On moduli spaces of flat connections with non-simply connected structure
  group

Abstract:
  We consider the moduli space of flat G-bundles over the twodimensional torus,
where G is a real, compact, simple Lie group which is not simply connected. We
show that the connected components that describe topologically non-trivial
bundles are isomorphic as symplectic spaces to moduli spaces of topologically
trivial bundles with a different structure group. Some physical applications of
this isomorphism which allows to trade topological non-triviality for a change
of the gauge group are sketched.

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\begin{document}\version\versionno

\begin{flushright}  {~} \\[-23 mm] {\sf hep-th/9611092} \\
{\sf UCB-PTH-96/53} \\ {\sf LBNL-39585}
\\[1 mm]{\sf November 1996} \end{flushright} \vskip 2mm

\begin{center} \vskip 12mm

{\Large\bf ON MODULI SPACES OF FLAT CONNECTIONS } \vskip 0.3cm
{\Large\bf WITH NON-SIMPLY CONNECTED} \vskip 0.3cm
{\Large\bf STRUCTURE GROUP} \vskip 12mm
{\large Christoph Schweigert}\\[5mm] {\small Department of Physics,
University of California, Berkeley, CA 94720, USA}\\
{\small and} \\
{\small Theoretical Physics Group, Lawrence Berkeley National Laboratory}\\
{\small Berkeley, CA 94720, USA}
\end{center}

\vskip 15mm

\begin{quote} {\bf Abstract.} \\
We consider the moduli space of flat $\mbox{$G$}$-bundles over the two-di\-men\-si\-o\-nal\ torus, where
$\mbox{$G$}$ is a real, compact, simple Lie group which is not simply connected. We
show that the connected components that describe topologically non-trivial\
bundles are isomorphic as symplectic spaces to moduli spaces of topologically
trivial bundles with a different structure group. Some physical applications
of this isomorphism which allows to trade topological non-triviality for a
change of the gauge group are sketched.
\end{quote}

\vskip 12mm

\sect{Introduction}

In this letter we present an isomorphism between two different moduli spaces of
gauge equivalent classes of flat connections on principal bundles over the
two-di\-men\-si\-o\-nal\ torus $\Sigma$. Moduli spaces of flat connections over complex curves
have been the subject of intensive investigations, since
they play a key role both in Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theory\ in three dimensions and in two-di\-men\-si\-o\-nal\ conformal field theory.
In Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theory\ with structure group $\mbox{$G$}$ the moduli space ${\cal M}_{G}$ parametrizes the
space of inequivalent
classical solutions. Holomorphic quantization allows to associate a finite-dimensional\
complex vector space to ${\cal M}_{G}$, the space of conformal blocks
\cite{axdw,witt27}. Appropriate sesquilinear combinations
of elements of these spaces describe the correlation functions of the WZW model\
based on \mbox{$G$}.
The latter constitute an important subclass of two-di\-men\-si\-o\-nal\ conformal field theories; they also
serve as the building blocks of many other conformal field theories, e.g.\ via the coset
construction.

In the present note, we consider the case when the structure group
$\mbox{$G$}$ is a compact, connected, finite-dimensional Lie group, which is not simply
connected. The corresponding moduli spaces arise naturally, e.g.\ in the
description of WZW model s or Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theories\ based on these groups. Another important
application of these spaces is
the resolution of field identification fixed points in conformal field theories \cite{fusS4}. In
the algebraic approach, the solution of this problem has given rise to a
surprisingly
rich structure, both in the case of coset conformal field theories\ \cite{fusS4} and integer spin
simple current modular invariants \cite{fusS6}. It has been argued \cite{hori}
that in order to describe the resolution
of field identification fixed points in these models in a Lagrangean framework,
non-simply connected\ structure groups $\mbox{$G$}$ have to be considered. The result of this
letter is therefore a first step towards a geometric understanding of the
results of \cite{fusS4,fusS6}.

If the structure group $\mbox{$G$}$ is non-simply connected, the moduli space ${\cal M}_{G}$ consists of
different connected components, which typically have different dimensions.
Writing \mbox{$G$}\ as the quotient of $\mbox{$\tilde G$}$, the universal covering group
of $\mbox{$G$}$, by a subgroup $Z$ of the center of $\mbox{$\tilde G$}$,
\begin{equation} \mbox{$G$} \cong  \mbox{$\tilde G$} / Z \, , \end{equation}
the connected components of ${\cal M}_G$ are labeled by the finite abelian group
$Z$:
\begin{equation} {\cal M}_{G} = \bigcup_{\omega\in Z} {\cal M}_{G}^\omega \, . \end{equation}
If $\omega\in Z$ is not the identity, ${\cal M}_{G}^\omega$ is said to describe a
topologically non-trivial\ sector of the theory.

The main result of this note is that the moduli space ${\cal M}_{G}^\omega$
describing a topologically non-trivial\ sector is isomorphic
to the moduli space for some {\em other} Lie group\ $\mbox{$G$}^\omega$, which
describes the topologically {\em trivial} sector:
\begin{equation} {\cal M}_{G}^\omega \cong {\cal M}^{{\bf1}}_{G^\omega} \, . \labl{claim}
The Lie group\ $\mbox{$G$}^\omega$ is again simple, finite-dimensional\ and compact.
This isomorphism allows us to trade
topological non-triviality for some other structure group and to reduce
calculations to calculations in the topologically trivial sector only.
The moduli spaces ${\cal M}_{G}$ can be obtained as a symplectic
quotient of the infinite-dimensional symplectic space of all gauge potentials;
as a consequence, a smooth dense open subset of them is a symplectic
manifold with a symplectic form $\Omega$. The symplectic structure $\Omega$
plays an important role, in particular for holomorphic quantization; we will
see that the isomorphism \erf{claim} respects $\Omega$.

Before we describe how the Lie group\ $\mbox{$G$}^\omega$ is obtained from $\omega$ and
$\mbox{$G$}$, it is helpful to discuss the implications of this result for the
quantized Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theory. To apply the method of holomorphic quantization to
the spaces ${\cal M}^{{\bf1}}_{G}$ of topologically trivial connections one picks
a complex structure on the torus $\Sigma$, parametrized by some complex number
$\tau$ in the complex upper half plane. This turns $\Sigma$ into a complex
surface $\Sigma_\tau$, and also induces a complex structure on ${\cal M}_{G}$.
Next, one chooses a holomorphic line bundle $\cal L$ over
${\cal M}^{{\bf1}}_{G}$ such that its curvature is given by $2\pi{\rm i} \Omega$,
where $\Omega$ is the symplectic form on  ${\cal M}^{{\bf1}}_{G}$.
After fixing a positive integral value, the level $k$ (which plays the role
of a coupling constant for the field theory),
the quantization ${\cal B}^{\bfe,k}_G$ of ${\cal M}^{{\bf1}}_{G}$ is obtained as the
finite-dimensional\ vector
space of holomorphic sections of the $k$-th tensor power of $\cal L$:
\begin{equation} {\cal B}^{\bfe,k}_G := H^0({\cal M}^{{\bf1}}_{G}, {\cal L}^{\otimes k}) \, . \labl{qspace}

If the underlying Riemann surface is a torus $\Sigma_\tau$, there is a
distinguished basis for ${\cal B}^{\bfe,k}_G$:
denote the Lie algebra\ of $\mbox{$G$}$ by $\bar {\sf g}$, and consider the
untwisted affine Lie algebra\ ${\sf g}=\bar {\sf g}^{(1)}$ based on $\bar {\sf g}$. For fixed level $k$,
there are finitely many unitarizable irreducible highest weight representation s
$\mbox{$\calH$}_\Lambda$ of {\sf g}. The character
\begin{equation} \chi_\Lambda(\tau, h) := {\rm Tr}_{{\cal H}_\Lambda} {\rm e}^{2\pi{\rm i}\tau(L_0
- c/24)} {\rm e}^{{\rm i} h} \, , \labl{15}
of any irreducible representation\ is a function depending on $\tau$ and a variable $h$ that takes
values in the Cartan subalgebra\ $\bar {\sf g}_0$ of $\bar {\sf g}$. (After a choice of basis
in $\bar {\sf g}_0$, $h$ can be described by its components, the so-called Cartan
angles.)
The characters of all irreducible representation s at level $k$ form a basis of the space ${\cal B}^{\bfe,k}_G$
\cite{axdw,emss}.

