% Berezin Toeplitz quantization of compact Kaehler manifolds
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% Date: 11.1.96
% Rev. 28.3.96
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\define\Pmm{{\Cal P}(M)}
\define\Pmw{{(\Cal P}(M),\w)}
\define\Cim{C^{\infty}(M)}
\define\Po{{\Cal P}(\P^1)}
\define\ghm{\Gamma_{hol}(M,L^{m})}
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\redefine\L{\frak L}
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\define\Tgm{T_g^{(m)}}
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\topmatter
\title
Berezin-Toeplitz Quantization of compact
K\"ahler manifolds
\endtitle
\rightheadtext{Berezin-Toeplitz Quantization}
\leftheadtext{M. Schlichenmaier}
\author Martin Schlichenmaier
\endauthor
\address
Department of Mathematics and Computer Science,
University of Mannheim
D-68131 Mannheim, Germany
\endaddress
\email
schlichenmaier\@math.uni-mannheim.de
\endemail
\date January 96
\enddate
\keywords
geometric quantization, Berezin-Toeplitz quantization, K\"ahler
manifolds, star product deformation, Toeplitz operator
\endkeywords
\subjclass
58F06, 81S10, 32J81, 47B35, 17B66
\endsubjclass
\abstract
In this lecture results are reviewed obtained by the author
together with Martin Bordemann and Eckhard Meinrenken on the
Berezin-Toeplitz
quantization of compact K\"ahler manifolds. Using global
Toeplitz operators, approximation results for the
quantum operators are shown.
{}From them it follows that the
quantum operators have the correct classical limit.
A star product deformation of the Poisson algebra is constructed.
\endabstract
\endtopmatter
%
%
{\sl Invited lecture at the XIV${}^{th}$ workshop
on geometric methods in physics, Bia\l owie\D za, Poland,
July 9-15, 1995
}
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%
\newref rAAGM % Ali, Antoine coherent states
\newref rADo % Ali Doebner Prime quantization
\newref rBFFLS % Bayen, Flato etc. Ann.
\newref rBeC % Berceanu
\newref rBeTQ % Berezin alle
\newref rBeCo % Berger Coburn
\newref rBHSS % Bordemann, Hoppe, Schaller, Schlichenmaier: gl(infty)
\newref rBMS % Bordemann, Meinrenken, Schlichenmaier: Toeplitz
\newref rBLU % Borthwick, Lesniewski, Upmeier
\newref rBGTo
\newref rCGR
\newref rCaIe % Calabi Isometric embeddings
\newref rCoDe % Coburn Deformation estimates
\newref rCoXi % Toeplitz und Rieffel
\newref rDeLe %Deligne letter
\newref rDeWL
\newref rFedTQ % Fedosov
\newref rFlSt
\newref rGHPA
\newref rGuCT
\newref rHoePD
\newref rKaMa % Karasev, Maslov ASymptotics
\newref rKaMaP % Karasev, Maslov Poissonbracket
\newref rKlLeQr
\newref rMor % Moreno Starproducts on kaehler
\newref rMorOr % Starproducts on S2, etc.
\newref rOdCs
\newref rOMY % Omori, Maeda Y., Yoshioka, A.
\newref rRiDQ % MArc Rieffel, Deformation quantization
\newref rSCHLRS
\newref rSnGQ %Sniaticky Geometric Quantization
\newref rTuQM
\newref rUnUp % Unterberger, Upmeier
\newref rWeFe %Weinstein Bourbaki Exp. Fedosov
\newref rWoGQ
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\document
%\input bia1.tex % 1. Introduction
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%%% Start of Section 1.
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%% Date: 21.11.95
%%%%%
\head
1. Introduction\hfill\hbox{ }
\endhead
\def\kn{1}
Let me start with some mathematical aspects of quantization.
As a mathematician,
especially as an algebraic geometer, I find the following concepts
very fascinating.
Dear reader if you are a physicist or a fellow mathematician working
in a different field (e.g\. in measure theory) you will probably
prefer other aspects of the quantization. So please excuse if these
other important concepts are not covered here.
The arena of classical mechanics is as follows. One starts with
a phase space $M$, which locally should represent position and momentum.
We assume $M$ to be a differentiable manifold.
The physical observables are functions on $M$. One needs a
symplectic form $\w$, a non-degenerate antisymmetric closed
2-form, which roughly speaking opens the possibility to introduce
dynamics.
This form defines a Poisson structure on $M$ in the following
way. One assign to every function $f$ its Hamiltonian vector field
$X_f$ via
$$f\in\Cim \ \mapsto\ X_f,\quad\text{with }\ X_f\ \text{\ defined by}\quad
\w(X_f,.)=df(.)\ .$$
A Lie algebra structure on $\ \Cim\ $ is now defined by the product
$$\{f,g\}:=\w(X_f,X_g)\ .$$
The Lie product
fulfils the compatibility
$$
\text{for all}\quad f,g,h\in \Cim:\quad
\{f\cdot g,h\}=f\cdot\{g,h\}+\{f,h\}\cdot g\ .
$$
This says that $\ (\Cim,\,\cdot\,,\{..\,,..\})\ $ is a Poisson algebra.
