_M(\varphi(A)) = sol(\varphi(A)\wedge\alpha(A){=}M)$$ Note that $M\not\in\alpha(sol(\varphi(A)))\Rightarrow
sol_M(\varphi(A))=\emptyset$. Hence, for ${{\cal M}}\supseteq
\alpha(sol(\varphi(A)))$, $$\label{eq:approx}
sol(\varphi(A)) =
\bigcup_{M\in{{\cal M}}} sol_M(\varphi(A))$$ Equation (\[eq:approx\]) implies that, using any over-approximation ${{\cal M}}\supseteq \alpha(sol(\varphi(A)))$, we can compute the solutions to a graph coloring problem by computing the independent sets $sol_M(\varphi(A))$ for each $M \in {{\cal M}}$. This facilitates the computation of $sol(\varphi(A))$ for three reasons: (1) The problem is now broken into a set of independent sub-problems for each $M\in{{\cal M}}$ which can be solved in parallel. (2) The computation of each individual $sol_M(\varphi(A))$ is now directed using $M$, and (3) Symmetry breaking is facilitated.
There are two sources of symmetries when solving $\varphi(A)$. First, we compute ${{\cal M}}$ to consist of canonical degree matrices, sorted lexicographically by rows and by columns. Second, we impose an additional symmetry breaking constraint ${\textsf{sb}}^*_\ell(A,M)$ as explained below.
Consider a computation of all solutions of the constraint in the right side of Equation (\[eq:solM\]). Consider a permutation $\pi$ of the rows and columns of $A$, such that $\alpha(\pi(A))=\alpha(A)=M$. Then, both $A$ and $A'$ are solutions and they are weakly isomorphic. The following equation $$\label{eq:scenario1}
sol_M(\varphi(A)) = sol(\varphi(A)\wedge (\alpha(A){=}M) \wedge{\textsf{sb}}^*_\ell(A,M))$$ refines Equation (\[eq:solM\]) introducing a symmetry breaking constraint similar to the (partitioned lexicographic) symmetry break predicate introduced by Codish [[*et al.*]{}]{} in [@DBLP:conf/ijcai/CodishMPS13] for Boolean adjacency matrices. $$\label{eq:sbdm}
{\textsf{sb}}^*_\ell(A,M) =
\bigwedge_{i<j} \left(\begin{array}{l}
\big(M_i=M_j\Rightarrow A_i\preceq_{\{i,j\}} A_j\big)
\end{array}\right)$$ where $s\preceq_{\{i,j\}}s'$ denotes the lexicographic order on strings $s$ and $s'$ after simultaneously omitting the elements at positions $i$ and $j$.
To justify that Equations (\[eq:solM\]) and (\[eq:scenario1\]) both compute $sol_M(\varphi(A))$, modulo weak isomorphism, we must show that whenever ${\textsf{sb}}^*_\ell(A,M)$ excludes a solution then there is another weakly isomorphic solution that is not excluded. To this end, we introduce a definition and then a theorem.
Let $A$ be an adjacency matrix with a lexicographically ordered degree matrix $\alpha(A) = M$. We say that permutation $\pi$ is *degree matrix preserving* for $M$ and $A$ if $\alpha(\pi(A)) = M$.
Let $A$ be an adjacency matrix with a lexicographically ordered degree matrix $\alpha(A) = M$. Then, there exists a degree matrix preserving permutation $\pi$ such that $\alpha(\pi(A)) = M$ and ${\textsf{sb}}^{*}_\ell(\pi(A),M)$ holds.
If the rows of $M$ are distinct, then the theorem holds with $\pi$ the identity permutation. Assume that some rows of $M$ are equal. Denote by $P$ the set of degree matrix preserving permutations for $M$ and $A$. Assume the premise and that no $\pi\in
P$ satisfies ${\textsf{sb}}^{*}_\ell(\pi(A),M)$. Let $\pi\in P$ be such that $\pi(A) = \min{\left\{~\pi'(A) \in P \left|
\begin{array}{l}\pi'
\in P\end{array}
\right. \right\}}$ (in the lexicographical order viewing matrices as strings). From the assumption, there exist $i<j$ such that $M_i=M_j$ and $\pi(A)_i \not\preceq_{\{i,j\}} \pi(A)_j$. Hence there exists a minimal index $k\notin\{ i,j\}$ such that $\pi(A)_{i,k} > \pi(A)_{j,k}$. Let $A'$ be the matrix obtained by permuting nodes $i$ and $j$ in $\pi(A) $. Since $M_i = M_j$ it follows that $\alpha(A') = M$. Thus there is a $\pi' \in P$ such that $\pi'(A) = A'$. If $k < i$ : for $1\leq l < k$ we have $\pi(A)_l=A'_l$. Thus $k$ is the first row for which $A'$ and $\pi(A)$ differ. Permuting nodes $i$ and $j$ changes row $k$ by simply swapping elements $\pi(A)_{k,i}$ and $\pi(A)_{k,j}$. Since $\pi(A)_{k,i} >
\pi(A)_{k,j}$, clearly $A'_k \prec \pi(A)_k$ hence $A' \prec
\pi(A)$ which is a contradiction. Similarly if $k > i$ the same argument applies to show that $i$ is the first row for which $A'$ and $\pi(A)$ differ, thus obtaining the same contradiction for row $i$.
The following corollary clarifies that if a solution $A$ is eliminated when introducing the symmetry break predicate ${\textsf{sb}}^{*}_\ell(A,\alpha(A))$ to a graph coloring problem then there always remains an isomorphic solution $A'$ which satisfies the predicate ${\textsf{sb}}^{*}_\ell(A',\alpha(A'))$.
Let $A$ be an adjacency matrix. Then there exists $A'$ isomorphic to $A$ such that $\alpha(A')$ is lex ordered and ${\textsf{sb}}^{*}_\ell(A',\alpha(A'))$ holds.
Let $M = \alpha(A)$. From Corollary \[cor:lexM\] we know that $M \equiv
lex(M)$, thus there exists $A''$ isomorphic to $A$ such that $\alpha(A'') = lex(M)$. From Theorem 3 it follows that there exists a degree matrix preserving permutation $\pi$ such that $\alpha(\pi(A'')) = lex(M)$ and ${\textsf{sb}}^{*}_\ell(\pi(A''),\alpha(\pi(A'')))$ holds. If $A' = \pi(A'')$ then $A'$ is isomorphic to $A$, $\alpha(A')$ is lex ordered and ${\textsf{sb}}^{*}_\ell(A',\alpha(A'))$ holds.
Computing Degree Matrices for $R(3,3,3;13)$ {#sec:adm1}
===========================================
This section described how we compute a set ${{\cal M}}$ of degree matrices that approximate those of the solutions of Constraint \[constraint:r333\]. We apply a strategy in which we mix SAT solving with brute-force enumeration as follows. The computation of the degree matrices is summarized in Table \[tab:333\_computeDMs\].
In the first step, we compute bounds on the degrees of the nodes in any $R(3,3,3;13)$ coloring.
\[lemma:db\] Let $A$ be a $R(3,3,3;13)$ coloring then for every vertex $x$ in $A$, and color $c\in\{1,2,3\}$, $2\leq deg_{c}(x)\leq 5$.
By solving Constraint \[constraint:r333\] together with ${\textsf{sb}}^*_\ell(A,M)$ seeking a graph with minimal degree less than 2 or maximal degree greater than 5. The CNF encoding is of size 13672 clauses with 2748 Boolean variables and takes under 15 seconds to solve and yields an UNSAT result which implies that such graph does not exist.
In the second step, we enumerate the degree sequences with values within the bounds specified by Lemma \[lemma:db\]. Recall that the degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees. Not every non-increasing sequence of integers corresponds to a degree sequence. A sequence that corresponds to a degree sequence is said to be graphical. The number of degree sequences of graphs with 13 vertices is 836[,]{}315 (see Sequence number `A004251` of The On-Line Encyclopedia of Integer Sequences, published electronically at<http://oeis.org>). However, when the degrees are bound by Lemma \[lemma:db\] There are only 280.
\[lemma:ds\] There are 280 degree sequences with values between $2$ and $5$.
By straightforward enumeration using the algorithm of Erdos and Gallai [@ErdosGallai1960].
In the third step, we test each of the 280 degree sequences identified by Lemma \[lemma:ds\] to determine how many of them might occur as the left column in a degree matrix.
\[lemma:ds2\] Let $A$ be a $R(3,3,3;13)$ coloring and let $M$ be the canonical form of $\alpha(A)$. Then, (a) the left column of $M$ is one of the 280 degree sequences identified in Lemma \[lemma:ds\]; and (b) there are only 80 degree sequences from the 280 which are the left column of $\alpha(A)$ for some coloring $A$ in $R(3,3,3;13)$.
By solving Constraint \[constraint:r333\] with each degree sequence from Lemma \[lemma:ds\] to test if it is satisfiable. This involves 280 instances with average CNF size: 10861 clauses and 2215 Boolean variables. The total solving time is 375.76 hours and the hardest instance required about 50 hours. These instances were solved in parallel on the cluster described in Section \[sec:intro\].
In the fourth step we extend the 80 degree sequences identified in Lemma \[lemma:ds2\] to obtain all possible degree matrices.
\[lemma:dm\] Given the 80 degree sequences identified in Lemma \[lemma:ds2\] as potential left columns of a degree matrix, there are 11[,]{}933 possible degree matrices.
By straightforward enumeration. The rows and columns are lex sorted, must sum to 12, and the columns must be graphical (when sorted). We first compute all of the degree matrices and then select the smallest representatives under permutations of rows and columns. The computation requires a few seconds.
In the fifth step, we test each of the 11[,]{}933 degree matrices identified by Lemma \[lemma:ds2\] to determine how many of them are the abstraction of some $R(3,3,3;13)$ coloring.
\[lemma:dm2\] From the 11[,]{}933 degree matrices identified in Lemma \[lemma:dm\], 999 are $\alpha(A)$ for a coloring $A$ in $R(3,3,3;13)$.
By solving Constraint \[constraint:r333\] together with a given degree matrix to test if it is satisfiable. This involves 11[,]{}933 instances with average CNF size: 7632 clauses and 1520 Boolean variables. The total solving time is 126.55 hours and the hardest instance required 0.88 hours. These instances were solved in parallel on the cluster described in Section \[sec:intro\].
