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diff --git a/macros/latex/contrib/springer/svjour/cmp/example.tex b/macros/latex/contrib/springer/svjour/cmp/example.tex new file mode 100644 index 0000000000..d79cad296b --- /dev/null +++ b/macros/latex/contrib/springer/svjour/cmp/example.tex @@ -0,0 +1,1148 @@ +\documentclass[cmp]{svjour} %envcountsame +\usepackage{amsmath} +\usepackage{amsfonts,amssymb} +\usepackage{psfig} + +\journalname{Communications in Mathematical Physics} + +\newenvironment{bew}[2]{\removelastskip\vspace{6pt}\noindent + {\it Proof #1.}~\rm#2}{\par\vspace{6pt}} +\newlength{\Taille} + + + +\newcommand{\oh}{O(h^\infty)} +\newcommand{\dx}{\partial_x} +\newcommand{\dy}{\partial_y} +\newcommand{\dt}{\partial_t} +\newcommand{\dz}{\partial_z} +\newcommand{\dxi}{\partial_\xi} +\newcommand{\deriv}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\ddt}{\frac{d}{dt}} +\newcommand{\lie}{{\cal L}} +\newcommand{\pscal}[2]{\langle #1,#2\rangle} +\newcommand{\ham}[1]{\mathcal{X}_{#1}} +\newcommand{\Lie}[1]{\mathfrak{#1}} +\newcommand{\fourier}{\mathcal{F}_h} +\newcommand{\fouriero}{\mathcal{F}} +\newcommand{\re}{\mathfrak{R}} +\newcommand{\im}{\mathfrak{I}} + +\newcommand{\phy}{\varphi} +\newcommand{\epsi}{\varepsilon} +\newcommand{\bep}{\mbox{\boldmath{$\epsilon$}}} +\newcommand{\bc}{\mathbf{c}} +\newcommand{\om}{\omega} +\newcommand{\al}{\alpha} +\newcommand{\la}{\lambda} + +\newcommand{\ssi}{\Longleftrightarrow} +\newcommand{\impliq}{\Rightarrow} +\newcommand{\fleche}{\rightarrow} +\newcommand{\inject}{\hookrightarrow} +\newcommand{\restr}{\upharpoonright} +\newcommand{\trsp}{\raisebox{.6ex}{${\scriptstyle t}$}} +\newcommand{\limi}[1]{\displaystyle \lim_{#1}} +\newcommand{\tr}{\textrm{tr}\;} +\newcommand{\demo}[1][$\!\!$]{\noindent\textbf{Proof }\textsl{#1}. } +\newcommand{\egdef}{\stackrel{\mathrm{def}}{=}} +\newcommand{\flechediagbas}[1]{ + \settowidth{\unitlength}{\mbox{$ ~ #1 ~$}} + \begin{array}{r}\begin{picture}(0.1,0.5)(0,0) + \put(-0.4,0.5){\vector(1,-1){1}} + \end{picture}\\ #1 \end{array}} +\newcommand{\flechebas}[1]{ + \settoheight{\unitlength}{\mbox{$#1$}} + \settowidth{\Taille}{\mbox{~${\scriptstyle #1}$}} + \addtolength{\unitlength}{4ex} + \begin{picture}(0,1) + \put(0,1){\vector(0,-1){1}} + \put(0,0.5){\makebox(0,0){${\scriptstyle #1}$ \hspace{\the\Taille}}} + \end{picture}} +\newcommand{\flechehaut}[1]{ + \settoheight{\unitlength}{\mbox{$#1$}} + \settowidth{\Taille}{\mbox{~${\scriptstyle #1}$}} + \addtolength{\unitlength}{4ex} + \begin{picture}(0,1) + \put(0,0){\vector(0,1){1}} + \put(0,0.5){\makebox(0,0){\hspace{\the\Taille}${\scriptstyle #1}$ }} + \end{picture}} +\newcommand{\flechedroite}[1]{ + \settowidth{\unitlength}{\mbox{$#1$}} + \settoheight{\Taille}{\mbox{${\scriptstyle #1}$}} + \addtolength{\Taille}{1ex} + \addtolength{\unitlength}{4ex} + \raisebox{0.5ex}{ + \begin{picture}(1,0) + \put(0,0){\vector(1,0){1}} + \put(0.5,0){\makebox(0,0){${\scriptstyle #1}$ \vspace{\the\Taille}}} + \end{picture}}} +\newcommand{\flechegauche}[1]{ + \settowidth{\unitlength}{\mbox{$#1$}} + \settoheight{\Taille}{\mbox{${\scriptstyle #1}$}} + \addtolength{\Taille}{1ex} + \addtolength{\unitlength}{4ex} + \raisebox{0.5ex}{ + \begin{picture}(1,0) + \put(1,0){\vector(-1,0){1}} + \put(0.5,0){\makebox(0,0){${\scriptstyle #1}$ \vspace{\the\Taille}}} + \end{picture}}} +\newcommand{\vecteur}[1]{\settowidth{\unitlength}{\mbox{$#1$}}\addtolength{\unitlength}{-0.5ex}\settoheight{\Taille}{\fbox{$#1$}}\raisebox{\Taille}{\begin{picture}(0,0) + \put(0,0){\vector(1,0){1}} + \end{picture}}\!#1} +% cette commande peut remplacer \overrightarrow, qui ne marche pas +% bien (??) en 11pt. Attention,produit une erreur si la fleche est +% petite !!! (utiliser \vec alors) +% pour 10pt, remplacer -0.35pt par -0.07ex ou reciproquement +\newcommand{\cutevector}[1]{\,\settowidth{\unitlength}{\mbox{$#1$}}\addtolength{\unitlength}{-1.3ex}\settoheight{\Taille}{\mbox{$#1$}}\addtolength{\Taille}{0.5ex}{\raisebox{\Taille}{\begin{picture}(0,0) + \put(0,0){\line(1,0){1}} + \put(1,0){\raisebox{-0.07ex}{\makebox(0,0){${\scriptstyle + \rightarrow}$}}} + \end{picture}}\!#1}\addtolength{\Taille}{0.4ex}\rule{0cm}{\Taille}} +% devrait faire a peu pres la meme chose, mais sans erreur si trop +% petit (mais sans etre plus beau que \vec: ca va depasser) +\newcommand{\cutevectorbis}[1]{\makebox[0.6ex][l]{$#1$}\settowidth{\unitlength}{\mbox{$#1$}}\addtolength{\unitlength}{-0.5ex}\settoheight{\Taille}{\mbox{$#1$}}\addtolength{\Taille}{0.4ex}\raisebox{\Taille}{\makebox[\unitlength][l]{\hrulefill\raisebox{-0.35ex}{$\!{\scriptstyle + \rightarrow}$}}}} +\newcommand{\tinyvector}[1]{\cutevector{\mbox{${\scriptscriptstyle #1}$}}} +\newcommand{\cqfd}{\hfill $\square$} +\newcommand{\finex}{$\diamond$} +\newcommand{\finrem}{\hfill $\oslash$} +\newcommand{\dens}{\Omega_{\frac{1}{2}}} +\newcommand{\Cinf}{C^\infty} +\newcommand{\COT}[1]{T^* #1 \setminus\{0\}} +\newcommand{\intint}{\int\!\!\!\int} +\newcommand{\gener}[1]{\langle #1 \rangle} +\newcommand{\ind}{\text{\it {\em Ind }}} +\newcommand{\coker}{\text{\it {\em Coker }}} +%\newcommand{\im}{\text{\it {\em Im }}} +\newcommand{\ssub}{\sigma_{\mathrm{sub}}} +\newcommand{\spec}{\text{\it {\em Spec }}} +\renewcommand{\mod}{\textrm{ mod }} + +\newcommand{\fio}{Fourier integral operator} + +\newcommand{\pdo}{pseudo-differential operator} +\newcommand{\das}{asymptotic expansion} +\newcommand{\cdv}{Colin de Verdi\`ere} +\newcommand{\cis}{completely integrable system} +\newcommand{\mi}{microlocal} +\newcommand{\ouf}{\vspace{3mm}} + +\newcommand{\AAAA}{\fbox{** \`A COMPL\'ETER **}} + +\newcommand{\RM}{\mathbb{R}} +\newcommand{\ZM}{\mathbb{Z}} +\newcommand{\QM}{\mathbb{Q}} +\newcommand{\NM}{\mathbb{N}} +\newcommand{\CM}{\mathbb{C}} + +\newcommand{\T}{\mathbb{T}} + +\newcommand{\PM}{\mathbb{P}} +\newcommand{\LM}{\barre{L}} + +\newcommand{\B}{{\cal B}} +\newcommand{\F}{{\cal F}} +\newcommand{\K}{{\cal K}} +\newcommand{\A}{\mathcal{A}} + + +\newcommand{\ff}{\emph{focus-focus}} +\newcommand{\U}{\mathcal{U}} +\newcommand{\M}{\mathcal{M}} +\renewcommand{\L}{\mathcal{L}} +%\renewcommand{\tinyvector}[1]{\overrightarrow{\scriptscriptstyle #1}} +\newcommand{\bmu}{\mbox{\boldmath{$\mu$}}} +%\newcommand{\parag}[1]{{\textbf #1} } + +\begin{document} + + +\title{{Quantum Monodromy in Integrable Systems}} +\titlerunning{Quantum Monodromy in Integrable Systems} + +\author{San V\~u Ng\d oc\inst{1}\fnmsep\inst{2}} +\institute{Institut Fourier UMR5582, B.P. 74, + 38402 Saint-Martin d'H\`eres, France.