Let us now turn to the case of our interest, when $\mbox{$G$}$ is
not simply connected. The center of the universal covering group
$\mbox{$\tilde G$}$ can be identified with a subgroup of the symmetries of the Dynkin diagram\ of the
affine Lie algebra\ {\sf g}. Any symmetry $\dot\omega$ of order $N$ of the Dynkin diagram of ${\sf g}$
induces an automorphism $\omega$ of the affine Lie algebra\
which acts like $\omega(E^i_\pm) :=  E^{\omd i}_\pm$ on the step operators
corresponding to the simple roots and $\omega(H^i) := H^{\omd i}$ for the
generators of the Cartan subalgebra. (The action on a full basis of the centrally extended
loop algebra can be found in section 6 of \cite{fusS3}.) The automorphism
$\omega$ preserves the
triangular decomposition of ${\sf g}$ and in particular the Cartan subalgebra; hence the dual map
$\omega^\star$ restricts to an isomorphism of the weight space ${\sf g}_0^\star$
of ${\sf g}$.

The automorphism $\omega$ gives rise \cite{fusS3} to `twisted intertwiner maps',
i.e.\ linear maps between irreducible highest weight representation s of {\sf g}\
\begin{equation} {\tau_\omega} \, :\quad \mbox{$\calH$}_\Lambda \to \mbox{$\calH$}_{\omega^\star(\Lambda)} \end{equation}
which obey
\begin{equation} {\tau_\omega} \, x = \omega(x) \, {\tau_\omega} \quad\mbox{for all} \, \, x\in {\sf g}\,   \end{equation}
and which map the highest weight vector of $\mbox{$\calH$}_\Lambda$ to the highest weight
vector of $\mbox{$\calH$}_{\omega^\star(\Lambda)}$. Those weights for which
$\omega^\star(\Lambda)=\Lambda$, so-called {\sl fixed points} of
$\omega^\star$, are of particular interest: in this case ${\tau_\omega}$ is an
endomorphism, and one can
insert ${\tau_\omega}$ in the trace \erf{15} to obtain a new set of functions on the
Cartan subalgebra, the so-called {\sl twining characters}
\begin{equation} \chi^\omega_\Lambda(\tau,h)
:= {\rm Tr}_{{\cal H}_\Lambda} {\tau_\omega} {\rm e}^{2\pi{\rm i}\tau(L_0 - c/24)} {\rm e}^{{\rm i} h}
\, . \end{equation}
The twining characters are dominated by the ordinary characters,
and hence they converge wherever the ordinary characters converge.
We will be interested in those symmetries of {\sf g}\ which are associated to
elements of the center of $\mbox{$\tilde G$}$; these describe \cite{scya6}
the action of a so-called simple current in the corresponding WZW theory.
The twining characters are close relatives of the index in a supersymmetric
theory, where $(-1)^F$, $F$ the fermion number, plays the role of ${\tau_\omega}$.
Indeed, in any rational superconformal field theory the supercurrent is a simple
current.

It was shown in \cite{fusS3} (see also \cite{furs}) that the twining character
is identical, in a sense to be made precise later, to the character of some
other Lie algebra, the {\sl orbit Lie algebra} $\mbox{$\brev{\g}$}$. (The orbit Lie algebra\ depends on both {\sf g}\ and
$\omega$; for the ease of notation, we do not mark the dependence on
$\omega$ explicitly.)
The Dynkin diagram\ of this Lie algebra\ is obtained by
folding the Dynkin diagram\ of {\sf g}\ according to the symmetry $\dot\omega$. More explicitly,
the Cartan matrix of $\mbox{$\brev{\g}$}$ is defined as follows:
Denote the Cartan matrix of ${\sf g}$ by $A=(a_{ij})_{i,j\in I}$,
where the index set is $I=\{0,1,\ldots, {\rm rank} \, \bar {\sf g}\}$. The symmetry
$\dot\omega$ of the Dynkin diagram\ organizes $I$ into orbits of different length $N_i$; we
choose a set $\hat I$ of representatives in $I$ from each $\dot\omega$-orbit.
The Cartan matrix of the orbit Lie algebra\
is then labelled by the subset $\breve{I}$ of the set of orbits $\hat I$:
  \begin{equation} \breve{I}:=\{i\!\in\!{\hat I}\mid\sum_{l=0}^{N_i-1}\!a_{i,\dot\omega^li} > 0 \} \,. \end{equation}
For any orbit we denote by $s_i$ the number
 \begin{equation} s_i:=\cases{{a_{ii}/\sum_{l=0}^{N_i-1}\!a_{i,\dot\omega^li}}\,,&if $i\!\in\!\breve{I}$ and $a_{ii}\not=0\,,$\cr
  \noalign{\vskip 2pt} \ 1\,,&otherwise$\,$,\cr}  \end{equation}
which is either $1$ or $2$. The elements of the Cartan matrix
${\breve A}=({\brev{a}_{ij}})_{{i,j}\in {\breve I}}$ of the orbit Lie algebra\ $\mbox{$\brev{\g}$}$ are then
given by
  \begin{equation} \brev{a}_{ij}:=s_j\sum_{l=0}^{N_j-1}\!a_{i,\dot\omega^lj}. \end{equation}
Note that $\breve{I}$ can be the empty set, in which case the orbit Lie algebra\ is the trivial
Lie algebra.  One can show that the orbit Lie algebra\ of an affine Lie algebra\ is again an
affine Lie algebra, unless it is trivial.
\futnote{ All results on orbit Lie algebra s and twining characters are valid in a much
more general context: they hold for arbitrary generalized Kac\hy Moo\-dy algebra s \cite{furs}.}

We emphasize that the orbit Lie algebra\ $\mbox{$\brev{\g}$}$ is not constructed as a subalgebra of {\sf g};
in particular, the orbit Lie algebra\ is in general not isomorphic to the subalgebra of
{\sf g}\ that is fixed under $\omega$. There is however a natural map $\mbox{\sc p}^{}_{\!\omega}$ from
the subspace ${\sf g}_0^{(0)}$ of the Cartan subalgebra\ that is fixed under $\omega$ to the
Cartan subalgebra\ $\mbox{$\brev{\g}$}_0$
of the orbit Lie algebra\ \cite{fusS3}. It is a bijection and the  invariant bilinear
forms on ${\sf g}_0^{(0)}$ and $\mbox{$\brev{\g}$}_0$ are related by
  \begin{equation} (h\mid h') = \frac 1N\, (\mbox{\sc p}^{}_{\!\omega}(h)\mid \mbox{\sc p}^{}_{\!\omega}(h')) \, , \labl{normrel}
for all $h,h'\in \hat\g_\circ^{(0)}$. (Recall that $N$ is the order of $\omega$.)
The dual relation for weights reads
  \begin{equation}  (\lambda\mid\mu) = N\cdot (\mbox{\sc p}^{\star-1}_{\!\omega}(\lambda)\mid \mbox{\sc p}^{\star-1}_{\!\omega}(\mu)) \, , \labl{nrw}
where we have assumed that the invariant bilinear form on weight space
is normalized such that the highest root of the horizontal subalgebra
has length squared 2. With this notation
the statement that the twining characters are given by the characters
of the orbit Lie algebra s can be made precise \cite{furs,fusS3}:
  \begin{equation} \Chi_\Lambda^{[\omega]}(\tau,h) = \brev\Chi_{\mbox{\sc p}^{\star-1}_{\!\omega}(\Lambda)}(\tau,\mbox{\sc p}^{}_{\!\omega}(h)) \, . \labl{thm}
Our results show that this theorem and the isomorphism \erf{claim} are
closely related by the procedure of holomorphic quantization.

