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Richard J. Szabo
Equivariant Cohomology
and Localization of Path Inte
4
-
-
Springer
ra s
Author Richard J. Szabo The Niels Bohr Institute
Blegdamsvej 17 2100 Copenhagen 0, Denmark and
Department of Physics University of Oxford i
-
Theoretical
Physics
Keble Road
Oxford OX1 3NP, United
Kingdom
Library of Congress Cataloging-in-Publication Data. Die Deutsche Bibliothek
-
CIP-Einheitsaufnahme
Szabo, Richard J.:
Equivariant cohomology and localization of path integrals / Richard J. Szabo. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan ; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in physics : N.s. M, Monographs ; 63)
-
ISBN 3-540-67126-9 ISSN 0940-7677 (Lecture Notes in Physics. Monographs) ISBN 3-540-67126-9 Springer-Verlag Berlin Heidelberg New York
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5 43
2 10
Preface
This book reviews equivariant localization techniques for the evaluation of Feynman path integrals. It develops systematic geometric methods for study-
ing the semi-classical properties
of
phase
space
path integrals for dynamical
systems, emphasizing the relations with integrable and topological quantum field theories.
Beginning with a detailed review of the relevant mathematbackground equivariant cohomology and the Duistermaat-Heckman theorem, it demonstrates how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. The book emphasizes the common symmetries that such localizable models always possess and uses these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemarm surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented and used to discuss the general relationship between topological field theories and more conventional physical models. ical
-
Copenhagen, September
Acknowledgements:
1999
Richard J. Szabo
I would like to thank I.
Semenoff for advice and encouragement book and for
helpful discussions.
I
during
Kogan,
G.
Landi,
F. Lizzi and G.
various stages of the
writing
the various calculational and
grateful conceptual aspects of Section 7,
portant historical remarks
the manuscript, and to 0. Tirkkonen for
on
am
to L. Paniak for his
of this
participation
in
to E. Gozzi for im-
illuminating
VIII
Preface
discussions. I would also like to thank D. Austin and R.
suggestions supported Canada.
on some
in part
of the
more
Douglas for
comments and
mathematical aspects of this book. This work
was
by the Natural Sciences and Engineering Research Council of
Contents
1.
Introduction
2.
..............................................
1.2 1.3
Outline
............
1
..........................
5
...............................................
Equivariant Cohomology Principle 2.1 Example: The Height Function of the Sphere 2.2 A Brief Review of DeRham Cohomology 2.3 The Cartan Model of Equivariant Cohomology and the Localization
2.4
Fiber Bundles
2.5
The
2.6
The
and
3.
7
............................
11
...............
11
..................
13
.............
19
....................
24
....................
33
.............................
38
Equivariant Characteristic Classes Equivariant Localization Principle Berline-Vergne Theorem
Fin ite-Dimensional Localization for 3.1 3.2 3.3
Theory Dynamical Systems Symplectic Geometry Equivariant Cohomology on Symplectic Manifolds Stationary-Phase Approximation
...................................
43
...................................
44
..........
47
and the Duistermaat-Heckman Theorem 3.4 3.5
3.7
...................
51
.....................
56
...................................
58
and Kirwan's Theorem
Theory Examples: The Height Function Morse
of 3.6
a
Riemann Surface
Equivariant Localization Degenerate Version
and Classical
of the Duistermaat-Heckman Theorem
4.
I
Path
Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory Equivariant Localization Theory
1.1
....................
67
70
............................
74
The Witten Localization Formula
3.9
The Wu Localization Formula
Quantum 4.1
62
.........................
3.8
Localization
Theory Space Path Integrals Phase Space Path Integrals
for Phase
Integrability
.........
...........................
77
..............................
78
X
Contents
4.2
4.3 4.4
Example: Path Integral Derivation of the Atiyah-Singer Index Theorem Loop Space Symplectic Geometry and Equivariant Cohomology Hidden Supersymmetry and the Loop Space Localization Principle
......................
84
............................
95
.................
99
..........................
105
4.5
The WKB Localization Formula
4.6
Degenerate Path Integrals
4.7
Connections with the Duistermaat-Heckman
and the Niemi-Tirkkonen Localization Formula
4.8 4.9 4.10 5.
.............
Integration Formula Equivariant Localization and Quantum Integrability Localization for Functionals of Isometry Generators Topological Quantum Field Theories
107
....................................
112
........
114
.........
117
......................
121
Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems 127 5.1 Coadjoint Orbit Quantization .........................................
and Character Formulas 5.2
5.3
5.4 5.5
5.6 5.7
.................................
Isometry Groups of Simply Connected Riemannian Spaces Euclidean Phase Spaces and Holomorphic Quantization Coherent States on Homogeneous Kiffiler Manifolds and Holomorphic Localization Formulas Spherical Phase Spaces and Quantization of Spin Systems Hyperbolic Phase Spaces Localization of Generalized Spin Models
..................
137
...........................
146
...................
153
........................................
158
................................
169
and Hamiltonian Reduction 5.8 5.9 6.
Quantization Quantization
of on
130
..............................
Isospin Systems Non-Homogeneous Phase Spaces
..........................
...........
Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations 6.1 Isometry Groups of Multiply Connected Spaces 6.2 Equivariant Hamiltonian Systems in Genus One 6.3 Homology Representations and Topological Quantum Field Theory 6.4 Integrability Properties and Localization Formulas 6.5 Holomorphic Quantization and Non-Symmetric Coadjoint Orbits 6.6 Generalization to Hyperbolic Riemann Surfaces
171 180 191
...............
203
............
205
............
207
...................
210
..........
213
.....................
217
............
226
Contents
7.
Beyond 7.1
the Semi-Classical
................
233
..................................
234
Geometrical Characterizations of the
Loop Expansion
7.2
Conformal and Geodetic Localization
7.3
Corrections to the Duistermaat-Heckman Formula:
Symmetries
..........
244
..................................
253
..............................................
259
A Geometric
Approach
7.4
Examples
7.5
Heuristic Generalizations to Path
Supersymmetry Breaking 8.
Approximation
Integrals:
................................
Equivariant Localization in Cohomological. Field Theory Two-Dimensional Yang-Mills Theory: Equivalences 8.1 Between Physical and Topological Gauge Theories 8.2 Symplectic Geometry of Poincar6 Supersymmetric Quantum Field Theories 8.3
8.4
269
..........
270
.........................................
276
Supergeometry and the Batalin-Radkin-Vilkovisky Formalism Equivariant Euler Numbers, Thom Classes and the Mathai-Quillen Formalism The Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles
A: BRST
.............
Appendix
10.
Appendix B: Other Models of Equivariant Cohomology 10.1 The Topological Definition 10.2 The Weil Model
.........................
295
...............................
299
...............................
299
......................................
300
304
..................................
306
....................................................
309
Loop Space
References
287 291
........................................
10.3 The BRST Model
281
...................
Quantization
9.
10.4
266
...........................
........................
8.5
XI
Extensions
1. Introduction
In this book
shall review the systematic situations where
approaches and applications of a Feynman path integrals of physical systems can be evaluated exactly leading to a complete understanding of the quantum physics. These mathematical formalisms are in large part motivated by the symmetries present in integrable systems and topological quantum field theories which make these latter examples exactly solvable problems. Besides providing conceptual understandings of the solvability features of these special classes of problems, this framework yields geometric approaches to evaluating the quantum spectrum of generic quantum mechanical and quantum field theoretical partition functions. The techniques that we shall present here in fact motivate an approach to studying generic physical problems by relating their properties to those of integrable and topological field theories. In doing so, we shall therefore also review some of the more we
theory which investigates
modern quantum field theoretical and mathematical ideas which have been at the forefront of theoretical physics over the past two decades.
1.1 Path
Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory The idea of as a
novel
integral
path integration was introduced by Feynman [46] in the 1940's approach to quantum theory. Symbolically, the fundamental path
formula is IC (q', q;
eiTL[Cqqll
T)
(1. 1)
Cqqt where the 'sum' is
over all paths Cqq, between the points q and q, on the configuration space of a physical system, and L[Cqqll is the length of the path. The quantity on the left-hand side of (1.1) represents the probability amplitude for the system to evolve from a state with configuration q to one with configuration q' in a time span T. One of the great advantages of the path integral formulation is that it gives a global (integral) solution of the quantum problem in question, in contrast to the standard approach to quantum mechanics based on the Schr8dinger equation which gives a local (differential) formulation of the problem. Of utmost significance at the time was Feynman's
R. J. Szabo: LNPm 63, pp. 1 - 9, 2000 © Springer-Verlag Berlin Heidelberg 2000
1. Introduction
generalization of the path integral
to quantum electrodynamics from which systematic derivation of the famous Feynman rules, and hence the basis of most perturbative calculations in quantum field theory, can be carried out a
[75]. The problem of quantum integrability, i.e. the possibility of solving analytically for the spectrum of a quantum Hamiltonian and the corresponding energy eigenfunctions, is a non-trivial problem. This is even apparent from the point of view of the path integral, which describes the time evolution of wavefunctions. Relatively few quantum systems have been solved exactly and even fewer have had an exactly solvable path integral. At the time that the functional integration (1.1) was introduced, the only known examples where it could be evaluated exactly were the harmonic oscillator and the free particle. The path integrals for these 2 examples can be evaluated using the formal functional analog of the classical Gaussian integration formula [176] n
fl dx -00
k
(2-7r e' .7r/2)
el1E.
i
e
I
Ei,jAi(M-YjAj
v1det M
k=1
(1.2) where M
=
[Mij]
is
a
non-singular symmetric
matrix. In this way, be evaluated formally for any field theory n x n
Feynman propagator (1.1) can quadratic in the field variables. If this is not the case, then one can expand the argument of the exponential in (1.1), approximate it by a quadratic form as in (1.2), and then ,take the formula (1.2) as an approximation for the integral. For a finite-dimensional integral this is the well-known stationary phase (otherwise known as the saddle-point or steepest-descent) approximation [64]. In the framework of path integration, it is usually referred to as the Wentzel-Kramers-Brillouin (or WKB for short) approximation [101, 147]. Since the result (1.2) is determined by substituting into the exponential integrand the global minimum (Le. classical value) of the quadratic form and multiplying it by a term involving the second variation of that form (i.e. the fluctuation determinant), this approach to functional integration is also called the semi-classical approximation. In this sense, (1.1) interprets quantum mechanics as a sum over paths fluctuating about the classical trajectories (those with minimal length L[Cqqll) of a dynamical system. When the semi-classical approximation is exact, one can think of the Gaussian integration formula (1.2) as a 'localization' of the complicated looking integral on the left-hand side of (1.2) onto the global minimum of the quadratic form the
which is at most
there. For
long time, these
the
only examples of exactly solvable path a path integral describing the [146] precession of a spin vector was given exactly by its WKB approximation. This was subsequently generalized by Dowker [38] who proved the exactness of the semi-classical approximation for the path integral describing free geodesic a
integrals.
motion
were
In 1968 Schulman
on
compact
found that
group manifolds. It
was
not until the late 1970's that
1.1 Path
Integrals
3
general methods, beyond the restrictive range of the standard WKB method, were developed. In these methods, the Feynman path integral is calculated rigorously in discretized form (i.e. over piecewise-linear paths) by a careful regularization prescription [93], and then exploiting information provided by functional analysis, the theory of special functions, and the theory of differential equations (see [31] and references therein). With these tricks the list of exactly solvable path integrals has significantly increased over the last 15 years, so that today one is able to essentially evaluate analytically the path integral for any quantum mechanical problem for which the Schr6dinger equation can be solved exactly. We refer to [91] and [62] for an overview of these methods and a complete classification of the known examples of exactly solved quantum mechanical path integrals to present date. The situation is somewhat better in quantum field theory, which represents the real functional integrals of interest from a physical standpoint. There are many non-trivial examples of classically integrable models (i.e. ones whose classical equations of motion are 'exactly solvable'), for example the sine-Gordon model, where the semi-classical approximation describes the exact spectrum of the quantum field theory [176]. Indeed, for any classically integrable dynamical system one can canonically transform the phase space variables so that, using Hamilton-Jacobi theory [55], the path integral can be formally manipulated to yield a result which if taken naively would imply the exactness of the WKB approximation for any classically integrable system [147]. This is not really the case, because the canonical transformations used in the phase space path integral do not respect the ordering prescription used for the properly discretized path integral and consequently the integration measure is not invariant under these transformations [30]. However, as these problems stem mainly from ordering ambiguities in the discretization of the path integral, in quantum field theory these ordering ambiguities could be removed by a suitable renormalization, for instance by an operator commutator ordering prescription. This has led to the conjecture that properly interpreted results of semi-classical approximations in integrable field theories reproduce features of the exact quantum spectrum [176]. One of the present motivations for us is to therefore develop a systematic way to implement more
realizations of this conjecture. Another class of field theories where the
path integral is exactly solvable in most cases is supersymmetric topological quantum field theories for concise a review). Topological field theories have lately been of (see [22] much interest in both the mathematics and physics literature. A field theory is topological if it has only global degrees of freedom. This means, for example, that its classical equations of motion eliminate all propagating degrees of freedom from the problem (so that the effective quantum action vanishes). In particular, the theory cannot depend on any metric of the space on which theories and
the fields
are
fore describe
defined. The observables of these quantum field theories theregeometrical and topological invariants of the spacetime which
1. Introduction
computable by the conventional techniques of quantum field theory and of prime interest in mathematics. Physically, topological quantum field theories bear resemblances to many systems of longstanding physical interest and it is hoped that this special class of field theories might serve to provide insight into the structure of more complicated physical systems and a testing ground for new approaches to quantum field theory. There is also a conjecture that topological quantum field theories represent different (topological) phases of their more conventional counterparts (e.g. 4-dimensional Yang-Mills theory). Furthermore, from a mathematical point of view, these field theories provide novel representations of some global invariants whose properties are frequently transparent in the path integral approach. Topological field theory essentially traces back to the work of Schwarz [148] in 1978 who showed that a particular topological invariant, the RaySinger analytic torsion, could be represented as the partition function of a certain quantum field theory. The most important historical work for us, however, is the observation made by Witten [166] in 1982 that the supersymmetry algebra of supersymmetric quantum mechanics describes exactly the DeRham complex of a manifold, where the supersymmetry charge is the exterior derivative. This gives a framework for understanding Morse theory in terms of supersymmetric quantum mechanics in which the quantum partition function computes exactly the Euler characteristic of the configuration manifold, i.e. the index of the DeRharn complex. Witten's partition function computed the so-called Witten index [167], are
are
the difference between the number of bosonic and fermionic states. In order for the
of
supersymmetry
to be broken in the
zero
energy
ground
state
supersymmetric model, the Witten index must vanish. As supersymmetry, i.e. a boson-fermion symmetry, is not observed in nature, it is necessary to have some criterion for dynamical supersymmetry breaking if supersymmetric theories are to have any physical meaning. Witten's construction was subsequently generalized by Alvarez-Gaum6 [5], and Friedan and Windey [48], to give supersymmetric field theory proofs of the Atiyah-Singer index theorem [41]. In this way, the partition function is reduced to an integral over the configuration manifold M. This occurs because the supersymmetry of the action causes only zero modes of the fields, i.e. points on M, to contribute to the path integral, and the integrals over the remaining fluctuation a
modes
are
Gaussian. The
resulting integral encodes topological information a huge reduction of the original infinite-
about the manifold M and represents dimensional path integration. This field
began to draw more attention around 1988 when Witten introtopological field theories in a more general setting [169] (see also [170]). A particular supersymmetric non-abelian gauge theory was shown by Witten to describe a theory with only global degrees of freedom whose observables duced
are
the
the Donaldson invariants, certain differential invariants which are used for study of differentiable structures on 4-manifolds. Subsequent work then
1.2
put these ideas into
a
Equivariant Localization Theory
general framework
oretic structures of Witten's actions
are
so
that
today
the formal field the-
well-understood
[22]. Furthermore,
because of their topological nature, these field theories have become the focal point for the description of topological effects in quantum systems using
quantum field theory, for instance for the description of holonomy effects in physical systems arising from the adiabatic transports of particles [152] and extended objects such as strings [181 (i.e. Aharonov-Bohm type effects). In this way the functional integral has in recent years become a very popular tool lying on the interface between string theory, conformal field theory and topological quantum field theory in physics, and between topology and algebraic geometry in mathematics. Because of the consistent reliability of results that path integrals of these theories can produce when handled with care, functional integration has even acquired a certain degree of respectability among mathematicians.
1.2 The
Equivariant Localization Theory feature of
topological field theories is that their path integrals exactly by the semi-classical approximation. It would be nice to put semi-classically exact features of functional integrals, as well as the features which reduce them to integrals over finite-dimensional manifolds as described above, into some sort of general framework. More generally, we would are
common
described
like to have certain criteria available for when
we
expect partition functions
of quantum theories to reduce to such
simple expressions, or 'localize'. This motivates an approach to quantum integrability in which one can systematically study the properties of integrable field theories and their conjectured semi-classical "exactness" that we mentioned before. In this approach we focus on the general features and properties that path integrals appearing in this context have in
large
number of
Foremost among these is the existence of a (super-) symmetries in the underlying dynamical theory, so
that these functional
common.
integrals reduce
sent finite-dimensional
integrals'.
to Gaussian ones and essentially repreThe transition between the functional and
finite-dimensional
integrals can then be regarded as a rather drastic localoriginal infinite-dimensional integral, thereby putting it into a form that is useable for extracting physical and mathematical information. The mathematical framework for describing these symmetries, which turn out to be of a topological nature, is equivariant cohomology and the approach discussed in this paragraph is usually called 'equivariant localization ization of the
The exact solvability features of path integrals in this context is similar to the solvability features of the Schr6dinger equation in quantum mechanics when there is a large symmetry group of the problem. For instance, the 0(4) symmetry of the 3-dimensional Coulomb problem is what makes the hydrogen atom an exactly
solvable quantum system
[1011.
1. Introduction
theory'. This approach introduces an equivariant cohomological framework as a tool for developing geometric techniques for manipulating path integrals and examining their localization properties. Historically, this subject originated in the mathematics literature in 1982 with the Duistermaat-Heckman theorem
[39], which established the exactness
of the semi-classical
approximation for finite-dimensional oscillatory integrals (i.e. finite-dimensional versions of (1.1)) over compact symplectic manifolds
in certain instances. The Duistermaat-Heckman theorem
applies
to classical
systems whose trajectories all have a common period, so that the symmetry responsible for the localization here is the existence of a global Hamiltonian torus action
on
the manifold.
Duistermaat-Heckman localization ization
property of
action in the
case
Atiyah
and Bott
[9]
showed that the
special case of a more general localequivariant cohomology (with respect to the torus group was a
of the Duistermaat-Heckman
theorem).
This fact
was
used
by Berline and Vergne [19, 20] at around the same time to derive a quite general integration formula valid for Killing vectors on general compact Riemannian manifolds.
The first infinite-dimensional
generalization of the Duistermaat-Heckman Atiyah and Witten [8], in the setting of a supersymmetric path integral for the index (i.e. the dimension of the space of zero modes) of a Dirac operator. They showed that a formal application of the Duistermaat-Heckman theorem on the loop space LM of a manifold M to
theorem is due to
the partition function of N
.1
supersymmetric quantum mechanics (i.e. a 2 supersymmetric spinning particle in a gravitational background) reproduced the well-known
=
Atiyah-Singer
index theorem
correctly.
The crucial idea
was
the interpretation of the fermion bilinear in the supersymmetric action as a loop space symplectic 2-form. This approach was then generalized by Bismut
[23, 24], within a mathematically rigorous framework, to twisted Dirac oper(i.e. the path integral for spinning particles in gauge field backgrounds), and to the computation of the Lefschetz number of a Killing vector field V (a measure of the number of zeroes of V) acting on the manifold. Another nice ators
infinite-dimensional generalization of the Duistermaat-Heckman theorem was suggested by Picken [139] who formally applied the theorem to the space of loops over a group manifold to localize the path integral for geodesic motion on the group, thus establishing the well-known semi-classical properties of these systems. It was the beautiful paper by Stone [156] in 1989 that brought the Duistermaat-Heckman theorem to the attention of a wider physics audience. Stone for
presented a supersymmetric derivation of the Weyl character formula SU(2) using the path integral for spin and interpreted the result as a
Duistermaat-Heckman localization. This supersymmetric derivation was exby Alvarez, Singer and Windey [4] to more general Lie groups using
tended
fiber bundle very
closely
theory,
and the
supersymmetries in both of these approaches are cohomology. At around this time, other im-
related to equivariant
1.3 Outline
7
portant papers concerning the quantization of spin appeared. Most notably,
[1171 (see also [50]) viewed the path integral for spin geometrical point of view, using as action functional the solid angle swept out by the closed orbit of the spin. This approach was related more closely to geometric quantization and group representation theory by Alekseev, Faddeev and Shatashvili [2, 3], who calculated the coadjoint orbit path integral for unitary and orthogonal groups, and also for cotangent bundles of compact groups., Kae-Moody groups and the Virasoro group. The common feature is always that the path integrals are given exactly by a semi-classical Nielsen and Rohrlich from
a more
localization formula that resembles the Duistermaat-Heckman formula. The connections between supersymmetry and equivariant
cohomology
in
the quantum mechanics of spin were clarified by Blau in [26], who related the Weinstein action invariant [165] to Chern-Simons gauge theory using the
Duistermaat-Heckman
integration formula. Based
on
this interpretation, and
[57]-[59] of the hidden supersymmetry underclassical dynamical system, Blau, Keski-Vakkuri and Niemi [30] lying any introduced a general supersymmetric (or equivariant cohomological) frame-
the observation of Gozzi et al.
investigating Duistermaat-Heckman (or WKB) localization formulas generic (non-supersymmetric) phase space path integrals, leading to the fair amount of activity in this field which is today the foundation of equivariant localization theory. They showed formally that the partition function for the quantum mechanics of circle actions of isometries on symplectic manifolds localizes. Their method of proof involves formal techniques of Becchi-RouetStora-Tyupin (or BRST for short) quantization of constrained systems [16]. BRST-cohomology is the fundamental structure in topological field theories, and such BRST supersymmetries are always the symmetries that are responwork for for
sible for localization in these models.
1.3 Outline
primarily explore the geometric features of the localiza, phase space path integrals. In particular, we shall focus on how these models can be used to extract information about integrable and topological quantum field theories. In this sense, the path integrals we study for the most part can be thought of as "toy models" serving as a testing ground for ideas in some more sophisticated field theories. The main idea behind this reasoning is that the localization in topological field theories is determined by their kinematical properties. The path integrals we shall focus on allow us to study their kinematical (i.e. geometrical and topological) aspects in isolation from their dynamical properties. These models are typically dynamically linear (i.e. free field theories) in some sense and the entire non-triviality of their path integrals lie in thelarge kinematical non-linearity that these theories possess. The appropriate relationship between topological field theories and more conventional, physical interacting quantum field In this Book
we
shall
tion formalism for
1. Introduction
(which are kinematically linear but dynamically highly non-linear) then, in principle at least, allow one to incorporate the approaches and techniques which follow to generic physical models. Indeed, one of the central themes in what follows will be the interplay between physical, integrable and topological field theories, and we shall see that the equivariant localization formalism implies connections between these 3 classes of models and thus a sort of unified description of functional integration which provides alternative approximation techniques to the usual perturbative expansion in quantum field theory. We shall therefore approach the localization formalism for path integrals in the following manner. Focusing on the idea of localizing a quantum paxtition function by reducing it using the large symmetry of the dynamical system to a sum or finite-dimensional integral in analogy with the classical Gaussian integration formula (1.2), we shall first analyse the symmetries responsible for the localization of finite-dimensional integrals (where the symmetry of the dynamics is represented by an equivariant cohomology). The main focus of this Book will then be the formal generalizations of these ideas to phase space path integrals, where the symmetry becomes a "hidden" supersymmetry of the dynamics representing the infinite-dimensional analog of equivariant cohomology. The subsequent generalization will be then the extension of these theories
would
notions to both Poincar6
supersymmetric quantum field theories (where the symmetry is represented by the supersymmetry of that model) and topological quantum field theories (where the symmetry is represented by a gauge
symmetry).
The hope is that these serve as testing grounds for the more sophisticated quantum field theories of real physical interest, such as quantum chromodynamics (QCD). This gives a geometric framework for studying quantum integrability, as well as insights into the structure of topological and supersymmetric field theories, and integrable models. In particular, from this analysis we can hope to uncover systematically the reasons why some quantum problems are exactly solvable, and the reasons why others aren't. Briefly, the structure of this Book is as follows. In Chapter 2 we go through the main mathematical background for localization theory, i.e. equivariant cohomology, with reviews of some other mathematical ideas that will be important for later Chapters as well. In Chapter 3 we present the DuistermaatHeckman theorem and its generalizations and discuss the connections they imply between equivariant cohomology and the notion of classical integrability of a dynamical system. Chapter 4 then goes through the formal supersymmetry and loop space equivariant cohomology arguments establishing the localization of phase space path integrals when there is a Riemannian structure on the phase space which is invariant under the classical dynamics of the system. Depending on the choice of localization scheme, different sets of phase space trajectories are lifted to a preferred status in the integral. Then all contributions to the functional integral come from these preferred paths along with a term taking into account the quantum fluctuations about these
1.3 Outline
9
selected
loops. Chapters 5 and 6 contain the main physical and mathematical applications of equivariant localization. There we use the fundamental isometry condition to construct numerous examples of localizable path integrals. In each case we evaluate and discuss the localization formulas from both physical and mathematical standpoints. Here we shall encounter numerous examples and gain much insight into the range of applicability of the localization formalism in general. We will also see here many interesting features of the localizable partition functions when interpreted as topological field theories, and
discuss in detail various other issues
(e.g. coherent state quantization coadjoint orbit character formulas) which are common to all the localizable examples that we find within this setting and which have been of interest in the more modern approaches to the quantization of dynamical systems. Chapter 7 then takes a slightly different approach to analysing localizable systems, this time by some geometric constructs of the full loop-expansion on the phase space. Here we shall discuss how the standard localization symmetries could be extended to more general ones, and we shall also show how the localization ideas could be applied to the formulation of a geometric approach to obtaining corrections to the standard WKB approximation for non-localizable partition functions. The analysis of this Chapter is a first step towards a systematic, geometric understanding of the reasons why the localization formulas may not apply to a given dynamical system. In Chapter 8 we turn our attention to field theoretical applications of equivariant localization and discuss the relationships that are implied between topological field theories, physical quantum field theories, and the localization formalism for dynamical systems. For completeness, 2 Appendices at the end of the book are devoted to an overview of some ideas in the BRST quantization formalism and some more mathematical ideas of equivariant cohomology, all of which play an important role in the development of the ideas in the main body of we
and
this Book. We close this
introductory Chapter with some comments about the style Although we have attempted to keep things self-contained and at places where topics aren't developed in full detail we have included ample references for further reading, we do assume that the reader has a relatively solid background in many of the mathematical techniques of modern theoretical physics such as topology, differential geometry and group theory. All of the group theory that is used extensively in this Book can be found in [162] (or see [53] for a more elementary introduction), while most of the material discussing differential geometry, homology and cohomology, and index theoof this Book.
rems can
For
be found in the books
a more
[61, 32, 111]
detailed introduction to
and the review articles
algebraic topology,
see
[98].
[22, 411.
The basic
reference for quantum field theory is the classic text [75). Finally, for a discussion of the issues in supersymmetry theory and BRST quantization, see
[16, 22, 69, 118, 155]
and references therein.
Equivariant Cohomology and the Localization Principle 2.
2.1
The
Example:
Function of the
Height
Sphere
some of the abstract and technical formalism which follows, by considering the evaluation of a rather simple integral. Consider 3 the 2-sphere S2 of unit radius viewed in Euclidean 3-space R as a sphere and centered at z a symmetrically about standing on end on the xy-plane the z-axis. We introduce the usual spherical polar coordinates x sin 0 cos 0, for 0 the of the 0 sin and sin 0 z cos a sphere in 3-space embedding y
To
help
we
start
motivate
=
=
=
as
=
S2
=:
f (x, y, z)
0 height function 0 <
< 27r.
E R
The z
-
3
X2
:
height
in R
3
y2
+
of the
ZO (T)
=
ff 0
measure
parameter. It is
11,
where 0
xy-plane
is
<
0 <
and
7r
given by
the
S2, =
dO
a
-
cos
(2.1)
0
on
do
sin 0
e
iTho (0, 0)
(2.2)
0
(1.1).
version of
the standard volume form of the
=
27r
ir
'toy'
a)2
off of the
oscillatory integral
We want to evaluate the
a
-
sphere
restricted to
ho(O, 0)
which represents
(z
+
S2, 3
dx
dy dz of R straightforward
The
integration
measure
i.e. that which is obtained
in
(2.2)
is
by sphere, and T is some real-valued carry out the integration in (2.2) to get
restriction
to the to
+1
Zo(T)
=
dcosO
27r
e
iT(a-cos 0)
(2.3)
1
2,7r
T
Although
this
integral
e- iT(I+a)
is
simple
_
e
iT(l-a)
)
to evaluate
47r =
T
e
-iTa
explicitly,
sin T
it illustrates 2 im-
portant features that will be the common theme throughout our discussion. The first characteristic is the second equality in (2.3). This shows that Zo(T) can
be
expressed as a sum of 2 terms height function (2.1)
extrema of the
R. J. Szabo: LNPm 63, pp. 11 - 42, 2000 © Springer-Verlag Berlin Heidelberg 2000
which -
one
correspond, respectively, to the 2 pole 0 7r, which
from the north
=
12
Equivariant Cohomology and the Localization Principle
2.
is the maximum of
(2-1),
and the other from the south
pole
0
=
0, which
is its
minimum. The relative minus
sign between these 2 terms arises from the fact that the signature of the Hessian matrix of ho at its maximum is negative while that at its minimum is positive, i.e. the maximum of ho is unstable in the 2 directions
along
the
sphere, each of which, heuristically, constributes a (2.3). Finally, the factor 2-7ri/T can be understood as the contribution from the 1-loop determinant when (2.1) is expanded to quadratic order in (0, 0) and the standard WKB approximation to the integral is used. In other words, (2.3) coincides exactly with the Gaussian integral factor of i to the
formula
sum
(1.2), except
in
that it
sums over
both the minimum and maximum of
the argument of the exponential in (2.2). The second noteworthy feature here is that there is
a
symmetry
respon-
simple evaluation of (2.2). This symmetry is associated with the interplay between the globally-defined (i.e. single-valued) integration measure and the integrand function (2.1) (see the first equality in (2.3)). Both the height function (2.1) and the integration measure in (2.2) are independent of the polar coordinate 0 of S2 This is what led to the simple evaluation of (2.2), and it means, in particular, that the quantities integrated in (2.2) are invariant under the translations 0 --+ 0+00, 00 E [0, 27r), which correspond to rigid rotations of the sphere about the z-axis. These translations generate the circle group S1 U(1). The existence of a group acting on S' which serves as a mechanism for the 'localization' of ZO(T) onto the stationary points of ho gives us hope that we could understand this feature by exploiting non-trivial global features of the quotient space S21S1 within a mathematically rigorous framework. Our hopes are immediately dashed because the 2-sphere with its points identified under this (continuous) circular symmetry globally has the same properties as the mathematically trivial interval [0, 1] (the space where sible for the
.
-
0/,7r lives),
i.e.
S21U(j) We shall
see
that the
reason we
_
[0, 1]
(2.4)
cannot examine this space in this way is
because the circle action above leaves fixed the north pole, at 0 south pole, at 0 0, of the sphere. The fixed points in this case
=
=
extrema, of the we
should
use
height
function
ho. The
7r, and the
are
at the 2
correct mathematical framework that
to describe this situation should take into proper account of
this group action
this is 'equivariant cohomology'. on the space Equivariant cohomology has over the past few years become an increasingly popular tool in theoretical physics, primarily in studies of topological models such as topological gauge theories, topological string theory, and topological gravity. This theory, and its connection with the ideas of this Section, will be the topic of this Chapter. Beginning with a quick review of the DeRham theory, which has for quite a while now been at the forefront of many of the developments of modern theoretical physics (see [41] for a comprehensive review), we shall then develop the framework which describes the topology of a
space when there is
-
an
action of
some
Lie group
on
it. This is reminiscent of
2.2 A Brief Review of DeRham
how
Cohomology
13
changes ordinary derivatives to gauge-covariant ones in a gauge field theory to properly incorporate local gauge invariance of the model. We shall ultimately end up discussing the important localization property of integration in equivariant cohomology, and we will see later on that the localization theorems are then fairly immediate consequences of this general formalism. We close this Section with a comment about the above example. Although it may seem to serve merely as a toy model for some ideas that we wish to pursue, we shall see that this example can be considered as the classical parone
tition function of
generalization
spin system (i.e.
a
a
classical
rotor).
A quantum mechanical quantization of spin.
of it will therefore be associated with the
sphere as the Lie group quotient space SU(2)IU(1), then, extensively later on, this example has a nice generalization to the so-called 'homogeneous' spaces of the form GIT, where G is a Lie group and T is a maximal torus of G. These sets of examples, known as 'coadjoint orbits', will frequently occur as non-trivial verifications of the If
we
as we
think of the
shall discuss
localization formalisms.
2.2 A Brief Review of DeRham To introduce
some
notation and to
Cohomology
provide
a
basis for
some
of the
more
abstract concepts that will be used throughout this Book, we begin with an elementary 'lightning' review of DeRham cohomology theory and how it
probes the topological features of a space. Throughout we shall be working on an abstract topological space (i.e. a set with a collection of open subsets which is closed under unions and finite intersections), and we always regard 2 topological spaces as the same if there is an invertible mapping between the 2 spaces which preserves their open sets, i.e. a bi-continuous function or 'homeomorphism'. To carry out calculus on these spaces, we have to introduce
a
smooth structure
differentiable
-
or
to the notion of
a
on
them
short)
(i.e.
one
that is
infinitely-continuously by turning
which is done in the usual way differentiable manifold. C' for
C' manifold of dimension n, i.e. M is a paracompact Hausdorff topological space which can be covered by open sets Ui, M Ui Ui, Let M be
a
=
each of which is
homeomorphic
to n-dimensional Euclidean space
R' and the
homeomorphisms so used induce C' coordinate transformations on the overlaps of patches in R This means that locally, in a neighbourhood of any point on M, we can treat the manifold as a copy of the more familiar Rn, but globally the space M may be very different from Euclidean space. One way to characterize the global properties of M, i.e. its topology, which make it quite different from R is through the theory of homology and its dual theory, cohomology. Of particular importance to us will be the DeRham theory [321. We shall always assume that M is orientable and path-connected (i.e. any 2 points in M can be joined by a continuous path in M). We shall usually assume, unless otherwise stated, that M is compact. In the non-compact case, local
n
.
n
14
Equivariant Cohomology and the Localization Principle
2.
shall
we
certain
assume
regularity conditions
the compact case hold there as well. Around each point of the manifold copy of R'. In R'
infinity
at
choose
we
so
that results for
open set U which is
an
a
have the natural notion of tangent vectors to a point, and so we can use the locally defined homeomorphisms to define tangent vectors to a point x E M. Using the local coordinatization provided by the
homeomorphism denoted
we
onto
R',
a
general linear combination V
where
throughout
we
use
=
49
V"(X)
(2.5)
09xtL
the Einstein summation convention for
upper and lower indices. A linear combination such to here
as
and
specified by
are
a
vector field. Its
V"(x)
components
as
are
(2.5)
repeated
will be refered
C' functions
the introduction of local coordinates from R'.
smooth function
on a
of tangent vectors is
as
f(x),
the quantity
V(f)
=-
V049tf
M
on
Acting
is the directional
derivative of f in the direction of the vector components fV/'J. The local derivatives span an n-dimensional vector space over R which is ax" A called the tangent space to M at x and it is denoted by T.,M. The disjoint union of all
tangent
spaces of the
manifold,
TM
11
=
(2.6)
T, M
xEM
is called the
Any
tangent bundle of M.
vector space W has
linear functionals
TXM
is called the
defined
a
HOMR(W, R) cotangent
dual vector space W* which is the space of on W --+ R. The dual of the tangent space
(5 T 9
dx t'
The
Tx*M
space
and its basis elements dx4
are
by
disjoint
union of all the
cotangent T*M
=
J-'-
(2.7)
spaces of
11
M,
Tx*M
(2.8)
XEM
is called the
cotangent bundle of M.
The space (Tx*M) (Dk is the space of n-multilinear functionals x TxM whose elements are the linear combinations T
The
object (2.9)
functions of
x
E
(x) dx
=
is called
M.
a
rank-(k, 0)
Similarly,
" (9
...
on
TxM
0 dx "'
tensor and its
x
(2.9) components
the associated dual space
(TxM)oe
are
C'
consists
of the linear combinations
t
=
tAl
...
At
(X)
a
a -
,OXA'
0
...
&
09X/J-,
(2.10)
2.2 A Brief Review of DeRham
which
are
called
(k, t)
(O,t)
called
tensors. The elements of
tensors and
one
along
ax",
a
09XA Oxl,\
imply that the components of transform
x'(x), (2.9)
--+
=
and
dx'A
TX731
generic rank (k,t)
a
(T__M)O
are
(2.10)
(2.11)
tensor field
T,"', -VA.'(x)
as
Tp'A,"",\"(X,) ...
ax"\
09x"\" ax"
aXA1
axttf ax,pi
axVk
YX,-P, T"'
...
Pk
Such local coordinate transformations
(2.11)
x
C')XA
dx"
-
'OXA
0
analogously to the C' change of coordi-
local
a
M represented by the diffeomorphism with the usual chain rules on
a
(T*M)Ok
15
define tensor bundles
can
tangent and cotangent bundles above. Under nates
Cohomology
can
be
thought
Vk
of
the tangent and cotangent spaces. We are now ready to define the DeRham complex of
as
W changes
(2.12) of bases
on
a
manifold M. Given
the tensor
product of copies of the cotangent bundle as above, we define a multi-linear anti-symmetric multiplication of elements of the cotangent bundle, called the exterior or wedge product, by dxA1 A
...
=J:
A dxt"
sgn(P)dxl'P(l)
(2)
...
(9 dxl'P(';)
(2.13)
PESk
where the of
sum
defined
P, example,
For
is
as
k and sgn(P) is the sign all permutations P of number the of transpositions in P. where is t(P) (-1)'(P) over
for 2 cotangent basis vector elements
dx A
dy
=
dx &
dy
-
dy
The space of all linear combinations of the basis elements 1 a
is the
T-,a,-,,
=
antisymmetrization
...
..
(x)dxl"
A(Tx*M)O'
A
(2.14)
o dx
...
A dx Ak
(2.13), (2.15)
of the k-th tensor power of the cotan-
gent bundle. The disjoint union, over all x E M, of these vector spaces is called the k-th exterior power A kM of M. Its elements (2.15) are called differential k-forms whose components are Cm functions on M which Notice that by the completely antisymmetric in their indices Al 7 Ak. -
-
are
an-
-
0 tisymmetry of the exterior product, if M is n-dimensional, then A kM smooth of functions the for all k > n. Furthermore, A'M CI(M), space T*M is the cotangent bundle of M. on M --+ R, and A'M The exterior product of a p-form a and a q-form 3 is the (p + q)-form 1 A dXttp+q with local components a A)3 (a A)3),,,...,p+q(x)dxA' A :F (p+q)! =
=
=
=
(a
...
(x)
=
E sgn(P)a,,,(,)
...
JIP(p)
(X)'311P(p+1)***AP(p+q) (X)
PESp+q
(2.16)
16
2.
Equivariant Cohomology and the Localization Principle
The exterior
product
of differential forms makes the direct
sum
of the exterior
powers n
(
AM
AkM
(2.17)
k=O
into
graded-commutative algebra called the exterior algebra of M. In product of a p-form a and a q-form 0 obeys the gradedcommutativity property
AM,
a
the exterior
Ce
A
0
On the exterior
(_I)pqa A a
=
algebra (2.17),
we
d: A kM on
k-forms
a
E
Al"M,O
define
a
linear operator
E
AqM
(2.18)
Ak+1M
--->
(2.19)
(2.15) by =
E sgn(P),Oip(,)allp(2)*.*IJP(k+l)(X)
(2.20)
PESi.+i
T1
and da A A dxAk+l The operator d is called (k+l)! (da)Aj***Ak+1 (x)dx,41 the exterior derivative and it generalizes the notion of the differential of a =
...
-
function
df to
=
49XA
dx"
f
generic differential forms. It is
a
E
AOM
=
C'(M)
graded derivation,
(2.21)
i.e. it satisfies the
graded Leibniz property
d(a AO) and it is
=
da A
0
+
(-l)Pa A dp
a
,
E
APM, 0
E
AqM
(2.22)
nilpotent, d2
which follows from the
0
(2.23)
of
multiple partial derivatives of C' one to generalize the common
=
commutativity
functions. Thus the exterior derivative allows
notion of vector calculus to more general spaces other than R'. The collection of vector spaces JAkMjn and nilpotent derivations d form what is called k= 0 the DeRham complex A*(M) of the manifold M.
There
are
2
important subspaces of the
exterior
the map d is concerned. One is the kernel of
ker d whose elements
are
called closed
imd=
whose elements ker d. Thus
we
are can
=
Ja
E
AM: da
forms,
JOE AM:,3=da
algebra (2.17)
=
01
far
as
(2.24)
and the other is the
for
as
d,
some
called exact forms. Since d is
image
of
aEAMj
nilpotent,
(2.25)
we
consider the quotient of the kernel of d
d,
have im d C
by
its
image.
2.2 A Brief Review of DeRham
Cohomology
17
The vector space of closed k-forms modulo exact k-forms is called the k-th DeRharn cohomology group (or vector space) of M,
Hk(M; R)
=
The elements of the vector space
dlAkM/im dlAk-'M
ker
(2.26)
(2.26)
the equivalence classes of differenequivalent if and only if they differ only by an exact form, i.e. if the closed form a E AkM is a representative of the cohomology class [a] E Hk (M; R), then so is the closed form a + do for any differential form)3 E Ak-1M. tial k-forms where 2 differential forms
are
are
One important theorem in DeRham
cohomology is Poincar6's lemma. This star-shaped region S of the manifold M (i.e. one in which the affine line segment joining any 2 points in S lies in S), then dO in that region for some other differential form 0. Thus one can write w each representative of a DeRharn cohomology class can be locally written as an exact form, but globally there may be an obstruction to extending the form 0 over the entire manifold in a smooth way depending on whether or not [w] 54 0 in the DeRham cohomology group. The DeRham cohomology groups are related to the topology of the manifold M as follows. Consider the following q-dimensional subspace of Rq+', states that if dw
=
0 in
a
=
q
'Aq
(X0' X1, ...,Xq)
=
R q+1
E
:
O'EXi
Xi
=
(2.27)
1
i=O
which is called the standard
q-simplex. Geometrically, Aq is the convex hull generated by the vertices placed at unit distance along the axes of Rq+'. We define the geometric boundary of the standard q-simplex as q
E(_I)i,,3qW
(2.28)
i=O
where
', q
is the
(q
-
I)-simplex generated by
all the vertices of zAq except
the i-th one, and the sum on the right-hand side is the formal algebraic sum of simplices (where a minus sign signifies a change of orientation). A singular
q-simplex
of the manifold M is defined to be
A formal
algebraic
a
continuous map
a :
Aq
M.
of
q-simplices with integer coefficients is called a q-chain, and the collection of all q-chains in a manifold M is called the qsum
th chain group Cq(M) of M. It defines an abelian group under the formal addition. The boundary of a q-chain is the (q I)-chain -
q
au
=
1>1yo,
which is
easily verified
to
give 49:
a
(2.29) M
i=O
nilpotent homomorphism
cq(M)
-+
Cq-l(M)
(2.30)
18
Equivariant Cohomology
2.
and the Localization
Principle
of abelian groups. The collection of abelian groups fCq(.A4)lqcz+ and nilpohomomorphisms 0 form the singular chain complex C,, (M) of the man-
tent
ifold M.
Nilpotency
of the
boundary
(2.30)
that every
q-chain in the q-boundaries of M, lies as well in the kernel of Olc,, whose elements are called the q-cycles of M. The abelian group defined as the quotient of the group of all q-cycles modulo the group of all q-boundaries is called the q-th (singular) homology group of M, image
of
alc,,+,,
map
the elements of which
Hq (M; Z)
=
ker
are
means
called the
91 c, /im
aI
(2.31)
cq+ ,
These groups are homotopy invariants of the manifold M (i.e. invariant under continuous deformations of the space), and in particular they are topological
diffeomorphism invariants (i.e. invariant under C' invertible mappings of M). As such, they are invariant under local deformations of the space and depend only on the global characteristics of M. Intuitively, they measure whether or not a manifold has 'holes' in it or not. If Hq(M; Z) 0, then every q-cycle (intuitively a closed q-dimensional curve or surface) encloses a q + 1-dimensional chain and M has no 'q-holes'. For instance, if M is simply-connected (i.e. every loop in M can be contracted 0. A star-shaped region, such as a simplex, is to a point) then H, (M; Z) simply-connected. Given the abelian groups (2.31), we can form their duals using the uniinvariants and
bi-continuous
=
=
versal coefficient theorem
Hq (M;
Z)
_-
Homz (Hq(M; Z), Z)
E)
Extz(Hq-I(M; Z), Z)
(2.32)
which is called the
q-th singular cohomology group of M with integer coeffiHomZ (Hq (M; Z), Z) Hq (M; Z) is the free part of the cohomology group, and ExtZ is the torsion subgroup of Hq(.M; Z). The DeRham theorem then states that the DeRharn cohomology groups are naturally isomorphic to the singular cohomology groups with real coefficients, *
cients. Here
=
Hq (M; where the tensor
R)
product
=
Hq (M;
Z) OR=Hq(M;R)*
with the reals
means
that Hq is considered
abelian group with real instead of integer coefficients, i.e. R (this eliminates the torsion subgroup in (2.32)). The
crux
of the
I
dw
proof =
jw
,
map
P, c)
an
over
theorem,
CECq+ I (M)
(2.34)
integral of an exact (q + 1)-form over a smooth (q + 1) -chain integral over the closed q-dimensional boundary ac of c. The R defines a natural duality f w on Hq (M; R) 0 Hq (M; R)
which relates the in M to
as an
vector space
ac
C
c
a
of DeRham's theorem is Stokes'
wEAqM
(2.33)
-+
C
2.3 The Cartan Model of
Equivariant Cohomology
pairing between Hq (M; R) and Hq (M; R) and isomorphism. In particular, (2-34) generalizes to theorem,
I
dw
w
m
which relates the
(n
closed
-
w
c-
19
is the basis of the DeRham the
global
version of Stokes'
A'-'M
(2.35)
am
integral
of
an
exact form
over
M to
an
integral
over
the
I)-dimensional boundary
ifold is defined
by
aM of M. Here integration over a manpartitioning the manifold up into open sets homeomorphic
n
integrating a top form (i.e. a differential form of highest degree n on M) locally over Rn as usual', and then summing up all of these contributions. In this way, we see how the DeRham cohomology of a manifold measures its topological (or global) features in an analytic way suited for the differential calculus of C' manifolds. We refer to [98] and [32] for a more complete and leisurely introduction to this subject. to R
,
2.3 The Cartan Model of We shall
now
where there is whose group
the constructions of the last Section to the
generalize a
Equivariant Cohomology
Lie group
(i.e.
multiplication
a
continuous group with
is also
smooth) acting
case
smooth structure
the space. Then the (i.e. structures that are
on
construction of topological invariants for these spaces
the
a
homeomorphic spaces) will be the foundation for the derivation general integration formulas in the subsequent Chapters. Many situations in theoretical physics involve not only a differentiable manifold M, but also the action of some Lie group G acting on M, which we denote symbolically by GxM-->M same
for
of
(2.36)
(g, X)
-
g
-
X
By a group action we mean that g x x, Vx G M if g is the identity element G, and the group action represents the multiplication law of the group, i.e. 91'(92'X) (9192) X7 V91 92 C- G. We shall throughout assume that G is connected and that its action on M is smooth, i.e. for fixed g E G, the function x -+ g x is a diffeomorphism of M. Usually G is taken to be the symmetry group of the given physical problem. The common (infinite-dimensional) example in topological field theory is where M is the space of gauge connections of a gauge theory and G is the group of gauge transformations. The space M -
=
of
=
*
,
-
modulo this group action is then the moduli space of gauge orbits. Another is in string theory where M is the space of metrics on a Riemann sur-
example face
(a
The
be
connected orientable
integral
zero.
over
M of
a
2-manifold)
p4orm. with
and G is the semi-direct
p < dim M is
product
of
always understood here
to
20
the
2.
Equivariant Cohomology
and the Localization
Principle
Weyl
and diffeomorphism groups of that 2-surface. Then M modulo this action is the moduli space of the Riemann surface. In such instances group we are interested in knowing the cohomology of the manifold M given this action of the group G. This
cohomology is known as the G-equivariant cohoon M, the space of orbits MIG is the set mology of equivalence classes where x and x' are equivalent if and only if x' g x for some g E G (the topology of MIG is the induced topology from M). If the G-action on M is free, i.e. g x x if and only if g is the identity element of G, Vx E M, then the space of orbits MIG is also a differentiable manifold of dimension dim M dim G and the G-equivariant cohomology is defined the as simply cohomology of the coset space MIG, of M. Given the G-action
=
-
=
-
-
Tjk
G(M)
However,
H
k
(MIG)
(2.37)
if the group action is not free and has fixed points become singular. The dimension of the orbit G
MIG GI of a point
space
=
can
-
M is dimG
M, the
on x
=
fg
-
x :
dimGx, where Gx fg E G: g x x} is the isotropy subgroup of x. Consequently, in a neighbourhood of a fixed point x, the dimension dim M dim G x of MIG can be larger than the dimension dim M dim G of other fixed-point free coordinate neighbourhoods (because then the isotropy subgroup Gx of that fixed point x is non-trivial), and there is no smooth notion of dimensionality for the coset MIG. A singular quotient space MIG is called an orbifold. In such instances, one cannot define the equivariant cohomology of M in a smooth way using (2.37) and more elaborate methods are needed to define this cohomology. This is the "right" cohomology theory that properly accounts for the group action and it is always defined in a manner such that if the group action is trivial, the cohomology reduces to the usual cohomological ideas of the classical DeRham theory. There are many approaches to defining the equivariant cohomology of M, but there is only one that will be used extensively in this Book. This is the Cartan model of equivariant cohomology and it is defined in a manner similar to the analytic DeRham cohomology which was reviewed in the last Section. However, the other models of equivariant cohomology are equally the Weil algebra formulation relates the algebraic models to as important the topological definition of equivariant cohomology using universal bundles of Lie groups [9, 33, 81, 99], while the BRST model relates the Cartan and g E
X
E
=
-
-
-
=
-
-
-
Weil models and
moreover
is the basis for the superspace formulation of
topological Yang-Mills theory in 4-dimensions and other cohomological field theories [34, 81, 1291. These other models are outlined in Appendix B. We begin by generalizing the notion of a differential form to the case where there is a %roup action on M as above. We say that a map f : M, -* M2 between 2 manifolds with G-actions on them is equivariant with respect to the group action if
f(g-x) =g.f(x)
VXEM1
,
VgEG
(2.38)
2.3 The Cartan Model of
Equivariant Cohomology
21
Wewant to extend this notion of equivariance to differential forms. Consider the symmetric polynomial functions from the Lie algebra g of G =- exp(g) into the exterior algebra AM of the manifold M. These maps form the algebra
S(g*) (&AM,
where
S(g*)
is called the
symmetric algebra over the dual vector
space g* of g and it corresponds to the algebra of polynomial functions g. The action of g E G on an element a E S(g*) 0 AM is given by
(g a) (X) -
where X
Here
g
=
(a (g-'Xg))
-
have used the natural
on
(2.39)
coadjoint
action of G
on g* by the tensor transformation law (2.12) with x'(x) g x. From this it follows immediately that the equivariance condition (2.38) is satisfied for the polynomial maps a: g --> AM in the G-invariant subalgebra
E g.
we
and the induced G-action
AM from that
on
=
A,M where the
(2.40)
are
Elements of G
are
g are
as
dictated
0
AM)'
(infinitesimal)
(2.40) G-invariant part. The
called equivariant differential forms
represented through the exponential map,
where &
M
-
(S(g*)
G denotes the
superscript
elements of
=
on
constants and Xa
are
[19, 21].
in terms of elements of the Lie
=
ec
aXa
algebra
g
(2.41)
the generators of g
obeying the
Lie bracket
algebra
[Xa, Xb] with
ing
fabc
we
the
shall
=
fabcXc
(2.42)
antisymmetric structure constants of g. Here and in the followimplicit sum over the Lie algebraic indices a, b, c,.
assume an
...
The space where the ca,s in (2.41) lie defines the group manifold of G. The generators Xa can be written as Xa gl,=o and so the Lie algebra g =
be
a
ac,
the tangent space to the identity on the group manifold regarded of the Lie group G. The strucutre constants in (2.42) define a natural representation of G of dimension dimG, called the adjoint representation, whose can
as
(Hermitian) generators
have matrix elements
The smooth G-action
on
M
can
be
(ad
Xa )be
=
represented locally
if abc. as
the continuous
flow 9t
-
X
=
X(t)
t E
R+
(2.43)
where gt is a path in G starting at the identity gt=o. The induced action differential forms is defined by pullback, i.e. as
(gt a)(x) -
For
==
a(x(t))
on
(2.44)
example, we can represent the group action on C' functions by diffeomorphisms on M which are connected to the identity, i.e.
2.
22
Equivaxiant Cohomology and the Localization Principle
(gt f ) (x)
=
-
The action where
V(x)
element. It
f (x (t))
etv ('(t)) f (x)
=
f
,
AO M
E
(2.45)
(2.45) represents the flow of the group on C' functions on M, VA(x) ax" a is a vector field on M representing a Lie algebra is related to the flows (2.43) on the manifold by =
- 'W which defines
a
set of
=
VIWO)
in M which
curves
we
(2.46)
will refer to
as
the
integral
curves
of the group action. If VI is the vector field representing the generator X1 of g, then the Lie algebra (2.42) is represented on C' functions by
[Va' Vb] (h)
fabcVc(h)
=
Vh
E
AOM
(2.47)
represented by the ordinary commutator bracket. This derepresentation of G by vector fields in the tangent bundle TM. In this setting, the group G is represented as a subgroup of the (infinite-dimensional) connected diffeomorphism group of M whose Lie algebra is generated by all
with Lie bracket fines
a
vector fields of M with the commutator bracket.
The infinitesimal
(t
--*
0)
action of the group
on
AOM
can
be
expressed
as
V(f)
(2.48)
ivdf
=
where
AkM
---+
operator,
or
iv is the
nilpotent
contraction
to V and it is defined
iva
locally I
=
(k
-
1)!
:
on
V111
interior
k-forms
Wcellvl
Ak-IM
(2.49) multiplication, with respect
(2-15) by
...
M,
(X)dXIA2
A
...
A dx Ilk
(2.50)
The operator iv is a graded derivation (c.f. (2.22)) and the quantity ivT represents the component of a tensor T along the vector field V. The infinitesimal G-action the Lie derivative
on
along
the
higher-degree differential forms
is
generated by
V
Lv: A kM
--+
AkM
(2.51)
where
Lv
=
generates the induced action of G
Lva(x(O)) This
can
be verifed
using (2.12)
and
by
direct
(2.46),
and
(2.52)
div + ivd
on
AM, d
=
i.e.
wt- a(x(t))
lt=O
computation from expanding (2.44) about by noting that
(2.53) t
=
0
2.3 The Cartan Model of
[,CV. "CVb ] (a) Thus the Lie derivative in
=
Equivariant Cohomology
f,b,,CV. (a)
general defines
Va
,
a
E
23
AM
(2.54)
representation of G
on
AM.
The local components of LvT for a general (k, t) tensor field T are found by substituting into the tensor transformation law (2.12) the infinitesimal
coordinate field W
=
change x'A(x)
39 axl,
WA (x)
(LVW? Furthermore, on
=
xP(t)
=
xg +
tVl'(x).
For
example,
on a
vector
have
we
=
W'09,V"
V,09,WA
-
the Lie derivative Lv is
an
=
[WI VIII
(2.55)
ungraded derivation
and its action
contractions is
[iVa,,CVbl (a)
=
fabc iv. (a)
(2.56)
We are now ready to define the Cartan model for the G-equivariant cohomology of M [21, 33, 99]. We assign a Z-grading2 to the elements of (2.40) by defining the degree of an equivariant differential form to be the sum of its ordinary form degree and twice the polynomial degree from the S(g*) part. G Let 10aldimG be a basis of g* dual to the basis IXa}dim of g, so that a=, a=1
Oa(Xb) With the above
grading,
=
jab
the basis elements
(2.57) Oa
have
degree
2. We define
a
linear map k
D9 :A G M on
the
--+
Ak+1M G
(2.58)
algebra (2.40) by
DgOa
=
Dga= (10d-OaOiV. )a
0
;
(2.59)
aEAM
The operator Dg is called the equivariant exterior derivative and it is derivation. Its definition (2.59) means that its action on forms a E
a
graded
S(g*)
(9
AM is
(Dga)(X) where V X in
=
(d
-
iv)(a(X))
(2.60)
&Va is the vector field
on M representing the Lie algebra element However, unlike the operators d and iv, Dg is not nilpotent general, but its square is given by the Cartan-Weil identity =
=
CaXa E g.
D
2 =
9
_Oa
(&
(diVa+ iv.d)
=
_Oa (&,CV.
(2.61)
Thus the operator Dg is nilpotent on the algebra AGM of equivariant differential forms. The set of G-invariant algebras lAkGM}kEz+ and nilpotent A
Z-grading is usually refered to as a 'ghost number' in the physics literature. equivalence between the 2 notions will become clearer when we deal with path integrals in Chapter 4 see Appendices A and B for this algebraic correspondence. The
-
24
2.
Equivariant Cohomology and the Localization Principle
derivations D 9 thereon defines the
G-equivariant complex A*(M) G
of the
manifold M.
Thus, just
as
in the last
Section,
we can
proceed
to define the
of the operator Dg. The space of equivariantly closed modulo the space of equivariantly exact forms, i.e. a
G-equivariant cohomology TTk "G(M)
With this
definition,
free G-action space
MIG,
known
(2.60)
as
=
in
(2.37).
Dg,8,
Dga
=
0,
is called the
DgIA M/im DglAk-im
cohomology
k a
(2.62)
a
of the operator
(2.62) [21, 33, 99]. Note
The definition
the Cartan model
resembles
ker
=
cohomology
i.e.
M,
M reduces to the DeRham
on
as
the
group of
forms,
Dg
for
a
fixed-point
cohomology
of the quotient of equivariant cohomology is that the definition of
Dg
in
gauge-covariant derivative. We close this Section with a few remarks concerning the above construction. First of all, it follows from these definitions that HGk (M) coincides with the ordinary DeRham cohomology of M if G is the trivial group consisting of only the identity element (i.e. V =- 0 in the above), and that the Gequivariant cohomology of a point is the algebra of G-invariant polynomials on g, HG(pt) S(g*)G, of the given degree. Secondly, if a form a c AGM is equivariantly exact, a Dg,8, then its top-form component a(') E AIM a
=
=
ordinary DeRham sense. This follows because the iv part of Dg lowers the form-degree by I so there is no way to produce a toP-form by acting with iv. Finally, in what follows we shall have occasion to also consider of AGM to include arbitrary G-invariant smooth the C' extension AIM G functions from g to AM. In this extension we lose the Z-grading described above, but we are left with a Z2-grading corresponding to the differential form being of even or odd degree [21] (Z2 Z/2Z is the cyclic group of order is exact in the
=
2).
2.4 Fiber Bundles
and
Equivariant Characteristic Classes
The 'bundles'
examples
we
introduced in Section 2.2
(tangent, cotangent, etc.)
are
all
general geometric entity known as a fiber bundle. The geometry and topology of fiber bundles will play an important role in the development of equivariant localization theory, and in this Section we briefly review the essential features that we shall need (see [22, 41, 341 for more detailed discussions). A fiber bundle consists of a quadruple (E, M, F,7), where E is a topological space called the total space of the fiber bundle, M is a topological space called the base space of the fiber bundle (usually we take M to be a manifold), F is a topological space called the fiber, and 1 7r : E --> M is a suriective continuous map with 7r- (x) F, Vx E M, which is called the projection of the fiber bundle. A fiber bundle is also defined so of
a more
=
2.4 Fiber Bundles and
that
locally it is trivial, neighbourhood U C M
Equivariant Characteristic Classes
25
locally the bundle is a product U x F of an open fibers, and 7r : U x F -+ U is the 3. the In first coordinate onto the case of the tangent bundle, for projection the fibers T R' F and the projection map is defined -_ are .,R' instance, on TM --+ M by 7r: T ,M --+ x. In fact, in this case the fibration spaces are vector spaces, so that the tangent bundle is an example of a vector bundle. If the fiber of a bundle is a Lie group G, then the fiber bundle is called a principal fiber bundle with structure group G. It has a right, smooth and i.e.
of the base and the
=
free action of G
the total space E and the base M which gives
on
representation of the group in the fibers. This action also embeds the group'G inside E. The vector and tensor fields introduced in Section 2.2 should be defined
cisely s
:
M
-+
Although
as
'sections' of the associated
E which take
shall be
a
point
bit abusive in
a
local
copy of
more
pre-
i.e. smooth maps
M into the fiber
E
X
bundles,
a
7r-'(x)
over
discussion
by considering M, for simplicity and ease of notation, it should be "kept in mind that it is only locally where these objects admit such a functional interpretation. Thus, for instance, the tangent bundle is TM J(x, V) : x E M, V E TxMJ and locally a section of TM can be x.
these
as
we
a
genuine functions
our
on
=
(x, V4 (x) .0. ). (i.e. bases) on the tangent bundle TM form a principal GL(n,R)-bundle over M, called the frame bundle, whose points are (x; (ei,..., en)) where x E M and (ei,..., en) is a linear basis for TxM. If M has a.metric (i.e. a globally defined inner product on each tangent space TxM), then we can restrict the basis to an orthonormal basis and obtain a principal 0 (n N, N)-bundle, where (n N, N) is the signature of the metric. written
as
X
The set of frames
-
-
If M is furthermore
orientable, then we can further restrict to an oriented ordefined basis, by the equivalence classes with respect to the equivthonoi'tmal alence relation e =- M f where det M > 0, and get a principal SO (n N, N)bundle. When M is a space-time manifold, the Lie group SO (n N, N) is -
-
-
as the local Lorentz group of M (or of the tangent bundle The associated spin group spin(n N, N) is defined as a double cover of the local Lorentz group, i.e. SO(n N, N) -- spin(n N, N)/Z2 (for
then referred to
TM).
-
-
instance, spin(2) bundle
over
=
U(1)
and
spin(3)
M whose fibers form
orthqnormal
frame bundle is called
structure
the manifold M.
on
Conversely,
to any
principal'
=
double
a
a
-
SU(2)).
principal spin(n
A
cover
spin bundle and
G-bundle P
--+
-
N, N)-
of those of the oriented is said to define
M there is
an
a
spin
associated
vector bundle. Let W be the
G. Since G acts
representation space for a representation p of smoothly and freely on the right of P, locally on P x W
(p, v) bundle (P
there is the G-action
associated vector 3
The
topology
taken
as
on
(p g 1, p(g) v) where g E G. This defines the x W)IG PIG for the representation p, which
--+
-
-
the total spaces E
the induced
topology
-
-4
=
ILEM 7r-'(x)
from the erection of
of fiber bundles is
points from M.
usually
26
2.
Equivaxiant Cohomology and the Localization Principle
has fiber the vector space W. For instance, for the trivial representation p, (P x W)IG PIG x W. In this way, we can naturally identify sections (e.g. =
(P x W)IG with equivariant functions f : P W, i.e. p(g-1) f (p). Notice that for a vector bundle E M, the bundle
differential
f (p g) -
=
forms)
on
--+
-+
-
of differential forms
on
M with values in E is defined
Ak (M, E)
=-
as
A kM 0 E
(2.63)
products and direct (Whitney) sums of bundles are defined locally by corresponding algebraic combination of their fiber spaces. Intuitively, a fiber bundle 'pins' some geometrical or topological object where tensor
the
over
each point of
bundle).
a
manifold M
vector space W
bundle is
=
(e.g.
a
vector space in the
case
of
a
vector
R', then the tangent bundle TR' associates the R' to each point of R'. In fact in this case, the tangent
For instance, if M
=
globally given by TR'
R'
=
x
W, the product
of its base and
fibers. We then say that the bundle is trivial, in that the erecting of points into vector spaces is done without any 'twistings' of the fibers. However, a
general
vector bundle is
only locally
trivial and
globally
the fibers
can
twist
very complicated fashion. One way to characterize the non-triviality of fiber bundles is through special cohomology classes of the base manifold M
in
a
[104].
called characteristic classes
signifies
sense
As
the
non-triviality
A non-trivial characteristic class in this
of the vector bundle.
shall see, all of these notions can be generalized to the case of the equivariant cohomology of a manifold which signifies the non-triviality of an we
equivariant bundle. First,
we
define what
we mean
by
an
equivariant bundle
[19, 20]. We say that a fiber bundle E ") M is a G-equivariant bundle if there are G-actions on both E and M which are compatible with each other in the
sense
that g
This
means
.7r(x)
=
that the bundle
ir(g x)
VxEE
-
projection
7r
is
a
,
(2.64)
VgEG
G-equivariant
map. The action
of the group G on differential forms with values in the bundle is by the Lie derivatives LV..
generated
ordinary DeRham case, when there are 'twists' in the given bundle specify how to 'connect' different fibers. This is done using a connection.V which is a geometrical object (such as a 1-form) defined over M with values in E whose action on sections of the bundle specifies their parallel transport along fibers, as required. The parallel transport is generated by the covariant derivative associated with F, In the
one
needs to
V
:--
d +.V
(2.65)
The derivative operator (2.65) is a linear derivation which associates to each section of the given vector bundle a 1-form in Al (M, E). If V is a tangent vector on M, its action on a section s is defined in local coordinates by
2.4 Fiber Bundles and
(VS)"M where
(.P,,)Pdx"
the
path
is
1-form
a
E).
space indices in
on
If x(t) is
a
=
Equivariant Characteristic Classes
V109AS"
+
(FOPSO)
M with values in
path
(2.66)
End(E) (a,,3
M, then: (t)
in
27
is
a
are
tangent
the vector
vector
along
and the equation
Ts)(: M) determines
parallel transport along
the
=
(2.67)
0
path allowing
us
to connect different
fibers of the bundle. The first order differential equation (2.67) can admit topologically non-trivial solutions if either the space M is multiply-connected
(Hi (M; Z) 54- 0), or if the connection F has non-trivial curvature F 54 0 (see below). The latter condition characterizes the non-triviality of the bundle, so that F 0 on a trivial bundle and the solutions to (2.67) are straight lines. At each point p of the total space E, there is a natural vertical tangent space Vp in the tangent space TpE along the fiber of E. A choice of connection above is just equivalent to a choice of horizontal component Hp in the tangent space so that TpE Vp Hp. =
=
When the bundle P
that this
(i.e. Hp
splitting
Hp.,
--*
connection.P is r E
A' (P,
g).
Horizontality
The horizontal
and
ker subspace can then be taken to be Hp F that satisfies G-equivariance mean, respectively, =
=
Cvl'
X on
P
adjoint finite, total adjoint
r,.g
on
briefly
G and G
look at
then ---
pt
=
a
=_
E g
-[x, ri and
action of the group
Ad(g-')r
=
(2.68)
ad(X)
on
can
r
denotes the
be
exponenti-
as
g-irg
(2.69)
examples. If G is a matrix group (e.g. SU(N), regard matrix elements of g E G as functions principal G-bundle. Then the unique solution to
some
as
a
(2.68)
for
a
Lie group is called the Cartan-Maurer
1-form F
For
ad(x)r
F.
can
we
the connection conditions
(matrix)
-
representing X
ated to give the
us
=
action of X. This infinitesimal action
SO(N), etc.),
require further G-equivariant
we
-
where V is the vector field
Let
principal G-bundle,
a
under the action of g E G). In this case, Vp g so that the 1-form with values the Lie in a globally-defined algebra g, i.e.
iVI'
infinitesimal
M is
-->
into horizontal and vertical components be
general G-bundle
P
=
") M, in
g-ldg a
(2.70)
local trivialization U
x
G
-*
ir-'(U),
the connection must look like
r(.,,g) where A which is
=
A,-,dxA
usually
=
A'Xa
g-ldg + g-'A,,gdx4 dxA is
a
Lie
refered to in this context
as
algebra a
(2.71) valued 1-form
gauge connection
on
or
M,
gauge
28
Equivariant Cohomology and the Localization Principle
2.
field. The transformation laws
by
A
-->
local
patch boundaries on M, labelled (the automorphisms of the bundle), act on
across
G-valued transition matrix g gauge connections via pull-back, a
Ag
_=
g-'Ag + g-ldg
(2.72)
which is the familiar form of the gauge transformation law in a gauge field theory [22]. Another example is where the bundle is the tangent bundle TM equipped with a Riemannian metric g. Then _V is the (affine) Levi-
Civita-Christoffel connection P
gv
(g)
associated with g. Its transformation by (2.72) when the
law under local
changes
diffeomorphisms
of the tangent bundle
of coordinates is determined
regarded
are
as
the
automorphisms
of the associated frame bundle. In this case, the parallel transport equation (2.67) determines the geodesics of the Riemannian manifold (M, g) (i.e. the
"straight lines", geometry
or
of minimal
paths
distance,
The failure of the covariant derivative V a
complex
is measured
by
The curvature 2-form
=_
(2.73)
V2
is
=
a
and
so
principal G-bundle
to define
[A A, A]/2
dA +
(2.73)
horizontal, iVF
local trivialization of the
sentation of
on a
its curvature
F
and in
with respect to the curved
g).