\\ \email{San.Vu-Ngoc@ujf-grenoble.fr} \and + Mathematics Institute, P.O. Box 80010, 3508 TA Utrecht, The Netherlands} +\authorrunning{S. V\~u Ng\d oc} + +\date{Received: 21 April 1998 / Accepted: 8 December 1998} +\communicated{H. Araki} + +\maketitle +\begin{abstract} + Let $P_1(h),\dots,P_n(h)$ be a set of commuting self-adjoint + $h$-pseudo-differen\-tial operators on an $n$-dimensional manifold. If the joint principal + symbol $p$ is proper, it is known from the work of Colin de + Verdi\`ere~\cite{colinII} and Charbonnel~\cite{charbonnel} that in a + neighbourhood of any regular value of $p$, the joint spectrum + locally has the structure of an affine integral lattice. This leads + to the construction of a natural invariant of the spectrum, called + the quantum monodromy. We present this construction here, and show + that this invariant is given by the classical monodromy of the + underlying Liouville integrable system, as introduced by + Duistermaat~\cite{duistermaat}. The most striking application of + this result is that all two degree of freedom quantum integrable + systems with a \emph{focus-focus} singularity have the same + non-trivial quantum monodromy. For instance, this proves a + conjecture of Cushman and Duistermaat~\cite{duist-cushman} + concerning the quantum spherical pendulum. +\end{abstract} + +\section{Introduction} +Obstructions to the existence of global action-angle coordinates for +completely integrable systems are well known since Duistermaat's +article \cite{duistermaat}. It was then natural to raise the question +about the impact of these obstructions on \emph{quantum} integrable +systems, at least for the (semi)-classical pseudo-differential +quantisation on cotangent bundles. The first attempts in this +direction were \cite{duist-cushman} and \cite{guillemin-uribe}, both +of them concerning the monodromy invariant for the example of the +spherical pendulum. This system is indeed one of the simplest (along +with the Champagne bottle \cite{bates}) that exhibits a non-trivial +monodromy. The first of these articles \cite{duist-cushman} +proposed a particularly interesting way of detecting the monodromy by +observing a shift in the lattice structure of the joint spectrum. It +is the purpose of this article to state, prove and explain this idea. + +Surprisingly enough, this idea of quantum monodromy has been sleeping +for ten years, before new interest resulted in its experimental +discovery in the spectrum of excited water molecules +\cite{child,tennyson}. + + +Back to mathematics, it turns out that, in the framework of +semi-classical microlocal analysis (developed for integrable systems +in \cite{charbonnel}), there is a natural way of defining an invariant +of the joint spectrum away from singularities of the principal +symbols, that precisely describes the obstruction to the existence of +a \emph{global} lattice structure for the spectrum. The organisation +of this article is as follows: we first extract the relevant +properties of joint spectra, and define the \emph{quantum monodromy} +invariant for any set that shares these properties +(Sect.~\ref{sec:construction}). Then we prove in Sect.~\ref{sec:classical} +that, for spectra, the quantum monodromy is precisely given by the +classical monodromy of the underlying classical Hamiltonian system. +The result is applied in Sect.~\ref{sec:ff} to the particularly +interesting case of systems admitting a \ff\ singularity. The last +Sect.~\ref{sec:detect} finally shows how to read off the monodromy +from a picture of the spectrum. As an example, we use the spectrum of +the Champagne bottle computed by Child \cite{child}. + +\section{Construction of the Quantum Monodromy} +\label{sec:construction} +Let $\U$ be an open subset of $\RM^n$, let $H$ be a set of positive +real numbers accumulating at $0$, and for any $h$ in $H$ let +$\Sigma(h)$ be a discrete subset of $\U$. + +If $B$ is an open subset of\ $\U$, a family $(f(h))_{h\in H}$ of smooth +functions on $B$ with values in $\RM^n$ is called a \emph{symbol} (of +order zero) if it admits an asymptotic expansion of the form +\[ f(h) = f_0 + hf_1 + h^2f_2 +\cdots \] +for smooth functions $f_i : B\fleche\RM^n$. More precisely we require +that for any $\ell\geq 0$, for any $N\geq 0$, and for any compact +$K\subset B$, there is a constant $C_{\ell,N,K}$ such that for all +$h\in H$, +\[ \left\| f(h)-\sum_{k=0}^N h^kf_k \right\|_\ell \leq C_{\ell,N,K}h^{N+1},\] +where $\|.\|_\ell$ denotes the $C^\ell$ norm in $K$. The symbol $f(h)$ +is \emph{elliptic} if its principal part $f_0$ is a local +diffeomorphism of $B$ into $\RM^n$. The value of $f(h)$ at a point +$c\in B$ will be denoted by $f(h;c)$. + +A family $(r(h))_{h\in H}$ of elements of a finite dimensional +vector space is said to be $\oh$ if for any $N\geq 0$ there is a +constant $C>0$ such that $\|r(h)\|\leq Ch^N$, uniformly for all $h\in +H$. If $S(h)$ is any family of sets depending on $h$, then the +notation $f(h)\in S(h)+\oh$ means that the function +$\mathrm{dist}(f(h),S(h))$ is $\oh$. + +We will say that $\Sigma(h)$ has the structure of an ``asymptotic affine +lattice'' whenever it can be described with a locally finite set of +``asymptotic affine integral charts'', in the following sense: + +\begin{figure} + \begin{center} + \leavevmode + \begin{picture}(0,0)% +\psfig{file=424-1.eps}% +\end{picture}% +\setlength{\unitlength}{2763sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(5476,2857)(3263,-3061) +\put(3751,-3061){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$U$}}} +\put(6076,-1186){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$f(h)$}}} +\put(7576,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$h$}}} +\put(8026,-2986){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$h\ZM^n$}}} +\put(3901,-1186){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$B$}}} +\end{picture} + \caption{An asymptotic affine