The key property of the orbit Lie algebra\ $\mbox{$\brev{\g}$}$ is as follows: the dual map $\omega^\star$
acts as a linear map on the weight space of ${\sf g}$, on which also
the Weyl group $W$ of ${\sf g}$ acts. It can
be shown \cite{furs} that the subgroup $\hat W$ of $W$ that consists of all
elements of $W$ that commute with $\omega^\star$ is isomorphic to the
Weyl group
$\brev{W}$ of the orbit Lie algebra\ $\mbox{$\brev{\g}$}$. This fact enters crucially in the proof of
\erf{thm}; it will also be used in the present letter.
Indeed, the simply connected Lie group\ $\mbox{$G$}^\omega$ appearing in \erf{claim} is
just the simply connected compact Lie group\ whose Lie algebra\ is the horizontal
subalgebra of $\mbox{$\brev{\g}$}$.

The rest of this letter is organized as follows: in Section 2 we derive an
explicit
description of the moduli space ${\cal M}_{G}^\omega$, which is used to set up a map
realizing the isomorphism \erf{claim}. In Section 3 we check that this map
preserves the symplectic structure and derive a condition on the level which
is necessary for the existence of a quantization. In the last section we
comment on applications of our result and present the conclusions.

\sect{The isomorphism}

In this section we will derive an explicit description of ${\cal M}_{G}^\omega$. To
this end we use
the description of moduli spaces of flat connections
in terms of monodromies around non-trivial\ cycles: the group $\mbox{$G$}$ acts on
the space of all group homomorphisms from the fundamental
group $\pi_1(\Sigma)$ to $\mbox{$G$}$ by conjugation. This action is just the action
of gauge symmetries on the monodromies; the moduli space is then isomorphic
to the quotient ${\rm Hom} (\pi_1(\Sigma), \mbox{$G$}) / \mbox{$G$} $.

The fundamental group of the torus is ${\bf  Z}^2={\bf  Z}\times{\bf  Z}$, and for
simply connected $\mbox{$G$}$ we have to classify all solutions of the equation
\begin{equation} \mbox{$g_a$} \mbox{$g_b$} \mbox{$(g_a)^{-1}$} \mbox{${\bar\g}^{}_-$} = {\bf1} \, .  \labl{11}
for $\mbox{$g_a$},\mbox{$g_b$}\in\mbox{$G$}$, up to a simultaneous conjugation of $\mbox{$g_a$}$ and $\mbox{$g_b$}$.
In the case of non-simply connected\ $\mbox{$G$}$ we prefer to work with elements of the universal
covering group $\mbox{$\tilde G$}$ rather than with elements of $\mbox{$G$} \cong \mbox{$\tilde G$}/Z$. Thus we
have to find all solutions $(\mbox{$g_a$},\mbox{$g_b$})$, with $\mbox{$g_a$},\mbox{$g_b$}\in \mbox{$\tilde G$}$, of the equation
\begin{equation} \mbox{$g_a$} \mbox{$g_b$} \mbox{$(g_a)^{-1}$} \mbox{${\bar\g}^{}_-$} = \omega  \, , \labl1
where $\omega\in Z$ labels the topological sector. Again we have to
identify solutions that are related by a simultaneous conjugation
with some element of $\mbox{$\tilde G$}$.

In the topologically trivial sector equation \erf1 tells us that $\mbox{$g_a$}$ and
$\mbox{$g_b$}$ commute. For any two commuting elements of the real compact Lie group\ $\mbox{$\tilde G$}$
there is a maximal torus
containing both elements. The maximal torus is isomorphic to the Cartan subalgebra\
divided by the coroot lattice $\bar L^{\scriptscriptstyle\vee}$; the intersection of the orbits
of conjugation with a maximal torus $T$ are just the orbits of the
Weyl group $\bar W_T$.
As a consequence, the moduli space in the topologically trivial sector is
\begin{equation} {\cal M}_{G}^{{\bf1}} \cong  (\bar {\sf g}_0/\bar L^{\scriptscriptstyle\vee} \times \bar {\sf g}_0/\bar L^{\scriptscriptstyle\vee} ) / \bar W_T \, ,
\labl{standard}
where $\bar W_T$ acts diagonally.

The analogous analysis in the case of topologically non-trivial\ sectors is
more involved; for the ease of the reader we present the result immediately.
Consider the untwisted affine Lie algebra\ ${\sf g}=\bar {\sf g}^{(1)}$ with horizontal
subalgebra $\bar{\sf g}$. To any element $\omega$ of the center of $\mbox{$\tilde G$}$ is
associated a diagram automorphism of
${\sf g}$ corresponding to a simple current. (These correspond to the
symmetries of the
Dynkin diagram\ of {\sf g}\ which are not already symmetries of the Dynkin diagram\ of $\bar {\sf g}$.)
Upon identifying the Lie algebra\ ${\sf h}_T$ of $T$ (which is a Cartan subalgebra\ for the
Lie algebra\ $\bar {\sf g}$ of $\mbox{$\tilde G$}$) and
the horizontal projection of the Cartan subalgebra\ of ${\sf g}^{(1)}$, $\omega$ gives rise to
an affine map on ${\sf h}_T$. This map can be expressed in terms of an
element $w_0$ of the Weyl group $\bar W_T$ of $\bar {\sf g}$ and a shift by an
element $p^{\scriptscriptstyle\vee}$ of the coweight lattice
\begin{equation} \omb(h) = p^{\scriptscriptstyle\vee} + w_0(h) = p^{\scriptscriptstyle\vee}+  a_{w_0} h a_{w_0}^{-1} \quad . \labl2
Here we have chosen a group element $a_{w_0}$ in $\mbox{$\tilde G$}$ to implement the action
of the Weyl group element $w_0$; the element $a_{w_0}$ is only determined up to
an element of the maximal torus $T$, and we will have to fix some convenient
choice for $a_{w_0}$.  The
map $\omb$ leaves the (horizontal projection of the) fundamental affine Weyl
chamber invariant. We will show that
any solution of equation \erf1 is conjugate to a solution of the form
\begin{equation} \mbox{$g_a$} = \exp({\rm i} h) a_{w_0}  \qquad\mbox{and}\qquad \mbox{$g_b$} = \exp({\rm i} (h_0+ h') )
\, \labl{res}
where $h_0,h$ and $h'$ are elements of the Cartan subalgebra\ ${\sf h}_T$ that obey
$w_0(h) = h$, $w_0(h')=h'$ and $\omb(h_0)= h_0$, respectively.
Using the map $\mbox{\sc p}^{}_{\!\omega}$ the elements $h$ and $h'$ can be identified
with elements of the Cartan subalgebra\ of the orbit Lie algebra\ \mbox{$\brev{\g}$}. We are interested in group
elements and therefore any two solutions for which $h$ and $h'$
differ by elements of the coroot lattice should be identified. We will see
that the elements $\beta^{\scriptscriptstyle\vee}$
of the coroot lattice of $\bar {\sf g}$ respecting the conditions
$w_0(\beta^{\scriptscriptstyle\vee})=\beta^{\scriptscriptstyle\vee}$ are
in one-to-one correspondence to elements of the coroot lattice of
$\overline{\mbox{$\brev{\g}$}}$. Next,
we also have to take into account the effect of simultaneous conjugation with
elements of $\mbox{$\tilde G$}$ that preserve the conditions on $h$ and $h'$. We will see
that this is described by the diagonal action of the subgroup $\hat{\bar W}$
of the Weyl group $\bar W$ of $\bar {\sf g}$ that commutes with $\omb$. This subgroup,
however, is isomorphic to the Weyl group of the horizontal subalgebra\ of the
orbit Lie algebra, and comparing with \erf{standard} we obtain the isomorphism \erf{claim}.