=
(2.74)
0
bundle,
F transforms in the
adjoint
repre-
G,
g-1Fjv(x)gdx1' A dx'
F
--+
it
can
be
regarded
=
(g-1X'g)F1,', 0 dxA A dxv
element of A 2 (M, Ad
as an
P),
(2.75)
where Ad P is the
vector bundle associated to P more, from its definition
by the adjoint representation of G. Further(2.73), the curvature F obeys the Bianchi identity
[V, F] When the bundle
=
dF
+[A
being considered
that the covariant derivative
(2.65)
is
I
(2.76)
=0
a G-equivariant bundle, G-invariant,
is
[V,,Cv-l Mimicking the equivariant
F]
A
we assume
0
(2.77)
exterior derivative
(2.58), we define the equivariant
=
covariant derivative
Vg
=
1 (& V
-
Oa
as an
differential forms
M with values in E. In
U C
M,
this
algebra
operator
looks like
on
the
a
(2.78)
iV.
algebra Ag (M, E)
which is considered on
(&
of
equivariant U x W,
local trivialization E
=
2.4 Fiber Bundles and
AG(U, E)
Equivariant Characteristic Classes
(S(g*)
=
0 AU o
Recalling the Cartan-Weil identity (2.61), of the connection
we
29
W)G
(2.79)
define the equivariant curvature
(2.78) R
g
(Vg)2
=
+
Oa
(&
Ev.
(2.80)
which, using (2.77), then satisfies the equivariant Bianchi identity
[Vg, Fg]
(2.81)
0
=
Notice that if G is the trivial group, these identities reduce to the usual notions of curvature, etc. discussed above. Expanding out (2.80) explicitly
using (2.77) gives
Fg
=
1 (9 F + y
(2.82)
where /_1
=
Oa
(9
Lv-
[Oa
-
iV.,
(2)
1 (&
V]
(2.83)
is called the moment map of the G-action with respect to the connection V. The moment map /,t is a G-equivariant extension of the ordinary curvature
(2.73)
2-form
equivariant
from
one
a
When evaluated V E TM 0
W,
covariantly-closed 2-form,
in the
we
sense
on
an
of
in the
sense
of
(2.76),
to
an
(2.81).
element X E g,
represented by
a
vector field
write
Fg (X)
=
F + /-t (X)
-=
F + /-t v
-=
Fv
(2.84)
where mv
=
generates the induced G-action map in this way
CV
on
-
[iV, V1
(2.85)
the fibers of the bundle. The moment
be
regarded locally as a function 1L : AU 0 W g*. Furthermore, using the equivariant Bianchi identity (2-81) we see that it obeys the important property can
--*
Vttv so
that
a
non-trivial moment map
the curvature of the connection V
=
ivF
produces
(2.86)
a non-zero
(c.f. (2-74)).
vertical component of we shall encounter
Later on,
2 important instances of equivariant bundles on M, one associated with a Riemannian structure, and the other with a symplectic structure. In the latter case the moment map is associated with the Hamiltonian of a dynamical
system. Now
we
class. First, [104]. Given
are we
a
ready
to define the notion of
equivariant characteristic
Lie group H with Lie algebra h, we say that a real- or complexan invariant polynomial on h if it is invariant under the
valued function P is natural
an
recall how 'to construct conventional characteristic classes
adjoint
action of H
on
h,
30
2.
Equivariant Cohomology and the Localization Principle
P(h-'Yh)
=
P(Y)
Vh E
H,VY
E
(2.87)
h
An invariant
polynomial P can be used to define characteristic classes on principal fiber bundles with structure group H. If we consider the polynomial P in such a setting as a function on h-valued 2-forms on M, then the Hinvariance (2.87) of P implies that
dP(a) where
is the
r
degree
curvature 2-form
is
locally
an
a
=
h-valued
=
rP(Va)
of P. In F
=
E
A 2M 0 h
(2.88)
particular, taking the argument a to be the the principal H-bundle E ") M (which
V2
2-form),
a
on
have
we
dP(F)
=
(2.89)
0
identity (2.76) for F. This means that P(F) (DeRham) cohomology class of M. What is particularly remarkable about this cohomology class is that it is independent of the particular connection V used to define the curvature F. To see this, consider the simplest case where the invariant polynomial is tr an 4, with tr the invariant Cartan-Killing linear form of the just P(a) Lie algebra h (usually the ordinary operator trace). Consider a continuous V2t one-parameter family of connections Vt, t E R, with curvatures Ft as a
consequence of the Bianchi
defines
a
=
=
Then d -
dt and
applying
Ft
=
,
d
tr
n
Ft
polynomial
n
tr
n
tr
tr
Ftn gives
( Ft) [Vt, ( Vt) ( Vt) d
=
(2.90)
dt
this to the invariant
Tt
[,Vt, dVt] n-1
Ft
dt
d
=
=
d tr
n-1
d
dt
n-1
Ft
where d is the exterior derivative and in the last
(2.88).
This
(2.91)
Ft
dt
equality
we
have
applied
that any continuous deformation of the 2n-form tr F1 exact form, so that the cohomology class determined by it is
means
changes it by an independent of the
general, the invariant polynomial given H-bundle. This notion and construction of characteristic classes can be generalized almost verbaturn to the equivariant case [21]. Taking instead the Gequivariant curvature (2.80) as the argument of the G-invariant polynomial P, (2.89) generalizes to
P(F)
E
Occasionally, forms
choice of connection. In
AM is called
as
for
ease
a
characteristic class of the
of notation,
ordinary multiplication.
we
shall denote exterior products of differential
For
instance,
we
define
ce
An
=
Cen.
2.4 Fiber Bundles and
DgP(Fg) and
Equivariant Characteristic Classes
rP(VgFg)
-
=
31
(2.92)
0
the
resulting equivariant characteristic classes P(Fg) of the given G-equivariant bundle are elements of the algebra AGM. These are denoted by Pg(F), or when evaluated on an element X E g with associated vector field V E TU (9 W, we write now
Pg(F)(X)
P(Fv)
=:
=_
PV(F)
(2.93)
The equivariant cohomology class of Pg(F) is independent of the chosen connection on the bundle. Consequently, on a trivial vector bundle M x W choose
we can
flat connection 5, F
a
Pmxw(Fg)(X)
=
0, and then
=
PMxW(I-tv)
=
P(p(X))
(2-94)
where p is the representation of G defined by the G-action on the fibers W. There are 4 equivariant characteristic classes that commonly appear in the localization formalism for be understood and
as
extensively discussed polynomial tr ec, and is used one
topological field theories, all of which are to completion A M. These can all be found
elements of the
are
in which the fibers
are
in
for
[211.
The first
one
is related to the invariant
G-equivariant complex vector bundles (i.e. over the complex numbers C). It is
vector spaces
called the G-equivariant Chern character
chg(F) The other 3
axe
equivariant real
tr
=
eFg
(2.95)
given by determinants of specific polynomials. On we define the equivariant Dirac A-genus
a
G-
vector bundle
Ag(F)
det sinh(!Fg)] 2
=
F
9
(2.96)
2
where the inverse of
always
an
inhomogeneous polynomial
of differential forms is
to be understood in terms of the power series
(1
+
X)-1
=
1:(_I)kXk
(2-97)
k=O
On
a
is the
complex fiber bundle, the complex equivariant Todd class
tdg(F) 5
A n6n-trivial vector bundle with
a
can
non-trivial curvature F
applications
to
topological
:34
=
det
always
[
version of the
eFg
-
11
be considered
0. This
point of
gauge theories.
equivariant
A-genus
(2.98) as a
view is
trivial
one
quite useful
endowed
in certain
32
2.
Equivariant Cohomology
and the Localization
Principle
When G is the trivial group, these all reduce to the conventional characteristic classes [104] defined by replacing Fg --+ F in the above. Just as for the
ordinary A-genus and Todd classes, their equivariant generalizations inherit multiplicativity property under Whitney sums of bundles,
the
A E(DF
AEA F
=
9
Finally,
td EEDF
,
9
9
orientable real bundle
on an
alization of the Euler
(or
Pfaff M
E 9
td
F
(2.99)
9
define the equivariant gener-
we can
Pfaff (Fg)
=
Salam-Mathiews
[Mij]
metric matrix M
td
class,
Eg (F) where the Pfaffian
=
9
is defined
determinant)
of
a
2N
x
2N
antisym-
as
Mi2N-1i2N
Mili2
E
(2.100)
N
1:
_N N!
sgn(P)
(2.101)
11 MP(2k-1),P(2k) k=1
PES2N
with the property that
det M
=
(Pfaff M)
2
(2.102)
The sign of the Pfaffian when written as the square root of the determinant as in (2.102) is chosen so that it is the product of the upper skew-diagonal
eigenvalues
(2.101),
in
a
skew-diagonalization
Eil***iN is the antisymmetric
integration
antisymmetric matrix M. In C
123
...
N
-
will see, as fermionic determinants from naturally, of fermion bilinears in supersymmetric and topological field
+1. Pfaffians arise the
of the
tensor with the convention
as we
change the
theories. Transformations which
orientation of the bundle
change
the sign of the Pfaffian. R is the Riemann curvature 2-form associated with the tangent When F bundle TM (which can be regarded as a principal SO(2n)-bundle) of a closed =
manifold M of
even
dimension 2n, the
integral
over
M of the
ordinary Euler
class is the integer
X(M)
=
(47r)nn!
f E(R)
(2.103)
M
where 2n
X(M)
=
T(_l)k dimRHk (M; R)
(2.104)
k=O
is the famous
topological
ifold M. That a
(2.104)
invariant called the Euler characteristic of the
can
be written
celebrated result of differential
rem.
The
as an
known
as
to the case of
an
topology
generalization of (2.103)
integral
of
a
density
in
man-
(2.103)
is
the Gauss-Bonnet theo-
arbitrary
vector bundle
2.5 The
E(F)
with Euler class
of
Equivariant Localization Principle
curvature 2-form F E
a
A'(M, E)
33
is called the
Gauss-Bonnet-Chern theorem. In that case, the cphornology groups appearing in the alternating sum (2.104) get replaced by the cohomology groups
Hk (M; E) of the twisted derivative operator V : Ak (M, E) Ak+ 1 (M, E). FA the curvature of a gauge connection A on a Similarly, with F principle H-bundle over a 2k-dimensional manifold M, the integral over M of the k-th term in the expansion of the conventional version of the Chern --+
=
(2.95) (which
class
defines the k-th Chern
Ck(M)
class)
(_ )k f
is the number
I
==
2?Ti
tr
FAk
(2.105)
M
which is
a topological invariant of M called the k-th Chern number of M (or, precisely, of the complex vector bundle (E, M, W, 7r)) The Chern number is always an integer for closed orientable manifolds. Thus the equivariant; characteristic classes defined above lead to interesting equivariant generalizations of some classical topological invariants. In the next Chapter we will see that their topological invariance in both the ordinary DeRham and the equivariant cases are a consequence of the topological invariance of the integrations there. We shall see later on that they appear in most interesting ways within the formalism of localization formulas and topological field theory functional integration. more
.
2.5 The We
now
Equivariant Localization Principle
discuss
a
very
interesting property of equivariant cohomology which
is the fundamental feature of all localization theorems. It also introduces
the fundamental
constraint that will be
'geometric
in what follows. In most of
following
situation. Let M be
ary and let V be
circle group G
0
E
S(u(l)*), U(1),
bra of
a
=
a
vector field
U(1)
-
which is
will not be
a
applications
our
S'
we
one
of the issues of focus
will be concerned with the
compact orientable manifold without boundM corresponding to some action of the
over
M. In this
on
linear functional
case on
the role of the
multiplier algefollows. Indeed,
the 1-dimensional Lie
important for the discussion that
regard 0 as just some external parameter in'this case and 'localize' -1. As shown in [9] (see also Appendix B), algebraically by setting 0 the operations of evaluating 0 on Lie algebra elements and the formation of equivariant cohomology commute for abelian group actions, so that all results below will coincide independently of the interpretation of 0. In particular, for we can
=
a
free
U(1)-action
on
(S(u(l)*) so
that in this
case
M
0
we
have
AM)U(1)
the
=
S(u(l)*)
(9
A(MIU(1))
(2.106)
multipliers 0 play no cohomological role and the restricts to the cohomology of the quotient space
equivariant cohomology just
34
Equivariant Cohomology and
2.
MIU(1).
The
the Localization
corresponding equivariant
Principle
exterior derivative is
now
denoted
as
D,,,(:L) and it is
now
considered
as an
AvM It
was
Atiyah
Dv
=
operator
fa
=
[9]
and Bott
-=
d + iv
on
the
algebra
AM:,Cva
E
(2.107)
01
=
and Berline and
(2.108)
Vergne [19, 20] who first point locus
noticed that equivariant cohomology is determined by the fixed of the G-action. In our simplified case here, this is the set
MV
fX
=
E
M:
V(X)
=
01
(2.109)
This fact is at the very heart of the localization theorems in both the finite dimensional case and in topological field theory, and it is known as the equivariant localization
principle.
analytic ways. For a the Weil algebra and the in 2
see
In this Section
shall establish this property description of this principle using we
algebraic topological definition
more
of
equivariant cohomology,
[9]. Our first argument for localization involves
an explicit proof at the level of integral fM a over M of an equivariantly closed differential form a E AVM, Dva 0, we wish to show that this integral depends only on the fixed-point set (2.109) of the U(I)-action on M. To show this, we shall explicitly construct a differential form A on M MV a. This is just the equivariant version of the Poincar6 satisfying DVA lemma. Thus the form a is equivariantly exact away from the zero locus MV, and we recall that this implies that the top-form component of a is exact. Since integration over M picks up the top-form component of any differential 0 by hypothesis here, it follows from Stokes' theorem form, and since o9M (2.35) that the integral fm a only receives contributions from an arbitrarily small neighbourhood of Mv in M, i.e. the integral 'localizes' onto the smaller subspace MV of M. To construct A, we need to impose the following geometric restriction on the manifold M. We assume that M has a globally-defined U(1)-invariant Riemannian structure on it, which means that it admits a globally-defined
differential forms. Given
an
=
-
=
=
metric tensor I 9
which is invariant under the
=
2
g,,, (x) dx" 0 dx'
U(1)-action generated by V, LVg
or
in local coordinates
on
(2.110)
=
i.e. for which
(2.111)
0
M,
gM,X-q,V-1 + g ,,Xa,,VA + Oaxgl"
=
0
(2.112)
2.5 The
Alternatively,
Equivariant Localization Principle
this Lie derivative constraint
g"XV1.V1'
gA,\VVV,1
+
(2.65)
where V is the covariant derivative
be written
can
35
as
(2.113)
0
=
constructed from the Levi-Civita-
Christoffel connection
_VA
/w
associated with g
on
2
9" (.9Ag" + '9vg"
(2.114)
'9PgA')
-
the tangent bundle TM. Here gAV is the matrix inverse on the vector field V in the usual
of gA, and the covariant derivative acts way
as
V A VV
19AVV
=
VM VA
+ -PA
(2.115)
I
plus sign for (0, k)-tensors and a minus sign for (k, 0)-tensors in front in (2.115). Notice that by construction the Levi- Civita-Christoffel r,\ and it is compatible with the the metric connection is torsion-free, rA V117 AV 0, which together mean that V preserves the inner product in g, V,\gAv
with
a
of F,
as
=
=
the fibers of the tangent bundle. The equivalent equations (2.111)-(2.113) and in this
case we
say that V is
a
Killing
are
called the
Killing equations
vector field of the metric g. Since
the map V --+ LV is linear, the space of Killing vectors of a Riemannian manifold (M, g) generate the Lie algebra of a Lie group acting on M by
diffeomorphisms
isometry group of (M, g). We shall deChapters 5 and 6. The Killing equations here
which is called the
scribe this group in detail in are assumed to hold globally
over
the entire manifold M. If both M and G
are compact, then such a metric can always be obtained from an arbitrary Riemannian metric h on M by averaging h over the group manifold of G in its (G-invariant) Haar measure, i.e. g = D (j h). However, we shall have
fG
occasion to also consider
more
general
-
vector field flows which aren't
neces-
or when the manifold M isn't compact, as are the cases in many applications. In such cases the Lie derivative constraint (2.111) is a
closed
sarily physical very
stringent
one on
the manifold. This feature of the localization
that the manifold admit
globally
a
formalism,
defined metric with the property (2.111) are globally-defined C' functions on M,
whose components gAv (x) g,, (x) is the crux of all finite- and infinite-dimensional localization formulas and will =
be
analysed
in detail later
on
in this Book. For now,
we
content ourselves with
assuming that such a metric tensor has been constructed. Any metric tensor defines a duality between vector fields and differential 1-forms, i.e. we can consider the metric tensor (2.110) as a map g: TM
which takes
a
--+
T*M
(2.116)
vector field V into its metric dual 1-form
#
=-
g (V,
-)
=
gA, (x) V' (x) dxA
(2.117)
36
2.
Equivariant Cohomology and the Localization Principle
Non-degeneracy, det g(x) :7 0, VX an isomorphism between The I-form,6 satisfies
M, of the
E
defines
D 2,3 V
=
fVa
=
LVV Killing equivariant differential 1-form. Furthermore, since
=
0 and V is
globally-defined Kv
(2.118)
=
Kv
+
we
that 3 is
means
an
have
Ov
(2.119)
CI-function
g(V, V)
=
implies that this
0
vector of g. This
a
Dv,3 where KV is the
metric tensor
the tangent and cotangent bundles of M.
=
g,,, (x) V" (x) V'(x)
(2.120)
and
Ov
=
d
=
dg(V,
(2.121)
is the 2-form with local components
(Q0,11 Consequently,
away from
part KV of DVO is
Again with
we
can
-
g'AVI'V),
(2.122)
locus MV of the vector field
zero
V, the 0-form DV,3 is invertible on M MV. an inhomogeneous differential form
and hence
non-zero
understand here the inverse of
non-zero
We
94AVIVII
=
-
analogy with the formula (2.97). inhomogenous differential form by
scalar term in
now-,define
an
=,3(Dv,3)-l on
M
-.A4v, which
(2.118) since
a
satisfies
of 3. Thus
=
I and
Lv
=
0
owing
to the
define
equivariance a, and
we can an equivariant differential form A is equivariantly closed it follows that
a
Thus,
Dv
(2.123)
=
I
-
a
=
(Dv )a
=
=
DV(6a)
(2.124)
claimed
above, any equivariantly closed form is equivariantly exact MV, and in particular the top-form component of an equivariantly closed form is exact away from MV. This establishes the equivariant as
away from
localization property mentioned above. The other argument we wish to present here for equivariant localization is less explicit and involves cohomological arguments. First, consider an or-
dinary
closed form w, dw
=
0. For any other differential form
j (w
+
dA)
M
by of
Stokes' theorem a
closed form
w
(2.35)
f
we
have
(2.125)
W
M
since o9M
depends only
=
A,
on
0. This the
means
cohomology
that the
integral fm W by w, not
class defined
2.5 The
a
linear map
to
a
37
particular representative. Since the map w ---+ fm w in general defines 6nk R, it follows that this map descends on A kM --> An-k(pt)
the
on
Equivariant Localization Principle
on
map
=
Hn (.A4;
R)
for
-+
HO (pt; R)
=
R. The
same
is true for
equivariant
integration. Since, general M, integration of a differential form picks up the top-form component which for an equivariantly exact form is exact, for any equivariantly-closed differential form a we can again invoke G-action
a
on
Stokes' theorem to deduce
f (a
DgA)
+
(2.126)
a
M
M
that the
integral of an equivariantly closed form depends only on the equivcohomology class defined by it, and not on the particular representative. Note, however, that equivariant integration for general Lie groups G takes a far richer form. In analogy with the DeRham case above, the integration of equivariant differential forms defines a map on HG(M) HG(pt) S(g*)G. This we define by so
ariant
-->
I
(X)
a
I a(X)
=
(2.127)
X E g
,
M
with
the AM part of
ordinary DeRham. sense. algebra elements 01 in a more 'dynamical' situation where they are a more integral part of the cohomological description above. We shall see then how this definition of integration should be accordingly modified. In any case, the arguments below which lead to the equivariant localization principle generalize immediately to the non-abelian integration
Later on,
case as
we
over
in the
a
shall also consider the dual Lie
well.
Given that the class defined
by
a,
fm a depends only on the equivariant cohomology
integral we can
choose
particular representative of the cohomolTaking the equivariant differential integral
a
ogy class making the localization manifest. form,3 defined in (2.117), we consider the
Z(S)
f
=
a
e- Dv,6
(2.128)
M
viewed of
S E
as a
function of
R+ and that
its
s
E
s
--+
R+.
We
0 and
s
-*
oo
that
(2.128)
is
a
regular
limits exist. Its
s
--
assume
function
0 limit is the
interest, fM a, while from the identities (2.119) and (2.120) we see integrand of (2.128) is an increasingly sharply Gaussian peaked form around Mv C M as s --+ oo. The crucial point here is that the equivariant differential form which is the integrand of (2.128) is equivariantly cohomologous to a for all s E R+. This can be seen by applying Stokes' theorem to
integral
that the
get
of
38
Equivariant Cohomology and the Localization Principle
2.
d
a(Dv,3) e-sDv)3
Z(S) M
f f Dv(ao e-sDvO) +,3Dv (a e-sDv,6)1
(2.129)
M
8f a,3(Lv)3) e-,DvO
=
0
M
where
have used the fact that
a is equivariantly closed and the equivari0. Therefore the integral (2.128) is independent of the parameter s E R+, and so its s 0 and s ---* oo limits coincide. Hence, we may evaluate the integral of interest as ance
we
property
(2.118)
of
--+
f
a
=
lim S
00
M
a
e-,Dv,8
(2.130)
M
which establishes the localization of It should be
f
fm a to
Mv.
though that there is nothing particularly unique about the choice of 3 in (2.130). Indeed, the same steps leading to (2.130) can be carried out for an arbitrary equivariant differential form 0, i.e. any one with the property (2.118). In this general case, the localization of fm a is onto the subspace of M which is the support for the non-trivial equivariant 0. Different cohomology of a, i.e. fm a localizes to the points where DO choices of representatives 0 for the equivariant cohomology classes then lead to potentially different localizations other than the one onto MV. This would lead to seemingly different expressions for the integral in (2.130), but of course these must all coincide in some way. In principle this argument for localization could also therefore work without the assumption that V is a Killing vector for some metric on M, but it appears difficult to make general statements in that case. Nonetheless, as everything at the end will be equivariantly closed by our general arguments above, it is possible to reduce the resulting expressions further to MV by applying the above localization arguments once more, now pointed
out
=
to the localized
expression. We shall examine situations Killing vector field in Chapter 7.
necessarily
a
2.6 The
Berline-Vergne
The first
in which V isn't
Theorem
general localization formula using only the general equivariant cohomological arguments presented in the last Section was derived by Berline and Vergne [19, 20]. This formula, as well as some of the arguments leading to the equivariant localization principle, have since been established in many different contexts suitable to other finite dimensional applications and also to path integrals [9, 11, 21, 23, 24]. The proof presented here introduces a
2.6 The
Berline-Vergne Theorem
39
method that will generalize to functional integrals. For now, we assume the fixed-point set MV of the U(1)-action on M consists of discrete isolated dim M 6. We points, i.e. MV is a submanifold of M of codimension n shall discuss the generalization to the case where MV has non-zero dimension later on. If we assume that M is compact, then MV is a finite set of points. We wish to evaluate explicitly the right-hand side of the localization formula (2.130). To do this, we introduce an alternative way of evaluating integrals over differential forms which is based on a more algebraic description of the exterior bundle of M. We introduce a set of nilpotent anticommuting (fermionic) variables qA, jj 1, n, =
=
-
-
-
,
77 A77,
(2.131)
-77,77
=
which generate the exterior algebra AM. The variables q/1 are to be identified with the local basis vectors dxl' of A'M T*M with the exterior product of =
ordinary product of the 'q11 variables with the algebra (2.131). The k-th exterior power AkM is then generated by the products ql" 77"k and this definition turns AM into a graded Grassmarm algebra with the generators q" having grading 1. For instance, suppose the differential forms
the
replaced by
...
differential form
a
is the
a
=
a(0)
a
a(')
+... +
a(k) the k-form component
with is
+
sum
Cl-function
on
of
a
a(n)
and
a(k)
a(0) (x)
E
AkM
its 0-form
M. The k-form component of
a
(2.132)
component which
for k > 0 then has the
form
(X, q)
a
and from this
point
with local coordinates Berezin
a.,... t4k (X)nA,
functions
are
the 2n-dimensional
now
(2.133)
k>O
of view differential forms
exterior bundle which is
The
-
a(x,,q)
on
the
0 AM
supermanifold M
(x,,q).
integration of a differential form is now defined by introducing the rules for integrating Grassmann variables [17],
f d77A
IR
A
=
f dql'
1
1
=
(2.134)
0
71"'s are nilpotent, any function of them is a polynomial in 77" consequently the rules (2.134) unambiguously define the integral of any function of the anticommuting variables 77". For instance, it is easily verified Since the and
that with this definition of We shall 7
If
we
introduce
formula
here that
assume
but it will allow
integration is
even.
have 7
This restriction is
by
no means
necessary
of the arguments in this Section. second independent set 1 1' I of Grassmann variables, then the
us
a
n
we
to shorten
(2.135) generlizes
to
some
arbitrary (not necessarily even) dimensions
n as
40
2.
Equivariant Cohomology
and the Localization
dnq el'7"M111"7' d?7' dq'-1
_-
Principle
Pfaff M
(2.135)
d771. Note that under a local change of basis qA algebra the antisymmetry property (2.131) and the Berezin rules (2.134) imply that f dni7 det A f dnq. It follows from this that the Berezin integral in (2.135) is invariant under similarity transformations. (2.135) is the fermionic analog of the Gaussian integration formula (1.2). The differentiation of Grassmann variables, for which the integration in (2.134) is the antiderivative thereof, is defined by the anticommutator where
dn,
=
...
A Vlq' of the Grassmann
--+
19
aqA With these
,77V
I
=
R
(2.136)
A
+
definitions, the integration by parts formula
f dq"
d
d
d7711
f(ntt)
=
0
f dqA f(,qli) &71A gqji). definitions, we can now alternatively write the integral of any differential form over M as an integral over the cotangent bundle M OA1M. Thus given the localization formula (2.130) with the I-formfi in (2.117) and always holds,
since
=
Given these
the identities
j
a
=
(2.119)-(2.122),
M
d'x
have
d',q a(x, 77)
MOAIM x
where the the
j
lim 5--+oo
we
exp,
measure
measures
d'x
(2.137) S
S9tIV (X) V, (X) V, (X) d'x =
dlq
dxl
other. To evaluate the
on
A
...
large-s
(&A' M
M
-
2
Wn"n')
is
coordinate-independent because dnq transform inversely to each
A dXn and
limit of
(QV) w,
(2.137),
we
use
the delta-function
representations 5 (V)
lim.
=
s
Jim 5-00
as can on
be
seen
directly
00
S Ir
)
n/2
1
( -S)-n/2 Pfaff
is
2
e
from the respective
integrations in local coordinates Killing equations (2.113), the matrix
given by
(S'2v),,, Thus
S?v V
(2.138)
(2.139)
-
-
M and A' M. Notice that 'from the
(Qv),,,
Vld-etg e-'9- V" V'
using (2.138) and (2.139)
=
2g7,,\V,,VA
we can
write
(2.140) (2.137)
n
f j1d#4 d?714 jA=1
e-fI"M-`7'
=
detM
as
2.6 The
f
a
f
(-,r)n/2
=
M
dnx
dnq a(x, 71)
Pfaff f2v (
M(DA1M
where
VFd-etg(x)
A'M
Theorem
6(V(x))6(,q)
41
(2.141)
Sn/2 between (2.138) and
note the cancellation of the factors of
we
(2.139).
Berline-Vergne
(2-141)
kills off all k-form components of the form a except its Cl-function part a(') (x) =- a(x, 0), while the integration over M localizes it onto a sum over the points in MV. This yields The
integration
J
over
in
(0)(p) Pfaff S?v(p) (-Ir)n/2 E IdetdV(p)l VFd-etg(p) pEMv C,
a
=
M
(2.142)
I det dV(p) I comes from the Jacobian of the coordinate transV(x) used to transform J(V(x)) to a sum of delta-functions EPEM, J(x -p) localizing onto the zero locus MV. Substituting in the idendV(p), the tity (2.140) and noting that at a point p E Mv we have VV(p) expression (2.142) reduces to
where the factor formation
x
--
=
1
a=
PEMv
M
where
we
emphasize the
a
(-2-7r)n/2
manner
(0)
Pfaff
in which the
(p) W(p)
(2.143)
dependence of orientation
in the
Pfaffian has been transfered from the numerator to the denominator in going from (2.142) to (2.143). This is the (non-degenerate form of the) Berline-
Vergne integration formula, a
and it is
localization formula. It reduces the
space M to
example of what we shall call original integral over the n-dimensional our
first
discrete set of points in M and it is valid for any differential form a on a manifold with a globally-defined
a sum over a
equivariantly-closed
diffeomorphism. Killing vector). In general, the localization formulas we shall encounter will always at least reduce the dimensionality of the integration of interest. This will be particularly important for path integrals, where we shall see that localization theory can be used to reduce complicated infinitedimensional integrals to finite sums or finite-dimensional integrals. We close this Chapter by noting the appearence of the operator in the denominator of the expression (2.143). For each p E MV, it is readily seen circle action
(and
generator is
a
Riemannian metric for which the associated
that the operator dV(p) appearing in the argument of the Pfaffian in (2.143) is just the in vertible linear transformation Lv(p) induced by the Lie derivative the tangent spaces TpM, i.e. by the induced infinitesimal group the tangent bundle (see (2.55)). Explicitly, this operator is defined vector fields W W"(x) .9x" Jx=p E TpM by
acting
on
action
on
on
2.
=
Lv(p)W Note however that
right
on
dV(p)
=
aV"(p)W'(p)
is not covariant in
the tangent space
TpM
a ax"
(2.144) X=P
general and so this is only true general on the entire tangent
and not in
42
2.
Equivariant Cohomology and the Localization Principle
bundle TM. A linear transformation
on the whole of TM can only be induced by introducing a (metric or non-metric) connection FA of TM and inducing an operator from VV, as in the matrix (2.140). We AV shall return to this point later on in a more specific setting.
from the Lie derivative
3. Finite-Dimensional Localization
for
Theory
Dynamical Systems
We shall
now
proceed
to
study
a
certain class of
integrals
ered to be toy models for the functional integrals that terested in. The advantage of these models is that they
that
we are
are
can
be consid-
ultimately
in-
finite-dimensional
rigorous mathematical theorems concerning their behaviour
and therefore
be formulated. In the infinite-dimensional cases, although the techniques used will be standard methods of supersymmetry and topological field theory, can
a
lot of rigor is lost due the ill-definedness of infinite-dimensional manifolds integrals. A lot can therefore be learned by looking closely at
and functional some
finite-dimensional
cases.
We shall be interested in certain
resenting
the
Fourier-Laplace
manifold M in terms of
a
oscillatory integrals
transform of
some
smooth
smooth function H. The
fm dp
e
measure
common
iTH
dy
repon
a
method of eval-
uating such integrals is the stationary phase approximation which expresses large-T the main contributions to the integral come from the critical points of H. The main result of this Chapter is the DuistermaatHeckman theorem [39] which provides a criterion for the stationary phase approximation to an oscillatory integral to be exact. Although this theorem was originally discovered within the context of symplectic geometry, it turns out to have its most natural explanation in the setting of equivariant coho-
the fact that for
mology and equivariant characteristic classes [91,[19]-[21]. The DuistermaatHeckman theorem, and its various extensions that we shall discuss towards the end of this Chapter, are precisely those which originally motivated the localization theory of path integrals. For physical applications, we shall be primarily interested in a special class of differentiable manifolds known as 'symplectic' manifolds. As we shall see in this Chapter, the application of the equivariant cohomological ideas to these manifolds leads quite nicely to the notion of a Hamiltonian from a mathematical perspective, as well as some standard ideas in the geometrical theory of classical integrability. Furthermore, the configuration space of a topological field theory is typically an (infinite-dimensional) symplectic manifold
(or phase space) [22]
and
we
shall therefore restrict
the remainder of this Book to the localization over
symplectic manifolds.
R. J. Szabo: LNPm 63, pp. 43 - 75, 2000 © Springer-Verlag Berlin Heidelberg 2000
theory
for
our
attention for
oscillatory integrals
44
1 Finite-Dimensional Localization
3.1
Theory
for
Dynamical Systems
Symplectic Geometry
Symplectic geometry
is the natural mathematical
setting for the geometristudy of classical integrability branches of physics, such as geometri-
cal formulation of classical mechanics and the
[1, 6].
It also has
applications in other elementary classical mechanics [55], one is introduced to the Hamiltonian formalism of classical dynamics as follows. For a dynamical syscal
[65].
optics
In
tem defined
on some manifold M (usually R') with coordinates (ql, , qn), introduce the canonical momenta p,,, conjugate to each variable q1' from the Lagrangian of the system and then the Hamiltonian H(p, q) is obtained by .
.
.
we
Legendre transformation of the Lagrangian. In this way one has a descripdynamics on the 2n-dimensional space of the (p, q) variables which is called the phase space of the dynamical system. With this construction the phase space is the cotangent bundle M (9) A' M of the configuration manifold M. The equations of motion can be represented through the time evolution of the phase space coordinates by Hamilton's equations. For most elementary dynamical systems, this description is sufficient. However, there are relatively few examples of mechanical systems whose equations of motion can be solved by quadratures and it is desirable to seek other more general formulations of this elementary situation in the hopes of being able to formulate rigorous a
tion of the
theorems about when
motion,
a
classical mechanical system has solvable equations of
'integrable'. Furthermore, the above notion of a 'phase space' is very local and is strictly speaking only globally valid when the phase space is 2n R a rather restrictive class of systems. Motivated by the search for more non-trivial integrable models in both classical and quantum physics, theoretical physicists have turned to the general theory of symplectic geometry which encompasses the above local description in a coordinate-free way suitable to or
is
,
the methods of modern differential geometry. In this Section we shall review the basic ideas of symplectic geometry and how these descriptions tie in with the
more
familiar
A
2n
symplectic together with
ones
of
elementary
manifold is a
a
globally-defined non-degenerate closed 1 W
symplectic
form of M.
By closed dw
or
even
dimension
2-form.
w,,, (x) dxA A dxv
=
2
called the
classical mechanics.
differentiable manifold A4 of
=
(3.1)
we mean as
usual that
0
(3.2)
in local coordinates
9mwv,\ Thus
w
defines
a
DeRham
+
19VWA1Z
+
a,\Wvv
cohomology class
=
0
in H 2(M;
(3.3) R). By non-degenerate
that the components w,,,(x) of w define an invertible 2n symmetric matrix globally on the manifold M, i.e.
we mean
x
2n anti-
3.1
det w (x)
:
Symplectic Geometry
(3.4)
VX E M
0
45
The manifold M of
together with its symplectic form w defines the phase space dynamical system, as we shall see below. Since w is closed, it follows from the Poincar6 lemma that locally there
a
exists
1-form
a
0
0,,(x)dx"
(3.5)
dO
(3.6)
=,900, -,9,00
(3.7)
=
such that
or
in local coordinates
wo, The
locally-defined
1-form 0 is called the
symplectic potential or canonical 1generated globally as above by a symplectic potential 0 it is said to be integrable. Diffeomorphisms of M that leave the symplectic 2-form invariant are called canonical or symplectic transformations. These are determined by C'-maps that act on the symplectic potential as form of M. When
w
is
F
0 or
OF:= 0
(3.8)
+ dF
in local coordinates F
O1'(X) so
)
that
OFO(x)
)
by nilpotency of the transformations,
=
O.(x)
+
OF(x)
exterior derivative it follows that
(3.9) W
is invariant
under such
w
The function
F(x)
=
dO
is called the
F )
wF
=
dOF
(3.10)
_= W
generating function of the canonical
trans-
formation. The
AOM
-+
symplectic 2-form determines a bilinear function I., J, AOM called the Poisson bracket. It is defined by
If, gl,, or
w-
=
1
(df, dg)
f,g
E
AOM
:
AOM
(3.11)
in local coordinates
If, gl",
=
W", (490f (X),9,g(x)
(3.12)
where wA' is the matrix inverse of wjAv. Note that the local coordinate functions themselves have Poisson bracket I
The Poisson bracket is
IXA' X'},,,
=
anti-symmetric,
W/" (X)
(3.13)
46
3. Finite-Dimensional Localization
If, 0", it
obeys
Theory
for
Dynamical Systems
1 g' f 1"'
(3.14)
the Leibniz property
I f, ghl,, and it satisfies the Jacobi
gf f, hj,,
=
+
hf f, gl,,
+
I h, I f, gj,, j,,,
(3.15)
identity
I f, Ig, hl,, J,,
+
I g, I h, f j,, 1,,
=
(3.16)
0
This latter property follows from the fact (3-3) that w is closed. These 3 properties of the Poisson bracket mean that it defines a Lie bracket. Thus the Poisson bracket makes the space of C'-functions we call the Poisson algebra of (M, w)
which
M into
on
a
Lie
algebra
-
The connection with the
elementary formulation of classical mechangiven by a result known as Darboux's theorem [65], which states that this connection is always possible locally. More precisely, Darboux's theorem states that locally there exists a system of coordinates (pj, qll)'=, on M in which the symplectic 2-form looks like ics discussed above is
w
so
that
they have
dpl,
=
A
Poisson brackets
fpj,,p,j,,,=fqA,q"j,=0 These coordinates
(3.18)
from
we see
are
called canonical
that
they
fp/,, q'l,,,
,
0
(3.8)
0
where
(P,,, Ql')'=, A
=
are
pjdqA
J"
(3.18)
A
Darboux coordinates
or
position variables on the phase the symplectic potential is
and the transformation
=
on
M and
be identified with the usual canonical
can
mentum and nates
(3.17)
dql'
space M
[551.
pjdqA
=
mo-
In these coordi-
(3.19)
becomes F )
0 + dF
=
OF
=
also canonical coordinates
(3.20)
PldQ" according
to
(3.10).
It fol-
lows that
pjzdqA
-
PjdQ1'
=
(3.21)
dF
position vari(p,,, qA) (Pl,,-Qll) are the usual canonical form is of transformation determined on a (3.21) the function F by generating [55]. Smooth real-valued functions H on M (i.e. elements of AOM) will
where both ables
and
canonical momentum and
M.
be called classical observables. Exterior
products
mine non-trivial closed 2k-forms
(i.e.
[Wk]
E
on
M
H2k (M; R)). In particular, the 2n-form
of
non-zero
W
with itself deter-
cohomology
classes
3.2
Equivariant Cohomology
dl-ZL defines
a
=
wn /n!
natural volume element
v/-det (X)
d2nX
W
=
on
Symplectic Manifolds
on
47
(3.22)
M which is invariant under canonical
transformations. It is called the Liouville measure, and in the local Darboux coordinates (3.17) it becomes the familiar phase space measure [55]
(_ 1)n(n-l)/2,n /n!
3.2
=
dpl
A
...
Equivariant Cohornology
In this Section
dPn
A
on
A
dql
A
-
-
-
A
dqn
(3.23)
Symplectic Manifolds
specialize the discussion of Chapter 2 to the case a symplectic manifold of dimension 2n. Consider the action of some connected Lie group G on M generated by the vector fields VI with the commutator algebra (2.47). We assume that the action of G on M is symplectic so that it preserves the symplectic structure, we
shall
where the differentiable manifold M is
'CV-W or
in other words G acts
closed this
means
on
M
-+
potential
M be
a
(3.24)
0
by symplectic
transformations. Since
W
is
that
div.w Let L
=
complex
=
(3.25)
0
line bundle with connection 1-form the
symplectic
0. If 0 also satisfies
Lv,O
=
(3.26)
0
G-invariant, and aca G-equivariant bundle. By definition (see Section 2.4) the structure group of this symplectic line bundle acts by canonical transformations. As such, w represents the first Chern class of this U(I)-bundle, and, if M is closed, it defines an integer 2 cohomology class in H (M; Z) (as the Chern numbers generated by w are then the associated covariant derivative V
cording
then
to the
general
=
d + 0 is
discussion of Section 2.4 this defines
integers).
The associated moment map H : M --+ g* evaluated on a Lie algebra element X E g with associated vector field V is called the Hamiltonian
corresponding
to
V, Hv
From
(3.6)
and
(3.26)
=
Cv
-
[iv, V]
equivalently
map since
w
ivO
=
W'Op
this follows from the
=
-ivw
(3.28)
general property (2.86) of the moment coordinates, this
is the curvature of the connection 0. In local
last equation reads
(3.27)
it then follows that
dHv or
=
48
3. Finite-Dimensional Localization
,01,Hv(x) In
particular,
Theory
for
Dynamical Systems
V'(x)w,,,(x)
==
(3.29)
the components H' of the moment map H
Oa
=
(9 Ha
(3.30)
-iv.w
(3.31)
satisfy dHa
=
Comparing symplecticity (3.25) on the group action, we see that this is equivalent to the statement that the closed 1-forms iv.w with the
exact. If
condition
H'(M; R)
0 this is
certainly true, but in the following we phase spaces as well. We therefore multiply this from the exactness onset on the action of G on M, impose requirement i.e. the equivariance requirement (3.26) on the symplectic potential 0. When such a Hamiltonian function exists as a globally-defined C'-map.on M, we are
=
will want to consider
connected
shall say that the group action is Hamiltonian. A vector field V which satisfies (3.28) is said to be the Hamiltonian vector field associated with HV, and we
triple (M, w, HV), i.e. a symplectic manifold with a Hamiltonian it, a Hamiltonian system or a dynamical system, The, integral curves (2.46) defined by the flows (or time-evolution) of a Hamiltonian vector field V as in (3.29) define the Hamilton equations of shall call the G-action
on
motion
V'(t)
=
W "(X(t))a'HV(X(t))
JX"' HVJ"'
=
(3.32)
The Poisson bracket of the Hamiltonian with any other function f determines the (infinitesimal) variation (or time-evolution) of f along the classical
trajectories of the dynamical system (compare with (2.55)),
f f, Hvl,,
=
LVf
In the canonical coordinates defined
4A which
are
o9H
d
=
Tt f (X (0)
by (3-17)
the
(3.33) t=O
equations (3.32) read
9H
.
=
I
api"
PA
=
(3.34)
-
o9qA
the usual form of the Hamilton equations of motion encountered in
elementary classical mechanics [55]. Thus we see that the above formalisms for symplectic geometry encompass all of the usual ideas of classical Hamiltonian mechanics in a general, coordinate-independent setting. The equivariant curvature of the above defined equivariant bundle is given
by
the equivariant extension of the W
and evaluated
on
X E g
we
9
symplectic 2-form,
(,,+Oa
=
o H
a
(3.35)
have
(Dgwg) (X)
=
(d
-
iv) (w
+
Hv)
=
0
(3.36)
3.2
which is
equivalent
Equivariant Cohomology
(3.28)
to the definition
Symplectic Manifolds
on
49
of the Hamiltonian vector field
V. In fact, the extension (3.35) is the unique equivariant extension of the symplectic 2-form w [129], i.e. the unique extension of W from a closed 2-form
equivariantly-closed one. Thus, we see that finding an equivariantlyequivalent to finding a moment map for the G-action. If w defines an integer cohomology class [w] E H 2 (M; Z), then the line bundle L M introduced above can be thought of as the prequanturn line bundle of geometric quantization [172], the natural geometric framework (in terms of symplectic geometry) for the coordinate independent formulation of quantum mechanics. Within this framework, the equivariant curvature 2-form WV to
an
closed extension of w is
--3,
=
wg (X) about
above is refered to some
of the
general
(3.26) (or (3.27))
that if
the prequantum operator. We shall say more ideas of geometric quantization later on. Notice
as
does
potential for the equivariant From
(3.31)
hold, then 0
is also the
(3.35),
equivariant symplectic
i.e. Wg DgO. it follows that the Poisson algebra of the Hamiltonians HI is extension
=
given by
Ha, Hb
W
(Va, Vb
,Va,ttVb,v
W
=
b b Va,l'a,,H =,Cv-H
=
-LVbHa
(3.37) From the Jacobi identity (3.16) it follows that the map H' -- Va is since morphism of the Lie algebras (AOM, .1,,) --+ (TM,
VIH',H bJ,, However,
[Va' Vb]
a
homo-
(3.38)
the inverse of this map does not
necessarily define a homomorphism. corresponds to the commutator of 2 group Poisson bracket of the pertinent Hamiltonian
The Hamiltonian function which
generators may differ from the functions
as
fH
H
blo
_Cba is C(Xa, Xb) [77] (see Appendix A), Le.
where Cab of G
a
=
a
=
CQX1 X217 X3) i
+
C([x2) X31 X1)
=
2-cocycle
+
i
fab'H'
+
Cab
(3.39)
in the Lie
C([X3, X11, X2)
algebra cohomology
VX1, X2, X3
0
=
E g
(3.40) If H2 (G)
=
0 then
homomorphism functions
on
we can
set
Cab
between the Lie
=
0 and the map Xa
algebra
--+
Ha determines
g and the Poisson
algebra
a
of Cl-
M.
The appearence of the 2-cocycle Cab in (3.39) is in fact related to the non-invariance of the symplectic potential under G (c.f. eq. (3.26)).
possible
From the
symplecticity (3.24) ,Cv-0
of the group action and
=
(iv-d + diVa)O=
locally in a neighbourhood JV in M locally-defined linear functions consistency condition
the
=
it follows that
(3.41)
a
w
=
dO and Va
9(Xa)
=
-H a +
wherein a
d
(3.31)
34
0. Here
iVaO obey the
50
3. Finite-Dimensional Localization
fH(Xi), g(X2)},,, which follows from
-
IH(X2), g(X,)I,,,
(3.39). However,
Theory
=
for
Dynamical Systems
g([Xi, X2])
if there exists
VXI, X2
(3.42)
E g
locally-defined function
a
f such that
ga then
Of
we can remove
0 +
=
1-form
df
=
fH a, fj"
the functions 9 a
by the canonical transformation 0 --+ symplectic potential Of is G-invariant. Indeed, the
that the
so
Of obeys LV-Of
which
implies
where C is a
a
(3.44)
0
=
H a+C
(3.45)
constant. This constant is irrelevant here because
we can
intro-
function K in M such that
fH KJ,, a
,
and
=
neighbourhood JV,
that in the
jVaOf duce
(3.43)
dim G
a
defining
F
=
f
+ CK
we
Va
=
'ajK
=
(3.46)
1
find
iVaOF
=
Ha
(3.47)
However, notice that the G-invariance (3.47) of the symplectic potential general holds only locally in M, and furthermore the canonical transformation 0 --+ Of above does not remove the functions ga for the entire Lie algebra g, but only for a closed subalgebra of g which depends on the function f and on the phase space M where G acts [65, 123]. In this subspace, the symplectic potential is G-invariant and the identity (3.27) relating the Hamiltonians to the symplectic potential by Ha iv-0 holds (so that 0 is a local solution to the equivariant Poincar6 lemma). In general though, on the entire Lie algebra g, defining ha =-iv.dF in the above we have in
=
iv-0 and then the Poisson bracket
(3.39)
is
only when &b isomorphically to
Thus it is
on
(3.37) implies
that the
=
fab'h'
-
Lv- h b+ LVbh
2-cocycle appearing
in
algebra
H 2(Sl)
=
(3.49)
a
0 for all a, b that the G-action of the vector fields
the Poisson action of the
corresponding on
Hamil-
the Cartan subal-
g (i.e. its maximal commuting subalgebra), since 0. We shall see in Chapter 4 that the dynamical
systems for which the equivariance condition
special
(3.48)
M. Notice that this is certainly true
of the Lie
H 2(U(j))
=
=
tonians H a
gebra
H a+ ha
given by
&b Va lifts
==
class of quantum theories.
(3.27)
holds determine
a
very
3.3 The Duistermaat-Heckman Theorem
3.3
51
Stationary-Phase Approximation
and the Duistermaat-Heckman Theorem We
start examining localization integrals. We shall concentrate
now
space
abelian circle action also
assume
on
for the time
M,
as we
that the Hamiltonian H defined
function. This
means
by dH(p)
are
=
the manifold
formulas for
0,
did in Section 2.6. We shall in the last Section is
a
Morse
that the critical points p of the Hamiltonian, defined isolated and the Hessian matrix of H,
7i(x) at each critical
as
specific class of phase being on the case of an
a
point
p is
a
=
laX1149XI'l
(3.50)
non-degenerate matrix, det
7i(p) 54
i.e.
(3.51)
0
by (3.29) and it represents the phase space M. We shall assume S1. Later here that the orbits (2.46) of V generate the circle group U(1) on we shall consider more general cases. Notice that the critical points of H coincide with zero locus MV of the vector field V. There is an important quantity of physical interest for the statistical mechanics of a classical dynamical system called the partition function. It is constructed as follows. Each point x of the phase space M represents a classical state of the dynamical system which in canonical coordinates is specified by its configuration q and its momentum p. The energy of this state is determined by the Hamiltonian H of the dynamical system which as usual is its energy function. According to the general principles of classical statistical mechanics [1441 the partition function is built by attaching to each point iTH(') and X E M the Boltzmann weight e 'summing' them over all states of the system. Here the parameter iT is 'physically' to be identified with -,3lkB where kB is Boltzmann's constant and,3 is the inverse temperature. However, for mathematical ease in the following, we shall assume that the parameter The
Hamiltonian
action of
some
vector field V is defined
1-parameter
group
on
the
-
T is real. In the canonical
position and
momentum coordinates
we
would
just simply integrate up the Boltzmann weights. However, we would like to obtain a quantity which is invariant under transformations which preserve the
(symplectic)
volume of the phase space M (i.e. those which preserve the (3.32) and hence the density of classical states),
classical equations of motion
and so we integrate using the Liouville measure (3.22) to obtain a canonically invariant quantity. This defines the classical partition function of the dynamical system,
Z(T)
=
f M
,n
n!
e
iTH
d2nX M
-
Vd-et (x) w
e
iTH(x)
(3.52)
52
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
The partition function determines all the usual the
thermodynamic quantities of energies and specific heats, as well in the canonical ensemble of the classical system.
dynamical system [144],
such
as
its free
all statistical averages However, it is very seldom that
as
one can actually obtain an exact closed partition function (3.52) as the integrals involved are usually rather complicated. But there is a method of approximating the integral (3.52), which is very familiar to both physicists and mathematicians, called the stationary-phase approximation [65, 72, 172]. This method is often employed when one encounters oscillatory integrals such as (3-52) to obtain an idea of its behaviour, at least for large T. It works as follows. Notice that for T oo the integrand of Z(T) oscillates very rapidly and begins to damp to 0. The integral therefore has an asymptotic expansion in powers of 11T. The larger T gets the more the integrand tends to localize around its stationary values wherever the function H(x) has extrema (equivalently where 0)1. To evaluate these contributions, we expand both H and the dH(p) Liouville density in (3.52) in a neighbourhood Up about each critical point p E MV in a Taylor series, where as usual integration in Up can be thought of as integration in the more familiar R 2n We expand the exponential of all derivative terms in H of order higher than 2 in the exponential power series, and in this way we are left with an infinite series of Gaussian moment integrals with Gaussian weight determined by the bilinear form defined by the Hessian matrix (3.50) of H at p. The lowest order contribution is just
form for the
--+
=
.
the normalization of the Gaussian are
down
by
powers of
(see (1.2)), while the k 11T compared to the leading
k-th order moments term.
Carrying
out
these Gaussian integrations, taking into careful account the signature of the Hessian at each point, and summing over all points p E MV, in this way we
obtain the standard lowest-order
stationary-phase approximation
to the
integral (3.52), 2,7ri
Z(T)
T
)
n
(-i)' (P)
e
iTH(p)
det (p) et
pEMv
(p)
+
O(11T n+1)
(3.53)
I
A(p) is the Morse index of the critical point p, defined as the number negative eigenvalues in a diagonalization of the symmetric Hessian matrix of H at p. We shall always ignore a possible regular function of T in the large-T expansion (3.53). The higher-order terms in (3.53) are found from the higher-moment Gaussian integrals [157] and they will be analysed in Chapter 7. For now, we concern ourselves only with the lowest-order term in the stationary-phase series of (3.52). where of
Usually
argues that the
one
is minimized
for T
--->
dH(p)
=
order of
oo.
phase will
(the ground state)
concentrate around the
points where
However, the localization
is
properly determined by all points where
0 since the contribution from other extrema turn out to be of the
magnitude
as
H
since this should be the dominant contribution
those from the minima
[72].
same
3.3 The Duistermaat-Heckman Theorem
53
was essentially born in 1982 general class of Hamiltonian the stationary-phase approximation
The field of equivariant localization theory [39] found a
when Duistermaat and Heckman
systems for which the leading-order of gives the exact result for the partition function
the
O(I/Tn+l)
the
(3.53)
correction terms in
Duistermaat-Heckman theorem goes
all
as
(3.52) (i.e. for which vanish). Roughly speaking,
follows. Let M be
a
compact sym-
plectic manifold. Suppose that the vector field V defined by (3.29) generates the global Hamiltonian action of a torus group T (S')' on M (where we 1 for simplicity) Since the critical point set of shall usually assume that m the Hamiltonian H coincides with the fixed-point set MV of the T-action on M we can apply the equivariant Darboux theorem to the Hamiltonian sys=
=
[65].
tem at hand
This
-
of Darboux's theorem tells
generalization
us
that not
local canonical system of coordinates in a neighbourhood of each critical point in which the symplectic 2-form looks like (3.17), but
only
find
can we
a
these coordinates coordinate
can
0 of the so that the origin p,, q1' the fixed point p of the given compact that in these canonical coordinates the torus
further be chosen
=
=
neighbourhood represents
group action
M. This
on
means
(locally) linear and has the form (rotations in each (p,,, ql') plane) [39]
action is
V
(P
=
a -
/t
i qA
-
ql-t
of
n
canonical rotation generators
PEMV
apl,
(3.54)
A,(p) are weights that will be specified shortly. From the Hamilton equations (3.29) it follows that the Hamiltonian near each critical point p
where
can
be written in the
quadratic
form n
H(x)
=
H(p)
+
2
(P2
+
q 2)
(3.55)
by the Hamilton equations of mot e"\, about the critical points, which tion (3.34) are the circles p,, (t), qt(t) gives an explicit representation of the Hamiltonian T-action locally on M and the group action preserves the Darboux coordinate neighbourhood. Thus each neighbourhood integration above is purely Gaussian and so all higher-order terms in the stationary-phase evaluation of (3.52) vanish and the partition function is given exactly by the leading term in (3.53) of its stationary-phase series2. This theorem therefore has the potential of supplying a large class of dynamical systems whose partition function (and hence all thermodynamic and statistical observables) can be evaluated exactly. Atiyah and Bott [9] pointed out that the basic principle underlying the Duistermaat-Heckman theorem is not that of stationary-phase, but rather of In these coordinates the fiows determined -
2
Of T
course
the
proof
contributing
is
in this
completed by showing case
to
(3.53)
-
that there is
for details
we
no
refer to
regular
[39].
function of
54
3. Finite-Dimensional Localization
the
Theory
for
Dynamical Systems
general localization properties of equivariant cohomology that we Chapter. Suppose that the Hamiltonian vector field V generates a global, symplectic circle action on the phase space M. Suppose further that M admits a globally defined Riemannian structure for which V is Killing vector, as in Section 2.5. Recall from the last Section that the symplecticity of the circle action implies that w + H is the equivariant extension of the symplectic 2-form. w, i.e. Dv(w + H) 0. Since integration over the 2n-dimensional manifold M picks up the 2n-degree component of any differential form, it follows that the partition function (3.52) can be written more
discussed in the last
=
as
Z(T)
(3.56)
a
M
where
a
is the
inhomogeneous a
=
17
tT
differential form n
e
iT(H+w)
(iT)k
eiTH
(iT)n
k!
k=O
W
k
(3.57)
whose 2k-form component is a(2k) e iTHWk /(iT)n-k M. Since H + w is 0. Thus we can apply the Berlineequivariantly closed, it follows that Dva =
=
Vergne
localization formula
(2.143) 27ri
Z(T) In the
T
(3.29)
dV(p) so
we
see
to
get
eiTH(p)
E pEMv
which
=
integral (3.56)
Pfaff
(3.58)
(3.58)
dV(p) at
a
critical point p is found
give
w-'(p)li(p)
(3.59)
how the determinant factors appear in the formula
However,
we
choice of
sign when taking
Pfaffian Pfaff
)n
at hand the denominator of
case
from the Hamilton equations
and
to the
have to remember that the Pfaffian also encodes
dV(p)
can
the square root determinant. The by examining it in the
be determined
a
(3.53). specific
sign of the equivariant
Darboux coordinates above in which the matrix
w(p) is skew-diagonal with skew-eigenvalues 1 and the Hessian H(p) which comes from (3.55) is diagonal with eigenvalues iA,,(p) each of multiplicity 2. It follows that in these coordinates the matrix dV(p) is skew-diagonal with skew-eigenvalues iA,(p). Introducing the eta-invariant 77(7-t(p)) of 'H(p), defined as the difference between the number of positive and negative eigenvalues of the Hessian of H at p, i.e. its spectral asymmetry, we find n
,q(li(p))
=
2
sgn
iA,,,(p)
which is related to the Morse index of H at p
by
(3.60)
3.3 The Duistermaat-Heckman Theorem
77('H(p)) Using
the
identity
1
eif(11-1)
=
2n
=
55
(3.61)
2A(p)
-
it follows that
n
sgn Pfaff
14 sgn iA,, (p)
dV(p)
e'i -1'q (71 (p)) 2
=
-
2
n)
-i)'\(P)
=e-'12F\(P)
JL=1
(3.62) substituting (3.59) and (3.62) into (3.58) Duistermaat-Heckman integration formula and
so
27ri
Z(T)
T
)n
(-i)\(P)
e
we
iTH(p)
pEMv
Recall from Section 2.6 that
dV(p)
finally
arrive
at the
d-et(p) aet
(3.63)
(P)
is associated with the
anti-self-adjoint
linear operator LV (p) which generates the infinitesimal circle (or torus) action on the tangent space TpM. From the above it then follows that the complex numbers
4(p)
the Cartan
generators)
are just the weights (i.e. eigenvalues of complex linear representation of the circle (or
(3.54)
introduced in of the
action in the tangent space at p and the determinant factors from appear in terms of them as the products
torus) (3.58)
n
e(p)
as
=
(_,),\(p)/2
if each unstable mode contributes
a
(3.64)
At, (p)
factor of i to the
for
integral
Z(T)
fact, dV(p) which appears in (3.58) is none other Pfaff dV(p) (see than the equivariant Euler characteristic class Ev(Yp) critical each of in M normal bundle the of point p E MV. The Arp (2.100)) the Pfaffian Pfaff
above. In
=
the bundle of points normal to the directions Mv, so that in a neighbourhood near MV we can
normal bundle is defined of the critical
point
set
as
write the local coordinates
By
its
construction,
the terms in
(3-58)
Pfaff define
x
as
dV(p)
=
p + pj_ with p E
is taken
over
MV and
A(p (see
Section
equivariant cohomology class
an
A(p
in
p
i
2.6).
(-=
Arp.
Thus
T-T2n,) U( (.A4).
..'
through nontrivial irreducible representations and we can therefore decompose the normal bundle at p E MV into a direct (Whitney) sum of 2-plane bundles with respect to this group action, From
(3.54)
it follows that the induced circle action
on
is
n
Ar,P
NO')
=
P
(3.65)
A=J
is simply Ev (Np(t,)) (3.54) then implies that the equivariant Euler class of N(t') P of the orientation into account JVp induced by the proper iA1, (p) /2. Taking
Hamiltonian vector field
near x
=
p and the Liouville measure, and
using the
56
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
multiplicativity
of the Euler class under Whitney sums of bundles find that the equivariant Euler class of the normal bundle at p is
[21],
we
n
Ev(N(")) P
Ev(A(p)
=_
e(p)
(3-66)
g=1
which is just the weight product (3-64). Thus, for Hamiltonians that generate circle actions, the I-loop contribution to the classical partition function (i.e. the Duistermaat-Heckman formula in the form
(3.58)) describes the equivphase space with respect to the Hamiltonian circle action on M. The particular value of the Duistermaat-Heckman formula depends on the equivariant cohomology group H 2n(,) (M) of the manifold M. All the localization formulas we shall derive in this Book will be represented by equivariant characteristic classes, so that the partition functions of the physical systems we consider provide representations for the equivariant cohomology of the phase space M. This is a consequence of the cohomological localization principle of Section 2.5. The remarkable cohomological derivation of the Duistermaat-Heckman formula above, which followed from the quite general principles of equivariant cohomology of the last Chapter, suggests that one could try to develop more general types of localization formulas from these general principles in the hopes of being able to generate more general types of integration formulas for the classical partition function. Moreover, given the localization criteria of the last Chapter this has the possibility of expanding the set of dynamical systems whose partition functions are exactly solvable. We stress again that the crucial step in this cohomological derivation is the assumption that the Hamiltonian flows of the dynamical system globally generate isometries of a metric g on M, i.e. the Hamiltonian vector field V is a global Killing vector of cohomology
ariant
g
(equivalently,
out
a
as we
torus T in
M).
dynamical systems be
one
of
our
of the
main
will see, for M compact, the classical flows x(t) trace This geometric condition and a classification of the
for which these localization constraints do hold true will
topics
in what follows. The extensions and
of the Duistermaat-Heckman localization formula and the
of equivariant
cohomology for dynamical systems will Chapter.
applications general formalism
be the focus of the
remainder of this
3.4 Morse There is
Theory
and Kirwan's Theorem
interesting and useful connection between the Duistermaattheory determined by the non-degenerate Hamiltonian H. Morse theory relates the structure of the critical points of a Morse function H to the topology of the manifold M on which it is defined. We very briefly now review some of the basic ideas in Morse theory (see [111] a
very
Heckman theorem and the Morse
Theory and
3.4 Morse
for
a
comprehensive introduction).
Given
Kirwan's Theorem
Morse function H
a
as
57
above,
we
define its Morse series
t' (P)
MH(t)
(3.67)
PEMv
which is
a
finite
because the
sum
non-degeneracy of H implies
that its critical
points are all discrete and the compactness of M implies that the critical point set MV is finite. The topology of the manifold M now enters the the Poincar6 series of
problem through
M, which
is defined
by
2n
PM (t; IP)
=
E dimiF Hk(M; Ip)tk
(3.68)
k=O
where IF is Morse
algebraic field (usually inequality
some
theory
R
or
C).
The fundamental result of
is the
(3.69)
MH (t) ! PM (t; IF)
(3.69) for all fields IF, then we say that H inequality (3.69) leads to various relations perfect between the critical points of H and the topology of M. These are called the Morse inequalities, and the only feature of them that we shall really need in the following is the fact that the number of critical points of H of a given Morse index k > 0 is always at least the number dimR H k(M; R). This puts a severe restriction on the types of non-degenerate functions that can exist as C'-maps on a manifold of a given topology. -1 in the Morse Another interesting relation is obtained when we set t for all fields IP. If is
equality
holds in
Morse function. The
a
=
and Poincar6 series. In the former series
we
get
sgndetH(p)
MH(-l)
(3.70)
pEMv
while
(2.104) shows that in the latter series the result is the Euler charX(M) of M. That these 2 quantities are equal is known as the
acteristic
Poincar6-Hopf theorem, and employing further orem
(2.103)
we
1: PEMv
with
E(R)
the Gauss-Bonnet-Chern the-
find
sgndet?i(p)
=
(41r)nn!
the Euler class constructed from
f E(R)
(3.71)
M a
Riemann curvature 2-form R
M. This relation gives a very interesting connection between the structure of the critical point set of a non-degenerate function and the topology and on
one can also define equivariusing the topological definition of equivariant cohomology [111] which is suitable to the equivariant cohomological ideas that we formulated earlier on. These equivariant generalizations
geometry of the phase
space M. We remark that
ant versions of the Morse and Poincar6 series
58
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
which localize
topological integrals such as (3.71) onto the zero locus of a Mathai-Quillen formalism and its application to the construction of topological field theories [27, 29, 34, 81, 99, 121]. We shall discuss some of these ideas in Chapter 8. In regards to the Duistermaat-Heckman theorem, there is a very interesting Morse theoretical result due to Kirwan [881. Kirwan showed that the only vector field is the basis of the
Morse functions for which the stationary phase approximation can be exact those which have only even Morse indices A(p). This theorem includes the cases where the Duistermaat-Heckman integration formula is exact, and are
under the assumptions of the Duistermaat-Heckman theorem it is a consequence of the circle action (see the previous Section). However, this result is
stronger
even
phase uniformly
series in
as
-
it
means
that when
one
constructs the full
described in the last Section
11T
to the exact
stationary-
[134], if that series partition function Z(T), then the
converges
Morse in-
dex of every critical point of H must be even. From the Morse inequalities mentioned above this furthermore gives a relation between equivariant localization and the
topology of the phase space of interest if the manifold M has cohomology groups of odd dimension, then the stationary phase series diverges for any Morse function defined on M and in particular the Duistermaat-Heckman localization formula for such phase spaces can never give the exact result for Z(T). In this way, Kirwan's theorem rules out a large number of dynamical systems for which the stationary phase approximation could be exact in terms of the topology of the underlying phase space where the dynamical system lives. Moreover, an application of the Morse lacunary principle [111] shows that, when the stationary-phase approximation is exact so that H has only even Morse indices, H is in fact a perfect Morse function and its Morse inequalities become equalities. We shall not go into the rather straightforward proof of Kirwan's theorem here, but refer to [88] for the details. In the following we can therefore use Kirwan's theorem as an initial test using the topology of the phase space to determine which dynamical -
non-trivial
systems will localize in the
sense
of the Duistermaat-Heckman theorem. In
Chapter 7 we shall see the direct connection between the higher order terms in the saddle-point series for the partition function and Kirwan's theorem, and more generally the geometry and topology of the manifold M.
Examples: The Height Rinction
3.5
of We
a
Riemann Surface
some concrete examples of the equivariant localization presented above. One of the most common examples in both Morse theory and localization theory is the dynamical system whose phase
present
now
formalism space is
with g
a
compact Riemann surface Z9 of genus g
'handles')
and whose Hamiltonian hZg is the
(i.e. height
a
closed surface
function
on
Z9
3.5
Examples: The Height
Function of
a
Riemann Surface
59
[29, 85, 111, 153, 157].
For instance, we have already encountered the case S2 in Section 2.1 with the height function h_ro sphere ZO given by (2.1). The symplectic 2-form. is the usual volume form of the Riemann
=
wzo
dcos 0 A
=
(3.72)
do
metric g,,, = J,,, of R in 3-dimensional space. The partition function
induced
by the Euclidean
Z_,o (T)
f
=
e
wzo
3
from the
of S2
embedding
iThzo
(3.73)
ro
is
given by the expression (2.3) which
is
precisely the
value
anticipated
from
the Duistermaat-Heckman theorem. The relative minus sign in the last line -7r of (2.3) comes from the fact that the Morse index of the maximum 0 =
is 2 while that of 0 action
on
=
S2 associated with rigid rotations
corresponding
generating the compact group of the sphere is V a,-, and the ao
0 is 0. The vector field
moment map is
The Poincar6 series of the
just hzo. 3 2-sphere is
2
PS2 (t; 3F)
1: dimiF H (S2; IF)tk k
=
=
I +
t2
(3.74)
k=O
which coincides with the Morse series consistent with Kirwan's
with
even
theorem,
(3.67)
we see
for the
that
hzo
height function h-Vo. Thus, a perfect Morse function
is
Morse indices. Notice that the Hamiltonian vector field V
-
-2ao
here generates an isometry of the standard round metric dO(gdO+sin 20 doodo induced by the flat Euclidean metric of R 3. The differential form (2.123) with this metric is
=
do, which
Now the partition function
as
can
Z_ro (T)
expected be written I
=
-
T
f
d
is ill-defined at the 2
poles
of
S2
as
(
e
iTh zrO
do)
(3.75)
zo
0, 7r, the endreceiving contributions from only the critical points 0 the with for in the of cos explicit eval0, agreement points integration range uation in Section 2.1. The partition function (2.3) represents the equivariant 4 cohornology classes in
thus
3
In
=
general,
if M is
path-connected,
Z and if M is closed, then 4
as we
H2,,(M; Z)
=
always
assume
here, then Ho (M; Z)
Z. The intermediate homology groups
depend on whether or not M has 'holes' in it or not. Equivariant cohomology groups are usually computed using so-called classifying bundles of Lie groups (the topological definition of equivariant cohomology) see [111], for example.
-
60
3. Finite-Dimensional Localization TT2
11u(j) (S
Intuitively, (3.76)
2
)
are
=
for
Dynamical Systems
Z (D Z
(3.76)
follows from the fact that the
single
Lie
algebra generator linearly independent, for any 2 functions fl, f2 E S(u(l)*), the equivariant cohomology classes spanned by the linearly independent generators f, (fl (wzo + Ohzo) and
P E R and the invariant volume form i.e.