The pair $\ (M,\w)\ $ is called a symplectic manifold.
A Hamiltonian system $\ (M,\w,H)\ $ is given by fixing
a function $H\in\Cim$, the
so called Hamiltonian function.
The first part of quantization
(and only this step will be discussed here)
consists in replacing the
commutative algebra of functions
by something noncommutative.
But there is the fundamental requirement, that the classical situation
(including the Poisson structure) should be recovered again as
``limit''.
There are some methods to achieve at least partially this goal.
I do not want to give a review of these methods. Let me just
mention a few.
There is the ``canonical quantization'', the
deformation quantization using star product,
geometric quantization,
Berezin quantization using coherent states
and Berezin symbols, \BT\ quantization,
and so on. I am heading here for \BT\ quantization which has
relations to the more known geometric quantization
as introduced by Kostant and Souriau.
In the following section I will recall some necessary definitions
for the case I will consider later on.
For a systematic treatment see
\cite\rSnGQ, \cite\rWoGQ.
%%%
% End of Section 1.
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%\input bia2.tex % 2. Geometric quantization
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%%% Start of Section 2.
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%% Date: 11.1.96
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\head
2. Geometric Quantization\hfill\hbox{ }
\endhead
\def\kn{2}
Here I will assume $\ (M,\w)\ $ to be a K\"ahler manifold, i.e\.
$M$ is a complex manifold and $\w$ a K\"ahler form.
This says that $\w$ is a positive, non-degenerate closed 2-form of
type $(1,1)$.
If $\dim_\C M=n$ and $z_1,z_2,\ldots,z_n$ are local holomorphic
coordinates then it can be written as
$$\w=\i\sum_{i,j=1}^ng_{ij}(z)dz_i\wedge d\zb_j\ ,\qquad
g_{ij}\in C^\infty(M,\C)\ ,$$
where the matrix $\ (g_{ij}(z))$ is
for every $z$ a positive definite hermitian matrix.
Obviously $(M,\w)$ is a symplectic manifold.
A further data is $\ (L,h,\nabla)\ $, with $L$ a holomorphic line bundle,
$h$ a hermitian metric on $L$
(conjugate-linear in the first argument), and $\nabla$ a connection which is
compatible with
the metric and the complex structure.
With respect to local holomorphic coordinates
and with respect to a local holomorphic frame of the bundle
it can be given as
$\ \nabla=\d +\d\log h+\db$.
The curvature of $L$ is defined as
$$F(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}\ .$$
\definition{Definition}
The K\"ahler manifold $(M,\w)$ is called quantizable, if there
is such a triple $(L,h,\nabla)$ with
$$F(X,Y)=-\i\w(X,Y)\ .\tag 1$$
\enddefinition
The condition (1) is called the prequantum condition.
The bundle $\ (L,h,\nabla)\ $ is called a (pre)quantum line bundle.
Usually we will drop $h$ and $\nabla$ in the notation.
\example{Example 1}
The flat complex space $\C^n$ with
$$
\w=\i\sum_{j=1}^ndz_j\wedge d\zb_j\ .
$$
\endexample
\example{Example 2}
The Riemann sphere, the complex projective line,
$\P(\C)=\C\cup \{\infty\}\cong S^2$. With respect to the quasi-global
coordinate $z$ the form can be given as
$$
\w=\frac {\i}{(1+z\zb)^2}dz\wedge d\zb\ .$$
The quantum line bundle $L$ is the hyperplane bundle.
For the Poisson bracket one obtains
$$\{f,g\}=\i(1+z\zb)^2\left(\Pfzb f\cdot\Pfz g-\Pfz f\Pfzb g\right)\ .
$$
\endexample
\example{Example 3}
The (complex-) one dimensional torus $M$.
Up to isomorphy it can be given as
$M = \C/\Gamma_\tau$ where $\ \Gamma_\tau:=\{n+m\tau\mid n,m\in\Z\}$
is a lattice with $\ \im \tau>0$.
As K\"ahler form we take
$$
\w=\frac {\i\pi}{\im \tau}dz\wedge d\zb\ ,$$
with respect to the coordinate $z$ on the
covering space $\C$.
The corresponding quantum line bundle is the theta line bundle
of degree 1, i.e\. the bundle whose global sections are
multiples of the Riemann theta function.
\endexample
\example{Example 4}
A compact Riemann surface $M$ of genus $g\ge 2$.
Such an $M$ is the quotient of the open unit disc $\Cal E$ in
$\C$ under the fractional linear transformations of
a Fuchsian subgroup of $SU(1,1)$.
