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+\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$}}
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+ \addtolength{\unitlength}{4ex}
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+ \end{picture}}
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+ \settowidth{\unitlength}{\mbox{$#1$}}
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+ \begin{picture}(1,0)
+ \put(0,0){\vector(1,0){1}}
+ \put(0.5,0){\makebox(0,0){${\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}}
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+ \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}
+
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+\end{document}