Theory
f2 (!P)
of S2
(3.72)
are
-
As
later on, the above example for the Riemann sphere is essentially only Hamiltonian system to which the geometric equivariant localization constraints apply on a simply-connected phase space (i.e. we
shall
see
the
H,(M;Z)
=
0).
The situation is much different
on
a
multiply-connected
phase space, which as we shall see is due to the fact that the non-trivial first homology group of the phase space severely restricts the allowed U(1) group actions on it and hence the Morse functions thereon. For example, consider the case of a genus I Riemann surface [85, 153, 157], i.e. Z' is the 2-torus T 2= S1 x S'. The torus can be viewed as a in the
parallelogram
complex edge the line segment from 0 to 1 along the real axis and the other slanted edge the line segment from 0 to some complex number -r in the complex plane. The number plane
-r
with its
opposite edges identified. We take
as
is called the modular parameter of the torus and
the upper
horizontal
we can
take it to lie in
complex half-plane C+
Geometrically,
-r
=
fz
C: Im
E
>
z
01
(3.77)
determines the inner and outer radii of the 2 circles of the
torus, and it labels the inequivalent complex structures of Z' '. We view the torus embedded in
3-space as a doughnut standing on end on xy-plane and centered symmetrically about the z-axis. If (01) 02) are the angle coordinates on S' x S', then the height function on Z' can be written
the
as
hEi (01, 02) where ri IRe -rl + Im radii of the torus. The =
:.-::
that there is
one
=
do,
a
A
d02
(3.79)
would therefore say that
C+
is the Teichmiiller space
a
in genus 0. This is
Riemann uniformization theorem. We refer to introduction to Teichmiiller spaces in can
(3.78)
simply-connected Riemann surface is a unique complex structure (i.e. a unique way of defin-
ing complex coordinates)
treatment
01) COS 02
2-form
of the torus. The TeichmUller space of so
TCOS
andr2 IRe -rl + 2 Im -r label the inner and outer symplectic volume form on T 2 is just that induced Z' as a parallelogram in the plane with its opposite
WD
point,
+ IM
-r
by the identification of edges identified, i.e. the Darboux
5In algebraic geometry
(r,
r2
be found in
[74].
a
consequence of the celebrated
[111]
and
[145]
for
algebraic geometry, while
an
elementary
a more
extensive
3.5
Examples:
The
Height Function
The associated Hamiltonian vector field for this
of
a
Riemann Surface
61
dynamical system has
com-
ponents T,r 1
,,_r,=-(rj+Im-rcos0j)sin02
Tr2l =Im-rsin0jC0S02 ,z
i
(3.80)
The Hamiltonian
(3.78) has 4 isolated non-degenerate critical points on (011 02) (0, 7r) (top of the outer circle), a minimum at (0, 0) (bottom of the outer circle), and 2 saddle points at (ir, 0) and (ir, ir) (corresponding to the bottom and top of the inner circle, respectively). The S1
x
S1
-
a
maximum at
=
Morse index of the maximum is 2, that of the minimum is 0, and those of the 2 saddle points are both 1. According to Kirwan's theorem, the appearence of odd Morse indices, or equivalently the fact that
H,(Zl; Z)
=
Z E) Z
(3.81)
with each Z
labelling the windings around the 2 independent non-contractable loops associated with each Sl-factor, implies that the Duistermaat-Heckman integration formula should fail in this case. Indeed, evaluating the right-hand side of the Duistermaat-Heckman formula (3.63) gives eiTh_,l (p)
27ri
e(p)
T
PEMv_,, 27ri =
T vII-m
-r
[r
1/2
(1+ e 2iTr2)
+
which for the parameter values iT value 21r
On the other
hand,
e3
an
( v'32
IRe rl-112
e
2iTImr
e
2iTIRe-rl
)1 (3.82)
=
1 and
I + i
-r
sinh 3 + 2 cosh 1
explicit evaluation
gives the numerical
(3.83)
1849.33
of the
partition function gives
21r 21r
ZZ, (T)
=
I I dol d02 0
e
iTh_,l (01,02)
0
(3.84)
27r =
27r
e
r2j do, Jo (iT(r,
+ Im
-r cos
01))
0
with J0 the
regular Bessel function of order 0 [60]. For the parameter values 2117.13 6 contradicting integration in (3.84) gives Z_ri the result (3.83). Thus even though in this case the Hamiltonian h_rl is a above,
a
numerical
All numerical
software
-
,
integrations
package
in this Book
MATHEMATICA.
were
performed using the mathematical
3. Finite-Dimensional Localization
62
perfect Morse function, space here. This argument
can
it doesn't
connected
sum
for
generate any
be extended to the
Riemann surface
hyperbolic is the g-fold
Theory
case
Dynamical Systems
torus action
where the
on
phase
the
phase
space is
a
For g > 1, Z-9 Z1# #_T1 of 2-tori and therefore its first homology group
Z9,
g > 1
[153].
=
...
is 2g
H, (Z9; Z)
=
(
(3-85)
Z
i=1
3
as g doughnuts stuck together on end and standing xy-plane. The height function on Z9 now has 2g + 2 critical points consisting of I maximum, I minimum and 2g saddle points. Again the maximum and minimum have Morse indices 2 and 0, respectively, while those of the 2g saddle points are all 1. As a consequence the perfect Morse function hZg generates no torus action on Z9. The above non-exactness of the stationary-phase approximation (and even worse the divergence of the stationary-phase series for (3.78)) is a consequence of the fact that the orbits of the vector field (3.80) do not generate a global,
It
on
can
be viewed in R
the
compact group action
on
V. Here the orbits of the Hamiltonian
vector field
bifurcate at the saddle points (like the classical trajectories of the simple pendulum which cross each other in figure eights), and we shall see explicitly
generate isometries of any metric on Z1 how this makes the stationary phase series diverge. The extensions
in
Chapter
as
well
as
7
why
its flows cannot
principle to non-compact group actions and to non-compact phase spaces are not always immediate [29]. A version of the Duistermaat-Heckman theorem appropriate to both abelian and non-abelian group actions on non-compact manifolds has been presented by Prato and of the equivariant localization
Wu in
[141].
assumes
This non-compact version of the Duistermaat-Heckman theorem a component of the moment map which is regular and
that there is
bounded from below
(so that the Fourier-Laplace transform Z(T) exists).
The
examples illustrate the strong topological dependence of the dynamical systems to which equivariant localization is applicable. The height function above
restricted to
a
compact Riemann surface
can
only be used
for Duistermaat-
Heckman localization in genus 0, and the introduction of more complicated topologies restricts even further the class of Hamiltonian systems to which the
localization constraints
apply. We shall investigate this phenomenon in
detailed geometric setting later
on
when
we
a more
consider quantum localization
techniques.
3.6
Equivariant Localization and Classical Integrability
In this Section
we
discuss
an
localization formalism and
interesting
integrable dynamical system we mean
connection between the
equivariant
Hamiltonian systems [84, 85]. By an this in the sense of the Liouville-Arnold
integrable
3.6
theorem which is
a
Equivariant Localization and Classical Integrability
generalized,
coordinate
Liouville theorem that dictates when
63
independent version of the classical
given Hamiltonian system will have whose solutions can be explicitly found by integrating a
equations of motion by quadratures [35, 55]. The Liouville-Arnold theorem version of Darboux's theorem and it states that
if
one can
find
defined almost
everywhere
H(1)
functional of
=
is
themselves
essentially a global integrable
a
are
supposed
the
on
phase
JV
=
(3.86)
A
M,
space
such that the Hamiltonian
only [6]. The action variables functionally-independent and in involution, the action variables
to be
f 1,,, 1, J"'
=
i,, (t)
=
(3.87)
0
and from the Hamilton equations of motion
so
a
canonically conjugated action-angle variables (I,,,
f 1'4' OV 10 H
is
Hamiltonian is
(3.32)
f I,,, H (1) J,,
=
it follows that
(3.88)
0
that the time-evolution of the action variables is constant.
Consequently,
that the action variables generate a Cartan subalgebra (SI)l of the Poisson algebra of the phase space, and the 1,, therefore label a set of
(3.87) implies
canonically therefore
on the phase space which are called Liouville tori. is constrained to the Liouville tori, and the system is
invariant tori
The motion of
H(I)
integrable
in the
that
have found
n independent degrees of problems are conserved simple 1,,,'s quantities such as the total energy or angular momentum which generate a particular symmetry of the dynamics, such as time-independence or radial symmetry. The symplectic 2-form in the action-angle variables is
sense
we
freedom for the classical motion. The
w
and the as
=
dI1,
in
do"
A
(3.89)
corresponding symplectic potential which generates the Hamiltonian a global U(1) group action on M as in (3.47) is
the moment map of
OF The connection between
=-
0 + dF
=
integrability
11,do" and
(3.90)
equivariant localization
now
becomes rather transparent. The above integrability requirement that H be a functional of some torus action generators is precisely the requirement of the Duistermaat-Heckman theorem. The
equations of
motion in this
I,, W where
wl'(I)
tem therefore
=
=
1,, (0)
49I,,H(1).
move
along
global
solutions to the Hamilton
case are
1
00 (t)
The classical
=
00 (0)
+ W/' (I)t
trajectories
of the
the Liouville tori with constant
(3.91)
dynamical Sysangular velocity
64
3. Finite-Dimensional Localization
11
=
wt'
(equivalently they
share
Theory
a common
for
Dynamical Systems
period)
which represents
large
a
a
assosymmetry of the classical mechanics. The Hamiltonian vector field ciated with the action variable I,, generates the jL-th circle action component
of the full torus action
on
M, and consequently
any Hamiltonian which is
a
linear combination of the action variables will generate a torus action on M and meet the criteria of the Duistermaat-Heckman theorem. For quadratic functionals of the action variables the associated Hamilto-
higher-order
and
general generates a circle action which does not have a angular velocity on the phase space and the Duistermaat-Heckman formula will not hold. We shall see, however, that modified versions of the Duistermaat-Heckman localization formula can still be derived, so that any integrable model will provide an example of a partition function that localizes. For the height function of S' above, the action-angle variables are a cos 0, 0' Il hZo 0. We shall see some more general (higherin Chapter 5. dimensional) examples nian vector field in constant
=
=
Recall that above
=
-
of the primary assumptions in the localization framework
one
phase space admit
Riemannian metric g which is globally a Riemannian geomeU(1) under the classical is invariant which dynamics of a given Hamilglobally try tonian system is a very strong requirement. A U(I)-invariant metric tensor was
that the
invariant under the
action
on
a
it. The existence of
always exists locally in the regions where H has no critical points. this, introduce local equivariant Darboux coordinates (pi, pn, q1, .
in that
q1.
region
This
.
.
,
To .
.
.
,
see
qn)
in which the Hamiltonian vector field
means
that H
=
p, is taken
as
generates translations in the radius of this equivariant Darboux
coordinate system. The U(1)-invariant metric tensor can then be taken to be any metric tensor whose components are independent of the coordinate q1 which follows from the Killing equations (2.112). However, (e.g. g,,, there may be global obstructions to extending these local metrics to metrics defined globally on the entire phase space in a smooth way. This feature =
just equivalent to the well-known fact that any Hamiltonian system is locally integrable. This is easily seen from the local representation (3.54),(3.55) where we can define p,, I,, sin OP. Then H En=J 12 and 1,, cos OA, q" is
=
V
_
En=J A
a
..
generates
=
rotations of the local coordinate sor
-
angle variables 0" (rigid locally the metric tendepend only on the action variables
translations in the local
neighbourhood).
components g,,, should be taken
to
Then
radially symmetric in the coordinate neighbourhood). However, local integrability does not necessarily ensure global integrability. For the latter to follow, it is necessary that the neighbourhoods containing the conserved charges I,, be patched together in such a way as to yield a complete set of conserved charges defined almost everywhere on the phase space M. Furthermore, global integrability also implements strong requirements on
I,, (i.e.
g is
the behaviour of H in the vicinity of its critical points. As on, the so
isometry
that the
group of
global
a
we
shall
see
later
compact Riemannian manifold is also compact,
existence of
an
invariant metric tensor in the above for
a
3.6
Equivariant Localization and Classical Integrability
compact phase space is equivalent
global
action of
a
(or
circle
65
to the
requirement that H generates the generally a torus). This means that the Cartan element of the algebra of isometries
more
Hamiltonian vector field V is
a
of the metric g (or equivalently H is a Cartan element of the corresponding Poisson algebra). In other words, H is a globally-defined action variable (or a
functional
thereof), so that the applicable Hamiltonians within the framework
equivariant localization determine integrable dynamical systems. Thus it isometry condition that puts a rather severe restriction on the Hamiltonian functions which generate the circle action through the relation (3.28). These features also appear in the infinite-dimensional generalizations of the of
is the
localization formalism above and
Chapters
they
will be discussed at greater
length
in
5 and 6.
We note also that for
an
integrable Hamiltonian
H
we can
explicit representation of the function F which appears in above. Indeed, the function K in (3.46) can be constructed the critical point set of H that
by assuming
that
a
construct
(3.47) locally
action variable
given
and
outside of
I,,
be realized
can
and the condition
(3.47)
(3.93)
becomes
SWOF which is satisfied
explicitly by
aH)-' (ail,
K(I,
is such
(3.92)
0
In this case, the function K
an
(3.90)
Itz
=
aH
ail,
+
JH, Fj,,
H
(3.94)
G(I)
(3.95)
=
by F=K.
(H-11,, 19111 OH)
+
G(I) is an arbitrary function of the action variables. Consequently, in neighbourhood where action-angle variables can be introduced and where H does not admit critial points, we get an explicit realization of the function F in (3.47) and thus a locally invariant symplectic potential OFIn fact, given the equivariantly closed 2-form Kv + f2v introduced in (2.119), we note that f2v is a closed 2-form. (but not necessarily non-
where a
degenerate)
and that the function KV satisfies
dKv as a
consequence of
(2-121) W'
and =
=
(3.96)
-ivf2v
(2.118), respectively.
S-21"49,Kv V
=
wl"c),H
It follows that
(3-97)
66
3. Finite-Dimensional Localization
and
so
the classical
(M, w, H)
and
equations of motion f2v, (M, Kv) coincide 7
for
Dynamical Systems
for the 2 Hamiltonian systems
,
V' (t) This
Theory
=
Jx4, HJ,,
Jx4, KV JS2,
=
(3.98)
that these 2
dynamical systems determine a bi-Hamiltonian strucare interesting consequences of this structure. The first follows from the fact that if H H(I) as above is integrable, then these action-angle variables can be chosen so that in addition KV KV (I) is an integrable Hamiltonian. We can therefore replace H everywhere in (3.92)-(3.95) by the function KV and w by S?V, and after a bit of algebra we find that the 1-form O(V) above which generates S?V satisfies F means
ture. There
2
=
=
Kv
+
Qv
=
DvO(V) F
(3.99)
and likewise H +
(3.100)
DvOF
Since both H +
w and KV + S2V are equivariantly closed, we see that for integrable bi-Hamiltonian system we can solve explicitly the equivariant version of the Poincar6 lemma. The global existence of the 1-forms OF and O(V) is therefore connected not only to the non-triviality of the DeRharn coF homology of M, but also to the non-triviality of the equivariant cohomology
an
associated with the equivariant exterior derivative DV. Note that this derivation could also have been carried out for
an arbitrary equivariant differential 0 with the definition (2.119) (c.f. eq. (2.124)). This suggests an intimate relationship between the localization formalism, and more generally equivariant cohomology, and the existence of bi-Hamiltonian structures for a given phase space. Furthermore, it is well-known that the existence alone of a bi-Hamiltonian system is directly connected to integrability [6, 35]. If the symplectic 2-forms w and QV are such that the rank (1, 1) tensor
1-form
L is
non-trivial, then
one can
=
Qv w-1
straightforwardly L
=
Vi'a,,L
(3.101)
-
=
show
[841
that
[L, dV]
(3.102)
which is just the Lax equation, so that (L, dV) determines a Lax pair [35]. Under a certain additional assumption on the tensor L it can then be shown
[84]
that the
Here
quantities
we assume
that Qv is
non-degenerate
on
M except
ifolds of M of codimension at least 2, since when it is
equations
in
(3.96)
should be considered
as
constraints. On these
the Hamiltonian KV must then vanish in order to
non-singular [6].
possibly degenerate
keep
the
on
subman-
some
of the
submanifolds,
equations of
motion
3.7
Degenerate
Version of the Duistermaat-Heckman Theorem
4, give variables which
are
1
(3.103)
tr D'
=
A
in involution and which
commute with the Hamiltonian H. If these
67
are
quantities
i.e. which
conserved, are
in addition
com-
plete, i.e. the number of functionally independent variables (3.103) is half the phase space dimension, then the Hamiltonian system (M, w, H) is integrable in the
sense
of the Liouville-Arnold theorem. We refer to
[84]
for
more
de-
tails of how this construction works. Therefore the
equivariant localization formalism for classical dynamical systems presents an alternative, geometric approach to the problem of integrability.
3.7
Degenerate Version
of the Duistermaat-Heckman Theorem
Chapter we shall quickly run through some of generalizations of the Duistermaat-Heckman theorem which can be applied to more general dynamical systems. The first generalization we consider is to the case where H isn't necessarily non-degenerate and its critical point
In these last 3 Sections of this the
set M V is
now a
submanifold of M of co-dimension
[9, 19, 21, 23, 24, 39, 121].
In this
case some
evaluation of the canonical localization
r
=
modifications
integral (2.137)
dim M are
which
-
dim M V
required was
in the
used in the
Berline-Vergne theorem with the differential form a given in 0 now vanishes everywhere on MV (because dH everywhere on MV), but we assume that it is non-vanishing in the directions normal to the critical submanifold MV [111]. This defines the normal bundle JVV of MV in M, and the phase space is now locally the disjoint union derivation of the
(3.57).
The Hessian of H
=
M so on
that in
M
a
neighbourhood
near
=
MV IIJVV
MV
we can
(3.104)
decompose the
XA
-
-
A X0 + XA 1
(3.105)
where xO are local coordinates on MV, i.e. V(xo) coordinates on Arv. Similarly, the tangent space at any
=
be
local coordinates
as
decomposed
0, and xi- are local point x near MV can
as
TxM
=
TxMv G TxNv
TxA(v is the space of vectors orthogonal to those in TxMV. decompose the Grassmann variables qt' which generate the algebra of M as
(3.106)
where
We
therefore
exterior
77
where,q`0
generate the exterior
A
770
+,qA1
algebra AMV andqA1 generate AJVV.
can
(3.107)
68
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
Under the usual assumptions used in deriving the equivariant localization principle, it follows that the tangent bundle, equipped with a Levi-Civitaa U(1)-invariant metric tensor g as vector bundle. Recall that the Lie derivative Lv
Christoffel connection r associated with in
(2.114),
induces
is
an
equivariant
non-trivial action of the group on the fibers of the tangent bundle which is mediated by the matrix dV. More precisely, this action is given by a
,Cv and
so
W'al,
dV"iv,.
+
the moment map associated with this
moment map
mann
=
-
(3.108)
dV
equivariant bundle
is the Rie-
[21] /.IV
=
(3.109)
VV
which
as always is regarded as a matrix acting on the fiber spaces. Given the Killing equations for V, this moment map is related to the 2-form S?V by
(S?V)i,,
=
2g,,,\(AV),A
(3.110)
and the equivariant curvature of the bundle is
Rv
=
(3.111)
R + /-tv
where the Riemann curvature 2-form of the tangent bundle is 1
R',',
2
R'
(3.112)
and
RPI,,,
=
0'
are
01,FP,
-
al_ P,
+ FP F\ 14A vo-
-
V'\ FP VA
(3.113)
tta
-
the components of the associated Riemann curvature tensor R dr+rA the normal bundle from the inherits a decomposition that, (3.106), =
.P. Note
U(I)-invariant
TM, and the curvature and moment map on corresponding objects defined on TM. Given these features of the 2-form S?V, it follows that the generators 7710' AMV satisfy
TjVV of
are
connection from
just the
restrictions of the
(f2V)I,v(xo)?7v0
=
2(gI,.\o9vV\)(xo)77v0
=
(3.114)
0
Mv. For large s E R+ in a neighbourhood of Mv, exponentially and so, in the linearization (3.105) of the coordinates perpendicular to MV wherein we approximate this neighbourhood with a neighbourhood of the normal bundle JVV, we can extend the integration over all values of x 1 there. We now introduce the scaled change of integration variables dxI lie ql0 0 the integral (2.137)
since
xg
-
=
in
a
direction cotangent to
will localize
xA xA1 --4x' + xI 0 0 + 1 /-vFs
to
,
q"
=
q' 0 +,ql 1
-->
ql0
+
77'1 1- Fs
(3.115)
expand the argument of the large-s exponential in (2.137) using the decompositions (3.115). The Jacobian determinants from the anticommuting and
Degenerate Version of the Duistermaat-Heckman Theorem
3.7
69
variables and the commuting x1f variables cancel each other, and so the integral (2.137) remains unchanged under this coordinate rescaling. A tedious but straightforward calculation using observations such as (3.114) shows that the large-s expansion of the argument of the exponential in (2.137) is given by [121]
nl-z
S
S_+00
-(S?V)A,(xo)nl L n _L 2
V
2
+ 0 (1 /
I
S-00
SnV
-
2
(Qv)O,(xo)R' P(xo)xl'x',q' no
+
VA
I
1
0
(3.116)
"FS)
G'V)1(Xo)(S?V)P'(Xo)X"X' I I +0(11 'Fs) 1A
expanded the C'-functions in (3.116) in their respective Taypoints. Thus with the coordinate change (3.115), the integration over the normal part of the full integration domain
where
have
we
lor series about the critical
A'M
M 0 i.e.
over
(x 1711 )
The result is
an
in
=
(Mv (& A'Mv)
(2.137),
integral
over
(Xv (& A'Arv)
is Gaussian and
(3.117)
be carried out
can
explicitly.
the critical submanifold
27ri
Z(T)
II
d'xo d'770 eiT(H(xo)+w(xo,?7o))
T MvOA'Mv X
Pfaff S?v (xo)
.\/de-t QV (xo) (MV (xo)
+
(3.118)
R(xo, no))
chv(iTw)jm, Ev (R) JArv
27ri
T MV
where
we
have identified the
equivariant Chern and Euler characters (2.95)
and (2.100) of the respective fiber bundles. In (3.118) the equivariant Chern and Euler characters are restricted to the critical submanifold MV, and the determinant and Pfaffian there
taken
are
over
the normal bundle
jvv. Note Mv is
that the above derivation has assumed that the critical submanifold
connected. If MV consists of several connected components, then the formula (3.118) means a sum over the contributions from each of these components. Notice also the role that the
large equivariant cohomological symmetry
of the
dynamical system has played here it renders the Jacobian for the rescaling transformation (3.115) trivial and reduces the required integrations to Gaus-
ones. This symmetry appears as a sort of supersymmetry here symmetry between the scalar x1' and Grassmann 779 coordinates).
sian
There
are
several comments in order here. First of
of discrete isolated
points,
so
that
r
=
2n, then,
all, if MV
(i.e.
a
consists
since the curvature of the
70
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
normal bundle of
a point vanishes and so the Riemann moment map /-IV coincides with the usual moment map dV on TM calculated at that point, the formula (3.118) reduces to the non-degenerate localization formula (3.58) and hence to the Duistermaat-Heckman theorem. Secondly, we recall that
the equivariant characteristic classes in
(3.118) provide representatives of the equivariant cohomology of M and the integration formula (3.118) is formally independent of the chosen metric on M. Thus the localization formulas are topological invariants of M, as they should be, and they represent types of 'index theorems'. This fact will have important implications later on in the formal applications to topological field theory functional integration. Finally, we point out that Kirwan's theorem generalizes to the degenerate case above [88]. In this case, since the Hessian is a non-singular symmetric matrix along the directions normal to MV, we can orthogonally decompose the normal bundle,
with the aid of
some locally-defined Riemannian metric on MV, into positive- and negative-eigenvalue eigenspaces of R. The dimension of the latter subspace is now defined as the index of MV and a
direct
sum
of the
Kirwan's theorem
now
states that the index of every connected
component
of
MV must be even when the localization formula (3.118) holds. The Morse inequalities for this degenerate case [111] then relate the exactness or failure of
(3.118)
as
before to the
dynamical system
homology
of the
to which the formula
function of the torus when the torus is
underlying phase space M. One could be applied is the height viewed in 3-space as a doughnut
(3.118) now
sitting on a dinner plate (the xy-plane). This function has 2 extrema, but they are now circles, instead of points, which are parallel to each other and one
is
minimum and the other is
a
T 2 in this
case
a
maximum. The critical submanifold of
consists of 2 connected
components, T2 V
=
S1 II S1.
3.8 The Witten Localization Formula We have thus far
only applied the localization formalism to abelian group M. The first generalization of the Duistermaat-Heckman theorem to non-abelian group actions was presented by Guilleman and Prato [63] in actions
the
on
where the induced action of the Cartan
subgroup (or maximal torus) finite number of isolated fixed points pi and the stabalizer fg E G : g pi pi I of all these fixed points coincides with the Cartan subgroup. The Guilleman-Prato localization formula reduces the integrals over the dual Lie algebra g* to integrals over the dual of the Cartan subalgebra, using the so-called Weyl integral formula [29]. With this reduction one can case
of G has
only -
a
=
apply
the standard abelian localization formalism above. This procedure of abelianization thus reduces the problem to the consideration of localization theory for functions of Cartan elements of the Lie group G, i.e. integrable Hamiltonian systems. Witten [171] proposed a more general non-abelian lo-
calization formalism and used it to study 2-dimensional
Yang-Mills theory.
3.8 The Witten Localization Formula
In this Section
71
shall outline the basic features of Witten's localization
we
theory. a Lie group G acting on the phase space M, we wish to evaluate partition function with the general equivariant extension (3.35),
Given the
f
ZG
Wn
e-40aOHa
n!
(3.119)
4
where
as
usual the Boltzmann
map of the G-action
on
weights
M. There
are
are
given by the symplectic the dual
regard
2 ways to
moment
algebra
give the 0' fixed values, regarding them as of elements of the values S(g*) acting on algebra elements, i.e. the 01 are complex-valued parameters, as is unambiguously the case if G is abelian [9] (in which case we set 0 -iT in (3.119)). In this case we are integrating with a fixed element of the Lie group G, i.e. we are essentially in the abelian case. We shall see that various localization schemes reproduce features of character functions
01
in
(3.119).
We
can
=
formulas for the action of the Lie group G on M at the quantum level. The other possibility is to regard the Oa as dynamical variables and integrate over them. This case allows a richer intepretation and is the basis of nonabelian localization formulas and the localization formalism in
field
theory. employ
this latter
To
topological
interpretation for the symmetric algebra elements,
we a definition for equivariant integration. The definition (2.127) gives a map on AG.A4 --, S(g*)G, but in analogy with ordinary DeRham integration we wish to obtain a map on AGM --+ C. The group G has a natural
need
G-invariant
isomorphic Haar
it, namely its Haar measure. Since g is naturally tangent space of G at the identity, it inherits from the
measure on
to the
measure a
the definition
natural translation-invariant
we
1
a
=
lim 00
8
a
E
dimG
vol(G) f R 1
d0a
-_L e
2
AGM, where vol(G)
=
Given this measure,
[171]
(0a)2
a=1
m0g*
for
measure.
take for equivariant integration is
f
(3.120)
a
M
fg* r1a doa/21r
fG Dg
is the volume of
the group G in its Haar measure. The parameter s E R+ in (3.120) is used The definition to regulate the possible.divergence on the completion AIM. G
AGM -+ C, and the 01's in it can be regarded on g* such that the measure there coincides e1g in (3.120), with the chosen Haar measure at the identity of G. Setting a with wg the equivariant extension (3.35) of the symplectic 2-form of M, and performing the Gaussian oa-integrals, we arrive at Witten's localization formula for the partition function (3.119),
(3.120)
as
indeed gives
a
map
on
local Euclidean coordinates
=
ZG
-lim
(T7r) S
dim
G12
vol(G) 1 1
M
n
n!
e-9
E.(H' )2
(3.121)
72
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
The right-hand side of (3.121) localizes onto the extrema of the square of the moment map E.(H')'. The absolute minima of this function are the solutions to H 0. The contribution of the absolute minimum to 0' 0 H' =
=
ZG (the dominant contribution for formula
s
oo)
--+
is
given by
a
[171] (2-7r) di G12 vol(G) 'm
ZNin
lim
=
f
-
s-oo
IMO
e' +
Mo
where Mo
=
H-1(0)IG
(or symplectic quotient)
projection we
that takes
is the Marsden-Weinstein reduced
phase space [97] global minima onto A40 is integration in (3.120) (as one can
X
E H-
have assumed that G acts
the over
1
(0)
freely
into its on
-4
H-1(0)
it is the
H- 1 (0) IG, with bundle
equivalence class [x]
H- 1 (0), and (9 is
cohomology group H 4(Mo ; R) that is defined the O's in (3.120) we note that E.(O,)2 /2
restricted to
(3.122)
and the localization of the
a consequence of the G-equivariance of the then integrate over the fibers of the bundle H- 1 (0) -7r
simple cohomological
pullback
of
Mo).
E
Here
certain element of
follows. In
as
E
a
HG4(M),
(9 E H4 (MO;
integrating
so
that when
R).
Therefore the equivariant cohomology class of Ea (0a)2 /2 E HG4(M) is determined by this form e which then serves as a characteristic class of the principal Gbundle H-1(0) ---> Mo. The Witten localization formula can in this way be used to describe the
cohomology
given symplectic G-action
on
some
of the reduced
M. We refer to
[171]
phase
space
Mo of the
for further details of this
construction.
However, the contributions from the other local extrema of Ea(Ha)2, which correspond to the critical points of H as in the Duistermaat-Heckman integration formula, are in general very complicated functions of the limiting parameter s E R+. For instance, in the simple abelian example of Sections 2.1 and 3.5 above where G S2 and H U(1), M hzo is the height function (2.1) of the sphere, the Witten localization formula (3.121) above =
=
=
becomes +1
ZZO
lim
=
s-oo
1/2
_0 f (2-7r
e-s(a-cos 0)2 /2
dcosO
=
liM 8__ 00
(1
_
I+ (S)
_
I- (s))
1
where
(3.123) we
functions
have assumed that
Jal
<
1, and 1(s)
are
the transcendental
error
[60] 00
1/2
1(S)
dX
e-'(a-X)2 /2
(3.124)
The 3 final terms in
(3.123) are the anticipated contributions from the 3 critical points of h20 (cos 0 a)2 the absolute minimum at cos 0 a Z contributes +1, while the local maxima at cosO 1 contribute negative terms -I to the localization formula. The complicated error functions arise =
-
-
=
=
3.8 The Witten Localization Formula
because here the critical point at
cos
0
=
a
is
a
73
degenerate critical point of
The appearence of these error functions is in marked contrast with the elementary functions that appear as
integral
the canonical localization
in
(2-137).
the contributions from the critical points in the usual Duistermaat-Heckman formula.
Another interesting application of the Witten localization formalism is can be used to derive integration formulas when the argument of the Boltzmann weight in the partition function is instead the square of the that it
moment map. This
by reversing the arguments which
be done
can
led to the
(3.121), and further localizing the Duistermaat-Heckman type integral (3.119) using the localization principle of Section 2.5. The result (for finite s) is then a sum of local contributions E,,, Z,,'(s), where the functions Z (s) can only be determined explicitly in appropriate instances [78, 171, 1731 (see the simple abelian example above). Combining these ideas localization formula
..
together,
we
arrive at the localization formula
iT)
I
vol(G)
dim
G12
ir
n
e
n!
(H a)2
iT
M dim
1
vol(G)
=
f
W
-
vol(G)
Gd Oa
II
gj.
e-
I
UT
a=1
--c-
g*
n!
e-
M
j
lim
n
dimG
11 a=1
do'
e-
4iT
27r
I
n
n!
e- OIOH'-sD,,A
M
(3.125) applied the localization principle to the Duistermaat-Heckman type integral over M on the right-hand side of the first equality in (3.125). The localization 1-form. A is chosen just as before
where A E
AlG M and
we
have
using a G-invariant metric on M and the Hamiltonian vector field associated with the square of the moment map. In Chapter 8 we shall outline how the formal infinite-dimensional generalization of this last localization formula can be used to evaluate the
partition
function of 2-dimensional
Yang-Mills theory
[1711. Finally, we point out the work of Jeffrey and Kirwan [78) who rigorously derived, in certain special cases, the contribution to Zg from the reduced H- 1 (0) IG in (3.122). Let Hc c G be the Cartan subgroup phase space MO of G, and assume that the fixed points p of the induced Hc-action on M are isolated and non-degenerate. Then for any equivariantly-closed differential form a of degree dim MO in AGM, we have the so-called residue formula [781 =
1
.A40
Res
almo PEMHC
e-O'OH'.(p)
(11,,0) a(0) (p) e(p)
(3.126)
74
3. Finite-Dimensional Localization
Theory
for
Dynamical Systems
where 0 are the roots associated to He C G (the eigenvalues of the generators of He in the adjoint representation of G), and Res is Jeffrey-Kirwan-Kalkman
residue, defined
as
the coefficient of
V1.0 where 0 is the element of the symmetric
algebra S(g*) representing the induced He-action on M (see [781 and (821 for its precise definition). This residue, whose explicit form was computed by Kalkman [82] (for some more recent results see [135, 163]), depends on the
MHc of the He-action on M and it can be expressed weight determinants e(p) in (3.64) of the He-action and 0, (& Ha(p). It is in forms similar to (3.126) that the H(p)
fixed-point
set
in terms of the
the values
=
generalizations of the Duistermaat-Heckman theorem due to Guilleman and Prato appeared [631. The residue formula can explicitly be used to obtain information about the cohomology ring of the reduced phase space Mo above [78, 82]. This is particularly useful in applications to topological gauge theories (see Chapter 8). first non-abelian
3.9 The Wu Localization Formula
generalization of the Duistermaat-Heckman theorem that
The final
present here is an interesting application, due to Wu ization formula in the form (3.125) when applied to
[173], a
we
shall
of Witten's local-
global U(I)-action
on
M. This yields a localization formula for Hamiltonians which are not themselves the associated symplectic moment map, but are functionals of such an observable H. This is
ZU (1) (T)
=
I
,n
n!
accomplished eiTH
via the localization formula
2
27r
1/2
Jim
47riT
I
do
0
e-
477
o2
I
n
n!
e- OOH-aD,.,(1)A
M
(3.127) right-hand side of (3.127) is just that which appears in the canonical localization integral (2.128) used in the derivation of the Duistermaat-Heckman formula. Working this out just as before and performing the resulting Gaussian 0-integral yields Wu's localization formula for The final
integral
circle actions
on
the
[1731 00
ZU(1) (T)
=
(27r)n (n 1)! -
E Fp) Ids sn-1 pEMV I
.
-
0
6iT(3+1H(p) 1)2
+
I
e'+iF/4T
IMO
Mo
(3.128) where F
=
abelian gauge connection on the (nonH-'(0) --+ Mo. The formula (3.128) can be
dA is the curvature of
trivial) principal U(I)-bundle used to determine the
symplectic
an
volume of the Marsden-Weinstein reduced
3.9 The Wu Localization Formula
phase
space
Mo
[173].
This gives
an
alternative localization for Hamiltonians
which themselves do not generate an isometry of some metric g on are quadratic in such isometry generators. As we shall see, the path
generalizations problems.
of Wu's formula
75
are
rather
important for
certain
M, but integral physical
Quantum Localization Theory for Phase Space Path Integrals
4.
In quantum mechanics there are not too many path integrals that can be evaluated explicitly and exactly, while the analog of the stationary phase ap-
proximation, i.e. the semi-classical approximation, can usually be obtained quite readily. In this Chapter we shall investigate the possibility of obtaining some path integral analogs of the Duistermaat-Heckman formula and its generalizations. A large class of examples where one has an underlying equivariant cohomology which could serve as a structure responsible for localization is provided by phase space path integrals, i.e. the direct loop space analogs of (3.52). Of course, as path integrals in general are mathematically awkward
objects, the localization formulas that we will obtain in this way really definite predictions but rather suggestions for what kind of results to expect. Because of the lack of rigor that goes into deriving these localization formulas it is perhaps surprising then that some of these results are not only conceptually interesting but also physically reasonable. Besides these there are many other field-theoretic analogies with the functional integraJ generahzation of the Duistermaat-Heckman theorem, the common theme being always some underlying geometrical or topological structure which is ultimately responsible for localization. We have already mentioned one of these in the last Chapter, namely the Witten localization formula which is in principle the right framework to apply equivariant localization to a cohomological formulation of 2-dimensional quantum Yang-Mills theory (see Chapter 8). Another large class of quantum models for which the are
not
Duistermaat-Heckman theorem metric quantum mechanics
[8].
seems
to make
This formal
sense
is N
=
application, due
.1 2
supersym-
Atiyah and a path integral Chapter exists. to
Witten, was indeed the first encouraging evidence that such generalization of the rigorous localization formulas of the last Strictly speaking though, this example really falls into the category of the Berline-Vergne localization of Section 2.6 as the free loop space of a configuration manifold is not quite a symplectic manifold in general [21]. More generally, the Duistermaat-Heckman localization can be directly generalized to the infinite-dimensional case within the Lagrangian formalism, if the loop space defined
over
the
configuration space has on it a natural symplectic example, for geodesic motion on a Lie group of based loops is a Kiihler manifold [142] and the space
structure. This is the case, for
manifold, where the
R. J. Szabo: LNPm 63, pp. 77 - 125, 2000 © Springer-Verlag Berlin Heidelberg 2000
78
4.
Quantum Localization Theory for Phase Space Path Integrals
stationary phase approximation
is well-known to be exact
[38, 146].
This
by Picken [139, 140]. We will discuss these specific applications in more detail, but we are really interested in obtaining some version of the equivariant localization formulas available which can be applied to non-supersymmetric models and when the partition functions cannot be calculated directly by some other means. The Duistermaat-Heckman theorem in this context would now express something like the exactness of the one-loop approximation to the path integral. These functional integral formulas, and their connections to the finite-dimensional formulas of Chapter 3, will be discussed at length in this Chapter. The formal techniques we shall employ throughout use ideas from supersymmetric and topological field theories, and indeed we shall see how to interpret an arbitrary phase space path integral quite naturally both as a supersymmetric and as a topological field theory partition function. In the Hamiltonian approach to localization, therefore, topological field theories fit quite naturally into the loop space equivariant localization framework. As we shall see, this has deep connections with the integrability properties of these models. In all of this, the common mechanism will be a fundamental cohomological. nature of the model which can be understood in terms of a supersymmetry allowing one to deform the integrand without changing the integral. formal localization has been carried out
4.1 Phase We
Space Path Integrals
begin this Chapter by deriving the quantum mechanical path integral for
bosonic quantum system with no internal degrees of freedom. For simplicity, 1 degree of freedom in Darboux cowe shall present the calculation for n a
=
plane R 2. The extension to n > I will then be immediate, and then we simply add the appropriate symplectic quantities to obtain a canonically-invariant object on a general symplectic manifold M to ensure invariance under transformations which preserve the density of states. To transform the classical theory of the last Chapter into a quantum mechanical one (i.e. to 'quantize' it), we replace the phase space coordinates (p, q) with operators (p, 4) which obey an operator algebra that is obtained by replacing the Poisson algebra of the Darboux coordinates (3.18) by allowing the commutator bracket of the basis operators (p, 4) to be simply equal to the Poisson brackets of the same objects as elements of the Poisson algebra of Cl-functions on the phase space, times an additional factor of ih where ordinates
on
M,
i.e.
we
essentially
carry out the calculation
on
the
h is Planck's constant,
[fi, 4]
=
ih
(4.1)
with the canonical commutation relation (4.1) make the The operators space of Cl-functions on M into an infinite-dimensional associative opera,
4.1 Phase tor
algebra
called the
Space Path Integrals
Heisenberg algebra'. This algebra
can
be
79
represented
the space V(q) of square integrable functions of the configuration space coordinate q by letting the operator 4 act as multiplication by q and P as the on
derivative
P
=
a
i
(4.2)
,Oq
This representation of the Heisenberg algebra is called the Schr6dinger picture and the elements of the Hilbert space V(q) are called the wavefunctions or
physical quantum states of the dynamical system'. The eigenstates of the (Hermitian) position and momentum operators denoted by the usual Dirac bra-ket notation
41q) These states
are
they obey
PIP)
=
6(q
=
-
q')
(PIP')
=
the momentum and position space 00
f dp 1p) (pI f dq lq)(ql -00
(4.2) are
on
identity operator by
6(p
-
P')
(4.4)
completeness relations
=
(4.5)
1
-00
on
the respective space. In the representation
L 2-functions the momentum and
related
(4.3)
00
=
with 1 the
PIP)
orthonormal,
(qlq') and
q1q)
=
are
configuration
space
representations
the usual Fourier transformation 00
1q)
=
f
e-ipqlh 1P)
(4.6)
00
which identifies the matrix element
(p1q)
=
(qlp)*
h v/'2-7r
e-ipq/h
(4.7)
The basis operators have the matrix elements
(pIdIq) All observables tian operators
=
(i.e.
&Iq)
(p1p1q)
=
p(p1q)
real-valued C'-functions of
acting
on
the Hilbert space. In
=
ih
(p, q))
a
-5q
(p1q)
now
(4.8)
become Hermi-
particular, the Hamiltonian
of
precisely, the operators (fi, 4) generate the universal enveloping algebra algebra which is usually identified as the Heisenberg algebra. 2Strictly speaking, these function spaces should be properly defined as distribution spaces in light of the discussion which follows. More
of
an
extended affine Lie
80
4.
Quantum Localization Theory for Phase SPace Path Integrals
dynamical system
the
now
becomes
H(P, 4)
Hermitian operator
a
with
the matrix elements
(pIft1q) and the
=
H(p, q) (p1q)
H(p, q)
=
e
-ipqlh
(4.9)
v/-2-rh 7
eigenvalues of this operator determine the energy levels
of the
physical
system. The time evolution of any quantum operator is determined by the quanmapping above of the Hamilton equations of motion (3.32). In particular,
tum
the time evolution of the position operator is determined '
qW
which may be solved
=
;,-h
by
ft
(4.10)
formally by
d(t)
=
e'ftt/hd(O) e-'kt/h
(4.11)
by a unitary transformation of the position operator d(O). Schr6dinger representation, we treat the operators as time-independent quantities using the unitary transformation (4.11) and consider the time-evolution of the quantum states. The configuration of the system at a time t is defined using the unitary time-evolution operator in (4.11) acting on an initial configuration 1q) at time t 0, so
that the time evolution is determined In the
=
1q, t) which is
an
=
e%Ht1h 1q)
(4.12)
eigenstate of (4.11) for all t. physical quantity is the quantum propagator
An important
IC(q', q; T)
=
(q', Tjq, 0)
=
(q'I e-iftT/h 1q)
(4.13)
which, according to the fundamental principles of quantum mechanics [101], represents the probability of the system evolving from a state with configuration q to one with configuration q' in a time interval T. The propagator (4.13) satisfies the Schr6dinger wave equation ih
a
o9T
IC(q, q; T)
=
kIC(q', q; T)
(4.14)
where the momentum operators involved in the Hamiltonian ft on the righthand side of (4.14) are represented in the Schr6dinger polarization (4.2). The
Schr6dinger equation
is to be solved with the Dirac delta-function initial
condition
IC(q', q; T
=
0)
=
S(q'
-
q)
(4.15)
The function IC (q', q; T) acts as an integration kernel which determines the time-evolution of the wavefunctions as
4.1 Phase
Space
Path
Integrals
81
00
!P(ql; T)
f dq JC(q,
=
T)TI(q; 0)
q;
(4.16)
00
TI(q; t)
where
=-
(q, tITI)
are
the
time-dependent configuration
space represenof the system. Thus the propagator represents the fundamental quantum dynamics of the system and the stationary state solutions to the Schr6dinger equation (4-14) determine the energy eigen-
physical
tations of the
states
ITI)
values of the
dynamical system. phase space path integral provides
The
a functional representation of the quantum propagator in terms of a 'sum' over continuous trajectories on the phase space. It is constructed as follows [147]. Between the initial and final configurations q and q' we introduce N 1 intermediate configurations -
qO,
.
.
.
,
qN with qo
=_
q and qN
q',
=_
and each
,At
Introducing
separated by the
time interval
TIN
=
(4.17)
intermediate momenta p,.... 7 PN and
inserting the completeness
relations 00
f dqj-l dqj dpj Jqj)(qjJpj)(pjJqj_1)(qj_1J
N
(4.18)
'cc
into the matrix element
)C (q', q; 00
T)
=
(q'I (
e-
(4.13)
obtain
we
iftAt1h) N I q)
N
fldqj-l
e- ft`1t14Jpj)(pjJqj_1)(qj_1Jq)
dqj dpj (q'lqj)(qjl
(4.19)
j=1 00
f -C,
N-1
11
N
dqj
j=1
11 j=1
dpj 27rh
N
i
exp,
E h
-
i=1
xJ(qo where
we
-
(Pi
qi
qi_1 -
-at z
q)J(qN
have used the various identities
-
-
H(pi, qi)
At
q')
quoted above.
In the limit N
-->
00,
equivalently At 0, the discrete points (pj, qj) describe paths (p(t), q(t)) in the phase space between the configurations q and q', and the -sum in (4.19) or
--
becomes the continuous limit of time
a
Riemann
sum
representing
a
discretized
integration. Then (4.19) becomes 00
IC (q', q;
T)
=
lim N--).oo
f -,,
N-1
11
N
dp "3
*
dqj
j=1
xJ(q(O)
11 2,7rh
exp
j=1 -
q)J(q(T)
-
1
T
-i h
f
dt
(p(t)4(t)
-
H(p, q))
0
q') (4.20)
82
Quantum Localization Theory
4.
for Phase
Space Path Integrals
exponential in (4.20) is just the classical acdynamical system, because its integrand is the usual Legendre transformation between the Lagrangian and Hamiltonian descriptions of the classical dynamics [55]. Notice also that, in light of the Heisenberg uncer21rh, the normalization factors 27rh there can be tainty principle AqAp physically interpreted as the volume of an elementary quantum state in the phase space. The integration measure in (4.20) formally gives an integral over all phase space paths defined in the time interval [0, T]. This measure is denoted by Note that the argument of the
tion of the
-
N
[dp dq]
=
N-1
- P-j
11 27rh fI
lim N--+oo
j=1
and it is called the
Feynman as
a
tE[O,T]
3=
is to be understood
on
2,7rh
dq (t)
equality
The last
measure.
'measure'
dp (t)
H
dqj
(4.21)
means
that it
the infinite-dimensional functional
trajectories (p(t), q(t)), where for each fixed time slice [0,T], dp(t) dq(t) is ordinary Riemann-Lebesgue measure. Being an infinite-dimensional quantity, it is not rigorously defined, and some special care must be taken to determine the precise meaning of the limit N -+ 00 above. This has been a topic of much dispute over the years and we shall make no attempt in this Book to discuss the ill-defined ambiguities associated with the Feynman measure. Many rigorous attempts at formulating the path integral have been proposed in constructive quantum field theory. For instance, it is possible to give the limit (4.21) a somewhat precise meaning using the which assumes that the paths which so-called Lipschitz functions of order .1 2
phase
space of
t
space
E
(4.20)
contribute in are
called Wiener
(4.21)
measure
is
grow
no
integrals)'.
supported
faster than
O(vt) (these
functional
integrals
assume that the integration space paths and that the quantum
We shall at least
on
C"O
phase
mechanical propagator given by (4.20) is a tempered distribution, i.e. it can diverge with at most a polynomial growth. This latter restriction on the path
integral
is
part of the celebrated Wightman axioms for quantum field theory
rigorous manipulations theory However, a physicist will typically proceed without worry and succeed in extracting a surprising amount of information from formulas such as (4.20) without the need to investigate in more detail the implications of the limit N oo above. To actually carry out functional integrations such as (4.20) one uses formal functional analogs of the usual rules of Riemann-Lebesgue integration in the straightforward sense, where all time integrals are treated
which allows
to at least carry out certain formal of distributions.
one
from the
--
as
continuous
regarded
as a
sums
on
the functional space
continuous
trajectories
can
the time parameter t is
multiple integral representation in (4.19) to phase space paths requires that these approximated by piecewise-linear functions.
Note that the transition from the the representation
(i.e.
index).
(4.20),(4.21)
at least be
in terms of
4.1 Phase
If
set q
we
=
q'
and integrate
over
Space Path Integrals
all q, then the left-hand side of
83
(4.20)
yields
i dq (ql e-ikTlhlq)
e-ifIT/hll
11
tr
_=
dE
e-iET/h
(4.22)
00
where E
are
the energy eigenvalues of ft and the symbol will be used to matrix of interest is considered as an infinite dimensional
emphasize that the one over
either the Hilbert space of physical states hand, the right-hand side of
space. On the other
or
the functional
(4.20)
trajectory
becomes
T
Z(T)
[dp dq]
(p(t)d(t)
dt
exp
-
H(p, q))
6(q(O)
-
q(T)) (4.23)
0
which is called the quantum partition function. From (4.22) we see that the quantum partition function describes the spectrum of the quantum Hamiltonian of the
dynamical system and that the poles
of its Fourier transform
00
G(E)
I
=
dT
eiETIhZ(T)
(4.24)
0
give the bound
state
[101].
spectrum of the system
The quantity
(4.24)
is
ft)-111
other than the energy Green's function G(E) which trJJ(E is associated with the Schr6dinger equation (4.14). Thus the quantum partition function is in some sense the fundamental quantity which describes the none
=
quantum dynamics
Finally,
(i.e.
the energy
dimension 2n is
spectrum)
of
a
Hamiltonian system.
arbitrary symplectic manifold (M, W) of immediate. The factor pd becomes simply p,,41' in higher
generalization
the
-
to
an
dimensions, and, in view of (3.19), the canonical form of this is 0,, (x): 4 in an arbitrary coordinate system on M. Likewise, the phase space measure dp A dq according to (3.23) should be replaced by the canonically-invariant Liouville measure (3.22). Thus the quantum partition function for a generic dynamical system (M, w,
H)
is defined
as
[dAL(x)] e'slxl
Z(T) LA4
[d2nXI
J1 V/det 11w(x(t))JI tE
LM
e'slxl (4.25)
[0,T]
where T
S[X]
f
dt
(0,,(x):V'
-
H(x))
(4.26)
0
is the classical action of the Hamiltonian
shall set h
=-
I for
simplicity, and
system. Here and in the following
the functional
integration
in
(4.25)
we
is taken
4.
84
over
the
x(t)
:
Quantum Localization Theory
for Phase
Space Path Integrals
M, i.e. the infinite-dimensional space of paths obeying periodic boundary conditions x11(O) xA(T). Although much of the formalism which follows can be applied to path integrals over the larger trajectory space of all paths, we shall find it convenient to deal mostly with the loop space over the phase space. The partition function (4.25) can be regarded as the formal infinite-dimensional analog of the classical integral (3.52), or, as mentioned before, the prototype of a topological field theory functional integral regarded as a (0 + l)-dimensional quantum field theory. In the latter application the discrete index sums over y contain as well integrals over the manifold on which the fields are defined. Notice that the symplectic potential 0 appearing in (4.25),(4.26) is only locally defined, and so some care must be taken in defining (4.25) when W is not globally exact. We shall discuss this procedure later on. Note also that the Liouville measure in (4.25), which is defined by the last equality in (4.21),
loop
[0, T]
space LM of
---
M
differs from that of
=
(4.23)
in that in the latter
case
there is
one
extra
mo-
integration in the phase space Feynman measure (4.21), so that the endpoints are fixed and we integrate over all intermediate momenta. Thus one must carefully define appropriate boundary conditions for the integrations in (4.25) for the Schr8dinger path integral measure in order to maintain mentum
a
formal
elaborate
analogy between on
the finite and infinite dimensional
cases.
We shall
Chapter. Further discussion of this and ordering prescriptions that are needed to de-
this point in the next
the proper discretizations and fine the functional integrations that appear above
can
be found in
[147]
and
[93].
Example: Path Integral Derivation of the Atiyah-Singer Index Theorem
4.2
'
we did at the start of Chapter 2 above, we shall motivate the formal manipulations that will be carried out on the phase space path integral (4.25) with an explicit example which captures the essential ideas we shall need. At the same time, this particular example sets the stage for the analogies with topological field theory functional integrals which will follow and will serve as a starting point -for some of the applications which will be discussed in Chapter 8. We will consider the derivation, via the evaluation of a path integral for supersymmetric quantum mechanics [5, 48], of the Atiyah-Singer index theorem which expresses the fact that the analytical index of a Dirac operator is a topological invariant of the background fields in the quantum field theory in which it is defined. This theorem and its extensions have many uses in quantum field theory, particularly for the study of anomalies and the fractional fermion number of solitons [41, 125, 158].
As
Consider
a
Dirac operator
iY1
on an
even-dimensional compact orientable
Riemannian manifold M with metric g of Minkowski signature,
4.2
Example: Path Integral
Derivation of the
i-Y" Here
-y"(x)
49,4
Atiyah-Singer
1Wttjk [,Yj,
+
yk]
8
Index Theorem
the Dirac matrices which generate the Clifford
are
17 /J,^/' +
-Y,-Yij'
(4.27)
iAj,
+
algebra
of
M,
(4.28)
2g"'(x)
=
85
on a principal fiber bundle E --* M (i.e. a gauge field). simplicity that the structure group of the principal bundle M. The spinis G U(1), so that A is a connection on a line bundle L connection wi, is defined as follows. At each point x E M we introduce a local basis of orthonormal tangent vectors ei(x), called a vielbein, where y labels dim M parametrizes the fibers the basis components in TM and i 1, of TM (i.e. the local rotation index in the tangent space). Orthonormality means that g1"(x)e1,,(x)e3,(x) 71 j is the flat Minkowski metric in TM, or
A,,
and
is
We shall
a
connection
assume
for
--*
=
=
-
-
-
,
,
equivalently 77ij e,, (x) e3, (x)
=
(4.29)
g,,, (x)
-y' ei1(x)-yA(x) in (4.27) and the spin-connection spin bundle SM of M, defined by the dim Mdimensional spinor representations of the local Lorentz group of the tangent
In this vielbein formalism,
(i.e.
connection
bundle)
on
=
the
is
e'V (i9,,Ev +
w"
1-43
Ei"(x)
and
(4.30)
is
under
a
a
are
j.FA,,\&) 3
the inverse vielbein fields, i.e.
Ej"O,,
(4.30)
=
Jj3.
local Lorentz transformation
e' (x) A
-+
spin-connection tangent bundle, i.e.
The
gauge field of the local Lorentz group of the
A (x)ej (x), A(x) A
3
E
SO(2n
-
1, 1), on the frame bundle of M, the gauge field w,, transforms in the usual 1 ajA A- 1. It is defined so that the covariant derivative way as w,, -- AwijAin (4.27) coincides with the Levi-Civita-Christoffel connection, i.e. V,,e'V -
-
% It 71 0. The covariant derivative in (4.27) is in general F,,,,e,\ + w...0L, regarded as a connection on the bundle TM 0 SM (9 L which together define *
i
,9,,e,,
=
-
the twisted
spin complex of M (the 'twisting' being associated with the
presence of the gauge field
A).
representation of the Dirac matrices is that in which the chiral2n 1 2 1 and (-YC)t with the properties (,,c)2 -Y', i-y y ity matrix -yc ^t with commutes Dirac is diagonal. Since the -yc, in this repoperator (4.27) resentation of the Clifford algebra (4.28) these 2 operators can be written in The chiral
=
=
,
the block forms
'Y
The
analytical
=
(D
(0 -1)
index of
iY7
is then defined
dimensions of the kernel and co-kernel of the
index(iV )
=-
dim ker D
-
dim coker D
E)
0
0
1
C
=
as
t
(4.31)
0
the difference between the
elliptic operator D,
dim ker 'D
-
dim ker
Dt (4.32)
86
Quantum Localization Theory
4.
(4.31),
In the chiral representation
the Dirac
SPace Path Integrals
the Dirac spinors, i.e. the solutions Tf of
equation
('Dt Do TTff+ 0
iv Tf are
for Phase
(4.33)
ETI
by their positive and negative chirality spin components P. DTV_ zero mode solutions, E 0, satisfy DtTf+ 0, the index just the difference between the number of positive and negative
determined
Since the
(4.32)
is
=
=
=
chirality zero-mode solutions of the Dirac equation (4.33),
index(i)F)
=
i.e.
(4.34)
rL E=O^fc
Moreover, since [i)7 -ye] 0, the chirality operator provides a one-to-one mapping between positive and negative non-zero energy states. Thus the index (4.34) can be written as a trace over the full Hilbert space 71 spanned by the Dirac spinors as ==
index(i)7) where tion
we
trHII-y'
=
-
e
T(VVt +Vt V)
(4.35)
1
satisfying the eigenvalue Schr6dinger equations
have used the fact that the spinors
(4.33)
also
obey
the
DD4+
=
E
2
DtDP_
T1+
=
2
equa-
(4.36)
f E 0-
The parameter T > 0 in (4.35) regulates the operator trace. The representation (4.34) of the index of a Dirac operator is known
the Witten index
chirality spinors operator
can
[22, 166, 167].
as
bosons and
be written
as
7r-
ciated with
a
can
then be identified with
a
supersymmetry
mapping between fermions and bosons, assosupersymmetric theory with Hamiltonian given by the graded
BRST commutator
(see (4.35) above) H
as
can
(-l)F where F is the fermion (or ghost) num-
=
ber operator. The operator Df generator Q, which provides a
as
identify the positive and negative fermions, respectively, and then the chirality We
is standard in
a
=
JQ,QtJ
(4.37)
> 0
supersymmetric model. This
ment above that bosonic and fermionic states of
is
equivalent
non-zero
to the state-
energy in the
su-
QQt and QtQ are positivealways paired. persymmetric theory definite Hermitian operators, the zero modes 10) of H are supersymmetric, QtJO) 0, and thus they provide a (trivial) 1-dimensional represenQJO) Since
are
=
tation of the
tational.and
=
supersymmetry. Small perturbations of the background gravi0 states, but bosonic: gauge fields g and A may excite the E =
always
and fermionic states must
be lifted in pairs.
Consequently, the Witten
index
index(i)F)
=
tr-Hll(-I)' e"11
=-
stril e"11
(4.38)
4.2
Example: Path Integral Derivation of the Atiyah-Singer Index Theorem
is
topological
a
invariant that is
independent
87
spin and gauge since M is compact by
of the choice of
connections. Here str denotes the supertrace
and,
assumption, there are only finitely many modes which contribute in (4.38). quantities are independent of the parameter T. In the low temperature limit (T --+ oo) only zero modes contribute to (4.38) according to their chirality. Thus all these
physical relevence of the Witten index is immediate. As the zero engeneral need not be paired, the non-vanishing of the Witten index (4.38) implies that there is at least one zero energy state which is then an appropriate supersymmetric ground state of the underlying supersymmetric theory. Thus the non-vanishing of (4.38) is a sufficient condition for the presence of supersymmetric ground states. Conversely, a necessary criterion for dynamical supersymmetry breaking is that TrE=O (_l)F should vanish. Using the standard path integral techniques of the last Section, it is straightforward to write down a path integral representation of (4.38) [5]. The collection of all fields P will clearly involve both bosonic and fermionip degrees of freedom which will be connected by a supersymmetry, i.e. the appropriate path integral representation will be that of a supersymmetric field theory. Furthermore, the integral over the function space of fields on M will be restricted to fields which satisfy periodic boundary conditions for both The
ergy states in
the space and time coordinates, P(t + T) 0(t). This restriction is necessary the reason for this condition in and discussed of states for the pairing above, =
the time direction for the fermionic fields is because of the presence of the Klein operator (-l)F in the supertrace. The path integral representation of the index
(4.38)
[5, 48]
is then
index(iy)
=
f
2n [d 2nX] [d V)] eiT11/2[X,1P1
(4.39)
LMOLA1M
where
0/'(t)
Section
2.6,
are can
anticommuting periodic paths be taken to lie in
LA'M)
on
with
M (which, according to path integration defined
using functional analogs of the Berezin integration rules discussed in Section 1 2.6. The action in (4.39) is that of N supersymmetric quantum 2 (Dirac) mechanics, i.e. the invariant action for a spinning particle in background 4 gravitational and gauge fields =
11/2 [X 01
1 ( 1g1,,V':t' +V'A,, d-r
=
2
1 +
-
2
2
S1
(4.40) 4
In
general
an
N-component supersymmetric model
contains N fermion chiral
with 2N associated superpartner bosonic fields Fi, and conjugate pairs N corresponding supersymmetry charges (Qit, Qi) which mix the fields with their
superpartners.
88
Quantum
4.
Localization
Theory
for Phase
Space Path Integrals
01' are the Grassmann (superpartner) coordinates for the particle configurations x ' E M, and we have rescaled the time by T so that time inte-
where
grations lie
on
the unit circle S1. Here
V,W"(x(-r))
aW"(x(,r))
=
+
r','
(4.41)
along the loop x(-r) induced by the Riemannian M, and F,,, 91,A, o%Aj, is the gauge field strength tensor. In (4.40), the particle current :V1 is minimally coupled to the gauge field A/-,, and its spinor degrees of freedom couple to the electromagnetic field of A, by the usual Pauli magnetic moment interaction. The action (4.40) has is the covariant derivative connection V
the
on
=
-
(infinitesimal) supersymmetry Sx"(-r)
(4.40)
The action
symmetric model in and superfields. us
briefly
(4.39) (see [71] resented of M is
SV)A(T)
arises from the standard
Zumino) sigma-model,
Let
V)"(T)
=
and
a more
we
see
in
OT
details).
always 04
=
Ei"O'.
therefore be realized
S
=
Chapter
8 how to write this super-
=
gl"(X)
(4.43)
The zero-mode equation for the Dirac operator the graded constraint equation
as
a,,
10A
path integral representation algebra of the spin bundle is rep77'j, so that the Clifford algebra
arrives at the
one
by the anticommutator [V)', V5j] + represented is represented as
as
can
(4.42)
supersymmetric non-linear (Wess-
The local Dirac
=
where
e(T)
conventional fashion using superspace coordinates
describe how
for
shall
=
1wjjjO'Oj + iA,,
+
(4.44)
0
4
where the supersymmetry generator S associated with 1 5 supersymmetry algebra (graded) N
(4.27) generates
the
2
IS, S1
=
H
=
g"'
91,
+
_1wjjjO'Oj + iAj, 4 I +
2
IS, HI
19v
+
1Wvk4kO' + iA,
4
O"Flvov =
JH' HI
0
(4.45)
(4.45)
have used the various symmetry properties of the Riemann curvature tensor. Notice that the Hamiltonian H vanishes on phys-
In
arriving
ical 5
at
we
(supersymmetric) ground states,
See
Appendix
so
that there
A for the convention for the
graded
are no
local propagating
commutator
f -, .1.
4.2
Example: Path Integral
degrees
Atiyah-Singer
Derivation of the
of freedom and the model
can
Index Theorem
89
only describe the global topological
characteristics of the manifold M, Le. this supersymmetric model defines topological field theory. The constraint
algebra (4.45)
contains first class
constraints,
a
i.e. it defines
(see Appendix A)
such that H is supersymmetric under the infinitesimal supersymmetry transformations generated S 0 ensure the reparametrization invariance of by S. The constraints H a
closed
algebra
between H and S
=
=
trajectories x"(,r). It is straightforward to now construct the BRST gauge fixed path integral associated with this constraint algebra (in the proper time gauge). Since the various ghost degrees of freedom only couple to world line quantities and not to the metric structure of M, (4.39) coincides with the canonical BRST gauge-fixed path integral describing the propagation of a Dirac particle on the configuration space M (with the identification p,, '91, as the canonical momentum conjugate to xA). The gauge-fixed quantum action (4.40) is written only modulo the ghost field and other contributions that decouple from the background metric of M, as these fields only contribute to the overall normalization in (4.39). The necessity to use periodic boundary of conditions in the path integral follows from the identification the
-
the Dirac matrices. There are several ways to evaluate explicitly the supersymmetric path integral (4.39). The traditional method is to exploit the T-independence and use, in the high-temperature limit (T --> 0), either a heat kernel expansion of the trace in (4.35) [41] or a normal coordinate expansion to evaluate the partition function (4.39) [5, 48]. Here, however, we wish to emphasize the observation of Atiyah and Witten [8] (and the later generalizations to twisted Dirac operators by Bismut [23, 24] and Jones and Petrack [80]) that the path .1 quantum mechanics admits a formal integral for N 2 supersymmetric equivariant cohomological structure on the superloop, space LM (9 LA'M. To see how this structure arises, we introduce a geometric framework for manipulating the path integral (4-39). These geometric manipulations will be the starting point for the general analyses of generic phase space path integrals which will follow. Given any functional F[x] of closed paths in the loop space LM, we define functional differentiation, for which functional integration is the anti-derivative thereof, by the rule =
J
JxA(-r)
F[x(,r')]
=
5(-r
-
-r')F'[x(-r')]
and the rules for functional differentiation of the
paths by the
periodic Grassmann-valued
anti-commutator
[ J'OA('r) J
The crucial
(4.40)
(4.46)
point
,
OV (Ir
1+
is that the fermionic
is bilinear in the fermion fields
gration induces
a
=
so
JI', 5 (-r
-
-r')
(4.47)
part of the supersymmetric action that the functional Berezin inte-
determinant factor det 1/2 11 b1l which makes the remaining
90
4.
Quantum Localization Theory for Phase Space Path Integrals
integration over LM in (4.39) resemble the phase space path integral (4.25). More precisely, the loop space fermionic bilinear form appearing in (4.40) is Q [x,,Ol
j
=
d-r
10" (-r) (gm, V,
F,,, (x (-r))) 0'(-r)
-
2
(4.48)
S1
which,
[d 2nx]
after Berezin
integration, induces
loop
a
space Liouville
measure
Vdet jj Yjj- Introducing the nilpotent graded derivative operator D
d-r
01' (r)
6
(4.49)
6xll(r)
S1
(4.48)
that
we see
be
can
expressed
S?[x, 0]
as a
=
D-exact
quantity
DZ[x, 0]
(4.50)
where Z [X,
V)]
=:idr
A,,(x(-r))} 0"(-r)
+
-=i
S1
d-r
-0,,(X(T))OA(7-
S1
(4.51) The functional
(4.48)
be
interpreted as a loop space symplectic structure. Strictly speaking though, it is properly termed a 'pre-symplectic' structure because although it is D-closed, Db[x,01 0, it is not necessarily non-degenerate on the loop space. It is this interpretation of supersymmetric theories in general that makes infinite-dimensional generalizations of the equivariant localization formalisms of Chapter 3 very powerful tools. .1 In particular, the N can be represented by a 2 supersymmetry (4.42) derivative loop space equivariant operator. To see this, introduce the nilpotent graded contraction operator can
=
=
d-r b "
1
(-r)6 6
(4.52)
V) A (-r)
S1
and define the
Dj
=
corresponding graded equivariant
D +
1,b
(01' (-r)
d-r
exterior derivative operator
+ &" (-r)
S1
Then the supersymmetry
(4.42)
is
6
immediately recognized
the derivative D, S on LM 0 LA1M. The square of Dj of time translations on the superloop space LM (9 LA'M, -
D? X
=
j (.t d-r
S1
A
(-r)
6 -
JxA(-r)
+
0 11 (T)
6
To _('r)
)
(4.53)
as
the action of
5011(r)
) =j S1
d'r
is the generator
d
77
(4.54)
Example: Path Integral Derivation of the Atiyah-Singer Index Theorem
4.2
so
that its action
D?W[x,01
on a
loop d
d-r
=
X
W[x, 0]
space functional
91
is
W[X,'O]=W[X(I),0(1)]-W[X(O),'O(O)J
(4.55)
S1
Consequently, (4.53) is a nilpotent operator provided we restrict to single1 loop space functionals W[x, 01. Hence the action of N 2 supersymmetric quantum mechanics defines an equivariant structure on LM (9 LAIM and on the basis of the general arguments of Chapter 2 we expect its path integral to localize to an integral over M, the zero locus of the vector field x(O) E M Vt). This is well-known to il'(-r) (i.e. the constant paths x(t) valued
=
=
be the In
case
[5, 48].
fact, the full
II/2[X7'01
=
i
d-r
(4.40)
action
is
D.6,-exact,
t,, (x(-r))V' (-r) + f2 [x, 0]
=
1,,b '[x, 0] + h [x, 0]
=_
D, 3 [x,,O]
S1
(4.56) and its bosonic part resembles the general phase space action functional (4.26) with H _= 0 there. As mentioned before, the vanishing of the Hamiltonian is the topological feature of such supersymmetric field theories. Now the equivariant localization principle applied to the case at hand would imply on its own that the path integral
index(iyl)
=
f
[d 2nx] [d2no] eiTDjt[x,,0]
(4.57)
LM(&LAlM
is
formally independent
V'(-r) (4.57) because
onto
=
0
(for
T
of the parameter T, and thus it manifestly localizes 0 in -oo). Of course, we cannot simply set T =
the bosonic integration would yield oo while the fermionic one would give 0, leading to an ill-defined quantity. In any case, if we think of the coefficient T in front of the action as Planck's constant h, then this is just
seeing that the semi-classical approximation is exact. The Tindependence can be understood from the point of view that if we differentiate the right-hand side of (4.57) with respect to T, then we obtain the vacuum expectation value (01D.. _010) in the supersymmetric quantum field theory another way of
above. If the the
model,
vacuum
then D.