\bigskip
In order to prove that any solution of \erf1 is indeed conjugate to
\erf{res}, we fix a maximal torus $T$ of $\mbox{$\tilde G$}$
that contains $\mbox{$g_b$}$ (for any element of a compact real Lie group\ such a torus
exists); we can then write
\begin{equation} \mbox{$g_b$} = \exp({\rm i} h'') \,  \end{equation}
with $h''\in {\sf h}_T$. The element $\omega$ of the center can be written as
$\omega=\exp(-{\rm i}p^{\scriptscriptstyle\vee})$, where $p^{\scriptscriptstyle\vee}\in {\sf h}_T$ is an element of the co-weight
lattice $\mbox{$\overline L^{\rm w {\scriptscriptstyle\vee}}$}$ of \mbox{$\tilde G$}\ relative to $T$. Moreover, by adding elements of
the co-root lattice and after choosing the convention for dividing the roots
into positive and negative roots appropriately,
we can assume that $h''$ is an element of the (closure of the) fundamental
affine Weyl chamber. Without loss of generality
we can write $\mbox{$g_a$}$ as
\begin{equation}        \mbox{$g_a$}= r a_{w_0}     \, , \end{equation}
where $r$ is some element in \mbox{$\tilde G$}. At the present stage, we will fix some
arbitrary choice for $a_{w_0}$; later on, we will determine a canonical choice
for $a_{w_0}$.

Equation \erf1 then becomes
\begin{equation} \mbox{$g_b$} \exp(-{\rm i} p^{\scriptscriptstyle\vee}) = r a_{w_0} \exp({\rm i} h'') a_{w_0}^{-1} r^{-1}
= r \exp({\rm i} w_0(h'')) r^{-1}
= r \exp(\omb ({\rm i} h'')) \exp(-{\rm i} p^{\scriptscriptstyle\vee})  r^{-1} \, .  \labl{pp}
The element $\omega = \exp(-{\rm i} p^{\scriptscriptstyle\vee})$ is in the center of $\mbox{$\tilde G$}$,
equation \erf{pp} is therefore equivalent to
\begin{equation} \exp(\omb({\rm i} h'')) = r^{-1} \exp({\rm i} h'') r \, . \labl{commut}
We now observe that $\omb$ preserves the Cartan subalgebra\ of $\bar {\sf g}$, hence the left hand
side is in the maximal torus again. Since the orbits of conjugation on the
maximal torus are the orbits of the Weyl group $\bar W_T$, we find that the
right hand side is equal to $\exp({\rm i} w(h''))$ with $w$ a suitable element of
the Weyl group $\bar W_T$ of $\bar {\sf g}$. However, $\omb$ preserves the fundamental
affine Weyl chamber, and the
only Weyl group element that does the same is the identity.
{}From this we learn that $\omb$ leaves $h''$ fixed, $\omb(h'')=h''$, and as a
consequence, \erf{commut} shows that $r$ and $\mbox{$g_b$}=\exp({\rm i} h'')$ commute. Now
any two commuting elements of the real compact Lie group\ $\mbox{$\tilde G$}$ are contained in
some maximal torus $\tilde T$, and since all maximal tori of $\mbox{$\tilde G$}$ are
conjugate, we can find $g\in\mbox{$\tilde G$}$ such that
\begin{equation} \tilde T = g^{-1} T g \, . \end{equation}
Denoting by ${\sf h}_{\tilde T}$ the Cartan subalgebra\ of $\bar {\sf g}$ that is the Lie algebra\ of
$\tilde T$, we have
\begin{equation} \mbox{$g_b$} = g \exp({\rm i}\eta'') g^{-1} \quad\mbox{and}\quad r = g \exp({\rm i}\eta)
g^{-1} \end{equation}
with $\eta,\eta''\in {\sf h}_{\tilde T}$. This allows us to rewrite $\mbox{$g_a$}$ as
\begin{equation} \mbox{$g_a$} = r a_{w_0}  = g ({\rm e}^{{\rm i}\eta} g^{-1} a_{w_0} g ) g^{-1} \, . \end{equation}
Notice that $\tildea_{w_0} := g^{-1} a_{w_0} g$ represents the element of the Weyl
group $\bar W_{\tilde T}$ for the new maximal torus $\tilde T$ that corresponds
to the same abstract Weyl group element as the one in $\bar W_T$ described by
$a_{w_0}$.

Since we are interested in solutions of \erf1 only up to conjugation, we can
drop the tildes and find that any solution of \erf1 is conjugate to
\begin{equation} \mbox{$g_a$} = \exp({\rm i} h) a_{w_0} \quad\mbox{and}\quad \mbox{$g_b$} = \exp({\rm i} h'') ,\labl{110}
where $\exp({\rm i} h)$ and $\exp({\rm i} h'')$ are elements of the same maximal
torus $T$ of $\mbox{$\tilde G$}$, and $h'$ is in the fundamental Weyl chamber. By
conjugation with $g$, $\omb$ gives also rise to an analogous affine map on
the Lie algebra\ of the new maximal
torus; $h''$ is invariant under the analogue of $\omb$ on the new maximal torus.

The space ${\cal A}$ of $\omb$-invariant elements of the Cartan subalgebra\ is an affine space
relative to the vector space ${\cal F}:= \ker(1-w_0)$.
The map $\mbox{\sc p}^{}_{\!\omega}$ gives an isomorphism
between the affine space ${\cal A}$ and the horizontal projection of the
weight space of the orbit Lie algebra; moreover, it also provides us with a distinguished
base point in ${\cal A}$: $h_0 := \mbox{\sc p}^{-1}_{\!\omega}(0)$; we write
\begin{equation} \mbox{$g_b$} = \exp(h_0 + h')\, , \quad\mbox{where}
\quad h'\in{\cal F}=\ker(1-w_0) \, . \end{equation}

We now have now
to give a more detailed description of the element $h\in \bar {\sf g}_0$
in \erf{110}: we remark that simultaneous conjugation of $\mbox{$g_a$}$ and
$\mbox{$g_b$}$ with an element $\exp({\rm i}\tilde h)$ of the maximal torus does not change
$\mbox{$g_b$}$. The action on $\mbox{$g_a$}$ can be computed as follows:
\begin{equation} \begin{array}{lll}
\mbox{$g_a$}=\exp({\rm i} h) a_{w_0} & \mapsto & \exp({\rm i} \tilde h) \exp({\rm i} h) a_{w_0}
\exp(-{\rm i} \tilde h)     \\[1em]
              &=& \exp({\rm i} (h+\tilde h - w_0(\tilde h))) a_{w_0}  \, .
\end{array}\end{equation}

Hence we always change $h$ by a conjugation to
$h \mapsto h + \tilde h - w_0(\tilde h)$ and obtain an equivalent solution, i.e.\
we are free to add elements of the subspace
${\rm range}(1-w_0)$. Using the fact that $w_0$ is an orthogonal transformation,
this subspace can be expressed as
\begin{equation}
{\rm range}(1-w) = (\ker (1-w_0^t) )^\perp = (\ker (1-w_0^{-1}) )^\perp
            = (\ker (1-w_0))^\perp  \, . \end{equation}
Hence we can assume, after conjugating $\mbox{$g_a$}$ and $\mbox{$g_b$}$ simultaneously with a
suitable element of the form $\exp({\rm i}\tilde h)$, that both $h$ and $h'$ are
in the kernel ${\cal F}$.

We have shown that the solutions
of \erf2 are all conjugated to a solution of the form
\begin{equation} \mbox{$g_a$}= \exp({\rm i} h) a_{w_0} \quad\mbox{and}\quad \mbox{$g_b$} = \exp({\rm i}(h_0 + h'))
= \exp({\rm i} h_0) \exp({\rm i} h') \, \labl{cond}
where $h$ and $h'$ are in ${\cal F}$, and $h_0 + h'$ is in the (horizontal
projection of the) fundamental affine
Weyl chamber. Conversely, it is easy to check that any such pair of elements
gives indeed a solution of \erf1.