If $\ R=\pmatrix a&b\\ \overline{b} &\overline{a}\endpmatrix
\ $ with $\ |a|^2-|b|^2=1\ $ (as an element of $SU(1,1)$) then
the action is
$$z\mapsto R(z):=\frac {a z + b} {\overline{b} z + \overline{a}}\ .$$
The K\"ahler form
$$\w=\frac {2\i}{(1-z\zb)^2}dz\wedge d\zb\ $$
of $\Cal E$ is invariant under the fractional linear
transformations. Hence it defines a K\"ahler form on $M$.
The quantum bundle is the canonical bundle, i.e\. the bundle
whose local sections are the holomorphic differentials.
Its global sections can be identified with the automorphic forms
of weight $2$ with respect to the Fuchsian group.
\endexample
\example{Example 5}
The complex projective space $\P^n(\C)$. This generalizes Example 2.
The points in $\P^n(\C)$ are given by their homogeneous coordinates
$\ (z_0:z_1:\ldots:z_n)\ $. In the affine chart with $z_0\ne 0$ we
take $\ w_j=z_j/z_0\ $ with $j=1,\ldots, n$ as holomorphic
coordinates. The K\"ahler form is the Fubini-Study fundamental
form
$$
\w_{FS}:=
\i\frac
{(1+|w|^2)\sum_{i=1}^ndw_i\wedge
d\wb_i-\sum_{i,j=1}^n\wb_iw_jdw_i\wedge d\wb_j} {{(1+|w|^2)}^2}
\ .$$
The quantum line bundle is the hyperplane bundle $H$, i.e\. the
line bundle whose global holomorphic sections can
be identified with the linear forms in the $n+1$ variables $z_i$.
\endexample
\example{Example 6}
Projective K\"ahler submanifolds.
Let $M$ be a complex submanifold of $\P^N(\C)$ and denote
by
$\ i:M\hookrightarrow \P^N(\C)$ the embedding, then
the pull-back of the Fubini-Study form
$i^*(\w_{FS})=\w_M$ is a K\"ahler form on $M$ and
the pull-back of the hyperplane bundle $i^*(H)=L$ is a quantum line
bundle for the K\"ahler manifold $(M,\w_M)$.
Note that by general results $i(M)$ is an algebraic
manifold.
\endexample
There is an important observation.
If $M$ is a compact K\"ahler manifold which is quantizable then from
the prequantum condition (1)
we get for the Chern form of the line bundle
the relation
$$c(L)=\frac {\i}{2\pi}F=\frac {\w}{2\pi}\ .$$
This implies that $L$ is a positive line bundle. In the terminology
of algebraic geometry it is an ample line bundle.
By the Kodaira embedding theorem $M$ can be embedded (as algebraic
submanifold) into projective space $\P^N(\C)$ using
a basis of the global holomorphic
sections $s_i$ of a suitable tensor power
$L^{m_0}$ of the bundle $L$
$$z\ \mapsto\ (s_0(z):s_1(z):\ldots:s_N(z))\in \P^N(\C)\ .$$
These algebraic manifolds can be described
as zero sets of homogeneous polynomials.
Note that the dimension of the space $\ghmo$
consisting of the global holomorphic sections of $L^{m_0}$, can be determined
by the Theorem of Grothendieck-Hirzebruch-Riemann-Roch, see
\cite\rGHPA, \cite\rSCHLRS.
So even if we start with an arbitrary K\"ahler manifold the
quantization condition will force the manifold to be an algebraic manifold
and we are in the realm of algebraic geometry.
This should be compared with the fact that there are ``considerable more''
K\"ahler manifolds than algebraic manifolds.
This tight relation between quantization and algebraic geometry can
also be found in the theory of coherent states as explained
by A. Odijewicz \cite\rOdCs\ and S. Berceanu \cite\rBeC.
Here a warning is in order. With the help of the embedding into
projective space we obtain by pull-back of the Fubini-Study form
another K\"ahler form on $M$ and
by pull-back of the hyperplane bundle
another quantum bundle on $M$.
As holomorphic bundles the two bundles are the same, but in general
the K\"ahler form and the metric of the bundle and hence the
connection will be different.
Essentially, these data will only coincide
if $M$ is a K\"ahler submanifold, or
in other words if the embedding is an isometric K\"ahler
embedding.
The situation is very much related to Calabi's diastatic function
\cite\rCaIe, \cite{\rCGR, 2nd ref.}, see also Section 4.
Now we have to deal with the functions and how to assign
operators to them.
In geometric quantization such an assignment
is given by
$$
P:(\Cim,\{.\,,.\})\to \End(\gul,[.\,,.]),\quad
f\mapsto P_f:=-\nabla_{X_f}+\i f\cdot id\ .
$$
Here $\gul$ is the space of differentiable global sections of the
bundle $L$.
Due to the prequantum condition this is a Lie homomorphism.