10)
vanishes. It is these
itself is invariant under the N =
0 and the
same
=
-1 2
supersymmetry of
expectation value of this operator sorts of arguments which establish the topological vacuum
path integrals (also known as cohomological or Witten-type topological field theories) in general [22]. The above connection between the formalisms of the previous Chapters and the Atiyah-Singer index theorem is the usual intimate connection between standard supersymmetric invariance of BRST-exact
models
(for
instance those which arise in the Duistermaat-Heckman
tation of the quantum mechanics of spin
[26].
[156, 4])
and
interpreequivariant cohomology
92
Quantum Localization Theory
4.
for Phase
Space Path Integrals
We now use the fact that (4.57) can be evaluated for T --* oo and use a trick similar to that in Section 3.7. We introduce based loops on LMOLAIM,
X" (-r) with
(X0, ?PO)
X, JP (r) 0 +
=
M(&AIM
E
01(-r)
by6
[d2nX] [d
+
000
the constant modes of the fields and
constant fluctuations about these measure
b'0
=
2n
(4.58) the
non-
zero-modes, and define the path integral
11
d2nXO d2nV)O
d2n,
(-r)
d2n (,r)
(4.59)
-rES1
We then rescale the non-constant modes
00 / V'T
(-0 With this
T
rescaling,
(4.40)
action
find after
we
000 some
-
0(-r)1V'T
algebra that
in the limit T
(4.60) 00
the
becomes
11/2[X7 01
-
as
T-+oo
` V
d7
(0 X (-0
X
2
+
2
i 00'qij a,, j 00
S1
-
2
O1'Fj,,(xo)O'+ '4(-r)X""(,r) 0 0 0 0 _2 Rjjjv(xo)O'Ojx
+O(11V'T) -
(4.61) where
we
(4.58)
and
have
Taylor expanded
(4.60).
the quantities in (4.40) about (xo,'Oo) using The Jacobians for the scaling by v/T- cancel out from the
bosonic and fermionic tional
integrations
integration
over
measures
non-constant modes
in
(4.59),
are
and the
remaining func-
Gaussian. This illustrates the
strong role that supersymmetry plays here in reducing the complicated integrations in (4.39) to Gaussian ones.
Evaluating
these Gaussian
integrations
in
(4.39)
d 2n xo d2n Oo
index(iy)
leads to
'
0 0 ew F, ,(xo)V)Nk'
(4.62)
MOAIM
(det'jjPo9,
x
where
we
functional on
R"(xo, '00),,)-1/2 V
ignored (infinite) constant factors arising from the Gaussian integrations and normalized the U(1) connection. Here the prime
have
the determinant
riodic
-
V
boundary
In Section 4.6
means
that it is taken
conditions we
general superloop
shall be space
(i.e. a
bit
more
LM(DLA'M.
this to evaluate the index.
over
the fluctuation modes with pezero modes excluded).
the determinant with
precise about this decomposition For now,
we
over
a
arejust concerned with using
4.2
Example:
Path
Integral
Derivation of the
Atiyah-Singer Index Theorem
93
The exponential factor in (4.62) is Chern character ch(F) of the given
immediately seen to be the (ordinary) complex line bundle L -- M, while the functional determinant coincides (modulo overall signs to be discussed below)
with the Euler form of the normal bundle to M in LM
(this
is the bundle
spanned by the non-constant modes of x(-r)). Thus (4.62) coincides with a formal application of the degenerate Duistermaat-Heckman integration formula (3.118) (more precisely the degenerate version of the Berline-Vergne theorem) to the infinite-dimensional integral (4.39). Finally, we discuss how to calculate the Euler form in (4.62). A regularization scheme in general must always be chosen to evaluate infinite-dimensional determinants
[1021.
Notice first that here the infinite-dimensional Pfaffian
arising from the fermionic integration cancels from the result of the infinitedimensional Gaussian integral over the bosonic fluctuation modes. Thus, just as in the finite-dimensional case, the sign dependence of the Pfaffian gets transfered to the inverse square root of the determinant. The spectral asymmetry associated with the sign of the infinite-dimensional Pfaffian (see (3-62)) has to be regulated and is given by the Atiyah-Patodi-Singer eta-invariant [41, 158] of the Dirac operator 0, R, -
R)
=
lim s--+O
t
dA
sgn(A)IAI-'
+ dim
ker(a,
-
R)
CO
=
Jim 8-0
f
I
_r
where the integration and
(8+21)
dt
t(s-1)12
(and/or sum)
is
over
tr
all
(c),
-
R) e-t(a-,
non-zero
-R)21 1
eigenvalues
A of
(4.63)
'9,
-
R
00
J
F(X)
dt
tx-1 e-t
X
> 0
(4.64)
0
is the Euler
Next,
gamma-function.
we
ularizations
evaluate the determinant
using standard supersymmetry
reg-
[5, 48]
for first-order differential operators defined on a circle. The most convenient such choice is Riemann zeta-function regularization. The non-constant single-valued eigenfunctions of the operator '9, on S' are
e2,jrikr ,where k
are non-zero integers. Since the matrix R is antisymmetric, skew-diagonalized into n 2 x 2 skew-diagonal blocks RU) with skew eigenvalues Aj, where j 1,...'n. For each such block RU), we get the
it
can
be
=
formal contribution to the determinant in
det'1149,
-
RW 11
=
(4.129),
11 (21rik + Aj) (21rik
-
Aj)
k:00 =
g(Aj/27ri)g(-Aj/2iri)
11 (21ri )2 k96O
(4.65)
4.
94
where
we
Quantum Localization Theory for Phase Space Path Integrals have defined the function
g(z)
g(z)
=
as
the formal
product
rl (k + z)
(4.66)
koo
We
determine the
can
logarithmic
function with
g'(z)1g(z) [48].
simple poles
g'(z) Ig (z)
take
=
ir
cot,7rz
This is, k of residue 1 at z
we
have normalized
as
g(z)
sin,7rz
=
so
g(z) by examining function of
z
that
e
g(O)
bz
a
we
(4.67)
/7rZ
=
its
C,
E
integer. Thus b and integrating this we get
1 /z +
-
a
a non-zero
=
g(z) where
form of the function
regulated
derivative
1. The
arbitrary phase in (4.67)
k E Z which appears because the zeroes of the function (4.66) occur at z determine it up to a function without zeroes, i.e. an exponential function. =
(4.65),
When substituted into
it is related to the
and hence to the eta-invariant
(4.63).
sign
of the
In certain instances
determinant,
(see
Section
5.4)
regularization of this phase [102] (i.e. a choice for b). In our case here, however, the phase b will cancel out explicitly in (4.65) and so we can neglect its effect. The infinite prefactor in (4.65) is regularized using the Riemann zetait is necessary to make
a
specific
choice for the
function 00
C(s)
E
=
1
(4.68)
k=1
which is finite for
s
with
> 0
fl (27ri )2
fl (2,7ri )4
kiW
k>0
=
C(O)
=
-1/2 [60].
(2,7ri )4(E-
'
k= I
and thus the block contribution
(4.65)
TT
)1-0
We find that
=
(21ri )4
(O)
=
(2-7r i)-2 (4.69)
to the functional determinant in
(4.62)
is
det'l 1 o9,
-
R(i)
i
T2
(
2
( )2 1
=
Aj
-
21ri
det
[sinh -I -I RU) 2
1RO)
(4.70)
2
Multiplying the blocks together we see that the fluctuation determinant ap0 A-genus (2.96) with pearing in (4.62) is just given by the ordinary V =
respect
to the curvature
R, and thus the index
index(i)F )
=
I ch(F)
A
is
A(R)
(4.71)
M
The result (4.71) is the celebrated Atiyah-Singer index theorem for a twisted spin complex [41]. We see then that a formal application of the Berline-Vergne theorem yields the well-known Atiyah-Singer geometrical
4.3
Loop Space Symplectic Geometry and Equivariant Cohomology
iV
representation of the index of clude the
coupling
This result
.
of fermions to
be
can
generalized
95
to in-
non-abelian gauge field A on a vector bundle E --> M. Now the functional above is no longer closed (F dA + [A A, A] /2 obeys the Bianchi identity), but the construction above a
b[x,,O]
=
still be carried
through using the coadjoint orbit representation of the principal fiber bundle [3, 71] (see Section 5.1). The above representation of the Witten index in terms of a supersymmetric path integral can also be generalized to other differential operators, not just the can
structure group of the
Dirac operator
(4.27).
For instance, the Witten index for the DeRham,
ex-
terior derivative operator d describes the DeRham
[166].
M
The index is
path integral
metric
now
complex of the manifold Euler characteristic of M, the supersym-
the
is that of N
=
1
(DeRham) supersymmetric
quantum
mechanics,
and the localization formula reproduces the Gauss-Bonnet-Chern theorem. An equivariant generalization then yields the Poincar6-Hopf theof classical Morse theory [166]. We shall elaborate on some of these ideas, as well as how they extend to infinite-dimensional cases relevant to topological field theories, in Chapter 8. Finally, we point out that the equivariant cohomological interpretation above is particularly well-suited to reproduce, the Callias-Bott index theorems [70], i.e. the analog of the Atiyah-Singer index theorem for a Dirac operator on orem
an
is
odd-dimensional non-compact manifold [41]. The supersymmetric model that of N 1 supersymmetric quantum mechanics with background
now
=
monopole and soliton configurations. The the Witten index true that the
-
in this
case
be
trace
over zero
infinite-dimensional,
modes
representing simply not
and it is
partition function is independent of the parameter T (the index
being obtained the T
can
for T
--->
oo)
so
that
one
cannot
simplify
0 limit. The canonical realization of the N
by considering an equivariant by a larger mixing of
defined
structure
over
an
matters
=
1
by taking
supersymmetry
extended
bosonic and fermionic
superloop space, coordinates, preserves
the contributions of the
zero modes which would otherwise be lost [107, 108] and the localization tricks used above become directly applicable. We shall discuss this a bit more in Chapter 8. Furthermore, the index in these cases
can be computed from a higher-dimensional Atyah-Singer index theorem by introducing a simple first class constraint (i.e. one that is a symmetry of the Hamiltonian, or equivalently a constant of the motion) that eliminates the extra dimensions. We refer to [70] for more details about this approach to index theorems in general.
4.3
and The
Loop Space Symplectic Geometry Equivariant Cohomology
example of the last Section has shown that a formal generalization of symplectic geometry and equivariant cohomology to the loop space of a physical problem can result in a (correct) localization formula in the same spirit
96
as
4.
Quantum Localization Theory for Phase SPace Path Integrals
those of
Chapters
2 and 3. The localization
principle
in this context
was
just manifestation of the supersymmetry of that model. It has also provided us with some important functional space tools that will be used throughout a
this
Book,
well
hints
how to
proceed to loop space generalizations of Chapters. Following these lessons we have learned, we shall now focus on developing some geometric methods of determining quantum partition functions of generic (not necessarily supersymmetric) dynamical systems. Given the formulation of the path integral in Section 4.1 above on a general symplectic manifold, we wish to treat the problem of its exact evaluation within the geometric context of Chapter 3. As exemplified by the example of the previous Section, for this we need a formulation of exterior and symplectic differential geometry on the loop space LM over the phase space M. This will ultimately lead to a formal, infinite-dimensional generalization of the equivariant localization priniciple for path integrals, and thus formal conditions and methods for evaluating exactly these functional integrations which in general are far more difficult to deal with than their classical counterparts. As with the precise definition of the functional integrals above, we shall be rather cavalier here about the technicalities of infinitedimensional manifolds. The loop space LM --+ M is an infinite-dimensional the fiber over a point x E M is the space of all loops x(t) vector bundle as
as
on
the results of the earlier
-
based at x, x(T) group with group
=
x(O)
=
x, which is
an
infinite-dimensional non-abelian
multiplication of loops (XIX2)(t) defined by first traversing the loop x1(t), and then the loop X2(-t) in the opposite direction. These quantities should therefore be properly defined using Sobolev completions of the infinite-dimensional groups and spaces involved. This can always be done in an essentially straightforward and routine manner [22]. We define the exterior algebra LAM of the loop space by lifting the Grassmann generators 7711 of AM to anti-commuting periodic paths 77A(t) which generate LAM and which loop space 1-forms. With this, we
to be identified
are
can
define
loop
as
the basis
dx"(t)
of
space differential k-forms
T ce
=
f
dt,
...
dtk
1
k!
ap,- ..I,,
[X;tl,
-
-
.)tkJ?1/-z'1(t1)
*
'
'77"(4)
(4.72)
0
and the
loop space exterior derivative phase space M,
is defined
by lifting the
exterior deriva-
tive of the
T
dL
dt
no(t)
(4.73)
6XI, (t)
0
The loop space symplectic geometry plectic 2-form
is determined
by
a
loop
space sym-
T
dt dt'
S? 0
1
S?,,, [x; t, t],q" (t),q' (t')
(4.74)
Loop Space Symplectic Geometry and Equivariant Cohomology
4.3
97
which is closed
dLO or
x"(t)
in local coordinates
j JXA (t) Thus
S?"\ [X; tt, t// ]
j
Q'\j, [X; tf, t// ]
JX" (t)
(4.75)
0
LM,
on
6 +
-=
+
6XII W
S?,,, [x; t', t"]
=
0
(4.76)
apply the infinite-dimensional version of Poincar6's lemma to locally in terms of the exterior derivative of a loop space 1-form
we can
represent Q
T
V
=
I
?9, [x; tjq" (t)
dt
(4.77)
0 as
S?
We further
assume
that
(4.74)
the
loop
space.
is invertible
on
The canonical choice of the
loop
space Liouville
from the
symplectic
symplectic
measure
diagonal
loop
(4.78)
structure
introduced in
phase
structure of the
in its
dL?9
non-degenerate,
is
Qj" [X; t' t'j which is
=
on
space indices
J?,,, [x; t, t']
LM which coincides with
(4.25)
is that which is induced
space,
Wt" (X(t))J(t
=
i.e. the matrix
-
t')
t, t. We shall
(4.79) use
similar
liftings
of
other quantities from the phase space to the loop space. In this way, elements a(x) of LA.,,M (or LTxM) at a loop x E LM are regarded as deformations of the
loop,
that
a[x; t]
over
LM
i.e. E
are
as
elements of AM
(or TM)
A.,(t)M (or Tx(t)M).
This
restricted to the
means
loop xA(t) such
that these vector bundles
infinite-dimensional spaces of sections of the
pull-back
of the
phase space bundles to [0, T] by the map x(t) : [0, T] -+ M. In particular, we define loop space canonical transformations as loop space changes of variable
F[x(t)]
that leave S-2 invariant. These ?9
Thus in the context of the
-F 4 19F
loop
are
=
the transformations of the form
V +
space
(4.80)
dLF
symplectic geometry determined
by (4.79),
the quantum partition function is an integral over the infinitedimensional symplectic manifold (LM, 0) with the loop space Liouville measure
by
there determined
exterior
by the canonically-invariant closed form products of S? with itself,
[d[tL(X)l The
loop
:--
on
LM given
[d2nXj V/det 11f2j[
space Hamiltonian vector field associated with the action
has components
(4.81)
(4.26)
98
Quantum Localization Theory
4.
for Phase
Space Path Integrals
T
Vs, IX; t]
=f
dt' f2l" [x; t, t']
bA W
=
Jxv (t,)
V"(X(t))
-
(4.82)
0
with VA of
=
wl"o%H
as
usual the Hamiltonian vector field
on
M. The
zeroes
Vs LMs
are
the extrema of the action
of the
(4.83)
E
(4.26)
and coincide with the classical trajectories
LM:
Vs[x(t)]
01
Jx(t)
=
=
i.e. the solutions of the classical Hamilton
dynamical system, loop space
of motion. The
Wl'[x; t]
vector field
is
contraction operator with
respect
to
a
equations
loop
space
given by T
iw
=
I
dt W" [x;
6
t]
(4.84)
J771, (t)
0
Thus
we can
define
a
loop
space
exterior derivative
equivariant
Qw whose square is the Lie derivative
(4.85)
dL+iW
=
along
the
space vector field
loop
W,
T
Q2W
=
dLiW
+
iwdL
=
dt
f-w
(W"
6
aW"71'
+
-
JXt1
J
6'qA
(4.86)
0
When W
Vs is the loop space Hamiltonian vector field, corresponding operators above as ivs =- is, etc.
=
denote the
we
shall for
ease
The partition function can be written as in the finite-dimensional case using the functional Berezin integration rules to absorb the determinant factor into the exponential in terms of the anti-commuting periodic fields qA (t), T
Z(T)
=
f
2n
[d2nx] [d q]
i'4SIX] +
exp
2
f
dt
0
LMOLAIM
f
,
[d 2nX] [d
2n
q] e4s x]+01','11)
LMOLA'M
(4.87) that in this way Z(T) is written in terms of an augmented action S + 0 on the super-loop space LM (9 LA'M. From this we can now formally describe so
the
SI-equivariant cohomology The operator
Qs
is
of the
nilpotent
LAsM
=
on
Ja
E
loop space. subspace
the
LAM :,Csa
=
01
(4.88)
4.4 Hidden
of
Supersymmetry
Loop Space
and the
Localization
Principle
99
equivariant loop space functionals. The loop space observable S[x] defines loop space Hamiltonian vector field through
the
dLS from which it follows that the
(4.87)
is
so
(4.89)
-iSS?
integrand
of the quantum partition function
equivariantly closed,
Qs(S + 0) and
=
the
augmented
(dL
=
action S + Q
ariant exterior derivative of
a
+
can
iS)(S be
Q)
+
=
(4.90)
0
locally represented
as
the equiv-
1-form T
S + Q
=
Qs
f (Vsl' ,, -1 f2j,,77477') dt
+
(4.91)
2
0
From
(4.90)
find that
we
Q2S =,CS and
so
lies in the
subspace (4.88).
=
If Ps is
(4.92)
0
some
globally defined loop
space
0-form with f s (dOS)
then
we see
under the
that
loop
is not
=
(4.93)
0
unique but the augmented
action
(4.91)
is invariant
space canonical transformation --+
+
dLOS
(4.94)
Thus the partition function (4.87) has a very definite interpretation in terms of the loop space equivariant cohomology HS(LM) determined by the operator
QS
on
LASM.
Supersymmetry Loop Space Localization Principle
4.4 Hidden
and the
The fact that the
integrand of the partition function above can be interpreted loop space equivariant cohomology suggests that we can localize it by choosing an appropriate representative of the loop space equivariant cohomology class determined by the augmented action S + Q. However, the
in terms of
a
arguments which showed in the finite-dimensional
cases that the partition integral is invariant under such topological deformations cannot be straightforwardly applied here since there is no direct analog of Stokes' theorem for infinite-dimensional manifolds. Nonetheless, the localization priniciple can be established by interpreting the equivariant cohomological structure on LM as a "hidden" supersymmetry of -the quantum theory. In this way
function
100
Quantum Localization Theory for Phase Space Path Integrals
4.
has
one
sort of Stokes' theorem in the form of
a
with this supersymmetry
(as
was
the
identity
Ward
a
in Section 4.2
case
associated
above),
where
we
interpret the fundamental localization property (2.126) as an infinitesimal change of variables in the integral. The partition function (4.87) can be interpreted as a BRST gauge-fixed path integral [22] with the 771'(t) viewed as fermionic ghost fields and x11(t) as the fundamental bosonic fields of the model. The supersymmetry is suggested by the ungraded structure of QS on LASM which maps even-degree, commuting loop space forms (bosons) into odd-degree, anti-commuting forms (fermions)7. Since the fermion fields q/(t) .1 appear by themselves without a conjugate partner, this determines an N 2 LS implies that Q2S supersymmetry. The N 2 supersymmetry algebra QS is a supersymmetry charge on the subspace LASM, and the augmented 0. Thus here LASM coincides with action is supersymmetric, QS(S + 0) the BRST complex of physical (supersymmetric) states, and the BRST trans=
=
=
formations of the fundamental bosonic fields
77/'(t) QS
xl'(t)
action of the infinitesimal
given by the
are
and their superpartners supersymmetry generator
8
Qsq" W
QSXIT) =,q1*) This formal identification of the
Vs" [X; tj
=
(4.95)
equivariant cohomological
structure
as
supersymmetry allows one to interpret the (non-supersymmetric) quantum theory as a supersymmetric or topological field theory. It was Blau, Keski-Vakkuri and Niemi [30] who pointed out that a quite general localization principle could be formulated for path integrals using rather formal functional techniques introduced in the BRST quantization of first class cona
hidden
strained systems In this
In these theories
interpretation the form degree
that the
8
[118].
physical
a
can
observables of the system
the smooth functions
on
analogy between Qs
=
BRST transformation
be
thought
(i.e.
of
produces
ghost number, so ghost number 0) are
as a
those with
M. Furthermore, at this point it is useful to recall the
dL + is and the gauge-covariant derivative in
a
gauge
theory for the following analogies with BRST quantization of gauge theories. See Appendix A for a brief review of some of the ideas of BRST quantization. In supersymmetric quantum field theories the BRST transformations of operators and fields are represented by a graded BRST commutator JQs, .1. This commutator in the case at hand can be represented by the Poisson structure of the
phase
conjugate
to
space
xl'(t)
as
follows. We introduce
and anticommuting
periodic trajectories \,(t)
periodic paths ij,(t) conjugate
in LM
to
77"(t),
i.e.
which
A, (t)
are -
to be identified 5
6XII
(t)
and
,,(t)
as
the Poisson 8
-
.5?7 1, M
acting
algebra realization
of the operators
in the usual way. This
gives
a
Poisson
bracket realization of the actions of the operators dL and is, and then the action of
Qs
can
is
keep
with
represented by in mind this
the BRST commutator
JQs, +,.
representation which maintains
supersymmetric theories.
a
In the
complete
following, one analogy
formal
4.4 Hidden
a
Supersymmetry and the Loop Space
super-Jacobian
on
Principle
101
space LM 0 LAM whose corrections
super-loop
the
Localization
related to anomalies and BRST supersymmetry breaking. The arguments below are therefore valid provided that the Qs-supersymmetry above is not are
broken in the quantum theory. The argument for infinite-dimensional localization Consider the 1-parameter
Z(A)
I
=
family
of
[d 2nx] [d
phase
space
proceeds path integrals
follows.
as
q] e (S1'1+O1x,'71+'\QsO1x,7D
2n
(4.96)
LM(&LAIM
V)
where \ E R and
E
LAIS M
is
gauged
a
fermion field which is
homotopic
supersymmetry transformation generated by QS (i.e. V) where 0,,, s E [0, 11, is a 1-parameter family of gauge fermions with 0 and 0,=o 0). As in the finite-dimensional case, we wish to
to 0 under the
Q2S 03
=
=
\-independence of this path integral, i.e. that (4.96) depends cohomology class determined by the augmented action, only on of A : 0 amounts to a choice of representative of S + S? choice that so a in its loop space equivariant cohomology class and different choices of nontrivial representatives then lead to the desired localization schemes. Consider
establish the
the BRST
an
0
infinitesimal variation A
--+
V)
JO
+
--.>
(4.96),
A + JA of the argument of
60
=
JA
-
(4.97)
0
and consider the infinitesimal supersymmetry transformation loop space parametrized by the gauge fermion JV) E LAIS M,
X" --+.t"
=
x" + Jx"
W,
=
771,
77
Since
tt
__'
QS(S
i.e. let
with
+
R)
=
+
=
577,
Lso
=
X4 +
6V) QSX1,
77,
60 QS771' =,ql
+
-
=
-
X1, + +
on
the super-
JO -77 ' 50
-
Vs'
(4.98)
0, the argument of the path integral (4.96) is
=
BRST-invariant.
However, the corresponding super-Jacobian arising in the Feynman in (4.96) on LM 0 LAIM is non-trivial and it has precisely the
sure
functional form tinent
as
that in
super-Jacobian
standard BRST transformation
a
here is
[1181.
measame
The per-
given by the super-determinant J.T 6x,;-
[d2nt] [d 2nq]
Tx =
sdet
5-7)
jq jq
[d 2nx] [d
2n
(4.99)
Tx TO and the
path integral (4.96) is invariant under arbitrary smooth changes J,\, the identity
of
variables. For infinitesimal
tr
log 11AII
=
log det 11AII
(4.100)
102
4.
Quantum Localization Theory
implies that the super-determinant
for Phase
in
the super-trace, the super-loop space I + strIJAII. This gives as sdetjjAjj
(4.99)
Space
Integrals
be computed in terms of diagonal entries in (4.99),
can
of the
sum
Path
=
T
2n
[d j ] [d 2n ]
1 +
f
dt
6,qA
6XII
(60)Vs")
[d 2nX] [d2n.]
0
T
1
-
f
dt
JXA
+
VS1
6
6,q A
)
60
[d2nX] [d2n,,]
0 ==
(I
-
QsJO)[d2nX] [d2n,,]
-
e-5A*QsP[d2n x] [d2n,,] (4.101)
substituting the change of variables (4.98) with super-Jacobian (4.101) into the path integral (4.96) we immediately see that Thus
Z(A)
=
Z(,\
JA)
-
(4.102)
path integral (4.96) under homosubspace (4.88). This proof topically of the A-independence (or the V)-independence more generally) of (4.96) is a specialization of the Fradkin-Vilkovisky theorem [13, 14, 118] to the supersymmetric theory above, which states that local supersymmetric variations of gauge fermions in a supersymmetric BRST gauge-fixed path integral leave it invariant. Indeed, the addition of the BRST-exact term QSO can be regarded as a gauge-fixing term (the reason why 0 is termed here a 'gauge fermion') which renormalizes the theory but leaves it invariant under these perturbative deformations. The addition of this term to the action of the quantum theory above is therefore regarded as a topological deformation, in that it does not 0 limit change the value of the original partition function which is the A of (4.96) above. This is consistent with the general ideas of topological field theory, in which a supersymmetric BRST-exact action is known to have no local propagating degrees of freedom and so can only describe topological invariants of the underlying space. We shall discuss these more topological aspects of BRST-exact path integrals, also known as Witten-type topological field theories [22], in due course. In any case, we can now write down the loop space localization principle which establishes the
independence of
the
trivial deformations which live in the
--
Z(T)
=
I
lim A-00
[d2nX] [d
2n
q] e'(S1'1+21','1J+AQs'P1x,7D
(4.103)
LM(DLAIM so
that the quantum
fermion field
partition function localizes
like to
zeroes
of the gauge
0.
Given the localization property now
onto the
pick
a
suitable
(4.103)
of the quantum
representative 0 making
theory,
we
would
the localization manifest.
4.4 Hidden
Supersymmetry and the Loop Space Localization Principle
103
As in the finite dimensional cases, the localizations of interest both physically and mathematically are usually the fixed point locuses of loop space vector fields W introduce
on a
LM. To translate this into
metric tensor G
the
on
a
space differential
loop
space and
loop
takeO
form,
we
to be the associated
metric-dual form T
0
=
f
G,,, [x; t, t'] W1 [x; t],q'(t')
dt dt'
(4.104)
0
0 is loop space vector field W. The supersymmetry condition Lso additional the and 0 the to requireLsG Killing equation equivalent
of the then ment
=
=
LsW
=
0
on
W
1,
where T
'CSW
=
I (Tt 'CV(X(t))) d
dt
W [X;
_
(4.105)
t]
0
In
principle
but
we
there
are
many useful choices for W
shall be concerned
mostly with those
W" [x; where the parameters r, scheme.
s are
t]
=
r.V'(t)
-
obeying such
which
can
a
restriction,
be summarized in
(4.106)
sV" (x (t))
chosen appropriate to the desired localization
As for the metric in (4.104), there are also in principle many possibilities. However, there only seems to be 1 general class of loop space metric tensors to which general arguments and analyses can be applied. To motivate these, we note first that the equivariant exterior derivative QS can be written as
QS
=
Q
-
iv
=
dL
+
ii
-
(4.107)
iV
and the square of the operator Qj, is just the generator of time translations T
T
Q?x
=
dt
C,
(V'(t) 6XIL(t)
+
"(t)
6
JWIM
we assume
that the
We also
require that
d
Tt
(4.108)
loop space Hamiltonian vector parametrized by a parameter
field generates an S'-flow on loop space, ,r E [0, 1] so that the flow is x11 (t) --+ x1'(t;,r) with x" (t; the
9
dt
0
0
This operator arises when
f
the combination
(4.104)
0)
=
x1' (t;
1),
be such that it determines
such a
ho-
topological motopically effects into the path integral (4.1,03) when evaluated on contractable loops. For the most part, we shall be rather cavalier about this requirement and discuss it trivial element
as
above,
only towards the end of this Book.
so
that it introduces
no
extra
104
4.
Quantum Localization Theory for Phase Space Path Integrals
loop space coordinates xA(t) the flow parameter loop (time) parameter t - t +,r. In this case we have
that in the selected shifts the
VS11 IX; t] -'9xA(t;'r) t9-r and the supersymmetry transformation
Q.+x11(t) =,q"(t) which
we
above. In
I-r=O= e(t)
(4.95)
-r
also
(4.109)
becomes
Q+77'(t)
=
b'(t)
(4.110)
recall is the infinitesimal supersymmetry discussed in Section 4.2 particular, the effective action is now (locally) of the functional
form T
S+S?=Idt (0p(x)V'+1w,,(x(t))774?7"
(dL
2
+
0
theory, i.e. the invariance of 0 of the subspace LA1S M, is according to (4.108) determined by arbitrary globally defined single-valued functionals on LM, i.e. 0(0) O(T). This form of the U(1)-equivariant the model-independent circle action. called the is loop space cohomology on and the
(4. 111)
topological
invariance of the quantum by elements
under BRST-deformations
=
We shall therefore demand that the localization functionals in
(4.104)
be
model-independent Sl-action on LM (i.e. rigid rotations the ---> of x (t) x (t + -r) loops). This requires that the loop space metric tensor above obey L., G 0, or equivalently that G,,, [x; t, t'] G,,, [x; t t'] is the is to describe the indices. Since its in diagonal loop space quantum theory the Hamiltonian for which know of we a given underlying dynamics system manifold M, the best way to pick the Riemannian structure on LM is to lift invariant under the
=
=
a
metric tensor g from M
so
that G takes the ultra-local form
G,,[x;t,t'] and its action
on
loop
-
=
g,,,(x(t))J(t
space vector fields is
-
t')
(4.112)
given by
T
G(Vl) V2)
=
I
dt gAv (,r(t)) V11A [X;
t] V2v [X; t]
(4-113)
0
Because of the
reparametrization
invariance of the
tensor G is invariant under the canonical flow
derivative condition
on
G is then
equivalent
on
integral (4-113), the metric generated by.' . The Lie
LM
to the Lie derivative condition
(2.111)
with respect to the Hamiltonian vector field V on M. Thus infinitedimensional localization requires as well that the phase space M admit a
globally-defined U(1)-invariant Riemannian structure on M with respect to the classical dynamics of the given Hamiltonian system. As discussed before,
4.5 The WKB Localization Formula
105
the condition that the Hamiltonian H generates an isometry of a metric 9 on M (through the induced Poisson structure on (M, w)) is a very restrictive
dynamics. Essentially it means that H must be of a group G on M, so that the classical mechanics generates a very large degree of symmetry. As mentioned before, the infinite-dimensional results above, in particular the evaluation of the superJacobian in (4.101), are as reliable as the corresponding calculations in standard BRST quantization, provided that the boundary conditions in (4.96) are also supersymmetric. Provided that the assumptions on the classical proper-
condition
on
the Hamiltonian
related to the
global
action
ties of the Hamiltonian
are
(2-36)
satisfied
(as
for the finite-dimensional
cases),
the
'CS is above derivation will stand correct unless the supersymmetry Q'S broken in the quantum theory, for instance by a scale anomaly in the rescal=
ing of the metric G,,, -+ A G,,, above. See Appendices A and B for the precise correspondence between BRST quantization, equivariant cohomology -
and localization.
4.5 The WKB Localization Formula We shall that
can
begin examining the various types of localization formulas be derived from the general principles of the last Section. The now
shall present is the formal generalization of the Duistermaat-Heckman integration formula, whose derivation follows the loop space versions of the steps used in Sections 2.6 first infinite-dimensional localization formula that
and 3.3. We
assume
that the action 5 has that the
(finitely-many) isolated and nonzero locus (4.83) consists of iso-
degenerate critical trajectories, so loops in LM, i.e. we assume that
lated classical
ated Jacobi fields arising from on
these classical
(4.106),
so
trajectories.
a
we
the determinant of the associ-
second-order variation of S is
Under these assumptions,
we
non-vanishing
set
r
=
s
=
1 in
that T
f
dt
g,,,VS"77'
0T QSO
f
(4.114) dt
[gt,,VS"VS' + 77" (gI,&t
-
gxi9jVA + VS'\8Ig,,\) 77v]
0
Proceeding just as in the finite-dimensional integral (4.103) gives
localization
case, the evaluation of the
106
4.
Quantum Localization Theory for Phase Space Path Integrals
f [d2nx] VFdet 11011 Vdet_JJJVSJJ J(VS) e slxl
Z(T)
LM
det 11 Jl',9t
[d2nx] Vdet 11 Q 11
a,
-
V
(4.115)
LM
w"'aH) e'slxl
x
_11w(x(t)) 11 ,Fdet
e'slxl
x(t)ELMs will be used following the symbol the determinants which into prefactors
where here and in the
absorption of infinite
-
to
signify
the
arise from the
functional Gaussian integrations. The functional determinant in the denom(4.115) can be evaluated in the same manner as in Section 4.2
inator of
above with R
--+
odic functions
on
2.7rik
--->
A-genus
2.7rik/T
instead of S1
[0, TJ (4.65).
in
-
The result
T
0,,V/'
using the ordinary
be written in terms of the Dirac
can
2n
27ri
det
mally the leading
-
term of
U(1)-action
(4-116)
here.
partition function paths and not just those S. If we reinstate the factors of h, then it is forthe stationary phase expansion of the partition
except that it is summed
which minimize the action
A(T dV) -2
12: dV
moment map for the
This result is the famous WKB
[147],
eigenfunctions of at are now the periso that the eigenvalues are replaced as
of the tangent bundle of M
J,"at
det
dV there and the
approximation
over
to the
all classical
function in powers of h as h --+ 0. The limit h -- 0 is called the classical limit of the quantum mechanics problem above, since then according to (4.1) the operators P and 4 behave as ordinary commuting c-numbers as in the classical theory. For h --* 0 we can naturally evaluate the path integral by
stationary-phase method discussed in Section 3.3, i.e. we expand the trajectories x(t) xo(t) + Jx(t) in the action, with xo(t) E LM,5 and 6x(t) the fluctuations about the classical paths xO(t) with 6x(O) 6x(T) 0, and then functional the these fluctuations. Gaussian out over integration leading carry introduced the path integral the this was Indeed, way Feynman originally the
=
=
to describe
quantum mechanics
as
a sum
over
=
trajectories which fluctuate
paths
of the system. This presentation of quantum mechanics thus leads to the dynamical Hamilton action principle of classical
around the classical
mechanics
[55],
i.e. the classical
paths of
motion of
a
dynamical system
are
those which minimize the action, as a limiting case. If the classical trajectories were unique, then we would only obtain the factors e'sfxl/r' above as h --+ 0.
Quantum mechanics can then be interpreted as implying fluctuations (the one-loop determinant factors in (4.115)) around these classical trajectories.
4.6 The Niemi-Tirkkonen Localization Formula
107
We should point out here that the standard WKB formulas are usually given for configuration space path integrals where the fluctuation determinant (det IlLs(x(t)) 11)-1/2 appearing in (4.115) is the so-called Van Vleck determinant which is essentially the Hessian of S in configuration space coordinates q. Here the determinant is the functional determinant of the Jacobi operator which arises from the usual 'Legendre transformation to phase space coordinates (p, q). This operator is important in the Hamilton-Jacobi theory
[6, 551, and this determinant can be interpreted as the The result (4.115) and the assumptions that trajectories. density went into deriving it, such as the non-vanishing of the determinant of the Ja, cobi fields and the existence of an invariant phase space metric, are certainly true for the classic examples in quantum mechanics and field theory where the semi-classical approximation is known to be exact, such as for the propagator of a particle moving on a group manifold [38, 139, 146]. The above localization principle yields sufficient, geometric conditions for when a given path integral is given exactly by its WKB approximation, and it therefore has the possibility of expanding the set of quantum systems for which the Feynman path integral is WKB exact and localizes onto the classical trajectories of classical mechanics of classical
of the system.
Degenerate Path Integrals
4.6
and the Niemi-Tirkkonen Localization Formula
approximation is unsuitable for whose clasa quantum mechanical path integral, such as a dynamical system desirable therefore is It sical phase space trajectories coalesce at some point. be which formulas can applied to seek alternative, more general localization to larger classes of quantum systems. Niemi and Palo [121] have investigated the types of degeneracies that can occur for phase space path integrals and
There
have
are
many instances in which the WKB
argued
trajectories
that for Harniltonians which generate circle actions the classical set of can be characterized as follows. In general, the critical point
xO(O) xO' lie on periodic solutions xO(T) this In M. the context, LMS is phase space a compact submanifold LMS of and it is in gensolutions classical of refered to as the moduli space T-periodic time T. of the values discrete for propagation eral a non-isolated set only some x1j, exist with solutions the of T xl'(O) For generic values xl'(T) periodic 0 H. Then Hamiltonian of the submanifold critical the x' if lies MV on only 0
the action S with non-constant
=
=
=
=
0 and so wl"aH VA equations of motion reduce to V1 the moduli space LMS coincides with the critical point set MV C M. Notice that in this case the functional determinants involving the symplectic 2-form in (4.115) cancel out and one is left with only the regularized determinant in
the classical
(4.116).
We shall
=
see some
With this in mind
specific examples derive a loop
we can
Duistermaat-Heckman formula of Section 3.7.
=
=
of this later
on.
analog of the degenerate We decompose LM and LA'M
space
Quantum Localization Theory for Phase Space Path Integrals
4.
108
into classical modes and fluctuations about the classical solutions and scale
the latter
by
X1, (t)
-:t(t)
V, (t) + X, (t) /,V"Af
LMs '(.t(t))=XtV'
where
Vs
=
11VA-,
E
kernel of the
loop
are
q '(t)
'(t)
=
(4.117)
f
the solutions of the classical equations of motion, i.e. 0, and 11(t) dV'(t) E AlLms span the =
-
space Riemann moment map,
(S?S)4-(-t)qV
=
(4.118)
0
where T
Ds
=
J
dLO
dt
6 JXA
(g Vx vs, ) n IIn
(4.119)
V
0
0 given they obey
with i.e.
in
(4.114).
particular, this implies
In
(JI, at
-
V
a, V1,(,t)) qV
(4.117) obey q f'(T) 0.
The fluctuation modes in
x" (T) f
=
The
that
q1' (t)
are
Jacobi
fields,
the fluctuation equation
0 and
qf1'(0)
super-loop
[d 2nX] [d2n,,]
=
space
=
the
=
(4.120)
0
boundary
conditions
xo(O) f
=
measure
d2n;;:; V (t)
with this
decomposition
is then
11 d2nXf (t) d2nnf M
d2n (t)
(4.121)
tE[O,T] where
as
usual the
change
of variables
(4.117)
has unit Jacobian because the
determinants from the bosonic and fermionic fluctuations cancel
(this
is the
manifestation of the "hidden" supersymmetry in these theories). The calculation now proceeds analogously to that in Section 3.7, so that
powerful
evaluating the Gaussian integrals over the fluctuation modes localizes the path integral to a finite-dimensional integral over the moduli space LMS of classical solutions, d2 nX,.-,- (t)
Z(T) LMs
V1 d et w (.:t) eisp]
PfaffllMt,'Ot
-
(1,ts)1'1'(-:t)
-
M(.t) 11
1JVLMS
(4.122)
' where ps S?s and R is as usual the Riemann curvature 2-form of the g metric g evaluated on LMS. In (4-122) the Pfaffian is taken over the fluctu-
=
ation modes
-
xl'(t) f
the normal bundle
about the classical trajectories tA(t) E LMs (i.e. along A(LMS in LM), and the measure there is an invariant
measure over the moduli space of classical solutions which is itself a symplectic manifold. The localization formula (4.122) is the loop space version of the
degenerate localization formula (3.118)
in which the various factors
can
be
4.6 The Niemi-Tirkkonen Localization Formula
109
interpreted as loop space extensions of the equivariant characteristic classes. particular, in the limit where the solutions to the classical equations of motion VA(x(t)) 0 become isolated and non-degenerate paths the integration S In
=
formula
(4.122)
However,
reduces to the standard WKB localization formula
(4.115).
degenerate localization formula (4.122) is hard to use in practise because in general the moduli space of classical solutions has a complicated, T-dependent structure, i.e. it is usually a highly non-trivial problem to solve the classical equations of motion for the T-periodic classical trajectories of a dynamical system". We would therefore like to obtain alternative degenerate localization formulas which are applicable independently of the the
structure of the moduli space
LMS above. Given the form of (4.122), we could some sort of equivariant characteristic classes of the manifold M. The first step in this direction was carried out by Niemi and Tirkkonen in [127]. Their localization formula can be derived by I in (4.106) so that 0, r setting s
then
hope
to obtain
=
a
localization onto
=
T
f
dt
gm, bmn'
0T
f
QsO
(4.123) V') +,q" (gl,,Ot
dt
bAg,,prP,),q']
+
0
locus of the vector field (4-106) consists of the constant loops points on M, so that the canonical localization integral will reduce to an integral over the finite-dimensional manifold M (as in Section 4.2), rather than a sum or integral over the moduli space of classical solutions Here the
zero
.tm
i.e.
as
0,
==
above.
right-hand side of (4.103) with (4.123), we use the standecompose LM and LA'M into constant modes and fluctuand scale the latter by 1/vfA,
To evaluate the
dard trick. We ation modes
X,Z(t)
=
X0, +v*)/VA
77,z (t)
not,
+
e (t) / VA-
dt
qm (t)
(4.124)
where T
xlj'
T
dt x" (t)
-
770
0
0
T
49tX/10
=
atqoll
=
0
f 0
(4.125)
T
dt -4-m (t)
f
dt
e(t)
=
0
0
Some features of the space of T-periodic classical trajectories for both energy conserving and non-conserving Hamiltonian systems have been discussed recently
by
Niemi and Palo in
[119, 1221.
110
Quantum Localization Theory
4.
decomposition (4.124) is essentially some complete sets of states JX1J'(t)JkEZ k The
(t)
k
k
Space Path Integrals
for Phase
a
Fourier
and
(t)
Feynman
so
in terms of
that
Uk"77k"(O
M
k560
and the
decomposition
f?7A(t)JkEZ) k
(4.126)
kOO
path integral
in the
measure
is then defined
just
as
before
as
[d 2nx] [d 2n.1
=
2n
d
xo d
2n
11
77o
d 2n,
(t) d2n (t)
tE[O,T] =
d
2n
xo
d2n 77o
11 d
2n
Sk
(4.127)
d2n Uk
kOO
resealing
With the
QSIP
(4.124)
in
of the fluctuation
modes,
the gauge
fixing
term
is T
QsV)
=
I 1 dt
x
it
((S?V)P'Vat
_
g,,,at2) Xv +
1Rpvxl'. bv +
2
Agjv-9t v
0
+ 0 (1
/,/-A-) (4.128)
where
we
integrated by parts
have
over
t and used the
periodic boundary
conditions. In
(4.128)
we see
mannian manifold
usual
the appearence of the equivariant curvature of the RieSince S?V and R there act on the fluctuation
(M, g).
they
be
interpreted
forming
the
equivariant curvature resealing the fluctuation and zero modes decouple in the localization limit A -+ oo, just as before. The integrations over the fluctuations are as usual Gaussian, and the result of these integrations is
modes,
as
can
as
of the normal bundle of M in LM. With the above
Z(T)
-
f chv(-iTw)
A
(det'11J, 'o9t
-
(Rv)','Il)
-1/2
(4.129)
M
This form of the partition function is completely analogous to the degenerate localization formula of Section 3.7, and it is also similar to the formula (4.122),
except that now the domain of integration has changed from the moduli space LMS of classical solutions to the entire phase space M. This makes the
(4.129) much more appealing, in that there is no further reference T-dependent submanifold LMS of M. Note that (4.129) differs from the classical partition function for the dynamical system (M, W, H) by a oneloop determinant factor which can be thought of as encoding the information due to quantum fluctuations. The classical Boltzmann weight e-iTH comes from evaluating the action S[x] on the constant loops x E M C LM, so that formula
to the
4.6 The Niemi-Tirkkonen Localization Formula
(4.129)
is another sort of semi-classical localization of the
ill
Feynman path
integral. The fluctuation determinant in
Section 4.2 and it
now
yields
the
(4.129)
be again evaluated just as in equivariantA-genus (2.96) with respect to can
the equivariant curvature RV. The localization formula
Z(T)
(4.129)
is therefore
f chv(-iTw) AAv(TR)
-
(4.130)
M
[127] and it expresses the partition (finite-dimensional) integral over the phase of equivariant characteristic classes in the U(1)-equivariant coho-
This is the Niemi-Tirkkonen localization formula
quantum space M
function
mology generated by tage of this formula
assumptions localization
as a
the Hamiltonian vector field V
over
appear to have gone into its derivation
constraints).
WKB localization
breaks down
(e.g.
on
M. The
huge
advan-
the localization formula of the last Section is that It thus
applies
not
only
(other
to the
no
than the standard
cases
by
covered
the
theorem, but also to those where the WKB approximation when classical paths coalesce in LM). Indeed, being a lo-
time-independent loops it does not detect degenerate types of phase space trajectories that a dynamical system may possess. In fact, the localization formula (4.130) can be viewed as an integral over the equivariant generalization of the Atiyah-Singer index density of a Dirac operator with background gravitational and gauge fields, and it therefore represents a sort of equivariant generalization of the Atiyah-Singer index theorem for a twisted spin complex. Indeed, when H, V 0 the effective action in the canonical localization integral is calization onto
--->
(S +
0 +
=
AQSV))IH=V=O
f 1Ag1,,V'e dt
+
01,V'
+
Ag/, ,77/JVt ,qV+
177 AWMV77V
2
0
On the other
hand, the left-hand side
of the localization formula
becomes
Z(T)IH=O
=
trJJ
e
-iftT11
IT=O= dimHM
(4.103) (4.132)
integer representing the dimension of the free Hilbert space assoS(H 0) and which can therefore only describe the topological characteristics of the manifold M. Recalling the discussion of Section 4.2, we see that the action (4.131) is the supersymmetric action for a bosonic field
which is
an
ciated with
x11(t)
=
and its Dirac fermion superpartner field 7711(t) in the background of a 1 Dirac 0,, and a gravitational field gl, i.e. the action of N
gauge field
=
2
supersymmetric quantum mechanics. Moreover, the integer (4.132) coincides with the V 0 limit of (4.130) which is the ordinary Atiyah-Singer index for =
twisted spin complex (the 'twisting' here associated with the usual sYmplectic line bundle L --* M). Thus the localization formalism here is just a
112
Quantum Localization Theory for Phase Space Path Integrals
4.
general case of the localization example of Section 4.2 above which reproduced quite beautifully the celebrated Atiyah-Singer index theorem We shall describe some more of these cohomological field theoretical aspects of equivariant localization in Section 4.10, and the connections between the a more
localization formalism and other supersymmetric quantum field theories in Chapter 8. The equivariant cohomological structure of these theories is consistent with the
logical topological
supersymmetric models (the basic toposee below) and they always yield certain of the underlying manifolds such as the Atiyah-Singer
topological
field theories
-
invariants
nature of
Section 4.10
index.
4.7 Connections with the Duistermaat-Heckman
Integration Formula In this Section
gral localization
we
shall
point
out
some
relations between the
path
inte-
formulas derived thus far and their relations to the finite-
dimensional Duistermaat-Heckman formula. Since the localization formulas are
all derived from the
same
fundamental geometric constraints, one would least, they are all related to each other. In par-
expect that, in some ticular, when 2 localization formulas hold for a certain quantum mechanical path integral, they must both coincide somehow. We can relate the various localization formulas by noting that the integrand of (4.130) is an equivariantly closed differential form on M (being an equivariant characteristic class) with respect to the finite-dimensional equivariant cohomology defined by the d + iv. Thus we can apply the Berlineordinary Cartan derivative DV form theorem Vergne (in degenerate compare with Section 3.7) of Section 2.6 to localize the equivariant Atiyah-Singer index onto the critical points of limits at
=
-
the Hamiltonian H to obtain
Z(T)
I chv(-iTw) (R) lvv
-
Ev
-
A
Av (TR)
MV
so
that
a
(degenerate)
MV the equivariant Euler
(4.133)
path integral Mv, and of the normal bundle A(V. Note finite-dimensional localization formula (3.118) only equivariant A-genus which arises from the evalua-
Hamiltonian gives
in terms of the
onto
Mv
a
localization of the
Chern class restricted to
equivariant class and A-genus
that this differs from the
in the appearence of the tion of the temporal determinants which
occur.
the quantum fluctuations about the classical
This factor therefore encodes
values, and
its appearence is
analogy, as well as the localization of the quantum paxtition function in general, requires that boundary conditions for the path integral be selected which respect the pertinent supersymmetry. We shall say more about this requirement This
later
on.
4.7 Connections with the Duistermaat-Heckman
Integration
Formula
113
quite natural according to the general supersymmetry arguments above (as DiracA-genus quite frequently arises from supersymmetric field theory path integrals). Furthermore, the localization formula (4.133) follows from
the
the moduli space formula (4.122) for certain values of the propagation time (see the discussion at the beginning of the last Section).
T
The connection between the WKB and Niemi-Tirkkonen localization fornow immediate if we assume that the critical point set MV of
mulas is
the Hamiltonian consists of Hamiltonian H is field
(4.106)
set
we can
r
only isolated and non-degenerate points (i.e. the
function).
Morse
a
=
0 and
s
Then in the canonical localization vector =
-I
so
that
T
f
ivg
dt
g1,,V1',q'
0
(4.134)
T
f [-I (S?v)m,?7mq'
QsO
dt
2
+
Vt'gi,, (-4-'
-
VL1)
0
We
use the rescaled decomposition (4.124) again which decouples the zero modes from the fluctuation modes because of the "hidden" supersymmetry. The Gaussian integrations over the fluctuation modes then yields
Z(T)
f
d 2n xo
e-
iTH
rT-etl
det
o9t
f2v -
Qv
e-iTH(p)
J(V) PEMv
v det_Qv
A(TQV)
(4.135) (ordinary) Dirac A-genus arises from evaluating the temporal determinant in (4.135) as described before and we recall that Qv (p) 2dV (p) where the
=
2w
-
can
1
(p)H (p)
=
critical point p E M V Thus under these circumstances we localize the partition function path integral onto the time-independent at
a
-
classical trajectories of the dynamical system, yielding a localization formula that differs from the standard Duistermaat-Heckman formula (3.63) only by the usual quantum fluctuation term. The localization formula
degenerate
formula
(4.133)
(4.135)
follow from the WKB formula
[26, 165]
which
of
course
also follows
in the usual way, and it
(4.115) using
probes the first cohomology
directly
be shown
can
from the
[851
to also
the Weinstein action invariant
group of the
symplectomorphism
group of the symplectic manifold (i.e. the diffeomorphism subgroup of canonical transformations). This latter argument requires that M is compact, the classical trajectories are non-intersecting and each classical trajectory can be
contracted to
(for
a
critical
point of H through
instance when H1 (M;
boundary
condition
xl'(0)
R)
=
a
family
of classical
trajectories
0), and that the period T is such that the x" (T) admits only constant loops as solutions to =
the classical equations of motion. The localization onto the critical points of the Hamiltonian is not entirely surprising, since as discussed at the beginning
114
4.
for Phase
Quantum Localization Theory
Space Path Integrals
of the last Section for Hamiltonian circle actions
MV
and
coincide.
(4.130)
dex
is
in this
particular,
tonian H is
[85].
We
see
arguments
a
that the
conclude from Kirwan's theorem that the Hamil-
case
perfect
Morse function that admits
(such
as
Kirwan's way that
exactly phase space integrals.
4.8
locuses LMS
zero
of
only
even
Morse indices
therefore that localization formulas and various Morse theoretic
the
in
M the
from the
Drawing
Duistermaat-Heckman theorem in
on
analogy (4.135) with the equivariant Atiyah-Singer in(i.e. its stationary phase approximation), one can, given exactly by
general
in
same
theorem) they
(formally)
follow
followed for
and
Equivariant Localization
path integrals ordinary finite-dimensional for
Quantum Integrability
Chapter we shall discuss some more formal features path integrals, as well as some extensions of it. We have shown in Chapter 3 that there is an intimate connection between classical integrability and the localization formalism for dynamical systems. With this in mind, we can use the localization formalism to construct an alternative, geometric formulation of the problem of quantum integrability [47, 120] (in the sense that the quantum partition function can be evaluated exactly) which differs from the usual approaches to this problem [35]. As in Section 3.6 we consider a generic integrable Hamiltonian which is a functional
For the remainder of this
of the localization formalism for
H
=
H(1)
of action variables Ja which
are
in involution
as
in
(3.87).
From
the point of view of the localization constraints above, the condition that H generates a circle action which is an isometry of some Riemannian geometry on
M
means
that the action variables Ja generate the Cartan subalgebra of (M, g) in its Poisson bracket realization on
the associated isometry group of
(M, W)
-
For such to write the
Z(T)
dynamical system, we use quantum partition function
a
=
exp
(
T
-i
I
dt H
[ iji(t)
0
1 if
set of
a
generating functionals ja(t)
as
)
e _jjj?jj [d 2nX] ,,/dt
LM
(4.136)
T
x
exp
dt
(Om, bl_t
-
jaja) J=O
0
To evaluate the
path integral in (4.136),
we
consider
an
infinitesimal variation
of its action
J(OA : /_t
-
jagA1a)
jaja)
(4.137)
with the infinitesimal Poisson bracket variation
jx/-t
=
ca
ga, xttj,"
=
_,a,ttvaVja
(4.138)
4.8
where &
are
Equivariant Localization and Quantum Integrability
infinitesimal
115
coordinate-independent parameters. The transforto the leading order infinitesimal limit of
(4.137),(4.138) corresponds
mation
the canonical transformation
x14
--*
f e-c"., x14 eca
x14 +
.1a
X,I+
0
1XI" P L,
b +2 ,a f fX/i, [a i,,
(4.139) ,
ib+
and it gives
j(OA&/j, after
an
integration by parts
-
over
jap)
=
- aja
(4.140)
time. Since the Liouville
measure
in
(4.136)
is invariant under canonical
transformations, it follows that the only effect of the variation (4.140) on the loop space coordinates in (4.136) is to shift the external sources as ja ja + p. Note that if we identify ja (t) as --,
the
temporal component Aa0 of a gauge field then this shift has the same functional form as a time-dependent abelian gauge transformation [120]. Thus if for some reason the quantum theory breaks the invariance of the Liouville measure
under these coordinate
transformations,
we
would expect to be able
to relate the non-trivial Jacobian that arises to conventional gauge anomalies
[151]. Thus if
we
Fourier
and fluctuation modes
decompose the fields ja(t) into their zero modes Joa ja (t) as in (4.124), we can use this canonical transfor-
'gauge' away the time-dependent parts of ja in (4.136) so that the path integral there depends only on the constant modes Joa of the generating functionals and the partition function is given by mation to
Z(T)
=
iTH
exp
1i
'9
1
[d2nX] V/det 11-011
OJO LM
(4.141)
T x
i
exp
f
dt
(OA &it
jaja) 0
-
0
Since the Hamiltonian group action
(4.130) Z(T)
on
M,
JOaP
we
can
in the action in
localize it
JO=O (4.141) generates
an
abelian
using the Niemi-Tirkkonen formula
to arrive at
(-iTH 1i ajol) f chi,-Ia (-iTw) AAj,.i.(TR) 1
-
exp
M
(4.142) JO=O
and so the path integral now localizes to a derivative expansion of equivariant characteristic classes. The localization formula (4.142) is valid for any inte-
grable Hamiltonian system whose conserved charges joaja generate a global isometry on M, and consequently the localization formalism can be used to establish the exact quantum solvability of generic integrable models.
Quantum Localization Theory for Phase Space Path Integrals
4.
116
examples of integrable models where valid, and this has led (4.115) to the conjecture that for a large class of integrable field theories a "proper" version of the semi-classical approximation should yield a reliable reproduction of the features of the exact quantum theory [1761. The formula (4.142) is one such candidate, and thus it yields an explicit realization of this conjecture. However, one may also hope that the localization principle of Section 4.4 Indeed,
there
are
several non-trivial
is known to be
the WKB localization formula
could be used to derive weaker versions of the localization formulas above for some
(in
dynamical systems
the
sense
For
this,
are
in involution
we
of motion
ja
which
not
are
necessarily completely integrable [85]
that the localization formalism above does not carry through). consider a Hamiltonian with r < n conserved charges Ia which
=
as
(3.87),(3.88),
in
and which have the classical equations
0. We then set T
T
V)
=
I
dt Ia a/, Jantt
,
QSV)
=I (fa )2
(4.143)
dt
0
0
integral (4.103). The cohomological relation 0 follows from the involutary property of the charges Ia. Q2,0 LSV) S Then the right-hand side of (4.103) yields a localization of the path integral onto the constant values of the conserved charges Ia, in the canonical localization =
=
Z(T)
-
I [d2nx]
-11p1l 11 j(ja ) ,Fdet
(4.144)
is
a
(4.144)
a=1
LM
The formula
e SM
weaker version of the above localization formulas
which is valid for any non-integrable system that admits conserved charges. It can be viewed as a quantum generalization of the classical reduction theorem [7] which states that conserved charges in involution reduce the dynamics onto the constant
symplectic subspace
(classical)
pletely integrable
this
original phase space determined by the integrals of motion Ia. When H is com-
of the
values of the
subspace coincides with the
discussed in Section 3.6. Thus localization formulas above
even
(e.g.
when there
the WKB
are
invariant Liouville tori
corrections to the various
approximation),
the supersym-
metry arguments of Section 4.4 can be used to derive weaker versions of the localization formulas. Notice that, as anticipated, the localization formula
(4.144)
does not presume any isometric structure
on
the
phase
space
(see
the
provide a natural geometric framework for understanding quantum integrability, and the localization formulas associated with general integrable models represent equivariant characteristic classes of the phase space. For more details about this and other connections between equivariant localization and integrability, discussion of Section
see
[47, 84, 85].
3.6). Equivariant cohomology might
therefore
4.9 Localization for Functionals of
4.9 Localization for Functionals of In the last Section were
considered
we
whether
which
or
117
Isometry Generators
particular class
functionals of action variables and
localization formula for these now
a
Isometry Generators
we were
of Hamiltonians which
able to derive
dynamical systems.
a
quite general explore
It is natural to
not localization formulas could be derived for Hamiltonians
general types of functionals. We begin with the case where a dynamical system is an a pTiori arbitrary functional of observable H which generates an abelian isometry through the an .F(H) Hamiltonian equations for H in the usual sense. Thus we want to evaluate the path integral [128] are more
the Hamiltonian of
T
Z(TI.F(H))
=
f [d2nx] \/det I 10 11
i
exp
LM
f
dt
(01, V'
-
JF(H))
(4.145)
0
We shall
see that such path integrals are important for certain physical appliNote, however, that although such functionals may seem arbitrary, we must at least require that F(H) be a semi-bounded functional of the observable H [159]. Otherwise, a Wick rotation off of the real time axis to imaginary time may produce a propagator tr 11 e-iT.F(H) 11 which is not a tempered distribution and thus eliminating any rigorous attempts to make the path integral a well-defined mathematical entity. The formalism used to treat path integrals such as (4.145) is the auxilliary field formalism for supersymmetric theories [71, 107, 108] which enables one to relate the loop space equivariant cohomology determined by the derivative QS to the more general model-independent S' loop space formalism, i.e. that determined by the equivariant exterior derivative Qj. We recall from Section 4.4 that in this formulation the path integral action is BRST-exact, as required for supersymmetric field theories. Here the auxilliary fields that are introduced turn out to coincide with those used to formulate generic Poincar6 supersymmetric theories in terms of the model-independent S' loop space equivariant cohomology which renders their actions BRST-exact. These supersymmetric models will be discussed in Chapter 8.
cations.
To start,
we
assume
Gaussian functional
exp
(
integral
T
-i
I
dt
F(H)
0
Because of the local
that there is
)
a
function
0( )
such that
17(H)
is
a
transformation of it,
1f
T
[< LR
exp
i
0
dt
2
-
0( )H)
(4.146)
integrability of F(H), locally such a function 0(6) can always be constructed, but there may be obstructions to constructing 0(6) globally on the loop space LM, for the reasons discussed before. The transformation 6 -+ 0 which maps the Gaussian in to a non-linear functional of 0 is just the Nicolai transformation in supersymmetry theory [22, 116), i.e. the
118
Quantum Localization Theory for Phase Space Path Integrals
4.
change into
a
of variables that maps the bosonic part of the supersymmetric action change of variables coincides
Gaussian such that the Jacobian for this
with the determinant obtained of the
supersymmetric
struct
a
by integrating
over
the bilinear fermionic part
action. This observation enables
one to explicitly conpath integral (4.145). Notice that when F(H) is either linear or quadratic in the observable H, the Nicolai transform is directly related to the functional Fourier transfor-
localization for the
F(H),
mation of
T
T
exp
-
if
f [do]
F(H)
dt
-i
exp
dt
P(O)
-i
exp
0
LR
0
f
T
f
dt
OH
0
(4.147) However, for more complicated functionals F(H) this straightforward. In particular, if we change variables
(4.146),
sian transformation
we
-i
f
in the Gaus-
find
T
exp
connection is less
T
dt
-97(H)
f [do]
=
LR
0
fj
'(O)
if ( dt
exp
tE [0,T]
1 2
2 (0)
-
OH)
0
(4.148) that the effect of this transformation is to isolate the isometry generator H and make it contribute linearly to the effective action in (4.145) (as we
so
did in the last
Section).
prescriptions of Section
Substituting (4.148)
This allows
one
to localize
(4.145) using
the
general
4.4 above. into
(4.145),
we
then carry out the
led to the Niemi-Tirkkonen localization formula
same
steps which
(4.130). However,
now
there
auxilliary, time-dependent field 0 which appears in the path integral action which must be incorporated into the localization procedure. These fields appear in the terms OH above and are therefore interpreted as the dynamical generators of S(u(l)*). We introduce a superpartnerq for the auxilliary field 0 whose Berezin integration absorbs the Jacobian factor in (4.148). The path integral (4.145) thus becomes a functional integral over an extended superloop space. As discussed in Appendix B, one can now introduce an extended BRST-operator incorporating the super-multiplet (0,'q) -such that the partiis
an
tion function is evaluated with
the
BRST-complex
of
physical
BRST-exact action whose argument lies in states and, as was the case in Section 4.2, the a
Niemi-Tirkkonen localization onto constant modes becomes manifest. This is the so-called Weil differential whose
cohomology cohomology [99, 129]. This more sophisticated technique is required whenever the basis elements oa of the symmetric algebra S(g*) are made dynamical and are integrated out, as extended
BRST-operator
defines the BRST model for the U (I)-equivariant
is the
case
here.
We shall not enter into the cumbersome details of this extended superspace evaluation of (4.145), but merely refer to [128] for the details (see also
4.9 Localization for Functionals of
Appendix
B for
sketch of the
a
idea).
Isometry Generators
The final result is the
119
integration for-
mula 00
doo 0` (0o) e iT 2 (0o)/2
Z(TJ,F(H))
0
f chp,,v(-iTw) AAoov(TR)
00
(4.149) where 00
the
modes of the
0. (4.149) is valid (formally) for any semi-bounded functional _F(H) of an isometry generator H on M. Thus even for functionals of Hamiltonian isometry generators the localization formula is a relatively simple expression in terms of equivariant characteristic classes. The only computational complication in these formulas are
zero
is the identification of the function
P(O)).
We note that
when.F(H)
=
auxilliary
(O) (or H,
we
field
the functional Fourier transform
have
0( )
consistently to the Niemi-Tirkkonen localization H 2, we find 0( ) portant special case.F(H)
(4.149)
1 and
(4.149)
formula.(4.130). (i.e. F(O)
=
localization formula
=
=
reduces
In the im-
02)
and the
becomes
00
Z(TIH 2)
f doo
_
e
iT020 /2
v
(- iTw)
Ao. v (TR)
A
(4.150)
M
00
which is the formal
f choo
path integral generalization of the Wu localization formula
(3.128). In fact, the above dynamical treatment of the multipliers 0 suggests a possible non-abelian generalization of the localization formulas and hence a path integral generalization of the Witten localization formula of Section 3.8 [161]. At the same time we generalize the localization formalism of Section 4.8 above to the
case
where the Hamiltonian is
a
functional of the generators
of the full
isometry group of (M, g), and not just simply the Cartan subgroup thereof. We consider a general non-abelian Hamiltonian moment map (3-30) where the component functions HI are assumed to generate a Poisson algebra realization of the isometry group G of some Riemannian metric g on M. As mentioned in Section 3.8, when the 0' are fixed we are essentially in the abelian situation above and this what follows. Here
case
will be discussed in
more
detail in
that the
multipliers 0' are time-dependent and we integrate over them in the path integral following the same prescription for equivariant integration introduced in Section 3.8. This corresponds to modelling the G-equivariant cohomology of M in the Weil algebra using the BRST formalism [129, 161] (see Appendix B). When the 0' are fixed we assume
parameters, the action functional
(4.26) generates the
action of
S'
on
LM in
the model
independent circle action described in Section 4.4 above. However, when the 0' are dynamical quantities, S generates the action of the semidirect
the
product LGDS', where the
loop parameter
t and LG
group G. These actions
are
action of
C' (S',
G)
S' corresponds
to translations of
loop group of the isometry generated, respectively, by the loop space vector =
is the
120
4.
Quantum Localization Theory for Phase Space Path Integrals
fields T
dt
Vsi
:V'(t)
JX/1 (t)
0
T
I
VLG
T
0'(t)w1"(x(t))
dt
(
j Jx- (t)
H')
j -
=
Jxt' (t)
0
j
dt
0'(t)V'(t)
0
(4.151) The commutator
algebra of the
(4.151)
vector fields
is that of
LG2DS'
on
LM, T
VS1, VLGI
dt
'H
[Va (t), Vb (tl)]
a
=
fabc Vc(t)J(t-t) (4.152)
0
The equivariant extension of the
symplectic
2-form S?
LM is therefore
on
S+Q.
multipliers Oa (now regarded as local coordinates on Lg*) are integrated directly, then the isometry functions H become constraints because the Oa appear linearly in the action and so act as Lagrange multipliers. In this case we are left with a topological quantum theory (i.e. there are no classical degrees of freedom) with vanishing classical action, in parallel to the If the
a
over
finite-dimensional F
=
F(Oa)
case
of Section 3.8.
Alternatively,
argument of the exponential
to the
we can
term in the
add
a
functional
partition function
+ F is equivariantly closed. We then introduce generalization of the procedure outlined above [161] (see Appendix B for details). Introducing an extended equivariant BRST operator QT for the semi-direct product action of LGs)Sl on LM (the non-abelian version of that above), it turns out that S + Q + F is equivariantly closed with respect to QT only for either F 0 or F !(Oa)2' where the latter is 2 the invariant polynomial corresponding to the quadratic Casimir element of G. Note that this is precisely the choice that was made in our definition of equivariant integration in Section 3.8. As shown in Appendix B, within this framework we can reproduce loop space generalizations of the cohomological formulation of Section 3.2 for the Hamiltonian dynamics. The rest of the localization procedure now carries through parallel to that above and in the Niemi-Tirkkonen localization, and it yields the localization formula [161]
such that the a
quantity S + Q
non-abelian
=
Z(T)
-
j 9
which is
a
and is the
.
dim G
11 a=1
d0a0
e
iT(,Oo' )2 /2
f chko-
=
v-
(-iTw)
AAo.-v-(TR)
(4.153)
M
quadratic localization formula (4.150) path integral generalization of the Witten localization formula
non-abelian version of the
Topological Quantum
4. 10
presented
Field Theories
121
12
in Section 3.8
Notice that the primary difference between this non-abelian localization and its abelian counterpart is that in the latter the functional F(O) is a pHoTi arbitrary.
4.10
.
Topological Quantum Field Theories
In this last Section of this
Chapter
basis elements of
fixed numbers. We wish to
S(g*)
are
we
return to the
case
where the dual
study the properties
of the quantum theory when the effective action is BRST-exact as in (4.91) locally on the loop space [85, 123]. In this case the quantum theory is said to be
topological, in that there are no local physical degrees of freedom and the remaining partition function can only describe topological invariants of the space on which it is defined [22]. We shall see this explicitly below, and indeed
have
we
integral for this,
already
seen hints of this in the expressions for the path equivariant characteristic classes above. To get a flavour first consider a quantum theory that admits a model independent
in terms of we
circle action vector field
globally
generates
the loop space, i.e. whose loop space Hamiltonian global constant velocity U(1) action on LM, so that given locally by (4.111). In this case, the determinant
on
a
its action functional is
that appears in the denominator of the WKB localization formula
L=0= det JjJ(f2 .:k)ll I
11j2S11
det
where the localization is
determinants det
11,9t1j, only
on
the
the
now
detjjf2c9tjj
side of
modes of
WKB localization formula in this
i9t
X.
(4.115)
can
case
loops
now
X0 E
M. Since the
cancel modulo the factor
contribute. Thus the
(degenerate)
becomes
d2nXO VdetjjS?ja,_ojj
Z(T)
is
(4.154)
X=XO
onto the constant
right-hand
zero
=
(4.115)
lx=xo
(4.155)
M
and only the zero modes of the symplectic 2-form contribute. Since this path integral yields the topological Witten index of the corresponding supersymmetric theory [167], the localization formula identifies the loop space characteristic class which corresponds to the Witten index of which the ensuing
Atiyah-Singer This is
one
index counts the
of the
new
zero
modes of the associated Dirac operator. into supersymmetric theories from the
insights gained
equivariant localization formalism.