Step
------- -------------------------------------------------------------- -------------------- ------------ ------------
compute degree bounds (Lemma \[lemma:db\]) \#Vars \#Clauses
(1 instance, unsat) 2748 13672
enumerate 280 possible degree sequences (Lemma \[lemma:ds\])
test degree sequences (Lemma \[lemma:ds2\]) 16.32 hrs. \#Vars \#Clauses
(280 instances: 200 unsat, 80 sat) hardest: 1.34 hrs 1215 (avg) 7729(avg)
[4]{} enumerate 11[,]{}933 degree matrices (Lemma \[lemma:dm\])
test degree matrices (Lemma \[lemma:dm2\]) 126.55 hrs. \#Vars \#Clauses
(11[,]{}933 instances: 10[,]{}934 unsat, 999 sat) hardest: 0.88 hrs. 1520 (avg) 7632 (avg)
: Computing the degree matrices for ${{\cal R}}(3,3,3;13)$ step by step.[]{data-label="tab:333_computeDMs"}
Computing $R(3,3,3;13)$ from Degree Matrices {#sec:dm2}
============================================
We describe the computation of the set of all $(3,3,3;13)$ colorings starting from the 3805 degree matrices identified in Section \[sec:adm1\]. Table \[tab:333\_times\] summarizes the two step experiment reporting the computation on three different SAT solvers: MiniSAT [@minisat; @EenS03], CryptoMiniSAT [@Crypto], and Glucose [@Glucose; @AudemardS09].
#### **step 1:**
For each degree matrix we compute, using a SAT solver, all corresponding solutions of Equation (\[eq:scenario1\]), where $\varphi(A)$ is constraint (4) and $M$ is one of the 999 degree matrices identified in (Lemma \[lemma:dm2\]). These instances were solved in parallel on the cluster described in Section \[sec:intro\]. This generates in total 129[,]{}188 $(3,3,3;13)$ Ramsey colorings. Table \[tab:333\_times\] details the total solving time for these instances and the solving times for the hardest instance for each SAT solver. The largest number of graphs generated by a single instance is 3720.
#### **step 2:**
The 129[,]{}188 $(3,3,3;13)$ colorings from step 1 are reduced modulo weak-isomorphism using `nauty`[^2] [@nauty]. This process results in a set with 78[,]{}892 graphs.
Step
------- ------------------------------------------------- -------------------- -------------------- -------------------
compute all Ramsey $(3,3,3;13)$ MiniSAT CryptoMiniSAT Glucose
colorings per degree matrix total: 308.23 hr. total: 136.31 hr. total: 373.2 hr.
(999 instances, 129[,]{}188 solutions) hardest:9.15 hr. hardest:4.3 hr. hardest:17.67 hr.
[2]{} reduce modulo $\approx$. (78[,]{}892 solutions)
: Computing ${{\cal R}}(3,3,3;13)$ step by step.[]{data-label="tab:333_times"}
#### **Does $\mathbf{R(4,3,3)=30}$?**
Note that, in order to prove that there are no $(4,3,3;30)$ colorings with degrees $(13,8,8)$ using the embedding approach, we would need to check all embedding instances that contain one of the $(3,3,3;13)$ colorings, a $(4.2.3;8)$ coloring and a $(4,3,2;8)$ coloring. Since there are $78{,}892$, $3$ and $3$ of these colorings respectively, we can prove there are no $(4,3,3;30)$ colorings by showing that these $78,892\times 3\times 3$ embedding instances are unsatisfiable. We expect that this is the case and in the past three months have shown that 50% of the instances are indeed unsatisfiable. Ongoing computation is proceeding in order to complete the proof.
Conclusion
==========
We have applied SAT solving techniqes to show that any $(4,3,3;30)$ Ramsey coloring must be $(13,8,8)$ regular in the degrees of the three colors. In order to apply the same technique to show that there is no $(13,8,8)$ regular $(4,3,3;30)$ Ramsey coloring we would need to make use of the set of all $(3,3,3;13)$ colorings. We have computed this set modulo weak isomorphism. To this end we applied a technique involving abstraction and symmetry breaking to reduce the redundancies in the number of isomorphic solutions obtained when applying the SAT solver. Ongoing computation is proceeding to prove that $R(4,3,3)=30$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Stanislaw Radziszowski for his guidance and comments which helped to improve the presentation of this paper. In particular Stanislaw proposed to show that our techniique is able to find the $(4,3,3;29)$ coloring depicted as Figure \[embed\_12\_8\_8\].
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[^1]: Supported by the Israel Science Foundation, grant 182/13. Computational resources provided by an IBM Shared University Award (Israel).
[^2]: Note that `nauty` does not handle edge colored graphs and weak isomorphism directly. We applied an approach called $k$-layering described at <https://computationalcombinatorics.wordpress.com/2012/09/20/canonical-labelings-with-nauty>.
|
---
author:
- Sebastian Deffner and Steve Campbell
title: |
Quantum Thermodynamics\
An introduction to the thermodynamics of quantum information
---
|
---
author:
- 'Ž. Bošnjak'
- 'D. Götz'
- 'L. Bouchet'
- 'S. Schanne'
- 'B. Cordier'
bibliography:
- 'bosnjak\_integral.bib'
date: 'Received September ; accepted '
subtitle: 'results of the joint IBIS/SPI spectral analysis'
title: 'The spectral catalogue of [*INTEGRAL*]{} gamma-ray bursts:'
---
Introduction
============
Over the past two decades Gamma-Ray Bursts (GRBs) have been observed by several missions, providing a wealth of spectral and temporal data. The properties of the prompt gamma-ray emission have been studied over a broad energy range, from keV to GeV energies. Prompt GRB spectra are commonly described by two power laws smoothly connected around the spectral peak energy typically observed at a few hundred keV [@band93; @preece00]. The values of the spectral parameters, i.e. the slopes of the low- and high-energy power laws and the peak energy, are associated with the radiative mechanisms governing the emission and with the energy dissipation processes within the relativistic jet, and therefore impose important constraints on the theoretical models for prompt GRB emission. To date the most complete catalogues of spectral GRB properties comprise the events observed by BATSE (Burst And Transient Source Experiment) on board the [*Compton Gamma Ray Observatory*]{} in operation from 1991 to 2000 [@fishman89; @gehrels94], by the [*Swift*]{} satellite launched in 2004 [@gehrels04], and by the [*Fermi*]{} satellite launched in 2008 [@meegan09; @gehrels13].
[*INTEGRAL*]{} has contributed important discoveries to the GRB field, including the detection and observation of GRB 031203 associated with SN 2003lw [@malesani04], the polarization measurements from GRB 041219 [@gotz09] and GRB 061122 [@gotz13], and the discovery of the (inferred) population of low-luminosity GRBs [@foley08]. In this paper we present a catalogue of gamma-ray bursts detected by the [*INTEGRAL*]{} satellite. In the period between December 2002 and February 2012 [*INTEGRAL*]{} observed 83 GRBs (the low number of events with respect to other GRB missions, e.g. 2700 GRBs observed by BATSE in a nine year period, is mainly due to the small field of view of the IBIS instrument, $\sim$ 0.1 sr). We report the results of spectral analysis of 59 out of 83 GRBs. The spectral parameters were derived by combining the data from the two main instruments on board [*INTEGRAL*]{}, the spectrometer SPI nominally covering the energy range 18 keV - 8 MeV, and the imager IBIS with spectral sensitivity in the range 15 keV - 10 MeV. To date, the systematic spectral analysis of [*INTEGRAL*]{} GRBs has been performed in a limited energy range using only the data from the IBIS instrument [@vianello09; @foley08; @tierney10]. @foley08 reported the results of the spectral analysis using SPI data for nine GRBs, but, with one exception, the analysis of the IBIS data and the SPI data has been performed independently. In addition to the spectral analysis performed over a broad energy range for the complete sample of [*INTEGRAL*]{} GRBs, we have derived the IBIS light curves and durations for the previously unpublished 28 events observed between September 2008 and February 2012.
The paper is organised as follows. In Section 2 we discuss the catalogues of GRBs detected by BATSE, [*Fermi*]{} and [*Swift*]{}, and the possible biases in the results due to the instrumental differences. We compare the instrumental properties of different missions with those of the [*INTEGRAL*]{} instruments. The timing analysis and the spectral extraction technique we developed are presented in Section 3. The basic properties of [*INTEGRAL*]{} GRB sample are discussed in Section 4; we compare the global properties of our sample with the large GRB samples obtained by [*CGRO*]{} BATSE, [*Fermi*]{} Gamma-Ray Burst Monitor (GBM) and [*Swift*]{} Burst Alert Telescope (BAT) instruments. We report the results of the spectral analysis in Section 5, and make a statistical comparison of our results with respect to BATSE and [*Fermi*]{}/GBM samples. The summary of our results is presented in Section 6.
GRB samples
===========
A systematic spectral analysis of a sub-sample of 350 bright GRBs selected from the complete set of 2704 BATSE GRBs, on the energy range $\sim$ 30 keV - 2 MeV observed by BATSE was performed by @kaneko06 (see also @preece00). Five percent of the bursts in this sample are short GRBs (with durations less than 2 seconds). @kaneko06 found that the most common value for the low-energy slope of the photon spectra is $\alpha \sim$ –1, and therefore the distribution of the low-energy indices is not consistent with the value predicted by the standard synchrotron emission model in fast cooling regime, –3/2 [@sari98]. The distribution of the peak energies of the time-integrated spectra of long BATSE GRBs has a maximum at $\sim$ 250 keV, and a very narrow width $\lesssim$ 100 keV. @ghirlanda04 found for a sample of short GRBs observed by BATSE that their time-integrated spectra are harder than those of long GRBs spectra, mainly due to a harder low-energy spectral component ($\sim$ –0.6). @goldstein12 (see also @bissaldi11, @nava11) reported the results of the spectral analysis of 487 GRBs detected by [*Fermi*]{}/GBM operating in the energy band $\sim$ 8 keV – 40 MeV, during its first two years of operation. They found that the distribution of spectral peak energies has a maximum at $\sim$ 200 keV for the time integrated spectra, and also reported several GRBs with time-integrated peak energies greater than 1 MeV. The properties of the sample of 476 GRBs observed by [*Swift*]{}/BAT on 15–150 keV energy range were reported by @sakamoto11. They distinguish the classes of long duration GRBs (89%), short duration GRBs (8%), and short-duration GRBs with extended emission (2%). Their GRB sample was found to be significantly softer than the BATSE bright GRBs, having time-integrated peak energies around $\sim$ 80 keV.
To test the emission models using the observed spectral parameters or to deduce some global properties of a GRB sample, it is necessary to take into account the possible biases in the results of the spectral analysis:
1. When we consider the low energy portion of the spectrum, the data may not approach low energy asymptotic power law within the energy band of the instrument [@preece98; @kaneko06]: e.g. if the spectral peak energy is close to the lower edge of the instrumental energy band, lower values of $\alpha$ are determined. @preece98 introduced as a better measure of the low energy spectral index the effective value of $\alpha$, defined as the slope of the power law tangent to the GRB spectrum at some chosen energy (25 keV for BATSE data).
2. There may be biases in the results when the analysis is performed only on a sample of the brightest events, since there is a tendency for bright GRBs (having higher photon fluxes) to have higher spectral peak energies than faint GRBs [@borgonovo01; @mallozzi95]. For example, @kaneko06 burst selection criteria required a peak photon flux on energy range 50–300 keV greater than 10 photons s$^{-1}$ cm$^{-2}$ or a total energy fluence in 20–2000 keV energy range larger than 2.0 $\times$ 10$^{-5}$ erg cm$^{-2}$. @nava08 have extended the spectral analysis of BATSE GRBs to the fainter bursts (down to fluences $\sim$ 10$^{-6}$ erg cm$^{-2}$) and found a lower value for the average spectral peak energy, $\sim$ 160 keV, with respect to the @kaneko06 results.
