Table Of ContentProgress in Mathematics
Volume 198
Series Editors
H. Bass
J. Oesterle
A. Weinstein
Quantization
of Singular Symplectic
Quotients
N. P. Landsman
M. Pt1aum
M. Schlichenmaier
Editors
Springer Basel AG
Editors:
N.P. Landsman
Universiteit van Amsterdam
Korteweg-de Vries Instituut voor Wiskunde M. Ptlaum
Plantage Muidergracht 24 Humboldt-Universităt zu Berlin
10 18 TV Amsterdam Mathematisch-Naturwissenschaftliche
The Netherlands Fakultăt II
Institut fUr Mathematik
e-mail: [email protected] Unter den Linden 6
10099 Berlin
M. Schlichenmaier Germany
Universităt Mannheim
Fakultăt fUr Mathematik und Informatik e-mail: [email protected]
D7,27
68131 Mannheim
Germany
e-mail: [email protected]
2000 Mathematics Subject Classification 53D20, 53D30, 81S1O, 35S35, 58A35
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C.,
USA
Deutsche Bibliothek Cataloging-in-Publication Data
Quantization of singular symplectic quotients / N. P. Landsman ... ed .. - Basel ; Boston; Berlin:
Birkhăuser,2001
(Progress in mathematics ; VoI. 198)
ISBN 978-3-0348-9535-4 ISBN 978-3-0348-8364-1 (eBook)
DOI 10.1007/978-3-0348-8364-1
This work is subject to copyright. AII rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, re-use of illustrations,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any
kind ofuse whatsoever, permission from the copyright owner must be obtained.
© 2001 Springer Basel AG
Originally published by Birkhăuser Verlag in 2001
Softcover reprint ofthe hardcover Ist edition 2001
Printed on acid-free paper produced of chlorine-free pulp. TCF
00
ISBN 978-3-0348-9535-4
987654321 www.birkhauser-science.ch
Contents
Preface ................................................................... vii
J. E. Marsden and A. Weinstein
Some comments on the history, theory, and applications
of symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
M.-T. Benameur and V. Nistor
Homology of complete symbols and non-commutative geometry 21
M. Braverman
Cohomology of the Mumford quotient................................ 47
A. Cattaneo and G. Felder
Poisson sigma models and symplectic groupoids ...................... 61
B. Fedosov
Pseudo-differential operators and deformation quantization. . . . ... . . . . 95
J. Huebschmann
Singularities and Poisson geometry of certain
representation spaces ................................................ 119
N.P. Landsman
Quantized reduction as a tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
R. Lauter and V. Nistor
Analysis of geometric operator on open manifolds:
a groupoid approach ................................................. 181
M. Pflaum
Smooth structures on stratified spaces.... ... . .. . . . ... . .. . ... . . .. .. . .. 231
M. Schlichenmaier
Singular projective varieties and quantization ........................ 259
V. Schomerus
Poisson structure and quantization of Chern-Simons theory.......... 283
B.J. Schroers
Combinatorial quantization of Euclidean gravity
in three dimensions .................................................. 307
A. Sengupta
The Yang-Mills measure and symplectic structure
over spaces of connections ........................................... 329
Preface
The purpose of this volume is to present new techniques and ideas that have a di
rect significance for the description of stratified (symplectic) spaces and their quan
tization. The book grew out of a Research-in-Pairs Workshop held at Oberwolfach
from 2-6 August, 1999, organized by the editors with Martin Bordemann. They
are grateful to the Volkswagen-Stiftung and to the Mathematisches Forschungszen
trum Oberwolfach, particularly to its director, Matthias Kreck, for financial and
other support.
The papers by Cattaneo and Felder, Huebschmann, Landsman, Pflaum, Schli
chenmaier, Schomerus, Schroers, and Sengupta are based on talks given at the
workshop. To obtain a more complete picture of the field, the editors invited a
number of outside contributions as well. Thus they are happy to include the papers
by Benameur and Nistor, Braverman, Fedosov, and Lauter and Nistor. All papers
were refereed. The opening article by Marsden and Weinstein provides a historical
and personal overview of the subject.