lattice} + \label{fig:chart} + \end{center} +\end{figure} +\begin{definition} + \label{def:chart} + $(\Sigma(h),\U)$ is an ``asymptotic affine lattice'' if for + any $c\in \U$, there exists a small open ball $B\subset \U$ around + $c$, and an elliptic symbol $f(h)~: B\fleche \RM^n$ of order zero +such that, for any family $\lambda(h)\in B$~: + \begin{itemize} + \item $\lambda(h)\in \Sigma(h)\cap B + O(h^\infty) \ssi f(h;\lambda(h))\in + h\ZM^n + O(h^\infty)$ + \item if $\lambda(h)$ and $\lambda'(h)$ are in $\Sigma(h)\cap B$, + then $\lambda'(h)-\lambda(h) = O(h^\infty)$ if and only if for small $h$, + $\lambda'(h)=\lambda(h)$. + \end{itemize} +\end{definition} + +Intuitively this means that zooming by a factor of $\frac{1}{h}$ +inside $B$ makes $\Sigma(h)\cap B$ converge to the standard lattice as +$h$ tends to zero. The issue here is to see what prevents $\Sigma(h)$ +from \emph{globally} converging to a lattice. Of course, the reason +for this definition is that, under suitable hypothesis, the joint +spectrum of a set of $n$ commuting $h$-\pdo s on an $n$-dimensional +manifold is indeed an ``affine asymptotic lattice'' (see the next +section). + +For short, a symbol $f(h)$ satisfying Definition \ref{def:chart} will +be referred to as an ``affine chart'' of $\Sigma(h)$. + +The main point is that the transition functions associated to these +charts are elements of the affine group $GA(n,\ZM)$ (following Berger +\cite{berger-affine}, we denote by $GA(n,\RM)$ the group of invertible +affine transformations of $\RM^n$, which is the semi-direct product of +the linear group $GL(n,\RM)$ by the normal subgroup of +translations. Some authors use the notation $\textrm{Aff}_n(\RM)$ +instead. The subgroup $GA(n,\ZM)$ consists then of elements $A\in +GA(n,\RM)$ such that $A$ and $A^{-1}$ leave $\ZM^n$ globally +invariant). +\begin{proposition} + \label{prop:GA} + Let $f(h)$ and $g(h)$ be two affine charts of $\Sigma(h)$, both + defined on a ball $B$. Then there is a unique + $A\in GA(n,\ZM)\subset GA(n,\RM)$ such that +\[ \left(\frac{g(h)}{h}\right)\circ\left(\frac{f(h)}{h}\right)^{-1} = + A_{\restr f(h)(B)/h} + O(h^\infty).\] +\end{proposition} + +Suppose now that $\U$ is covered by a locally finite union of balls +$B_\alpha$ on each of which is defined an affine chart $f_\alpha(h)$ +of $\Sigma(h)$. Proposition \ref{prop:GA} yields a family of affine +linear maps $A_{\alpha\beta}$ such that on non-empty intersections +$B_\alpha\cap B_\beta$, +\[ \frac{1}{h}f_\alpha(h) = +A_{\alpha\beta}\left(\frac{1}{h}f_\beta(h)\right).\] +This in turn defines a 1-cocycle $\M$ in the \v Cech cohomology of $\U$ +with values in the non-Abelian group $GA(n,\ZM)$. +\begin{definition} + \label{def:monodromy} + The class $[\M]\in\check{H}^1(\U,GA(n,\ZM))$ of the cocycle defined by + $A_{\alpha\beta}$ is called the {\bf quantum monodromy} of + $(\Sigma(h),\U)$. +\end{definition} + +Let $L$ be the canonical homomorphism, whose kernel is the group of +translations: +\[ L : GA(n,\RM) \fleche GL(n,\RM). \] +Let $\iota$ be the inclusion of $GL(n,\RM)$ into $GA(n,\RM)$ such that +for any $M\in GL(n,\RM)$, $\iota(M)$ leaves the origin $0\in\RM^n$ +invariant. Then $\iota$ is an injective homomorphism that depends on +the choice of the origin $0$, satisfying $L\circ\iota=Id$. Any $A\in +GA(n,\RM)$ can be written in a unique way +\[ A = \tau(k) \circ \iota(M),\] +(which is usually written $A = M + k$), where $M=L(A)\in GL(n,\ZM)$ +and $\tau(k)$ is translation by the vector $k\in\ZM^n$. + +The exact sequence of group homomorphisms +\[ 0\flechedroite{}\ \ZM^n \flechedroite{\tau}\ GA(n,\ZM) +\flechedroite{L}\ GL(n,\ZM) \flechedroite{}\ 1 \] gives rise to the +following sequence of maps (which are not homomorphisms, since +cohomology sets with values in a non-abelian group have no natural +group structure -- see \cite[p. 38]{hirzebruch}): +\[ \check{H}^1(\U,\ZM^n) \flechedroite{\tau_*}\ \check{H}^1(\U,GA(n,\ZM)) +\flechedroite{L_*}\ \check{H}^1(\U,GL(n,\ZM)) \flechedroite{}\ 1. \] +This sequence is ``exact'' in the sense that $L_*$ is surjective, and +if $L_*([\M])=1$, then there is an integer cocycle $[\omega]\in +\check{H}^1(\U,\ZM^n)$ such that $[\M]=\tau_*([\omega])$. The +surjectivity of $L_*$ is due to the existence of the cross section +$\iota$, which gives rise to the map +\[ \check{H}^1(\U,GA(n,\ZM)) \flechegauche{\iota_*}\ +\check{H}^1(\U,GL(n,\ZM)) \] such that $L_*\iota_*=Id$. For the second +point, we remark that if the cocycle $L(A_{\alpha\beta})$ is a +coboundary, then it can be written $M_\alpha M_\beta^{-1}$. Therefore +the cocycle $\iota(M_\alpha^{-1})A_{\alpha\beta}\iota(M_\beta)$ (which +is equivalent to $A_{\alpha\beta}$) has a linear part equal to the +identity, hence is a translation. + +\begin{remark} + The lack of injectivity for $\tau_*$ is measured by + $\check{H}^0(\U,GL(n,\ZM))$~: one can check that two cocycles $[k]$ + and $[k']$ in $\check{H}^1(\U,\ZM^n)$ yield the same element of\linebreak + $\check{H}^1(\U,GA(n,\ZM))$ if and only if there is an $M\in + \check{H}^0(\U,GL(n,\ZM))$ such that $[k']=[M\cdot k]$. +\end{remark} +Let us now give various interpretations of the quantum monodromy $\M$. + +The action of $GA(n,\ZM)$ on $\ZM^n$ being effective, it is a standard +fact that the cohomology set $\check{H}^1(\U,GA(n,\ZM))$ classifies +the isomorphism classes of fibre bundles over $\U$ with structure +group $GA(n,\ZM)$ and fibre $\ZM^n$ (see for instance +\cite[pp.40--41]{hirzebruch}). Let $\L$ be such a lattice bundle +associated to $\M$. The elements $A_{\alpha\beta}$ just define the +transition functions between two adjacent trivialisations of $\L$. + +Since these trivialisation functions are locally constant, there is a +naturally defined parallel transport $\gamma.p$ of a point $p\in\L_c$ +along a path $\gamma$ in the base $\U$. This defines the holonomy of +$\L$, as a map from $\pi_1(\U,c)$ into $GA(\L_c)$. We will always +identify the latter with $GA(n,\ZM)$ by choosing an affine basis of +$\L_c$. + +The choice of such a basis is equivalent to that of a trivialisation +$f$ of $\L$ above $c$ that sends this basis to the canonical basis of +$\ZM^n$; the holonomy $\bmu_f$ is then defined by~: +\begin{equation} + \label{equ:holonomy} + f(\gamma.p) = \bmu_f(\gamma)f(p). +\end{equation} +Finally, this is also equivalent to the choice of an affine chart +$f(h)$ of $\Sigma(h)$ around $c$. +If $\M$ is any cocycle associated to this +trivialisation, then +\begin{equation} + \label{equ:integral} + \bmu_{f}(\gamma) = A_{1,\ell}\circ\cdots\circ A_{3,2}\circ A_{2,1}, +\end{equation} +where $A_{i,j}$ denotes the transition element corresponding to a +pair of intersecting open balls $(B_i,B_j)$, and $B_1,\dots,B_\ell$ +enumerate elements of a cover of $\U$ encountered by $\gamma(t)$ when +$t$ runs from $0$ to $1$. + +We shall always assume that $\U$ is connected, so that $\bmu_f$ does +not depend on the base point $c$. Note that since +$(\gamma'\gamma).p=\gamma.