3. The instrumental selection effects (e.g. the integration time scale for the burst trigger) may also affect the properties of the GRB samples obtained by different gamma-ray experiments. For example, @sakamoto11 (see @qin13 for [*Fermi*]{}/GBM results) found that the distribution of long GRB durations from [*Swift*]{}/BAT sample is shifted towards longer times ($\sim$ 70 s) with respect to BATSE ($\sim$ 25 s, @kouveliotou93), coherently with the longer BAT triggering time scales. The lack of short GRBs in imaging instruments (such as [*Swift*]{}/BAT) with respect to non-imaging instruments (like BATSE and GBM) on the other hand, is attributed to the requirement for a minimum number of photons needed to build an image with a coded-mask instrument.
[*INTEGRAL*]{} instruments
--------------------------
[[*INTEGRAL*]{}]{}[@winkler03] is an ESA mission launched on October 17, 2002 dedicated to high resolution imaging and spectroscopy in the hard X-/soft $\gamma$-ray domain. It carries two main coded-mask instruments, SPI [@vedrenne03], and IBIS [@ubertini03].
SPI is made of 19 Ge detectors[^1], working in the 20 keV–8 MeV energy range, and is optimized for high resolution spectroscopy ($\sim$2 keV @ 1 MeV), in spite of a relatively poor spatial resolving power of $\sim$2$^{\circ}$. IBIS is made of two pixellated detection planes: the upper plane, ISGRI – [*INTEGRAL*]{} Soft Gamma-Ray Imager [@lebrun03], is made of 128$\times$128 CdTe detectors and operates in the 15 keV–1 MeV energy range. ISGRI has an unprecedented point spread function (PSF) in the soft $\gamma$-ray domain of 12 arc min FWHM. The lower detection plane, PICsIT – PIxellated CsI Telescope [@dicocco03], is made of 64$\times$64 pixels of CsI, and is sensitive between 150 keV and 10 MeV. For our analysis we used only ISGRI data from IBIS. Due to satellite telemetry limitations, PICsIT data are temporally binned over the duration of [[*INTEGRAL*]{}]{}pointing (lasting typically 30–45 minutes) and hence they are not suited for studies of short transients like GRBs.
Despite being a non GRB-oriented mission, [[*INTEGRAL*]{}]{}can be used as a GRB experiment: the GRBs presented in this paper have all been detected in (near-)real time by the [[*INTEGRAL*]{}]{}Burst Alert System (IBAS; @mereghetti03). IBAS is running on ground at the INTEGRAL Science Data Centre (ISDC; @courvoisier03) thanks to the continuous downlink of the [[*INTEGRAL*]{}]{}telemetry. As soon as the IBIS/ISGRI data are received at ISDC, they are analysed in real-time by several triggering processes running in parallel. The triggering algorithms are of two kinds: one is continuously comparing the current sky image with a reference image to look for new sources, and the second one examines the global count rate of ISGRI. In the latter case, once a significant excess is found, imaging is used to check if it corresponds to a new point source or if it has a different origin (e.g. cosmic rays or solar flares). The nominal triggering energy band for IBAS is 15–200 keV, and different time scales are explored, from 2 ms up to 100 s.
As a comparison, in Fig. \[fig:sensi\] we show the sensitivity of some past and current GRB triggering experiments[^2], as a function of the GRB peak energy. The sensitivity is presented as the threshold peak photon flux (in 1–1000 keV energy band) detected at a 5.5 $\sigma$ signal-to-noise ratio, during accumulation time $\Delta$t = 1 s (cf. @band03). It can be seen that the IBAS system is expected to be the most sensitive experiment provided that the peak energy is larger than $\sim$ 50 keV. This has allowed us to investigate the GRB spectral properties for faint (down to fluences of a few 10$^{-8}$ erg cm$^{-2}$) GRBs, see Fig. \[fig:fluence\].
Data analysis {#sec:analysis}
=============
Due to their short durations, GRBs usually do not provide a large number of counts, especially above $\sim$ 200 keV where the IBIS/ISGRI effective area starts to decline rapidly. In order to provide a broader energy coverage and a better sensitivity for the [[*INTEGRAL*]{}]{}GRB spectra, we combined the data from the IBIS/ISGRI and the SPI instruments.
The previously published catalogue of [[*INTEGRAL*]{}]{}GRBs [@vianello09] comprises the analysis of the IBIS/ISGRI data only, providing the GRB spectra on 18–300 keV band. Most of the spectra were fitted with a single power law model over this limited energy range. The spectral energy peak was determined for only 9 out of 56 bursts in their sample. @foley08 used SPI and IBIS/ISGRI data to analyse 9 out of 45 GRBs. They performed the spectral analysis using the data from each of the two instruments independently; in four cases the spectrum was fitted by a different model for IBIS and for SPI data.
We combined the data from both instruments, SPI and IBIS/ISGRI: in this way the low energy portion ($\lesssim$ 200 keV) of the GRB spectrum is explored by ISGRI where its sensitivity is highest, while the high energy portion of the spectrum ($\gtrsim$ 200 keV) is better investigated by the SPI data. Joint spectral analysis, using the data from both instruments (see Fig. \[fig:GRB061122\]), allows us to analyze the spectra consistently and to exploit the maximum potential of each instrument. The SPI data can provide better spectral information at energies where IBIS/ISGRI effective area becomes low, and therefore are suitable to determine the GRB spectral peak energy (typically at $\sim$ a few 100 keV).
The spectra were analysed using the C-statistic [@cash79], which is commonly used for experiments with a low number of counts. In order to fit the spectra of both instruments simultaneously, we used the XPSEC 12.7.1 fitting package [@arnaud10]. For the C-statistic to be applied, we needed to provide on-burst spectra and background spectra separately for every GRB. This cannot be obtained by the [[*INTEGRAL*]{}]{}standard Off-line Scientifc Analysis software (OSA), and therefore we developed additional tools to extract the spectra in the required format.
In order to maximize the sensitivity of both instruments, SPI and ISGRI spectra were extracted in the range (40 keV - 1 MeV) and (20 - 200 keV) respectively, because outside this energy range, the effective areas of the corresponding instruments decrease very rapidly (see @ubertini03 [@vedrenne03]). First, we computed the ISGRI light curves for each GRB (Figs. A.1 - A.5). To enhance the signal-to-noise ratio, we selected only the events that were recorded by the pixels having more than 60% of their surface illuminated by the GRB.
Based on these light curves we selected on-burst intervals for spectral analysis (see dashed lines in Figs. A.1 - A.5). The off-burst intervals for the spectral analysis were determined by selecting the times before and after the GRB, excluding intervals of $\gtrsim$ 10 s close to the event in order to ensure that the off-burst intervals were not contaminated by the GRB counts. For the SPI instrument, a spectrum for each of the 19 (where applicable) Ge detectors was computed. The net individual GRB spectra (i.e. on-burst – off-burst spectra) have the advantage (with respect to the *global* spectra produced by OSA software) of being more accurate since the background spectra were computed for each GRB and each detector, taking into account the local spectral and temporal background evolution. The OSA software, on the contrary, computes the SPI background from a model template and the net spectrum is derived from the sky deconvolution process, which introduces more uncertainties than a simple subtraction of the number of background counts per detector. For each SPI detector an individual response function was calculated, taking into account the GRB direction (either as determined by IBIS/ISGRI or by more precise X-ray or optical follow-up observations). The response function takes into account the exposed fraction of each detector given the GRB direction. This means that for detectors that are completely shadowed by the SPI mask the corresponding net spectrum is consistent with zero (see e.g. @mcglynn09), and it is automatically neglected in the analysis since the effective area is also consistent with zero. For the IBIS/ISGRI spectra, due to the large number of detectors, we decided not to compute individual pixel spectra. We selected only the pixels that were fully illuminated by the GRB, in order to compute the off-burst and on-burst spectra. A corresponding ancillary response function (ARF) was computed, taking into account the reduced ($\sim$ 30%) area of the detector plane we used. For each GRB we computed and fitted the time-integrated spectrum, using all the available SPI spectra and one ISGRI spectrum. In order to account for SPI/IBIS inter-calibration and especially for IBIS count losses due to telemetry limitations (see e.g. Fig. A.1 last panel), we allowed a constant normalization factor between ISGRI and SPI. On the other hand we assumed that the differences among the individual SPI detectors are all accounted for by the ad-hoc generated response matrices. An example of a simultaneous fit is shown in Fig. \[fig:GRB061122\].
We determined the T$_{90}$ duration for sample of GRBs observed after September 2008 (Table \[tab:t90\]). The T$_{90}$ duration of prompt emission measures the duration of the time interval during which 90% of the observed counts are accumulated [@kouveliotou93]. The start and the end of this interval are defined by the time at which 5% and 95% of the counts are accumulated (see Fig. \[fig:t90\]), respectively.
The GRB durations were determined using the IBIS/ISGRI light curves (see Figs. A.1 - A.5) obtained for 20–200 keV energy band. The background rate was determined by fitting a linear or constant function to the data, in the time intervals before and after burst. The time intervals for background fitting lasted typically for $\gtrsim$100 s, and they were separated by $\sim$10 s from the burst. We show the background-subtracted ligthcurves for GRBs detected between September 2008 and February 2012 in Figs. A.1 - A.5. The errors on T$_{90}$ were calculated using the method developed by @koshut96. They define the total net (i.e. background subtracted) source counts observed for a single event as
$$S_{tot}=\int_{-\infty}^{+\infty} \frac{dS}{dt}dt$$
where $dS/dt$ is the source count rate history. The time $\tau_{f}$ during which a given fraction $f$ of the total counts is accumulated is defined as the time at which
$$\frac{\int_{-\infty}^{\tau_{f}} \frac{dS}{dt} dt} {S_{tot}} = f.$$
Hence T$_{90}$ is defined as $\tau_{95}-\tau_{5}$, see Fig. \[fig:t90\]. $S_{f}=\int_{-\infty}^{\tau_{f}} \frac{dS}{dt} dt$ represents the value of the integrated counts $S(t)$ when $f$ of the total counts have been detected. The uncertainties on $S_{f}$ consist of two contributions: $(dS_{f})_{cnt}$, due to the uncertainty in the integrated counts $S(t)$ at any time $t$ (see Eq. (12) in @koshut96), and $(dS_{f})_{fluc}$, due to the statistical fluctuations (see Eq. (14) in @koshut96) with respect to the smooth background model:
$$(dS_{f})_{tot}=\sqrt{(dS_{f})_{cnt}^{2}+(dS_{f})_{fluc}^{2}}.$$
The times $\tau_{f-}$ and $\tau_{f+}$ are the times at which $S_{f}-(dS_{f})_{tot}$ and $S_{f}+(dS_{f})_{tot}$ counts have been reached respectively. In this case, one can define
$$\Delta\tau_{f}=\tau_{f+}-\tau_{f-}$$
and the statistical uncertainty in T$_{90}$ is given by
$$\delta T_{90}=\sqrt{(\Delta\tau_{5})^{2}+(\Delta\tau_{95})^{2}}.$$
![An example of T$_{90}$ calculation using the time integrated IBIS/ISGRI light curve of GRB 081003B. The dashed vertical lines represent the times when the GRB integrated counts exceed the 5% (S$_{5}$) and 95% (S$_{95}$) of the maximal integrated flux value, respectively.[]{data-label="fig:t90"}](Fig3.eps){width="50.00000%"}
In Table \[tab:t90\], we report also the peak fluxes of the GRBs. They were calculated over 1 s for long GRBs, using the standard OSA v10.0 software and IBIS/ISGRI data only. We did not use our new spectral extraction method since due to the low statistics over such a short time interval the spectral fitting does not require more sophisticated models than a simple power law to estimate the peak flux. In this case, the large IBIS/ISGRI effective area is adapted to provide a fair measure of the peak flux. We also recalculated, using the latest available calibration files, the positions and the associated 90% c.l. errors. They were computed using ISGRI only, due to its higher positional accuracy with respect to SPI.