(4.155)
purely cohomological reprephysical information. Next, consider the more general case of an equivariantly-exact action (4.91). Note that this is precisely the solution to the problem of solving is
sentative of the manifold M which contains
The
procedure outlined
above could also be
a
no
employed
in the discussion of the
Witten localization formalism in Section 3.8. This has been
carrying
out the
implicitly
equivariant integrations there (see Appendix B).
done in
122
4.
Quantum Localization Theory for Phase Space Path Integrals
loop space equivariant Poincar6 lemma for S + 0. If we assume that the symplectic potential is invariant under the global U(I)-action on M, as in (3.26), then the Hamiltonian is given by H iVO and the loop space 1-form ,0 in (4.91) is given by the
=
T
f
dt
(4.156)
0,,(x(t))?71'(t)
0
loop space localization principle naively implies that the resulting path integral should be trivial. Indeed, since the 1-form (4.156) lies in the subspace (4.88), the partition function can be written as The
Z(T)
f
=
2n [d2nx] [d 77] e'AQs
(4.157)
LMOLA'M
independent of the parameter A E R. In particular, it should be independent of the action S. However, the above argument for the triviality of the path integral assumes that 0 is homotopic to 0 in the subspace (4.88) under the supersymmetry generated by QS, i.e. that (4.91) holds globally for all loops. For the remainder of this Chapter we will assume that the manifold M is simply 0. Then the above argument presumes that connected, so that HI (M; R) 0 is trivial. If this is not the second DeRham cohomology group H 2(M; R) the careful about be the case, then one must A-independence of the arguing the of Consider symplectic 2-forms family path integral (4.157). and it is
=
=
w(A) associated with the action in
=
AdO
(4.157).
parametrized by
phase
space M
Since
by assumption -I(x) implies that the kinetic
is the
theorem
=
(4.158)
Aw
We consider
closed
a
loop -Y(x)
periodic trajectory x(t) boundary of a 2-surface Zi the
AO in
term
(4.157)
:
[0, T]
in
can
M,
in the --+
M.
Stokes'
be written
as
T
dt
-Y W
0
For
W(,\)
0 (A)
0('\)(x(t)) bO(t)
(4.159)
El
consistency of the path integral (4.157), which
closed
loops
is expressed as a sum over M, the phase (4.159) must be independent of the represenZ, spanning -y(x), owing to the topological invariance of the
in
tative surface
partition function Z(T) (of opposite orientation surface
have
(sphere)
we introduce another surface Z2 boundary -y(x) and let Z be the closed divided into 2 halves Z, and Z2 by 7(x), then we
over
to
which is
LM. Thus if
Zi)
with
Topological Quantum Field Theories
4. 10
e
and
consequently
the
e
integral
of
f
w(A)
over
M must satisfy a version of the Dirac tization condition [168]
or
A
1 =
21r
f
e
27r E
W(X)
123
(4.160)
2
any closed orientable surface Z in
Wess-Zumino-Witten
Jw
E
(flux)
quan-
(4.161)
Z
Z
means that w(,\) is an integral element of H 2(M; R), i.e. it defines an 2 integer cohomology class in H (M; Z), which is possible only for certain
This
discrete values of A
E
R. It follows that
a
continuous variation JA of A cannot
path integral (4.157) invariant and it depends non-trivially on the localization 1-form 0 =-,O and thus also on the action S. Thus the path integral (4.157) defines a consistent quantum theory only when the symplectic 2-form (4.158) defines an integral curvature on M. However, if we introduce a variation 0 --* 0 + JO of the symplectic potential in da a trivial (4.157) corresponding to a variation w --+ w + 6w with Jw element of H 2(M; R) in the subspace (4.88), then the localization principle implies that the path integral remains unchanged (using Stokes' theorem for Jw in (4-161)). Thus the path integral depends only on the cohomology class of w in H 2(M; R), not on the particular representative w dO, which means that the partition function (4.157) determines a cohomological topological quantum field theory on the phase space M. Furthermore, we note that within the framework of the Niemi-Tirkkonen localization formula, the BRST-exact term Qs(AO + ), with given by (4.156) and 0 given in (4.123), gives the effective action in the canonical localization integral (4.96). We saw earlier that the Qb-exact piece of this action corresponds to the Atiyah-Singer index of a Dirac operator iY1 in the background of a U(1) gauge field 0,, and a gravitational field g,,,. The remaining terms there, given by the iv-exact pieces, then coincide with the terms that one expects in a supersymmetric path integral representation of the infinitesimal Lefschetz number (also known as a character index or equivariant leave the
=
=
G-index)
indeXH(iY; T) generated by
=
lim A
trJJ
e
iTH
( e-Al)tl)
-
e-\VV)11
*00
the Hamiltonian H
[21, 23, 24, 120, 127].
(4.162)
This follows from
arguments similar to those in Section 4.2 which arrived at the supersymmetric path integral representation of the (ordinary) Atiyah-Singer index. In general, when the Dirac operator is invariant under the action of the isometry group G on A4, [VI, iV] 0, then the eigenstates of i)F which correspond to a fixed =
eigenvalue E define a representation of the Lie algebra of G. It is possible to show just as before that the right-hand side of the Lefschetz number (4.162) is independent of A c: R+, and, therefore, when either DtD or VDt has no
124
4.
zero
i)F
Quantum Localization Theory
modes,
we can
contribute to
take the limit A
for Phase
--+
(4.162). Consequently,
SPace Path Integrals
0 there and
the
only
the
zero
modes of
equivariant index coincides with
the character
indeXH (iV; T) of the
irreducible) representation by the zero modes of i)F.
(reducible
G determined
strR
=
or
e
iTH
(4.163)
R of the Cartan element H of
LVO Consequently, in the case of Hamiltonian systems for which LVg 0, the Niemi-Tirkkonen localization formula (4.130) reproduces the Lefschetz fixed point formulas of Bismut [23, 24] and Atiyah, Bott and Singer [41), =
provided that boundary conditions for the path integral have been properly selected. Thus a purely bosonic theory can be related to the properties of a (functional) Dirac operator defined in the canonical phase space of the bosonic theory, and this analogy leads one to the hope that the above localization prescriptions can be made quite rigorous in a number of interesting infinite-dimensional cases. Note also that the path integral (4.157) has the precise form of a Witten-type or cohomological quantum field theory, which is characterized by a classical action which is BRST-exact with the BRST charge QS representing gauge and other symmetries of the classical theory. These types of topological field theories are known to have partition functions which are given exactly by their semi-classical approximation more precisely, they admit Nicolai maps which trivialize the action and restrict to the moduli space of classical solutions [22]. Thus the topological and localization properties of supersymmetric and topological field theories find their natural explanation within the framework of loop space equivariant localiza-
tion.
Of course, the above results rely heavily on the G-invariance condition (3.26) for the symplectic potential 0. In the general case, we recall from Section 3.2 that
we
have the relation
(3.47)
which holds
locally
in
a
neighbour-
dO. In this hood JV in M away from the critical points of H and in which w and solution the Poincar6 lemma to equivariant although case, (3.47) gives a =
the action is
locally BRST-exact, globally
the quantum
theory
is non-trivial
and may not be given exactly by a semi-classical approximation. Then the path integral (4.96) has the form of a gauge-fixed topological field theory,
Schwarz-type or quantum topological field theory [221, charge representing the gauge degrees of freedom. With QS ?9 as in (4.156), the loop space equivariant symplectic 2-form can be written in the neighbourhood LA( as
otherwise known with
as a
the BRST
T
S + Q
Qs(
=
+
dLF)
-
f dF(x(t))
(4.164)
0
and the
path integral
Z(T)
=
can
I LM(&LAlM
be
represented locally
2n
[d x] [d
2n
as
( +dLfl-i fl(x) dF q] e'Qs
(4.165)
4.10
If
Topological Quantum
Field Theories
125
that M is simply connected, so that H1 (M; R) 0, then, by theorem, the dF term in (4.165) can be ignored for closed trajectories the phase spacell. Since from (4.164) we have
we assume
=
Stokes' on
,Cs(?9 it follows that
+
dLF
equivariantly-exact The non-triviality
E
in the
that
particular, function
dLF)
=
Qs(S + Q)
=
(4.166)
0
LAIS M and the effective classical action S neighbourhood LA(.
+ Q is
of the
when the local
occurs
+
path integral now depends on the non-triviality neighbourhoods Af above are patched together. In
invoke the above argument to conclude that the partition depends only on the cohomology class of w in H 2(M; R), in
we can
(4.165)
addition to the critical point set of the action S. Thus the partition function in the general case locally determines a cohomological topological quantum field
theory.
From the discussion of Section 3.6
with the fact that the
we see
that this is consistent
theory locally integrable outside of the critical point set of H. We recall also from that discussion that in a neighbourhood A(
where
is
action-angle variables
any critical points, F above and hence
can
be introduced and where H does not have
explicit realization of the function topological quantum theory (4.165). For integrable models where action-angle variables can be defined almost everywhere on the phase space M, the ensuing theory is topological, i.e. it can be represented by a topological action of the form (4.164) almost everywhere on the loop space LM. Notice that all of the above ar0. In Chapter 6 we guments stem from the assumption that H'(M; R) shall encounter a cohomological topological quantum field theory defined on a multiply-connected phase space which obeys all of the equivariant localization criteria. We also remark that in the general case, when W is not globally exact, the Wess-Zumino-Witten prescription above for considering the action (4.26) in terms of surface integrals as in (4.159) makes rigorous the definition of the partition function on a general symplectic manifold, a point which up until now we have ignored for simplicity. In this case the required consistency condition (4.161) means that w itself defines an integral curvature, which is consistent with the usual ideas of geometric quantization [172]. We shall see how this prescription works on a multiply-connected phase space in Chapter we can
an
construct
explicit
an
realization of the
=
6.
This term is
theory
analogous to the instanton term F A F in 4-dimensional Yang-Mills can be represented in terms of a locally exact form and is therefore
which
non-trivial
only
for
space-times which have non-contractable loops [151].
Equivariant Localization on Simply Connected Phase Spaces: Applications Quantum Mechanics, Group Theory and Spin Systems 5.
to
When the phase space M of a dynamical system is compact, the condition that the Hamiltonian vector field V generate a global isometry of some Riemannian geometry on M automatically implies that its orbits must be closed circles (see ahead Section 5.2). This feature is usually essential for the finite-dimensional localization theorems, but within the loop space localizaframework, where the arguments for localization are based on formal su-
tion
persymmetry arguments on the infinite-dimensional manifold LM, the flows generated by V need not be closed and indeed many of the formal arguments of the last Chapter will still apply to non-compact group actions. For instance, if we wanted to apply the localization formalism to an n-dimensional R 2n,then we potential problem, i.e. on the non-compact phase space M would expect to be allowed to use a Hamiltonian vector field which generates non-compact global isometries. As we have already emphasized, the underlying feature of quantum equivariant localization is the interpretation of an equivariant cohomological structure of the model as a supersymmetry among the physical, auxilliary or ghost variables. But as shown in Section 4.3, this structure is exhibited quite naturally by arbitrary phase space path integrals, so that, under the seemingly weak conditions outlined there, this formally results in the equivariant localization of these path integrals. This would in turn naively imply the exact computability of any phase space path integral. Of course, we do not really expect this to be the case, and there is therefore the need to explore the loop space equivariant localization formalism in more detail to see precisely what sort of dynamical systems will localize. In this Chapter we shall explore the range of applicability of the equivariant localization formulas [40, 159] by presenting a more detailed analysis of the meaning and implications of the required localization symmetries, and we shall work out numerous explicit mathematical and physical applications of the formalisms of the previous Chapters. As we shall see, the global isometry condition on the Hamiltonian dynamics is a very restrictive one, essentially meaning that H is related to a global group action (2.36). The natural examples of such situations are the harmonic oscillator and free particle Hamilto2n nians on R (the trivial Gaussian, free field theories), and the quantization of spin [117] (i.e. the height function on the sphere), or more generally the quantization of the coadjoint orbits of Lie groups [4, 26, 85, 128, 156, 159] and =
R. J. Szabo: LNPm 63, pp. 127 - 201, 2000 © Springer-Verlag Berlin Heidelberg 2000
128
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
the equivalent Kirillov-Kostant geometric quantization of homogeneous phase space manifolds [2, 3]. Indeed, the exactness of the semi-classical approxima-
(or
formula) for these classes of phase space important inspirations for the development of quantum localization theory and these systems will be extensively studied in this Chapter, along with some generalizations of them. We shall see that the Hamiltonian systems whose phase space path integrals can be equivariantly localized essentially all fall into this general framework, and that the localization formulas in these cases always represent important group-theoretical Xd invariants called characters, i.e. the traces trR9 evaluated in an trR ec' irreducible representation R of a group G which are invariant under similarity transformations representing equivalent group representations, and they reproduce, in certain instances, some classical formulas for these characters tion
the Duistermaat-Heckman
path integrals
was one
of the most
=
[87].
In
our case
structure
on
As it is
the group G will be the group of isometries of
essentially
the
isometry
structure of the Hamiltonian
work,
a
Riemannian
M. group G that determines the
integrable
system in the equivariant localization frame-
study the localization formalism from the point of view of possible isometries can be for a given phase space manifold. A detailed analysis of this sort will lead to a geometrical characterization of the integrable dynamical systems from the viewpoint of localization and will lead to topological field theoretical interpretations of integrability, as outlined in Section 4.10. It also promises deeper insights into what one may consider to be the geometrical structure of the quantum theory. This latter result is a particularly interesting characterization of the quantum theory because the partition functions considered are all ab initio independent of any Riemannian geometry on the underlying phase space (as are usually the classical and quantum mechanics). Nonetheless, we shall see that for a given Riemannian geometry, the localizable dynamical systems depend on this geometry in such a way so that they determine Hamiltonian isometry actions. Strictly speaking, most of this general geometric analysis in this Chapter and the next will only carry through for a 2-dimensional phase space. The reason for this is that the topological and geometrical classifications of Riemann surfaces is a completely solved problem from a mathematical point we
shall
what the
of view. We may therefore invoke this classification scheme to in turn classify the Hamiltonian systems which fit the localization framework. Such a neat mathematical characterization of
higher
dimensional manifolds is for
the most part an unsolved problem (although much progress has been made over the last 7 years or so in the classification of 3- and 4-manifolds), so
that
a classification scheme such as the one that foll6ws does not generalize higher-dimensional models. We shall, however, illustrate how these situations generalize to higher dimensions via some explicit examples which
to
will show that the 2-dimensional classifications do indeed tell
us
about the
properties of general localizable dynamical systems. In particular,
we
shall
5.
see
Equivariant Localization
on
Simply Connected
Phase
Spaces
129
that from certain points of view all the localizable Hamiltonians repre"generalized" harmonic oscillators, a sort of feature that is anticipated
sent
from the
previous integrability arguments and
the local forms of Hamiltoni-
which generate circle actions. These seemingly trivial behaviours are a reflection of the large degree of symmetry that is the basis for the large reans
duction of the
analyse
complicated functional integrals
to Gaussian
ones.
We will also
in full detail the localization formulas of the last
Chapter, which will therefore give explicit examples of the cohomological and integrable models that appear quite naturally in (loop space) equivariant localization theory. This analysis will also provide new integrable quantum systems, as we shall see, which fall into the class of the generalized localization formulas (e.g. the Niemi-Tirkkonen formula
(4.130)),
but not the
approximation. Such examples represent
more
traditional WKB
major, non-trivial advance of localization theory and illustrate the potential usefulness of the localization formulas
as
At the
a
reliable calculational tools.
same
time
we can
address
some
of the issues that arise when deal-
phase space path integrals, which are generally regarded as rather disreputable because of the unusual discretization of momentum and configuration paths that occurs (in contrast to the more conventional configuration space (Lagrangian) path integral [147)). For instance, we recall from Section 4.1 that the general identification between the Schr6dinger picture path integral and loop space Liouville measures was done rather artificially, basically by drawing an analogy between them. For a generic phase space path integral to represent the actual energy spectrum of the quantum Hamiltonian, one would have to carry out the usual quantization of generic Poisson brackets jx ', x'J,, 0" (x). However, unlike the Heisenberg canonical commutation relations (4.1), the Lie algebra generated by this procedure is not necessarily finite-dimensional (for M compact) and so the representation problem has no straightforward solution when the phase space is not a cotangent bundle M (& AIM [961, as is the case for a Euclidean phase space. This approach is therefore hopelessly complicated and in general hardly consistent. One way around this, as we shall see, is to use instead coherent state path integrals. This enables one to obtain the desired identification above while maintaining the original phase space path integral, and therefore at the same time keeping a formal analogy between the finite-dimensional and loop space localization formulas. Furthermore, because of their classical properties, coherent states are particularly well-suited for semi-classical studies of quantum dynamics. We shall see that all the localizable dynamical systems in 2-dimensions have phase space path integrals that can be represented in terms of coherent states, thus giving an explicit evaluation of the quantum propagator and the connection with some of the conventional coadjoint orbit models. In this Chapter we shall in addition confine our attention to the case of a simply-connected phase space, leaving the case where M can have noncontractible loops for the next Chapter. In both cases, however, we shall ing
with
=
130
focus a
Equivaxiant Localization
5.
on
Simply Connected
on
Phase
Spaces
the construction of localizable Hamiltonian systems starting from space metric, which will illustrate explicitly the geometrical
geneTic phase
dependence of these dynamical systems and will therefore give a further probe geometrical nature of (quantum) integrability. In this way, we will get a good general idea of what sort of phase space path integrals will localize and a detailed description of the symmetries responsible for localization, as well as what sort of topological field theories the localization formulas will into the
represent.
5.1
Coadjoint Orbit Quantization
and Character Formulas There is
a very interesting class of cohomological quantum theories which quite naturally within the framework of equivariant localization. These will set the stage for the discussion of this Chapter wherein we shall focus on the generic equivariant Hamiltonian systems with simply connected phase spaces and thus present numerous explicit examples of the localization formalism. For a (compact or non-compact) semi-simple Lie group G (i.e. one whose Lie algebra g has no abelian invariant subalgebras), we are interested in the coadjoint action of G on the coset space MG GlHc Jghc : g E GJ, where HC (S')' is the Cartan subgroup of G. The coset obtained by quotienting a Lie group by a maximal torus is often called a 'flag manifold'. The coadjoint orbit
arise
=
=
-
0A1
fAd*(g)A':
=
g E
GJ
is the orbit of maximal
joint
action of G
(Ad*(g)A') (-y) and h is the Cartan
---+
(5.1)
(5.1)
V-y
of g. The natural
(5.2)
E g
isomorphism
in
(5.1)
flag manifold MG GIHC and the coadjoint orbit 0A1 is with the maximal torus HC identified as the stabalizer Ad*(g)A'
group of the
There is
A'(g-1-yg)
subalgebra
between the
gHC
=
A'E h*
,
of G. Here Ad* (g) A' denotes the coad-
dimensionality A, i.e.
on
MG
-
a
=
point A'
E h*. We assume
natural G-invariant
henceforth that H1 (G)
symplectic
structure
which is defined
the point A E
g*
is
by the Kirillov-Kostant given by WA
where T is
a
2
A T
on
2-form
the
[2, 3].
algebra
H2 (G)
coadjoint
=
0.
orbit
This 2-form at
A, T])
1-form with values in the Lie
=
(5.3) g which satisfies the
equation
dA(-y)
=
ad* (T)A(-y)
-=
A([-I, 7])
V-y
E g
(5.4)
5.1
and
ad*(T)
Coadjoint Orbit Quantization and Character Formulas
131
denotes the infinitesimal
The 2-form
(5.3)
is closed and
coadjoint action of the element T E g. non-degenerate on the orbit (5.1), and by OA, by symplectic (canonical) transforma-
construction the group G acts on respect to the Kirillov-Kostant 2-form. Its main characteristic is
tions with
that the Poisson
algebra
(5.3) isomorphically represents
with respect to
the
G,
group
f X1 (A), X2 (A) where Xi E g
are
A(Xi). Alekseev,
regarded
as
[Xi, X2] (A)
linear functionals
Faddeev and Shatashvili
on
[2, 3]
(5.5)
the orbit
0A1 with Xi (A) =_ phase space
have studied the
path integrals for such dynamical systems with Hamiltonians defined on the coadjoint orbit (5.1) (e.g. Cartan generators of g) and have shown that, quite generally, the associated quantum mechanical matrix elements correspond to matrix elements of the Hamiltonian
representation of the
generator of g in
group G. We shall
some
irreducible
this feature
explicitly later on. However, for our purposes here, there is a much nicer description of the orbit space (5.1) using its representation as the quotient space MG GIRC [68]. As a smooth space, MG is an example of a complex manifold of complex dimension n (real dimension 2n), i.e. a manifold which is covered by open sets each homeomorphic to (Cn and for which the coordinate transformations on the overlap of 2 open sets are given by holomorphic functions. Here the complexification of the group G is defined by exponentiating the complexification g (9 C of the finite-dimensional vector space g. Let us quickly review some facts about the differential geometry of complex manifolds. In local coordinates x (zl,..., Zn) E Cn, we can define the tangent space Tx(0'1)MG at 9 X E MG as the complex vector s p ace spanned by the 2 derivatives 1 Z T In 'a=j) see
=
=
and
Tx("O)MG is the complex vector space spanned by the The key feature is that barred and unbarred vectors f --!2-}n=,. .9z"
analogously
derivatives
A
do not mix under sense
phic
(globally) and
a
holomorphic change of coordinates,
anti-holornorphic
(p, q)
and therefore it makes
to consider tensors with definite numbers k and t of holomor-
We refer to these
type
z
as
indices of either covariant
tensors of
is denoted
A(p,q) MG
or
contravariant type.
(k, t).
The vector space of (p + q)-forms of and the exterior algebra Of MG now refines
type
to n
A (p,q) MG
(5.6)
The DeRharn exterior derivative operator d now decomposes into holomorphic and
ql' ax" : A kMG anti-holomorphic exterior
AMG
=
p,q=O
A k+ 1MG
derivative operators
as
d
where o9
azt,
'9
-
A(p,q) MG
-4
=
o9 + 0
A(P+',q)MG (,ql'
(5.7) -
dzA)
and
qA--La2A
:
A(p,q) MG A(p,q+l) MG (qT' d.P). The anti-holomorphic exterior derivative 0 is called the Dolbeault operator, and the nilpotency of d now translates --
into the set of conditions
-
132
Equivariant Localization
5.
=,92
0
Finally,
let
fined rank
T(0,1)MG)
us
note that
(1, 1)
Simply Connected
on
=
02
a6
=
Phase
Spaces
0,9
+
(5.8)
complex manifold always possesses a globally de(i.e. an endomorphism of the space T(',O) MG 6) It can be defined locally by
a
tensor field J
with j2
=
_1.
J'A
jr"a
ij/1 V
=
i jrA,
(5.9)
with all other components vanishing, and it is known as a complex structure. Given this important property of the coadjoint orbit, we now introduce local complex coordinates (zA,,P') on Mc which are generated by a complex
(5. 1)
0 and topological features H1 (M G; Z) dim HC is the rank of G and Z' r corresponds to the lattice of roots of Hc [162]. The cohomology classes in H 2(MG; Z) are then represented by r closed non-degenerate 2-forms of type structure J. The orbit
H 2 (Mg.
Z)
=
H1 (Hc; Z)
has the
=
Z',
=
where
=
(1,1) [68] &A 29 b (z, )
w
The components g,,r, of
non-degeneracy T(0,1)MG by
(5.10)
condition
holomorphic
and
on
d F'
(i)
(5.10)
define Hermitian matrices,
implies that they define
9
The closure condition
A
=
g,*I,
metrics
=
on
g,'F', and the
T(1,0)MG
ED
g
(5.10)
the 2-forms
can
anti-holomorphic components
be written in terms of the
of the exterior derivative
(5.7)
as
aw(i)
OW(i)
=
=
(5.12)
0
analogue of the Poincar6 lemma for the Dolbeault operator 6 is the Dolbeault-Grothendieck lemma. Since the 2-forms w(') in the case at hand
The
are
closed under both c9 and
that
locally they
can
be
6,
the Dolbeault-Grothendieck lemma
expressed
w(')
in terms of C'-functions
or
=
F(')
On
implies MG as
-06F(')
(5.13)
in local coordinates
W
4AZ" O In
general,
is called
a
a
complex manifold
=
192 F(')(z,.
)
(5.14)
-
azt'09' Fl with
a
symplectic
Kdhler manifold. The closed 2-forms
structure such
(5.10)
are
as
(5. 10)
then refered to
as Kiffiler classes or Khhler 2-forms, the associated metrics (5.11) are called Khhler metrics, and the locally-defined functions F(') in (5.14) are called Kdhler potentials. For an elementary, comprehensive introduction to com-
plex manifolds and Kiffiler structures,
we
refer to
[41]
and
[61].
In the
case
5.1 at hand
here, the above
which act
on
133
yields a G-action on MG by symplecholomorphic functions f (z) on MG the Khhler potentials by
(canonical)
tic
Coadjoint Orbit Quantization and Character Formulas construction
transformations
[68],
i.e.
F(')(z,, )
(5.15)
This follows from the fact that the cotangent bundle of G is
T*G
=
G
(5.16)
g*
x
on T*G is g A) (g j, A). w() define G-invariant integral symplectic 2 structures on MG- Since H (G) 0, the 2-cocycles in (3.39) vanish and this G-action determines group isomorphisms into the Poisson algebras of M. This also follows directly from the property (5.5) of the Kirillov-Kostant 2-form above. Notice that the only non-vanishing components (up to permutation of indices and complex conjugation) of the Riemannian connection
so
that the natural
Consequently,
symplectic
action of G
=
the closed 2-forms
=
(5.11)
and curvature associated with the Kdhler metric
r"'A
=
RA
gAP'9Vg'X'0
g
=
-49-IA P
-
-
h(
by
(5.17)
JZV
AVP
The Cartan basis of g is defined
are
the root space
decomposition
(ED g,,,)
(5.18)
ci
of g, where tan
a
=
generators
subspaces of
(a,,..., a,)
are the roots of g (i.e. the eigenvalues of the Caradjoint representation of G) and g,, are one-dimensional [53, 162]. In this basis, the generators have the non-vanishing
in the
g
Lie brackets
N,,,,,aE,,+,3
[Hi, E,,]
where
a,,3
=
are
aiE,,
,
[E, E,3]
the roots of g,
Hi
=
a
+,3 54
0
(5.19)
r
3
aiHi
Hil,
i
_-
1,
.
.
.
,
r,
are
=
-a
the generators of the
El are the step operators of subalgebra h 0 C of g (9 C, and E,, which, for each a, span g,, in (5.18) and which act as raising operators by a > 0 (relative to some Weyl chamber hyperplane in root space) on the representation states I A) which diagonalize the Cartan generators (the weight states), i.e. HiJA) oc JA) and E,,,IA) oc IA+a) for a > 0. The unitary irreducible r. For 1, representations of G are characterized by highest weights Ai, i each i, Ai is an eigenvalue of Hi whose eigenvector is annihilated by all the E, for a > 0. Corresponding to each highest weight vector A (A,.... Ar) we introduce the G-invariant symplectic 2-form Cartan
=
,
g0C
=
=
.
.
.
,
I
134
5.
Equivaxiant Localization
on
Simply Connected
Phase
Spaces
r
w(A) The
symplectic potentials associated
O(A) To construct
=
E ( \i
i9F(') azil
Aiw(') with
dz"
(5.20)
(5.20)
are
i9F(') -
a
A
+ dF
(5.21)
topological path integral from this symplectic structure, we a Hamiltonian satisfying (3.27), i.e. a Hamiltonian which is given by generators of the subalgebra of g 0 C which leave the symplectic potential (5.21) invariant. These are the canonical choices that give welldefined functions on the coadjoint orbit (5.1). As remarked at the end of Section 3.2, there usually exists a choice of function F(z,, ) in (5.21) for which this subalgebra, contains the Cartan subalgebra h 0 C of g (9 C. Let a
need to construct
H(A)
be the generators of h 0 C in the representation with
highest weight
vector A. Then the Hamiltonian
H(A)
hiH A)
(5.22)
satisfies the
required conditions and the corresponding path integral will adtopological form (4.157). Note that this is also consistent with the integrability arguments of the previous Chapters, which showed that the localizable Hamiltonians were those given by the Cartan generators of an isometry group G. Thus the path integral for the above dynamical system determines a cohomological. topological quantum field theory which depends only on the second cohomology class of the symplectic 2-form (5.20), i.e. on the representation with highest weight vector A (A,, A,). To apply the equivariant localization formalism to these dynamical systems, we note that since the Kiihler metrics g(i) above are G-invariant, the mit the
=
.
.
.
,
metric
9('\) obeys
Aig(')
the usual localization criteria. We shall
soon see
(5.23) that these group theo-
implied by the localization constraints, in that they are the only equivariant Hamiltonian systems associated with homogeneous symplectic manifolds as above. Through numerous examples of such systems and others we shall verify the localization formulas of the last Chapter and retic structures
discuss the now,
common
however,
resent for the
space
in fact
are
we
features that these quantum theories all represent. For just explore what the localization formulas will rep-
will
propagators
path integral
tr
eiTH( ') assuming
that they admit the phase Chapter with the above symplectic strucimportant group theoretic notions, and later
form of the last
ture. This will introduce
some
Coadjoint Orbit Quantization
5.1
we
shall show
precise1v
more
tions and discuss
and Character Formulas
path integral representaevaluating the localization
how to arrive at these
of the intricacies involved in
some
135
formulas.
apply the Niemi-Tirkkonen localization formula (4.130) to the dynamabove, we first observe that the tangent and normal bundles of related by [21] in are 0A1 g* To
ical system
TA, g*
=
Rom the construction of the
MON
TON (D MON
coadjoint
=
ON
(5.24)
9*
X
orbit it follows that the normal bundle
on the fibers, and the with the coadjoint action of G g* g* in the fibers. Then using (2.94) and the multiplicativity property (2.99), we can write the G-equivariant A-genus of the orbit ON as
g* product ON in
is
a
trivial bundle with trivial G-action is
x
a
Av
trivial bundle
over
rdet[ sinh(ad Y) I ad X
=
-1
-
=
-
where ad X is the Cartan element X E h in the g. We
now
vector
A,
choose the radius of the orbit to be the
i.e. A'
=
Weyl regarded as or
adjoint representation of Weyl shift of the weight
A + p where
P
is the
(5.25)
V/37(-adX)
2
(5.26)
a
"'>0
half-sum of positive roots of G), where A and a are linear functions on g by returning the total value of the weight
vector
(the
root associated to X E h. Then the localization formula
other than the celebrated Kirillov character formula
tr,\
eiTX
n!
(4.130)
is
none
[87, 1401 e
iTH(A)
(5.27)
0X+P
where tr,\ denotes the trace in the representation with highest weight vector A and H01) is the Hamiltonian (5.22) associated with the Cartan element X E h. If
we
further
apply
the finite-dimensional Duistermaat-Heckman theorem
to the Fourier transform of the orbit
localization formula
(4.135))
we
on
the
right-hand
side of
(5.27) (i.e.
the
arrive at the famous Harish-Chandra formula
[21, 67, 140]. The
resulting
character formula associated with the Harish-Chandra for-
mula for the Fourier transform of the orbit is the classical
Weyl
character
N(Hc) lHc W(Hc) [2]-[4], [51, 87, 117, 143, 156, 1621. Weyl group of Hc, where N(Hc) is the normalizer subgroup of Hc, i.e. the subgroup of g E G with hgHC gHC, Vh E HC, so that N(HC) is the left of HC on the orbit MG action of the fixed of GlHc. points subgroup Let
formula of G
=
be the
=
=
136
Equivariant Localization
5.
Given
w
=
nHC
W(Hc),
E
with
Simply Connected Phase Spaces
on
n
=
the respective adjoint representation formula can then be written as
tr.x
iN
E
N(Hc),
e x(_)
WEW(Hc)
a>O
let
X(')
=
n-'Nn be
n-1 e'xn. The Weyl character
eiT(A+p)(X('))
eiTX
a(X(w))
e
2i sin
2:a(X(-)) 2
(5.28)
XW. We shall how these character formulas arise from the equivariant localization formulas of the last Chapter, but for now we simply note here the deep group theoretical significance that the localization formulas will where see
are
explicitly later
the roots associated to the Cartan elements
on
represent for the path integral representations of the characters tr,\ e iTX in that the equivariant localization formalism reproduces some classical results of group theory. Note that the Weyl character formula writes the character a Cartan group element as a sum of terms, one for each element of the
of
Weyl
group, the group of
symmetries of the roots of the Lie algebra g. In the Chapter 4, the Weyl character formula will follow
context of the formalism of
from the
coadjoint orbit path integral over LO,\+p. It was Stone [156] who first Weyl character formula to the index of a Dirac operator from a supersymmetric path integral and hence to the semi-classical WKB evaluation of the spin partition function, as we did quite generally in Section 4.2 above. The path integral quantization of the coadjoint orbits of semi-simple Lie groups is essential to the quantization of spin systems. One important feature of the above topological field theories is that there is a one-to-one correspondence between the points on the orbits GIHC and the related this derivation of the
so-called coherent states associated with the Lie group G in the representation highest weight vector A [137]. The above character formulas can therefore be represented in complex polarizations using coherent state path integrals.
with
We shall discuss these and other aspects of the path integral representations as we go along in this Chapter.
of character formulas
Rom the
point of
performing Weyl Weyl shift problem a
view of
shift A
---
path integral quantization, the necessity of unsatisfactory. This
A + p in the above is rather
has been a point of some controversy in the literature shall [49]. see, the Weyl character formula follows directly from the WKB formula for the spin partition function [156], and a proper discretization
As
we
(5.27) really does give the path integral over the orbit 0,\ [3, 117]. The Weyl shift is in fact an artifact of the regularization procedure [3, 102, 117, 143, 160] discussed in Section 4.2 in evaluating the fluctuation
of the trace in
determinant there which led to the Niemi-Tirkkonen localization formula (4.130) and which leads directly to the Kirillov character formula (5.27). As
A-genus is inherently related to tangent bundles of real manifolds, the problem here essentially is that the regularization discussed in Section 4.2 does not respect the complex structure defined on the orbit. We shall see later on how a coherent state formulation avoids this problem and leads to a
the
5.2
Isometry Groups of Simply Connected Riemannian Spaces
correct localization formula without the need to introduce
an
137
explicit Weyl
shift.
Isometry Groups Simply Connected
5.2
of
Riemannian
Spaces
large class of localizable dynamical systems of the last Section topological and group theoretical properties, we now turn to an opposite point of view and begin examining what Hamiltonian systems in general fit within the framework of equivariant localization. For this we shall analyse the fundamental isometry condition on the physical theory in a quite general setting, and show that the localizable systems "essentially" all fall into the general framework of the coadjoint orbit quantization of the last Section. Indeed, this will be consistent with the integrability features implied by the equivariant localization criteria. We consider a simply-connected, connected and orientable Riemannian manifold (M, g) of dimension d (not necessarily symplectic for now) and with Given the
and their novel
metric g of Euclidean signature, for definiteness. The isometry group _T(M, g) is the diffeomorphism subgroup of C' coordinate transformations x ---> x'(x)
which preserve the metric distance on M, i.e. for which The generators VI of the connected component of I(M, field Lie
g)
=
g,,,(x').
form the vector
algebra
IC(M, g) and
g,,(x')
obey
=
IV
E
the commutation relations
TM: Lvg
(2.47).
For
=
a
(5.29)
0}
generic simply-connected
space, the Lie group T(M, g) is locally compact in the compact-open topology induced by M [681. In particular, if M is compact then so is _T(M, g). When
dim IC(M,
g) 0 0,
symmetric
space.
we
shall say that the Riemannian manifold
(M, g)
is
a
quickly run through some of the-basic facts concerning simply-connected Riemannian manifolds, all of whose proofs can be found in [42, 43, 68, 159, 164]. First of all, by analysing the possible solutions of the first order linear partial differential equations Lvg 0, it is possible to show that the number of linearly independent Killing vectors (i.e. We shall
now
isometries of
=
generators of
(5.29))
is bounded
as
dim IC(M,
g)
:5
d(d + 1)/2
(5.30)
d, so that the infinitesimal isometries of (M, g) are therefore characterized by finitely-many linearly independent Killing vectors in IC(M, g). There are 2 important classes of metric spaces (M, g) characterized by their possible isometries. We say that a metric space (M, g) is homogeneous if there exists infinitesimal isometries V that carry any given point x E M to any other point in its immediate neighbourhood. (M,g) is
when M has dimension
138
Equivaxiant Localization
5.
on
Simply Connected Phase Spaces
said to be
isotropic about a point X E M if there exists infinitesimal isomepoint x fixed, and, in particular, if (M, g) is isotropic about all of its points then we say that it is isotropic. The homogeneity con-
tries V that leave the
dition
that the metric g must admit Killing vectors that at any given on all possible values (i.e. any point on M is geometrically like any other point). The isotropy condition means that an isotropic point x0 of M is always a fixed point of an -T(M, g)-action on M, V(xo) 0 for means
point of M take
=
V E
some
IC(M, g),
subject only
but whose first derivatives take
to the
on
all
possible values,
0. Killing equation Lvg dim M homogeneous metric space always admits d linearly independent Killing vectors (intuitively generating translations in the d directions), and a space that is isotropic about some point admits d(d 1) /2 Killing vector fields (intuitively generating rigid rotations about that point). The connection between isotropy and homogeneity of a metric space lies in the fact that any metric space which is isotropic is also homogeneous. The spaces which have the maximal number d(d + 1)/2 of linearly independent Killing vectors enjoy some very special properties, as we shall soon see. We shall refer to such spaces as maximally symmetric spaces. The above dimension counting shows that a homogeneous metric space which is isotropic about some point is maximally symmetric, and, in particular, any isotropic space is maximally symmetric. The converse is also true, i.e. a maximally symmetric space is homogeneous and isotropic. In these cases, there is only one orbit under the -T(M, g)-action on M, i.e. M can be represented as the orbit M I(M, g) -x of any element x E M, and the space of orbits M/T(M, g) consists of only a single point. In this case we say that the group I(M, g) acts transitively on
It follows that
=
a
=
-
=
M.
Conversely, if a Lie group G acts transitively on a C'-manifold M, then a homogeneous space and the stabalizer Gx Ig E G g x x} of h x defines any point x E M is a closed subgroup of G. The map hG,, a homeomorphism GlGx -- M with the quotient topology on GlGx induced by the natural (continuous and surjective) projection map ir : G GlGx. On the other hand, if G is locally compact and H is any closed subgroup of G, then there is a natural action of G on GIH defined by g -7r(h) =,7r(gh), M is
=
=
-
-
--+
g, h E
G, which is transitive and for which H is the stabalizer of the point words, homogeneous spaces are essentially coadjoint orbits of Lie groups [68], with H GA the stabalizer group of a point A E g* under the coadjoint action OA Ad*(G)A C g* of G on g*. A sufficient
7r(l). GIH
In other
condition for the coset space GIH fgH : g E GI to be a symmetric space a reductive decomposition, i.e. an orthogonal decomposition =
is that g admit g
=
h (D
h-L such that [h
introduce Khhler structures
1 ,
h-L]
(the
C
h
[68]. Furthermore,
it is
possible
to
Kirillov-Kostant 2-form introduced in the previous Section) on the group orbits for which G is the associated isometry group. These spaces therefore generalize the maximal coadjoint orbit models of the last Section where H was taken as the Cartan subgroup HC and which
Isometry Groups of Simply Connected Riemannian Spaces
5.2
maximally symmetric [68]. We shall
in fact be shown to be
can
examples
later
We shall
see
139
explicit
on.
now
manifold if and
maximally symmetric spaces. uniquely characterized by a special curva(M, g) is a maximally symmetric Riemannian
describe the rich features of
It turns out that these spaces ture constant K. Specifically,
only
curvature tensor of g
are
if there exists can
R,\p,,
a
be written
=
9AMRp1,,,
constant K E R such that the Riemann
locally
=
In dimension d > 3, Schur's lemma
everywhere
almost
K(g,pg,\,
-
as
(5.31)
gpg,\,)
states that the existence of such
[68]
a
the constancy of K. For d 2, however, this is not the case, and indeed dimension counting shows that the curvature of a Riemann surface always takes the form (5.31). In form for the curvature tensor
automatically implies
-
=
this
(M, g)
K is called the Gaussian curvature of
case
implies
not constant. The above result
and it is in
general
that the Gaussian curvature K of
maximally symmetric simply connected
a
Riemann surface is constant.
amazing result here is the isometric correspondence between maximally symmetric spaces. Any 2 maximally symmetric spaces (M 1, gi) and The
(M2) 92)
of the
same
dimension and with the
same
curvature constant K
are
M2 between the 2 isometric, i.e. there exists a diffeomorphism f : MI Thus manifolds relating their metrics by gi (x) given any maxi92 (f (x)). onto it isometrically any other one with mally symmetric space we can map --+
=
the
same
curvature tensor
(5.31).
We
can
therefore model
maximally
sym-
metric spaces by some "standard" spaces, which we now proceed to describe. Consider a flat (d + l)-dimensional space with coordinates W', z) and metric I
?ld+ 1
dx,,
=
JKJ
(9 dxP +
1
K
(5.32)
dz (9 dz
real-valued constant. A d-dimensional space can be embedded into this larger space by restricting the variables x" and z to the surface of
where K is
a
a
(pseudo-)sphere,
Using (5.33) induced
on
(dx,-, JKJ (dx,, 1
1
9K
=
(5.33)
1
z(x) and substituting by this embedding is then
0 dx"
for
XAX, &A (& dxv X2
1
-
for
(5.32),
the metric
K > 0
-
xxv2 dxl' (& dx'
I
-
for
(5.34)
K < 0
X
K =0
represent, respectively, the standard metrics
of radius K- 1/2, the curvature
Z2
this into
(& dx" +
jdx,,0dxA cases
+
to solve for
the surface
k
These 3
sgn(K)x2
hyperbolic Lobaschevsky
K, and Euclidean d-space
R
d
space
on
the
d-sphere Sd
'Hd.,of constant negative
with its usual flat metric 77Ed
-
140
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
(5.33) and the manifest invariances of the it is straightforward to show that the above (5.32) space all admit a + d(d 1)/2-parameter group of isometries. These consist spaces about the of d(d rotations and d (quasi-)translations. The rigid origin 1)/2 Rom the
embedding
embedding
condition
geometry
-
first set of isometries:
always leave
points
some
on
the manifold
fixed, while
the second set translate any point on M to any other point in its vicinity. The 3 spaces above are therefore the 3 unique (up to isometric equivalence) maxi-
mally symmetric
spaces in
d-dimensions,
and any other
maximally symmetric
0, space will be isometric to one of these spaces, depending on whether K K > 0 or K < 0. It is this feature of maximally symmetric spaces that allows =
complete
the rather
vector fields that
VK
isometric
correspondence which will follow. The Killing are, respectively,
generate the above stated isometries
(Q,Axv
+ a '
sgn(K)X2]1/2 )
[I
09 aXA
for
K
:A
0
(5.35)
=
(s2AXV V
where S?A V
=
-S?,v,
+
a") -5 X/"
and a"
are
for
K
=
0
real-valued parameters. These
Killing
vectors
generate the respective isometry groups
_T(Sd)
=
SO (d +
1)
,
_T(lid)
=
SO (d,
1)
,
T(R d)
=
E
d
(5.36)
where E d denotes the Euclidean group in d-dimensions, i.e. the semi-direct product of the rotation and translation groups in R d' SO(d + 1) is the rotad+1 and SO (d, 1) is the Lorentz group in (d + l)-dimensional tion group of R ,
Minkowski space. Rom this we see therefore what sort of group actions should be considered within the localization framework for maximally symmetric spaces. Note that the maximal
symmetry of the spaces Sd and 'Hd are acSd can be regarded as the one-point
that of R d, because
tually implied by d R U fool (also known as stereographic compactification of Rd, i.e. Sd projection), and Hd can be obtained from Sd by Wick rotating one of its coordinates to purely imaginary values. ==
The next situation of interest is the
case
where
(M, g)
is not itself maxi-
mally symmetric, but contains a smaller, dH-dimensional maximally symmetric subspace Mo (e.g. a homogeneous but non-isotropic space). The general theorem that governs the structure of such spaces is as follows. We can distinguish Mo from M Mo by d dH coordinates V", and locate points within -
-
subspace Mo with dH coordinates u . It can then be shown [68, 1641 that is possible to choose the local u-coordinates so that the metric of the entire
the it
space M has the form
g
=
Ig,,,, (x) dx" 0 dx'
2
2
g,,O(v)dv'
0
dv'9
+
If(v)jij(u)du 0dui
2
(5.37)
where g,,o (v) and f (v) are functions of the v-coordinates alone, and ij (u) is a function of only the u-coordinates that is itself the metric of Mo. As (MO, j)
Isometry Groups of Simply Connected Riemannian Spaces
5.2
141
a dH-dimensional maximally symmetric space, it is isometric to one of the 3 standard spaces in dH-dimensions above and can be represented in one of the forms given in (5.34) depending on the curvature of the maximally
is
symmetric subspace Mo.
general result concerning Killing vectors on generic d-dimensional
Our final
connected manifolds is for the
simply
where the isometry group of
cases
(M, g) has the opposite feature of maximal symmetry, i.e. when _T(M, g) is 1-dimensional. Consider a 1-parameter group of isometries acting on the met-
8
V" (x) ric space (M, g). Let V E TM be a generator of I(M, g), and ax" let X/-'(x) be differentiable functions on M such that the change of variables =
x"j,
X/-(x)
=
has non-trivial Jacobian
det
For /.t
=
2,..., d
-
d-1 .
diffeornorphisms XA(x)
=
VV49,X/'
constant coordinate lines
given by the from R
LVXI'
=
The functions
(5.38)
0
to in addition be
solutions of the first order linear
1
V(X1)
=A
aXV
choose the
we can
linearly independent partial differential equation
the d
I']
X/(x)
0
=
IL
X/(x)
for M
=
(5.39)
2,..., d
constant embedded into M
2,...,d also have
=
homogeneous
an
invertible
Jacobian matrix since then
I I "X/'
rank2
owes
to the existence of
=
aX11
paths under
as
implied by (5.39) and If we
now
(5.40)
1
dx d
dX2
(5.41)
equation (2.46).
the flow
choose the function
-
the flow of the isometry group
such that
dxl
d
X1 (x)
the coordinate transformation x"
---+
so
=
Za;X1
0 for y 1.... d, then x : ax" XA (x) changes the components
that
xP (x)
=
)
of the vector field V to
V'P It follows from nents
x'
--+
(5.39)
=
V,
a
x/j,
'9X1'
that in these
V" 54 0 and V'A
=
0 for y
new
=
2,
.
(5.42)
d
A
x'-coordinates V therefore has compo..'
d. Now further
change
coordinates
x" defined by
X/11
dx" V/1 (X/)
X/1/'=X1A=XP
for
fz=2,...'d
(5.43)
142
5.
Equivariant Localization
on
Simply Connected
where x0 is a fixed basepoint in M. In this way case of a 1-parameter isometry group action on
we
Spaces
Phase
have shown
(M, g),
system of x"-coordinates defined almost everywhere on M the Killing vector of the isometry group has components
Vill
=
1
V"I'
7
=
for
0
p
=
that,
there exists -
MV
in the a
local
in which
(5.44)
2,..., d
Furthermore, an application of the Killing equation (2.112) shows that (M, g) admits a Killing vector if and only if there are local coordinates x" on M in which the metric tensor components g'-',(X") are independent of the coordiI nate x" ,
a9 11v(x /1)
=
09Xl/l
integral
and then the
isometry and
curves
(5.45)
0
of x"l parametrize the
of the finite total
isometry according
paths of the infinitesimal to (2.46). Moreover, the
above derivation also shows that 2 distinct isometries V, and V2 of
(M, g)
path, since they can be independently chosen to have V//2 the single non-vanishing components Vill 0. These results mean 1 2 that locally any isometry of g looks like translations in a single coordinate, and this therefore gives the representation of a 1-parameter isometry as an explicit Rl-action on (M, g) (which is either bounded or is a U(l)-action when cannot have the
same
=
compact).
M is
,
=
We shall refer to this system of coordinates a Killing vector field V.
prefered
as a
set
of coordinates with respect to
now concentrate on the cases where M is a simply symplectic manifold with metric g. Notice that the standard Cauchy-Riemann equations of complex analysis imply that a Riemann surface is always a complex manifold. The advantage of this insofar as
For
simplicity,
we
shall
connected 2-dimensional
the localization formalism is concerned is that the Riemann uniformization
[74, 111, 145] tells us that gl,,(x) has globally only I independent component. This situation is therefore amenable to a detailed analysis of the equivariant localization constraints in terms of the single degree of freedom of I the metric g. Defining complex coordinates z, x iX2, we can represent theorem
the metric
as
g
=
A(dz
+
pd2)
o
(d2 +,adz)
(5.46)
where A
tr g +
2Vdetg)14
A
=
(911
-
922 +
2igl2)/4A
Orientation-preserving diffeomorphisms of M which only change A > 0 above the
complex
an
0.
the function
called conformal transformations. The function [t determines structures of M, and therefore the set of inequivalent complex
are
structures of M is in
equivalence
(5.47)
a
one-to-one
classes of metrics
on
correspondence with the space of conformal M. A complex coordinate w is said to be
isothermal coordinate for g if g p dw & d Cv for some function p > for g, it follows from (5.46) that the law transformation tensor Using =
5.2
Isometry Groups of Simply Connected Riemannian Spaces
143
if the Beltrami
partial
isothermal coordinate
an
only
for g exists if and
w
differential equation
has
a
M
w(z,. ).
C'-solution
function can
is
az
Such
solution
a
as
(5.48) always
jj/_ijj,,,
< 1.
provided
exists
globally'
g
that the
A complex structure
therefore be identified with the conformal structure
phism. and Weyl rescaling
on
represented by
2-manifold M implies that via a diffeomorewg of the coordinates the metric can be put
a
--+
into the isothermal form
g,,,(x) W(x)
refer to
aw
5=z uniformly bounded
the Riemannian metric g. The simple-connectivity of
where
aw
as
is
a
=
ew(x)J,,,
g
or
0*,2)dz
=
globally-defined real-valued
function
the conformal factor of the metric. This
unique complex
structure
on
on
(5.49)
0 d,
M which
means
we
shall
that there is
the Riemann surface M which
we
can
a
define
1 iX2 Notice that these x complex coordinates z, 2 then the metric has additional H1 because remarks are not true if (M; Z) : 0, the torus example in Section degrees of freedom from moduli parameters (see 3.5), i.e. (5.49) should be replaced by
by
the standard local
=
g
where
-r
.
(5.50)
el (-r)
=
degress of freedom of the metric. We phase spaces in the next Chapter. V1 iV2 for any vector define V',2
labels the additional modular
shall discuss the
case
of multiply-connected
complex structure we i a ). The'Killing equations (2.112) and field V, we set o9, 0 1 (Do axT T- -a-x-"T 2 these complex coordinates can be written as With this
=
=
5Vz
=
19V2
The first set of
=
gvz + W +
0 in
equations
(5.51)
are
Vz&P
+
V25V
=
0
in
(5.51)
Cauchy-Riemann equations and Killing vector field Vz is a equation is a source equation for Vz Killing fields in terms of the single the
that in these local coordinates the
they imply holomorphic function on M. The other and V2 that explicitly determines the degree of freedom of the metric 9 (i.e. the conformal factor 0). The Gaussian curvature scalar K(x) of (M, g), which is always defined by (5.31) in 2 dimensions, can be written in these isothermal coordinates as
K(x) The fact that this holds
Riemann-Roch theorem,
61, 111].
1 =
-
2
globally
or
the
e-w(x) V2 WW
application of the classical Atiyah-Singer index theorem [41,
follows from
more
modern
(5.52)
an
144
Equivariant Localization
5.
where V
=
g4'V,,V,
A
=
associated with the metric
non-vanishing
Simply
on
Connected Phase
is the 2-dimensional scalar
(5.49).
This follows from
=
F222
'9W
Laplacian on M noting that the only
(5.49)
connection coefficients of the metric
r,
Spaces
are
(5.53)
W
(M, g) then uniquely characterizes the isometry acting on the phase space. If K is constant, then (M, g) is maximally symmetric with 3 linearly independent Killing vectors. Moreover, in this case (M, g) is isometric to either the 2-sphere S', the Lobaschevsky plane H' or The Gaussian curvature of group
2
plane R We shall soon examine these 3 distinct Riemannian Z1' is a compact Riemann spaces in detail. Notice, however, that if M surface of genus h, then the Gauss-Bonnet-Chern theorem (2.103) in the case the Euclidean
.
=
at hand reads
f dvol(g(x)) K(x)
=
4ir(i
-
h)
(5.54)
Zh
dvol(g(x))
where
d2X
=
V'-detg(x)
is the metric volume form of
(M,g).
maximally symmetric compact Riemann surface of constant negative curvature must have genus h > 2. It follows, under the simple-connectivity 0 or K > 0 the phase space assumption of this Chapter, that when K M can be either compact or non-compact, but when K < 0 it is necessarily Thus
a
=
non-compact. The other extremal
case
is where
(M, g)
general
of isometries. FYom the above
X1
there exist 2 differentiable functions
x1
on
09
09X/1
X2 (X1, X2)
and in these coordinates the 0.
ator
1-parameter
group case
and X 2on M and local coordinates
Moreover,
V,
on a
912("rf) f
Xf2
0
=
Killing
the characteristic
the initial data surfaces of the
be chosen to be
can
lie
=
by
fined
a
M such that VIZ
1, V/2
only
admits
discussion it follows that in this
i.e.
we can
orthogonal
to the
0. Thus in this g
and from
(5.45)
2(X1,X2)
vector field has curves
(5.55)
components V`
of the coordinate X'2
case
partial paths defined by the isometry means
the metric
glldx" 1
=
0
dx"
==
differential equation in
choose the initial conditions for the solutions of
non-characteristic surface. This =
X
=
+
can
that in these
be written
9212dx
/2
0 dx
new
locally
X2, de(5.55), gener-
(5.55)
to
coordinates
as
/2
(5.56)
it follows that gil and 92'2 are functions only of X'2 The phase a surface of revolution, for example a cylinder or .
space therefore describes
the
'cigar-shaped' geometries
that
are
described in
typical black hole theories
[164]. The
only other
case
left to consider here is when
dimensional isometry group. In this
case we
have 2
(M, g)
independent
has
a
2-
vector fields
Isometry Groups of Simply Connected Riemannian Spaces
5.2
145
obey the'Lie algebra (2.47) with possibilities 1, algebra either the isomea, b, it is or is fabc 0, abelian, non-abelian, fabc 4 0 for some a, b, c. try group Since V, and V2 cannot have the same path in M, we can choose paths for V,
=
c
8
Vi" axi,
and V2
2. There
=
V21 8XA
=
which
for this Lie
2
are
-
=
the constant coordinate lines
V12
that
so
commutativity
[V1 V21 implies that T
V2 2
rl
l
ric
0. In the abelian case, the
(5.57)
V22
is a function only Of X2 everywhere on M in which these coordinates, the Killing equations imply that the met-
V11
is
1. In
a
x1 alone and
function of
.
choose local coordinates almost
we can --
=
0
=
7
As above,
V21
=
of V, and V2
components g,,, (x)
are
all constant. Thus in this
case
(M, g)
is isometric
to flat Euclidean space, which contradicts the standard maximal
symmetry
arguments above. In the non-abelian case, we can choose linear combinations of the so that their Lie algebra is
isometry
generators V, and V2
[V1 V21 i
which
implies
2
V22
1 and
V11
a291,tv
=
a2 109 V11
0
1/
choose local coordinates almost
so we can
=
(5.58)
V1
that
'91V2 and
7--
=
e-X2
191911
=
Killing equations
The
.
=
0
191912
7
=
911
V22
(5.59)
everywhere
on
M in which
then become
7
191922
=
2g12
(5.60)
which have solutions gil
=
a
,
912
=
aX
1
+0
922
1
=
a(xl)2
+
20xl
where a, 6 and -y are real-valued constants. It is then compute the Gaussian curvature of g from the identity
K(x)
=
-R1212(x)/det;g(x)
which gives K(x) as the constant K a/()32 maximal symmetry theorems quoted above. =
Thus
a
2-dimensional
phase
-
space is either
+ -y
(5.61)
straightforward
to
(5.62)
a-y), again contradicting
the
maximally symmetric with
a
3-dimensional isometry group, or it admits a I-parameter group of isometries (or, equivalently, has a single 1-dimensional maximally symmetric subspace), because the above arguments show that it clearly cannot have a 2-dimensional isometry group. The fact that there are only 2 distinct classes of isometries in 2 dimensions is another very appealing feature of these cases for the analysis which follows. For the remainder of this Chapter we shall analyse the
146
Equivaxiant Localization
5.
Simply Connected
on
Phase
Spaces
equivariant Hamiltonian systems which can be studied on the various isometric types of spaces discussed in this Section and discuss the features of the integrable quantum models that arise from the localization formalism. We shall primarily develop these systems in 2 dimensions, and present higherdimensional examples in Sections 5.7 and 5.8. This will provide a large set of explicit examples of the formalism developed thus far and at the same time
clarify
some
other issues that arise within the formalism of
path integral
quantization.
5.3 Euclidean Phase
and
Spaces Holornorphic Quantization
begin
We
study of general localizable Hamiltonian systems with the case 0. The conformal factor phase space M is locally flat, i.e. K (5.49) and (5.52) then satisfies the 2-dimensional Laplace equation our
where the in
=
v2 whose
general solutions
aaw(z" )
(M, g)
(5.63)
0
are
O(Z' o ) where
=
=
fW
+
A0
(5.64)
f (z)
is any holomorphic function on M. The Riemannian manifold is isometric to the flat Euclidean space (R 'nE2) and from the metric
2,
tensor transformation law it follows that this coordinate
the metric
aw a'CV az
(5.49)
09' aw +
_FZ
=
-5-z- wz=
It follows from
change
z --+ w
taking
to dw 0 dfv- satisfies
(5.65)
ew(z,2)
=
ef (z) el(2)
49W &cv
aw afv=
,
-
19Z az
a' ;
_FZ
=
0
(5.65)
that this isometric transformation is the 2-dimensional
conformal transformation
z -4 wf (z) (i.e. an analytic dard flat Euclidean metric of the plane) where
Wf W
=
f d6
ef (
rescaling
of the stan-
)
(5.66)
C.
and Cz C M is
a
simple
curve
(eq. (5.35)) we complex coordinates (w, fv-)
from
fixed
basepoint in M to z. Rom 2 Killing vectors of (R ,77E2) in the general form
some
the last Section
know that the
the
take
VRW2 where S?
E
R and
a
=
E
-iS?W + C
are
C1
on
VJ:2
constants. The
=
(5.67)
iJ?fV + 5V
Killing
vectors
(5.67)
follow
0 there, and they generate the groups of 2directly (5.51) with W dimensional rotations w --- eis?w and translations w ---> w + a whose semidirect product forms the Euclidean group E 2 of the plane.
from
=
5.3 Euclidean Phase
In these local
Spaces and Holomorphic Quantization
147
2 complex coordinates on R the Hamiltonian equations dH
-ivw take the form aH
Z
=
-w(w, iv-)Vw
o9H
,
2
=
-
-w(w, fv- )VW
(5.68)
2
where w
The
symplectic:
2-form
(5.69)
=
-w(w, Cv)dw 2
can
be
A
(5.69)
dCv
explicitly
by recalling symplectic so that
determined here
that the Hamiltonian group action on the phase space is 0. In local coordinates this means that LVw =
'9j'(V1'W"\) -,9,(V'\W,,,\)
=
(5.70)
0
for each p and v. Requiring this symplecticity condition for the full isometry 2 2 group action of E on R , we substitute into (5.70) each of the 3 linearly
0, represented by (5.67) (corresponding to Q 0 there). The differential equations (5.70) for the function 2 easily imply that it is constant on R with these substitutions.
independent Killing a'
=
0 and a2
(w, fv-) now Thus w(w, Cv)
w
vectors
=
=
is the Riemannian volume
R 2.
(and
in this
case
the
Darboux)
2-
w(w, -) globally on Substituting Killing vectors (5.67) into the Hamiltonian equations above and integrating them up to get H(w, fv-), we see that the most general equivariant Hamiltonian on a planar phase space M is form
Ho(z,, ) where CO E R is formation
(5.66)
the Darboux value
=
S?wf (z),Cvf (2)
+
dwf (z)
+
afv-f
iv
+
Co
=
1 and the
(5.71)
constant of integration and wf (z) is the conformal transfrom the flat Euclidean space back onto the original phase
a
space.
uniquely determined to be the phase space geometry is a general feature of any homogeneous symplectic manifold. Indeed, when a Lie group G acts transitively on a symplectic manifold there is a unique G-invariant measure [68], i.e. a unique solution for the d(d 1)/2 functions w,', from the d(d 1) d(d + 1)/4 differential equations (5.70). Thus Wn/n! is necessarily the maximally symmetric volume form of (M, g) and the phase space is naturally a Kdhler manifold, as in Section 5. 1. We shall soon see the precise connection between maximally symmetric phase spaces and the coadjoint orbit models of Section 5. L In the present context, this is one of the underlying distinguishing features between the maximally symmetric and inhomogeneous cases. In the latter case w is not uniquely determined from the requirement of symplecticity of the isometry group action on M, leading to numerous possibilities for the equivariant Hamiltonian systems. In the case at hand here, the Darboux 2form on R2 is the unique 2-form which is invariant under the full Euclidean The fact that the
symplectic
2-form here is
volume form associated with the
-
-
-
148
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
group, i.e. invariant under rotations and translations in the plane, and it is the Kdhler form associated with the Kdhler metric (5.49) and
The form
on
M
(5.64).
(5.71)
planar equivariant Hamiltonian systems illustrates how the integrable dynamical systems which obey the localization criteria depend on the phase space geometry which needs to be introduced in this formalism. These systems are all, however, holomorphic: copies of the same initial dynamical system on R 2 defined by the Darboux Hamiltonian for the
HOD( z,. )
=
Qz.
+ dz + a. +
CO
;
(5.72)
C
E
z
or identifying z,. iq with (p, q) canonical momentum p variables, these dynamical Hamiltonians are of the form =
D
H0
Thus the in the
(p, q)
dependence
on
=
p(P
2
+ q
phase
the
2)
+ CelP + C92q +
and position
(5.73)
CO
space Riemannian
geometry is trivial
that these systems all lift to families of
holomorphic copies of the planar dynamical systems (5.72). This sort of trivial dependence is to be expected since the (classical or quantum) dynamical problem is initially independent of any Riemannian geometry of the phase space. It is also anticipated from the general topological field theory arguments that we presented earlier. Nonetheless, the general functions Ho(z,, ) in (5.71) illustrate how the geometry required for equivariant localization is determined by the different dynamical systems, and vice versa, i.e. the geometries that make these dynamical systems integrable. This probes into what one may consider to be the geometry of the classical or quantum dynamical system, and it illustrates the strong interplay between the Hamiltonian and Riemannian symmetries that are responsible for localization. Thus essentially the only equivariant Hamiltonian system on a planar symplectic manifold is the displaced harmonic oscillator Hamiltonian sense
HOD and in this
=
Q(z
case
action with the
+
a)(.
+
a)
=
Q
J(p + a, )2 + (q + a2 )21
+
(5.74)
CO
we can replace the requirement that H generate a circle requirement that it generate a semi-bounded group action.
To compare the localization formulas with
ementary quantum mechanics,
we
some
well-known results from el-
note that the Hamiltonian
(5.73)
can
only
describe 2 distinct 1-dimensional quantum mechanical models. These are the 1 harmonic oscillator Iz. (P2+ q2) wherein we take Q .12 and a 0 in 2 2 (5.72) and apply either the WKB or the Niemi-Tirkkonen localization formu=
las of the last a
=
Chapter,
112V2- in (5.72)
=
and the free
and
where particle jp2 2
we
=
take S?
=
0 and
apply quadratic localization formula (4.150) (or In fact, these are the original classic examples, which the
equivalently (4.142)). were for a long time the only known examples, where the Feynman path integral can be evaluated exactly because then their functional (and classical statistical mechanical) integrals are Gaussian. For the same reasons, these
Spaces and Holomorphic Quantization
5.3 Euclidean Phase
are
also the basic
exact
examples
where the WKB
149
is known to be
approximation
[147]. straightforward
It is
(4.130)
to
verify the Niemi-Tirkkonen localization formula r eiO with r C R+ polar coordinates z
for the harmonic oscillator. In
and 0 E
[0, 2-r],
that the
integral
we
in
have w,O
=
=
(S?V)O,
r,
=
-2r and R
0
=
on
flat R 2,
so
(4.130) gives 00
Zharm(T)
T
dr
can
be
1,
2
k E
2i sin
seen
by the Schr6dinger equation Z+ [1011, so that
determined k +
1
-iTr2/2
2
0
That this is the correct result
e
2 sin 1:
(5.75)
T 2
by noting that the energy spectrum for the harmonic oscillator is Ek 00
trJJ
e-
iT(02+42)/211
E
e-
iTEk
=
E
2 e- iT(k+.!)
2i sin
k=O
k
(4.115)
This result also follows from the WKB formula
regularized jectories determined by the flows
fluctuation determinant
orbits on as
z
(t)
=
z
(0)
it/2 e
as
of the vector field V'
only
loop space LC is to regard z(t) independent complex variables. This the
=
should be evaluated in
a
working
after
out the
described there. Here the classical tra-
Note that the
.
(5.76)
T 2
=
iz/2
way these orbits it/2 and e
z(O)
(t)
means
are
the circular
can
=
be defined i(T-t)/2 e
2(T)
that the functional
holomorphic polarization.
integral
We shall return to this
point shortly.
Alternatively, we note that for T: 27rn the only T-periodic critical trajecdynamical system are the critical points z,2 0 of the harmonic oscillator Hamiltonian z2 and (5.76) also follows from (4.135) which gives
tories of this
=
Pfaff For the discretized values T to
1
1
Zharm(T)
=
(
1
2i
0 -1
1)
sin
2i sin
2
(5.77)
T 2
0
2-7rn any initial condition z(O) E C leads space of critical trajectories is non-
T-periodic orbits, and the moduli
2
R In that case phase space M the degenerate path integral formula (4.122) yields the correct result. These results therefore all agree with the general assertions made at the beginning of Section 4.6 concerning the structure of the moduli space of T-periodic classical trajectories for a Hamiltonian circle action on the phase space. For the free particle partition function, we have R OV 0, and so The 001. the localization formula contributes in term theA-genus (4.150) and find Gaussian in trivial thus is one we a integral (4.150)
isolated and coincides with the entire
=
=
00
Zftee(T) 00
J -00
=
00
00
dp dq
.
doo e' T02-iTOop/2v 2_ 0
,
f dp dq -00
e-'TP2/2 (5.78)
150
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
which also coincides with the exact propagator trII e-iTp2 /2 11 in the phase 2 2 space representation. In this case the Hamiltonian 1p is degenerate on R 2
that the WKB localization formula is unsuitable for this
dynamical system (4.122) by noting 2 that LMs R in this case. Notice also that (5.78) coincides exactly with the classical partition function of this dynamical system as there are no quantum so
and the result
(5.78)
follows from the
degenerate
formula
=
fluctuations. There is another way to look at the path integral quantization of the Darboux Hamiltonian system (5.72) which ties in with some of the general ideas of Section 5.1 above. The Heisenberg-Weyl algebra 9HW [101] is the
algebra generated by
the usual harmonic oscillator creation and annihilation
operators
a, ait in the canonical
(4.1).
ata
6, and
=
The Lie
I(p2
2
+
42
(infinite-dimensional)
space R
2
and the
algebra generated by the operators with the commutation relations 1) 9HW is
_
[at,a] The
(5.79)
quantum theory associated with the phase
operator algebra
at,
id)
,,F2
=
(5.80)
1
Hilbert space which defines
a
representation of
these operators is spanned by the bosonic number basis In), n E Z+, which form the complete orthonormal system of eigenstates of the number operator with eigenvalue n,
NIn) and
which tit and 6, act
on
at In) We
now
=
as
=
ataln)
aln)
define the canonical coherent states
Iz)
=_
Heisenberg-Weyl Zn
e'at 10) n=0
These states
are
normalized
-=
Iz)t,
and
(5-81)
V/n_In
-
[45, 101, 137]
group
In)
=
;
GHW
z
E
1)
(5.82)
associated with
as
C
v n.
(5.83)
as
(z I z)
(zl
n1n)
raising and lowering operators, respectively,
v/n-_+1 In+ 1)
this representation of the
with
=
=
e'2
they obey the completeness relation
(5.84)
5.3 Euclidean Phase
d2Z
d2 Z
IZ)XZ1
21r
21r
Spaces and Holomorphic Quantization Zn,
e-'2
m
Vn--!m!