At this point it seems as if there were an asymmetry between $\mbox{$g_a$}$ and $\mbox{$g_b$}$.
However, the situation is indeed symmetric: both $h$ and $h'$ are in
${\cal F}$, and hence $a_{w_0}$ commutes with $\exp({\rm i} h)$ and $\exp({\rm i} h')$.
This shows that we
can find a second maximal torus ${\tilde T}$, which contains $a_{w_0}$, $\exp({\rm i} h)$
and $\exp({\rm i} h')$, but not $\exp({\rm i} h_0)$. (Note that $h$ and $h'$ are fixed
under $w_0$ and therefore so-called singular elements of the Cartan subalgebra;
hence their exponentials can be indeed contained in two different maximal
tori.)

The trivial rewriting of \erf1
\begin{equation} \mbox{$g_b$} \mbox{$g_a$} \mbox{${\bar\g}^{}_-$} \mbox{$(g_a)^{-1}$} = \omega^{-1}  \, , \end{equation}
allows us to change the roles of $\mbox{$g_a$}$ and $\mbox{$g_b$}$ in the above considerations,
provided we replace $w_0$ by $w_0^{-1}$ and $\omb$ by $\omb^{-1}$.
In particular, we can write $\mbox{$g_a$}= \exp({\rm i} h) a_{w_0}$ as the exponential of some
element $\eta$ of the Cartan subalgebra\ $\tilde{\bar\g}_0$ belonging to ${\tilde T}$, $\mbox{$g_a$}=\exp({\rm i} \eta)$,
where $\eta$ is invariant under the map $\ombsec$ acting on $\tilde{\bar\g}_0$.
Again, there is a natural base point for this affine space:
$\eta_0 := \tilde{\mbox{\sc p}}^{-1}_{\!\omega}(0)$, where $\tilde{\mbox{\sc p}}^{-1}_{\!\omega}$ is the analogue of $\mbox{\sc p}^{-1}_{\!\omega}$ for
$\tilde T$. Now recall that the element $a_{w_0}$ implementing the Weyl
group transformations is only specified up to an element of the maximal
torus $T$. The group element $\exp({\rm i} \eta_0)$ differs from $a_{w_0}$
only by an element of the form $\exp({\rm i} \tilde h)$ with $\tilde h\in{\cal F}$,
which is an element in the intersection of $T$ and ${\tilde T}$. Hence we are
free to replace $a_{w_0}$ by $\exp({\rm i} \eta_0)$ for our considerations.

Having found two distinguished base points, we can now describe any
solution of the form \erf{cond} in a natural way in terms of the orbit Lie algebra:
use the isomorphisms $\mbox{\sc p}^{}_{\!\omega}$ and $\tilde{\mbox{\sc p}}^{}_{\!\omega}$ to associate to it the pair
$(\mbox{\sc p}^{}_{\!\omega}(h_0+h), \tilde{\mbox{\sc p}}^{}_{\!\omega}(\eta_0+h')) $ in the weight space $\breve{\cal A}\times\breve{\cal A}$
of the orbit Lie algebra.

We are only interested in group elements rather than Lie algebra\ elements, and
we should identify solution of \erf{cond} for which $h$ and $h'$ differ
by elements of the coroot-lattice. However, we must preserve the condition
that $h$ and $h'$ are fixed under $w_0$; so we are only allowed to add
elements $\beta^{\scriptscriptstyle\vee}$ of the coroot lattice that are fixed under $w_0$
themselves. Then the translation by $\beta^{\scriptscriptstyle\vee}$ is an element of the affine
Weyl group of ${\sf g}$ that commutes with $\omb$; hence it is in the subgroup
$\hat W$
and corresponds to an element in the affine Weyl group of the orbit Lie algebra\ which is
a translation by a coroot of $\bar{\mbox{$\brev{\g}$}}$. This shows that, after applying
$\mbox{\sc p}^{}_{\!\omega}$ and $\tilde{\mbox{\sc p}}^{}_{\!\omega}$, we simply have to project modulo the coroot lattice of
the horizontal subalgebra\ of the orbit Lie algebra\ $\mbox{$\brev{\g}$}$:
\begin{equation} \breve{\cal A}\times \breve{\cal A} \to
\breve{\cal A}/ {\breve L}^{\scriptscriptstyle\vee}  \times\breve{\cal A}/ {\breve L}^{\scriptscriptstyle\vee} \, . \end{equation}

The only freedom we are left with now is simultaneous conjugation of $\mbox{$g_a$}$ and
$\mbox{$g_b$}$ with a group element $g\in\mbox{$\tilde G$}$, such that the relations in \erf{cond} are
preserved: then along with
$\mbox{$g_b$}=\exp({\rm i} (h_0+ h')$ we have $g \mbox{$g_b$} g^{-1} = \exp({\rm i} \tilde h') $.
The two elements $h_0+ h'$ and $\tilde h'$ are related by some element of the
Weyl group $\bar W$; moreover, both $\tilde h'$ and $h_0+h'$
are fixed under
$\omb$.  It was shown in the proof of proposition
3.3.\ in \cite{furs} that then the Weyl group element $\hat w$ relating $h_0+ h'$
and $\tilde h'$ can be chosen in $\hat{\bar W}$:
\begin{equation} g\exp({\rm i} (h_0+ h')) g^{-1} = \exp({\rm i} \hat w(h_0+ h')) \, . \labl{hh}
The isomorphism $\mbox{\sc p}^{}_{\!\omega}$ intertwines the action of $\hat{\bar W}$ and the one of the
affine Weyl group of the orbit Lie algebra. Hence $\hat{\bar W}$ leaves the base point fixed and
therefore
\begin{equation} g\exp({\rm i} (h_0+ h')) g^{-1} = \exp({\rm i} (h_0+ \hat w h') \, . \end{equation}
Analogous considerations can be applied to $\mbox{$g_a$}$, using this time the other
maximal torus ${\tilde T}$; we reach the analogous conclusions. Combining the
two results, we find that the remaining
redundancies are taken into account by the diagonal action of the Weyl group
$\hat{\bar W}$:
\begin{equation} {\cal M}_{G}^\omega \cong  ( \breve{\cal A}/ {\breve L}^{\scriptscriptstyle\vee}  \times\breve{\cal A}/ {\breve L}^{\scriptscriptstyle\vee} )
/ \hat{\bar W}  \, . \end{equation}

Comparing this explicit description of the moduli space ${\cal M}_{G}^\omega$
with the standard description \erf{standard} of the moduli space of the
orbit theory in the topologically trivial sector, we obtain the isomorphism
\erf{claim}.

\sect{The symplectic structure}

In this section we want to extend the isomorphism \erf{claim} to include
also the symplectic structure on a smooth open subset of ${\cal M}_{G}^\omega$.
To this end we construct explicitly a topologically non-trivial\ $\mbox{$G$}$-bundle
and a connection on it with the appropriate monodromies. We fix from now
on a complex structure on the torus, which is parametrized by a complex
number $\tau=\tau_1+{\rm i}\tau_2$ with positive imaginary part $\tau_2> 0$.