(\gamma'.p)$, we have +\[ \bmu_f(\gamma'\gamma) = \bmu_f(\gamma)\bmu_f(\gamma').\] + + +It should be noticed that the bundles considered here have discrete +fibres, so that we could reduce the discussion to the theory of +coverings. The fibre bundle formulation seems however to be more +natural when it comes to comparing them with objects arising in +Hamiltonian systems. Nevertheless, the covering approach will be used +in Sect.~\ref{sec:detect}. + +Other geometric interpretations of $\M$ will also be discussed in +Sect.~\ref{sec:detect}. For the moment just notice that the +non-triviality of $[\M]$ is equivalent to the non-triviality of the +lattice bundle $\L$ and to the fact that there is no globally defined +symbol $f(h)$ on $\U$ sending $\Sigma(h)$ to the straight lattice +$h\ZM^n$. + +\begin{bew}{of Proposition \ref{prop:GA}} There are no surprises in + this quite elementary proof. Let $c\in \U$, and $f(h)$, $g(h)$ be +two affine charts of $\Sigma$ defined on a ball $B$ around +$c$. Because of Definition \ref{def:chart}, any open ball around $c$ +contains, for $h$ small enough, at least one element of +$\Sigma(h)$. Therefore, there exists a family $\lambda(h)\in +\Sigma(h)\cap B$ such that +\[ \lim_{h\fleche 0} \lambda(h) = c.\] +Let $k\in\ZM^n$ and let $\lambda'(h)$ be a family of elements of +$\Sigma(h)\cap B$ such that +\[ f(h;\lambda(h)) = f(h;\lambda'(h)) + hk + O(h^\infty).\] +Then, as $h$ tends to zero, $\frac{\lambda'(h)-\lambda(h)}{h}$ tends +towards a limit $v\in\RM^n$ satisfying +\[ k = df_0(c)v \] +(recall that $f_0$ denotes the principal part of $f(h)$). + +Since $\lambda(h)$ and $\lambda'(h)$ are in $\Sigma(h)$, there is a +family $k'(h)\in\ZM^n$ such that +\[ \left(\frac{g(h;\lambda'(h))-g(h;\lambda(h))}{h}\right) = k'(h) + +O(h^\infty).\] The left-hand side of the above equation has limit +$dg_0(c)v$ as $h\fleche 0$. Therefore $k'(h)$ is equal to a constant +integer $k'$ for small $h$, and we have +\[ k' = dg_0(c)(df_0(c))^{-1}k, \] +which implies that $dg_0(c)(df_0(c))^{-1}\in GL(n,\ZM)$. Since $GL(n,\ZM)$ +is discrete, there is a constant matrix $M\in GL(n,\ZM)$ such that for +all $c\in B$, $dg_0(c)= M\cdot(df_0(c))$; this in turn implies the existence +of a constant $k\in\ZM^n$ such that, on $B$, +\[ g_0 = M\cdot f_0 + k. \] +But $k$ is necessarily zero~: indeed, applying the above equality to +$\lambda(h)$ gives a sequence $k'(h)\in\ZM^n$ such that +\[ hk'(h) \egdef g(h;\lambda(h))-M\cdot f(h;\lambda(h)) = k + O(h).\] +Therefore $k'(h)$ must tend to zero, and hence must equal zero for +small $h$, implying that $k=0$. + +We have proved the existence of a +smooth symbol $F(h)$ such that +\[ M\cdot f(h)-g(h)= hF(h).\] +Because $F(h;\lambda(h))\in\ZM^n+O(h^\infty)$ and $\lim_{h\fleche + 0}F(h;\lambda(h))=F_0(c)$, we must have $F_0(c)\in\ZM^n$. So +\[ F_0 = const \in \ZM^n \textrm{ in } B.\] +This easily implies that all lower order terms in $F(h)$ must vanish +on $B$, so we are left with +\[ F(h) = k + O(h^\infty), \textrm{ for a }k\in\ZM^n.\] + +This gives $g(h)=M\cdot f(h)-hk + O(h^\infty)$, which reads +\[ \frac{1}{h}g(h)=A(\frac{1}{h}f(h)) + O(h^\infty),\] +with $A\in GA(n,\ZM)$ defined by $A(p)=M\cdot p-k$, $p\in\ZM^n$.\qed +\end{bew} + +\begin{remark} + Because of the discreteness of $GA(n,\ZM)$, Proposition + \ref{prop:GA} implies that there is an $h_0>0$ such that the + transition element $A$ is uniquely defined by + $(g(h_0)/h_0)$ $(f(h_0)/h_0)^{-1}$ acting on a finite subset of + $\ZM^n$. Therefore, when restricted to any open subset of $\U$ with + compact closure in $\U$, the cocycle $[\M]$ + is really a \emph{quantum} object, in the sense that ``you don't + need to let $h$ tend to zero'' to define it. +\end{remark} + +\section{Link with the Classical Monodromy} +\label{sec:classical} +Let $P_1(h),\dots,P_n(h)$ be a set of commuting self-adjoint $h$-\pdo +s on an $n$-dimensional manifold $X$. They will be assumed to be classical +and of order zero, in the sense that in any coordinate chart their +Weyl symbols $p_j(h)$ have an asymptotic expansion of the form +\[ p_j(h;x,\xi) = p^j_0(x,\xi) + hp^j_1(x,\xi) + h^2p^j_2(x,\xi) + +\cdots . \] +Because the principal symbols $p_0^1,\dots,p_0^n$ commute with respect +to the symplectic Poisson bracket on $T^*X$, the map +\[ T^*X \ni (x,\xi)\flechedroite{p}\ +(p_0^1(x,\xi),\dots,p_0^n(x,\xi))\in \RM^n \] +is a momentum map for the local Hamiltonian action of $\RM^n$ on $T^*X$ +defined by the Hamiltonian flows of the $p_0^j$. We will always assume +that $p$ is \emph{proper}, so that the level sets +\[ \Lambda_c = p^{-1}(c)\] +are compact. Moreover, we ask that these level sets be +\emph{connected}. Conclusions for non-connected $\Lambda_c$ can be +obtained by separately studying the different connected components. + +Let $U_r$ be the open subset of regular values of the momentum map +$p$, and let $\U$ be an open subset of $U_r$ with compact closure. + +It follows from the Arnold-Liouville theorem that $p_{\restr \U}$ is a + smooth fibration whose fibres are Lagrangian tori. The structure of + this fibration is semi-globally (\emph{i.e.