[*INTEGRAL*]{} GRB sample: global properties
============================================
We present an updated version of the currently published [*INTEGRAL*]{} GRB catalogues in Table \[tab:t90\] (for the previous versions, see @vianello09 [@foley08]). The basic properties of the 28 events observed in the period September 2008 - February 2012 are reported. The information on the counterpart observations are adopted from [[*INTEGRAL*]{}]{}GRB archive [^3]. The histogram of the duration T$_{90}$ is shown in Fig. \[fig:t90distr\], bottom panel. For comparison, we also presented the distributions of T$_{90}$ for GRB samples from BATSE[^4] catalogue, as well as [*Fermi*]{}/GBM[^5] and [*Swift*]{}/BAT[^6] GRBs observed to date. The durations T$_{90}$ of BATSE and [*Fermi*]{}/GBM bursts were determined using the light curves in 50–300 keV energy band, while [*Swift*]{} GRB durations were determined using light curves obtained in 15–150 keV energy band. The maximum of the T$_{90}$ distribution for [*INTEGRAL*]{} GRB sample is at $\sim$ 30 s, which is comparable to the samples obtained by BATSE and [*Fermi*]{}/GBM. The distinct property of the distribution of T$_{90}$ durations of [*INTEGRAL*]{} GRBs is the very low number of short gamma-ray bursts with respect to the total number of observed events: only 6% of the GRBs in the sample have durations $<$ 2 s, compared to 24%, 17% and 9% of short bursts observed by BATSE, [*Fermi*]{}/GBM, and [*Swift*]{}/BAT respectively. The paucity of the short events is expected for the imaging instruments, as e.g. the [*Swift*]{}/BAT, where a minimum number of counts is required to localize an excess in the derived image, making confirmation of real bursts with fewer counts (like the short ones) difficult or impossible even if they are detected by count rate increase algorithms.
In order to make a comparison with the results from the other GRB missions, we present the cumulative fluence distributions for different instruments in Fig. \[fig:fluence\]. We calculated the fluences for the [[*INTEGRAL*]{}]{} set of bursts in two different energy bands, 50–300 keV and 15–150 keV, to compare with the data published for the BATSE and [*Fermi*]{}/GBM GRB samples, and [*Swift*]{}/BAT GRB sample, respectively. We applied the Kolmogorov–Smirnov (KS) test [@press92] and found that the fluence sample of GRBs observed by [*INTEGRAL*]{} is consistent with the distributions of fluences observed by [*Swift*]{}/BAT and [*Fermi*]{}/GBM, with the respective significance probabilities P$_{KS}$ = 0.76 and 0.27. A larger difference is found when [[*INTEGRAL*]{}]{} GRB sample is compared with the distribution corresponding to BATSE sample (P$_{KS}$ = 0.02). Due to the larger sensitivity of the IBIS instrument, one would expected that the larger number of faint GRBs are observed with respect to e.g. [*Swift*]{}/BAT and [*Fermi*]{}/GBM missions; however, since [[[*INTEGRAL*]{}]{}]{} points for 67% of its time at low Galactic latitude ($\vert$b$\vert<$20$^\circ$) targets, its sensitivity is affected due to the background induced by the Galactic sources.
[c c c c c c c c c c c c c]{} GRB & t$_{start}$ & R.A. & Dec. & Pos.error & X & O & T$_{90}$ & Peak flux$^*$\
& (UTC) & (deg) & (deg) & (arcmin) & & & (s) & (ph cm$^{-2}$ s$^{-1}$)\
\
081003 & 13:46:01.00 & 262.3764 & 16.5721 & 1.6 & Y & - & 25$^{+3}_{-3}$ & $<$0.32\
081003B & 20:48:08.00 & 285.0250 & 16.6914 & 1.3& - & - & 24$^{+6}_{-6}$ & 3.20$^{+0.10}_{-0.10}$\
081016 & 06:51:32.00 & 255.5708 & -23.3300 & 0.7& Y & - & 32$^{+5}_{-5}$ &$>$3.30\
081204 & 16:44:56.00 & 349.7750 & -60.2214 &1.9 & Y & - & 13$^{+6}_{-6}$ & 0.60$^{+0.40}_{-0.40}$\
$^a$081226B & 12:13:11.00 & 25.4884 & -47.4156 & 1.7& - & - & 0.55$^{+0.40}_{-0.40}$ & 0.60$^{+0.50}_{-0.50}$\
090107B & 16:20:38.00 & 284.8075 & 59.5925 & 0.7 & Y & - & 15$^{+3}_{-3}$ &1.50$^{+0.20}_{-0.20}$\
090625B & 13:26:21.00 & 2.2625 & -65.7817 & 1.5 & Y & - & 10$^{+5}_{-5}$ &2.10$^{+0.10}_{-0.10}$\
090702 & 10:40:35.00 & 175.8883 & 11.5001 & 2.0 & Y & - & 19$^{+8}_{-8}$ & $<$0.20\
090704 & 05:47:50.00 & 208.2042 & 22.7900 & 2.5 & - & - & 76$^{+17}_{-17}$ & 1.30$^{+0.10}_{-0.20}$\
090814B & 01:21:14.00 & 64.7750 & 60.5828 & 2.9 & Y & - & 51$^{+12}_{-12}$ & 0.60$^{+0.10}_{-0.10}$\
090817 & 00:51:25.00 & 63.9708 & 44.1244 & 2.6 & Y & - & 225$^{+7}_{-7}$ & 2.10$^{+0.10}_{-0.10}$\
091015 & 22:58:53.00 & 306.1292 & -6.1700 & 2.9 & - & - & 338$^{+77}_{-77}$ & $<$2.37\
091111 & 15:21:14.00 & 137.8125 & -45.9092 & 2.3 & Y & - & 339$^{+92}_{-92}$ & $<$0.11\
091202 & 23:10:08.00 & 138.8292 & 62.5439 & 2.5 & Y & - & 40$^{+23}_{-23}$ &$<$0.21\
091230 & 06:26:53.00 & 132.8875 & -53.8925 & 2.5 & Y & Y & 235$^{+36}_{-36}$ & 0.76$^{+0.02}_{-0.03}$\
100103A & 17:42:38.00 & 112.3667 & -34.4825 & 1.1 & Y & - & 35$^{+8}_{-8}$ & 3.40$^{+0.10}_{-0.10}$\
100331A & 00:30:23.00 & 261.0625 & -58.9353 & 2.5 & - & - & 20$^{+6}_{-6}$ & 0.70$^{+0.20}_{-0.20}$\
100518A & 11:33:38.00 & 304.7889 & -24.5435 & 1.3 & Y & Y & 39$^{+13}_{-13}$ & 0.80$^{+0.20}_{-0.20}$\
$^{b}$100703A & 17:43:37.37 & 9.5208& -25.7097 & 2.6 & - & - & 0.08$^{+0.04}_{-0.04}$ & $<$0.40\
100713A & 14:35:39.00 & 255.2083 & 28.3900 & 2.1 & Y &- & 106$^{+11}_{-11}$ & $<$0.50\
100909A & 09:04:04.00 & 73.9500 & 54.6544 & 2.0 & Y&Y & 70$^{+8}_{-8}$ & $<$0.88\
100915B & 05:49:36.00 & 85.3958 & 25.0950 &1.5 & -& - & 6$^{+4}_{-4}$& 0.50$^{+0.10}_{-0.10}$\
101112A & 22:10:14.00 & 292.2183 & 39.3589 &0.7 &Y &Y & 24$^{+5}_{-5}$ & $>$1.60\
$^{c}$110112B &22:24:54.70 &10.6000 &64.4064 &2.6 &- &- & 0.40$^{+0.15}_{-0.15}$ & 4.60$^{+0.20}_{-0.20}$\
110206A &18:07:55.00 &92.3417 &-58.8106 & 1.9 &Y & Y& 35$^{+14}_{-14}$ &1.60$^{+0.20}_{-0.20}$\
110708A & 04:43:26.00 & 340.1208 & 53.9597 & 1.2 & Y & - & 79$^{+14}_{-14}$ & 0.80$^{+0.10}_{-0.10}$\
110903A & 02:39:34.00 & 197.0750 & 58.9803 & 0.8 & Y & - & 349$^{+5}_{-5}$ & $>$3.00\
120202A & 21:39:59.00 & 203.5083 & 22.7744 & 1.6 & - & - & 119$^{+6}_{-6}$ & $<$0.20\
\
[Peak fluxes are calculated for short GRBs on the time interval of $^a$ 0.30 s, $^{b}$ 0.08 s, and $^{c}$ 0.10 s.\
$^*$The lower limits correspond to GRBs that are heavily affected by telemetry losses.]{}\
![Distribution of the duration T$_{90}$. [*Top:*]{} Distribution of durations derived from BATSE (violet) and [*Fermi*]{}/GBM (red) light curves in the 50-300 keV band [c.f. @kouveliotou93; @paciesas12]. [*Middle:*]{} The T$_{90}$ durations derived using [*Swift*]{}/BAT instrument on 15-150 keV [c.f. @sakamoto11]. [*Bottom:*]{} Distribution of durations for 20-200 keV light curves obtained from IBIS/ISGRI.[]{data-label="fig:t90distr"}](Fig4.eps){width="50.00000%" height="9cm"}
Spectral analysis {#sec:spec}
=================
We performed spectral analysis on the whole sample of bursts observed between December 2002 and February 2012. Out of 83 GRBs in the initial sample, we report the results for 59 bursts: for 23 GRBs the data do not provide sufficient signal above background for accurate spectral analysis (three of these events, GRB 021219, GRB 040624 and GRB 050129, were analysed using only IBIS/ISGRI data by @vianello09). One gamma-ray burst was observed only during the part of its duration (GRB 080603). The photon spectra were fitted with three models: a single power law, the empirical model for prompt GRB spectra proposed by @band93, and a cutoff power law model. The counts spectrum, $N(E)$, is given in photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$, and E is in units of keV. The power-law model is described by N(E) = AE$^\lambda$, and usually characterizes the spectra for which the break energy of the two component spectrum lies outside the instrument energy band, or the spectra for which the signal at high energies is weak and the break energy could not be accurately determined. The other two models we tested allow the determination of the spectral peak energy: the empirical model introduced by @band93,
$$\begin{aligned}
N(E) & = & A (E/100)^\alpha \exp \left( - {E \over E_0} \right); \quad
{\rm for} \ E \leq \left( \alpha - \beta \right) E_{0} \nonumber \\
& = & A (E/100)^{\beta} [(\alpha-\beta)(E_0/100)]^{\alpha-\beta} \exp(\beta-\alpha);
\nonumber \\
& & {\rm for} \;\; E \geq \left( \alpha - \beta \right) E_{0} \end{aligned}$$
and power-law model with a high-energy exponential cutoff: $$\begin{aligned}
N(E) = A E^{\alpha} \exp(-E/E_0).\end{aligned}$$
$E_0$ is the break energy in the Band model; in cutoff power-law model it denotes the [*e*]{}-folding energy. The cutoff power law model is often used for GRB spectra for which the high energy power law slope $\beta$ in Band model is not well defined due to the low number of high energy photon counts. Using this notation, the peak of the $\nu F_{\nu}$ spectrum for the spectra described by Band or cutoff power-law model is given by $E_{\rm peak}=(\alpha +2)E_{0}$. In Tables \[tab:spec\] and \[tab:spec2\] we report the results of the spectral analysis (the best-fit spectral model) of the time-integrated spectra. The fluences in (20–200) keV energy band reported in the table are calculated using the best-fit spectral model. We report also the value of C-statistic and the number of degrees of freedom for the XSPEC spectral analysis.