In) (ml 21r
00
f
dr
r
e-r
2
n+m
E Vn--!m! f dO ei(n-m)0In)(mI
n, m
0
0
00
f
151
dr
r
e-
r2
E
E In)(nI
-21rJnmIn)(mI
_n _ ! M!
n,m
0
00
rn+m
=
1
n=0
(5.85) where
we
have
as
usual written
=
r
e'0 and
Iz)IV(zlz)
Iz)) are
z
e-
=
z2/21Z)
(5.86)
the normalized coherent states. The normalized matrix elements of the
algebra generators
in these states
are
((ZIataIZ)) Thus the 3
=
independent
Z
((ZI&'tIZ))
((ZIetIZ))
,
terms in the Darboux Hamiltonian
=
(5.87)
Z
(5.72)
are none
other than the normalized canonical coherent state matrix elements of the group generators. These 3 observables represent the PoisLie group action of the Euclidean group E 2 on the coadjoint orbit GHWIHc = GHWIU(I) C' with the Darboux Poisson bracket
Heisenberg-Weyl son
=
IZI *"D
=
(5.88)
1
algebra representation of the Heisenberg-Weyl algebra correspondence with the coset space GHWIHC and the general framework of Section 5.1 is not entirely surprising, since homogeneous symplectic manifolds are in general essentially coadjoint orbits of Lie groups [681, i.e. they can be represented as the quotient of their isometry groups by a maximal torus according to the general discussion of the last Section. The integrable Hamiltonian systems in this case are functionals of Cartan elements Of 9HW (e.g. the harmonic oscillator at & or the free particle (a + at)2) which is the Poisson
(5.80).
This
.
The canonical coherent states
(5.83)
are
those quantum states which min' Aq -Ap ! [101], because they
Heisenberg uncertainty principle diagonalize the annihilation operator 6,, 6.1z) imize the
eralized to
arbitrary Lie
groups
[137],
as
we
2
=
z1z),
shall
and
soon
can be genThe Darboux
they
see.
2-form Z
WD
=
2
globally on C and, since R 2is contractable, and hence be be generated globally by the symplectic potential
is defined
0, it
can
OD
(5.89)
-dz A d,
2
(2dz
-
zd2)
H 2 (R
2;Z) (5.90)
152
Equivariant
5.
The canonical 1-form on
R
2
can
Localization
(5.90)
on
and the
=
dz (9 d,
11 d1z)) 11
=
globally-defined
2
tr
e-M
f
=
Spaces
11 d1z)) 11
(9
Kdhler
(5.83)
(5.89)
as
((zjdjz)) -
(5.91)
((zjdjz))
(9
((zjdjz))*
potential associated with (5.90)
FR2 (Z, ) The path integral path integral
Phase
and the flat Kdhler metric associated with
be written in terms of the coherent states
OD 9D
Simply Connected
=
(5.92) is
(5.93)
Z
here then coincides with. the standard coherent state
d2Z
((zl e`? I Z))
2-7r
T
dz (t) d, LR2 tE[O,T]
(t)
exp
27r
i
f
dt
(z'!-Z
-
i
-
H (z,
2))
0
(5.94) where
H(z,. )
=
((zJ7 jz))
is the coherent state matrix element of
some
(5.95)
operator 7
=
7 (6, at)
on
the
underlying representation space of the Heisenberg-Weyl algebra. The derivation of (5.94) is identical to that in Section 4.1 except that now we use the completeness relation (5.85) for the coherent state representation. of describing the quantum dynamics goes under the names of holomorphic, coherent state or Kdhler polarization. One of its nice features in general is that it provides a natural identification of the path integral and loop space Liouville measures. We recall from (4.21) that in the former measure there is one unpaired momentum in general and, besides the periodic boundary conditions, there is a formal analog between the measures in (4.21) and (4.25) only if the initial configuration of the propagator depends on the position variables q and the final configuration on the momentum variables p, or vice versa. In the holomorphic polarization above, however, the initial configuration depends on the z variables, the final one on the ' variables, and the dzi d, j/27r. Since path integral measure is the formal N -* OC) limit of IIN j= 1 modified This
manner
the number of
z
and,
integrations
are
the same,
we
obtain the desired formal
analog between the path integral localization formulas and the Duistermaat-Heckman theorem and its generalizations, this enables one to also ensure that the loop space supersymmetry encountered in Section 4.4, which is intimately connected with the definition of the path integral measure (as are the boundary conditions for the propagator), is consistent with the imposed boundary conditions. identifications. Besides
providing
one
with
a
formal
5.4
Holomorphic Localization Formulas
153
Thus on a planar phase space essentially the only equivariant Hamiltonian systems are harmonic oscillators, generalized as in (5.71) to the inclusion of a generic flat geometry so that the remaining Hamiltonian systems are merely holomorphic copies of these displaced oscillators defined by the analytic coordinate transformation (5.66). These systems generate a topological quantum theory of the sort discussed in Sections 4.10 and 5.1, with the Darboux Hamiltonian
topological
symplectic potential (5.90) by the usual HOD 'VR2 OD reflecting the fact that (5.90) is invariant of the rotation group of the plane. It is not, however, invari-
(5.72)
related to the
condition
under the action
:_
ant under the translation group not determine
a
action,
so
Witten-type topological
lator Hamiltonians do. This
means
that the translation generators do theory like the harmonic oscil-
field
that there
the
i.e. it is
E 2-invariant
are no
to find
impossible potentials plane, gives an invariant potential simultaneously for all on
a
function F in
3 of the
symplectic
(5.21)
independent
that
gener-
(5.72).
The harmonic oscillator nature of these systems is consistent with their global integrability properties. The holomorphic polarization of ators in
theory associates the canonical quantum theory above with the topological coadjoint orbit quantum theory of Section 5.1 and the coherent state path integral (5.94) yields character formulas for the isometry group of the phase space. This will be the general characteristic feature of all localizable systems we shall find. In the case at hand, the character formulas are associated with the Cartan elements of the Heisenberg-Weyl group. the quantum
5.4 Coherent States
and
Holomorphic
Before
carrying
on
with
on
Homogeneous
Kdhler Manifolds
Localization Formulas our
geometric determination of the localizable dy-
namical systems and their path integral representations, we pause to briefly discuss how the holomorphic quantization introduced above on the coadjoint orbit R2
be
can
generalized
to the action of
an
arbitrary semi-simple
Lie group
[17, 90, 1371. This representation of the quantum dynamics proves to be the most fruitful on homogeneous spaces GIHC, and later on we shall generG
alize this construction to apply to non-homogeneous phase spaces and even non-symmetric multiply connected phase spaces. As the coherent states are those which are closest to "classical" states, in that they are the most tightly peaked ones about their locations, they are the best quantum states in which to study the semi-classical localizations for quantum systems. We shall also see that they are related to the geometric quantization of dynamical systems
[1721. Given any irreducible unitary representation D(G) of the group G and normalized state 10) in the representation space, we define the (normal-
some
ized)
state
1g) by 1g)
=
D(g)10)
(5.96)
154
5.
Equivariant Localization
on
Simply Connected Phase Spaces
Dg denotes Haar measure of G, then Schur's lemma [162] and the pleteness of the representation D(G) implies the completeness relation If
dimD(G) vol(G)
f Dg Ig)(gl
=
com-
(5.97)
1
G
Following
the derivation of Section 4.1, it follows that the
associated with can
be
operator 7
an
acting
on
partition function
the representation space of
D(G)
represented by the path integral
f Dg (gl e-M Ijg)
trD(G) e-iT) i
G
(5.98)
T
LG
dimD(G) Dg(t) Vo l(G)
tE[O,T]
i
exp,
j (gldlg) if -
dt
(gJ7 jg)
0
-t(g)
10) to be a simultaneous eigenstate of the generators weight state), then the 'coherent' states 1g) associated with any one coset of GIRC are all phase multiples of one another. Thus the set of coherent states form a principal Hc-bundle L -- MG over GIHC and the coherent state path integral (5.98) is in fact taken over paths in the homogeneous space GIHC. This geometrical method for constructing irreducible representations of semi-simple Lie groups as sections of a line bundle L --4MG associated to the principal fiber bundle G -+ GIHC is known as the Borel-Weil-Bott method [142]. The holomorphic sections of this complex line bundle (the coherent states) form a basis for the irreducible representation. What is most interesting about the character representation (5.98) is that it is closely related to the Kiffiler geometry of the homogeneous space GIHC. To see this, we first define the Borel subgroups B of G which are the exponentiations of the subalgebras B spanned by Hi E h 0 C and E,, for a > 0 and a < 0, respectively (see (5.18)). The complexification of the coadjoint orbit MG is then provided by the isomorphism GIHC GIIB, where G' is the complexification of G [162]. Almost any g E G can be factored as a Gauss decomposition (+h(_ g (5.99) However, of HC c
if
we
G
take
(i.e.
a
-
=
where h E
Hb
and
(+ Here z' E C, weight state, states
are
in
=
and if we then a
(_
eF-,->o z'E,,,
eE
"
z'E,:,
(5.100)
apply the representation operator D(g) to a lowest as the identity and the set of physically distinct correspondence with the coset space G'IB+. Since
now
acts
one-to-one
7
5.4
B+
Holomorphic Localization Formulas
155
closed
subalgebra of g 0 C, the parameters z' E C define a complex MG. In this way, we can now write down coherent state path integral representations of the character formulas of Section 5.1 above. The is
a
structure
on
choice of
10)
=-
JA)
above
as a
weight
lowest
state
ensures
that the coherent
states
1Z) holomorphic.
are
E-,,Iz)
=
za1z)
for
Note that their a
+IA)
(5.101)
coherency follows from the fact that
> 0.
The coherent state
representation
can
be used to
provided
the one-to-one
identification of the given representation space with the coadjoint orbit of Section 5.1. This is provided by the injective mapping on g --+ R defined
by X ---> (gjXjg). The Kirillov-Kostant mapped onto the flag manifold MG as w,\
=
2-form
(AID(g-'dg)
A
on
the
coadjoint orbit
D(g-1dg)tjA)
is then
(5.102)
and likewise for the canonical G-invariant orbit metric g,\
=
(AID(g-ldg)
(9
D(g-1dj)tjA)
g-ldg
in terms of the Cartan-Maurer 1-form
in the
(5.103)
representation D(G).
(5.102) coincides with the appropriate symplectic potential 0,\ associated with the kinetic term (gj ) (gl 1g) in (5.98). Furthermore, the matrix
Tdt dt (gJ7 jg) in (5.98) for 7i E D(Hc) generates the Hamiltonian flow LG [138]. e" g for the action of the direct product LS' 0 Hc =
element
(g, t)
-+
on
identity iv(g-ldg) the admits usual topological form i (5.98) thus given by the normalizations This follows from the
e
F (h)
(Z' g)
=
(Z I Z)
=
g-17 g,
+vO.\.
(AlgIA)
= -
=
so
that the action in
The Kiffiler
(AlhIA)
potentials
are
(5.104)
potentials F(') (z, 2), i 1, r, each associated with the Cartan in generator Hi (5.104) (compare eqs. (5.84) and (5.93)). From this it follows that the associated Kdhler metrics g(i) and symplectic potentials OW can be represented as coherent state matrix elements as in (5.91) and (5.92). In this way the kinetic term in the coherent state path integral (5.98) coincides with the usual one induced by the symplectic Khhler structures of the homogeneous space MG and the path integral measure becomes the loop space with the
Liouville
=
.
.
.
,
measure.
A
particularly important aspect of the holomorphic quantization scheme some ambiguities in the localization formulas when compared with the classical group character formulas [102]. For this, we consider the usual symplectic line bundle L(A) MG associated with the principal G-bundle G --+ MG with connection 1-form, OW given in (5.21). On above is that it resolves
--+
this line bundle
we
then define the twisted
(covariant)
Dolbeault derivative
156
5.
Equivariant Localization
on
-
Simply Connected Phase Spaces
O(A) 2
-
'q'\
=
'9 +
(5.105)
The Riemann-Roch-Hirzebruch index theorem
Singer index theorem
(21, 102, 156]
(the analog of the Atiyahelliptic Dolbeault exterior derivative) analytical index of (5.105) to the topological invari-
for the twisted
relates the
ant
ch(w(A)) A td(R(A))
index(,O,\)
(5.106)
O;k
Furthermore,
the index of the twisted Dolbeault
complex
is
n
1:(_I)k dim H(O k) (MG; L('X))
index(O,x)
(5.107)
k=O
recall that A (p,q)
(MG; L(,\)) =- A(p,q) MG 0 L(A). When the line trivial, the index (5.107) is called the arithmetic genus of the complex manifold MG and the expression (5.106) for it in terms of the Todd class is called the Riemann-Roch theorem [41, 111]. This is the complex analog of the Gauss-Bonnet theorem for the Euler characteristic (the index of the DeRham. complex) and it can also be generalized to higher-dimensional complex vector bundles. The crucial point here is that the twisted Dolbeault operator (5.105) where
we
bundle
L(,\)
--+
MG
is
annihilates the normalized coherent states
Iz))
=
Jz)1(zJz),
so
that
Iz))
E
H(OM (MG; L(,\)). An application of the Lichnerowicz vanishing theorem [4, 21] shows that all other cohomology groups H(O q) (MG; VA)) for q > 0 are trivial, so that the dimension of the representation R,\ with highest weight vector A coincides with dim HAO) (MG; L(,\)) which is just the index of the twisted Dolbeault complex, dim IZ,\
tr,\ 1
=
=
index(O,\)
(5.108)
[162]
Since
a(A + p) a(p)
dim IZ,\ .>O
where
a( )
(5.109)
-
I:ri=1 aiAi,
we see that the holomorphic polarization above naturally incorporates the Weyl shift A -- A + p of the highest weight vector A. Furthermore, in this case the Dirac operator 5, \ has no zero modes and according to the arguments used in Section 4.10 above the Lefschetz number =
indeXH(-\) (5A; T)
=
lim 0--+00
trIl
e
e-'49-\
iTH(A)
coincides with the character of the
zero
A
-
e-05-\'It) 11 A
mode representation defined
(5.110) by 5A,
i.e.
indeXHUQ (0,\; T)
=
str,\
e
iTHUO
(5.111)
Holomorphic Localization Formulas
5.4
157
The character of the representation R,\ is therefore the equivariant index of
the twisted Dolbeault
complex.
This identification with the
holomorphic properties
of the
complex
man-
ifold MG motivates another way to regularize the fluctuation determinants discussed in Section 4.2. The proper way in this case to carry out the regular-
procedure is to attach different signs to the factors of b in Section 4.2 (see (4.67)) corresponding to the holomorphic and antiholomorphic sectors in the regularization (4.65),(4.67) [102]. This restores the holomorphic properties of the path integral wherein the skew-eigenvalues (Aj, -Aj) of the block ization
R(j) V correspond, respectively,
to the
holomorphic
and
anti-holomorphic
com-
ponents of the equivariant curvature 2-form. The above argument implies that the correct way to treat the complex tangent bundle here is to then restrict to the holomorphic component of this curvature. In doing this, the fluctuation determinant in
(4.129)
is not under
a square root in this complex integration over complex Grassmann variables, i.e. the localization symmetries in a Kdhler polarization are determined now by a larger N 1 "hidden" supersymmetry. Now the determinant (4.65) for the case at hand is replaced with
case, because
now
it arises from Berezin
=
dethol at
-
R(j),)
fj k=AO
(5.112)
where the determinant in
sponding
to the
(
27rik
Aj
T
(5.112)
is taken
holomorphic indices.
To
only over those eigenvalues correregularize (5.112) properly we now
R(j) have to take into account that the Dirac operator at has an infinite V(A) number of negative eigenvalues, i.e. its spectral asymmetry must be regular-
ized
using the
eta-invariant
(4.63).
Thus
take the fluctuation
regulariza', explicitly takes into account the 2 spectral asymmetry determined by the Atiyah-Patodi-Siriger eta-invariant and maintains the original symmetry of the path integral under the "large gauge transformations" Rv(.\) --* Rv(>,) + 21rn/T, n E Z. The evaluation of the fluctuation determinant in (4.129) now leads instead to the equivariant Todd class (2.98) of the complex tangent bundle, tion factor of Section 4.2 to be b
-
we
which
n
1
deth'.111at
=
RV( 0
j
i
A)
11
11 -A3/2 sinh(TAi/2 )
eiTAj/2
3= 1 -
1 -
Tn
tdv(,\) (TR(A)) (5.113)
Then, using the usual Niemi-Tirkkonen localization prescription, an explicit Weyl shift,
we
arrive
at the Kirillov character formula without
tr,\ eiTX
_
j chv(.\) (-iTw(\)) OX
A
tdv(,\) (TR(\))
(5.114)
158
Equivariant Localization
5.
which
be derived
can
(as opposed
to
as
LO,\+p
coadjoint orbit path integral over LO,\ corresponding localization onto the H(,\) (c.f. Section 4.7), or equivalently the
well from the
as
in
(5.27) )2
critical points of the Hamiltonian
Weyl
Simply Connected Phase Spaces
on
The
.
group, is
e-iTH(A)(z, )
tr,\ eiTX Z
where
EMV(,\)
det+,dV(,\) (z,. ) ho
td(dV(A)(z,. ))
have identified the Pfaffian in the real
we
(5.115)
polarization with the dedV(,\). We recall
holomorphic eigenvalues Atiyah-Singer index contribution to (5.27) evaluates the spectral asymmetry of the zero mode representation of g determined by the pertinent Dirac operator, while the Lefschetz number coincides with the character of that representation of the spin complex. From the formulas (5.114),(5.110), however, we see that the character of a Lie group G is a Lefschetz number related to the G-index theorem of the holomorphic Dolbeault complex, rather than to the Atiyah-Singer index theorem of the spin complex. The localizations onto the (equivariant) Todd classes hold quite generally for any phase space path integral in a holomorphic representation. One just replaces theA-genus factors everywhere in Chapter 4 with the corresponding terminant
over
of the matrix
the
from Section 4.10 that the
Todd classes.
Spherical Phase Spaces Spin Systems
5.5
of We
are now
ready
hence work out
when the In this
phase
case
space M has
a
a
our
positive
=
-2K
completely integrable system [35]
O(Z" )
=
log
1 (T
By the essential uniqueness 2
case
(M, g)
of
0'('A whose
(5.116) general
af (Z)'Of (1 0
4
in this
constant Gaussian curvature K > 0.
the conformal factor solves the Liouville field equation
v2 p(z,. ) which is
Quantization
general isometric classification and explicit examples. The next case we consider is
to continue with
some more
and
+
A-^"M' ))
2
I
solutions
are
(5.117)
maximally symmetric spaces, we know that sphere S2 of radius K- 1/2 with its
is isometric to the
analog of the Weyl shift ambiguity for the coadjoint orbit models of the last 1 zero-point energy Eo 2 for the harmonic oscillator Hamiltonian at a in certain ordering prescriptions. See
The
Section is associated with the appearence of the correct
[102]
for details.
=
Spherical Phase Spaces
5.5
and
Quantization
Spin Systems
of
159
standard round metric given in (5.34). From the transformation law of the metric tensor g and (5.117) it is straightforward to work out the explicit
diffeomorphism (z,, ) ---> (w(z,. ), Cv(z,, )) which accomplishes this isometric correspondence. First of all, we rewrite the spherical metric in (5.34) in complex coordi1 nates w, fvx iX2, with x1' the spherical coordinates defined in (5.34), to =
get iV-
gS2
_K
1
+2
-
2+
2
wCv
)
W Cv
I
-
W
dw 0 dw +
WiD
1
W Cv
-
dw 0 AD
2
dCv & div-
(5.118)
I
3
sphere as centered in the x'y'-plane in R and symmetrically about the z'-axis, then we can map S2 onto the complex -1, plane via the standard stereographic projection from the south pole z'
where wCv < 1. If
we
view the unit
=
2w/ 1 + W/V
This
gives
a
i
-
1/1
/ =
I
diffeomorphism of S2
-
with the
WfV
-
W11-V1
compactified plane CU fool.
(5.117)
(5.118),
the metric tensor transformation law and
algebra
that the coordinate transformation above must
( aw,
1
(1
+
WliV-1)2
aw, afv-,
afv-, 92
+
'92
K
'9Z
(5.119)
1 + W'fV_1
e '(',2)
find after
we
From some
satisfy
af (Z)51(2)
K
(K4
+
f (Z) f 0
))
2
(5.120) From
(5.120)
(5.119) it (M' g) to S2
and
formation from
then follows that the desired coordinate trans-
(5.118)
with the standard round metric
by
W(Z" )
4K1 +
1/2f(Z)
is
given
(5.121)
4K-1f (z)l(. )
The mapping (5.121) is just a generalized stereographic projection from the south pole of S' where f (z) maps (M, g) onto the entire complex plane with the usual Kiffiler geometry of S2 defined
4AFS2 (Z,
gS2
WS2
205FS2 (Z,
where the associated Kiffiler
(w,,Fv)
E
)dz
17;7Z ) 2 2i
A J
space
dz (& d2
(5.122)
-dz A d,
J + Zj)2
10g(l
+
(5.123)
Z, )
diffeomorphism (5.121) obeys
,
(5.119),
is
S2 and that the Khhler metric
original phase
the coordinates in 4
0 d5
potential
FS2 Notice that the
)dz
by
wiv-
gS2 in
geometry (5.49) when f (z)
=
<
(5.122)
1,
-!K 1/2Z 2
as
required
for
coincides with the in
(5.117)
above.
160
Equivariant Localization
5.
Rom the
Killing VW S2
Connected Phase
general considerations of Section
(5.118)
vectors of the metric
a(l
-iS?w +
=
Simply
on
we
know that the
are
v) 1/2
V
wi
-
5.2 above
Spaces,
W
=
2
d(l
iQFD +
-
W fV-
) 1/ 2(5.124)
Killing vectors (5.124) generate the rigid rotations w --+ 02w of the W + a(l sphere and the quasi-translations w WfV-)112 (i.e. translations along the geodesical great circles of S2) and they together generate the Lie group SO(3). Requiring the symplecticity condition (5.70) again under the full SO(3) group action generated by (5.124) on the symplectic 2-form (5.69), we find after some algebra that the equations (5.70) are uniquely solved by The
__+
_
,
(W, fV)
WS2
This is
symplectic 2-form
is
11K(l
=
WfV)112
(5.125)
again the volume form associated with (5.118). It
non-trivial element of H 2 (S2;
a
_
Z and it coincides with the Kdhler Z) (5.122) in the stereographic coordinates (5.119). We now substi(5.124) and (5.125) into the Hamiltonian equations (5.68), which are =
classes in tute
easily solved on S2 in the w-coordinates above, and then apply the generalized stereographic projection (5.121) to get the most general equivariant Hamiltonian on a spherical phase space as S?
H+ (z,. )
=
( 4K K
7 +
fWA
O)
AZM2
+
af ( ) K 4
+
+
af (z)
AZW' )
+
(5.126)
CO
Thus, again the
(or
even
defined as
Riemannian geometry of the phase space M is realized determined) by the equivariant Hamiltonian systems which can be
on
usual
as
M. The transformation to Darboux coordinates those coordinates
A WS2= Mv 2
dV,
can
be found from the fact
form associated with
(2.12)
for
w.
M
are
defined
on
After
(5.117)
some
by
(v, V)
in which the
the
and
applying
M, defined locally
symplectic thatWS2 is the (Kiihler) volume the tensor transformation law
algebra diffeomorphism (z, 2) we
on
2-form is
find that the local Darboux coordinates -->
(v(z,. ),V(z, 2)),
where the
function
V(Z" )
-
_f W 1/2 (L4 + AZVOO)
(5.127)
IZ
(5.128)
-
maps M onto the unit disc
D
which is the Darboux
phase
2 =
(C: Z
C
<
11
space associated with
a
general spherical phase
geometry. Thus, applying the transformation (5.127) to (5.126), that the general Darboux Hamiltonians in the present case are space
H which
D( z,, )
=
correspond
S?z.
+
to the
(az + a2) (1
_
Z2)1/2 + C0
;
z
E D
quasi-displaced harmonic oscillators
2
we see
(5.129)
Spherical Phase Spaces and Quantization
5.5
H
D
(z,. )
=:
Q
[z
a(l
+
with
compactified position and
of
(compact)
_
Z )1/2
+
Zj(l
_
of
Spin Systems
Z )1/2
161
(5.130)
momentum ranges. Thus here the criterion
circle action cannot be
removed, in contrast to the case of planar geometries of Section 5.3 where the Darboux phase space was the entire complex plane C. Notice that all translations in the planar case become quasi-translations in the spherical case, which is a measure of the presence a
the
of
a
curved Riemannian geometry on M. mapping onto Darboux coordinates above shows that
The
once again general spherical Hamiltonians are holomorphic copies of each other, as they all define the same Darboux dynamics. We shall therefore focus our attention to the quantum dynamics defined on the phase space S2 (i.e. f (z) K 1/2 z/2 above), and for simplicity we normalize the coordinates so that now K 1, i.e. S2 has unit radius. First of all, we write the 3 independent observables appearing in (5.126) above as
all the
=
=
JW(Z" ) 3
-1 -3
=
-
Z2
AD + (z,. )
+ Z;
=
2
31
P) (z,. )
+ Z'
=
2j
Z
I + Z'
(5.131) where the parameter j will be Kdhler 2-form
specified
W(j) and
working I
IWO)
-
we
define the
(5.132)
jWS2
=
out the associated Poisson
AD JW
Using (5.122)
below.
algebra
of the functions
JW I JM' +
JM
-
=
WU )
(5.131)
2P) 3
(5.133)
realize the SU(2) (angular momentum) Lie algebra [162]. (5.131) therefore generate the Poisson-Lie group action of the isometry group SO(3) on the coadjoint orbit
shows that
they
The functions
S2
GlHc and
ing
we
obtain the usual
=
SU(2)IU(I)
coadjoint
the Hamiltonian to be
an
orbit
_
S31S1
=
S2
(5.134)
topological quantum theory by choossubalgebra u(1) of su(2).
element of the Cartan
S2 is often called the maghomogeneous space SU(2) --> SU(2)1U(1) bundle. the basic For C of U(1), monopole representation space W the associated vector bundle (SU(2) x C)/U(l) is the usual symplectic line bundle over S2 The Borel-Weil-Bott wavefunctions in the presence of a magThe
=
netic
=
.
monopole take values in this bundle. Notice that, comparing (5.131) with the stereographic
netic
coordinates
(5.119),
that these observables just describe the Larmor precession of a classical spin vector of unit length J 1. The coadjoint orbit path integral
we see
=
associated with the observables
dynamics
(5.131)
will therefore describe the quantum
classical spin system, e.g. the system with Hamiltonian H J3 describes the Pauli magnetic moment interaction between a spin J and a uniof
a
form magnetic field directed along the z-axis. Thus in this
=
case
S2 is actually
162
Equivariant Localization
5.
Simply Connected Phase Spaces
on
naturally the configuration space for a spin system [156], which has on it a sYmplectic structure and so the corresponding path integral can be regarded as one for the Lagrangian formulation of the theory, rather than the Hamiltonian one [139, 156]. This is also immediate from noting that the stereographic complex coordinates above can be written as natural
z
in terms of the usual
J3
in
(5.131)
to
an
additive
=
e- O tan(0/2)
spherical polar coordinates (0, 0), height function (2.1)
so
that the observable
of S2 with
coincides with the
a
=
I
(up
constant),
the Khhler geometry above becomes the standard and the kinetic term in the action is
round geometry of S2 ,
01, (X).t" after
(5.135)
=
Cos
O
(5.136)
integration by parts over t. The classical partition function, evaluated 2.1, yields the usual Langevin formula for the classical statistical mechanics of a spin system. Alternatively, the motion of the precessing spin can be reduced to that of a charged particle around a monopole which is isomorphic to the problem for the motion of an excited diatomic molecule where the electrons are in a state with angular momentum j about the axis joining the nuclei (i.e. a rigid rotator with fixed angular momentum j about an
in Section
axis) [156].
on the particle, dO of the monopole located at the magnetic field w center of the sphere, and the potential force on the moment, due to the real magnetic field, that leads to the characteristic Larmor precession about the
its
It is the balance between the the Lorentz force
due to the fictitious
=
direction of the field. To construct
Section
j
=
where .
.
we
topological
.
,
Hamiltonian
along the lines
of the
theory of
an irreducible spin-j representation of SU(2), where The state space for this representation with heighest
.1, 1, 3,2.... [162]. 2 2
weight 1,
5.1,
a
consider
spanned by the complete set of orthonormal basis states li,m), the magnetic quantum numbers with the range m -j, -j + 1, j. The SU (2) generators act on these states as
3 is
m are
j
-
J3 Ji7 M)
-=
=
M
1i) M)
J Jj7 m)
I
=
V(j
::F
m) (j
m
+
1) 1j, m 1) (5.137)
Following the last Section, we define the SU(2) coherent states by successive applications of the raising operator j+ to the lowest weight (vacuum) state
1j, -j) [45, 137], 3
Jz)
=
e-3P
e'J+ 1j, -j)
=
e-'jP
E M=-j
1/2Z
2j i
+
M
j+M
lj,m)
;
zEC
(5.138) where for n,
m
E
Z+ with
n
>
m
(n) M
the binomial coefficient is defined n! =
m! (n
-
m)!
by
(5.139)
5.5
Spherical Phase Spaces and Quantization
of
Spin Systems
p(z,, ) is an arbitrary phase which as we F(z, ) in (5.21). It is easily verified that
and where the function
is related to the function
SU( )
generators
J3, J
in the coherent states
(5.131)
are
(Z2 I ZI)
see
then the
the normalized matrix elements of the operators
=
(I
2)2j e j[P(Z2,22)-P(Z1,21)1
+ Zl'
(5.140)
have used the binomial theorem
we
n
(X
Y),
+
(n) Xkyn-k
F-
=
They obey
the
completeness relation
f dl_i(j)(z,. ) Iz))((zl 10)
(5.141)
k
k=O
where
shall
(5.138), respectively 3 (5.138) are normalized as
The coherent states
where
163
is the
identity operator
the coherent state
measure
in the
=
1(j)
(5.142)
spin-j representation of SU(2) and
is
2j+1
djL(j) (z,. ) -i
27r
(1
+
zf) 2
(5.143)
dz A J
symplectic 2-form of the spin system above. The identity (5.142) follows from a calculation analogous to that in (5.85). Note that, as explained in the last Section, the Kdhler structure is generated e2jFS2 (Z,2). through the identity (zlz) which coincides with the
=
We want to evaluate the propagator K (Z2 , zi; for
some
T)
SU(2) operator ? given
=
((Z21
e
-M
I Z1))
the one-to-one
(5.144)
correspondence between
S2 -- C U foo} and the the points on the coadjoint orbit SU(2)IU(l) SU(2) coherent states (5.138). Dividing the time interval in (5.144) up into =
N segments and letting N ---> oo, following the analogous steps as in Section 4.1 using the completeness relation (5.142) we arrive at the coherent state
path integral IC (Z2 zi;
T)
=
JV
Vdet 1100)
dz(t) d. (t)
LR2 tE 0,T]
x
exp
j 109(l
+
Z2f2)
+
i 109(l
+
Zlfl)
T
+if [ dt
ij I + Z'
(fi
-
Zzi-)
-
ii
(azLei
+
LP 02
Z) -
-
H(z, f) j
0
(5.145) 3The dimension dimRj
=
2j +
1 of the
spin-j representation of SU(2)
be derived from the index theorems of the last Section
(see (5.108)).
can
also
164
Equivariant Localization
5.
on
Simply Connected Phase Spaces
where N-1
11
M= lim IV--+OO
is
a
normalization constant and
(5.138).
H(z,. )
Here
(5.146)
2jir
k=1
the coherent states
+
denotes the matrix elements
(5.95)
in
again the formal equivalence of defined by the Kdhler polarization
we see once
the
path integral and Liouville measures above. In particular, the local symplectic potential generating the Kiihler
structures
(5.132)
are
0(j)
(. Wz
1 + Z'
-
zd. )
(5.147)
ijdp
-
they coincide with the standard coherent state canonical 1-forms (5.91). Similarly, the Khhler structure (5.122) can be represented in the standard and
coherent state form
(5.92).
The Wess-Zumino-Witten quantization condition (4.161) applied to WW implies that j must be a half-integer, since 4,7r, corresponding to the WS2
fS2
=
unitary irreducible representations of G SU(2) [162]. This is the topological of The spin. (or Dirac) quantization quantization of the magnetic quantum =
numbers
m
above then follows from
an
application
of the semi-classical Bohr-
a quantization [101] topological quantum theory (or equivalently an integrable quantum system) as described in Sections 4.10 and 5.1, we need to choose the phase function p(z,. ) in the above so that ivOW H. This problem was analysed in detail by Niemi and Pasanen [123] who showed that it is impossible to satisfy this integrability requirement simultaneously for all 3 of the generators in (5.131). Again, this means that there are no SU(2)-invariant symplectic potentials on the sphere S2 However, such 1-forms do exist on the cylindrical representation of SU(2) [123], i.e. the complex plane with the origin removed, which is conformally equivalent to the Kdhler representation of S2 above under the
Sommerfeld
condition
for the spin system. To construct
=
.
transformation In this latter
z
=
e'l +iS2
which maps
(s 1,82)
representation, the Hamiltonian
in
E R x
(5.145)
S'
to
can
z
E
C
-
101.
be taken to be
arbitrary linear combination of the SU(2) generators, and the coherent path integral (5.145) determines a topological quantum field theory with p 0 in (5.147). This is not true, however, in the Khhler representation above, but we do find, for example, that the symplectic invariance condition can be fulfilled by choosing the basis H(z,. J3(j) (z,. ) of the Cartan sub) .1 and The algebra u(1) p(z,, ) ensuing topological path integral 2 log(z/. ). (5.145) then describes the quantization of spin. To evaluate this spin partition function, we set p 0 above. Although the ensuing quantum theory now does not have the topological form in terms of a BRST-exact action, it still maintains the Schwarz-type topological form described in Section 4.10, since the Hamiltonian then satisfies (3.45) with C j and the function K in (3.46) is an
state
=
=
=
=
=
5.5
Spherical
Spaces and Quantization
Phase
i
K(z,. )
log
=
2
of
Spin Systems
(Z7)
165
(5.148)
-
(5.145) is a topological path integral of the form (4.165), i.e. the quantheory determines a Schwarz-type topological field theory, as opposed to a Witten-type one as above. We first analyse the WKB localization formula (4.115) for the coadjoint orbit path integral (5.145). We note first of all that the boundary conditions in (5.145) are z(O) 2. In particu(T) zi and lar, the final value z (T) and the initial value (O) are not specified, and the boundary terms in (5.145) ensure that with these boundary conditions there is no boundary contribution to the pertinent classical equations of motion so
that
tum
=
+ iz
In
general,
are no
sphere
if
z(t)
#)
and
are
.L.
0
=
Z
7
-
=
i'
=
(5.149)
0
complex conjugates of each other,
classical trajectories that connect z(O) zi with S2 But if we view the path integral (5.145) instead =
.
(T)
then there
=
as a
2
on
the
matrix ele-
configurations in different polarizations, then there is always following solution to the equations of motion (5.149) with the required boundary conditions for arbitrary z, and, 2, ment between 2
the
z(t) (5.150)
=
zi
e-'t
#)
7
e-i(T-t)
2
=
complex, and hence z(t) and
#)
(5.150)
regarded one independent the holomorphic quantization formalism that makes it suitable to describe topological field theories. The trajectories (5.150) are therefore regarded as describing a complex saddle-point of the path integral [44, 85, 143]. We shall The solution
is
as
see
other forms of this feature later
Substituting the
on.
(5.150)
the solutions
must be
of the characteristic features behind
variables. This is
into the WKB formula
(4.115)
we
find
propagator4 IC (Z2 , zi;
The exact propagator from
((Z21 e-iTJ3 I ZO)
a
+
Z04)j (1
+
+
(5.151)
Z2, 2)j
direct calculation is
1
(I
e-ijT
+ ZO 2 e- iT)2j
T)
Z121)j(l
+
Z222)j
M=
"j (i
2j +
M
) (zl, 2
e-
iT
)j+' e ijT (5.152)
which coincides with
(5.141). ing
In
particular, setting
the coherent state
In this
(5.151)
case
measure
upon zi
=
Z2
(5.143),
=
z
we
the fluctuation determinant in
non-periodic boundary conditions
of the binomial theorem
application
and integrating
over z
E
C
us-
find the partition function
(4.115)
is
discussed above.
regulated using the generic
166
5.
Equivariant Localization
f dy(i)(z,,
ZSU(2) (T)
on
Simply Connected
Phase
Spaces
) ((zl e-iTj3lZ))
00
dr
(2j
r
1)(1
+
+
r
2e-iT)2j
e
-ijT
sin -
(1
0
+
r2)2j
sin
i
e-iTm M=-j
e
1
-
2
(5.153)
which also coincides with the exact result
trj e-iTJ3
21 (2i + 1))
iTj
e-'Ti +
e-iTi
eiTi
(5.154)
The
right-hand side of (5.154) is precisely what one anticipates from the Weyl character formula (5.28). The roots of SU(2) are a 1 [162], and the Cartan subalgebra is u(1) consisting of the single element j3. The Weyl group is W Z2 and it has 2 elements, the identity map and the reflection map 4 -+ -j3. Thus the formula (5.154) is simply the Weyl character formula (5.28) for the spin-j representation of SU(2). Within the framework of the Duistermaat-Heckman theorem, the terms summed in (5.154) are each associated with one of the poles of the sphere S2, i.e. with the critical points of the height function on S2. Indeed, since this Hamiltonian is a perfect Morse function with even Morse indices, we expect that the Weyl character formula above coincides with the pertinent stronger =
=
(4.135) of the localization formulas. Because of the Khhler structure (5.122) on S' (see (5.53)), the Riemann moment map has the non-vanishing
version
components
G'Vw)'Z and
consequently
the Dirac
=
iJO) (Z" O/i 3
-(1LV(i))2
A-genus
A(TS?V(j))
(5.155)
is
JU) 3
T
2i
sin
(5.156)
(2j Jw) T
3
Substituting these into the localization formula (4.135) yields precisely the Weyl character formula (5.154). This localization onto the critical points of the Hamiltonian, as for the harmonic oscillator example of Section 5.3, agrees with the general arguments at the beginning of Section 4.6. Substituting the stereographic projection map (5.135) into the classical equations of motion (5.149) gives
sinO For T
=
0
+ 1
=
(5.157)
0
:A 2-7rn,
n E Z, the only T-periodic critical trajectories coincide with cos 0), i.e. 0 points of the Hamiltonian j (I 0, 7r, and in this case the critical point set of the action is isolated and non-degenerate. How27rn, n E Z, we find T-periodic classical solutions for any initial ever, for T
the critical
-
=
=
value of 0 and
0
in
(5.157)
and the critical point set of the classical action
5.5
Spherical
Phase
Spaces and Quantization of Spin Systems
167
original phase space S'. Thus the moduli space of classical S2 and the localization onto this moduli is LMS space is now easily verified from (4.122) to give the correct anticipated result above. From the discussion of Section 4.10, it also follows that the sum of the terms in (5.154) describes exactly the properly normalized period group of the symplectic 2-form w(j) on the sphere [85], i.e. the integer-valued surface integrals of w(j) as in (4.161). We shall see in the next Chapter that quantizations of the propagation time T as above lead to interesting quantum coincides with the solutions in this
case
==
,
theories in certain other instances of the localization framework. It is
an
instructive exercise to work out the Niemi-Tirkkonen localization
(4.130)
formula
dynamical system.
for the above
S2,
because of the Kiihler geometry of
For this
note
we
that, again
the Riemann curvature 2-form has the
non-vanishing components R'z and
so
combined with
(5.155)
AV(j) (TR)
The equivariant extension of
J(j) 3 where
-
we
W(j)
j
1
-
-iW(j)
-RzZ
=
we see
that the
J(j) 3
T =
(5.158)
equivariant A-genus here
-
W(j)
w(j)
sin
(-L2j (J(j) 3
-
)
2i
Z-)2
1 + Z'
=j
(I
have redefined the Grassmann variables
ZSU(2) (T)
(4.130)
f
i -
7rT
,W))
is
Z'
Tirkkonen localization formula
(5.159)
-
2j
is
can
dz d.
Z`
+ Z
ql"
-+
(5.160)
77
+
Vi7- n/.
then be written
The Niemi-
as
d77 d L(z, +,q )
(5.161)
R20AIR2
where T 2
L(y) sin
Using
1-Y I+y
exp
(T (1-y)) f
1+y
[-ijT (I Y)l
the Parisi-Sourlas integration formula
(5.162)
1 + Y
[136]
00
1 7r
1
d 2x
d77 d
L(X2
+
7A
R20A-IR2 we
obtain from
du
=
du
L(oo)
-
L(O)
(5.163)
0
(5.161)
the partition function
ZSU(2) (T)
-
sin(Ti) / sin(T/2)
(5.164)
168
5.
Equivariant Localization
Introducing
the
Weyl shift j
Phase
Spaces
in + .1 2
(5.164) then yields the correct Weyl (5.153) for SU(2) 5 Note that (5.163) shows explicitly localization in (5.161) comes directly from the extrema of the height j
---
character formula how the
function at
As
Simply Connected
on
z
=
oo
.
and
z
0.
=
final application for the above dynamical system, we examine the quadratic localization formula (4.150). Now the (degenerate) Hamiltonian is a
.F(j(j)) 3
Following written
the
same
steps
as
-
(j(j))2 3
above,
(1+ Zf) 1
j2
=
-
Z'
(5.165)
the localization formula
(4.150)
can
be
as 00
W
ZS U (2) (T I (J3' ) )
f doo f 00
i
2
-
-v
47riT
dz d2
dq d L(Oo, z2 +,q )
R20A1R2
-00
(5.166)
where T-00 1-Y 2 1+y
L(00, y) sin
and
we
formula
010 (1-0) 2
1+y
have redefined
n4
(5.163) again
and
ZSU(2) Prp(j))2) 3
exp
14002 0
-
ijToo
(I Y)] 1 + Y
(5.167)
V/i7/_0o -7f. Using the Parisi-Sourlas integration introducing the Weyl shift j we find j + .1, 2
--+
--->
T4T7ri J
doo e'T 020/4
sin[(j + 25')Too] sin(Too/2)
--
__
-00
3
E M=-i
FT_
V 41ri
-
00
f doo
e- iTmoo
e
iTO2/4 0
L
e-
iTM2
M=-j
-00
(5.168)
which is again the correct character trj e- iTj32 Thus on a spherical phase space geometry the equivariant Hamiltonian systems provide a rich example of the topological quantum field theories discussed in Section 4.10, and they are the natural framework for the study of the quantum properties of classical spin systems. The character formula path integrals above describe the quantization of the harmonic oscillator on the
sphere, and therefore the only integrable quantum system, up to holomorphic equivalence (i.e. modification by the general geometry of the phase space), that exists within the equivariant localization framework on a general spherical geometry is the harmonic oscillator defined on the reduced compact phase space D 2. 5
Of course,
we
could
alternatively obtain the Weyl character formula using instead
the G-index localization formula
shift
[102].
(5.114)
without
having
to
perform this Weyl
Hyperbolic Phase Spaces
5.6
5.6
Hyperbolic
The situation for the
Phase case
169
Spaces
where the
space is endowed with
phase
a
Rieman-
parallels that negative differthe essential of the last Section, and we only therefore briefly discuss ences [1591. The phase space M is now necessarily a non-compact manifold, and we can map it onto the maximally symmetric space V, the Lobaschevsky plane (or pseudo-sphere) of constant negative curvature, with its standard curved hyperbolic metric g'H2 [42, 43, 68]. The Killing vectors of this metric have the general form nian
V;
geometry of
constant
2
a(l
==
-iQw +
+
Gaussian curvature K < 0
WiV-)112
V4
2
=
iQiV +
d(l
+
WfV-)112 (5.169)
they generate the isometry group SO(2, 1). The rest of the analysis at the beginning of the last Section now carries through analogously to the case at hand here, where we replace the K factors everywhere by -JKJ and the K1/2 factors by JK11/2. In particular, with these changes the generalized stereographic coordinate transformation (5.121) is the same except that now the holomorphic function -1 1/2 f (z) there maps the phase space onto the Poincar6 disk of radius 2 JKJ and
i.e. the disk
D2 with the Poincar6 metric 4 gIH2 -
(5.170)
dz (& d;
Khhler geometry on the disk for which the associated sym2-form is the unique invariant volume form under the transitive
which defines
plectic
Z )2
a
stereographic projection image for the Lobaschevsky plane when we regard through its embedding in R 3as the it by pseudo-spherical coordinates pseudo-sphere, so that we can represent cosh -r. The x2 sinh -r sin 0 and z xi sinh -r cos 0, (-r, 0) E R x [0, 27r] as z' the center from taken -1, projection stereographic projection is again at to disc Poincar6 the and the boundary of infinity of the corresponds points hyperboloid H2. The pseudo-sphere itself is represented by the interior of the disk. The explicit transformation in terms of pseudo-spherical coordinates is
SO(2, l)-action.
The Poincar6 disk is the
it
=
=
=
=
W1 Z
T7Z
=
e-'O tanh(-r/2)
(5.171)
conformally equivalent to the i)/( + i) which ( plane C+ via the Cayley transform --+ z C+ onto the Poincar6 disk, and the Poincar6 metric (5.170) on the
We also note here that the Poincar6 disc is upper half
takes
E
=
(Poincar6) upper-half plane
is
g, 12
The
path integral
over
such
studies of quantum chaos
-
=
IM(6)-2d6 (9 d
hyperbolic geometries
[311.
(5.172) arises in
string theory and
170
5.
Equivariant Localization
The most
general
on
Simply Connected
localizable Hamiltonian in
Phase
Spaces
hyperbolic phase
a
space
geometry is therefore
(
S? H_ (z,
2)
=
IKI 4
f
+
(Z)1(2))
al(2)
+
IK-I- AZ)A O 4
]KI
_
4
The transformation to Darboux coordinates
df(Z)
+
(5.173)
f(Z)I(' ).+ CO
on
M is
now
accomplished by
diffeornorphism
the
f(Z)
V(Z,,9) IKI -
4
(5.174)
1/2
f(z)!(
which maps M onto the complement of the unit disk C general Darboux Hamiltonians are therefore
B
(z,. )
=
Qz2 +
(az + a. ) (1
We note that here there
+
Z )1/2
C
E
z
;
-
-
int(D 2)
in
R2
int(D 2)
The
.
(5.175)
inequivalent Hamiltonians, corresponding to "spacelike" Killing vectors, but the generic hyall Hamiltonians are holomorphic copies of one another, again perbolic again harmonic oscillator. However, given that the to a quasi-displaced reducing Darboux phase space is now non-compact, we can again weaken the requirement of a global circle action on the phase space to a semi-bounded group a
2
are
and "timelike"
choice of
action.
Considering therefore the quantum problem defined on the Poincar6 disk radius, we write the 3 independent observables in (5.173) as
of unit
S(") (z, ) 3
=
kI + Z2 1
-
,
Z2
S(k) + (Z" )
=
Z
2k
I
-
Z2
S(k) (Z" )
7
=2k
z
1
-
Z'
(5.176) kW'H2,we see that the associated Defining the Khhler 2-form w(k) algebra of these observables is just the SU(I, 1) Lie algebra =
f S(k) 3
1
S(k)
I
IS(k)
-
7
L" (k)
The Hamiltonians in
(5.173)
are
f S(k), S(k) IW(k) +
therefore functions
SU(I' I)IU(I) of the non-compact Lie group
-
-
-
,
on
the
2S(k) 3
Poisson
(5.177)
coadjoint orbit
H2
(5.178)
,
SU(1, 1), and the generators (5.176) are the SU(I, 1) generators in the SU(I, 1) coher-
normalized matrix elements of the ent states c'O
Jz)
=
e' S+
Ik,O)
=
E n=O
(2k+n+ 1)1/2 n
z' I k,
n)
;
z
c
int (D
2) (5.179)
5.7 Localization of Generalized
171
Spin Models and Hamiltonian Reduction
representation of SU(1, 1) characterized by k representation spaces are now infinite-dimensional
for the discrete irreducible
1,
[137]. -32 , 2, 55,... 2
The
=
because of the non-compactness of the group manifold of SU(1, 1), and the representation states I k, n) defined here are the eigenstates of the generator
S3 with eigenvalues
S31k, n) The coherent states
(5.179)
have the normalization
(Z21ZO where
we
(5.180)
(k + n)lk, n)
=
=
(1
_
Zl' 2)-2k
expansion
have used the binomial series 00 =
X)n
+
m
E
(5.181)
n
-
1
(5.182)
XM
M
Tn=0
n E Z+ and JxJ < 1. integrable Hamiltonian systems are obtained by taking H S(k) which is the height function on H2 and the corresponding coherent state 3 path integral describes the quantization of the harmonic oscillator on the these are the only open infinite space R' (and up to holomorphic equivalence is straightforward It a on space). phase hyperbolic general integrable systems to analyse the localization formulas for the coherent state path integral just
which is valid for
Again,
the
,
,
as
in the last Section. For
coadjoint
instance, the WKB localization formula for the
path integral
orbit
T
[d cosh -r] [do]
Zsu(,,,)(T)
exp
i
dt
-
k(l
+ cosh
-r))
0
LIV
can
I (k coshr
(5.183)
be shown to coincide with the exact
Weyl character formula for SU(1, 1)
[51, 143] 00
Zsu(,,,)(T)
=
trk
e
e-iT(k+n)
-iTS3 n=O
5.7 Localization of Generalized
=
2i
e
-iT(ksin
T
2
(5.184)
2
Spin Models
and Hamiltonian Reduction The
explicit examples
have given thus far of the localization formalism quantum cases have, for simplicity, focused on dy-
we
in both the classical and
namical systems with 2-dimensional phase spaces. Our main examples have been the harmonic oscillator, where the localization is trivial because the Hamiltonian is a quadratic function, and the spin partition function, where
172
Equivariant Localization
5.
Simply Connected Phase Spaces
on
the exactness of the
stationary-phase approximation is a consequence of the conspiracy between the phase space volume and energy which makes this dynamical system resemble a harmonic oscillator. In Chapter 8 we shall present some true field theoretical applications of equivariant localization, but in this Section and the next
we
wish to overview the results which
actness of the localization formulas for
concern
the
ex-
higher-dimensional coadjoint orbit models which can be considered as generalizations of the spin models of this Chapter (and the previous ones) to larger Lie groups. We have already established quite generally that these are always examples of localizable dynamical systems, and here we shall explicitly examine their features in some special instances. The generalization of the classical partition function for SU(2) is what is commonly refered to as the Itzykson-Zuber integral [76]
I[X, Y; TJ
f
=
some
DU
e
iT
tr(UXUtY)
(5.185)
U(N)
where IV
DU
N
11
=
dUij
J
i,j=l
is Haar
(or N
on
x
SU(N)
N matrices
(5.186)
JiJ
-
U-1 U(N) of N x N unitary matrices Ut Here Hermitian are (Xt, Yt) (X, Y) U(N)I(U(l) Z2)). (i.e. elements of the U(N) Lie algebra) and which can be the group
measure on =
E UikU*k k=1
=
X
=
eigenvalues xi, yi E R by unitary transformations (VtXV, WtYW). By the invariance of the Haar measure in (5.185) (X, Y) WUVt of U(N), we can thus assume without under the left-right action U loss of generality that the matrices X and Y in (5.185) are diagonal so that therefore
with
diagonalized
---*
--*
N
I[X, Y; TI
DU exp
iT
U(N)
The
E
X, Yj
1Uj12 i
(5.187)
i,j=l
Itzykson-Zuber integral is a fundamental object that appears in mastring theories, low dimensional quantum gravity and higher-
trix models of
dimensional lattice gauge theories [36, 92, 106]. The integration over unitary matrices in (5.185) the Duistermaat-Heckman theorem via the
following
can
be carried out
using
observation. If we define
the Hermitian matrix
A then
explicitly
we can
variables U
--
A
in
=_
UYUt
(5.188)
compute the Jacobian for the
(5.185)
to
get
change
of
integration
[36, 106]
N
DA
=-
11 dAii i=1
fl I<j
d Re
Aik
d Im
Ajk
==
' A [Y] 2DU
(5.189)
5.7 Localization of Generalized
Spin Models and Hamiltonian Reduction
173
where A [y]
_=
qe 1,3
1yi
ri
j-1
-
yi)
(5.190)
I
is the Vandermonde determinant. The be written
(Yi
Itzykson-Zuber integral
can
therefore
as
I[X, Y; T]
f
A ly?
DA eiT tr(XA)
(5.191)
OY
Notice that the
diagonal components of U do not act on a diagonal matrix Y under unitary transformation. The integration in (5.191) is therefore over the coadjoint orbit of Y under the action of the unitary group, and as such it is an integral over the symmetric space GIHC where Hc U(I)N is the Cartan subgroup of G U(N). This can also be noted directly from the definition (5.185) in which the integrand is unchanged if U is multiplied on the right by a diagonal matrix so that the integration is really over the coset space obtained by quotienting U(N) by the subgroup of diagonal unitary matrices (this extra integration then produces a factor [Vol(U(1))]N (27r) N in front of the coadjoint orbit integral). This coset space has (even) dimension =
=
=
dim
U(N)
-
dim
U(1)N
=
N
2 -
(5.192)
N
and DA is the standard symplectic measure on the coadjoint orbit. The integral (5.191) was explicitly evaluated in [3] using the so-called GelfandTseytlin parametrization of the orbit. To apply the Duistermaat-Heckman integration formula to the integral
(5.187),
we
note that the extrema of the Hamiltonian N
HI [U]
E X,YjIUjl2
=
(5.193)
i,j=l as a
Uij
function of U E
and setting it
U(N), given by differentiating (5.193)
equal
to zero,
satisfy
the
[X, UYUt]
with respect to
stationary conditions
=
(5.194)
0
It is easy to see that the solutions of (5.194) for X, Y diagonal are of the form U Ud P where Ud is a diagonal matrix and P is a permutation =
matrix that
-
permutes the diagonal entries of a matrix when acting by unitary
conjugation. Diagonal and thus the
sum over
write U
Then
=
over
E
-
Taylor expanding (5.193)
HI[U=P.
(5.193)
extrema in the Duistermaat-Heckman formula is
SN. To evaluate the pertinent fluctuation determinants, P eiL in (5.193) where L is an infinitesimal Hermitian matrix.
permutations P we
matrices do not contribute to the Hamiltonian
e
iLi
to
quadratic order 1
=
xiyp(i) + 2
Lij 12(X,
in L
_
we
find
Xj)(Yp(,)
_
yp(j))
+...
(5.195)
174
Equivariant Localization
5.
on
Simply Connected
and the Duistermaat-Heckman formula e
iT
Phase
Spaces
yields
J:, xiyp(j) N!
PESN M
f rl dLij
X
e
W
-00
(5.196)
N(N-1)12
2,7ri
N(N-1)12
2-7ri
1:
sgn (P)
detij
1
e
iT
E,
xi yp (i)
'AIXI'Alyl
PESN
[ iTxiyjl e
A[X],A[y]
N!
T we
1
N!
T
where
Eili JLij 12(X,_Xj)(YP(,)_YP(j))
(iT/2)
have used
fl(yp(i)
yp(j))
-
sgn(P) 11(yi
=
-
yj)
=:
sgn(P).A[y]
(5.197)
i<j
i<j
The sign of the permutation P in Hessian of (5.193). The localization formula
(5.197)
(5.196),
arises from the eta-invariant of the
which is
a
special
of the Harish-
case
Chandra formula, was discovered by Itzykson and Zuber [761 in the context of 2-matrix models which describe conformal matter coupled to 2-dimensional
[36] (e.g.
the Ising model on a random surface). It was using heat kernel and U(N) group character expansion methods [76], and orthogonal polynomial techniques for the corresponding 2-matrix model [36]. In matrix models, saddle-point approximations are always employed in the large-N limit where the models describe the relevant continuum physical theories. Notice that the classical spin partition function (2.3) is a special case of the above result where N 2 and X and Y are both proportional to the SU(2) Pauli spin matrix
quantum gravity
originally
derived
=
3
(defining
In the
0 -1
(5.198)
1
vector) spin- 2 representation, we can represent an arbiSU(2) (obtained from a 3-dimensional rotation matrix in
or
trary matrix D E SO (3))
1 0
as
D
--
ie1('P-0)sin22
ieYi(P+'0)cos22
(ie-i(0+0)cos2 ie-!iF('P-0)sin2) 2
where
0, 0
and
0
E
[0, 41r]
are
of the principal fiber bundle
(0, 0, 0)
E
S3
_-,
(0, 0)
E
recover
(2.2)
(2.3)
-->
now
into
angles.
SU(2)IU(l)
S2 which effectively
stituting these identifications and
the usual Euler
SU(2)
(5.199)
2
The
projection Hopf
is then the
map
map
0 in (5.199). Sub0 the Itzykson-Zuber integral (5.185), we
follows from
sets
(5.196) (for
=
a
=
0 in Section
2.1).
5.7 Localization of Generalized
There
which a
are
unitary
Spin Models and Hamiltonian Reduction
various extensions of the above
are
relevant in matrix model theories matrix
generalized
[92].
classical
First of
all,
175
spin model
we can
consider
integral of the form
I[A; T]
=
N
f
DU exp
iT
1:
Aij I Ujj 12
(5.200)
i,j=l
U(N)
The stationary conditions for the Hamiltonian in
(5.200)
are
N
EUjj(Ajj-Ajk)Utjk =0
i, k
(5.201)
N
=
j=1
whose solutions
expansion of
again the permutation matrices (5.200) thus yields (c.f. eq. (3.53)) are
U
=
P. The
saddle-point
(I
0(11T A))
eiT E. Ak,P(k)
I[A; T] PESN
r1j<j (Ai,p(j)
+
Aj,p(j)
-
Aj,p(j)
-
Aj,p(j))
+
-
(5.202) In
general, the higher-order corrections in (5.202) to the lowest-order stationary phase approximation do not vanish because the Hamiltonian in (5.200) is not defined on any coadjoint orbit in general but on the entire group manifold which is not even a symplectic space. The corrections do, however, vanish in some interesting exceptions where (5.200) is a coadjoint orbit integral. One case is that discussed above, namely when Aij is of rank 1, so that Aij xjyj. Another interesting case is when N 2, so that (5.200) is slight modification of the spin partition function. For the case of SU(2) one can check explicitly that the leading order term in (5.201) is the exact result for the integral so =
=
that
ISU(2) [A; T] The unitary matrix
eiT(Ajj+A22) All
+
A22
-
-
e
iT(A12+A21)
A12
-
(5.203)
A21
integrals of the form (5.200) are generating functions Itzykson-Zuber model (5.185),
for the correlation functions in the I [X,
Y; T]
k-j1j***kIII
=
f
DU
Uj1j1
...
U(N) x
which
U iPjP
(Ut)ki I,
...
(Ut)kplp (5.204)
eiT tr(UXUtY)
only non-vanishing correlators because of the U(1) phase ine"9Uij of the integral (5.185). The evaluation of the unitary matrix integrals (5.204) is very important for matrix models of induced gauge theories and string theories [92] and has been a difficult problem that has are
variance
the
Ujj
--+
received much attention in the last few years. For instance, using GelfandTseytlin coordinates on the group manifold, Shatashvili [1541 has shown that
176
Equivariant Localization
5.
some
(5.204)
of the correlators
mulas, such
Simply Connected Phase Spaces
on
explicitly given by
are
complicated for-
very
as
I[X,Y;T]'IJ1JJ2, "" k2,1...k,,l IJP *
Jil ki Ji2 k2
*
*
00
f! riN-1 R=1
Jjp kp
(iT)(N-2)(N-1)
6 [X1,1A (Y2 i
...
7
f
YN,
_cO
(k=1E N
X
iTy,
exp
N-I
Xk
E
-
k=1
Ak)
N-1
11
n
dAk
k=1
11 1=1
XjI)
-
-
XjI)
eM\iyj
det 1
IIN-I q=1 (Aq IJqOjI (Xq
1
1
(5.205) where the delta-functions in
(5.205)
arise from the
U(N)
gauge invariance
VUVt of the Itzykson-Zuber integral (5.185). From the point of view equivariant localization theory, these types of integrals fall into the category of the problem of developing a description of the corrections to the Duistermaat-Heckman formula in a universal way. This problem will be discussed in Chapter 7. There is also a more geometric generalization of the spin partition function in terms of higher-dimensional generalizations of the Kiihler structure of S2 These examples will also introduce another interpretation of the localization symmetries which is directly tied to the integrability properties of these spin systems. Consider the complex N-dimensional projective space (CpN, defined as the space of all complex lines through the origin in (C N+1 A hoS2N+llSl is obtained by mogeneous Khhler structure on the bundle CpN the symplectic reduction of that from S2N+1' i.e. the restriction of the unique Khhler structure of S2N+1 to (CpN (note that CP' S2) The embedding of S2N+I in (CN+l is defined by the constraint U
--+
of
.
.
-
=
.
N+1
zp' A
P(Z" ) and the
-
I
=
(5.206)
0
symplectic structure on the maximally symmetric by the Darboux 2-form
space
(CN+l
is
as
usual defined
N+1
W(N+I) D The standard dinates
complex =
on
CpN
WN+1
=
2iJ
A
(5.207)
dzl'
is defined
by the complex
=
coor-
=
(5-206). (5.207), the descendent symplectic (J)
d, 4
for it 2,..., N + 1, where z' : 0 solves the "zl Solving for z1 and substituting it along with z"
constraint
into
structure
zAlzl
2iJ
=
[
N
A=I
dtA +
A
N
d ll
EN '\=1
1,
=
structure
%
A'V=1
on
(CpN from (CN+l is
A WFI (I
+
A
d6'
EN A= I A X)2
(5.208)
5.7 Localization of Generalized
Spin Models and Hamiltonian Reduction
The Kdhler metric associated with
Fubini-Study
[41]
metric
of
(CpN,
F(') N+1
-
(5.208)
is
usually refered
to
177
as
the
and the associated Kdhler potential is
(
J log
N
E ve
1 +
(5.209)
tj=1
Since the symplectic 2-forms (5.208) define non-trivial elements of the cohomology group H'(CpN; Z), for N > I the spaces CP N are not homeomorphic: to any of the maximally symmetric spaces discussed in Section 5.2 above and this symplectic manifold leads to our first example of localization on a homogeneous space which is not maximally symmetric. This space for N > 1 has N independent isometries which are the rotations in each of the N 2-planes (see below). In the next Section we shall see how to explicitly relate the space CpN to a (non-maximal) coadjoint orbit of SU(N + 1) so that, physically, this example describes the d'ynamics of a particle with internal SU(N + 1) isospin degrees of freedom in an external magnetic field. There we shall also encounter other examples of this situation, i.e. where the phase space is not maximally symmetric but has a maximally symmetric subspace. For now, we shall concentrate
on
the properties of the SU(N + an integrable model.
1)
classical spin partition
function associated with
(Sl)N+l on CN+1 is given by z1' generated by the Darboux. Hamiltonian
An action of the torus TN+1 P C-
[0, 2-7r),
which is
e O"' ZA'
-+
=
N+1
-(N+l) BD (z
0)
=
2J
1: 01'ZA'
'a
(5.210)
that describes the
dynamics of N + 1 independent simple harmonic oscillators correspond to a conserved charge of this integrable system. The classical partition function is of course given by a trivial Gaussian integration
which each
N+1
ri
(T)
D
R2N+2
A=1
2J dzl' d, A e
iTH
(N+I)(Z,2;0)
D
(-iT)N+l
7r
01, rjN+1 t1=1
(5.211) which coincides with the Duistermaat-Heckman formula
the
only fixed point
interesting though
generated by (5.210)
of the torus action
is the Hamiltonian T N -action
on
always
as
is at z/'
CpN' "
--+
because
=
each tt, the Hamiltonian function which generates the circle action A-plane in (CpN is the conserved charge
2JO It4 1
and
an
integrable
functions
(5.212),
Hamiltonian
can
on
For
the
(5.212)
+EN V=1
be constructed
0. More
e'6" A.
as
the
sum
of the N
height-
178
5.
Equivaxiant Localization
on
Simply Connected
Phase
Spaces
N
(5.213)
HN
The conserved charges (5.212) are the action variables of the above integrable model and the associated angle variables OA c- [0, 27r] are the usual polar angles of the A-planes (as for the (CN+l example above). The LiouvilleArnold integrability of this dynamical system can be made more explicit by considering the generalized stereographic projection onto CpN in terms of the spherical coordinates (OA, 0") E S2N+1 (where 0 < 0/1 < ir),
tan(O'/2) COS(02 /2) e-'O'
2
,
N-l
=
tan(Ol/2) sin(02 /2)
...
N
=
tan(Ol/2) sin(02 /2)
...
=
tan(Ol/2) sin(02 /2) cos(O'/2)
2
e
sin(ON- 1/2) COS(ON /2) e-ioN-'
sin(ON- 1/2) sin(ON /2) e-ioN (5.214)
The action variables
It,
=
(5.212)
2J A sin 2(01/2) IN
...
in these
sin
are
2(0/'/2) COS2 (0"+1/2)
V N sin2(01/2)
=
spherical coordinates
sin
...
then
p < N
2(ON- 1/2) sin (ON /2) 2
(5.215)
straightforward to see that the symplectic 2-form (5.208) takes the z11,01 as usual). integrable form dI,, A dol' (for the oscillator above 1,, explicit torus action on (CpN generated by the Hamiltonian vector field
and it is usual The
-
associated with
(5.213)
(N
=
is thus
E i /-' 14=1
Vt
'9
a -
5p-A
E /I=
A
aol-t
(5.216)
integrable system on the phase space (CpN is therefore once again isomorphic to a linear combination of independent harmonic oscillators as above, generalizing the dynamics of our standard spin example. The relation between such generalized spin systems and systems of harmonic oscillators, and hence the exactness of the localization formulas, can be understood from a slightly different perspective than the usual localization symmetries provided by the underlying equivariant cohomological or supersymmetric structures of the dynamical system. To see this, we evaluate the classical partition function explicitly for the above dynamical system by extending the integration over (CpN to the whole of (CN+l via a Lagrange multiplier which enforces the constraint (5.206). This leads to This
5.7 Localization of Generalized
ZN (T)
Spin Models and Hamiltonian Reduction
2J dtf' d ' IIN p=j N(j + EN qrl)N+l
eiTHN( , ;O)
-
Ir
S2N+IISI
,,=,
00
f
dA
N+1
j
27r
d2F1 &A
ri
R2N+2
-00
(5.217)
7r
tj=1
1)
N+1 x
where the Gaussian
Tl.(N+ T "D
exp
179
1)
(Z
1
7
0)
E
+ j'X
angular parameters in (5.217) are related by OA integral over CN+1 in (5.217) can be performed 00
ZN (T)
f
=
01'
=
to
_
ON+1 The .
yield
N+1
dA
%+
e-'A
TOA
(5.218)
A
00
remaining integration in (5.218) can be carried out by continuing the integration over a large contour in the complex plane and using the residue theorem to pick up the N + 1 simple poles of the integrand at A -TOA, each of which have residue 1. Thus the classical partition function is The
=
N+1
ZN (T)
=
E
e
iT01'
j1=1
It is
readily verified
that
(5.219)
11 V014
i
T(Ov
-
(5.219)
P)
coincides the Duistermaat-Heckman integra(5.213) which has precisely N + 1 stationary
tion formula for the Hamiltonian
points [52]. (5.219) therefore represents the TN -equivariant cohomology of the manifold CpN. The
explicit evaluation above illustrates
an
interesting feature of the lo-
calization in these cases, as was first pointed out in [44, 52]. This is seen in (5.217) the dynamical system describing the SU(N + 1) isospin is the -
Hamiltonian reduction of
quadratic
Hamiltonian
on
a
larger dynamical system which is described by a 2N+2 and for symplectic manifold R
the Darboux
which the localization is therefore trivial. The reduction constraint function
(5.206)
commutes with the Darboux oscillator Hamiltonian
P(z,, ), HD( +')I N
so
that the constraint function
P(z,, )
=
(,+ )
0
(5.210),
i.e.
(5.220)
WD
determines
a
first class constraint
on
dynamical system and is therefore a symmetry (or conserved of the classical dynamics. This commutativity property is in fact the
the Darboux
charge)
correspondence between the integrations in (5.217), CN+1 is restricted to the symplectic subspace dedynamics termined by the constant values of the conserved charges P(z,, ) (according crucial mechanism for the as
then the
on
180
5.
Equivariant Localization
[7]
to the reduction theorem
-
on
see
Simply Connected Phase Spaces
(4.144)).
reduction mechanism here is the action of the usual matrix-vector
representation of
SU(N
1).
the Darboux system above
on
and hence restricts the
U(1)
defining (vector) symplectic Hamiltonian action leaves invariant the sphere S2N+1
multiplication by +
matrices in the
This defines
[441
dynamics
which
Another important feature of the SU(N + 1) on CN+1 defined by a
to this submanifold of R
invariance of the harmonic oscillator Hamiltonian
restricts further to the coset space
charges under (CN+l become bundle of the
U(1)
(CpN
=
on
S2N+11U(j)
2N+2
From the
.
(C N+1
above, this
and the conserved
the mapping by the symplectic constraint function P(z,. the components of the projection map of the generalized
S2N+1
_4
_-,
(CpN [44]. It is easily
quadratic (5.217) exactly with the poles in (5.218) Hamiltonian in
coincide
seen
in the space of the A and so
that
)
on
Hopf that the critical points
(5.219)
z
variables
coincides
precisely
with the Duistermaat-Heckman formula for this Darboux system. The localization of the isospin partition function is thus the reduction of that from R 2N+2
We recall that such
a
reduction method in equivariant localization is also
that which is used to derive the Callias-Bott index theorems from dimensional Section
4.2).