In the bulk of the book the reader may identify two fundamentally different
approaches. The first associates a commutative algebra of functions to a singular
space, preferably also equipped with a Poisson bracket, which one may subse
quently try to quantize. This generically involves techniques from algebraic and
differential geometry. Here the papers by Braverman, Cattaneo and Felder, Pflaum,
and Schlichenmaier are of a general nature, whereas Huebschmann, Schomerus,
Schroers, and Sengupta are specifically concerned with the moduli spaces M. (Cf.
the article by Marsden and Weinstein in this volume, §1, for a first description,
and [3, 6, 17] for details on these spaces. Apart from the method in [3], alternative
approaches to the symplectic structure of M are given in [9], [11, 14, 15), and
[1]. The structure of M as a stratified symplectic space was first described by
Huebschmann [13].)
The second approach follows the fundamental idea of Connes (5) that one
should describe singular spaces by noncommutative (operator) algebras from the
outset. Such algebras can be analysed with (co ) homological techniques, leading to
functors that tend to be invariant under Morita equivalence. In any case, the dis
tinction between a classical and a quantum description is now somewhat blurred.
The articles by Benameur and Nistor, Fedosov, Landsman, and Lauter and Nistor
follow this idea.
We now describe the contents of each paper in some detail, in alphabetical
order.
The paper by Benameur and Nistor deals with operator algebras on a man
ifold with corners. The authors introduce Hochschild (and hence cyclic/periodic
cyclic) homology of so-called "topologically filtered algebrafl". A salient feature is
viii Preface
that multiplication is no longer required to be continuous. Instead, one assumes
F::J.
separate continuity together with the existence of a "multi-filtration" that is
compatible with the multiplication and satisfies a number of technical conditions.
The Hochschild chain space is then defined as an inductive limit in terms of the
filtration. The authors compute the homologies for various algebras associated
to a a-compact manifold M with corners. Employing the theorem of Hochschild,
Kostant, and Rosenberg [12] as well as the method of Brylinski and Getzler [4],
the authors obtain descriptions of differential forms in terms of de Rham and
Poisson (co) homology. The main results concern (i) the periodic cyclic homology
of the algebras \[100(9)/\[1-00(9) and \[10(9)/\[1-00(9) (where \[100(9) means the
space of complete symbols on a differential groupoid Q with units M) and (ii)
the Hochschild and cyclic homology of an algebra of "Laurent complete symbols
vanishing to infinite order at infinity" .
The contribution of Braverman centers around the Guillemin-Sternberg con
jecture mentioned earlier. Let X be a smooth projective variety on which a com
plex reductive group G acts. Let L be a G-invariant positive line bundle L. As
explained in this volume by Schlichenmaier, using the global sections of a suitable
tensor power of the line bundle one obtains an embedding of X into projective
space. With respect to this embedding and the group action, the Zariski open
subset of semi-stable points XSs is defined. If the action of G on XSs is free, the
quotient space XSs /G carries a natural smooth variety structure, and LIXsS /G is a
holomorphic line bundle over XSs /G. The main result in the paper is the following
theorem on the dimensions of the cohomology:
where ( .. . )G refers to G-equivariants as appropriate. For j = 0 this is a classical
result of Guillemin and Sternberg [10]. Braverman's techniques have also been
extended to the case of singular symplectic reduction [21].
The article of Cattaneo and Felder relates to Kontsevich's remarkable preprint
on the formal deformation quantization of arbitrary Poisson manifolds [16]. In suc
cession to an earlier paper in which they explain Kontsevich's formula for the star
product in terms of a path-integral formulation of a certain topological ~-model,
they now provide a canonical Hamiltonian formulation of this model. This entails
an unexpected contribution to the "integration problem" of Poisson manifolds,
which asks for a symplectic groupoid whose base space is the given Poisson mani
fold. Namely, the reduced phase space of their model turns out to be a topological
groupoid, which in case that it is smooth is a symplectic groupoid integrating
the Poisson manifold defining the ~-model. In general, the groupoid fails to be a
manifold because of singularities, but it is smooth around its base space, so that
the authors provide a new proof that any Poisson manifold has a full symplectic
realization. Being the reduced phase space of a constrained Hamiltonian system,
the construction of the groupoid involves (singular) symplectic reduction in an
Preface ix
essential way. Its quantization would lead to a new understanding of Kontsevich's
work.