The corresponding torus $\Sigma_\tau$ can be obtained as the quotient of
the complex plane by the following action of ${\bf  Z}^2={\bf  Z}\times{\bf  Z}$:
\begin{equation} R(m,n)(z) :=  z+ m + n \tau\, , \, \mbox{where}\quad m,n\in{\bf  Z} \, . \end{equation}
The principal $\mbox{$G$}$-bundle is obtained by extending this action to an action
on the trivial principal $\mbox{$G$}$-bundle ${\bf  C}\times \mbox{$G$}$ over $\mbox{$G$}$: we fix an
element $h_0$ with $\omb(h_0) = h_0$ and set:
\begin{equation} R_{G}(m,n)(z,g)
:= (z+ m + n \tau, \, {\rm e}^{{\rm i} h_0 n} a_{w_0}^m g) \, . \labl{twist}
Here $g$ is an element of the non-simply connected\ group $\mbox{$G$}$, and we have written for
simplicity $a_{w_0}$ for the projection to $\mbox{$G$}$ of the element $a_{w_0}\in\mbox{$\tilde G$}$ of the
universal covering group we considered in the previous section.
This defines indeed an action of ${\bf  Z}^2$, as can be seen as follows:
the equation $w_0(h_0) = h_0 - p^{\scriptscriptstyle\vee}$ implies that one has in $\mbox{$\tilde G$}$
\begin{equation} {\rm e}^{{\rm i} h_0 n} a_{w_0}^m {\rm e}^{{\rm i} h_0 n'} a_{w_0}^{m'}
= {\rm e}^{-{\rm i} n' m  p^{\scriptscriptstyle\vee}} {\rm e}^{{\rm i} h_0 (n+n')} a_{w_0}^{m+m'} \, . \end{equation}
The projection of the first element to the non-simply connected\ group
$\mbox{$G$}$ is trivial, and hence we have indeed an action of ${\bf  Z}^2$.
$({\bf  C}\times \mbox{$G$})\, / \, {\bf  Z}^2$ is a topologically non-trivial
principal $\mbox{$G$}$-bundle over $\Sigma_\tau$.

Let us now fix $h,h'\in\bar {\sf g}_0$ such that $w_0(h)=h$ and $w_0(h')=h'$.
Introduce the element
\begin{equation} u := h' + \tau h \,  \end{equation}
of the complexification of $\bar {\sf g}_0$; the connection
\begin{equation} A(z) :=  \frac1{2\tau_2} ( -\bar u {\rm d} z + u {\rm d} \bar z)
\labl{pot}
on ${\bf  C}\times\mbox{$G$}$ is then invariant under the induced action of
$R_{G}(m,n)$: To see this, we remark that $a_{w_0}^{-1}\, u a_{w_0} = u$
and $\exp(-{\rm i} h_0) \, u \exp({\rm i} h_0) = u$.
Hence the induced action of $R(1,n)$ on the connection \erf{pot} is
\begin{equation} R(1,n) A(z)
= a_{w_0}^{-1} {\rm e}^{- {\rm i} h_0 n}  A(z-1-b\tau) {\rm e}^{{\rm i} h_0 n} a_{w_0}
= A(z) \, .  \end{equation}

The connection \erf{pot} therefore gives rise to a connection on the principal
$\mbox{$G$}$-bundle we constructed; this connection is flat.
Let us now verify that this connection reproduces the monodromies $\mbox{$g_a$}$ and
$\mbox{$g_b$}$. We parametrize the first homology cycle $C_a$
by $z(t) = t$ with $0\leq t \leq 1$ and see that
\begin{equation}
\int_{C_a} A = \int_0^1 {\rm d} t A(\frac\partial{\partial z} + \frac\partial{\partial \bar z})  = \frac1{2\tau_2} (u-\bar u)
= {\rm i} h \, . \end{equation}
Taking into account the additional twist by $a_{w_0}$ in \erf{twist}, the monodromy
around $C_a$ is indeed $g_a = \exp({\rm i} h) a_{w_0}$. The second generator $C_b$
of one-cycles can be parametrized as $z(t) = \tau t$; we find that
\begin{equation} \int_{C_b} A = \int_0^1 A(\tau\frac\partial{\partial z} +\bar\tau \frac\partial{\partial \bar z})  \
= \frac1{2\tau_2} (\bar\tau u-\tau \bar u) = {\rm i} h' \, . \end{equation}
Taking again into account the additional twist, we see that the monodromy
around $C_b$ has the correct value $\mbox{$g_b$} = \exp({\rm i} (h'+h_0))$ as well.

The symplectic structure is defined on the tangent space, which consists
of $\bar {\sf g}$-valued one-forms, which we can assume to be of the following form:
define $\delta u := \delta h' + \tau \delta h$ as a complex linear combination
of two elements $\delta h$ and $\delta h'$ of the Cartan subalgebra\ of the compact real
form. An arbitrary element of the tangent space is then of the form
\begin{equation} \delta A
= \frac1{2\tau_2} ( - \delta \bar u {\rm d} z + \delta u {\rm d} \bar z)\, , \end{equation}
and the symplectic form is given by
\begin{equation} \Omega_{\sf g}(\delta A_1, \delta A_2)
:= \frac12 \int {\rm d}^2z \, \kappa_{{\sf g}}(\delta A_1 \wedge \delta A_2) \, \end{equation}
where $\kappa_{\bar {\sf g}}(\cdot,\cdot)$ is the Killing form on the Lie algebra\ $\bar {\sf g}$.
Again, we adhere to the convention that Killing forms are normalized such that
the highest $\bar {\sf g}$-root has length squared two. A standard calculation gives
\begin{equation} \Omega_{\sf g}(\delta A_1, \delta A_2) = - \frac{{\rm i}}2
(\kappa_{\bar {\sf g}}(\delta h'_1, \delta h_2)
- \kappa_{\bar {\sf g}}(\delta h_1,\delta h'_2) \,)\, , \end{equation}
which shows that the symplectic form is real and
independent of the complex structure, which is parametrized by $\tau$.

The comparison of this symplectic form with the one on ${\cal M}_{G^\omega}$
therefore reduces to a comparison of the Killing forms on {\sf g}\ and $\mbox{$\brev{\g}$}$.
The relation \erf{normrel} shows that they just differ by a factor of $N$,
where $N$ is the order of $\omega$ in $Z$. Taking into account the level
$k\in\mbox{${\zet}_{>0}$}$, we see that
\begin{equation} k \Omega_{\sf g}(\delta A_1, \delta A_2) =
\frac k N \Omega_{\brev{\g}}(\delta \breve{A}_1, \delta \breve{A}_2) \,. \end{equation}
This shows that upon expressing the symplectic form in terms of quantities in
the orbit Lie algebra\ the level is divided by the order of the automorphism;
this is exactly the relation between the levels of the Lie algebra\ and its orbit Lie algebra\
that was derived in \cite{fusS3}. An important consequence is that
$k \Omega_G$
is an element of the {\em integral} cohomology only if the level is a multiple
of $N$. Only in this case the moduli space can be quantized: this is the
geometric counterpart of the fact that fixed points only occur at levels which
are multiples of the order of the automorphism.

\section{Applications and Conclusion}

In this letter we have proven an isomorphism which, in physical terms, allows
to trade topological non-triviality for a different gauge group. This result
has several applications: the moduli spaces ${\cal M}_{G}^\omega$ appear
naturally in the description of Chern$\mbox{-\hspace{-.66 mm}-}$ Simons theories\ or WZW-models on non-simply connected\ group
manifolds. For the latter theories (indeed, for any integer spin simple
current extension \cite{scya6} of
a conformal field theory) a formula for the modular matrix $S$ was derived in \cite{fusS6}.
The isomorphism \erf{claim} will be one ingredient to a rigorous proof of
this formula.
This formula in turn give a Verlinde formula for the dimension
of the space of conformal blocks with a non-simply connected\ structure group, a problem
that recently also has received attention in algebraic geometry \cite{beau2}.

Another application of the isomorphism \erf{claim} are coset
conformal field theories, in the description as gauged WZW theories: it has been argued \cite{hori}
that in these theories one actually has to gauge a non-simply connected\ group. This
observation has lead to the conjecture that the contributions
from the topologically non-trivial\ sectors account for the resolution of
field identification fixed points. This resolution can be written \cite{fusS4}
in terms of quantities of the orbit Lie algebra s. The isomorphism \erf{claim} therefore
lends evidence to the conjecture relating orbit theories and topologically
non-trivial\ sectors; it is also a first step towards a
better understanding of coset conformal field theories\ in the Lagrangean framework.