------------------- ------------------------- ------------------------- --------------------- ------------------------- --------------- -------------------------- --
GRB $\alpha$ $\beta$ $E_0$ $\lambda$ C-STAT/d.o.f. fluence (20-200 keV)
\[keV\] \[10$^{-7}$ erg/cm$^2$\]
\[0.5ex\]
030227 -1.03$^{+0.25}_{-0.24}$ - 97$^{+70}_{-30}$ - 115.8/73 6.1$_{-5.9}^{+3.5}$
\[0.5ex\] 030320 - - - -1.39$^{+0.01}_{-0.01}$ 2761.3/866 54.2$^{+13.3}_{-11.7}$
\[0.5ex\] 030501 -1.48$^{+0.08}_{-0.08}$ - 184$^{+63}_{-38}$ - 690.2/249 17.2$_{-3.1}^{+1.6}$
\[0.5ex\] 030529 - - - -1.61$^{+0.10}_{-0.10}$ 151.3/74 $<$ 5.3
\[0.5ex\] 031203 - - - -1.51$^{+0.03}_{-0.03}$ 477.4/162 10.6$_{-3.0}^{+2.7}$
\[0.5ex\] 040106 -1.27$^{+0.23}_{-0.18}$ - $>$135 - 265.5/117 9.5$_{-9.1}^{+2.3}$
\[0.5ex\] 040223 - - - -1.73$^{+0.06}_{-0.07}$ 126.1/75 27.2$^{+0.8}_{1.9}$
\[0.5ex\] 040323 -0.50$^{+0.09}_{-0.09}$ - 174$^{+44}_{-31}$ - 538.9/381 20.6$_{-2.9}^{+2.3}$
\[0.5ex\] 040403 -0.75$^{+0.38}_{-0.35}$ - 68$^{+56}_{-23}$ - 232.8/161 4.0$_{-3.7}^{+1.6}$
\[0.5ex\] 040422 -0.33$^{+0.30}_{-0.28}$ - 27$^{+5}_{-4}$ - 272.4/161 4.9$_{-3.6}^{+1.0}$
\[0.5ex\] 040730 - - - -1.25$^{+0.07}_{-0.07}$ 166.4/118 6.3$_{-3.3}^{+4.4}$
\[0.5ex\] 040812 - - - -2.10$^{+0.14}_{-0.15}$ 94.1/74 $<$ 6.9
\[0.5ex\] 040827 -0.34$^{+0.21}_{-0.20}$ - 54$^{+12}_{-9}$ - 668.7/293 11.1$_{-4.0}^{+2.8}$
\[0.5ex\] 041218 - - - -1.48$^{+0.01}_{-0.01}$ 768.5/250 58.2$_{-3.7}^{+3.5}$
\[0.5ex\] 041219 -1.48$^{+0.14}_{-0.11}$ -2.01$^{+0.05}_{-0.08}$ 301$^{+170}_{-105}$ - 280.7/304 867.3$_{-128.9}^{+5.4}$
\[0.5ex\] 050223 - - - -1.44$^{+0.06}_{-0.06}$ 342.5/161 $<$ 15.7
\[0.5ex\] 050502 -1.07$^{+0.13}_{-0.13}$ - 205$^{+132}_{-63}$ - 324.0/293 13.9$_{-4.0}^{+1.1}$
\[0.5ex\] 050504 - - - -1.01$^{+0.05}_{-0.04}$ 122.0/74 10.0$_{-4.5}^{+4.1}$
\[0.5ex\] 050520 - - - -1.45$^{+0.04}_{-0.03}$ 119.4/74 16.6$_{-5.0}^{+4.9}$
\[0.5ex\] 050525 -1.09$^{+0.04}_{-0.04}$ - 131$^{+12}_{-10}$ - 2511.6/733 153.9$_{-8.4}^{+5.7}$
\[0.5ex\] 050626 - - - -1.11$^{+0.13}_{-0.13}$ 33.3/31 6.3$_{-1.0}^{+0.4}$
\[0.5ex\] 050714 - - - -1.63$^{+0.10}_{-0.11}$ 99.3/74 $<$ 4.5
\[0.5ex\] 050918 - - - -1.50$^{+0.02}_{-0.02}$ 1476.5/866 30.2$_{-9.0}^{+10.5}$
\[0.5ex\] 051105B - - - -1.57$^{+0.11}_{-0.12}$ 290.5/250 2.8$_{-2.0}^{+1.5}$
\[0.5ex\] 051211B - - - -1.38$^{+0.04}_{-0.04}$ 304.7/162 16.1$_{-3.3}^{+4.6}$
\[0.5ex\] 060114 - - - -0.80$^{+0.07}_{-0.08}$ 89.0/31 16.0$_{-1.5}^{+0.7}$
\[0.5ex\] 060204 - - - -1.13$^{+0.11}_{-0.11}$ 217.2/162 4.8$_{-3.3}^{+2.4}$
\[0.5ex\] 060428C -0.90$^{+0.14}_{-0.12}$ -1.88$^{+0.14}_{-0.29}$ 108$^{+34}_{-26}$ - 179.0/116 18.6$_{-3.9}^{+2.2}$
\[0.5ex\] 060901 -1.11$^{+0.06}_{-0.05}$ - 265$^{+71}_{-49}$ - 540.7/293 62.2$_{-5.9}^{+3.5}$
\[0.5ex\] 060912B - - - -1.34$^{+0.08}_{-0.08}$ 174.5/162 12.0$_{-5.1}^{+5.8}$
\[0.5ex\] 061025 -0.53$^{+0.21}_{-0.20}$ - 87$^{+35}_{-20}$ - 227.8/117 10.1$_{-4.8}^{+1.3}$
\[0.5ex\] 061122 -1.30$^{+0.05}_{-0.05}$ - 263$^{+35}_{-30}$ - 685.3/513 155.1$_{-5.3}^{+3.4}$
\[0.5ex\] 070309 0.43$^{+0.78}_{-0.63}$ - 45$^{+39}_{-16}$ - 174.5/73 $<$ 12.6
\[0.5ex\] 070311 -0.84$^{+0.08}_{-0.15}$ - 266$^{+199}_{-88}$ - 449.9/205 23.6$_{-5.3}^{+1.7}$
\[0.5ex\] 070925 -1.06$^{+0.09}_{-0.08}$ - 317$^{+135}_{-80}$ - 497.6/337 36.1$_{-3.4}^{+1.7}$
\[0.5ex\] 071109 - - - -1.31$^{+0.08}_{-0.08}$ 166.5/118 3.6$_{-3.5}^{+4.0}$
\[0.5ex\] 080613 -1.00$^{+0.17}_{-0.12}$ - $>$202 - 187.0/117 12.3$_{-5.9}^{+1.7}$
\[0.5ex\] 080723B -1.01$^{+0.02}_{-0.02}$ - 326$^{+30}_{-26}$ - 535.2/293 396.4$_{-6.7}^{+6.7}$
\[0.5ex\] 080922 - - - -1.72$^{+0.03}_{-0.03}$ 274.8/162 17.3$_{-6.5}^{+6.9}$
\[0.5ex\] 081003B -1.31$^{+0.07}_{-0.04}$ - $>$435 - 598.1/381 26.2$_{-24.5}^{+2.0}$
\[0.5ex\] 081016 -1.09$^{+0.12}_{-0.12}$ - 135$^{+48}_{-29}$ - 509.8/425 22.0$_{-4.5}^{+1.4}$
\[0.5ex\] 081204 -1.34$^{+0.27}_{-0.25}$ - 110$^{+139}_{-42}$ - 504.0/249 5.1$_{-4.8}^{+5.1}$
\[0.5ex\] 090107B -1.20$^{+0.16}_{-0.15}$ - 217$^{+265}_{-81}$ - 304.1/205 12.4$_{-4.6}^{+1.3}$
\[0.5ex\] 090625B -0.47$^{+0.13}_{-0.13}$ - 104$^{+27}_{-18}$ - 405.5/205 12.4$_{-2.0}^{+1.2}$
\[0.5ex\] 090702 -1.19$^{+0.54}_{-0.72}$ - 46$^{+165}_{-25}$ - 247.1/117 $<$ 2.1
\[0.5ex\]
------------------- ------------------------- ------------------------- --------------------- ------------------------- --------------- -------------------------- --
------------------- ------------------------- --------- --------------------- ------------------------- --------------- -------------------------- --
GRB $\alpha$ $\beta$ $E_0$ $\lambda$ C-STAT/d.o.f. fluence (20-200 keV)
\[keV\] \[10$^{-7}$ erg/cm$^2$\]
\[0.5ex\]
090704 -1.19$^{+0.06}_{-0.06}$ - 447$^{+276}_{-135}$ - 1516.9/557 54.1$_{-8.0}^{+4.9}$
\[0.5ex\] 090814B - - - -0.94$^{+0.04}_{-0.04}$ 470.2/250 15.1$_{-2.4}^{+2.3}$
\[0.5ex\] 090817 - - - -1.39$^{+0.04}_{-0.05}$ 110.9/74 18.7$^{+10.9}_{-9.8}$
\[0.5ex\] 091015 - - - -1.36$^{+0.07}_{-0.07}$ 280.8/118 $<$ 30.2
\[0.5ex\] 091111 - - - -0.99$^{+0.07}_{-0.09}$ 286.4/250 $<$ 12.2
\[0.5ex\] 091202 - - - -1.07$^{+0.09}_{-0.09}$ 170.7/118 $<$ 4.2
\[0.5ex\] 100103A -0.85$^{+0.06}_{-0.06}$ - 222$^{+48}_{-35}$ - 731.1/381 52.5$_{-4.0}^{+2.1}$
\[0.5ex\] 100518A - - - -1.28$^{+0.05}_{-0.05}$ 410.9/162 5.2$_{3.8}^{+4.4}$
\[0.5ex\] 100713A - - - -1.44$^{+0.09}_{-0.09}$ 381.0/206 $<$ 4.5
\[0.5ex\] 100909A -0.38$^{+0.24}_{-0.21}$ - 181$^{+192}_{-69}$ - 218.2/73 $<$ 19.3
\[0.5ex\] 101112A -0.93$^{+0.14}_{-0.14}$ - 251$^{+279}_{-91}$ - 141.0/73 21.1$_{-7.4}^{+4.4}$
\[0.5ex\] 110708A -0.90$^{+0.11}_{-0.11}$ - 143$^{+48}_{-30}$ - 796.5/469 24.8$_{-4.6}^{+1.9}$
\[0.5ex\] 110903A -0.73$^{+0.04}_{-0.04}$ - 484$^{+165}_{-102}$ - 1490.7/469 148.0$_{-17.5}^{+11.9}$
\[0.5ex\] 120202A -1.09$^{+0.25}_{-0.17}$ - $>$130 - 455.7/425 8.0$_{-7.7}^{+2.1}$
\[0.5ex\]
------------------- ------------------------- --------- --------------------- ------------------------- --------------- -------------------------- --
{width="50.00000%"} {width="50.00000%"}
[*Distribution of E$_{peak}$*]{}. We show the histogram of the observed values of the spectral peak energy in Fig. \[fig:peak\], left panel. It contains the results obtained by fitting the Band or the cut off power law model to the time-integrated spectra of the [[*INTEGRAL*]{}]{} bursts. There were 30 GRBs that were fitted with the cut off power law model and 2 GRBs fitted with Band model in our sample. We compared these results with the results obtained by [*Fermi*]{}/GBM and BATSE detectors. It was shown by various authors that the observed E$_{peak}$ correlates with the burst brightness (e.g. @mallozzi95 [@lloyd00; @kaneko06; @nava08]); in order to account for the possible biases in the distribution of the spectral parameters, we made a comparison of the GRBs within the same fluence range (see also @nava11 for the comparison between the spectral properties of [*Fermi*]{}/GBM and BATSE gamma-ray bursts). We selected from the [*Fermi*]{}/GBM$^5$ and BATSE$^4$ databases the results of the analysis obtained for: (i) GRBs within same fluence range as [[*INTEGRAL*]{}]{} GRBs (i.e. fluence in 50-300 keV energy range $<$ 8.7 $\times$ 10$^{-5}$ erg cm$^{-2}$). The lower fluence limit is approximately the same for all three samples ($\sim$10$^{-8}$ erg cm$^{-2}$). Only the condition on the fluence limit was imposed since the peak fluxes were determined on 20–200 keV energy band for [*INTEGRAL*]{} GRBs, while the [*Fermi*]{}/GBM and BATSE databases contain the values of peak fluxes determined on 50–300 keV; (ii) GRBs with durations of T$_{90} >$ 2 s; (iii) GRBs for which the model that best fitted the time-integrated spectrum was the Band or the cutoff power-law model. We did not apply any additional condition based on the quality of the spectral fit (c.f. @goldstein12, @kaneko06). The histograms for [*Fermi*]{}/GBM bursts (red line) and BATSE bursts (violet line) are shown in Fig. \[fig:peak\]. We used the Kolmogorov-Smirnov test in order to establish the probability whether the distribution of the spectral parameters of the