Unfortunately one has too many degrees of freedom. The fields depend
locally on position and momentum. Physical
reasons imply that they should depend
only on half of them.
Such a choice of ``half of the variables'' is called a polarization.
In general there is no unique choice of polarization.
However, for K\"ahler manifolds there is
a canonical choice of coordinates: the splitting into
holomorphic and anti-holomorphic coordinates.
To obtain a polarization we consider only sections which depend
holomorphically on the coordinates.
This is called the K\"ahler (or holomorphic) polarization.
If we denote by
$$\Pi:\gul\to\gh,$$
the projection operator from the space of differentiable
sections onto the subspace consisting of holomorphic sections then
the quantum operators are defined as
$$
Q:\Cim\to\End(\gh),\qquad f\mapsto
Q_f=\Pi\, P_f\,\Pi \ .
$$
This map is still a linear map. But it is not a Lie homomorphism anymore.
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%% Date: 18.1.96
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\head
3. Berezin-Toeplitz Quantization\hfill\hbox{ }
\endhead
\def\kn{3}
Let the situation be as in the last section. We assume
everywhere in the following
that $M$ is compact.
We take $\ \Omega=\frac 1{n!}\w^n\ $ as volume form on $M$.
On the space of section $\gul$ we have the scalar product
$$
\langle\varphi,\psi\rangle:=\int_M h (\varphi,\psi)\;\Omega\ ,
\qquad
||\varphi||:=\sqrt{\langle \varphi,\varphi\rangle}\ .
\tag 2
$$
Let $\Lp$ be the L${}^2$-completion of the space of $C^\infty$-sections
of the bundle $L$ and
$\gh$ be its finite-dimensional closed subspace of holomorphic
sections.
Again let $\ \Pi:\Lp\to\gh\ $ be the projection.
\definition{Definition}
For $f\in\Cim$ the Toeplitz operator $T_f$
is defined to be
$$ T_f:=\Pi\, (f\cdot):\quad\gh\to\gh\ .$$
\enddefinition
In words: One multiplies the holomorphic section with the
differentiable function $f$. This yields only a differentiable section.
To obtain a holomorphic section again, we have to project it back.
The linear map
$$T:\Cim\to \End\big(\gh\big),\qquad f\to T_f\ ,$$
will be our \BT\ quantization. It is
neither a Lie algebra homomorphism nor
an associative algebra homomorphism,
because in general
$$T_f\, T_g=\Pi\,(f\cdot)\,\Pi\,(g\cdot)\,\Pi\ne
\Pi\,(fg\cdot)\,\Pi\ .$$
{}From the point of view of Berezin's approach \cite\rBeTQ, $T_f$
is the operator with contravariant symbol $f$
(see also \cite\rUnUp).
At the end of this section I will give some more references.
Due to the compactness of $M$ this defines a map
from the commutative algebra of functions to a noncommutative
finite-dimensional (matrix) algebra.
A lot of information will get lost. To recover this
information one should consider not just the bundle $L$ alone but
all its tensor powers $L^m$
and apply all the above constructions for every $m$.
In this way one obtains a family of
matrix algebras and maps
$$
\Tma {}:\Cim\to \End\big(\ghm\big),\qquad f\to \Tma f\ .$$
This infinite family should in some sense ``approximate'' the
algebra $\Cim$.(See \cite\rBHSS\
for a definition of such an approximation.)
For the Riemann sphere $\P(\C)$ we
obtain with the help of an integral kernel
the following explicit expression for
the Toeplitz operator
$$(\Tfm s)(z)=\frac {m+1}{2\pi}\int_{\C}\frac {(1+z\zeb)^m f(\ze)s(\ze)}
{(1+\ze\zeb)^m}
\frac {\i d\ze\wedge d\zeb}{(1+\ze\zeb)^2}\ .
$$
Here the function $s$ is representing a holomorphic section
of $L^m$.
The Toeplitz operator in our situation has always an integral kernel.
Let $k(m):=\dim\ghm$ and take
an orthonormal basis $s_i,\ i=1,\ldots, k(m)$ of the space
$\ghm$ then
$$(\Tfm s)(z)=
\int_M\sum_{i=1}^{k(m)} h^{(m)}\big(s_i(w),f(w)s(w)\big)\cdot
s_i(z)\;\Omega(w)\ .
\tag 3
$$
These Toeplitz operators are still complicated but they are easier to handle
than the quantum operators. For compact $M$
we have the following relation
$$
Q_f^{(m)}=\i\cdot T_{f-\frac 1{2m}\Delta f}^{(m)}=
\i\left(\Tfm-\frac 1{2m}T_{\Delta f}^{(m)}\right)\ .
$$
This is a result of Tuynman \cite{\rTuQM, Thm.2.1} reinterpreted
in our context, see also
\cite\rBHSS.