We finally mention that our results only concern principal bundles over a
two-di\-men\-si\-o\-nal\ torus; it would be interesting to unravel the implications of the
structures we found for Riemann surfaces of higher genus.

\vskip4em

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 \def{Cambridge University Press}    {{Cambridge University Press}}
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 \def{New York}     {{New York}}
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 \defQuantum\        {Quantum\ }
 \defquantum group      {quantum group}
 \defRepresentation     {Representation}
 \def{Sprin\-ger Verlag}     {{Sprin\-ger Verlag}}
 \defsym\-me\-tries    {sym\-me\-tries}
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\bigskip\bigskip
\small
\noindent{\bf Acknowledgement.} \ I am grateful to K.\ Gawedzki for helpful
discussions and to J.\ Fuchs for a critical reading of the manuscript. \\
This work was supported in part by the Director, Office of Energy Research,
Office of Basic Energy Sciences, of the U.S.\ Department of Energy under
Contract DE-AC03-76F00098 and in part by the National Science Foundation
under grant PHY95-14797.

\footnotesize


Deformation quantization of compact K\"ahler manifolds via
  Berezin-Toeplitz operators

Abstract:
  This talk reports on results on the deformation quantization (star products)
and on approximative operator representations for quantizable compact K"ahler
manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic
quantum line bundle the Berezin-Toeplitz operator associated to a
differentiable function on the manifold is the operator defined by multiplying
global holomorphic sections of the line bundle with this function and
projecting the differentiable section back to the subspace of holomorphic
sections. The results were obtained in (respectively based on) joint work with
M. Bordemann and E. Meinrenken. (Talk at the XXI Int. Coll. on Group
Theoretical Methods in Physics, 15-20 July, Goslar, Germany)

\section{The set-up}
Let $\ (M,\omega)\ $ be a compact (complex) K\"ahler manifold.
It should be considered as  phase space manifold $M$
with symplectic form given by the K\"ahler form $\omega$.
Note that by the assumed compactness  the ``free phase space''
is not included here. Nevertheless there exist important
examples of such compact situations. E.g.~they could
appear as
phase space manifolds for certain constrained systems and they could
appear after
dividing out certain symmetry group action in a
noncompact situation.
They also play a role in the quantization of
2-dimensional conformal field theory. Here the  space
which has to be quantized is the compactified moduli space of
stable holomorphic vector bundles (with additional structures)
on a Riemann surface.

Using the K\"ahler form
one can assign to every  differentiable function\footnote{
Differentiable always means arbitrary often
differentiable.} $f$ its Hamiltonian vector field $X_f$ and
to every pair of functions $f$ and $g$ its
Poisson bracket:
\begin{equation}\label{lPoi}
\omega(X_f,\cdot)=df(\cdot),\qquad
\{\,f,g\,\}:=\omega(X_f,X_g)\ .
\end{equation}
With the Poisson bracket the algebra of differentiable functions
$C^{\infty}(M)$
becomes a Poisson algebra ${\mathcal{P}}(M)$.
Assume $(M,\omega)$ to be quantizable. This says there exists an
associated quantum line bundle $(L,h,\nabla)$.
Here $L$ is a holomorphic line bundle, $h$ a Hermitian metric on $L$ and
$\nabla$ a connection compatible with the complex structure and
the Hermitian metric.
With respect to a
local complex coordinate patch and a
local holomorphic frame of the bundle the metric $h$  is given by
a function $\hat h$ and the connection as  ($d=\partial+\overline{\partial}$)
\begin{equation}\label {lcon}
\nabla_{|}=\partial+\partial\log\hat h+\overline{\partial}\  .
\end{equation}
The quantization condition  says that the
curvature $F$ of the line bundle and the K\"ahler form $\omega$ of the manifold
are related  as
\begin{equation}\label{lquant}
F(X,Y):=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}
=
-{\,\mathrm i\,}\omega(X,Y)\ .
\end{equation}
In terms of the metric $h$ this can be written as $\ {\,\mathrm i\,}\overline{\partial}\partial\log\hat h=\omega\ $.
The quantization condition  implies that $L$ is a positive (or ample)
line bundle. By the Kodaira embedding theorem  a certain
tensor power of $L$ is very ample, i.e. its  global
sections can be used to embed the phase space manifold $M$
into projective
space (with dimension given by the Riemann-Roch formula).
Hence, quantizable compact K\"ahler manifolds are projective
algebraic manifolds (and vice versa, see below).
\begin{example}
The Riemann sphere, the complex projective line,
 $\mathbb P (\mathbb C )=\mathbb C \cup \{\infty\}\cong S^2$. With respect to the quasi-global
coordinate $z$  the form $\omega$ can be given as
\begin{equation}
\omega=\frac {{\,\mathrm i\,}}{(1+z\overline{z})^2}dz\wedge d\overline{z}\ .
\end{equation}
The quantum line bundle  $L$ is the hyperplane  bundle.
For the Poisson bracket one obtains
\begin{equation}
\{f,g\}={\,\mathrm i\,}(1+z\overline{z})^2\left(\Pfzb f\cdot\Pfz g-\Pfz f\Pfzb g\right)\ .
\end{equation}
\end{example}
\begin{example}
The (complex-) one dimensional torus $M$.
Up to isomorphy it can be given as
$M\cong \mathbb C /\Gamma_\tau$, where $\ \Gamma_\tau:=\{n+m\tau\mid n,m\in\mathbb Z \}$
is a lattice with $\ \text{Im\kern1.0pt } \tau>0$.
As K\"ahler form we take
\begin{equation}
\omega=\frac {{\,\mathrm i\,}\pi}{\text{Im\kern1.0pt } \tau}dz\wedge d\overline{z}\ ,
\end{equation}
with respect to the coordinate $z$ on the
covering space $\mathbb C $.
The corresponding quantum line bundle is the theta line bundle
of degree 1, i.e. the bundle whose global sections are
the scalar multiples of the Riemann theta function.
\end{example}
\begin{example}
The complex projective space $\mathbb P ^n(\mathbb C )$ with
the Fubini-Study fundamental form as
K\"ahler form.
The associated quantum bundle is the hyperplane bundle.
By restricting the form and the bundle on
complex submanifolds (which are automatically algebraic)
all projective manifolds are quantizable.
As remarked  above every quantizable compact K\"ahler  manifold is
projective. Nevertheless the embedding into
projective space does not necessarily respect
the  K\"ahler form.
\end{example}
For the space of  (arbitrary) global sections of $L$ a scalar product
is defined by
\begin{equation}\label{lsca}
\langle s_1,s_2\rangle=\int_M h(s_1,s_2)(x)\Omega(x),\qquad
\Omega=\frac 1{n!}
\underbrace {\omega\wedge \omega\cdots\wedge \omega}_{\text{ $\dim_{\Bbb C} M$ times}}
\  .
\end{equation}
\section{Approximation results  for Toeplitz operators}
Let $\Gamma_{\infty}(M,L)$ be the  (infinite-dimensional) space of global
differentiable sections of the line bundle $L$,
${\mathrm{L}}^2(M,L)$ its L${}^2$-completion with respect to the scalar product
(\ref{lsca}), and
$\Gamma_{hol}(M,L)$ the finite-dimensional subspace
of global holomorphic sections.
Denote by $\Pi:{\mathrm{L}}^2(M,L)\to\Gamma_{hol}(M,L)$ the projection.
The Toeplitz operator associated to the function $f\in{\mathcal{P}}(M)$ is
defined as the composition
\begin{equation}\label{lToe}
T_f:\Gamma_{hol}(M,L)\xrightarrow{M_f}\Gamma_{\infty}(M,L)\xrightarrow{\Pi}\Gamma_{hol}(M,L),\quad
s\mapsto f\cdot s\mapsto \Pi(f\cdot s)=:T_f(s)\ .
\end{equation}
Here $M_f$ denotes  the multiplication of the sections by
the function $f$.