} in a neighbourhood of a + fibre) described with the help of action-angle coordinates. However, + the flat fibre bundle $H_1(\Lambda_c,\ZM)\fleche c\in\U$ (with fibre + $\ZM^n$) may have non-trivial monodromy, preventing the construction + of \emph{global} action variables on $p^{-1}(\U)$ (see Duistermaat + \cite{duistermaat}). We will denote by $[\M_{cl}]$ (classical + monodromy) the cocycle in $\check{H}^1(\U,GL(n,\ZM))$ associated to + this lattice bundle. + +On the other hand, let $\Sigma(h)$ be the intersection with $\U$ +of the joint spectrum of the operators $P_1(h),\dots,P_n(h)$. It is +known from \cite{charbonnel} that this spectrum is discrete and for small $h$ +is composed of simple eigenvalues. Moreover, the +following result holds: +\begin{proposition}[\cite{charbonnel}] +$\Sigma(h)$ is an asymptotic affine lattice on $\U$. +\end{proposition} +We denote by $[\M_{qu}]\in \check{H}^1(\U,GA(n,\ZM))$ the quantum +monodromy of the spectrum on $\U$, given by Definition +\ref{def:monodromy}. + +Recall that $\iota$ denotes the inclusion of $GL(n,\RM)$ into +$GA(n,\RM)$ such that for any $M\in GL(n,\RM)$, $\iota(M)$ leaves the +origin $0\in\RM^n$ invariant. + +The relation between $[\M_{qu}]$ and the classical monodromy +$[\M_{cl}]$ is then given by the following theorem~: +\begin{theorem} + \label{theo:main} + The quantum monodromy is ``dual'' to the classical monodromy in the + following sense: + \[ [\M_{qu}] = \iota_*(\trsp[\M_{cl}]^{-1}).\] + In other words, for any $c\in\U$ there exists a choice of + basis of $H_1(\Lambda_c,\ZM)$ and of an affine chart of + $\Sigma(h)$ such that the monodromy representations + \[\bmu^{cl} : \pi_1(\U,c)\fleche GL(n,\ZM) \] + and + \[\bmu^{qu} : \pi_1(\U,c)\fleche GA(n,\ZM) \] +defined by $[\M_{cl}]$ and $[\M_{qu}]$ satisfy~: +\[ \bmu^{qu} = \iota\circ(\trsp\bmu^{cl})^{-1}.\] +\end{theorem} +\begin{proof} Let $\alpha$ be the Liouville 1-form on $T^*X$. Let $c_0\in\U$ +and for $c$ near $c_0$ let $(\gamma_1(c),\dots,\gamma_n(c))$ be a smooth +family of loops on $\Lambda_c$ whose homology classes form a basis of +$H_1(\Lambda_c,\ZM)$. It is known from \cite{charbonnel,colinII} (see +also \cite{san2} for a viewpoint closer to this article) that one can +find an affine chart $f(h)$ for $\Sigma(h)$ around $c$ such that the +principal part $f_0$ is equal to the action integral associated to +$\gamma_1,\dots,\gamma_n$: +\[ f_0(c) = (\frac{1}{2\pi}\int_{\gamma_1(c)}\alpha, +\dots,\frac{1}{2\pi}\int_{\gamma_n(c)}\alpha).\] + +Because of Proposition \ref{prop:GA}, any other affine chart around +$c$ having the same principal part must equal $f(h)$ (modulo +$O(h^\infty)$). In this way, the choice of a local smooth basis of +$H_1(\Lambda_c,\ZM)$ determines an affine chart of +$\Sigma(h)$. If $(\gamma'_1(c),\dots,\gamma'_n(c))$ is another basis +of $H_1(\Lambda_c,\ZM)$ such that +\begin{equation} + \label{equ:bases} + (\gamma'(c)) = M(c)\cdot(\gamma(c)), +\end{equation} +for a matrix $M(c)\in GL(n,\ZM)$ depending smoothly on $c$, then the +corresponding affine charts $f(h)$ and $f'(h)$ of $\Sigma(h)$ satisfy~: +\[ f'(h;c) = M(c)\cdot f(h;c) + O(h^\infty).\] +Recall that the notation ``$M\cdot$'' here means matrix multiplication +by $M$, which is of course the same as affine composition by +$\iota(M)$. + +But formula (\ref{equ:bases}) says that if $k$ and $k'$ are +trivialisation functions of the bundle\linebreak $H_1(\Lambda_c,\ZM)\fleche c$ +associated to the basis $\gamma$ and $\gamma'$, then +$k'=\trsp M^{-1}k$. Therefore, if $\trsp M_{\alpha\beta}^{-1}$ are +transition elements for the lattice bundle $H_1(\Lambda_c,\ZM)\fleche +c$, then $\iota(M_{\alpha\beta})$ define a monodromy cocycle for +$\Sigma(h)$.\qed +\end{proof} + +\begin{remark} + The fact that the \emph{affine} nature of quantum monodromy is here + naturally reduced to an action of the \emph{linear} group + $GL(n,\ZM)$ is due the the global existence of a primitive of the + symplectic form on $T^*X$, namely the Liouville 1-form $\alpha$. +\end{remark} + +\section{Monodromy of a \emph{Focus-Focus} Singularity} +\label{sec:ff} +It is probably not worth discussing monodromy in arbitrary degrees of +freedom, for it is a typical phenomenon of 4-dimensional symplectic +manifolds (see \cite{zung}). + +More precisely, let $X$ be a 2-dimensional manifold, and let $P_1(h)$, +$P_2(h)$ be two commuting self-adjoint $h$-\pdo s on $X$. As before, +suppose that the momentum map $p=(p_0^1,p_0^2)$ defined by the +principal symbols is proper with connected level sets. + +We shall make the following hypothesis. There exists a critical point +$m\in T^*X$ of $p$ of maximal corank (\emph{i.e.} both $p_0^1$ and +$p_0^2$ are critical at $m$) such that, in some local symplectic +coordinates $(x,y,\xi,\eta)$, the Hessians $(p_0^1)''(m)$ and +$(p_0^2)''(m)$ (thereafter denoted by $\mathcal{H}(p_0^1)$ and +$\mathcal{H}(p_0^2)$) generate a 2-dimensional subalgebra of the +algebra $\mathcal{Q}(4)$ of quadratic forms in $(x,y,\xi,\eta)$ under +Poisson bracket that admits the following basis $(q_1,q_2)$: +\[ q_1 = x\xi + y\eta,\] +\[ q_2 = x\eta - y\xi.\] +Such a singularity $m$ is called a \emph{focus-focus} singularity. The +point $m$ is then isolated amongst critical points of $p$. Therefore, +we can choose $\U\subset U_r$ to be a small punctured disc around +$o=p(m)$. Finally, we shall always assume that $m$ is the only +critical point of the critical level set $\Lambda_0=p^{-1}(o)$. + +It is known (probably since \cite{zou}; see for instance \cite{san2} +or \cite{cushman-duist2} for discussions and more references on this +topic) that the fibration $p_{\restr \U}$ has non-trivial monodromy, +and can be described in the following way: + +Near $m$, we know from \cite{eliasson-these} that the integrable +Hamiltonian system $(p_0^1,p_0^2)$ can be brought into a normal form +given by $(q_1,q_2)$. In other words there exists a local +diffeomorphism $F:(\RM^2,0)\fleche (\RM^2,o)$ such that +\[ (p_0^1,p_0^2) = F(q_1,q_2).