Here the Laplacian $\Delta$ has to be calculated with respect to the
metric $\ g(X,Y)=\w(X,IY)\ $, where $I$ is the complex structure.
We see that
for $m\to\infty$ the quantum operator of geometric quantization
will asymptotically be equal to the quantum operator of
the \BT\ quantization.
For the following let us assume that $L$ is already very ample.
This says that its global sections will already do the
embedding.
If this is not the case we would have to start with a certain
$m_0$-tensor power of $L$ and the form $m_0\,\w$.
The following three theorems were obtained in
joint work with Martin Bordemann and Eckhard Meinrenken \cite\rBMS.
\proclaim{Theorem 1}
For every $\ f\in \Cim\ $ there is some $C>0$ such that
$$||f||_\infty-\frac Cm\ \le\ ||\Tfm||\ \le\
||f||_\infty\quad
\text{as}\quad m\to\infty\ .$$
Here $||f||_\infty$ is the sup-norm of $\ f\ $ on $M$ and
$||\Tfm||$ is the operator norm on $\ghm$.
In particular, we have
$ \ \lim_{m\to\infty}||\Tfm||=||f||_{\infty}$.
\endproclaim
\proclaim{Theorem 2}
For every $f,g\in \Cim\ $ we have
$$
||m\i[\Tfm,\Tgm]-\Tfgm||\quad=\quad O(\frac 1m)\quad
\text{as}\quad m\to\infty
\ .$$
\endproclaim
The proofs can be found in the above mentioned article \cite\rBMS.
I will give some ideas of them in the next section.
These theorems give two approximating sequences
of maps
$$(\Cim,||..||_\infty)\to (\frak g\frak l (n,\C), ||..||_m:=\frac 1m
||..||)\qquad
f\mapsto \i m \Tma f,\quad
f\mapsto m Q_f^{(m)}\ .$$
Restricted to real valued functions the maps take values in
$\frak u(k)$, for $k=\dim\ghm$. These families of maps are only linear
maps, not Lie homomorphism with respect to the
Poisson bracket. But by Theorem 1 they are
nontrivial and by Theorem 2 they are
approximatively Lie homomorphisms.
So every Poisson algebra of a K\"ahler manifold is a
$\frak u(k),\ k\to\infty$ limit.
This was a conjecture in \cite\rBHSS\ and our starting point
was the aim to prove
this conjecture.
In \cite\rBMS\ also a Egorov type theorem is presented.
If one puts $\hbar=\frac {1}m$ in Theorem 2 one can rewrite
it as
$$\lim_{\hbar\to 0}
||\frac{\i}{\hbar}[T^{(1/\hbar)}_f,T^{(1/\hbar)}_g]-
T^{(1/\hbar)}_{\{f,g\}}||
= 0\ .
$$
One should compare this with the definition of a
star product deformation of $\Cim$ (see \cite\rBFFLS, \cite\rWeFe)
based on the deformation theory of algebras as developed by Gerstenhaber.
Because there are different variants let me recall the
definition we are using.
Let $\Cal A=\Cim[[\hbar]]$ be the algebra of formal power series in the
variable $\hbar$ over the algebra $\Cim$. A product $*$ on $\Cal A$ is
called a (formal) star product if it is an
associative $\C[[\hbar]]$-linear product such that
\roster
\item
$\Cal A/\hbar\Cal A\cong\Cim$, i.e\.\quad $f*g \bmod \hbar=f\cdot g$,
\item
$\dfrac 1\hbar(f*g-g*f)\bmod \hbar = -\i\{f,g\}$.
\endroster
Note that
$\ f*g=\sum\limits_{i=0}^\infty C_i(f,g)\hbar^i\ $ with $\C$-bilinear maps
$C_i:\Cim\times\Cim\to\Cim$.
With this we calculate
$$C_0(f,g)=f\cdot g,\quad\text{and}\quad
C_1(f,g)-C_1(g,f)=-\i\{f,g\}\ .\tag 4$$
\proclaim{Theorem 3}
There exists a unique (formal) star product on $\Cim$
$$f * g:=\sum_{j=0}^\infty \hbar^j C_j(f,g),\quad C_j(f,g)\in
C^\infty(M),
\tag 5$$
in such a way that for $f,g\in\Cim$ and for every $N$ we have
$$||T_{f}^{(m)}T_{g}^{(m)}-\sum_{0\le j0\ \}\ \subset\ T^*Q
\setminus 0\ ,$$
and $\Pi $ is the above projection.
A (generalized) Toeplitz operator of order $k$ is an operator
$A:\Hc\to\Hc$ of the form
$\ A=\Pi\cdot R\cdot \Pi\ $ where $R$ is a
pseudodifferential operator
($\Psi$DO) of order $k$ on
$Q$.
The Toeplitz operators build a ring.
The (principal) symbol of $A$ is the restriction of the
principal symbol of $R$ (which lives on $T^*Q$) to $\Sigma$.