Atiyah-Singer
index theorems
[70] (see
higher-
comments at the end of
The above reduction
procedure is a standard method of inteby grating dynamical systems associating to a given Hamiltonian system a related one on a lower dimensional symplectic manifold [1]. In this scheme one
reduces the rank of the differential equations of motion
or
the number of
degrees of freedom using invariant relations and constants of the motion onto a coadjoint orbit as above. Furthermore, in the context of generic integrable models, there is the conjecture that every integrable system is the Hamiltonian reduction of a larger linear dynamical system by first class constraints [176]. If this conjecture were true, then the localization of generic integrable systems could be cast in another formalism giving an even stronger connection between the Duistermaat-Heckman theorem and the integrability properties of a dynamical system. Conversely, as the integration formulas we have encountered also always correspond to a sort of Hamiltonian reduction of the original dynamical model, they yield realizations of this reduction conjecture
as
-well
as
the quantum mechanical
conjecture mentioned earlier about description of the
the exactness of the semi-classical approximation for the quantum dynamics of generic integrable systems.
5.8 Let
Quantization of Isospin Systems us
now
quickly describe the quantum generalizations of the results of First, we note that, for a general group G, a point particle
the last Section. in
its
irreducible representation R of the internal symmetry group G has time-dependent isospin vectors R(t) living in a fixed orbit of G in the
an
adjoint representation [121. As described earlier, this fixed coadjoint orbit
Quantizat on
5.8
determines
a
of
Isospin Systems
181
unitary irreducible representation of G and if we let A' denote a fixed fiducial point on this orbit, then the internal isospin
(time-independent) vectors
be written
can
as
R(t) with
g(t)
Ad*(A')g(t)
=
=
g(t)A'g(t)-'
(5.221)
time-dependent group elements. In the absence of any "exter0. charged particle, the isospin vector is conserved, 7 (t) Thus the point particle action describing its motion in a generalized external magnetic field B B'R(X') is [12] E G
nal" motion of the
=
=
T
S[g, A]
=
I
dt
( tr(A'g-' )
+
tr(IZ B))
(5.222)
-
0
When A' is taken to be
Cartan
element, this dynamics of
generalizes the spin particle moving in both monopole and external fields (Hamiltonianly reduced from a free particle action on R 4) [12, 44]. Thus again we see that the kinetic term in (5.222) is of first order in time derivatives and consequently the action (5.222) is already cast in phase space. The phase space variables are the isospin vectors R(XI) and the Poisson algebra of them is just the Lie algebra of the internal symmetry group G [12]. Thus these generalized dynamical systems all fall into the class of those systems whose configuration and phase spaces coincide so that their path integral interpretations on the respective spaces are the a
action before which described the
action a
same.
To describe the quantum dynamics of these models, we start with the quantization of the classical SU(N + 1) isospin model of the last Section over the phase space (CpN. Since [143]
SU(N
+
1)
S2N+1
2,
x
it follows that this
phase
1)1(SU(N)
of
will be
a
x
U(1))
SU(N)
_
space is the
SU(N + 1) [68],
space is not
a
flag
_
(Sl)N
manifold
as
C
Aj(f)
and the
S2N-1
is
sum
10)
canonically
a
highest weight
over
is
vector
=
x
U(1) (so
associated with and
so
OJN,1J
that
10)
is
S3
X
...
(5.223)
coadjoint orbit OJN,1J SU(N an integrable Hamiltonian on =
SU(N
1)
+
+
it
which leave
Although the orbit possible to generalize the Section 5.4. They are constructed by invariant.
it is still
Ee neAe(f)
where nj is
vector of the fundamental
t is taken in such
SU(N)
I)ISU(N + 1)AI)
10)
X
and
OfN,1J
before,
construction of the coherent states in
taking the highest weight
group of
X
combination of the N Cartan elements of
the maximal torus HC
integer,
S2N+1
a
non-negative
a
representation,
way that the maximum stabalizer
that the
loop space path integral will be Ad* (SU(N + 1))A'-- SU(N + OA1 appropriate to the geometry of the given
=
-
coadjoint orbit (e.g. the normalization of the coherent Kdhler potentials of the orbits in the manner described
states
generates the
in Section
5.4).
The
182
5.
Equivariant Localization
existence of such
weight
a
state is in
Simply Connected
Phase
Spaces
general always guaranteed by the Borel-
[68, 142].
Weil-Bott theorem We
on
define the coherent states associated with the irreducible [2J] representation of the SU(N + 1) group, where 2J E Z+, 2J < N + 1 [53, 871
(i.e.
now
the representation with fundamental
highest weight J corresponding to Young tableau representation by a single column of 2J boxes). In this representation, the highest weight vector is denoted as I J; N + 1), there are N (i.e. roots which cannot be decomprecisely N simple roots a 1, 2, a
=
E,(,
N +1) of 2 other positive roots) with 1 J; N + 1) 0, and the Cartan generators in the Gell-Mann basis have the (diagonal) matrix elements
posed
[H,,,]ij
into the
sum
1 =
V2-m(m+ 1) -
(km=l1:
6ikJjk
-
=
MJi,m+lJj,m+l
M
N
=
!jab The 2
(5.224) where
have normalized the generators dimension of this representation is we
dim
The
1)
[2J]
=
( N+1)
so
that
tr(XaXb)
(N + 1)! (2J)!(N TI.- 2J)!
2J
.
(5.225)
generalized coherent states associated with this representation of SU(N+
are
1)
then
e
[137]
EN
-1
-E
(N+1)
IJ;N+l) 2J)! nN+11
nl+...+nN+1=2J
N)
=
(61)nl
...
C
CpN
(6N)nN JJnk 1N+1 k=1 ;N+1) (5.226)
These coherent states have the normalizations
(1 and
they obey
the
IV
+
2J
E V 2'5
(5.227)
a=1
completeness relation
dy(J)
1(j)
(5.228)
where
dl,P)
rIN I &P (2J + N)! (2J)! 7rN (I + EN
is the associated coherent state
C'=
measure.
Consider the quantum propagator
A
d6c'
(5.229)
Quantization
5.8
IC(J) ( 2, 61; T)
(( 21 e-M 161))
=
SU(N + 1) operator (5.224)
defined in terms of the invariant nation of the Cartan
generators
Isospin Systems
of
183
(5.230) that is
a
linear combi-
N
7
=
mHm
2J
(5.231)
2JCO
+
M=1
which leads
usual to
as
IC (J) (62 7
topological and integrable quantum theory. Then yields the coherent state path integral
a
the standard calculation
61; T) N
[dl-P) (6,
2J log
exp
1 +
L(S2N+IISI) T
N
+2J log
E V2 2ei a=1
1 +
E 61c'M
+ i
a=1
I
N
2U(4 6 11 1 + El'
1:
dt
-
CX=1
0
1"-) 61 6a -
qA
H(6, e)
(5.232) where the
boundary
path integral
conditions in the
(O)
are
and
(T)
and
2JCo
+ 2J
E
(5.233)
generalized height function (5.213) on (CpN. The WKB approximation path integral has been discussed extensively in [52, 131]. In fact, all of the localization formulas of Chapter 4 can be explicitly verified as the above dynamical system is just a multi-dimensional generalization of the Kdhler polarization for the spin propagator in Section 5.5 above. The classical equations of motion are is the
for the coherent state
+
2U ' '
whose solutions in the
periodic
=
2UP '
0
stereographic coordinates (5.214)
=
are
0
the
(5.234) conditionally
motions
0'(t)
=
0'(0)
0'(t)
generalizing the results of Section In any case,
1CM
we
=
0'(0)
+
2Jdt
(5.235)
5.5.
arrive at the propagator
+
T)
=
EN
1 1 '=
'Q
1
-2iJi"T
)2J
-
+
rk E,, 11 14) (1 + E,, 2 t2 ) J J
e
2iJCoT
(5.236)
184
Equivariant Localization
5.
on
Simply Connected
Phase
Spaces
and the associated quantum partition function
f dy(j)(C,t) (( j e_i? TI6))
ZSU(N+1)(T)
N+1
E
e-2iJO'T
a=1
11 '301a
(5.237)
1
I
-
e-2iJ(d#-d1)T
N+l =- Co. These generalize the previous results for SU(2) and, in particular, (5.237) coincides with the anticipated Weyl character formula for the [2J] representation of G SU(N + 1). What is also interesting here is where
=
that the Hamiltonian reduction mechanism of the last Section has
counterpart which
a
quantum
be used to interpret the quantum localization of this dynamical system. To see this, we recall from the last Section that the Hamiltonian (5.233) is the reduction of an (N + I)-dimensional simple harmonic oscillator
can
Hamiltonian,
which in the quantum
case
is the Hermitian operator
N+1
, j (D)
=
2J
E O'at"a"
(5.238)
a=1
Heisenberg-Weyl algebra E)N+1 gHW, where mutually commuting copies of the raising and lowering operators (5.79). Relating the coordinates on CN+1 and (CpN as usual via the constraint function (5.206), it is straightforward to show that the projection acting etta, bic,
on a
multi-dimensional
N + 1
are
operator in
(5.228)
is
[52]
27r
d'
pj
In,.... nN+1)(nj,...,nN+1I
27r nl+...+nN+1=2J
0
(5.239) where
Ini.... nN+l) ,
_(at)ni.... (,itN +J)nN+110) 1 -.nN+I! Vn-,!
-
-
are
the states of the orthonormal number basis for
the
weight
states of the
In,,..., nN+l) of
(5.240)
-
This identifies N+1 a=1 9HW.
SU(N+ 1) representation above as If nk jN+1; N+1) k=1 (5.240). This method
in terms of the bosonic Fock space states
constructing coherent
states is known
as
the
Schwinger
boson formalism
[49]-[52]. The trace formula
(5.237)
can now
be
represented
in terms of the canonical
coherent states N+'1
I z)) as
-=
e
=1
0) /
e'! 2
Z
=
(Zl'... zN+l) ,
E
(CN+l (5.241)
Quantization
5.8
ZSU(N+1)(T) Then
N+1
fl,,,=,
f
=
Isospin Systems
185
-
d.V dz"
-, j (D) T e-' I Z))
-
((zlPj
7rN+l
(5-242)
using the important property
I)q (D), f J] it is
of
straightforward
to show that
(5-243)
=0
(5.242)
becomes
[52]
ZSU(N+l) (T) 21r
dA
f Tir
e
d,01 (t) dzll (t)
-2iJ,
T x
rN+l
LR2N+2 tE[O,T]
0
exp
-
f
N+1
dt
we
states
(5.241).
e-
2iJTj'+i,\
(2aja
Z Za-La)
-
Z
a
a=1
0
where
(5.244)
have also used the resolution of The Gaussian functional
to carry out and
we
unity for the canonical coherent integration in (5.244) is now trivial
find 21r
ZSU(N+I)(T)
N+1
I
dA 2-7r
I
J1
e-2iJA
-
-2iJ61T+iA e
0
J
d aw 21r
W2J+N
we
integral
have transformed the
up N + I
simple poles each
w
-
e-2iJj11T
angular integration
S1. Carrying
the unit circle
over
J1 a=1
S1
where
(5.245)
N+1
in
(5-245)
out the contour
of residue I and leads to
into
a
contour
integration picks
Weyl character formula
(5.237). Thus the classical Hamiltonian reduction mechanism discussed in the last Section also
summed
implies localization
over
plectic reduction of CN+1 straint
at the
quantum level, and again the terms
in the WKB formula represent the
algebra (5.243)
coincides with the
to
CpN.
is determined
identity operator
singular points
In the quantum
case
of the sym-
the first class
by the projection operator
on
161
con-
which
the group representation space when as above. This Hamiltonian reduc-
restricted to the SU (N + 1) coherent states
always just a manifestation of the fact that these localizable dynamical are always in some way just a set of harmonic oscillators, because of the large "hidden" supersymmetry in these problems. The above analysis can also be straightforwardly generalized to the non-compact hyperbolic NJ space D (the complex open (N + l)-dimensional Poincar6 ball [68]) with the associated SU(N, 1) coherent states, generalizing the results of Section tion is
systems
186
5.
5.6 for
Weyl
Equivariant Localization
SU(1, 1) [52].
In
on
Simply Connected
Spaces
particular, the localization formulas all lead
to the
SU(N, 1)
character formula for
N
ZSU(N,I)(T)
=
1
]1
e-iCoKT
I
Oe=1
The
Phase
CpN model above
(5.246)
e-ij"T
-
special case of the more general Kdhler space Geometrically, this is defined as the k (N k) complex-dimensional space Gr(N, k) of k-planes through the origin of (CN (note that Gr(N, 1) CpN-1). Algebraically, it is the space of N x N Hermitian matrices obeying a quadratic constraint, called
a
is
a
Grassmann manifold.
-
-
=
Gr(N, k) from which it
lp : pt
=
be shown to be
can
=
pj p2
p,
=
isomorphic
tr P
to the
=
kj
(5.247)
U(N) coadjoint
orbit
[68, 143] Gr(N, k)
-
U(N)I(U(k)
x
U(N
that the
eigenvalues
k))
UPUt
U(N) (coadjoint) action P quadratic (5.247) implies
with the transitive
-
--->
(5.248) on
Gr(N, k).
Note
that the operators P have therefore be interpreted as fermionic occupation
constraint in
0
or
1.
They
can
number operators and the trace condition in (5.247) can be interpreted as the total number of fermions in a given state. The Grassmann manifolds are
intimately related to an underlying free fermion theory [143]. The symplectic structure on (5.247) is the associated Kirillov-Kostant 2-form
therefore
W
(N,k)
=
i
tr(PdP A dP)
(5.249)
explicit form can be written using the local coordinatization provided diffeomorphism (5.223) and U(N) -- SU(N) x S' [49]. Similarly, the non-compact hyperbolic analog of the Grassmann manifold is known as the Siegel disc D N,k [68]. It is defined algebraically as whose
by
the
D N,k
=
IfD:
j5t
=
16) -Pt?7(N,k)P
=
77(N,k)}
(5.250)
where ??(Nk) is the flat Minkowski metric with diagonal elements consisting k entries of -1 and k entries of +1. It is isomorphic to the coadjoint of N -
orbit D N,k
--
of the non-compact Lie group
.,(N,k) ':Z
=
The Grassmann and
tr
U (N,
k) / (U (N)
U(N, k),
x
U (k))
(5.251)
and its Kdhler structure is defined
(-Pn(N,k)df3 ?7(Nk)dA7(N,k)) A
by
(5.252)
Siegel spaces above are the representative spaces homogeneous manifolds [68]. The coherent state quantization and semiclassical exactness of these dynamical systems has been discussed extensively for
Quantization of Isospin Systems
5.8
in
[49, 143].
constant N
The x
integrable Hamiltonians
N Hermitian matrix X and
Hx(P) so
that the partition function algebra is
=
(5.247)
on are
parametrized by
are
defined
187
a
by
(5.253)
tr(XP)
generalizes the Itzykson-Zuber integral (5.185).
Their Poisson
f Hx, Hy J,,,(N, k) so
(5.254)
H[x,y]
=
that the associated partition functions define
topological
theories. The
Duistermaat-Heckman formula Gr(N,k)
Z.'I
2,7ri
(T)
k(N-k),
e
iT
Elk..
k
T
1
hil
(hj
-
hi,)
(5.255) eigenvalues of the Hermitian matrix X in (5.253) parametrizing the Hamiltonian, has been verified for this dynamical system and shown
with
hi
E
R the
to be associated with
a
Hamiltonian reduction mechanism
as
described above
of (CpN. The construction of coherent states via the
algebraic Schwinger boson formalisms parallels that above for the (CpN coherent states, although in the present case it is somewhat more involved. We shall not go into the technical details here, but refer to [49] where it was also shown for the
case
or
that the WKB localization formula e
Gr(N k)
Z (N)' (T)
-ikT
Ek
nk1=111joijl-
1
hi,
I= .1
iT(hj-hjI))
e-
(5.256)
e-iThinj
detij I A[ e-iThi]
yields the anticipated Weyl character formula in the representation defined by the coset space (5.248). Here k C: Z+ characterizes the highest weight of the pertinent U(N) representation, the non-negative integers ni > ni+1 are N i + bi where bi is the components of the vector A + p (defined as ni the number of boxes in the i-th row of the associated Young tableau representation), and here we have taken the basis of the Cartan subalgebra in =
which
(Hi)jk Jij6jk. Similarly, (5.250) are H, ,(P) =
bolic space real-valued
diagonal
the
integrable Hamiltonians
tr(77(N,k)-'C??(N,k)-P)
spaces
Ofn1,n2-.'n,eJ of dimension
on
with
the a
hyper-
constant
matrix.
These constructions of coherent states
joint orbit
-
can
also be
generalized
to the coad-
[66, 86, 130] =_
SU(N)I(SU(ni)
N2
SU(N) subgroup SU(ni)
n2,
where
x
x
...
x
...
SU(nt)
x
ni x
SU(nt) =
x
U(1)t-')
(5.257)
N and the dimension of the
U(1)i-1
is N
-
1. The
complex
188
Equivariant Localization
5.
Simply Connected Phase Spaces
on
on the orbit (5.257) is defined by the isomorphism Ojni,n2,-..'ntj SL(N,P/Pfn1,n2 ...... I Iwhere SL(N, C) is the complexification of SU(N) and Pfni,n2-.'n_,} is a parabolic subgroup of SL(N, C) which is the subgroup
structure
Ilt
of block upper
triangular matrices in the (n, + + nt) x (n, + nj) block decomposition of elements of SL(N, C). Note that for nj 1 nN the coadjoint orbit (5.257) is the SU(N) flag manifold (the maximal orbit) with Pf N 1 and n2 1 j,...jI the Borel subgroup BN, while for nj non-zero only, (5.257) coincides with the (CpN manifold discussed above (the minimal orbit). The isospin degrees of freedom on the coadjoint orbit (5.257) -
-
.
=
=
defined
are
as
[49, 130] (c.f. Q
where X is
a
=
diagonal
=
gXg-1
g E
matrix with entries xi E R
=
X2
=
-
SU(N)
satisfying
Xni > Xni+l Xni+2 Xn2 > The complexification of any element g E SU(N) N column vectors Zi E (CN, i N, with X1
=
(5.221))
eq.
Ad*(X)g
=
=
.
.
.
EN j=1 xi
=
0 and
Xnt-1 Xnt, be parametrized
=
can
(5.258)
=
by
=
Zil Zj The canonical I-form
=
on
det[Zl,..., ZNI
6ij
=
(5.259)
1
the orbit is then N-I
0Inij where Ji
=
x, +...
constraint in
tr
(Xg-ldg)
=
JiZtdZ.. i
i
(5.260)
+XN-1 ! 0 and we have used the determinant The associated Kdhler 2-form is
+2xi
(5.259).
N-1
.1nij
and the
=
dO {nil
isospin element (5.258)
can
N-1
Q
i
=
forward to
=
ji -
N
tr(QX')
A
dZj
(5.261)
as
1)
(5.262)
with the normalization of
above for the (CpN case, it is straightthat these functions generate the SU(N) Lie algebra in the
generators defined see
expressed
(Jizizil
Defining the isospin generators Q`
SU(N)
be
i
E i=1
the
JjdZj
-i
as
Poisson bracket generated by the set of orthonormal constraints in lations. These functions
symplectic
(5.259)
are
structure
(5.261)
once
the first
substituted into the above
re-
generalize the angular momentum generators in the SU(2) case (where the only coadjoint orbit is the maximal one). The constraint functions in (5.259) are easily seen to be first class constraint functions [49, 130] so that the coadjoint orbit is determined again as the symplectic
Quantization of Isospin Systems
5.8
reduction of
a
larger
SU(ni)
x
SU(ne) x U(1) -'
x
...
space. These constraint functions
of
189
generate the subgroup
SU(N).
The localization properties of the the orbit functions (5-262) can there-
topological Hamiltonians generated by fore be interpreted again in terms of generalized dynamical systems, the characteristic feature of
sorts of harmonic oscillator
the
equivariant localization
mechanism. The coherent states in these generic cases are defined analogously to those over CpN above, with the restrictions discussed at the beginning of this Section. The structure of the
integrable models defined
on
these coset spaces has
[66, 86]. The Poisson-Lie relations of the SU(N) symmetry of each orbit (5.257) lead to a maximal number N 2 N of mutually commuting functions, which is equal to half the (real) dimension of the maximal been discussed in
-
coadjoint orbit.
particular, the dynamical properties of the flag manifold theory of non-commutative integrability. The coherent state quantization of this flag manifold in the case In
SU(N) IU(I)N- 1
of
SU(3)
have been related to ideas in the
has also been worked out in detail in
that case, the Kdhler
FSU(3) (Zi 2)
:-*z
potential
109 K1
is
+ Zl, l +
[66, 86] (see
[79, 100]).
also
In
[139]
Z2, 2) P(I
+ Z3, 3 +
(Z2
-
Z1
Z3) (' 2
-
'l
3))'71 (5.263) Q3 and
where p and q are integers. Using the usual integrable Hamiltonians Q8 defined as above, it is straightforward to construct a coherent state
path integral representation for the quantum dynamics of this localizable system and verify the localization formulas of Chapter 4 above. For instance, the semi-classical approximation has been verified for the dynamical system with topological Hamiltonian function H Ei=:,: jQj, where Q Q3 , 'F3Q8 the circle actions symplectic generate (Z1 Z2) Z3) -+ ( e'01 zi, e'01 Z2 Z3) and ei02 e-i02 -(Zl) Z2) Z3) ( Z3), respectively [66, 86] (so that H generates Z1, Z2, the symplectic action of the 2-torus group T 2 S' x S' on SU(3) /U(1)2). =
=
i
7
=
The WKB localization formula for the quantum propagator in the coherent state representation for this integrable spin model can then be worked out to
be
ICSU(3) (. I, z; T)
=
(I +. jzj ei(i++j_)T X
(1
+
+
o 21Z2
eij+T)p
213 Z3 e-id- T+ (212 - 1, 31)(Z2
-
ZlZ3)
e'j+T)q (5.264)
which
can
be shown to coincide with the exact result from
Finally,
we
-point
out that
one can
also
formalisms of Sections 3.8 and 4.9 to these
a
direct calculation.
apply the nonabelian localization generalized spin models regarding
the associated partition functions as defined on the coordinate manifolds. As such, they can be applied to problems such as geodesic motion on group
manifolds, and
in
particular
the Hamiltonian framework
we can
[161].
reproduce the results of Picken [139] in precisely, we note that in these cases
More
there is the natural G-invariant metric
190
5.
Equivariant Localization g
Simply Connected Phase Spaces
on
tr(F
=
Ff)
0
(5.265)
h-ldh is the adjoint repredefined on the group manifold of G, where sentation of the Cartan-Maurer 1-form which takes values in the Lie algebra
geodesic motion on G, the Hamiltonian operator Schr6dinger polarization is given by the Laplace-Beltrami operator
g of G. For free
1
fi(O)
=
-
2
V2
(5.266)
9
with respect to the metric (5.265). The invariant be the Riemannian volume form associated with
trajectories
to show that the space of
manifold
G
over
in the
measure on
(5.265),
G is taken to
and it is
be made into
can
by absorbing the Riemannian volume form
into
an
possible a
Kdhler
effective action
[1391.
in the usual way
The generic classical trajectories generated by the Hamiltonian (5.266) are straight lines in the Weyl alcove A HcIW(Hc) of the Lie group G =
because of the
decomposition [162] 9192
1 =
vhv-
1
V91) 92
E
(5.267)
G
HC and the elements v can be parametrized by the coset space GlHc. The geodesic distance between g, and 92 defined by (5.265) is independent of v. Then the sum over extrema, in the semi-classical approximation is given by a sum over the lattice Z' in r-dimensional Euclidean root space generated by the simple roots of g. The time-evolution kernel is therefore a
where h E
function of the r-dimensional vector
formulas for it
IC (a;
T)
can
(
be shown to
2-7riT
analog
a(a
+
of the
tion formula
which is
a
21rk)
=
[96]
.
.
,
ar)
anticipated
E
A and the localization
result
the
[38, 139, 146, 161]
iE' (aj+21rkj)2/2T
(5.268)
+
Ej=, aj(aj
leads to over
.
2,7rk) '2 a (a + 21rk)
Weyl character
series
(a,,
k=(k-l,...,kr)EZ'
2 sin
ce>O
yield
=
the
G12
a(a
X
where
)
dim
a
+
27rkj). (5.268) is the configuration space (5.28). Applying the Poisson resumma-
formula
a spectral expansion of the quantum propagator unitary irreducible representations of G given by
IC(a; T)
E
dim R,\
(tr,\a) e-ic(,\)T
(5.269)
p?)
(5.270)
AEZ'
where
C(A)
+
p,)2
_
Quantization
5.9
on
Non-Homogeneous
Phase
Spaces
191
eigenvalues of the quadratic Casimir operator E,,(Xa)2 in the reprehighest weight vector A. Similar considerations also apply to n-spheres Sn SO(n + 1)/SO(n) [96, 100] and their hyperbolic counterparts Rn -_ SO(n 1, 1)/SO(n) obtained by the usual analytical continuation of Sn [31]. The case of S' we saw was associated with the Dirac monopole, that of S3 -- SU(2) describes the emergence of spin, and S4 corresponds to the BPST instanton-antiinstanton pair with 2 chiral spins [100]. Notice that these localizations also apply to the basic integrable models which are well-known to be equivalent to the group geodesic motion problems above, such as 2-dimensional Yang-Mills theory, supersymmetric quantum mechanics and Calegoro-Moser type theories. These describe the quantum mechanics of integrable models related to Hamiltonian reduction of free field theories [44, 56] and will be discussed again in Chapter 8. Note that these free theory reductions again illustrate the isomorphism the
are
sentation with
-
-
between the localizable models and
5.9
Quantization
on
(trivial)
harmonic oscillator type theories.
Non-Homogeneous Phase Spaces
Thus far in this
Chapter we have examined the localizable dynamical systems phase spaces which are maximally symmetric and those with multi-dimensional maximally-symmetric subspaces. This exhausts all spaces both those
on
with constant curvature and which
are
symmetric, and
we
outlined both the
physical
and group theoretical features of these dynamical systems. In this final Section of this Chapter we consider the final remaining possible class of Riemannian
geometries
K(x) /C(M,g)
which is
on
the
phase
space
M,
i.e. those with
a
Gaussian
non-constant function of the coordinates
on M, again to 2dimensional phase spaces. The geometries which admit only a single Killing vector are far more numerous than the maximally symmetric or homogeneous ones and it is here that one could hope to obtain more non-trivial applica-
curvature so
that dim
=
a
1. For
simplicity
we
restrict attention
tions of the localization formulas. Another nice feature of these spaces is that
the
corresponding
Hamiltonian Poisson
algebra will be abelian, so that the automatically be Cartan elements, in contrast to the previous cases where the Lie algebra IC(M, g) was non-abelian. Thus the abelian localization formulas of the last Chapter can be applied straightforwardly, and the resulting propagators will yield character formulas for the isometry group elements defined in terms of a topological field theory type path integral describing the properties of integrable quantum systems corresponding to Cartan element Hamiltonians. Given a I-parameter isometry group GM acting on (M, g), we begin by introducing a set of prefered coordinates (X11' X12 ) defined in terms of 2 Hamiltonians
so
obtained will
differentiable functions
x1
coordinates the
vector V has
the function
X1
Killing
and
X2
as
described in Section 5.2,
components V"
is any non-constant function
on
M,
=
so
1, V/2
but
we
that in these
=
0. For now,
shall
soon see
192
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
how,
once a given isometry of the dynamical system is identified, it can be fixed to suit the given problem. For a Hamiltonian system (M, W, H) which generates the flows of the given isometry in the usual way via Hamilton's
equations,
the
defining
(5.55)
condition
for the coordinate function X 2
now
reads
JH, X21,0
=
LVX2
=
(5.271)
0
which is assumed to hold away from the critical point set of H (i.e. the zeroes of V) almost everywhere on M. This means that X2 is a conserved charge of
the given dynamical system, i.e. a GM-invariant function of action variables. In higher dimensions there would be many such possibilities for the conserved
charges depending
the integrability properties of the system. However, requirement fixes the action variable to be simply a functional of the Hamiltonian H, on
in 2-dimensions this
X
and
so even
in the
2
(5.272)
non-homogeneous
cases we see
the intimate connection
here between the equivariant localization formalism and the
(classical
integrability
of
quantum) dynamical system. We note that this only fixes the requirement (5.271) that the coordinate transformation function be constant a
along
the
or
integral
Killing vector field V. The isometry consymplectic 2-form now only implies that, in the new xl-coordinates, w,,,(x') is independent of x" (just as for the metric). The Hamiltonian equations with V" 0 must be solved consistently 1, V/2 and associated now using (5.272) an symplectic structure. Notice that this construction is explicitly independent of the other coordinate transformation function x1 used in the construction of the prefered coordinates for V (c.f. dition
(5.70)
on
curves
of the
the
=
=
Section 5 .2).
Thus for
a
general
metric
(5.49)
that admits
a
sole isometry, the
general
"admissible" Hamiltonians within the framework of equivariant localization are given by the functionals in (5.272) determined by the transformation x
-+
x' to coordinates in which the
(circle
or
translation)
action of the
cor-
now arises because responding Killing explicit. the integrability condition LVw 0 for the Hamiltonian equations does not uniquely determine the symplectic 2-form w, as it did in the case of a homoge-
The rich structure
vector is
=
neous
symmetric geometry. The above
construction could therefore be started
with any given symplectic 2-form obeying this requirement, with the hope of being able to analyse quite general classes of Hamiltonian systems. This has
the
possibility
of
largely expanding the
known
examples
of quantum systems
where the Feynman path integral could be evaluated exactly, in contrast to the homogeneous cases where we saw that there was only a small number of
few-parameter Hamiltonians which fit the localization framework. However, it argued that the set of Hamiltonian systems in general for which the localization criteria apply is still rather small [40, 1591. For instance, we could 2 from the onset take w to be the Darboux 2-form on M R and hope to has been
=
Quantization
5.9
Non-Homogeneous
on
Phase
Spaces
193
obtain localizable
examples of 1-dimensional quantum mechanical problems potentials. These are defined by the Darboux Hamiltonians
with static
HQM (p, q) where
1P2 + U(q)
=
(5.273)
2
U(q)
is some potential which is a C'-function of the position q E Dykstra, Lykken and Raiten [40] who first pointed out that the formalism in Chapter 4 above, which naively seems like it would imply the exact solvability of any phase space path integral, does not work for arbitrary potentials U(q). To see this, we consider a generic potential U(q) which is bounded from below. By adding an irrelevant constant to the Hamiltonian (5.273) if necessary, we can assume that U(q) > 0 without loss of generality. We introduce a "harmonic" coordinate y E R and polar coordinates (r, 0) E R+ x S, by
R1.
It
was
p
where
we
further
=
r
in
=
Ir 2, 2
(5.274)
these
polar
so
and
that
assume
(5.273)
nates the Hamiltonian
form H
U(q)
rsinO
U(q)
is
IY2 2
1 =
2
r2 Cos 2o
(5.274)
monotone function. In these coordi-
takes the usual
integrable harmonic oscillator
X2 above defines the radial coordinate (5-272). The Hamiltonian vector field in
that the function
F(H)
vr2_H_
=
coordinates has the
in
single non-vanishing component
Vo The metric tensor
a
=
(5.49)
will have in
=
_dy
general have
gor under the coordinate transformation (2.112) become
Voo9ogoo + 2gooo9oVo
=
0
V00909rr
7
+
(5.275)
dq
(5.274),
190(groVO)
2goo9rVo
=
3 components grr, goo and
and the
+
Killing equations
900-9rVo
=
0
(5.276)
0
The 3 equations in (5.276) can be solved in succession and the general solution has the form
by integrating
them
0
goo
f (r)
gro
f (r) VO
I 00
dO' ar
( Vo,I-)+
h
Vo
grr
f (r)
+k(r)
(5.277) f (r), h(r) and k(r) are arbitrary C'-functions that are independent of the angular coordinate 0. Note that, as expected, there is no unique solution for the conformal factor W in (5.49), only the requirement that it be radially symmetric (i.e. independent of 0). However, the equations (5.277) impose a much stronger
where
194
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
requirement, this time on the actual coordinate transformation (5.274). If impose the required single-valuedness property on the metric components above, then the requirement that g,O (r, 0) g,O (r, 0 + 27r) is equivalent to
we
=
the condition 27r
a
dO
f V-0
09r
0
(5.278)
constant
(5.279)
=
0 or
equivalently
that 27r
f
dO
_q
=
dy
0
However,
only
the
solution to
(5.279)
is when the function
dy
is
independent
of the radial coordinate r, which from (5.274) is possible only when y so that U(q) -!q 2 and HQm is the harmonic oscillator Hamiltonian. =
=
-q,
Thus,
with the exception of the harmonic oscillator, equivariant localization fails for all 1-dimensional quantum mechanical Hamiltonians with static potentials which
are
bounded
satisfying the
below, due
to the non-existence of
Lie derivative constraint in this
a
single-valued
metric
case.
It is instructive to examine the localization formulas for the harmonic
oscillator, theory, to
which is considered trivial from the point of view of localization what role is played by the degree of freedom remaining in the
see
metric tensor which is not determined
generates
a
global SI-action on M
R
=
by
the
(5.275)
straints. The Hamiltonian vector field
2
equivariant localization con1 which case is Vo
in this
=
given by translations of the angle
co-
ordinate 0. Thus the localization formulas should be exact for the harmonic
radially symmetric geometry (5.49) to make manifest principle. This is certainly true of the WKB formula (4.115) which does not involve the metric tensor at all, but the more general localization formulas, such as the Niemi-Tirkkonen formula (4.130), are explicitly metric dependent through, e.g. theA-genus terms, although not manifestly so. Explicitly, the non-vanishing components of the metric tensor (5.49) under oscillator using any
the localization
the coordinate transformation grr
and it is
ew(-)
in the
goo
I
case
=
r
2
at hand
are
ew(r)
straightforward to work out the Riemann moment map and VO I lead to the non-vanishing components
tensor which with
where
=
(5.274)
we
(5.280) curvature
=
(00or
=
Roror
=
r
-(f2V)ro 1 -
2
S?V
=
2
d Or
dr
have introduced the function
ew(r) log
A (r)
2 +
r
dr
(5.281)
5.9
Quantization
A(r)
=
Non-Homogeneous
on
Phase
Spaces
e- '(')(Qv)o,/2r
195
(5.282)
Substituting the above quantities into the Niemi-Tirkkonen formula (4.130) r and working out the Grassmann and 0 integrals there, after some algebra we find the following expression for the harmonic oscillator partition function,
with w,O
=
00
Zharm(T)
-
-1 t
I
dr
d
sinTA(r)
dr
e-iTr 2/2
1 i
lim r,o
A(r) sin TA (r)
(5.283)
0
Comparing with (5.75),
we see
for the harmonic oscillator
behaves at the origin
r
=
0
that this result coincides with the exact result
partition function only if the function (5.282) as
lim r-O
which
isfy,
using (5.281) and (5.282)
in addition to the radial
(5.284)
2
that the
phase space metric must satthe additional constraint constraint, symmetry means
lim
d r
dr
r-O
The
A(r)
W(T)
=
(5.285)
0
requirement (5.285) means that the conformal factor W(r) of the Riegeometry must be an analytic function of r about r 0, and this
mannian
restriction
=
on
the
general
form of the metric
(5.49) (i.e.
on
the functional
properties of the conformal factor W) ensures that the partition function is independent of this phase space metric, as it should be. This analyticity requirement, however, simply means that the metric should be chosen so as to eliminate the singularity at the origin of the coordinate transformation to polar coordinates (r, 0) on the plane. That this transformation is singular at p 0 is easily seen by computing the Jacoq bian for the change of variables (5.274) with the harmonic oscillator potential (or by noting that w and g are degenerate at r 0 in these coordinates). Since the equivariant Atiyah-Singer index which appears as the Niemi-Tirkkonen formula for the quantum mechanical path integral is an integral over charac=
=
=
teristic
classes,
it is
manifestly invariant
under CI deformations of the metric
M. The transformation to polar coordinates is a diffeomorphism only on 2 the punctured plane R 101, which destroys the manifest topological invarion
_
of the partition function (at r 0 anyway). For A(O) 4 0, the metric a conical geometry [102] for which the parameter A(O) represents the tip angle of the cone. This example shows that for the localization ance
=
tensor describes
to work the choice of metric tensor is not
completely arbitrary,
since it has to
respect the topology of the phase space on which the problem is defined. As discussed in [40] and [159], this appears to be a general feature of the generalized localization
formalisms,
in that
they detect explicitly the topology
of the
196
Equivariant Localization
5.
on
Simply Connected Phase Spaces
phase
space and this can be used to eliminate some of the arbitrariness of the metric (5.49). Indeed, in the set of prefered coordinates for V it has no zeroes
and
so the critical points are "absorbed" into the symplectic 2-form W and general also the metric g. Thus the prefered coordinate transformation for V is a diffeomorphism only on M Mv in general. Nonetheless, this simple example illustrates that quite general, non-homogeneous geometries can still be used to carry out the equivariant localization framework for path integrals and describe the equivariant Hamiltonian systems which lead to topological quantum theories in terms of the generic phase space geometry. Although the above arguments appear to have eliminated a large number of interesting physical problems, owing to the fact that their Hamiltonian vector fields do not generate well-defined orbits on the O-circle, it is still possible that quantum mechanical Hamiltonians with unbounded static potentials could fit the localization framework. Such dynamical systems indeed do represent a rather large class of physically interesting quantum systems. The first such attempt was carried out by Dykstra, Lykken and Raiten [40]
in
-
who showed that the Niemi-Tirkkonen localization formula for such models be reduced to
can
a
relatively simple
contour
the equivariant localization formalism atom Hamiltonian [95]
Hh(p, q) The
=
applied
_1P2 2
integral.
For
example, consider hydrogen
to the 1-dimensional
1
(5.286)
_
JqJ
eigenvalues of the associated quantum Hamiltonian energies
form
a
discrete spec-
trum with
E,,
=
-1/2n
2
n
=
1,2....
(5.287)
which resembles the bound state spectrum of the more familiar 3-dimensional hydrogen atom [1011. What is even more interesting about this dynamical
system is that the classical bound state orbits all coalesce at the phase space 2 oo on R so that a localization onto classical trajectories points q 0, p =
(like
=
the WKB
,
formula)
highly unsuitable for this quantum mechanical problem could therefore provide an example wherein although the standard WKB approximation cannot be employed, the more general localization formulas, like the Niemi-Tirkkonen formula, which seem to have no constraints on them other than the usual isometry restrictions on the phase space M, could prove of use in describing the exact quantum theory of the dynamical system. The key to evaluating the localization formulas for the Darboux Hamiltonian (5.286) is the transformation to the hyperbolic coordinates (r, -r) with problem.
-oo
< r,
This
-r
< oo,
p so
is
=
Irl sinh T
that the Hamiltonian is again Hh
field has the
q =
=
2/rJrJ cosh'T
_1r2 and the Hamiltonian 2
single non-vanishing component
(5.288) vector
5.9
Quantization
on
Non-Homogeneous
1r3 cosh3
V
Phase
Spaces
(5.289)
-r
4
197
Killing equations have precisely the same form as in (5.276), with (r, 0) replaced by (r,,r) there, and thus the general solutions for the metric tensor have Precisely the same form as in (5.277). However, because of the non-compact range of the hyperbolic coordinate r in the case at hand, we do not encounter a single-valuedness problem in defining the components grr 2 and from (5.277) and (5.289) we find that it is given as C' functions on R the explicitly by perfectly well-defined function Now the
gr'r
_
12f (r)
=
sinh -r
V'r
In the context of
cosh
our
2
coordinates
are
X1
1arctan(sinh-r)
=
X"'
2
so
that.F(H)
is determined
the x'-coordinates in
(5.43)
=
we
h(r)
+
2
isometry analysis above,
nate transformation function X
coordinate function
+ _r
(5.290)
Vr
again choose the coordi-
,/----2H in (5.272).
by noting
from which
The other
that the above we
(r, -r)
wish to define the
prefered set of x"-coordinates for the Hamiltonian vector field V. There we identify (X11,X12) (-r, r) according to that prescription. Carrying out the x" r using (5.289), and then substituting in the over explicit integration transformation (5.288) back to the original Darboux coordinates, after some algebra we find =
=
X'(p,q)
2 _
jqj
P
-3/2
2
plql
1/2
2 _
jqj
P
2
P
+ 2 arctan 2
-
Jqj
P
2
11/2 (5.291)
Thus the Hamiltonian sor
(5.49)
(5.286)
is associated with the
phase
which is invariant under the translations
(5.291).
above shows
X1
space metric ten-
--+
x1
+ ao of the
explicitly that the phase
analysis space globally well-defined metric which is translation invariant in the variable (5.291). It is also possible to evaluate the Niemi-Tirkkonen localization formula for this quantum problem in a similar fashion as the harmonic oscillator example above. We shall not go into this computation here, but refer to [40] for the technical details. The only other point we wish to make here is that one needs to impose again certain regularity requirements coordinate
indeed does admit
on
The
a
(5.49).
the conformal factor of the metric
These conditions
are
far
more
complicated than above because of the more complicated form of the translation function (5.291), but they are again associated with the cancelling of the coordinate singularities in (5.288) which make the equivariant Atiyah-Singer index in (4.130) an explicitly metric dependent quantity. With these appropriate geometric restrictions it is enough to argue that the quantum partition function for the Darboux Hamiltonian
(5.286)
has the form
[40]
00
Zh (T)
-
E n=1
e
iT/2n2
(5.292)
198
5.
Equivariant Localization
which from
(5.287)
we see
1-dimensional
Simply Connected Phase Spaces
on
is indeed the exact
spectral propagator for the
hydrogen [95]. example shows that more complicated quantum systems can be studied within the equivariant localization framework on a simply connected phase space, but only for those phase spaces which admit Riemannian geometries which have complicated and unusual symmetries, such as translations in the coordinate (5.291) above. Thus besides having to find a metric tensor appropriate to the geometry and topology of a phase space, there is the further general problem as to whether or not a geometry can in fact possess the required symmetry (e.g. for Hamiltonians associated with bounded potentials, there is no such geometry). It is not expected, of course, that any Hamiltonian will have an exactly solvable path integral, and from the point of view of this Chapter the cases where the Feynman path integral fails to be effectively computable within the framework of equivariant localization will be those cases where a required symmetry of the phase space geometry does not lead to a globally well-defined metric tensor appropriate to the given topology. Nonetheless, the analysis in [40] for the 1-dimensional hydrogen atom is a highly non-trivial success of the equivariant localization formulas for path integrals which goes beyond the range of the standard WKB method. We conclude this Chapter by showing that it is possible to relate the path integrals for generic dynamical systems on non-homogenous phase spaces which fall into the framework of loop space equivariant localization to character formulas for the associated I-parameter isometry groups GM [159]. For this, we need to introduce a formalism for constructing coherent states assoatom
This
ciated with non-transitive group actions on manifolds [89, 159]. We consider (5.49) in the prefered x-coordinates for a Hamilto-
the isothermal metric nian vector field V
Using these coordinates, we define the complex analogy with the case where V defines a rotationally symmetric geometry (as for the harmonic oscillator) Let f (z, ) be a G(l)-invariant analytic solution of the ordinary differential equation coordinates
z
=
x"
on
M.
e x'l,
in
-
d
d
T(TZ) For the
ZZ_
d(z. )
log f (Z )
symplectic 2-form of the phase
volume form associated with
WM
0'(')
2
space,
we
take the
d
d
-
dz A
Z
whose associated
symplectic potential .
Om This definition turns the fold with Kiihler
=
(5.293) GM-invariant
(M, g),
log f (z. ) % -( zi) Z (Zi)) .
=
1
-1 2
phase potential
d
_F)
(5.294)
is
log f (z. ) (. dz
space into
a
-
zd. )
non-homogeneous
(5.295) Kiffiler mani-
5.9
Quantization
F( ') (z, 2) w(w) determines
such that
bundle L(w) Let
Np,
Taylor
the
Non-Homogeneous Phase Spaces
on
log f (z2)
=
199
(5.296) symplectic
the first Chern class of the usual
line
M.
--+
Nw
0 <
be the integer such that the function
< oo,
f (z2) admits
expansion
series
N,
E (Z2)'f"'
f (Z2)
(5.297)
n=O
and let
p(z2)
be
a
G(l)-invariant integrable function
whose moments
are
P
1 d(z2) (Z2)np(Z2)
0 <
T
n
<
(5.298)
Np
0
where P is
real number with 0 < P <
a
ation and annihilation
operators
Let &J and eb be bosonic
oo.
representation
on some
cre-
space of the isom-
etry group (as in Section 5.3 above), and let In), n E Z+, be the complete system of orthonormal eigenstates of the corresponding number operator,
ebt&.1n)
=
n1n).
The desired coherent states
are
then defined
as
N, -
I z)
=
E Vlf-n
z' I n)
(5.299)
F(11) (z,2)
(5.300)
n=O
The states
(5.299)
have the normalization
(zlz) and
they obey
invariant
a
=
completeness
f (z, )
=
relation
e
analogous
to
(5.142)
z; )
dz A d,
in the
isometry
measure
dp(w) (z 2) ,
wheree(x)
t =
21r
f (z. )p(z2)e(P
-
(5.301)
denotes the step function for X E R. The completeness of the (5.299) follows from a calculation analogous to that in (5.85)
coherent states
using the definitions (5.297)-(5.299) above.
f (z, )
Notice that for the functional values
(I
_
Z )-2k'
spin-j SU(2) Moreover, in that
case we
e' with P
p(z, ) and p(z. ) (5.293), as =
=
e', (1
=
+
Z. )2j
and
(5.299) reduces to, respectively, the Heisenberg-Weyl group, and level-k SU(1, 1) coherent states that we described earlier.
(2k
-
=
1)(1
consistently find, respectively, oo,
_
p(z. )
=
Z, )2(k-l)
(2j
Kiffiler
1)(1
geometries.
+
the
weight
Z, )-2(j+l)
functions
with P
=
oo,
anticipated from to the standard correspond (5.49)
with P
then the isothermal metrics in
maximally symmetric
+
=
1. This is
Here the
isometry
group acts
on
200
5.
the states ensures a
Equivariant Localization
(5.299)
that
a
as
h() Jz)
=
Simply Connected Phase Spaces
on
I e"z), h() E I(M, g) =- GM, -r E R', which (as we shall see explicitly below) such that
Hamiltonian exists
time-evolved coherent state remains coherent in this sense, regardless of (holomorphic) dependence of the non-normalized
the choice of p [891. The coherent state vectors Jz)
on
only the single complex variable z is, study of the isometry situation
what makes them amenable to the
(5.49)
Notice also that the metric tensor
at hand.
(5.295) can as (5.92) and (5.91),
and canonical 1-form
in the standard coherent state forms
usual be
usual,
as
represented respectively. Considering as usual the coherent state matrix elements (5.95) with respect to (5.299), using (5.295) and (5.301) we can construct the usual coherent state path integral
LM
tE[O,T) T
X
i
exp
1 [1 d(z, ) log f (z, ) (zZd
dt
-
2
fi)
-
F(H)
0
(5.302) where
we
have
again allowed
generator H. The observable
(5.294),
for
a
possible functional F(H)
H(z,, )
in
of the
(5.302)
written back in the x'-coordinates
can be found by substituting using the standard radial form 0 into the Hamiltoao, V/2
X12 e'x'l given in (5.280), and V" equations. Thus the equivariant localization constraints determine H in terms of the phase space metric as for
Z
=
=
=
nian
H( ') (z, )
=
ao
-
z d
d(z. )
log f (z )
+
isometry
Co
=
ao
-
((zl,06,1z))
+
in these
Co
=
cases
ivO') (5.303)
where the function that
(5.303)
f (z. )
is related to the metric
(5.49) by (5.293).
reduces t6the usual harmonic oscillator
height functions
Notice in the
5.3, 5.5 and 5.6 above. Thus (5.303) can be considered as the general localizable Hamiltonian valid for any phase space Riemannian geometry, be it maximally symmetric or otherwise (the same is true, of course, for the coherent state path integral (5.302)). This is
maximally symmetric
to be
expected,
cases
of Sections
because the localizable Hamiltonian functions in the
case
of
a
homogeneous symmetry are simply displaced harmonic oscillators, and these oscillator Hamiltonians correspond to the rotation generators of the isometry x" (this also agrees with the usual integroups, i.e. translations in arg(z) =
grability arguments). In fact, (5.303) shows explicitly that essentially just a harmonic oscillator Hamiltonian written generalized phase space geometry.
the function H is in terms of
some
The main difference in the present context between the homogeneous non-homogeneous cases lies in the path integral (5.302) itself In the
and
Quantization
5.9
former
the
case
the coherent state
Feynman
measure
in
on
Non-Homogeneous Phase Spaces
dy( O) (z,. )
measure
(5.302)
201
which must be used in
coincides with the volume form
(5.294),
mentioned earlier if the isometry group acts transitively on the Riemannian manifold (M, g) then there is a unique left-invariant measure
because
(i.e.
a
as
unique solution
Liouville
measure on
to
the
(5.70)) loop
and
so
dp(w)
=
w(w) yields
space LM. In the latter
case
the standard
dtLM : w(w),
(4.145) for the quantum partition loop space symplectic geometry. Nonetheless, by a suitable modification of the loop space supersymmetry associated with the dynamical system by noting that the coherent state measure in (5.301) is invariant under the action of the isometry group on M, it is still possible and
(5.302)
is not in the canonical form
function associated with the
appropriate versions of the standard localization formulas with the obvious replacements corresponding to this change of integration measure. Of course, we can alternatively follow the analysis of the former part of this
to derive
Section and
use
the standard Liouville
path integral
measure, but then
we
lose the formal analogies with the Duistermaat-Heckman theorem and its generalizations. It is essentially this non-uniqueness of an invariant symplectic 2-form in the case of non-transitive isometry group actions which leads to numerous
possibilities
for the localizable Hamiltonian
systems defined
on
everything homogeneous consistently makes the "natural" choice for w as the Kdhler 2-form (5.294), then indeed the only admissible Hamiltonian functions
such spaces, in marked contrast to the was
H
uniquely
are
fixed. If one
generalized
harmonic oscillators.
cases
where
Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations 6.
Chapter we deduced the general features of the localization formalsimply-connected symplectic manifold. We found general forms for the Hamiltonian functions in terms of the underlying phase space Riemannian geometry which is required for their Feynman path integrals to manifestly localize. This feature is quite interesting from the point of view that, as the quantum theory is always ab initio metric-independent, this analysis probes the role that the geometry and topology plays towards the understanding In the last ism
on a
of quantum integrability. For instance, we saw that the classical trajectories of a harmonic oscillator must be embedded into a rotationally-invariant geometry and that as such its orbits were always circular trajectories. For
complicated systems these quantum geometries are less familiar and endow the phase space with unusual Riemannian structures (i.e. complicated more
forms of the localization
supersymmetries).
In any case, all the localizable (e.g. the height function
harmonic oscillators
were essentially spherical phase space geometry) in some form or another, and their quantum partition functions could be represented naturally using coherent
Hamiltonians for
a
state formalisms associated with the Poisson-Lie group actions of the isom-
etry groups of the phase space. In the non-homogeneous cases we saw, in particular, that to investigate equivariant localization in general one needs to determine if
by
some
a
Riemannian geometry
can
the introduction of such
trivial, although
we saw
a
symmetries imposed dynamical system. In practice,
possess certain
rather ad-hoc restrictions from the
definite geometry into the
that it
was
possible
in
problem
some
is
highly
non-
non-trivial examples.
These results also impose restrictions on the classes of topological quantum field theories and supersymmetric models which fall into the framework of
geometric localization principles, as we shall discuss at greater length Chapter 8. In this Chapter we shall extend the analysis of Chapter 5 to the case when the phase space M is multiply-connected [153]. We shall primarily focus on the case where M is a compact Riemann surface of genus h > 1, again be-
these in
of the wealth of mathematical characterizations that are available for such spaces. We shall explore how the localization formalism differs from that on a simply-connected manifold. Recall that much of the formalism developed in Chapter 4, in particular that of Section 4.10, relied quite heavily cause
R. J. Szabo: LNPm 63, pp. 203 - 231, 2000 © Springer-Verlag Berlin Heidelberg 2000
204
Equivariant Localization
6.
this
Multiply Connected Phase Spaces
restriction. We shall
that
the
topological quanhomology group of the Riemann surface, and that it is completely independent of the geometrical structures that are used to carry out the equivariant localization on M, such as the conformal factors and the modular parameters. This is typically what a topological field theory should do (i.e. have only global features), and therefore the equivariant Hamiltonian systems that one obtains in these cases are nice examples of how the localization formalism is especially suited to describe the characteristics of topological quantum field theories on spaces with much larger topological degrees of freedom. Again the common feature will be the description of the quantum dynamics using a coherent state formalism, this time associated with a non-symmetric spin system and some ideas from geometric quantization [22, 172]. We shall in addition see on
topological
on
see
now
tum field theories that appear also describe the non-trivial first
that the coherent states span a multi- but finite-dimensional Hilbert space in which the wavefunctions carry a non-trivial representation of the discrete
homology group of the phase space. We shall verify the localization forChapter 4 in a slightly modified setting, pointing out the important subtleties that arise in trying to apply them directly on a multiply-connected phase space. Although we shall attempt to give a quite general description of the localizable dynamics on such spaces, most of our analysis will only be carried 2 S' x S1. In particular, we out explicitly for genus 1, i.e. on the 2-torus T shall view the torus in a way best suited to describe its complex algebraic geometry, i.e. in the parallelogram representation of Section 3.5, so that we can examine the topological properties of the quantum theory we find and get a good idea of the features of the localization formalism on multiplyconnected spaces in general. Another more explicit way to view the torus 3 b 2 on the is by embedding it in R by revolving the circle (y a) 2 + X2 2 xy-plane around the x-axis, where 0 < b < a, i.e. embedding T in 3-space b sin 01, y by x (a + b cos 01) Sill 02 and z (a + b cos 01) COS 02. The first
mulas of
=
=
-
=
b'do, of the
=
=
induced metric
on
the surface from the flat Euclidean metric of R
do, + (a + b cos 01 )2 d02 (& d02, and the modular parameter 2 parallelogram representation of T is (C. fSection 3.5)
(9
-r
If
3
we now
=
ib/v a2
-
b2
is then -r
E
C+
(6.1)
introduce the coordinate 01
0
=
do',
0(01)
b a
0
then it is
straightforward to verify that This defines
under translations in
w
=
(6.2)
01
02+0 is
an
isothermal coordinate
p(O) (d02 0 d02 + complex structure on T 2. Since this metric is invariant 02, we could heuristically follow the analysis of Section
for the induced metric
dOodO).
2
+ bcos
a
on
T
for which its isothermal form is
6.1
5.9 to deduce that
functions
Isometry Groups
of
Multiply Connected Spaces
205
class of localizable Hamiltonians are those which are In order that these Hamiltonians be well-defined globally
one
only of 01.
T2 S1 x S1, we require in addition that these be periodic functions 01. As we shall soon see, this is consistent with the general localizable dynamical systems we shall find. Topological invariance of the associated quantum theory in this context would be something like the invariance of it under certain rescalings of the modular parameter (6.1), i.e. under rescalings of the radius parameters a or b corresponding to a uniform 'shift' in the local 2 geometry of T A topological quantum theory shouldn't detect such shifts which aren't considered as ones modifying the topological properties of the torus. In other words, the topological quantum theory should be independent of the phase space complex structure. We shall see this in a more algebraic form later on in this Chapter. on
=
of
.
6.1
Isometry Groups
of
To describe the isometries of
Riemannian manifold
a
Multiply
Connected
Spaces
generic path connected, multiply-connected
(M, g), we lift these isometries up into what is known as
the universal
covering space of the manifold. The multiple-connectivity of M loops in it which cannot be contracted to a point (i.e. M has 'holes' in it). This is measured algebraically by what is called the fundamental homotopy group -7r,(M) of M, a similar but rather different mathematical entity as the first homology group Hi (M; Z). Roughly speaking, this group is defined as follows. We fix a basepoint xo E M and consider the loop space of periodic maps a : [0, 1] -+ M with c(O) a(l) xo. For any 2 loops a and -r based at xO in this way, the product loop a -r is defined to be the loop obtained by first going around a, and then going around 7% The set ,7r,(M) is the space of all equivalence classes [a] of loops, where 2 loops are equivalent if and only if they are homotopic to each other, i.e. there exists a continuous deformation between the loops. It can be shown that the above multiplication of loops then gives a well defined multiplication in 7r, (M) and turns it into a group with identity the homotopy class of the trivial loop [0, 1] --+ xo and with inverse defined by reversing the orientation of a loop. In general, this group is non-abelian and discrete, and it is related to the first homology group H, (M; Z) as follows. Let [G, G] denote the commutator subgroup of any group G, i.e. [G, G] is the normal subgroup of G generated by the products ghg-lh-1, g, h E G. The homology group Hj(M; Z) is then means
that it has
=
=
-
the abelianization of the fundamental group,
Hi (M; Z) If
7r,(M)
is itself
We refer to
homology,
[981
abelian,
for
in the
71
of
then the
-=
7ri
(M)/ fir, (M), ir, (M)]
(6.3)
homology and homotopy of M coincide. complete exposition of homotopy theory and how (6.3), is the natural approximation of homotopy.
a more
sense
(M) ab
206
Equivariant Localization
6.
Multiply Connected Phase Spaces
on
covering space of M is now defined as the smallest simply X4' covering M By a covering space we mean that there is a surjective continuous projection map 7r : M -+ M such that its restriction to any neighbourhood of M defines a local diffeomorphism. This means that locally on M we can lift any quantity defined on it to its universal cover and study it on the simply connected space M The manifold M and its universal covering space M are related by the homeomorphism The universal
connected manifold
-
-
M
where the fundamental group acts deck or covering transformations
X4'
such that
7r(u(x))
7r(x),
=
MI-7ri(M)
-
freely
[98],
Vx E
on
X4' through
i.e. the
A
(6.4) what
are
diffeomorphisms
known
a
:
M
as -4
Thus in this setting, the universal
principal fiber bundle where the total space A is locally regarded as the space of all pairs (x, [Cx]), where Cx is a curve in M from xO to x and [Cx] is its homotopy classl. The structure group of the bundle covering
is
space is
7r,(M)
curves
if it is
a
projection f4 'r) M takes a homotopy class of [Cx] --+ x. Clearly, M is its own universal cover i.e. ir, (M) 0. We shall see some examples in due
and the bundle
to their
endpoint, simply connected,
7r :
=
course. now a Riemannian metric g defined on M, and let 7r*g be its image under the canonical bundle projection of A onto M. Then (M, 7r*g) is a simply-connected Riemannian manifold, and from the analysis of the last Chapter we are well acquainted with the structure of its isometry groups. It is possible to show [94], from the principal fiber bundle interpretation (6.4) bove, that to every isometry h E _T(M, g) one can associate an isometry h E _T(X4,7r*g) which is compatible with the universal covering projection in the sense that
Consider
inverse
7r 0
To prove this
one
curves
is
a
X4' [94].
h
lifting
well-defined function
used for the definition of
(M,g)
(6.5)
o 7
needs to show that the
morphism of M which of
h
on
=-
ir*h gives
the
a
diffeo-
homotopy classes
Thus the isometries of the Rie-
simply connected space complete description from the last Chapter. It should be kept in mind though that there can be global obstructions from the homotopy of M to extending an isometry of M projected locally down onto M by the bundle projection 7r. We shall see how this works in the next mannian manifold
(M, ir*g)
of which
we
have
lift to isometries of the a
Section. a homotopy class of curves [C:,;] can be identified with an element of Irl (M) by choosing another basepoint xO and a grid of standard paths from xO to any other point in M. Then the associated homotopy class is represented by the loop
Here
IX0, XO]
U
Q,
U
[X, XO].
6.2
Equivariant Hamiltonian Systems
6.2 Our
Equivariant Hamiltonian Systems
prototypical
the 2-torus T 2 circle is
=
model for
S1
x
a
in Genus One
in Genus One
multiply-connected sYmplectic manifold
S' which
we
=
Z with the
integers labelling the
S1 'winds' around the circle, i.e. to each map homotopy class [a] E 7rl(Sl) we can associated an integer which we call the winding number of the loop a (where a change of sign signifies a change in the direction of traversing the loop). We can describe the homotopy of the torus by introducing 2 loops a and b, both fixed at the same basepoint on S1 x S1, with a encircling once the inner circle of the torus (i.e. a : S1 -+ (01, 0) E S1 x Sl) and b encircling once the outer circle (i.e. b : S1 --+ (07 02) E S1 X Sl) number of times that
a
o, :
S1
will be
first studied in Section 3.5. Notice that the
ir,(Sl)
multiply-connected with
207
--+
-
2
Since
clearly any other loop in T is homotopic to some combination of the loops a and b, it follows that they generate the fundamental group 7r, (T 2) of the torus, and furthermore they obey the relation aba-lb-1
(6.6)
1
easily seen by simply tracing the loop product in (6.6) around S1 x S1.
which is
(6.6)
=
that 7ri (T 2) is abelian and therefore coincides with the first hogroup (3.81). Thus the loops a and b defined above are also generators
means
mology
homology group H, (Zl; Z), and they will henceforth be refered to homology cycles of the torus. Note that any homology cycle in Z1 which defines the homology class of a (respectively b) can be labelled by the 01 angle coordinates (respectively 02). Thus any homology class of a genus 1 compact Riemann surface is labelled by a pair of integers (n, m) which represents the winding numbers around the canonical homology cycles of the first
the canonical
as
a
and b. Recall from Section 3.5 the
description of the torus as a parallelogram plane, and with modular parameter complex analytic structures on the torus (or equivalently the conformal equivalence classes of metrics on T 2) [111, 145]. This means that it can be represented as the quotient space
with its opposite edges identified in the -r E C+ which labels the inequivalent
Z'
=
C/(z
(6.7)
-rz)
where the quotient is by the free bi-holomorphic action of the lattice group Z E) -rZ on the simply-connected complex plane C. In other words, the lattice group is the discrete on
C z
automorphism 2 by the translations --+
z
+
21r(n
+
group of the
complex plane and
--+. +2,7r(n+;r-m)
-rm)
7r
under which the canonical bundle projection C plane is the universal cover of the torus is easily For
an
exposition of the
various
equivalent
)
;
n,mEZ
it acts
(6.8)
Z' is invariant. That the
seen
ways, such
compact Riemann surfaces in different geometric forms,
by observing that the as see
above, of describing
[110].
208
6.
real line .7r
(x)
=
Equivariant Localization
R1
is the universal
e21rix for
x
E
cover
on
Multiply Connected Phase Spaces
of the circle S' with the bundle
projection
R1.
With the identification (6.7), we can now consider the most general Euclidean signature metric on Z1. From our discussion in Section 5.2, we know that the most general metric on C can be written in the global isothermal form
(5.49).
general
The covering projection in (6.7) in this way induces the most on the torus, which can therefore be written in terms of a flat
metric
Kdhler metric
as
ew(zA 9-r or
in terms of the
=
Im
angle coordinates (01, 02) 02)
ew(Ol Im The Z
=
complex 01 + T02
structure
on
Z' is
01 +':r02 which
(6.8).
transformations
dz o d,
now
Z'
on
the normalization in
(6.9)
(i.e.
S'
X
Re
1
defined
T)
(6.10)
1,rl'
-r
by
the
complex coordinates
therefore considered invariant under the
The conformal factor
real-valued function
S'
E
(Re
'r
are
(6.9)
-r
W(z, )
is
now a
globally defined
invariant under the translations
is chosen for
(6.8)),
simplicity so that the associated
and
metric
volume of the torus
V019'r (ZI) is finite and
=
J
/d_et t7'
d 20
d
independent of the complex
2
0 0(01
structure of
volume parameter of the torus. The metric Gaussian curvature scalar
K(g,) which
2
02)
Im(,r)
(6.9)
e-I' V2 O
d20 V2(p(01' 02) o96
is the scalar
,V2
=
92
+
Z' with
=
v
(6.11)
E
R
fixed
a
is further constrained
by the Gauss-Bonnet-Chern theorem (5.54) for
where V2
(2 7r)2V
by
its
(6.12) genus h
I must
obey
(6.13)
0
Laplacian 2 Irl- 9202
+ 2
associated with the Kiihler structure in
Re(,r)IrI-2a01 a-02
(6-14)
(6.9).
Given this general geometric structure of the 2-torus, following the analysis of the last Chapter we would like to find the most general Hamiltonian
system
on
it which
obeys the localization
criteria. First of
all,
the condition
that the Hamiltonian H generates a globally integrable isometry of the metric (6.9) implies that the associated Hamiltonian vector fields VII(x) must be
6.2
Equivariant Hamiltonian Systems
in Genus One
209
single-valued functions under the windings (6.8) around the non-trivial homology cycles of El. This means that these functions must admit convergent 2-dimensional harmonic mode expansions 00
V '(01 02) 7
E
=
V. . e'(n-P1+M02)
(6.15)
n,m=-co
In other a
words, the components of V must be Cl-functions which admit 2-dimensional Fourier series plane wave expansion (6.15) appropriate to
globally-defined periodic functions on S' x S'. As we shall now demonstrate, topological restrictions from the underlying phase space severely limit the possible Hamiltonian systems to which the equivariant localization constraints apply. lFrom (2.112) it follows that the Killing equations for the metric (6.10) these
are
2ao, V1 2
a02V1
+
plane
(6.16),
we
Re(7)a02V1
Re(7-),gOJV2
+ Vpa01" (P
+ 2 17-12 '902 V2 +
Re(7*002 V2 +aOlVl)
Substituting the
+ 2
+
=
0
1,r12V/_'j90mW=0
1,rl2a01 V2
+
Re(7-)V"o90,W
0
in the harmonic
expansions (6.15) and using the completeness of there to equate the various components of the expansions in find after some algebra that (6.16) generates 2 coupled equations
(1,r 12 n(m
=
waves
for the Fourier components of the Hamiltonian vector
-
(6-16)
Re (7.),rn) V 1
n, m
Re(-r)n)V.',,,, n,
2
1,r 1 (M
Re
field,
Re(-r)n)V.2 n,m
Im(-r )n
-
Re(-r)m]
(6.17) n,m
which hold for all integers n and m. It is straightforward to show from the V2 m coupled equations (6.17) that for -r c: C+, V'm 0 unless n 0. m n, n, Thus the only non-vanishing components of the harmonic expansions (6.15) =
are
the constant
lations
only Killing
(by Vof4
=
=
modes,
V''. W and the
=
=
V0,
vectors of the metric
R) along
(6.9)
(6.18) are
the generators of trans-
independent homology cycles of Z1. Notice that this result is completely independent of the structure of the conformal factor p in (6.9), and it simply means that although the torus inherits locally 3 isometries from the maximally symmetric plane, i.e. local rotations and translations, only the 2 associated translations on El are global isometries. The independence of this result on the conformal factor is not too surprising, since this just reflects the fact that given any metric on a compact phase space we can make it invariant under a compact group action by averaging it over the group in its Haar measure. The above derivation gives an E
the 2
210
6.
Equivariant Localization
on
Multiply Connected
Phase
Spaces
explicit geometric view of how the non-trivial topology of Z' restricts the alglobal circle actions on the phase space, and we see therefore that the isometry group of any globally-defined Riemannian geometry on the torus is lowed
U(1)
X
U(1).
(5.70) for the symplectic structure can be solved by imposing the requirement of invariance, of w_ri independently under the 2 Killing vectors (6.18). This implies that the components WO,O,, must be constant functions, i.e. that w must be proportional to the Darboux 2-form The invariance condition
WD, and thus
we
take w-ri
v
dol
A
(6.19)
d02
Z' for the present Riemannian symplecreduction of that from the universal bundle projection C V. It is )
to be
an
associated metric-volume form
geometry (c.f. tic
=
(6.11)).
The
symplectic
on
structure here is thus the "
straightforward
to
now
(6.19),
we
find that the Hamiltonian
and
along
the
and
integrate
homology cycles
of
equations with (6.18) given by displacements
up the Hamiltonian
H_rl
is
Z1,
HZI (011 02)
=
001
+ h 202
(6.20)
where
h'
=
vVO
2
h
2 =
_VV
1
(6.21)
are real-valued constants. Note that, as anticipated from (6.7), the invariant symplectic structure here is uniquely determined just as for 2-dimensional maximally symmetric phase spaces which have 3 (as opposed to just 2 as above) linearly independent Killing vectors. Thus we see here that the localizable Hamiltonian systems in genus I are even more severely restricted by the equivariant localization constraints as compared to the simply-connected cases. Note that the Hamiltonian (6.20) does not determine a globally-defined single-valued function on Z1, a point which we shall return to shortly.