\] +This allows one to define transversal vector fields $\ham{1}$ and +$\ham{2}$ tangent to the fibres $\Lambda_c$ that are equal to +the Hamiltonian vector fields $\ham{q_1}$ and $\ham{q_2}$ near +$m$. Note that $\ham{2}$ is periodic of period $2\pi$. + +Around each $c\in\U$, we can now define the following smooth basis +$(\gamma_1(c),\gamma_2(c))$ of $H_1(\Lambda_c,\ZM)\simeq +\pi_1(\Lambda_c)$: +\begin{itemize} +\item $\gamma_2(c)$ is a simple integral loop of $\ham{2}$. +\item Take a point on $\gamma_2(c)$; let it evolve under the flow of + $\ham{1}$. After a finite time, it goes back on + $\gamma_2(c)$. Close it up on $\gamma_2(c)$. This defines $\gamma_1(c)$. +\end{itemize} +\begin{figure}[hbtp] + \begin{center} + \leavevmode +\begin{picture}(0,0)% +\psfig{file=424-2.eps}% +\end{picture}% +\setlength{\unitlength}{3947sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(4879,3143)(1859,-3217) +\put(3341,-2306){\makebox(0,0)[lb]{$\Lambda_c$}}% +\put(3291,-751){\makebox(0,0)[lb]{$\gamma_1(c)$}} +\put(4943,-2031){\makebox(0,0)[lb]{$\gamma_2(c)$}} +\end{picture} + \caption{The basis $(\gamma_1(c),\gamma_2(c))$} + \label{fig:basis} + \end{center} +\end{figure} +\begin{proposition}[\cite{zou}] + Let $c\in\U$. With respect to the basis $(\gamma_1(c),\gamma_2(c))$, + the action of the classical monodromy map $\bmu^{cl}$ on a simple + loop $\delta\in\pi_1(\U,c)$ enclosing $o$ is given by the matrix + \[ \bmu^{cl}(\delta) = \left(\begin{array}{cc} 1 & 0 \\ + \epsilon & 1\end{array}\right).\] + Here $\epsilon$ is the sign of + $\det M$, where $M\in GL(2,\RM)$ is the unique matrix such that~: + \[ (\mathcal{H}(p^1_0),\mathcal{H}(p^2_0)) = + M\cdot(\mathcal{H}(q_1),\mathcal{H}(q_2)). \] +\end{proposition} +Note also that $M=dF(0)$. + +This, together with Theorem \ref{theo:main}, proves the following +result: +\begin{theorem} + \label{theo:ff} + Let $P_1(h),P_2(h)$ be a quantum integrable system with a focus-focus + singularity. Then there exists a small punctured neighbourhood $\U$ + of the critical value $o$ such that for any $c\in\U$, if $f(h)$ is + an affine chart of the joint spectrum $\Sigma(h)$ around $c$ having + principal part + \[ \left(\frac{1}{2\pi}\int_{\gamma_1(c)}\alpha, + \frac{1}{2\pi}\int_{\gamma_2(c)}\alpha\right), \] + then the value of the quantum monodromy map + $\bmu_f^{qu}\in GA(2,\ZM)$ at a simple loop $\delta\in\pi_1(\U,c)$ + enclosing $o$ is given by the matrix +\[ \bmu_f^{qu}(\delta) = \iota\left(\begin{array}{cc} 1 & -\epsilon \\ + 0 & 1\end{array}\right).\] + Here $\epsilon$ is the sign of + $\det M$, where $M\in GL(2,\RM)$ is the unique matrix such that~: + \[ (\mathcal{H}(p^1_0),\mathcal{H}(p^2_0)) = + M\cdot(\mathcal{H}(q_1),\mathcal{H}(q_2)). \] +\end{theorem} + +\section{How to Detect Quantum Monodromy} +\label{sec:detect} +\subsection{Introduction} +Theorem \ref{theo:main} wouldn't be of much interest if one could not +``read off'' the quantum monodromy from a picture of the joint +spectrum. + +This is actually easy to do, at least in a heuristic way. The +rigorous mathematical formulation may however look slightly awkward. + +The first idea is the following. Given a straight lattice +$\ZM^n$, and any two points $A$ and $B$ in $\ZM^n$, there is a natural +parallel translation from $A$ to $B$ acting on $\ZM^n$, namely the +translation by the integral vector $\cutevector{AB}$. + +Now, the joint spectrum $\Sigma(h)$ locally around any point $c\in\U$ +looks like a lattice. If the points $A$ and $B$ in $\Sigma(h)$ are +close enough to $c$ and $h$ is small enough, one can still define a +parallel translation from $A$ to $B$, taking points of $\Sigma(h)$ +near $A$ to points in $\Sigma(h)$ near $B$. This allows us to pass +from one chart to another, and hence to define the notion of +parallel transport along any loop through $c$. This yields a +map from $\pi_1(\U,c)$ to $GL(n,\ZM)$ which is precisely the +linear part of the quantum monodromy $\bmu^{qu}$. + +\begin{figure}[hbtp] + \begin{center} + \leavevmode +\begin{picture}(0,0)% +\psfig{file=424-3.eps}% +\end{picture}% +\setlength{\unitlength}{3947sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(1824,1824)(5089,-2323) +\end{picture} + \caption{Parallel transport on $\Sigma(h)$} + \label{fig:connexion} + \end{center} +\end{figure} +This idea is made precise in Sect.~\ref{sec:parallel}. + +The problem can also be viewed the other way round. Roughly speaking, +$(\Sigma(h),\U)$ is an affine manifold, and hence can be defined by +the data of a local diffeomorphism $f(h)$ from the universal cover +$\tilde{\U}$ of $\U$ to $h\RM^n$ sending $\Sigma(h)$ to $h\ZM^n$, and +of the holonomy $\nu$ associated to it~: +\[ f(h;\gamma.\tilde{c}) = \nu_{\tilde{c}}(\gamma)f(h;\tilde{c}), +\quad \forall \gamma\in\pi_1(\U), \forall \tilde{c}\in\tilde{U}.\] Of +course, $\nu$ should be related to the quantum monodromy $\bmu_f$. +The diffeomorphism $f(h)$ can be seen as an ``unwinding'' of +$\Sigma(h)$ onto $\RM^n$. This viewpoint is developed in Sect.~\ref{sec:unwinding}. + +\subsection{Parallel transport on $\Sigma(h)$} +\label{sec:parallel} +We discuss here the notion of parallel transport on any asymptotic +affine lattice $(\Sigma(h),\U)$. + +\noindent 1.~ First suppose that there exists an affine chart $f(h)$ +of $\Sigma(h)$ defined +globally on $\U$. Since $f(h)$ is elliptic and sends elements of +$\Sigma(h)$ into $h\ZM^n + O(h^\infty)$, there is an $h_0>0$ such that +for any $h<h_0$, there is an injective map $\tilde{f}(h)$ sending +elements of $\Sigma(h)$ exactly into $h\ZM^n$ and such that +$\tilde{f}(h)-f(h)=O(h^\infty)$. + +Because $f(h)$ is of order zero, there is a fixed open ball +$\tilde{B}'\subset f(h;\U)$ such that $\tilde{B}'\cap(h\ZM^n)$ is +contained in $\tilde{f}(h;\Sigma(h))$. + +Then, one can find a smaller ball $\tilde{B}\subset \tilde{B'}$ such +that for any two points $\tilde{P}$, $\tilde{Q}$ in +$\tilde{B}\cap(h\ZM^n)$, the translation by the vector +$\cutevector{\tilde{P}\tilde{Q}}$ takes any point of +$\tilde{B}\cap(h\ZM^n)$ into $\tilde{B}'\cap(h\ZM^n)$ (Fig. +\ref{fig:translation}). +\begin{figure}[hbtp] + \begin{center} + +\begin{picture}(0,0)% +\hskip17mm\psfig{file=424-4.eps}% +\end{picture}% +\setlength{\unitlength}{2763sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(5779,4522)(1201,-5168) +\put(3151,-811){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$h$}}} +\put(1201,-3136){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$h\ZM^n$}}} +\put(6901,-1936){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{B}'$}}} +\put(6976,-3211){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{B}$}}} +\put(3826,-3211){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}$}}} +\put(4801,-2761){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{Q}$}}} +\put(4441,-2438){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{A}$}}} +\put(5394,-2048){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{A}'$}}} +\end{picture} + \caption{Parallel