Due to the  finite-dimensionality of $\Gamma_{hol}(M,L)$
information gets lost.
To recover this information we have to do everything
for each tensor power
$\ L^m=L^{\otimes m}$ of $L$.
This gives us
$\ (L^m,h^{(m)},\nabla^{(m)})$ and the  Toeplitz operators
$\Tmp f$.
The assignment
\begin{equation}\label{lBT}
T^{(m)}:{\mathcal{P}}(M)\to\text{End}(\Gamma_{hol}(M,L^{m})),\qquad f\mapsto T_f^{(m)}
\end{equation}
is called {\em Berezin-Toeplitz\ quantization} (of level $m$).
The theorems below will justify the use of  the term ``quantization''.
\begin{remark}
If one switches to the notation $\hbar=1/m$ then
the limit $m\to\infty$ corresponds to $\hbar\to 0$ as
``classical limit''.
\end{remark}
\begin{remark}
By a result of Tuynman
\cite{rTuyQ}
suitable  reinterpreted (see \cite{rBHSS} for a coordinate independent proof)
one obtains the relation
$\  \Qm f={\,\mathrm i\,}\Tmp {f-\frac {1}{2m}\Delta f}\ $,
where ${Q}^{(m)}$ is the more well-known operator
of
geometric quantization with respect to the
prequantum operator
$\ \Pm f= -\nabla_{X_f^{(m)}}^{(m)}+{\,\mathrm i\,} f\cdot id\ $ and
with K\"ahler polarization.
\end{remark}
\begin{remark}
In the example of 2-dimensional conformal field theory
mentioned at the beginning, there exists a natural quantum
line bundle $L$.
The spaces $\Gamma_{hol}(M,L^{m})$ are the {\em Verlinde spaces}.
\end{remark}

Assume $L$ to be already very ample. By a
rescaling of the K\"ahler form this always can be achieved.
The following two theorems are proved in
a joint paper with M. Bordemann and E. Meinrenken
\cite{rBMS}.
\begin{theorem}
For every  $\ f\in C^{\infty}(M)\ $ we have $\ ||T_f^{(m)}||\le||f||_\infty $
and
\begin{equation}\label{lThma}
\lim_{m\to\infty}||T_f^{(m)}||=||f||_{\infty}\ .
\end{equation}
Here $||f||_\infty$ is the sup-norm of $\ f\ $ on $M$ and
$||T_f^{(m)}||$ is the operator norm with respect to the
scalar product (\ref{lsca}) on $\Gamma_{hol}(M,L^{m})$.
\end{theorem}
\begin{theorem}
For every  $\ f,g\in C^{\infty}(M)\ $ we have
\begin{equation}\label{lThmb}
||m{\,\mathrm i\,}[T_f^{(m)},T_g^{(m)}]-T_{\{f,g\}}^{(m)}||\quad=\quad O(\frac 1m)\quad
\text{as}\quad m\to\infty
\ .
\end{equation}
\end{theorem}
By interpreting $1/m$ as $\hbar$ we see that up to order $\hbar^2$ the
quantization prescription
{\it ``Replace  the classical Poisson bracket
by the (rescaled) commutator of  the operators''}
is fulfilled. We obtain an approximative
operator representation.
\section{Deformation quantization (star products)}
By a refinement of the methods
for  the proofs of the above theorems it is possible to
construct a star product, resp. a  deformation
quantization for ${\mathcal{P}}(M)$.
Let me recall the definition of a star product.
Let $\mathcal{A}=C^{\infty}(M)[[\hbar]]$ be the algebra of formal power
 series in the
variable $\hbar$ over the algebra $C^{\infty}(M)$. A product $\star$
 on $\mathcal{A}$ is
called a (formal) star product if it is an
associative $\mathbb C [[\hbar]]$-linear product such that
\begin{enumerate}
\item
$\mathcal{A}/\hbar\mathcal{A}\congC^{\infty}(M)$, i.e.\quad $f\star g
 \bmod \hbar=f\cdot g$,
\item
$\dfrac 1\hbar(f\star g-g\star f)\bmod \hbar = -{\,\mathrm i\,} \{f,g\}$.
\end{enumerate}

We can always write
\begin{equation}\label{lcif}
 f\star g=\sum\limits_{j=0}^\infty C_j(f,g)\hbar^j\quad\text{with}
\quad C_j(f,g)\inC^{\infty}(M)\ .
\end{equation}
The conditions 1. and 2.  can
be reformulated as
\begin{equation}\label{lcifa}
C_0(f,g)=f\cdot g,\qquad
C_1(f,g)-C_1(g,f)=-{\,\mathrm i\,} \{f,g\}\ .
\end{equation}
\begin{theorem}
There exists a unique (formal) star product on $C^{\infty}(M)$
\begin{equation} \label{stareq}
f \star g:=\sum_{j=0}^\infty \hbar^j C_j(f,g),\quad C_j(f,g)\in
C^\infty(M),
\end{equation}
in such a way that for  $f,g\inC^{\infty}(M)$ and for every $N\in\Bbb N$  we have
\begin{equation}
||T_{f}^{(m)}T_{g}^{(m)}-\sum_{0\le j<N}\left(\frac 1m\right)^j
T_{C_j(f,g)}^{(m)}||=K_N(f,g) \left(\frac 1m\right)^N
\end{equation}
for  $m\to\infty$, with suitable constants $K_N(f,g)$.
\end{theorem}
Again this is joint work with M. Bordemann and E. Meinrenken.
The proof can be found in \cite{rSchlH}.
Also, it will appear  in a forthcoming paper.
\section{Some additional remarks}
Time  and space does not permit me to give even  a rough sketch of the
proof.
Let me only point out the following steps. We
use the embedding of $M$
via the  sections of $L$ (which is now assumed to be very ample)
into projective space.
For the embedded situation we have an associated  Toeplitz structure,
the generalized Toeplitz operators
(of arbitrary orders) and their symbol calculus as
introduced by Boutet de  Monvel and Guillemin \cite{rBoGu}.
The above defined $T_f^{(m)}$ can be considered as the ``modes''
(with respect to an $S^1$-action) of a global Toeplitz operator
 $\tilde T_f$.
The symbol calculus gives bounds for their norms.

With the help of the above  theorems it is possible to show
\cite{rBMS}
the
conjecture of Bordemann, Hoppe, Schaller  and Schlichenmaier
\cite{rBHSS} that for quantizable K\"ahler manifolds the Poisson
 algebra
is an $\mathfrak u(d(N))$, $N\to\infty$ quasi-limit with  a
strictly increasing integer valued function
$d(N)$  given by the Riemann-Roch formula.
For the  definition of quasi-limits
I refer to the above mentioned article.

References to related works
can be found in \cite{rSchlB}. Here I only want to quote some names:
F.A.~Berezin, B.V.~Fedosov, J.H.~Rawnsley, M.~Cahen, S.~Gutt,
S.~Klimek, A.~Lesniewski, L.A.~Coburn,...
Note also that the Berezin-Toeplitz quantization
can be considered in the frame-work of ``prime quantization''
as introduced  by S.T.~Ali and H.D.~Doebner.

The relation of Berezin-Toeplitz\
quantization  to Berezin's coherent states \cite{rBer} and symbols
(in the coordinate independent formulation due to Rawnsley)
is studied in \cite{rSchlH}. In particular, it is
shown that for compact K\"ahler manifolds the Toeplitz map
$\ f\mapsto T_f\ $ and Berezin's covariant symbol map
$\ B\mapsto \sigma(B)\ $ are adjoint maps with respect to
the Hilbert-Schmidt scalar product for the operators  and
the Liouville  measure on $M$ which is modified by the
 $\epsilon$-function of Rawnsley for the functions.