Note that $R$ is not fixed by $A$ but
Guillemin and Boutet de Monvel showed that the (principal) symbols
are well-defined and that they obey the same rules as the
symbols of $\Psi$DOs
$$
\sigma(A_1A_2)=\sigma(A_1)\sigma(A_2),\qquad
\sigma([A_1,A_2])=\i\{\sigma(A_1),\sigma(A_2)\}_\Sigma.
\tag 7
$$
Here we use the 2-form $\omega_0=\sum_i dq_i\wedge dp_i$ on
$T^*Q$ to define the Poisson bracket there.
We are only dealing with two Toeplitz operators:
\nl
(1) The generator of the circle action
gives the operator $D_\varphi=\dfrac 1{\i}\dfrac {\partial}
{\partial\varphi}$. It is an operator of order 1 with symbol $t$.
It operates on $\Hm$ as multiplication by $m$.
\nl
(2) For $f\in\Cim$ let $M_f$ be the multiplication operator on
$\Lqv$, i.e\. $M_f(g)(\la):=f(\tau(\la))g(\la)$.
We set $\ T_f=\Pi\cdot M_f\cdot\Pi:\Hc\to\Hc\ $.
Because $M_f$ is constant along the fibres, $T_f$
commutes with the circle action.
Hence
$\ T_f=\bigoplus\limits_{m=0}^\infty\Tfm\ $,
where $\Tfm$ is the restriction of $T_f$ to $\Hm$.
After the identification of $\Hm$ with $\ghm$ we see that these $\Tfm$
are exactly the Toeplitz operators $\Tfm$ introduced in Section 3.
In this sense we call $T_f$ also the global Toeplitz operator and
the $\Tfm$ the local Toeplitz operators.
$T_f$ is an operator of order $0$ and its symbol is just
$f$ pull-backed to $Q$ and further to $T^*Q$ (and restricted to $\Sigma$).
Let us denote by
$\ \tau^*_\Sigma:\Sigma\supseteq\tau^*Q\to Q\to M$ the composition
then we obtain for its symbol $\sigma(T_f)=\tau^*_\Sigma(f)$.
\nl
This is the set-up more details can be found in \cite\rBMS.
\demo{Proof of Theorem 2}
Now we are able to proof Theorem 2.
The commutator
$[T_f,T_g]$ is a Toeplitz operator of order $-1$.
Using $\ {\omega_0}_{|t\alpha(\lambda)}=-t\tau_\Sigma^*\omega\ $
for $t$ a fixed positive number, we obtain
\footnote{
Unfortunately, in \cite{\rBMS} the minus sign was missing.
This causes in Thm. 4.2 of that article also the wrong sign.}
with (7) that its
principal symbol is
$$\sigma([T_f,T_g])(t\alpha(\lambda))=\i\{\tau_\Sigma^* f,\tau_\Sigma^*g
\}_\Sigma(t\alpha(\lambda))=
-\i t^{-1}\{f,g\}_M(\tau(\lambda))\ .$$
Now consider
$$ A:=D_\varphi^2\,[T_f,T_g]+\i D_\varphi\, T_{\{f,g\}}\ .
$$
Formally this is an operator of order 1.
Using $\ \sigma(T_{\{f,g\}})=\tau^*_\Sigma \{f,g\}$
and $\sigma(D_\varphi)=t$ we see that its principal
symbol vanishes. Hence
it is an operator of order 0.
Now $M$ and hence $Q$ are compact manifolds.
This implies that $A$ is a bounded
operator ($\Psi$DOs of order 0 are bounded).
It is obviously $S^1$-invariant and we can write
$A=\sum_{m=0}^\infty A^{(m)}$
where $A^{(m)}$ is the restriction of $A$ on the space $\Hm$.
For the norms we get $\ ||A^{(m)}||\le ||A||$.
But
$$
A^{(m)}=A_{|\Hm}=m^2[\Tfm,\Tgm]+\i m\Tfgm.
$$
Taking the norm bound and dividing it by $m$ we get the claim of Theorem 2.
\qed
\enddemo
\demo{Proof of Theorem 3}
This proof is a modification of the above approach.
One constructs inductively
$C_j(f,g)\in\Cim$ such that
$$A_N=D_\varphi^N T_fT_g- \sum_{j=0}^{N-1}
D_\varphi^{N-j}T_{C_j(f,g)}$$
is a zero order Toeplitz operator. Because $A_N$ is $S^1$-invariant
and it is of zero order
its principal symbol descends to a function on $M$.
Take this function to be $C_N(f,g)$. Then
$A_N-T_{C_N(f,g)}$ is of order $-1$ and
$A_{N+1}=D_\varphi(A_N-T_{C_N(f,g)})$ is of order zero.
The induction starts with
$A_0=T_fT_g$ which implies $\sigma(A_0)=\sigma(T_f)\sigma(T_g)=f\cdot g=
C_0(f,g)$.