--------------------------------------------- ---------------------------------------------
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--------------------------------------------- ---------------------------------------------
![Distribution of the power law indices for the sub-sample of [*INTEGRAL*]{} GRBs that were fitted with the single power law model on the energy range 20-1000 keV (black line). BATSE results (violet) and [*Fermi GBM*]{} results (red) are shown for the gamma-ray bursts for which the best spectral model was a single power law. The results for the analysis of the time-integrated spectra are shown, using GRBs within the same fluence range as [*INTEGRAL*]{} GRBs.[]{data-label="fig:pl"}](Fig7.eps){width="50.00000%"}
![Correlations between spectral parameters. Energy fluence in 20-200 keV vs. spectral peak energy E$_{peak}$.[]{data-label="fig:epfl"}](Fig8.eps){width="50.00000%"}
[*INTEGRAL*]{} GRBs can be derived from the same parent distribution as [*Fermi*]{}/GBM or BATSE bursts. For the distribution of spectral peak energies, we found that our distribution is consistent with the distribution of the spectral peak energies of GRBs observed by [*Fermi*]{}/GBM (KS probability = 0.55), and not consistent with the distribution of BATSE GRBs (KS probability = 6 $\times$ 10$^{-3}$) in a given fluence range.
[*Distribution of $\alpha$.*]{} The distribution of the low energy spectral slopes for [*INTEGRAL*]{} GRBs is shown in Fig. \[fig:peak\], right panel. We compared this distribution with the parameters of the [*Fermi*]{}/GBM and BATSE GRB samples. The selection of GRBs from [*Fermi*]{}/GBM and BATSE samples was done in the same way as it was in the case of E$_{peak}$ distribution. The distribution of the low energy power law slopes obtained for ISGRI/SPI GRBs is consistent with both, [*Fermi*]{}/GBM (KS probability = 0.23) and BATSE (KS probability = 0.92) GRB samples.
[*Distribution of $\lambda$.*]{} In the [*INTEGRAL*]{} sample there were 27 GRBs for which the model that best fitted the data was a single power law with the slope $\lambda$ (see Table \[tab:spec\]). Fig. \[fig:pl\] shows the distribution of $\lambda$ for our sample. For the comparison we plotted the results obtained for the BATSE and [*Fermi*]{}/GBM sample of GRBs for which the best fitted model was a single power law. We selected only the long bursts within the same fluences range as the [*INTEGRAL*]{} GRBs. We find that the distribution obtained for the [*INTEGRAL*]{} GRBs is not consistent with the distribution corresponding to [*Fermi*]{}/GBM population (KS probability = 5 $\times$ 10$^{-6}$), and is consistent with the distribution of BATSE GRBs (KS probability = 0.05).
---------------------------------------- ----------------------------------------
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[*Correlations among spectral parameters*]{}. The empirical correlations among time-resolved spectral parameters were examined for BATSE and [*Fermi*]{}/GBM samples [@crider97; @preece98; @lloyd02; @kaneko06; @goldstein12]. The most significant correlation is found between E$_{peak}$ and low energy spectral index for the time resolved spectra of individual bursts. The correlations between the time-integrated parameters E$_{peak}$ and $\alpha, \beta$ and energy or photon flux/fluence were also investigated [@kaneko06; @goldstein12]. We show in Figs. \[fig:epfl\] and \[fig:corr\] energy fluence in 20–200 keV vs. E$_{peak}$ and the scatter plots $\alpha$ vs. E$_{peak}$, $\alpha$ vs. E$_{0}$. For reference, we show the parameters resulting from the spectral analysis of time-integrated spectra for [*Fermi*]{}/GBM and BATSE GRB samples. The general trend is that lower measured spectral peak energies (close to the lower end of the instrument energy band) increase the uncertainty of the low-energy power law index (c.f. @goldstein12 for the sample of [*Fermi*]{}/GBM GRBs). We calculated the Spearman rank-order correlation coefficient (r$_s$) and the corresponding significance probability P$_{rs}$ to test the existence of correlations among pairs of the time integrated parameters, low energy spectral index - E$_{peak}$ and low energy spectral index - E$_{0}$. We found no significant correlation in the first case, while for the latter one there exists a marginal negative correlation (r$_s$ = –0.44) with the associated significance probability P$_{rs}$ = 1.15 $\times$ 10$^{-2}$. Among the energy fluence - E$_{peak}$, see Fig. \[fig:epfl\], we found a weak positive correlation (r$_s$ = 0.50) with the associated significance probability P$_{rs}$ = 1.88 $\times$ 10$^{-2}$. @kaneko06 examining the correlations among time-integrated parameters also found only one significant correlation, namely between E$_{peak}$ and total energy fluence.
Summary
=======
We have presented a spectral catalogue of the GRBs observed by the [[*INTEGRAL*]{}]{}instruments in the period December 2002 - February 2012. We developed a new spectral extraction method especially suited for short transients where total number of counts is small. We are nevertheless able to probe the high spectral end of the [[*INTEGRAL*]{}]{}instruments’ energy range thanks to the use of the Cash statistic. This new method has been applied in a coherent way to the already previously published GRBs, as well as to the unpublished ones. It allowed us to measure the time integrated GRB peak energy in about 54% of the GRBs of our sample, while for the most complete [[*INTEGRAL*]{}]{}GRB sample published to date [@vianello09] this fraction was just 16% (for BAT data, this fraction corresponds to 17%, @sakamoto11). This has allowed us for the first time to fully compare the [[*INTEGRAL*]{}]{}sample to previous and current GRB dedicated experiments’ results in the spectral and temporal domain.
[c c c c]{} Low energy index & E$_{0}$ \[keV\] & E$_{peak}$ \[keV\] & $\lambda$\
\
–1.01$^{+0.28}_{-0.18}$&205$^{+97}_{-97}$ & 184$^{+110}_{-65}$& –1.39$^{+0.26}_{-0.12}$\
\
Our temporal analysis showed that the T$_{90}$ duration distribution of [[*INTEGRAL*]{}]{}GRBs is comparable to the one of BAT bursts, showing the paucity of the short GRBs with respect to the GRB samples detected by [*Fermi*]{}/GBM and BATSE. The maximum of the distribution of T$_{90}$ durations is at $\sim$ 30 s, which on the other hand makes it similar to the [*Fermi*]{}/GBM sample. The reason for that lies in the triggering time scales of the two instruments: in case of IBAS, [*Fermi*]{}/GBM, and BATSE it is of the order of tens of seconds, while [*Swift*]{}/BAT triggering time scales can be as long as tens of minutes.