translation} + \label{fig:translation} + \end{center} +\end{figure} +Let us denote by $B$ an open ball in $\RM^n$ such that $f(h;B)\subset +\tilde{B}$. Pulling back by $\tilde{f}(h)$, one thus defines the +``parallel transport'' $\tau_{\tinyvector{PQ}}(A)$ of a point +$A\in\Sigma(h)\cap B$ along the direction given by two points $P$ and +$Q$ in $\Sigma(h)\cap B$. When the composition is defined, we have +\begin{equation} + \label{equ:composition} + \tau_{\tinyvector{QR}}\circ\tau_{\tinyvector{PQ}} = + \tau_{\tinyvector{PR}}. +\end{equation} +Moreover, because translation in $\ZM^n$ is +an isometry, there exists a constant $C>0$, +independent of $h$, such that for any $A\in\Sigma(h)\cap B$ +\begin{equation} + \label{equ:bounded} + ||\cutevector{Q\tau_{\tinyvector{PQ}}(A)}|| < C||\cutevector{PA}||. +\end{equation} + +Because of Proposition \ref{prop:GA}, any other choice of affine +chart $f(h)$ gives the same parallel transport. + +\noindent 2.~ Now, let $(\Sigma(h),\U)$ be a general asymptotic +affine lattice. If $\gamma$ is any path in $\U$, one can cover its +image by open balls $B_i$ on which parallel transport is well defined +for $h$ less than some $h_i>0$. If $\overline{\U}$ is compact, as we +shall always assume, this can be done with a finite number of such +balls $B_1,\ldots,B_\ell$, ordered in a way that for each $1\leq +i<\ell$, $B_i\cap B_{i+1}\neq\emptyset$. + +In the following, take $h$ to be less than $\min_i h_i$. Let +$P\in\Sigma(h)\cap B_0$ and $Q\in\Sigma(h)\cap B_\ell$. For each +$i=1,\dots,\ell-1$, pick up a point $P_i\in \Sigma(h)\cap(B_i\cap +B_{i+1})$. For $h$ small enough, this set is not empty. Because of +the estimate (\ref{equ:bounded}), the mapping +\[ \tau_{\gamma,P,Q}\egdef +\tau_{\tinyvector{P_{\ell-1}Q}}\circ\cdots\circ +\tau_{\tinyvector{P_1P_2}} \circ \tau_{\tinyvector{PP_1}} \] is +well-defined when restricted to a sufficiently small ball $B_0$ around +$P$ (here again, $\Sigma(h)\cap B_0$ won't be empty if $h$ is small +enough). Equation (\ref{equ:composition}) shows that this map does +not depend on the choice of the intermediate points $P_i$. Therefore +it depends only on $P$, $Q$, and on the homotopy class of $\gamma$ (as +a path from a point in $B_1$ to a point in $B_\ell$). + +If $Q=P$, and $\gamma$ is a loop ($B_\ell\cap B_1\neq\emptyset$ and +$B_0\subset B_1$) then $\tau_{\gamma,P,P}$ is a map from +$\Sigma(h)\cap B_0$ to $\Sigma(h)\cap B_1$ leaving $P$ invariant. If +$f(h)$ is an affine chart for $\Sigma(h)$ on $B_1$, then +$\tilde{f}(h)\circ \tau_{\gamma,P,P} \circ \tilde{f}(h)^{-1}$ is a +locally defined map $\tilde{\tau}_{\gamma,f(h),P}$ from $h\ZM^n$ to +itself leaving $\tilde{f}(h;P)$ invariant. + +We know from Sect.~\ref{sec:construction} (formula +(\ref{equ:holonomy})) that the choice of such an affine chart allows +the quantum monodromy map $\bmu_f$ to take its values in +$GA(n,\ZM)$. Remember that $L$ denotes the natural homomorphism from +$GA(n,\RM)$ to $GL(n,\RM)$. +\begin{proposition} + \label{prop:parallel} + The map $\tilde{\tau}_{\gamma,f(h),P}$ is equal to the linearisation + at $\tilde{P}=\tilde{f}(h;P)$ of the quantum + monodromy along $\gamma$~: + \[ \forall \tilde{R}\in h\ZM^n, \quad + \cutevector{\tilde{P}\tilde{\tau}_{\gamma,f(h),P}(\tilde{R})} = + L(\bmu_f(\gamma))\cutevector{\tilde{P}\tilde{R}} , \] + whenever the left-hand side of the above is defined. +\end{proposition} +\begin{proof} If we choose affine charts $f_i(h)$ for $\Sigma(h)$ on each of +the $B_i$'s with $f_1=f$, and let $A_{i,i+1}$ be the transition +elements of the monodromy cocycle +\[ f_i(h)/h = A_{i,i+1}(f_{i+1}(h)/h) + O(h^\infty)\quad +(\textrm{convention } \ell+1\equiv 1), \] +then it is easy to check that +\[ \cutevector{\tilde{P}\tilde{\tau}_{\gamma,f(h),P}(\tilde{R})} = +L(A_{1,\ell})\cdots +L(A_{3,2})L(A_{2,1})\cdot\cutevector{\tilde{P}\tilde{R}},\] whenever +the composition is defined. Using (\ref{equ:integral}) finishes the +proof. \cqfd +\end{proof} + +As an application, one can easily ``read off'' from the spectrum of +the quantum Champagne bottle (Fig. \ref{fig:pendulum}) that the linear +part of the quantum monodromy is conjugate to the matrix +$\left(\begin{array}{cc} 1 & -1 \\ 0 & 1\end{array}\right)$. +\begin{figure}[hbtp] + \begin{center} + \leavevmode +\begin{picture}(0,0)% +\psfig{file=424-5.eps}% +\end{picture}% +\setlength{\unitlength}{2368sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(7524,5124)(169,-6073) +\put(7426,-3511){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$E_1$}}} +\put(4051,-1186){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$E_2=hn$}}} +\put(5612,-3898){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$P$}}} +\put(6032,-3411){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$R$}}} +\put(5140,-3306){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$R'$}}} +\put(2461,-4664){\makebox(0,0)[lb]{\smash{\SetFigFont{7}{8.4}{\rmdefault}{\mddefault}{\updefault}$\gamma$}}} +\end{picture} +\caption{Spectrum of the Champagne bottle. The gray disc + encloses the \ff\ critical value. $R'=\tau_{\gamma,P,P}(R)$} + \label{fig:pendulum} + \end{center} +\end{figure} + +\subsection{Unwinding the spectrum} +\label{sec:unwinding} +We keep here the notation of the previous paragraph. In particular, +$\Sigma(h)$ is any asymptotic affine lattice on $\U$, $\gamma$ is a +path in $\U$ whose image is covered by balls $B_i$ on which local +parallel translation is defined. We choose points $P\in +B_1\cap\Sigma(h)$, $Q\in B_\ell\cap\Sigma(h)$ and +$P_1,P_2,\dots,P_{\ell-1},P_\ell=Q$ such that for $i=1,\dots,\ell-1$, +$P_i\in B_i\cap B_{i+1}\cap\Sigma(h)$. + +Given an affine chart $f(h)$ on $B_1$, for $h$ small there is a unique +$k_1\in\ZM^n$ such that the map +$\tilde{f}(h)\circ\tau_{\tinyvector{PP_1}}\circ\tilde{f}(h)^{-1}$ is +just translation by $hk_1$. If $B_1,\dots,B_\ell$ are endowed with +affine charts $f_1(h)=f(h),f_2(h),\dots,f_\ell(h)$, in the same way we +define $k_i\in\ZM^n$ such that +\[ +\tilde{f_i}(h)\circ\tau_{\tinyvector{P_{i-1}P_i}}\circ\tilde{f_i}(h)^{-1} +\] +is translation