As a zero order operator $A_N$ is bounded, hence this is true
for the component operators
$A_N^{(m)}$. We obtain
$$
||m^N\Tfm\Tgm-\sum_{j=0}^{N-1}m^{N-j}T^{(m)}_{C_j(f,g)}||\le ||A_N||\ .$$
dividing this by $m^N$ we obtain the asymptotics (6) of the theorem.
Writing this explicitly for $N=2$ we obtain for the pair $(f,g)$
$$
||m^2 \Tfm\Tgm-m^2 T^{(m)}_{f\cdot g}-m T^{(m)}_{C_1(f,g)}||
\le K\ ,
$$
and a similar expression for the pair $(g,f)$.
By subtracting the corresponding operators,
using the triangle inequality,
dividing by $m$ and multiplying with $\i$ we obtain
$$||m\i (\Tfm\Tgm-\Tgm\Tfm)-T^{(m)}_{\i\big(C_1(f,g)-C_1(g,f)\big)}||
=O(\frac 1m)\ .$$
With Theorem 2 this yields
$\ || T^{(m)}_{\{f,g\}-\i\big(C_1(f,g)-C_1(g,f)\big)}||=O(\frac 1m)$.
But Theorem 1 says that the left hand side has as limit
$\ |\{f,g\}-\i(C_1\big(f,g)-C_1(g,f)\big)|_\infty\ $, hence
$\{f,g\}=\i(C_1(f,g)-C_1(g,f))$.
This shows equation (4). Uniqueness of the $C_N(f,g)$ follows
inductively in the same way from (6), again using Theorem 1.
The associativity follows from the definition by
operator products.
\qed
\enddemo
Unfortunately, Theorem 1 has a rather complicated proof using
Fourier integral operators, oscillatory integrals
and Berezin's coherent states.
(At least we have not been able to find a simpler one).
For the special situation
of projective K\"ahler submanifolds we have a much less involved proof,
using Calabi's diastatic function.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
Recall from Section 2 that a projective K\"ahler submanifold is
a K\"ahler manifold $M$ which can be embedded into projective space
$\P^N(\C)$ (with $N$ suitable chosen) such that the K\"ahler form of $M$
coincides with the pull-back of the Fubini-Study form.
The pull-back of the tautological bundle is the dual of
the quantum bundle. We denote this bundle by $U$. On the
tautological bundle we have the standard
hermitian metric $\ k(z,w):=\langle z,w\rangle
=\bar zw\ $ in $\C^{N+1}$. By pull-back
this defines a metric on $U$. Note that in this case
the pull-back is essentially just the restriction of all objects
to the submanifold.
The Calabi (diastatic) function
\cite{\rCaIe},\cite{\rCGR, 2nd ref.}
is defined as
$$D:M\times M\to
\R_{\ge 0}\cup\{\infty\},\qquad
D(\tau(\lambda),\tau(\mu))=-\log
|k(\lambda,\mu)|^2$$
(where we have to choose $\lambda$ and $\mu$ with
$k(\lambda,\lambda)=k(\mu,\mu)=1$
representing the points of $M$). It is well-defined,
vanishes only along the diagonal
and is strictly positive outside the diagonal.
\demo{Proof of Theorem 1 for this case}
First the easy part (which of course works in all cases).
Note that
$\ ||\Tfm||=||\rmm\,M_f^{(m)}\,\rmm||\le ||M_f^{(m)}||\ $ and
for $\varphi\ne 0$
$$\frac {{||M_f^{(m)}\varphi||}^2}{||\varphi||^2}=
\frac {\int_M h^{(m)}(f\varphi,f\varphi)\Omega}
{\int_M h^{(m)}(\varphi,\varphi)\Omega}
=
\frac {\int_M \overline{f(z)}f(z)h^{(m)}( \varphi,\varphi)\Omega}
{\int_M h^{(m)}(\varphi,\varphi)\Omega}
\le
||f||{}_\infty^2\ .$$
Hence,
$$
||\Tfm||\le ||M_f^{(m)}||=\sup_{\varphi\ne 0}
\frac {||M_f^{(m)}\varphi||}{||\varphi||}\le ||f||_\infty .$$
To proof the first inequality, let $x_0\in M$ be a point where $|f|$ assumes
its maximum, and fix a $\lambda_0\in \tau^{-1}(x_0)$ with
$k(\lambda_0,\lambda_0)=1$.
We define a sequence
of holomorphic functions
$\phtm$ by setting
$\ \phtm(\lambda):=k(\lambda_0,\lambda)^m
\ $.
Because
$ \phtm(c\la)=c^mk(\la_0,\la)^m=c^m\phtm(\la)\ $
this defines an element $\phm$ of $\ghm$.