The main goal in Fedosov's contribution is to compare his method of defor
mation quantization [7], here applied to the special case of a cotangent bundle,
with the symbol calculus of pseudo differential operators. He establishes a useful
and compact version of the change-of-variables formula for the transition of sym
bols of pseudo differential operators under diffeomorphisms. Using the exponential
function of a given torsion-free connection, he thereby relates formal symbols to
flat sections of the so-called Leibniz bundle of noncommutative algebras. This bun
dle consists fiberwise of symbols on the tangent spaces and possesses a canonical
noncommutative product, given fiberwise by the Leibniz product of symbols on
Euclidean space. Identifying symbols on the manifold with flat sections of the Leib
niz bundle, one thus arrives at a deformation quantization of the symbol space.
The resulting product coincides with the one inherited by pseudodifferential op
erators. As an application, Fedosov gives a new proof of the Atiyah-Singer index
theorem.
Huebschmann describes the local structure of the moduli spaces M(E, G).
Seen as Hom(7fl (E), G)/G, such a space can be decomposed according to orbit
types. By general principles, the orbit type decomposition of such a space is a
stratification. A suitable choice of a Poisson algebra of continuous functions en
dows this space with the structure of a stratified symplectic space, in such a way
that the choice of a complex structure on E induces a Kahler structure on each
stratum. When E has genus two and G = SU(2), the algebraic quotient as defined
in geometric invariant theory is isomorphic to the complex projective space ]P>3(C).
In particular, with the standard structure it is an ordinary smooth manifold. But
the symplectic structure of the symplectic quotient is not defined along the Kum
mer variety (embedded in jp3(C)) associated to the Jacobian of E. This gives an
example where the complex structure does not see the singularities of the symplec
tic quotient. The structure of the representation space as a stratified symplectic
space and its singularities are studied in detail. Finally, the relation between the
real Zariski tangent space (with respect to the non-standard real structure) and
the complex Zariski tangent space (with respect to the ordinary complex-analytic
structure) is examined. These two structures differ at the singularities of the real
structure, and are not related by complexification.
The paper of Landsman asks which operation in noncommutative geometry
corresponds to reduction in symplectic geometry. Here it helps to see symplectic
reduction as some sort of tensor product between dual pairs of Poisson manifolds.
If one regards C* -algebras as the quantum counterparts of Poisson manifolds, the
answer is that classical reduction corresponds to Rieffel's interior tensor product
between Hilbert (C*) bimodules [18]. If one uses von Neumann algebras instead,
the relative tensor product of Connes's correspondences [5] provides the answer.
This insight motivates categorical descriptions of Poisson manifolds and operator
algebras that emphasize the similarities between these structures. For example, in
x Preface
all cases Morita equivalence (as defined for Poisson manifolds by Xu [20] and for
operator algebras by Rieffel [18]) comes down to isomorphism of objects in the per
tinent category. Groupoids turn out to admit an analogous categorical description;
perhaps this explains their ubiquitous role in Poisson as well as noncommutative
geometry.
In the original examples of noncommutative geometry, Connes [5] applied his
strategy mainly to foliations. Here the operator algebra associated to the singular
leaf space is the convolution algebra defined by the holonomy groupoid of the foli
ation. In the article by Lauter and Nistor, groupoid techniques playa central role
as well, now in a more general setting, where besides foliations also manifolds with
corners, Lie groups, and the fundamental groupoid are incorporated. Following a
review of the necessary background material on Lie groupoids and pseudodifferen
tial operators on groupoids, they study (pseudodifferential) operators on singular
spaces defined by certain groupoids. This enables them to describe the above ex
amples and others. Subsequently, they discuss geometric differential operators, in
particular Dirac operators. Later sections are devoted to spectral properties. A
major application is a proof of a conjecture of Melrose on the essential spectrum
of the b-Laplacian on a manifold with corners.