by the vector $hk_i$. +We unwind the points $P,P_1,\dots,P_\ell$ onto $h\ZM^n$ using the following +procedure (see Fig. \ref{fig:unwinding}): +\begin{figure}[hbtp] + \begin{center} + \leavevmode +% \hspace{-0.7cm} +\begin{picture}(0,0)% +\psfig{file=424-6.eps,width=1.1\textwidth}% +\end{picture}% +\setlength{\unitlength}{2653sp}% +% +\begingroup\makeatletter\ifx\SetFigFont\undefined% +\gdef\SetFigFont#1#2#3#4#5{% + \reset@font\fontsize{#1}{#2pt}% + \fontfamily{#3}\fontseries{#4}\fontshape{#5}% + \selectfont}% +\fi\endgroup% +\begin{picture}(8649,5204)(1139,-5851) +\put(6931,-1379){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$h$}}} +\put(9268,-2896){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_{11}$}}} +\put(6928,-4381){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_5$}}} +\put(8278,-5206){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_8$}}} +\put(7603,-5881){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_7$}}} +\put(6883,-2161){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_2$}}} +\put(7378,-1456){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}_1$}}} +\put(9178,-1871){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{Q}$}}} +\put(7978,-2236){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$\tilde{P}$}}} +\put(5851,-886){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$f(h)$}}} +\put(4501,-4186){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_9$}}} +\put(4565,-3556){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_{10}$}}} +\put(4542,-2866){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_{11}$}}} +\put(3695,-2461){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$Q$}}} +\put(4201,-4786){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_8$}}} +\put(3414,-5386){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_7$}}} +\put(2457,-4786){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_6$}}} +\put(1670,-4111){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_5$}}} +\put(1557,-3436){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_4$}}} +\put(1670,-2836){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_3$}}} +\put(1782,-2161){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_2$}}} +\put(2570,-1861){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P_1$}}} +\put(3639,-2086){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$P$}}} +\put(5855,-3239){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$E_1$}}} +\put(3324,-914){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}$E_2=hn$}}} +\end{picture} + \caption{Unwinding of the points $P_i$. We deduce that + $y_{\tilde{P}}=4$, which allows us to locate the horizontal line + through the origin $0\in h\ZM^2$ (the dotted one)} + \label{fig:unwinding} + \end{center} +\end{figure} +\begin{itemize} +\item $\tilde{P}=\tilde{f}(h;P)$; +\item $\tilde{P}_1=\tilde{P}+hk_1 = \tilde{f}(h,P_1)$; +\item $\tilde{P}_2=\tilde{P}_1 + hL(A_{2,1})\cdot k_2$; +\item \ldots +\item $\tilde{Q}=\tilde{P}_\ell=\tilde{P}_{\ell-1} + +hL(A_{\ell,\ell-1})\cdots L(A_{2,1})\cdot k_\ell$. +\end{itemize} +Then one easily checks that +\[ \tilde{P}_i = hA_{1,2}\circ A_{2,3}\circ\cdots\circ +A_{i-1,i}(\tilde{f}_i(h;P_i)/h). \] +In particular, applying this procedure to a loop $\gamma$ ($P=Q$) proves the +following~: +\begin{proposition} + \label{prop:unwinding} + For $h$ small enough, the quantum monodromy $\bmu_f$ gives the end + point $\tilde{Q}$ of the unwinding of any loop $\gamma$ on $\U$ + through a point $P\in\Sigma(h)$ around which we are given an affine + chart $f(h)$ by the following formula~: + \[ \tilde{Q} = h(\bmu_f(\gamma))^{-1}(\tilde{f}(h;P)/h). \] +\end{proposition} +\begin{remark} + There is a unique symbol $g(h)$ defined on the universal cover + $\tilde{\U}$ of $\U$ that is an affine chart for $\Sigma(h)$ and + that coincides with $f(h)$ above $B_0$. Then $Q$ can be seen as the + lift $\gamma.P\in\tilde{\U}$. The point is now that + \[ g(h;Q) = \tilde{Q} + O(h^\infty). \] + For any $P\in\tilde{\U}$, and for any $\gamma\in\pi_1(\U)$, there is + a unique $\nu_P(\gamma)\in GA(n,\ZM)$ such that + \[ g(h;\gamma.P)/h = \nu_P(\gamma)(g(h;P)/h) + O(h^\infty). \] + By definition, we have + $\nu_P(\gamma\gamma')=\nu_{\gamma.P}(\gamma')\nu_P(\gamma)$. But one + can show that for any loop $\gamma$ such that $\gamma.P=Q$, then + \[ \nu_Q(\gamma') = \nu_P(\gamma)\nu_P(\gamma')\nu_P(\gamma)^{-1}. \] + Therefore, $\nu_P$ is actually a homomorphism. Proposition + \ref{prop:unwinding} just says that + \[ \nu_P=\bmu_f^{-1}. \] +\end{remark} +Applying this proposition together with Theorem \ref{theo:ff} to a \ff\ +singularity, we see that if the principal part of $f(h)$ is given by +the action integrals $\frac{1}{2\pi}\int_{\gamma_1}\alpha$ and +$\frac{1}{2\pi}\int_{\gamma_2}\alpha$ then, for a small loop +$\delta$ enclosing the critical value $o$, +\[ \nu(\delta) = \iota\left( + \begin{array}{cc} +1 & \epsilon \\ 0 & 1 + \end{array}\right). \] +In particular, the whole horizontal line through the origin consists +of fixed points. Of course, locating the origin on a diagram like +Fig. \ref{fig:unwinding} may require the computation of the action +at one point. However, given $\tilde{P}$ and its image $\tilde{Q}$, it +is easy to find the horizontal line through the origin, for +\[ \epsilon y_{\tilde{P}} = x_{\tilde{Q}} - x_{\tilde{P}}.\] +\begin{acknowledgements} One of the reasons for having written +this article is the enthusiasm of R. Cushman for the subject; I would +like to thank him for this. I would also like to thank my adviser +Y. Colin de Verdi\`ere, and J. J. Duistermaat, for stimulating +discussions. + +My research is supported by a Marie Curie Fellowship +Nr. ERBFMBICT961572. +\end{acknowledgements} + +\begin{thebibliography}{15} + +\bibitem{bates}Bates, L.M.: +{Monodromy in the {C}hampagne bottle}. Z. Angew. Math. Phys. + \textbf{6}, 837--847 (1991) + +\bibitem{berger-affine}Berger, M.: {\it G{\'e}om{\'e}trie}. Vol. \textbf{1}. Paris: +Cedic/Nathan, 1977 + +\bibitem{charbonnel}Charbonnel, A.-M.: +{Comportement semi-classique du spectre conjoint + d'op{\'e}rateurs pseudo-diff{\'e}rentiels qui commutent}. 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