Concerning the GRB fluence distribution of our sample, we found it statistically compatible with the [*Swift*]{}/BAT and [*Fermi*]{}/GBM ones. While the IBAS system is expected to be intrinsically more sensitive than [*Swift*]{}/BAT or [*Fermi*]{}/GBM (see Fig. \[fig:sensi\]), the fact that [[*INTEGRAL*]{}]{}spends most of its observing time pointing Galactic sources implies a diminished sensitivity due to the increased background induced by these sources.
In Table \[tab:res\] we report the median spectral parameter values and the dispersions for the distributions obtained for [[*INTEGRAL*]{}]{}GRB time-integrated spectra. The peak energy values we could determine are compatible with the ones obtained by the [*Fermi*]{}/GBM experiment, and not with the ones measured by BATSE, being systematically softer. This can be explained by the similar triggering threshold of the two former instruments (15 keV vs. 8 keV), which are both significantly lower than the nominal BATSE low energy threshold of 50 keV. The median of the peak energy distribution is at $\sim$180 keV, with a dispersion of $\sim$100 keV. The slopes of the low energy photon spectra are found to be consistent with both samples, [*Fermi*]{}/GBM and BATSE, having the median of the distribution at $\alpha$=–1 and a spread $\lesssim$0.3. When a single power law was fitted to the spectra, we found that the distribution of power law indices has its median at $\lambda$=–1.4, and a spread $\lesssim$0.3. [[*INTEGRAL*]{}]{}GRBs that are fitted with a single power-law are therefore harder when compared with [*Fermi*]{}/GBM sample, and consistent with the BATSE GRB sample. The correlations among the spectral properties (e.g. low energy spectral index vs. spectral peak energy) were investigated for the time-resolved spectra of the individual GRBs, and were not tested in this work due to the insufficient count number. We confirm that the analogous correlation among the time-integrated spectral properties does not hold for [[*INTEGRAL*]{}]{}GRB sample, as it was also found for e.g. BATSE data by @kaneko06. Weak correlations were found for low energy spectral index $\alpha$ vs. break energy E$_0$, and energy fluence vs. the observed spectral peak energy.
The GRB catalog we presented contains a limited number of events with respect to other missions’ databases. Our results allow however an important insight in the possible instrumental biases in spectral and temporal parameters distributions, and also provide the spectral analysis for a sample of faint GRBs with good statistics.
The authors thank Jochen Greiner for careful reading of the manuscript, and valuable comments on this work. The authors thank Thomas Maccarone, Patrick Sizun and Fabio Mattana for discussions on data analysis. ZB acknowledges the French Space Agency (CNES) for financial support. ISGRI has been realized and maintained in flight by CEA-Saclay/Irfu with the support of CNES. Based on observations with INTEGRAL, an ESA project with instruments and science data centre funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Switzerland, Spain), Czech Republic and Poland, and with the participation of Russia and the USA.
IBIS/ISGRI light curves of [*INTEGRAL*]{} GRBs
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[^1]: During the mission lifetime four SPI detectors have failed, and this has been accounted for in our spectral analysis.
[^2]: For updated sensitivity of [*Fermi*]{}/GBM, see @bissaldi09 and @meegan09.
[^3]: http://ibas.iasf-milano.inaf.it
[^4]: http://heasarc.gsfc.nasa.gov/W3Browse/cgro/batsegrbsp.html
[^5]: http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html
[^6]: http://swift.gsfc.nasa.gov/docs/swift/archive/grb$\_$table.html/
|
---
author:
- |
\
Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18,\
A-1050 Vienna, Austria\
E-mail:
title: 'Bethe–Salpeter-Motivated Modelling of Pseudo-Goldstone Pseudoscalar Mesons'
---
Interpretation of the Lightest Pseudoscalar Mesons as (Pseudo) Goldstone Bosons
===============================================================================
Spontaneous breakdown of the chiral symmetries of quantum chromodynamics (QCD) implies the presence of massless bosons, identified with the ground-state pseudoscalar mesons, with masses due to these symmetries’ further, explicit breaking. As a proof of concept, we study the treatment of such pseudo-Goldstone bosons by a simple approximation [@WLe; @WLp; @WLpa; @WLpb; @WL0] to the Bethe–Salpeter equation [@SB].
Bethe–Salpeter Equation for Fermion–Antifermion States in Instantaneous Limit {#I}
=============================================================================
Consider some boson bound state $|B(P)\rangle$ of mass $\widehat M_B$ and momentum $P,$ composed of a fermion and an antifermion with individual coordinates $x_{1,2}$, individual momenta $p_{1,2}$, center-of-momentum coordinate $X$, relative coordinate $x$, total momentum $P$, and relative momentum $p$, clearly related by$$x\equiv x_1-x_2\
,\qquad P\equiv p_1+p_2\ ,\qquad P^2=\widehat M_B^2\ .$$The Bethe–Salpeter formalism describes the bound state $|B(P)\rangle$ by its Bethe–Salpeter amplitude, in momentum space defined, in terms of the Dirac field operators $\psi_{1,2}(x_{1,2})$ of the two constituents, by$$\Phi(p,P)\equiv\exp({\rm i}\,P\,X)\int{\rm
d}^4x\,\exp({\rm i}\,p\,x)\,\langle0|{\rm T}(\psi_1(x_1)\,
\bar\psi_2(x_2))|B(P)\rangle\ .$$This Bethe–Salpeter amplitude satisfies the homogeneous Bethe–Salpeter equation [@SB] that, in turn, involves both the appropriate interaction kernel and the propagators of the bound-state constituents. In Lorentz-covariant settings, the full propagator $S_i(p)$ of any spin-$\frac{1}{2}$ fermion $i$ can be represented in terms of two Lorentz-scalar functions, *e.g.*, mass $M_i(p^2)$ and wave-function renormalization $Z_i(p^2)$, obtained as solutions to the Dyson–Schwinger equation for the fermion’s two-point Green function:$$S_i(p)=\frac{{\rm i}\,Z_i(p^2)}
{\slashed{p}-M_i(p^2)+{\rm i}\,\varepsilon}\ ,\qquad\slashed{p}
\equiv p^\mu\,\gamma_\mu\ ,\qquad\varepsilon\downarrow0\ ,\qquad
i=1,2\ .\label{p}$$
Some time ago, we devised a (Salpeter-equation-generalizing) three-dimensional reduction [@WLe] of the Poincaré-covariant Bethe–Salpeter equation, enabled by keeping in fermion propagators only terms linear in $p_0$. The latter, together with the assumption of instantaneity of all interactions among the bound-state constituents, suffices to formulate bound-state equations for Salpeter amplitudes [@SE]$$\phi(\bm{p})\propto
\int{\rm d}p_0\,\Phi(p,P)\ .$$In terms of its bound-state constituents’ free energies and projectors onto positive/negative energies,$$E_i(\bm{p})\equiv\sqrt{\bm{p}^2+M_i^2(\bm{p}^2)}\
,\qquad\Lambda_i^\pm(\bm{p})\equiv\frac{E_i(\bm{p})\pm\gamma_0\,
[\bm{\gamma}\cdot\bm{p}+M_i(\bm{p}^2)]}{2\,E_i(\bm{p})}\ ,$$and induced interaction kernel $K(\bm{p},\bm{q})$, our center-of-momentum-frame bound-state equation reads $$\begin{aligned}
\phi(\bm{p})=Z_1(\bm{p}^2)\,Z_2(\bm{p}^2)
\int\frac{{\rm d}^3q}{(2\pi)^3}&\left(\frac{\Lambda_1^+(\bm{p})\,
\gamma_0\,[K(\bm{p},\bm{q})\,\phi(\bm{q})]\,\Lambda_2^-(\bm{p})\,
\gamma_0}{\widehat M_B-E_1(\bm{p})-E_2(\bm{p})}\right.\nonumber\\&
\hspace{-.7ex}-\left.\frac{\Lambda_1^-(\bm{p})\,\gamma_0\,
[K(\bm{p},\bm{q})\,\phi(\bm{q})]\,\Lambda_2^+(\bm{p})\,\gamma_0}
{\widehat M_B+E_1(\bm{p})+E_2(\bm{p})}\right).\label{i}\end{aligned}$$
The normalization of the Salpeter amplitude $\phi(\bm{p})$ will, of course, reflect that of the state $|B(P)\rangle$ entering its definition. For the latter normalization, we adhere to the relativistically covariant choice$$\langle B(P)|B(P')\rangle=
(2\pi)^3\,2\,P_0\,\delta^{(3)}(\bm{P}-\bm{P}')\ .$$
Neglecting, for one reason or the other, the impact of the interaction kernel, this yields the condition (involving a trace over our bound-state constituents’ spinor, flavour, and colour degrees of freedom)$$\int{\rm d}^3p\,{\rm
Tr}\!\left[\phi^\dag(\bm{p})\,
\frac{\gamma_0\,[\bm{\gamma}\cdot\bm{p}+M_1(\bm{p}^2)]}
{E_1(\bm{p})}\,\phi(\bm{p})\right]=(2\pi)^3\,2\,P_0\ .$$
Application Suggesting Itself: Two Bound-State Constituents of Identical Flavour
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Now, let us adapt our general instantaneous Bethe–Salpeter formalism [@WLe] to just those physical systems we are actually interested in: bound states of a quark and an antiquark of precisely the same mass — tantamount, as far as only the strong interactions are taken into account, to bound states of a quark and its own antiquark. For this special case, we may drop the flavour-related subscript $i=1,2$ in our framework, whence the instantaneous Bethe–Salpeter equation (\[i\]) simplifies (a little bit) to$$\begin{aligned}
\phi(\bm{p})=
Z^2(\bm{p}^2)\int\frac{{\rm d}^3q}{(2\pi)^3}&
\left(\frac{\Lambda^+(\bm{p})\,\gamma_0\,
[K(\bm{p},\bm{q})\,\phi(\bm{q})]\,\Lambda^-(\bm{p})\,\gamma_0}
{\widehat M_B-2\,E(\bm{p})}\right.\nonumber\\&\hspace{-.7ex}
-\left.\frac{\Lambda^-(\bm{p})\,\gamma_0\,[K(\bm{p},\bm{q})\,
\phi(\bm{q})]\,\Lambda^+(\bm{p})\,\gamma_0}{\widehat
M_B+2\,E(\bm{p})}\right).\label{=}\end{aligned}$$Clearly, the spin–parity–charge-conjugation assignment of any pseudoscalar bound state formed by spin-$\frac{1}{2}$ fermion and spin-$\frac{1}{2}$ antifermion is given by $J^{P\,C}=0^{-+}$. The most general Salpeter amplitude $\phi(\bm{p})$ of any such state may be expanded into only two independent Lorentz-scalar components, say, $\varphi_{1,2}(\bm{p})$. Recalling its colour factor, for a bound state of quark and its antiquark this expansion reads$$\begin{aligned}
\phi(\bm{p})=\frac{1}{\sqrt{3}}\left[
\varphi_1(\bm{p})\,
\frac{\gamma_0\,[\bm{\gamma}\cdot\bm{p}+M(\bm{p}^2)]}{E(\bm{p})}+
\varphi_2(\bm{p})\right]\gamma_5\ ,&\\4\int{\rm d}^3p\,
[\varphi_1^\ast(\bm{p})\,\varphi_2(\bm{p})
+\varphi_2^\ast(\bm{p})\,\varphi_1(\bm{p})]=(2\pi)^3\,2\,P_0\
.&\end{aligned}$$
At that stage, the only element still lacking is the Bethe–Salpeter kernel $K(\bm{p},\bm{q})$, with regard to its Dirac structure and its dependence on the momenta $\bm{p}$ and $\bm{q}$. We tackle this problem in two steps.