Unlike smooth manifolds, singular spaces do not, in general, have a canon
ical structure sheaf of "smooth functions". On the other hand, this is essential
for studying analytic and geometric aspects of the space under consideration. The
article by Pflaum elaborates on the question of defining a structure sheaf on strat
ified spaces that is appropriate for geometric-analytic purposes, and in particular
for quantization. After a review of the basics of stratification theory, the author
introduces the notion of "smooth structure" on a stratified space. With the help of
such a structure, geometric objects on a stratified space can be defined. To men
tion a few examples, one now has stratified tangent spaces, Riemannian metrics,
Poisson bivectors, and so on. Finally, using the language of smooth structures, a
concept of deformation quantization of a symplectic stratified space is proposed.
The first part of Schlichenmaier's paper deals with the Berezin-Toeplitz quan
tization and the associated star-product of quantizable compact Kahler manifolds.
Such manifolds are automatically smooth projective varieties (over q, and it turns
out that the quantum Hilbert space (Le., the space on which the quantum oper
ators act) is the projective coordinate ring corresponding to the embedding. The
set-up also makes sense for singular projective varieties. The second part of the
paper is purely tutorial, introducing some concepts used by other authors of this
volume (e.g., Braverman and Huebschmann). A short review of the notion of a
projective variety is given. Particular attention is paid to the various definitions of
singular points on a variety, and to the hierarchy of "badness" of singularities in
terms of the algebraic properties of the local ring of the point. Finally, quotients
constructed by geometric invariant theory are discussed. Such quotients typically
appear in the construction of moduli spaces.
Preface xi
Schomerus discusses the Poisson structure and quantization of the moduli
space M(E, G) for compact G, seen as the reduced phase space of Chern-Simons
theory. He adopts an approach in which the original infinite-dimensional space of
connections is replaced by an intermediate finite-dimensional space of holonomies,
and reduction now takes place with respect to G rather than the original gauge
group. This leads to a combinatorial description, originally due to Fock and Rosly
[8] of the moduli space and its Poisson structure in terms of classical r-matrices
and the classical Yang-Baxter equation. The r-matrix description turns out to be
a good starting point for the quantization of the moduli space, which naturally
involves R-matrices and the quantum Yang-Baxter equation. This is a convincing
way to understand the emergence of quantum groups in the quantum theory of
the moduli space; in many other approaches this remains a mystery. The final step
in the quantization program then consist of finding representations of the "moduli
algebra" of quantum observables.
The paper of Schroers studies the quantization of the moduli space M(E, G)
for non-compact G. This is motivated by the observation that for suitable G such
a moduli space is the physical phase space of three-dimensional general relativity
on a spacetime E x JR., which thereby emerges as a topological field theory theory
[2, 19]. Depending on the signature of the metric and on the value of the cosmolog
ical constant, each of the choices G = SE(3), ISO(2, 1), SO(3, 1), SO(2,2), and
finally the compact SO(4) plays a role. Following the same approach as Schome
rus, Schroers finds that quantum groups enter the picture as well. In particular, for
G = ISO(3) the symmetries of the quantum moduli space are governed by Drin
feld's quantum double V(SU(2)), which emerges as a deformation of the group
algebra of ISO(3). A number of mathematical issues remain unsolved.
Sengupta first provides a technical review of the symplectic structure on the
moduli space M(E, G), including his own argument for the closedness of this form.
The case where ~ has a boundary, and the boundary holonomies are required to
lie in fixed conjugation classes, is covered in detail. In the spirit of constructive
field theory, he then constructs the "Yang-Mills measure" on the space of all con
nections (on a principal G-bundle over E) modulo gauge transformations. This
is the measure that in heuristic terms defines a path integral with respect to the
usual Euclidean Yang-Mills action. This is done using stochastic techniques that
go back to Gross and Driver. Finally, the two threads come together in the study
of the classical limit of the Yang-Mills measure. Rescaling the Yang-Mills action
by lit, one obtains a family of measures J..Lt, whose limit for t ---+ 0 generically
turns out to be concentrated on the moduli space of flat connections, where it is
proportional to the measure defined by the symplectic volume. In case that the
top stratum of the moduli space is of lower dimension than expected (e.g., where
E is a torus and G = SU(2)), a more subtle situation arises.