Note that
$$h^m(\phm,\phm)(x)
=\overline{\phtm(\la)}\phtm(\la)=
\overline{k(\la_0,\la)}^m k(\la_0,\la)^m=
\exp(-mD(x_0,x))\ $$
With
Cauchy-Schwartz's inequality
we obtain
$$\gather
||\Tfm||\ge \frac {||\Tfm\phm||}{||\phm||}
\ge \frac {|<\phm,\Tfm\phm>|}{<\phm,\phm>}
\\=
\frac {|\int_Mf(x)h^m(\phm,\phm)(x)\Omega(x)|}
{\int_Mh^m(\phm, \phm)(x)\Omega(x)}=
\frac {|\int_Mf(x)e^{-mD(x_0,x)}\Omega(x)|}
{\int_Me^{-mD(x_0,x)}\Omega(x)}\ .
\endgather
$$
We want to consider the $\ m \to\infty\ $ limit.
The part of the integral outside a small neighbourhood
of $x_0$ will vanish exponentially. For the rest
the stationary
phase theorem \cite\rHoePD\ allows one to
compute the asymptotics. The point $x=x_0$ is a zero of $D$ and it
is a non-degenerate critical point. Hence we obtain for the right hand
side the asymptotic
$$
\frac {|f(x_0)|+O(m^{-1})}{1+O(m^{-1})}=
|f(x_0)|+O(m^{-1}), $$
and hence
$$||\Tfm||\ge |f(x_0)|+O(m^{-1})=
||f||_\infty+O(m^{-1}) \ .\qed$$
\enddemo
%%%%%%%%%%%%%%%%%%
%%%%% end of Section 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\input biaref.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% Date: 11.1.96
%
% References
%
%\vskip 1cm
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Acknowledment.}
I like to thank the organizers of the conference and the audience
for the very lively atmosphere, the stimulating discussions
and the warm hospitality experienced
at the conference. My very special thanks go to
Anatol Odzijewicz and Aleksander Strasburger.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% Journals
%%%%%%%%%%%%%%%%%%
\def\Invent{Invent.~Math.}
\def\Einseig{Enseign.~Math.}
\def\PL{Phys\. Lett\. B}
\def\NP{Nucl\. Phys\. B}
\def\LMP{Lett\. Math\. Phys\. }
\def\CMP{Commun\. Math\. Phys\. }
\def\JMP{Jour\. Math\. Phys\. }
\def\Izv{Math\. USSR Izv\. }
\def\FA{Funktional Anal\. i\. Prilozhen\.}
\def\Pnas{Proc\. Natl\. Acad\. Sci\. USA}
\def\PAMS{Proc\. Amer\. Math\. Soc\.}
\def\DG{J\. Diff\. Geo\.}
\def\PRA{Phys\. Rev\. A}
\def\TAMS{Trans\. Amer\. Math\. Soc\.}
\def\PSPM{Proc\. Symp\. Pure Math\.}
\def\JFA{J\. Funct\. Anal\.}
\def\Adva{Advances in Math\.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\parskip=4pt
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Coherent states and their generalizations: A mathematical
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UCL-IPT-94-22, December 94
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Quantization
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Quantization in complex symmetric spaces
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General concept of quantization
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Toeplitz operators on the Segal-Bargmann space
\jour\TAMS \vol 301 \yr 1987\pages 813-829
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ref\key \rBHSS\by Bordemann, M., Hoppe J., Schaller, P.,
Schlichenmaier, M.
\paper $gl(\infty)$ and geometric quantization
\jour \CMP\vol 138\pages 209--244\yr 1991
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\ref\key \rBMS\by
Bordemann, M., Meinrenken, E.,
Schlichenmaier, M.
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$gl(N), N\to\infty$ limit
\jour \CMP\vol 165\pages 281--296\yr 1994
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Non-pertubative deformation quantization of Cartan domains.
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Princeton
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\moreref
\paper Quantization of K\"ahler manifolds II
\jour \TAMS\vol 337\yr 1993\pages 73--98
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\paper Quantization of K\"ahler manifolds III
\jour \LMP\vol 30\yr 1994\pages 291--305
\moreref
\paper Quantization of K\"ahler manifolds IV
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\jour Ann. Math. \vol 58
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\jour \LMP\vol 7 \pages 487--496
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Some topics of modern mathematics and their applications to
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A simple geometric construction of
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\jour \DG\vol 40\yr 1994\pages 213--238
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Flato, M, Sternheimer, D.
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Closedness of star products and cohomologies
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Lie theory and geometry, in honor of B. Kostant
\eds
Brylinski, J-L., Brylinski, R., Guillemin, V., Kac, V.
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\publ Birkh\"auser \yr 1994
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Klimek,~S., Lesniewski,~A.\paper
Quantum Riemann surfaces: I. The unit disc
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Quantum Riemann surfaces: II. The discrete series
\jour \LMP\vol 24\yr 1992\pages 125--139
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$*$-products on some K\"ahler manifolds
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\endRefs}
%
\enddocument
\bye