Dirac Structure of the Bethe–Salpeter Interaction Kernel by Sticking to Fierz Invariance
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We base the determination of the kernel $K(\bm{p},\bm{q})$ on our trust in Fierz symmetries and rely for its Dirac structure on a linear combination corresponding to an eigenstate under Fierz transformations: $$K(\bm{p},\bm{q})\,\phi(\bm{q})\propto
V(\bm{p},\bm{q})\left[\gamma_\mu\,\phi(\bm{q})\,\gamma^\mu
+\gamma_5\,\phi(\bm{q})\,\gamma_5-\phi(\bm{q})\right].\label{FI}$$Accordingly, all underlying effective interactions are subsumed by a single Lorentz-scalar potential function, $V(\bm{p},\bm{q})$. Assuming the latter to be of convolution type and to be compatible with spherical symmetry, that is, $V(\bm{p},\bm{q})=V((\bm{p}-\bm{q})^2)$, allows us to split off all reference to angular variables and to reduce our bound-state equation (\[=\]) to a system of equations for the radial factors $\varphi_{1,2}(p)$ of the independent components $\varphi_{1,2}(\bm{p})$, depending on the moduli $p\equiv|\bm{p}|,q\equiv|\bm{q}|$ of the momenta $\bm{p}$ and $\bm{q}$, with all interactions encoded by a yet to be found configuration-space central potential $V(r)$, $r\equiv|\bm{x}|$:
$$\begin{aligned}
&E(p)\,\varphi_2(p)+
\frac{2\,Z^2(p^2)}{\pi\,p}\int\limits_0^\infty{\rm d}q\,q\,{\rm
d}r\sin(p\,r)\sin(q\,r)\,V(r)\,\varphi_2(q)=\frac{\widehat M_B}{2}
\,\varphi_1(p)\ ,\label{ie}\\&E(p)\,\varphi_1(p)=\frac{\widehat
M_B}{2}\,\varphi_2(p)\ .\label{ae}\end{aligned}$$
\[e\]
Momentum Dependence of our Bethe–Salpeter Interaction Kernel by Utilizing Inversion
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The two (in general, coupled) Eqs. (\[e\]) constitute an eigenvalue problem, with the bound-state masses $\widehat M_B$ as eigenvalues, for bound states specified, in momentum-space representation, by the set of radial wave functions $\varphi_{1,2}(p)$. For *vanishing* eigenvalue, that is, for $\widehat M_B=0$, Eqs. (\[e\]) decouple: Eq. (\[ae\]) forces $\varphi_1(p)$ to vanish, *i.e.*, $\varphi_1(\bm{p})=0$. Thus, the corresponding Salpeter amplitude reads$$\phi(\bm{p})=
\frac{1}{\sqrt{3}}\,\varphi_2(\bm{p})\,\gamma_5\ .\label{0M}$$In configuration-space representation, denoting the free term by $T(r)$, Eq. (\[ie\]) then simplifies to a relation enabling us [@WLi; @WLia] to find the potential in action, $V(r)$, provided we know one solution $\varphi_2(r)$: $$T(r)+V(r)\,\varphi_2(r)=0\qquad\Longrightarrow
\qquad V(r)=-\frac{T(r)}{\varphi_2(r)}\ .\label{V}$$
In order to get hold of, at least, one of the desired solutions, we exploit the relationship between the full quark propagator $S(p)$ — obtainable as solution to the quark Dyson–Schwinger equation — and the Bethe–Salpeter amplitude $\Phi(p,0)$ of (flavour-nonsinglet) pseudoscalar mesons arising from the (renormalized) axial-vector Ward–Takahashi identity of QCD in the chiral limit [@MRT]: the sought relationship (in its Euclidean-space formulation indicated by underlined quantities) reads [@WLc; @WLca; @WLcb; @WLp; @WLr]$$\Phi(\underline{k},0)\propto
\frac{Z(\underline{k}^2)\,M(\underline{k}^2)}
{\underline{k}^2+M^2(\underline{k}^2)}\,\underline{\gamma}_5+
\mbox{subleading contributions}\ .\label{rs}$$
Just for the sake of illustration, let us follow the path sketched above by starting from a solution for the chiral-quark propagator found in Ref. [@MT] on the basis of a particular QCD-motivated ansatz for the *effective* interactions entering in the quark Dyson–Schwinger equation: the conversion of the propagator functions $M(\underline{k})$ and $Z(\underline{k})$ redrawn in Fig. \[MZ\], by means of Eq. (\[rs\]), to the massless-meson Salpeter amplitude of Fig. \[R\] entails, via the inversion (\[V\]), the interquark potential plotted in Fig. \[P\].
Basic Pseudoscalar-Meson Features in a Gell-Mann–Oakes–Renner-type Relation
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With the explicit behaviour of the effective interquark potential $V(r)$ at our disposal, we are in a position to embark on the intended simplified description of meson properties: for $\widehat M_B\ne0,$ inserting any of Eqs. (\[e\]) into the other, takes us to a single eigenvalue equation for eigenvalues ${\widehat M_B}^2$ [@WLs; @WLp; @WLpa; @WLpb; @WL0], which can be easily solved by expanding its solutions over suitable bases in function space [@WLlb; @WLst; @WLua; @WLnv; @WLtwr; @WLws; @WLh; @WLy].
Matching residues of pseudoscalar-meson poles in the axial-vector Ward–Takahashi identity of QCD gives a Gell-Mann–Oakes–Renner-resembling [@GOR] relation [@MRT] linking, besides meson mass $\widehat M_B$ and two quark masses, both *decay constant* $f_B$ and *in-hadron condensate* ${\mathbb C}_B$ of the pseudoscalar bound state $|B(P)\rangle$, defined, in terms of quark fields $\psi_f(x)$ (exhibiting the flavour index $f=1,2$), by$$\begin{aligned}
\langle0|
{:\!\bar\psi_1(0)\,\gamma_\mu\,\gamma_5\,\psi_2(0)\!:}|B(P)\rangle
={\rm i}\,f_B\,P_\mu\qquad\Longrightarrow\qquad f_B&\propto
\int{\rm d}^3p\,{\rm Tr}[\gamma_0\,\gamma_5\,\phi(\bm{p})]\ ,\\
\langle0|{:\!\bar\psi_1(0)\,\gamma_5\,\psi_2(0)\!:}|B(P)\rangle
\equiv{\mathbb C}_B&\propto\int{\rm d}^3p\,{\rm
Tr}[\gamma_5\,\phi(\bm{p})]\ .\end{aligned}$$Sticking still to the idealized case of bound-state constituents of equal mass $m$, this relation becomes $$f_B\,\widehat M_B^2=2\,m\,{\mathbb C}_B\ .$$
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![Dependence, on the Euclidean momentum $\underline{k}$, of both (a) mass function $M(\underline{k})$ and (b) wave-function renormalization function $Z(\underline{k})$ entering in the quark propagator (\[p\]) in the chiral limit, extracted and redrawn [@WLp; @WLpb] from Fig. 1 of Ref. [@PM], emerging as solution to the Dyson–Schwinger equation for the quark two-point Green function imitating the impact of truncated Dyson–Schwinger equations by QCD-inspired ansätze [@MT].\[MZ\]](LS-M.eps "fig:")
\[1ex\](a)
\[2ex\] ![Dependence, on the Euclidean momentum $\underline{k}$, of both (a) mass function $M(\underline{k})$ and (b) wave-function renormalization function $Z(\underline{k})$ entering in the quark propagator (\[p\]) in the chiral limit, extracted and redrawn [@WLp; @WLpb] from Fig. 1 of Ref. [@PM], emerging as solution to the Dyson–Schwinger equation for the quark two-point Green function imitating the impact of truncated Dyson–Schwinger equations by QCD-inspired ansätze [@MT].\[MZ\]](LS-Z.eps "fig:")
\[1ex\](b)
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![Nonvanishing radial component function determining the Salpeter amplitude (\[0M\]) of any massless pseudoscalar quark–antiquark bound state governed by our instantaneous Bethe–Salpeter equation (\[=\]) with *Fierz-symmetric* Dirac structure (\[FI\]), extracted from the quark propagator functions $M(\underline{k})$ and $Z(\underline{k})$ of Fig. [\[MZ\]]{}, shown in both (a) momentum-space representation, $\varphi_2(p)$, and (b) configuration-space representation, $\varphi_2(r)$.\[R\]](phipax.eps "fig:")
\[1ex\](a)
\[2ex\] ![Nonvanishing radial component function determining the Salpeter amplitude (\[0M\]) of any massless pseudoscalar quark–antiquark bound state governed by our instantaneous Bethe–Salpeter equation (\[=\]) with *Fierz-symmetric* Dirac structure (\[FI\]), extracted from the quark propagator functions $M(\underline{k})$ and $Z(\underline{k})$ of Fig. [\[MZ\]]{}, shown in both (a) momentum-space representation, $\varphi_2(p)$, and (b) configuration-space representation, $\varphi_2(r)$.\[R\]](phi2rax.eps "fig:")
\[1ex\](b)
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![Spherically symmetric effective quark–antiquark interaction potential $V(r)$ that upon insertion into our instantaneous Bethe–Salpeter equation (\[e\]) trimmed to describe pseudoscalar mesons reproduces, as the ground-state solution to that bound-state problem, the (starting-point) Salpeter component depicted in Fig. [\[R\]]{}. Its near flatness close to the origin and steep rise to infinity make $V(r)$ reminiscent of a *smoothed* square well.\[P\]](LSMartinaPo.eps)
Table \[F\] presents the very satisfactory outcomes of implementation of the potential of Fig. \[P\] into our bound-state approach (see, *e.g.*, Refs. [@GRHKL; @WL@; @C13] for corresponding recent Bethe–Salpeter results).
[rrccd[1.4]{}c]{} &&$f_B$&${\mathbb C}_B$&& $\overline{m}(2\;\mbox{GeV})$\
&& $[\mbox{MeV}]$ & $[\mbox{GeV}^2]$ && $[\mbox{MeV}]$ [@PDG]\
chiral quarks&6.8&151&0.585&0.0059&—\
$u$/$d$ quarks&148.6&155&0.598&2.85& $3.5^{+0.5}_{-0.2}$\
$s$ quarks&620.7&211&0.799&51.0& $95^{+9}_{-3}$\
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