6.3
and
Homology Representations Topological Quantum Field Theory
The Hamiltonian
(6.20)
but besides this feature from the
defines we see
a
rather odd
dynamical system
that the allotted Hamiltonians
on
as
the torus,
determined
geometric localization constraints are in effect completely indepenexplicit form of the phase space geometry and depend only on the topological properties of the manifold Z1, i.e. (6.20) is explicitly independent of both the complex structure -r and the conformal factor W appearing in (6.9). Rom the analysis of the last Chapter, we see that this is in marked contrast to what occurs in the case of a simply connected phase space, where the conformal factor of the metric entered into the final expression for the observable H and the equivariant Hamiltonian systems so obtained depended dent of the
6.3
Homology Representations
and
Topological Quantum Field Theory
on the phase space geometry explicitly. In the present tion with Hamiltonian (6.20) and symplectic 2-form
case
211
the partition funcobtained as the
(6.19)
unique solutions of the equivariant localization constraints
can be thought of defining a topological quantum theory on the torus which is completely independent of any Riemannian geometry on Z1. Furthermore, the symplectic potential associated with (6.19) is
in this way
as
V
OZ1 which
-
2
(0142
-
02dol)
(6.22)
only locally defined because it involves multi-valued funcZ. form, so that wEl is a non-trivial element of H 2(_pl; Z) The Hamiltonian (6.20) thus admits the local topological form H_rl iv, Ozi, so that the corresponding partition function defines a cohomological. field theory and it will be a topological invariant of the manifold Z1. To explore some of the features of this topological quantum field theory, we note first that (6-20) is not defined as a global C'-function on Z1. However, this is not a problem from the point of view of localization theory. Although for the classical dynamics the Hamiltonian can be a multi-valued function on Z1, to obtain a well-defined quantum theory we require singlevaluedness, under the windings (6.8) around the homology cycles of Z1, of we
note is
tions in this local
=
=
the time evolution operator (and also of the Boltzmann
e-iTk_,l weight
classical statistical
mechanics).
must be
i.e. h" E hZ for
quantized,
e
This
which defines the quantum propagator iTH_,l if we wish to have a well-defined
implies that the constants hl, in (6.20) h E R, and then time propagation in
some
this quantum system can only be defined in discretized intervals of the base time h-1, i.e. T NTh-1 where NT E Z+. Such quantizations of coupling =
parameters in topological gauge theories is a rather common occurence to ensure the invariance of a quantum theory under 'large gauge transformations' when the
underlying
space has non-trivial
topology [22).
In the quantum theory, the Hamiltonian (6.20) therefore represents the winding numbers around the homology cycles of the torus, and therefore to
each tem
is
homology class of Z' we can associate a corresponding Hamiltonian sysobeying the equivariant localization constraints. The partition function
now
denoted
as
NTh-1
Z i (k, t; NT)
-
j [d201 LEI
exp
i
f
dt
(02 1
+
h(kO,
+
42))
0
6.23) where k and
integers and we have integrated the kinetic term in (6.23) by parts. This path integral can be evaluated directly by first integrating over the loops 02(t), which gives are
212
6.
Z i (k,
f;
Equivariant Localization
on
Multiply Connected Phase Spaces NTh-1
[d0j] J(v j +hf) exp
NT)
I
i
dt
hkOj (t)
-
e-WNT2 /2v
0
LS1
(6.24) Thus the
partition function of this quantum system represents the non-trivial homology classes of the torus, through the winding numbers k and f and the time evolution integer NT. In fact, (6.24) defines a family of 1-dimensional unitary irreducible representations of the first homology group of ZI through the family of homomorphisms
Z j (-, .; NT): H,(Zl; Z)
--4
U(1)
(D
U(1)
(6.25)
homology group (3.81) into a multiplicative circle homologically-invariant quantum theory is trivial, in that the sum over all winding numbers of the partition function (6.24) vanishes, from the additive first
group. Notice that the associated
00
00
E
Z , (k, t; NT)
E
=
k, =-oo
e
k=_00
ikN2 /2v T
+ e
ikNT2/2v
-1
=0
(6.26) This
winding numbers is analogous to what one would do in 4Yang-Mills theory to include all instanton sectors into the quan-
sum over
dimensional
all
theory [151]. However, it is possible to modify slightly the definition of the quantum propagator on a multiply-connected phase space so that we obtain a partition function which is independent of the homology class defined by the Hamiltonian using a modification of the definition of the path integral over a multiply connected space [147]. In general, if the phase space M is multiply connected, i.e. irl(M) 7 0, then the Feynman path integral representation of the quantum
tum
propagator
can
contain parameters
X([a])
which
are
not
theory and which weight the homotopy classes [a] inequivalent time evolutions of the system3, classical
Zh.m (T)
I
X([O']) [a]Ewi(M)
x
Unitarity and completeness of (4.13)) yield, respectively, the phases,
[d2nx]
et 11011 Vd--
e
present in the
of
topologically
is[x]
(6.27)
(t) E [a]
the quantum theory (i.e. of the propagator constraints that the parameters XQU]) are
'
x([o-])*x([,-]) 3
(6.28)
applied to the full quantum propagator IC(x', x; T) phase space points. Then the sum in (6.27) is over all homotopy classes
This definition could also be between 2 of a
curves
[C,,x, ]
from
standard mesh of
x
to
paths.
x' which
are
identified with elements of
7r,
(M) using
Integrability Properties
6.4
and that
they
form
a
and Localization Formulas
213
1-dimensional unitary representation of 7r, (M),
X([UDX(10"D Note that the restriction of the
=
Vla aT
(6.29)
-
path integration
to
homotopy classes
as
in
(6.27)
makes well-defined the representation of the partition function action S with a local symplectic potential following the Wess-Zumino-Witten pre-
particular, we can invoke the argument there homotopy class [a] E -7r,(M), the path integral depends only on the second cohomology class defined by w. In the case at hand, the partition function (6.24) is regarded as that obtained by restricting the path integration in (6.23) to loops in the homology class labelled by (k, f) E Z2. In particular, we can add to the sum in (6.26) ia(k,i) for each the phases X(k, t) e (k, t) E Z2, which from (6.29) would then have to satisfy 4.10. In
scription of Section
to conclude that
over
each
=
a(k
+
k', t + t)
=
a(k, t)
+
a(k', t')
(6.30)
(6.30) means that the phase a(k, t) defines a u(l)-valued 1cocycle of the fundamental (or homology) group Z ED Z of Z1 (see Appendix A) as required for them to form a representation of it in the circle group S1. When they are combined with the character representation (6.24) and the resulting quantity is summed as in (6.26), we can obtain a propagator which is a non-trivial homological invariant of Z' and which yields a character formula for the non-trivial topological groups of the phase space. We shall see how to interpret these character formulas in a group-theoretic setting, as we did in the last Chapter, in the Section 6.5. Notice that, strictly speaking, the volume parameter v in (6.24) should be quantized in terms of h, k and t so that the partition function yields a non-zero result when integrated over the moduli space of T-periodic trajectories. In this way, (6.24) also represents the cohomology class defined by the symplectic 2-form (6.19) through the parameter v. We recall from Section 4.10 that for a simply-connected phase space, the localizable partition functions depend only on the second cohomology class defined by w. Here we find that the multiple-connectivity of the phase space makes it depend in addition on the first homology group of the manifold. Thus the partition function of the localizable quantum systems on the torus yield topological invariants of the phase space representing its The condition
(co-)homology
6.4 We
groups.
Integrability Properties and Localization Formulas now
turn to
a
discussion of the structure of the localization formulas for
these localizable Hamiltonian systems. Of course, since the canonical U(1) x U(1) - T 2 action on the torus generated by (6.20) has no fixed points, this means
that the classical
partition function
214
Equivariant Localization
6.
v
do, d02
e
Multiply Connected Phase Spaces
on
1
iTH_,, (01,02)
=
T2
e-27riTvVO2
T2VV0IV2
e27riTvV01 (6.31)
isn't
given by the Duistermaat-Heckman formula. The reason can be traced back to the Poisson algebra fOl -1/v which shows that the full 021w_,i =
7
Hamiltonian if
we
choose
(6.20)
a
functional of action variables in involution. Even Hamiltonian which is a Morse function on T 2 given by a funcis not
a
tional of the components of the localizable isometry generators on the torus (like the height function), Kirwan's theorem forbids the exactness of the sta-
tionary phase approximation We
for the associated classical partition functions. a Hamiltonian reduction as
also view this failure at the level of
can
discussed in Sections 5.7 and 5.8 above. For instance, consider the following not fulfill the Duistermaat-Heckman theorem,
partition function that does T2
Z f (T)
v
do, d02
e
iT(acosOj+bcosO2)
(21r )2Vjo(-iTa)Jo(-iTb) I
T2
(6.32) We on
can
write the left-hand side of
C2 by introducing
2
(6.32)
as a
complex coordinates
reduction from z,
larger integral Izil e'01 and Z2 jz2j e'02
=
a
=
with the constraints
P(Z, 2) for
=_
Z,
-
I
=
(6.33)
0
ZI Z2 Introducing Lagrange multipliers A, and A2 whose integration produces delta-functions enforcing these constraints, we can write the integration in (6.32) as Z
over
=
9
.
R
00
T2
Zj j (T)
=
v
f
d2Z,
dA, dA2
2Z2 exp
7r2 R4
00
+iA1(Z121
-
1)
+
iT (a (zl 2
i/\2(Z222
-
+
l) +
b 2
(Z2
+, 2))
1)j (6.34)
This shows
that, unlike
Hamiltonian in
(6.34)
the
mentioned in Sections 5.7 and 5.8, the from which the dynamical system in (6.32) is obtained
by Hamiltonian reduction
cases
is not
a
bilinear function and it does not
mute with the constraint functions defined
the reduction
by (6.33),
not first class constraints. These alternative
are
arguments therefore also
serve as an
com-
i.e. the constraints of
integrability
indication of the breakdown of the lo-
calization formalism when
applied to multiply connected phase spaces, in topological criteria provided by Kirwan's theorem. We Chapter 7 this sort of interplay between integrability and
addition to the usual shall
see
again
in
Kirwan's theorem. The situation is better for the quantum localizations, even as far as the of using functionals F(Hzi) of the isometry generator (6.20) for
possibilities localization
as
in Section 4.9. Here the arbitrariness of these functionals is
6.4
not
great
as
it
as
Integrability Properties and Localization Formulas
was
in the
simply connected
cases
of
Chapter
215
5. There
required generally only that F be bounded from below, while in the case at hand the discussion of Section 6.3 above shows that we need in addition
we
the requirement that F be formally a periodic functional of the observable (6.20). In general, this will not impose any quantization condition on the time translation
T,
it did before. For such
functionals, however,
general explicitly (4.148) required for the localization (4.149). Alternatively, one can try to localize the system using (4.142) and the above description of the quantum theory as a topological one, but then we lose the interpretation of the independent Hamiltonians in (6.20) as conserved charges of some integrable dynamical system with phase space the torus. These remarks imply, for example, that one cannot equivariantly quantize a free particle or harmonic oscillator (with compactified momentum and position ranges) on the torus, so that the localizable dynamical systems do not represent generalized harmonic oscillators as they did in the simply connected cases. The same is true of the torus height function (3.78), as anticipated. However, in these cases the periodicity as
rather difficult to determine
it is in
the Nicolai transform in
of the Hamiltonian function leads to
a
much better defined propagator in the
tempered distribution represented by a functional integral. Notice that this also shows explicitly, in a rather transparent way, how the Hamiltonian functions on T 2 are restricted by Kirwan's theorem, which essentially means in the above context that the localization formalism loses its interpretation in terms of integrability arguments on a multiply connected phase space. The topological field theory interpretations do, however, carry through from the simply-connected cases but with a much richer structure sense
of it
being
a
now.
expression (6.24) for the quantum partition function also follows disubstituting into the Boltzmann weight e s[-T] the value of the action in (6.23) evaluated on the classical trajectories Vl(t) V,", for the above quantum system, which here are defined by The
rectly
from
=
1(t) Thus the
=
1
VO
7
2(t)
path integral (6.23) (trivially) localizes
the WKB localization formula
(4.115), except
=
V02
(6.35)
onto the classical
that
now even
the
loops 1-loop
as
in
fluc-
tuation term vanishes and the
path integral is given exactly by its tree-level independently establishes the quantizations of the propagation time T and the volume parameter v, in that T-periodic solutions to the classical equations of motion with the degenerate structure of the Hamiltonian (6.20) only exist with the discretizations of the parameters hA and T above. This is consistent with the discussion at the beginning of Section 4.6 concerning the structure of the moduli space of classical solutions, and again for these discretizations the path integral can 'be evaluated using the degenerate localization formula (4.122) while for the non-discretized values value. This also
the critical
trajectory
set
(trivially)
coincides with the critical point set Mv
216
6.
Equivariant Localization
of the Hamiltonian.
Furthermore,
on
Multiply Connected Phase Spaces
the fact that the conformal factor W is not
involved at all in the solutions of the localization constraints just reflects the fact that the torus is locally flat (as is immediate from its parallelogram
representation) and any global 'curving' of its geometry represented by W in (6.9) can only be done in a uniform periodic fashion around the canonical homology cycles of Z' (c.f. eq. (6.13)). However, the Niemi-Tirkkonen formula (4-130) does depend explicitly on o. It is here that the geometry of the phase explicitly
space enters we
into the
quantum theory, as it did in Chapter 5, if (6.9) obey the appropriate regularity conditions
demand that the metric
and therefore make the
equivariant localization manifest. This
the localization formula
(4.130)
ensures
coincides with the exact result
(6.24),
that as
it
should. In the
at hand
case
Z i (k, f; NT)
(4.130)
becomes
chvl (-iNTwZi1h)
=Jd20 1
d 2n exp
[-
AAv,, (NTR,lh)
iNT h
(H_ri (k,
2
Zi
NT(2(V,)1,V ,
det
X
(R,),\PqA?7P)/4h
+
t,
__
-
sinh
(NT(2(V,),,V 1 (R,)1,\Pq-XqP)14h) +
(6.36) Again,
because of the Kdhler structure of
and curvature 2-form have the
-(Av,,)z
=
(6.9),
the Riemann moment map
non-vanishing components
V Ap+VI,5 o
,
RzZ
=
-R2
=
Im(-r)
e- ' V2
(6.37) We substitute
(6.18)-(6.21)
and
(6.37)
into
(6.36)
and carry out the Berezin expression with the exact one
integrations there. Comparing the resulting (6.24) for the partition function, we arrive after on
the conformal factor of the metric
I
e-iNT(kol+42)
d20
1
NT2 (V90, W 4V2
-
e
represents
at
a
condition
-
k'002 W)2 kc),p2
sinh2(NT (690, w 2v
-
(6.38)
-WNT2/2v
v TV
The Fourier series constraint a
algebra
(6.9),
-
El
2i
some
(6.38)
similar sort of metric
on
the metric is rather
regularity
condition that
complicated and we
it
encountered in
Section 5.9 before. It fixes the harmonic modes of the square-root integrand in (6.38) which should have an expansion such as (6.15). Notice, however, that
(6.38)
is
independent
of the
phase
space
complex
structure -r, and thus
6.5
it
Holomorphic Quantization and Non-Symmetric Coadjoint Orbits
only depends
the metric
(6.9).
representative of the conformal equivalence class of typical of a topological field theory path integral
the
on
217
This is
[22]. The condition
(6.38)
can
be used to check if
a
given phase
space metric
really does result in the correct quantum theory (6.24), and this procedure then tells us what (representatives of the conformal equivalence classes of) sense are applicable to the equivariant localization integrals on the torus. For example, suppose we tried to quantize a flat torus using equivariant localization. Then from (6.12) the conformal factor 0 globally on Z1. Since p is would have to solve the Laplace equation V2 (p assumed to be a globally-defined function on Z1, it must admit a harmonic mode expansion over Z' as in (6.15). From (6.14) and this Fourier series for W we see that the Laplace equation implies that all Fourier modes of V except the constant modes vanish, and so the left-hand side of (6.38) is zero. Thus a flat torus cannot be used to localize the quantum mechanical path integral (6.23) onto the equivariant Atiyah-Singer index in (4.130). This means that a flat Kdhler metric (6.9) on Z' does not lead to a hornotopically trivial localization 1-form iv,, g, on the loop space LZ1 within any homotopy class (c.f. Section 4.4). This simple example shows that the condition (6.38), along with the Riemannian restrictions (6.11) and (6.13), give a very strong probe of the quantum geometry of the torus. Moreover, when (6.38) does hold, we can represent the equivariant characteristic classes in (4.130) in terms of the homomorphism (6.24) of the first homology group of Z1.
quantum geometries in this
of path
=
6.5
and
Holomorphic Quantization Non-Symmetric Coadjoint Orbits
In this Section
we
shall show that it is
interpret the topological quantization novel sort of spin system described
possible
to
character formula associated with the
path integral (6.23) a coadjoint orbit corresponding to some by Z1, as was the situation in all of the simply connected cases of the last Chapter. For this, we examine the canonical quantum theory defined by the symplectic structure (6.19) in the Schr6dinger picture representation. We first rewrite the symplectic 2-form (6.19) in complex coordinates to get the Kdhler as a
of
structure
V
WEI
with
2i Im
dz A d2
=
-05F-ri
-r
(6-39)
corresponding local Kiffiler potential F_P1 (z, )
=
vz, /2
Im
-r
(6.40)
We then map the corresponding Poisson algebra onto the associated Heisenberg algebra by the standard commutator prescription (c.f. beginning of Section
5.1).
With this
we
obtain the quantum commutator
218
Equivariant Localization
6.
[i, ] We
can
Multiply Connected
on
Spaces
Im(-r)/v
2
=
Phase
(6.41)
represent the algebra (6.41) on the space Hol(Z'; -r) of holomorphic Tl(z) on Z' by letting i act as multipication by the complex coor-
functions dinate
7-02 and 22'
+
Z
the derivative operator
as
2 1m
c9
-r
(6.42)
'9Z
V
With this
holomorphic Schr6dinger polarization, the operators an d Z2' algebra (6.41) resemble the creation and annihilation operators (5.79) of the Heisenberg-Weyl algebra with the commutation rela, tion (5.80). In analogy with that situation, we can construct the correspondwith the commutator
ing coherent
states
1z) which
normalized
are
=
obey
the
?-)z2-
Im
z10)
e
=
(v/2
These coherent states
U(1)
U(1)
=
are
S'
it cannot be considered
orbit of
Z
E _r
1
(6.43)
=eF--1 (z,2)
(6.44)
completeness relation
1Z))((Z1
(2,r)2
x
-r)z;e
Im
d2z
orbit
;
as
(zlz) and
e( -v/2
x
(6.45)
associated with the quantization of the coadjoint S1. However, since Z' is a non-symmetric space,
as a
Khhler manifold associated with the
coadjoint
semi-simple Lie group, as was the case in the last Chapter. The orbits above are, however, associated with the action of the isometry group
U(1)
x
a
U(1)
on
Z1,
which has
an
interesting
Lie
algebraic
structure that
we
shall discuss below. In the
Schr6dinger representation (6.42), z: '-
we
consistently find the action holomorphic representation
Z "-Jz). (6.43) Hol(Zl; -r) in this context is then regarded as the space of entire functions TI(z) (zjT1) for each state ITI) in the span of the coherent states (6.43). An inner product on Hol(Z'; -r) is then determined from the completeness of the operator
on
the states
The
as
space
=
relation
(6.45)
(T11 I Tf2 )
and the normalization
d2Z
=I (27r) Zi
2
(T/1jZ)(Zjq/2) (ZIZ)
(6.44)
as
d2 _,
=
f (27r)
2
e
-(v/2
Im
-r)z2Tt1
Z
2
(Z)
El
(6.46) t is the adjoint product (6.46), we find that the operator z of ; , as it consistently should be. An operator ? acting on the space of coherent states (6.43) can now be represented on Hol(Z';-r) as usual by
With the inner
=
6.5
an
as in (5.95) with the identification of as the (6.42). Furthermore, the quantum propagator associated
integral kernel
operator an
Holomorphic Quantization and Non-Symmetric Coadjoint Orbits
219
derivative with such
operator
IC-ri (Y, z; T) determines the tions
corresponding
((z'l e-M I Z))
(6-47)
time-evolution of the coherent state wavefunc-
as
Tf(z; T) with
=
Tf(z; t)
=-
d2Z/ =
(z, tITI)
_2'7 2 and
Tl(z)
e-F_,l (z IX)lCzl
=-
Tf(z; 0).
z;
In the
T)Tf(z')
following
we
(6.48) shall build
up the initial states Tl(z), and then the associated time evolution determined by the localizable Hamiltonians on Z' (and hence the solutions of
the
Schr6dinger
wave
equation)
are
determined
for the propagator (6.47). The advantage of working with the
Hol(Zl; -r)
is that
we
by the path integral above
holomorphic representation space explicit structure of the Hilbert
shall want to discuss the
space associated with the localizable
quantum systems
we
found above. With
the Khhler structure defined
by the symplectic 2-form, wZ1 above, the Hilbert is then the space of holomorphic sections of the the of theory quantum space usual symplectic complex line bundle L --+ Z1, which in this context is usually called the prequantum, line bundle over Z1. As such, wzi represents the first Chern characteristic class of L, and so such a bundle exists only if WZ1 is an integral 2-form on Z1. This method of quantizing the Hamiltonian dynamics in terms of the geometry of fiber bundles is called geometric quantization [172] and it is equivalent to the Borel-Weil-Bott method of constructing coherent states that we encountered in the last Chapter. In light of the requirement of single-valuedness of the quantum propagator that we discussed in the Section 6.3, we require, from the point of view of equivariant localization, that the wavefunctions TI(z) change only by a unitary transformation under the winding transformations (6.8) on Z1, so that all physical quantities, such as the probability density T14, are well-defined C'-functions on the phase space Z' and respect the symmetries of the quantum theory as defined by the quantum Hamiltonian, i.e. by the supersymmetry making the dynamical system a localizable one. In this setting, the multivalued wavefunctions, regarded as sections of the associated line bundle L --+ Z' where the structure ZE)Z acts through a unitary representation, are single-valued group -7r, (El) functions on the universal cover C of the torus and so they can be thought of as single-valued functions of homotopy classes [a] of loops on Z1. This =
also
ensures
that the coherent states
evolution determined under the action of
by
(6.43)
remain coherent under the time
the localizable Hamiltonians of the last Section
I(El; g,))
which will lead to
a
(i.e.
consistent coherent state
path integral representation of (6.23). To explore this in more detail, we need a representation for the discretized equivariant Hamiltonian generators above of the isometry group 1(Z'; g,)
220
6.
Equivariant Localization
Hol(ZI; -r) [18, 153].
the space
on
the
on
automorphisms
Multiply
Connected Phase
Spaces
This group action then coincides with
of the
symplectic line bundle above in the usual way. Note that translations by a E C on z are generated on functions of z by " the action of the operator e z---, and likewise on functions of by 0-5'9-g. On the holomorphic representation space Hol(Zl; T), we represent the latter a
(v/2
1)61, in accordance with the coherent state representation above. Thus the generators of large U(1) transformations around the homology cycles of Z' in the holomorphic operator using the commutation relation
Schr6dinger polarization
U(n, m)
=
are
m-r) TZ
+
as
e
Im
the unitary quantum operators
09
27r(n
exp
above
(6.41)
'7rV
+
fm-T (n
+
m;r) z
n,
E
m
Z
(6.49)
which generate simultaneously both of the winding transformations in (6.8). By the above arguments, the quantum states should be invariant (up to uni-
tary equivalence) under their action on the Hilbert space. Solving this invariance condition will then give a representation of the equivariant localization symmetry constraints (i.e. of the pertinent cohomological supersymmetry) and of the coadjoint orbit system directly in the Hilbert space of the canonical quantum theory. In contrast with their classical counterparts, the quantum operators (6.49) do not commute among themselves in general and products of them differ from their reverse-ordered
A).
The
products by a u(l)-valued 2-cocycle (see Appendix Baker-Campbell-Hausdorff formula,
eX+Y
=
e- [X,Y]12
ex ey
when
[X, [X, Yfl
=
[Y, [X, Yj]
=
0
(6.50)
implies
ex ey
Applying (6.51) tion relation
to
(6.41)
products with
=
e1X,Y1 ey ex
(6.51)
(6.49) and using the commutathey obey what is called a clock
of the operators
(6.42),
we
find that
algebra,
U(nj,mj)U(n2,M2)
=
e
2-7riv(n2M1-niM2) U
(n2 M2) U (ni, mi) ,
(6.52)
To determine the action of the operators (6.49) explicitly on the wavefuncTl(z), we apply the Baker-Campbell-Hausdorff formula (6.50) to get
tions
U(n, m) so
Irv
=
exp
[Im.
that the action of
U(n, m)T/(z)
-r
(7rin + Mr12 + (n + m;r-)z)I
exp
27r(n+m-r)jfi Z-.
(6.53)
(6.53)
on
1IM
(7rjn +,Mrl2 + (n + m;r-)z)] T1(z + 21r(n + m-r))
Irv
=
e
-r
the quantum states of the
theory
is
(6.54) If the volume parameter v vol'q, (Z1)/(21r)2 is then it follows from the clock algebra (6.52) that the =
an
irrational
U(1)
number,
generators above
6.5
act
as
Holomorphic Quantization
and
Non-Symmetric Coadjoint Orbits
'
221
infinite-dimensional raising operators in (6.54) and so the Hilbert space case is infinite-dimensional. However, we recall the
of quantum states in this
quantization requirements for the parameters of the Hamiltonian a consistent quantum theory. With this in mind, we instead consider the case where the volume of the torus is quantized so that necessary
system required for
V
-`
V1/V2
VliV2 E
Z+
(6.55)
is rational-valued.
Alternatively, such a discretization of v is required in orsymplectic 2-form w_rl define an integer cohomology class, as (4.161). In this case, the cocycle relation (6.52) shows that the operator
der that the in
U(V2n, V2M) commutes with all of the other U(1) generators and the time evolution operator, and so they can be simultaneously diagonalized over the same basis of states. This means that their action (6.54) on the wavefunctions must
produce
defined
a
by Tl(z),
state that lies
on
the
same
ray in the Hilbert space
as
that
i.e.
U(V2n, V2M)T'(Z)
=
e'77(n,m)Tf(Z)
(6.56)
for some phases n(n, Tn) G S1. The invariance condition (6-56), expressing the symmetry of the wavefunctions under the action of the (non-simple) Lie group U(1) x U(1), is called a projective representation of the symmetry group. It must
obey
a
particular consistency condition.
for the group operations induces
U(V2(n, =
+
n2), V2(MI
+
a
The composition law composition law for the phases in (6.56),
M2))Tf(Z)
U(V2nl,V2Ml)U(V2n2,V2M2)Tf(Z)
(6.57)
ein(ni+n2,M1+M2)T/(Z)
=
x
exp,
[ijq(ni, mi) +,q(n2, M2) -q(ni
+ n2, M1 +
M2)11
If the last
phase in (6.57) vanishes, as in (6.30), then the projective phase 71(n, m) 1-cocycle of the symmetry group U(1) x U(1) and the wavefunctions carry a unitary representation of the group, as required [77]. The determination of these 1-cocycles explicitly below will then yield an explicit representation of the homologically-invariant partition function (6.27). Comparing (6.56) and (6.54), we see that the invariance of the quantum states under the U(1) action on the phase space can be expressed as is
a
TV (z + 27rV2 (n + =
The
exp
m-r))
[iq(n, m)
7rvl -
Im
-r
(7rV2 In + Mr12 + (n + m;r-) z) I
only functions which obey quasi-periodic conditions [61, 109, 145]
binations of the Jacobi theta functions
like
Tf (z)
(6.58)
(6.58)
are com-
222
Equivariant Localization
6.
Phase
Multiply Connected
on
Spaces
(C) [ZI.Ul
e(D)
d
(6.59)
E exp[iir(n'+c')Hep(n-+cP)+27ri(n +c )(z +dt)] In'61EZD
c , dj E [0, 1b. jztj E C
where
Siegal
upper
(6.59) are well-defined holomorphic funccomplex-valued matrices IT [111p] in the H > 0). They obey the doubly semi-periodic
The functions for D
tions of
D
x
half-plane (i.e.
Im
=
conditions
( ) [Z C
e(D)
exp
where
s
+
d
[27ric sf
Isil
=
and t
( ) [Z
e(D)
-
tily]
i7rOlItptP
-
=
+ a.U
d
exp
+ I.T
8
ftl
are
27ritt(zi + dt)] e(D)
-
(6.60)
(C)
zl IT]
d
and
integer-valued vectors,
till]
-
[_i7ra2ttH
ep tp
21riat'(z
-
+
de)]
19(D)
-
d
(6.61)
at) [Z1171
for any non-integer constant a E R. We remark here that the transformations in (6.60) can be applied in many different steps with the same final result, but successive
applications
of
context of the action of the are
differ
which forms
by
phase
a
action is
applied
uniquely
Tlp,,
Tl(z) (6.61).
the states
on
After
ambiguity,
before some
solved
() c
d
(z)
do not commute
unitary operators U(n, m) above, a
we
(6.58),
In the
the final results
representation of the clock algebra (6.52). To simply define the operators U(n, M) by their
e- (v/4
=
Im
r
-r) Z2 e(j)
(6.60)
=
(
21r2V2
1, 2,
=
cn
U(1) group acting (6.54) can be written
global U(1)
+27rvlP+V2r
vj.
The
) [vlzl27rvlV27'1
phases
x
-
,
d
(6.56)
(6.62) are
then
on
ZI here. Furthermore, the winding
as
1:'*'[U(n, m)],p P/=1
(6.63)
dm + IrVlV2nm
V2
Tn) T1p,
in
are
1-cocycles
,q(n, m)/21r
U (n,
[181.
when these
algebra, we find that the algebraic constraints (6.58) by the V1V2 independent holomorphic wavefunctions
=
transformations
(6.61)
with the convention that the transformation
where p 1, 2.... ) V2 and found to be the non-trivial
of the
and
in different orders in
applied
transformations
avoid this minor
(6.60)
( ) (Z) C
I
plr
d
(6.64)
Holomorphic Quantization and Non-Symmetric Coadjoint Orbits
6.5
where the finite-dimensional
[U(n, form
=
exp
[
223
unitary matrices
27ri
(cn
V2
-
dm +
I
7rvln(m + 2p)) Jp+m,v,
(6.65)
V2-dimensional projective representation, which is cyclic of period algebra (6.52). The projective phase here is the non-trivial U(1) 1-cocycle
a
V2, of the clock
U(1)
x
77(P)(n, m)/27r
(cn
=
-
dm +
.7rvin(m + 2p))/V2
which could also therefore be used to construct
function
as
in the Section 6.3
an
(6.66)
unambiguous partition
by 00
Zho.(T)
E ei?7(k,R)Z 1(k,E;NT)
=
(6.67)
k,t=-oo
Thus the Hilbert space is VlV2-dimensional and the quantum states carry a V2-dimensional projective representation of the equivariant localization symmetries via the clock algebra (6.52) which involves the u(l)-valued 2-cocycle
(nl, 7ni; n2, M2)/2ir
=
vi(n27nl
-
nIM2)/V2
(6.68)
of the U(1) x U(1) isometry group of Z1. This shows explicitly how the U(1) equivariant localization constraints and the topological toroidal restrictions are realized in the canonical quantum theory, as then these conditions imply that the only invariant operators on the Hilbert space here are essentially combinations of the generators (6.49). In particular, this implies, by construction, that the coherent state wavefunctions (6.62) are complete. This is much different than the situation for the coherent states associated with
simply-connected phase spaces where there are no such topological symmerespected for the supersymmetric localization of the path integral and the Hilbert space is 1-dimensional. Intuitively, the finite-dimensionality of the Hilbert space of physical states is expected from the compactness of the phase space Z1. Notice though that the wavefunctions (6.62) contain the 2 free parameters c and d. We can eliminate one of them by requiring that the Hamiltonian tries to be
(6.20)
in this basis of states does indeed lead to the correct
i.e. that
(6.24)
he
equal
finite dimensional vector space tr e- iNTft_,1(k,i)1h
spanned by the coherent
V2
V1
p=1
r=1
(Tlp,, I
=
where the coherent state inner
product,
it is
propagator
to the trace of the time evolution
straightforward
product
e-
is
states
(6.24), on
(6.69)
P ,
given by (6.46). With this
(6.62)
the
(6.62),
iNTfI_,1(k,t)1h1Tf 'r)
to show that the states
thonormal basis of the Hilbert space,
operator
define
inner
an or-
224
Equivariant Localization
6.
on
Multiply Connected Phase Spaces
Rvi,ri RP2,r2)
(6.70)
6PI,P2 41,r2
:--
Substituting the identity e-iNTfI_,1(k,t)1h using (6.64), (6.65) and (6.70) we find
=
[U(t, -k)]NTV2/2irv1
into
(6.69)
and
tr
e-iNrft_,, (k,t)lh
=
(_1)UN eiNT(ck+dt)1v1
(6.71)
T
Comparing the result (6.71) with the exact one (6.24), we find that the appearing in the wavefunctions (6.62) can be determined as
pa-
rameter d
dki
=
(UNT
-
2ck)/2t
(6.72)
Another way to eliminate the parameters c and d appearing in (6.62) is to regard the quantum theory as a topological field theory. The above construction produces a Hilbert space H' of holomorphic sections of a complex line bundle L' --+ Z' for each modular parameter -r. If we smoothly vary the com-
gives a family of finite-dimensional Hilbert spaces regarded as forming in this way a holomorphic vector bundle (i.e. one with a holomorphic projection map) over the Teichmiffier space C+ of the torus for which the projective representations above define a canonical projectively-flat connection. This is a typical feature of the Hilbert space for a Schwarz-type topological gauge theory [22]. Equivalent complex structures (i.e. those which generate the same conformal equivalence classes as (6.9)) in the sense of the topological field theory of this Chapter should be regarded as leading to the same quantum theory, and this should be inherent in both the homological partition functions of the Section 6.3 and in the canonical quantum theory above. It can be shown [111] that 2 toroidal complex structures 7,,7*1 E C+ define conformally equivalent metrics (i.e. g, pg,, for related if the if and smooth function some only they are by projective p > 0)
plex
structure r, then this
which
can
be
=
transformation 4 a-r
+,3
vith
Ir
'Y'r +
a,,3, -y, 6
E
Z
,
a6
-,8^1
=
1
(6.73)
C+. The transformations (6.73) generate the action of the group C+ SL(2, Z)/Z2 on C+, which is a discrete subgroup of the M6bius group SL(2, (C)/Z2 of linear fractional transformations of C wherein we take a,,3,'Y, 6 E C in (6.73). We call this discrete group the modular or mapping class group
on
--*
rZi of the Riemann surface Z' and it consists of the discrete automorphisms Z' (i.e. the conformal diffeomorphisms of Z' which aren't connected to
of
the
identity and
complex
cannot be
so
Teichmiiller space
structures
on
Z1,
representing each
as a
is
as
global
flows of vector
fields).
The
action, i.e. the space of inequivalent called the moduli space M-ri =_ C+/r-,i of Z1. group
are conformally isomorphic can be seen intuitively by parallelogram in the complex plane and tracing out this
That the 2 associated tori
transformation.
represented
C+ modulo this
Holomorphic Quantization
6.5
The
and
Non-Symmetric Coadjoint
Orbits
225
topological quantum theory above therefore should also reflect this sort topological invariance on the torus, because it is independent of the
of full
conformal factor W in (6.9). Under the modular transformation an
overall
as
[1091
e
C
d
phase,
(6.73),
it is
possible to show that,
the 1-dimensional Jacobi theta functions in
) [Z 1r)
,
C'
e
) fz'l-r')
(6.59)
V--I-r + J e"'21 (,7,+d) e (1)
=
up to
transform
( ) [Z I _rj C
d
(6.74) where
ZI is the
new
=
01 +T102
(but equivalent) complex
parameters c' and d'
c'
Using (6.74)
=
we
6c
are
-
=
Z/('Y'r + J)
(6.75)
structure defined
by (6-73) and the
new
given by
-yd
find after
-
-yJ12
some
algebra
d'
=
a0/2
ad
that the wavefunctions
form under the modular transformation of
(6.76)
(6.62)
isomorphic complex
trans-
structures
as
Tlp,
r(dc) (z) d
!P ,r
d , ) (Z
(6.77)
d'
with
a'
=
6c
It follows that
-
a
-yd
-
irv1V2'Y6
ad -)3c
-
(6.78)
irvlv2a)3
set of modular invariant wavefunctions
can
exist
only
when
the combination VIV2 is an even integer, in which case the invariance condition 0. For VIV2 an odd integer, we can take c, d E 10, .11, and d requires c =
=
2
then the
holomorphic wavefunctions carry a non-trivial spinor representation of the modular group as defined by (6.77). These choices of c and d correspond to the 4 possible choices of spin structure on the torus [61] (i.e. representations of the 2-dimensional spinor group U(1) in the tangent bundle of Z1) which are determined by the mod 2 cohomology H1 (Z'; Z) 0 Z2 Z2 E) Z2 [41, 61]. This increases the number of basis wavefunctions (6.62) by a factor of 4. It is in this way that one may adjust the parameters c and d so that the wavefunctions (6.62) are modular invariants, as they should be since the topological quantum theory defined by (6.23) is independent of the phase space complex structure. We note also that these specific choices of the parameters in turn then fix the propagation time integers NT by (6.72), so that these topological requirements completely determine the topological quantum field theory in this case. Thus one can remove all apparent ambiguities here and obtain a situation that parallels the topological quantum theories in the simply connected cases, although now the emerging topological and =
226
6.
Equivariant Localization
group theoretical structures ate choices of
path integral
ZZ, (k, f; NT)
ri LZi
2i Im.
I
-r
dt
Phase
Spaces
complicated. With these appropri-
more
(6.69)
then coincides with
dz (t)
d, (t) (2-7r) 2
tEJO,T)
NTh-1 exp
far
are
Multiply Connected
parameter values, the propagator
the coherent state
x
on
[
V1
-rk),
+ ih
2V2
-
(t
-
;r-k)z)
0
I
(6.79) path integral (6.79) models the quantization of some novel, unusual spin system defined by the Hamiltonians (6.20) which are associated with the quantized, non-symmetric coadjoint Lie group orbit U(1) x U(1) S' x S1. This abelian orbit is an unreduced one as it already is its own maximal torus, and we can therefore think of this spin system as 2 independent planar spins each tracing out a circle. The points on this orbit are in one-to-one correspondence with the coherent state representations above of the projective clock algebra (6.52) of the discrete first homology group of the torus. The associated character formula represented by (6.79) gives path integral representations of the homology classes of Z1, in accordance with the fact that it defines a topological quantum field theory, and these localizable quantum systems are exactly solvable via both the functional integral and canonical quantization formalisms, as above. In this latter formalism, the Hilbert space of physical states is finite-dimensional and the basis states carry a non-trivial projective representation of the first homology group of the phase space, in addition to the usual representation of H 2 (M; Z). The coherent state
=
6.6 Generalization to We conclude this
Hyperbolic
Chapter by indicating
Riemann Surfaces
how the above features of equivariant
localization could generalize to the case where the phase space is a hyperbolic Riemann surface [153], although our conclusions are somewhat heuristic and more care
needs to be exercised in order to
Since for h > can
1, _p hcan be regarded
as
by the 2h loops
bi,
be described
ai,
study these examples in detail. together, its homotopy h, where each pair ai, bi 1,
h tori stuck i
=
.
.
.
,
encircle the 2 holes of the i-th torus in the connected of
Zh The .
generalizes
(6.6)
constraint
on
the fundamental
sum
representation
homotopy generators
now
to h
liaibiai 'bi
1
(6.80)
i=1
and
so
the commutator
fundamental group of
a
subgroup of 7r, (_vh) for h > I is non-trivial and the hyperbolic Riemann surface is non-abelian. Its first
6.6 Generalization to
homology
group is
Hyperbolic
given by (3.85), and, using 1,
denote its generators as well by ai, bi, i basis of homology cycles for Zh.
an
Riemann Surfaces
227
abusive notation, we shall a canonical
h and call them
=
According to the Riernann uniformization theorem [111], there are only 3 (compact or non-compact) simply-connected Riemann surfaces the 2-sphere S2, the plane C and the Poincare upper half-plane H2, each equipped with -
their standard metrics
discussed in the last
Chapter. The sphere is its having a unique complex structure), while C is the universal cover of the torus. The hyperbolic plane 7j2 is always the universal cover of a Riemann surface of genus h > 2, which is represented as Zh =,H2/Fh (6.81) own
universal
as
(being simply-connected
cover
and
where Fh 7r, (_rh) is in this context refered to as a discrete FVchsian group. The quotient in (6.81) is by the fixed-point free bi-holomorphic action of =
Fh jj2
on
?j2 The
is
PSL(2,R)
group of
-
transformations
=
as
analytic automorphisms
of the upper
half-plane
SL(2,R)/Z2, the group of projective linear fractional in (6.73) except that now the coefficients a,,3,-Y,J are
7r,(Zh) is taken as a discrete subgroup of this -H2 and the different isomorphism. classes of complex analytic structures of _rh are essentially the different possible classes of discrete subgroups. Note that this generalizes the genus 1 situation above, where the automorphism group of C was the group PSL(2, C) of global conformal taken to be real-valued. Then
PSL(2,R)-action
on
transformations in 2-dimensions and 7rl (Zl)
was
taken to be the lattice sub-
Indeed, it is possible to regard Zh as a 4h-gon in the plane with edges identified appropriately to generate the h 'holes' in Zh. It is difficult to generalize the explicit constructions of the last few Sections because of the complicated, abstract fashion in (6.81) that the complex coordinatization of Zh occurs. For the various ways of describing the Teichmfiller space and Fuchsian groups of hyperbolic Riemann surfaces without the explicit introduction of local coordinates, see [74]. The Teichmiiller space of Zh can be naturally given the geometric structure of a non-compact complex manifold which is homeomorphic: to (CM-3, so that the coordinatization of _Vh is far more intricate for h > 2 because it now involves 3h 3 complex before. I to still it is as as parameters, opposed just Nonetheless, possible to deduce the unique localizable Hamiltonian system on a hyperbolic Riemann surface and deduce some general features of the ensuing topological quantum field theory just as we did above. We choose a complex structure on Zh for which the universal bundle projection in (6.81) is holomorphic (as for the torus), and then the metric gZh induced on Zh by this projection involves a globally-defined conformal group.
-
factor
p
as
in
(6.9)
and
a
constant
negative
curvature Khhler metric
(the
see Section 5.6). The condition now that the hyperbolic Poincar6 metric Killing vectors of this metric be globally-defined on _rh means that they must be single-valued under windings around the canonical homology cycles -
228
Equivariant Localization
6.
ae,
bt I h
E
1
Hi (_Th; Z),
Multiply Connected
on
equivalently
or
dV1'
dV"
Spaces
that
(6.82)
h
0
=
Phase
bt
at
Using this single-valued
condition and the g_Vh (dV,
-)
Killing equations
-iVdg_rh
=
(6.83)
deduce the general form of the Killing Hodge decomposition theorem [22, 32]
Zh. For this,
we can now
vectors of
apply
to the metric-dual 1-form
g(V, -)
the
we
Al _Th'
E
9(V7
dX
+
*d
+
(6.84)
4
where X and are Cl-functions on Zh and Ah is a harmonic solution of the zero-mode Laplace equation for 1-forms,
A'AlAh In the
above,
*
_=
(*d* d + d* d*)Ah
denotes the
i.e.
a
(6.85)
0
Hodge duality operator and,
on
a
general
(M, g),
d-dimensional Riemannian manifold
it encodes the Riemannian geinto the DeRham cohomology. It is defined as the map
ometry directly
*:
which is given
=
1-form,
AkM
Ad-kM
---*
(6.86)
locally by I
*ce
=
(d Xf
-
A1
k)! ...
Vdetg(X)
Ad-kil
...
ik
a l
gj,,X,
...
ik
W
-
(x) dxjl
9jd-kAd-k A
...
A
W
(6.87)
dXjd- k
and satisfies
(-,)(d-I)k on
AkM.
Using
product fm a A *,8 on each vector space AkM. product possible to show that a differential form Ah as harmonic if and only if It defines
an
inner
this inner
above is
(6.88)
it is
dAh
=
d
*
Ah
=
(6.89)
0
and the Hodge decomposition theorem (6.84) (which can be generalized to arbitrary degree differential forms in the general case) implies that the DeRham cohomology groups of M are spanned by a basis of harmonic forms. The Hodge decomposition (6.84) is unique and the components involved there are explicitly given by I X
=
V2Eh
*
d*
g(V,
V2Eh
dg (V,
(6.90)
6.6 Generalization to
Laplacians V2
where the scalar
*d
h
Hyperbolic
(6.90)
d in
*
Riemann Surfaces
229
assumed to have
are
modes removed. The 1-form Ah in (6.84) can be written as a linear combination of basis elements of the DeRham cohomology group H'(Zh; R)
their
zero
-
According choose which
an
are
to the
Poinear6-Hodge duality theorem [32],
orthonormal basis of harmonic 1-forms
f at
f at, &1h
homology
Poincar6-dual to the chosen canonical
H, (M; Z) above,
we can 1
basis
in
particular
H'(Z h; R) f at, bilht= 1 E
E
i.e.
at,
=
J 0j,
=
6et,
at,
=
bt
bt
f3t,
=
(6.91)
0
at
We remark here that the local parts of the decomposition (6.84) simply form the decomposition of the vector gZh (V7 -) into its curl-free, longitudinal and
divergence-free,
transverse
pieces
as
VZhX
+
VZh
We
can now
write the
general
. The harmonic part
x
Ah accounts for the fact that this 1-form may sit in cohomology class of H1 (Zh; R).
a
non-trivial DeRham
form of the isometries of Zh
_r hinherits 3 local isometries via the bundle
in
As before, (6.81) from the .
projection maximally symmetric Poincar6 upper half-plane. However, only the 2 quasitranslations on H2 become global isometries of Zh and they can be expressed in terms of the canonical homology basis using the above relations. This global isometry condition along with (6.82) and (6.83) imply that Hodge decomposition (6.84) of the metric-dual 1-form to the Hamiltonian vector field VZh is simply given by its harmonic part which can be written as ,
h
E (Vl'ae + 4,8f)
gZh (VZh,
(6.92)
t=1
decomposition (6.92) is the generalization of (6.18). Indeed, on the torus we can identify the canonical harmonic forms above as a do, /27r and 0 d02/2-7r. The Killing vectors dual to (6.92) generate translations along the homology cycles of _Th and the isometry group of Zh is 11 2hJ U(1). 0 on the symplectic 2-form of _Th The usual equivariance condition 'C Vv CO The harmonic
=
=
,
=
h
now
becomes h
div_,h where
O(x)
d
is the C'-function
(6.93) implies
that it is constant
Vl at
(D on
on
h
+
V2,Ot
defined
Zh7 just
as
by w,,,(x) in
(6.93)
0
=
CD(x),E1,
and
(6.19).
Integrating up the Hamiltonian equations we see therefore that the unique equivariant (Darboux) Hamiltonians have the form
H_r h (X)
=
E I (Mae + W2,3e) 1
t=1C.
(6.94)
230
6.
Equivariant Localization
hle
on
Multiply Connected
Phase
Spaces
real-valued constants and Q, C Zh is a simple curve from basepoint to x. The Hamiltonian (6.94) is multi-valued because it depends explicitly on the particular representatives at,0i of the DeRham 2h R As before, single-valuedness of the cohomology classes in H'(Zh; R) time-evolution operator requires that V W h for some W E Z and h E R, A A A and the propagation times are again the discrete intervals T NTh-'. Thus the Hamiltonian (6.94) represents the windings around the non-trivial homology cycles of'Eh and the partition function defines a topological quantum field theory which again represents the homology classes of Z h through a family of homomorphisms from 2h Z into U(j)(92h Again, the partition function path integral should be properly defined in the homologically-invariant form (6.27) to make the usual quantities appearing in the associated action S welldefined by restricting the functional integrations to homotopically equivalent loops. We note that again the general conformal factor involved in the metric gZh obeys Riemannian restrictions from the Gauss-Bonnet-Chern theorem where
.
are
fixed
some
=
.
=
7
=
.
and
a
ume
of
volume constraint similar to those in Section 6.2 above. When the vol-
parameter is quantized
physical
states will be
as
(6.55),
in
(VIV2)3h-3
we
expect that the Hilbert space
(one copy of the genus I Hilbert spaces for each of the 3h 3 modular degrees of freedom in this case) and the coherent state wavefunctions, which can be expressed in terms of D 3 dimensional Jacobi theta functions (6.59), will in addition 3h carry dimensional
-
=
-
(V2)3h-3
explicit
dimensional projective representation of the discrete first homol_ph (i.e. of the equivariant localization constraint algebra). The proofs of all of the above facts appear to be difficult, because of the
lack of
complex coordinatization
a
ogy group of
for these manifolds which is
the definition of coherent states associated with the
n2h i=1 U(1)
=
S' 112h i=1
on
the non-symmetric space
isometry Zh (S1 X =
required for group action
Sl)#h.
Thus the general feature of abelian equivariant localization of path integrals on multiply connected compact Riemann surfaces is that it leads to a
topological quantum theory whose
associated topologically invariant partition function represents the non-trivial homology classes of the phase space. The coherent states in the finite-dimensional Hilbert space also carry a multidimensional representation of the discrete first homology group, and the lo-
calizable Hamiltonians
these
phases spaces are rather unusual and even simply-connected cases. The invariant symplectic 2-forms in these cases are non-trivial elements of H 2(Zh; Z) Z, as in the maximally-symmetric cases, and it is essentially the global topological features of these multiply-connected phase spaces which leads to these rather more
on
restricted than in the
=
severe
restrictions. The coherent state
quantization of these systems shows
that the path
integral describes the coadjoint orbit quantization of an unusual spin system described by the Riemann surface. These spin systems are exactly solvable both from the point of view of path integral quantization on the loop space and of canonical holomorphic quantization in the Schr6dinger polarization. The localizable systems that
one
obtains in these
cases are
rather
6.6 Generalization to
Hyperbolic
Riemann Surfaces
231
trivial in appearence and are associated with abelian isometry groups acting on the phase spaces. However, these quantum theories probe deep geometric and
topological
features of the
phase
spaces, such
as
their
complex algebraic
geometry and their homology. This is in contrast with the topological quanwe found in the simply-connected cases, where at best topological path integral could only represent the possible non-trivial co2 homology classes in H (M; Z). It is not completely clear though how these path integral representations correspond to analogs of the standard character formulas on homogeneous symplectic manifolds which are associated with semi-simple Lie groups, since, for instance, the usual Khhler structure between the Riemannian and symplectic geometries is absent in these non-
tum field theories that
the
symmetric
cases.
Beyond
7.
In this
Chapter
ization
[1571.
the Semi-Classical
we
shall examine
We return to the
3 and consider
a
Approximation
a different approach to the problem of localgeneral finite-dimensional analysis of Chapter
Hamiltonian system whose Hamiltonian function is a Morse we will construct the full -Lexpansion for the Classical
function'. From this
T
partition function, as we described briefly in Section 3.3. A proper covariantization of this expansion will then allow us to determine somewhat general geometrical characteristics of dynamical systems whose partition functions localize, which in this context will be the vanishing of all terms in the perturbative loop expansion beyond I-loop order. The possible advantages of this analysis are numerous. For instance, we can analyse the fundamental isometry condition required for equivariant localization and see more precisely what mechanism or symmetry makes the higher-order terms disappear. This could then expand the set of localizable systems beyond the ones we have
already
predicted from localization theory, and at the same geometrical structures of the phase space probe deeper the whole thus or providing richer examples of topological dynamical system field theories. Indeed we shall find some noteworthy geometrical significances of when a partition function is given exactly by its semi-classical approximation as well as new geometric criteria for localization which expand the encountered that
are
into the
time
previous isometry conditions.
approach to the Duistermaat-Heckman integration formula using the perturbative loop-expansion has been discussed in a different context recently in [174] where the classical partition function was evaluated for the dynamical system describing the kinematics of thin vortex tubes in a 3-dimensional fluid This
whose Hamiltonian is similar to that considered at the end of Section 5.8 for
geodesic motion on group manifolds. For such fluid mechanics problems, the phase space M is neither finite-dimensional nor compact and the Hamiltonian flows need not be periodic, but the dynamical system admits an infinite sequence of constants of motion which
localizable and
one
[1741.
are
in involution
The standard localization
needs to resort to
The extension to
an
analysis
so
that it should be
analyses therefore do
not
apply
of the sort which will follow here. We
degenerate Hamiltonians
is
fairly straightforward.
In what fol-
lows all statements made concerning the structure of the discrete critical Point set
Mv of H will then apply
to the full critical submanifold.
R. J. Szabo: LNPm 63, pp. 233 - 268, 2000 © Springer-Verlag Berlin Heidelberg 2000
234
Beyond the Semi-Classical Approximation
7.
shall indeed find extensions of the localization formalisms which such
cover
certain
cases.
Recalling that the isometry condition can always be satisfied at least locally on M, we then present some ideas towards developing a novel geometric method for systematically constructing corrections to the DuistermaatHeckman formula. Given that a particular system does not localize, the idea is that we can "localize" in local neighbourhoods on M where the Killing equation can be satisfied. The correction terms are then picked up when these open sets are patched back together on the manifold, as then there are non-trivial singular contributions to the usual I-loop term owing to the fact that the Lie derived metric tensor cannot be defined globally in a smooth way over the entire manifold M. Recalling from Section 3.6 that the properties of such a metric tensor are intimately related to the integrability properties of the dynamical system, we can explore the integrability problem again in a (different) geometric setting now by closely examining these correction terms. This will provide a highly non-trivial geometric classification of the localizability of a dynamical system which is related to the homology of M, the integrability of the dynamical system, and is moreover completely consistent witl Kirwan's theorem. Although these ideas are not yet fully developed, they do provide a first step to a full analysis of corrections to path integral localization formulas
(e.g.
corrections to the WKB
approximation),
and to
uncovering systematically the reasons why these approximations aren't exact for certain dynamical systems [157]. The generalizations of these ideas to path integrals are not yet known, but we discuss the situation somewhat
heuristically
Chapter.
at the end of this
7.1 Geometrical Characterizations
of the
Loop Expansion
Throughout
this
Chapter
the Hamiltonian H is manifold M. For manifolds with
a
return to the situation of Section 3.3 where
we
Morse function
now we assume
boundary.
We
that 49M
now
phase series whose construction we first expand the C'-function H in point p E MV in a Taylor series
H(x)
=
H(p)
+
on a =
0,
(usually compact) symplectic but later
we
shall also consider
explicitly work out the full stationary briefly outlined in Section 3.3 [72]. We a neighbourhood U. of a given critical
+ g(x; p) 7i(p),,,x1'x'/2 P P
x
E
U,,
(7.1)
where xp x p E Up are the fluctuation modes about the extrema of H and g(x;p) is the Gaussian deviation of H(x) in the neighbourhood Up (i.e. =
-
all terms in the the
symplectic
Taylor
series
beyond quadratic order). The determinant (3.52) is similarly expanded in Up
2-form which appears in
of as
7.1 Geometrical Characterizations of the 00
VI'det (x) w
V-det (p)
-
=
+
w
1
!
where
x
/-t 1
ii XP
k=1
...
XAk o91,1
...
P
Loop Expansion
c9/,, V
_det (x) w
IX=P
235
(7.2)
U,.
E
We substitute
(7.1)
and
(7.2)
into
(3.52), expand the exponential function
there in powers of the Gaussian deviation function, and then integrate over each of the neighbourhoods Up R 2n. In this way we arrive at a series -
expansion of (3.52) for large-T in terms of Gaussian moment integrals over the P P fluctuations x., with Gaussian weight e iTW(V),,x1'x'12 associated with each , X11k) open neighbourhood Up for p E MV. The Gaussian moments (xAl ...
be found from the Gaussian integration formula (1.2) in the usual way '9 '9 to both sides of (1 -2) and then setting by applying the operator ax" a'\Ik all the A's equal to 0. The odd-order moments vanish, since these integrands can
-
-
-
odd functions, and the 2k-th order moment contributes
are
0(11Tn+k) Rearranging
terms
.
carefully, taking
of the Hessian at each critical point, and noting that for will localize around each of the disjoint neighbourhoods
standard
stationary-phase expansion 21ri
Z(T)
T
)n
a
term of order
into account the
signature
large-T the integral U, we arrive at the
2
w
i)
A (P)
eiTH(p)
pEMv
At(p)
E (-2T)t
(7.3)
i=O
where
A (p)
1 =
,,/det'H (p) x
and
7i(x)11v
jE
=0
(-1)3 2ij! (f + j)!
(g(x;p)jV1detw(x))
is the matrix inverse of
'H(x)Av
N(P) AV19AC9J,)t+3*
(7.4)
Jx=P -
stationary-phase series diverges (e.g. applying Kirwan's theorem in appropriate instances), then (7.3) is to be understood formally as an asymptotic expansion order by order in 1. Borrowing terminology from quantum T field theory, we shall refer to the series (7.3) as the loop-expansion of this zero-dimensional quantum field theory, because each of the 2f + I terms in (i.e. a loop) as(7.4) can be understood from pairing fluctuation modes x"xv p P sociated with each derivative operator there. Indeed, the expansion (7.3),(7.4) is just the finite-dimensional version of the perturbation expansion (for largeT) in quantum field theory. We shall call the O(11T1+i) contribution to the If the
.
series
(7.3)
the
(t + 1)-loop
dynamical 2
See
(72]
for the
function.
term.
shall be interested in extracting information about the system under consideration from the loop-expansion with the hope
In this Section
we
generalization
of this formula to the
case
where H is
a
degenerate
236
of
Beyond the Semi-Classical Approximation
7.
vanishing or non-vanishing of the k-loop congeometrical and topological features of the experience now with the Duistermaat-Heckman theo-
to understand the
being able
tributions for k > I in terms of
phase
space. Given
will
our
requirements on the flows of the Hamiltonian vector quite arbitrary for now. When these orbits describe tori, we already have a thorough understanding of the localization in terms of equivariant cohomology, and we shall therefore look at dynamical systems which do not necessarily obey this requirement. Thus any classification that we obtain below that is described solely by the vanishing of higher-loop contributions will for the most part be of a different geometrical nature than the situation that prevails in Duistermaat-Heckman localization. This then has the possibility of expanding the cohomological symmetries usually resposible rem,
we
any
remove
field and leave these
as
for localization.
The perturbative series before
we can
put
it to
(7.3), however, The
use.
must be
appropriately modified
(3.52) is invariant under (i.e. Z(T) is manifestly a be reflected order by order in the 1IT-
partition
function
arbitrary changes topological invariant), and this should expansion (7.3). This is explicitly observed in the lowest-order DuistermaatHeckman term Ao(p) above, but the higher-order terms (7.4) in the loop expansion are not manifestly scalar quantities under local diffeomorphisms of the coordinates. This is a result of having to pick local coordinates on M 2n At each order of the to carry out explicitly the Gaussian integrations in R coordinate should have a we independent manifestly quantity, '-expansion smooth
of local coordinates
M
on
.
i.e.
scalar. To write the contributions
a
manifestly a
invariant under local
Christoffel connection
r;, ,
(7-4)
in such
diffeomorphisms
of
a
M,
fashion
we
so
as
to be
have to introduce
of the tangent bundle of M which makes the
derivative operators appearing in (7.4) manifestly covariant objects, i.e. d + r. Because dH(p) write them in terms of covariant derivatives V =
we =
0
point p E Mv, the Hessian evaluated at a critical point is automatically covariant, Le. VVH(p) H(p). This process, which we shall call 'covariantization', will then ensure that each term (7.4) is manifestly a scalar. We note that the Morse index of any critical point is a topological at
a
critical
=
invariant in this
First,
we
sense.
cycle
out the
symplectic factors
V
Ae (p)
=
Ao (p)
1.y
1: 2ij!
+
i=O
j)!
in
(7.4)
to
get
(H(p)"'D,,EI,)'+i g(x; p)j
(7.5) X=P
where
Ao(p) is the Duistermaat-Heckman
=
(1-loop) D
=
(T __det(p) jet
contribution to
d+-y
(7.6)
(p)
(7.3),
and
(7.7)
7.1 Geometrical Characterizations of the
where
we
Loop Expansion
237
have introduced the one-component connection
hL'dhL
-y
(7.8)
and
-,Idetw(x)
hL(-'r) is the Liouville volume
(7.9)
density. The derivative operator D transforms like diffeomorphisms x --+ x'(x) of M,
an
abelian gauge connection under local A
Dy W
)
I DI,(x)
=
A'(x) [E),(x') A
+ tr
A-l(x')o9' A(x')]
(7.10)
where
A-1(x)
=
I
ax"
I
GL(2n, R)
E
(7.11)
is the induced
change of basis transformation on the tangent bundle. 1. We expand out the example, consider the expression (7.5) for t 3 terms there in higher-order derivatives of H and the connection -/, noting that only third- and higher-order derivatives of g(x; p) when evaluated at x p are non-vanishing. After some algebra, we arrive at For
=
=
A 1 (P)
Ao(p) 2
li(p) 'v
D,(x)7,(x)
+4-y,,(x)a,\a,apH(x)
+
-
12
4
o9,, a,\ o9, o9p H (x)
[3a,,a,\,9,,H(x)o9pao9,oH(x)
(7.12)
+2a,,i9,\49,,H(x),O,,api9,oH(x)]) I Jx=P readily checked, after some algebra, that this expression is indeed indiffeomorphisms of M. To manifestly covariantize it, we introduce an arbitrary torsion-free connection r ',\, of the tangent bundle TM.
It is
variant under local
For now, we need not assume that r is the Levi-Civita connection associated a Riemannian metric on M. Indeed, as the original dynamical problem
with
is defined
only
symplectic geometry, not a Riemannian geommanifestly covariant in its own right without reference to any geometry that is external to the problem. All that is required is some connection that specifies parallel transport along the fibers in terms of
etry, the expression
(7.12)
a
should be
of the tangent bundle and allows
us
to extend derivatives of
quantities
to
an
neighbourhood, rather than just at a point, in a covariant; way. The Hessian of H can be written in terms of this connection and the
entire
associated covariant derivative
7i(x),,, and, using d r, appearing in (7.12) in
=
as
V,,V,,H(x)
we can
+
rA (x)49,\H(x) Mv
(7.13)
write the third and fourth order derivatives
terms of V and r
by taking derivatives
of
(7.13).
238
7.
Beyond the Semi-Classical Approximation
Substituting
these
complicated derivative expressions into (7.12) and using 0 on MV, after a long and quite tedious calculation we arrive at a manifestly covariant and coordinate-independent expression for the 2-loop correction, the fact that dH
Aj(p)
=
H(P)"PH(PY113
7AW,
8
3
[3V,,V,V,\H(x)V,,VoVpH(x)
+2V,,V,\V,,H(x)V,VpV,3H(x)] +4
(VI, + A,,
-
-
H(p)APVpV,,V,V,\H(x)
X=P
H(p),\PVpV,,V,\H(x)) A, (x) + R,(r) I
(7-14) where
R,,, (r)
= -
R\gAv
-
aV r,\, /4"
-
a,\r,\ ttv
+
r, r,',A rc,A rc,, ,lov
(symmetric) Ricci curvature tensor of F and =A, (x)dxA with the local components
is the
we
(7-15)
have introduced the
1-form A
,6,, (x) It is
intriguing that the
=
-
r,, ,\ (x)
=
V,, log hL (X)
covariantization of the
replacing ordinary
volves
y,, (x)
(7.16)
2-loop expression simply inones V, non-covariant
derivatives d with covariant
connection terms -y with the 1-form A, and then the remainder terms from this process are simply determined by the curvature of the Christoffel connection 1' which realizes the covariantization. Note that if r is in addition chosen
as
the Levi-Civita connection compatible with a metric g, i.e. 0,, log vqe-tg and the 1-form components (7.16) become
Vg
=
0, then P\ JLA
A ,
=
a,, log Vdet(g-1 w).
=
-
The covariantization of the
higher-loop terms is hopelessly complicated. if, however, the 2-loop, correction Al (x) vanishes in an entire neighbourhood Up C M of each critical point p, then this is enough to imply the vanishing of the corrections to the Duistermaat-Heckman formula to all orders in the loop-expansion. To see this, we exploit the topological invariance of (7.14) and apply the Morse lemma [111] to the correction terms (7.5). This says that there exists a sufficiently small neighbourhood Up about each critical point p in which the Morse function H looks like a "harmonic oscillator", We note that
H(x) so
=
H(p) _(XI)2 _(X2)2_..._(X,\(p))2+(X,X(p)+1)2+...+(X2n)2
,
Up (7.17)
x
E
that the critical point p is at x 0 in this open set in M We shall call these coordinates', and this result simply means that the symmetric ma=
.
'harmonic trix
H(x)
critical nates
can be diagonalized constantly in point. Given that the quantity (7.5)
(although
not
manifestly),
we can
an
neighbourhood of the independent of coordi-
entire
must be
evaluate it in
a
harmonic coordinate
7.1 Geometrical Characterizations of the
system. Then the Gaussian deviation function the the
neighbourhood Up and only series (7.3) is simply 2-7ri
Z(T)
T
)n
E (-i))(P)
the j
e
=
iTH (P)
g (x;
p)
Loop Expansion vanishes
identically
Ao (p)
(
in
(7.5).
Thus
neighbourhood
of the
0 term contributes to
PEMv
239
e-
(7.18) It follows that if the
2-loop term vanishes in the (and not just at x p), i.e.
critical point p
=
'H(p)P'D,,(x)D,(x) then,
as
all
entire
-=
0
for
x
E
Up
(7.19)
higher-loop terms in these coordinates can be written as derivative on the 2-loop contribution A, (x) as prescribed by (7.18), all
operators acting
corrections to the semi-classical
approximation vanish. In the
case
of the
Duistermaat-Heckman theorem, it is for this reason that the Lie derivative condition Cvg 0 is generally required to hold globally on M. In general, =
not the vanishing of A, (p) implies the vanishing of all loop clear, because there is a large ambiguity in the structure of a function in a neighbourhood of MV which vanishes at each critical point p (any functional of VH will do). The above vanishing property in an entire neighbourhood therefore need not be true. It is hard to imagine though that the vanishing of the 2-loop correction term would not imply the vanishing of all higher-orders, because then there would be an infinite set of conditions that a dynamical system would have to obey in order for its partition function to be WKB-exact. This would then greatly limit the possibilities for localization. In any case, from the point of view of localization, we can consider the vanishing of the 2-loop contribution in (7.14) as an infinitesimal Duistermaat-
though,
whether
or
orders is not that
Heckman localization of the partition function. The expression (7.14) in genextremely complicated. However, besides being manifestly indepen-
eral is
dent of the choice of
coordinates, (7.14) is independent of the chosen connecV, because by construction it simply reduces to the original connectionindependent term (7.12). We can exploit this degree of freedom by choosing a connection that simplifies the correction (7.14) to a form that is amenable to explicit analysis. We shall now describe 2 geometrical characteristics of the loop-expansion above which can be used to classify the localizable or non-localizable dynamical systems 1157]. The first such general geometric localization symmetry is a symmetry implied by (7.14) between Hessian and metric tensors, i.e. that the Hessian essentially defines a metric on M. This is evident in the correction term tion
(7.14), form
where the inverse Hessians contract with the other tensorial terms to
scalars,
i.e. the Hessians in that expression act just like metrics. This suggests that the non-degenerate Hessian of H could be used to define a
metric which is
compatible with the
connection V used in
(7.14).
This in
7.
240
Beyond
the Semi-Classical
Approximation
globally on the manifold M, because the signature of 'H(x) varies over M in general, but for a C' Hamiltonian H it can at least be done locally in a sufficiently small neighbourhood surrounding each critical point. For now, we concentrate on the case of a 2-dimensional phase space. We define a Riemannian metric tensor g that is proportional to the covariant Hessian in the neighbourhood Up of each critical point p, general
cannot be done
where
!9(x)
is
some
globally-defined
(7.20)
=!Pg
VVH
C'-function
on
M, and for which the
connection F used in the covariant derivatives V is the Levi-Civita connection
for g. This
means
coupled
the
that, given
!g(x)g,,,(x) rA
MV
consistently
a
Hamiltonian H
system of non-linear
for g
2
=
partial
0,,OH(x)
gAP Oitgpv
+
M,
on
we
try
to
locally
solve
differential equations -
r\ (x)a,\H(x) MV
CUP/I
-
(7.21)
19p9tiv)
(or F).
This sort of "metric ansatz" may seem somewhat peculiar, and indeed impossible to solve in the general case. The covariant constancy condition on g in
(7.21) implies
that
')/,!g
c
where R
identity
=
gII`R,_,,
=
R,\a,\H M
=
is the scalar curvature of g.
for the Riemann curvature tensor R
V,,V,,V,\H
=
(7.22)
Ro9lH
V,V/-,V,\H
(7.22) =
follows from the
dF + F A
+ RP
A/m/
defining
-P,
VP H
(7.23)
H, (7.22) determines g locally in terms of g. This means that the can be written as an equation for the associated connection coefficients F\
Given
above ansatz
/4V
V,\V,,V,H The existence of
a
local solution to
=
Rl,,V,\H
(7.24)
'almost' all of the time
(7.24) can now
set of differential
equations in local isothermal coordinates (5.49) for the connection.P above [157]. Notice, in particular, that if the metric (7-20) actually has a constant curvature R, then the equation (7.22) can be integrated in each neighbourhood U., to give be
argued by analysing this
!P(x)
=
Co
+ R
-
H(x)
(7.25)
examples and other evidences for this sort of geometric structure throughout this Chapter. The main advantage of using the inductively-defined metric in (7.21) is that it allows a relatively straightforward analysis of the 2-loop, correction We shall
see
7.1 Geometrical Characterizations of the
Loop Expansion
terms to the Duistermaat-Heckman formula. With this
plies that and hence
definition, (7.22) impoint p E Mv,
the first order derivatives of 9 vanish at each critical so
do all third order covariant derivatives of H in
(7.14).
The fourth
order covariant derivatives contribute curvature terms
to
which
in
are
241
then cancelled
The final result is
by
according already present
the curvature tensor
(7.24), (7.14).
expression involving only the Liouville and Levi-Civita expressed in terms of the 1-form. A, which after some algebra we be written in the simple form an
connections
find
can
Aj(p) Requiring
9" IVA
=
29
+
AA W I A, W
this correction term to vanish in
an
entire
JX=P
(7.26)
neighbourhood
of each
critical point implies, from the definition (7.16), that the connection y ciated with the symplectic structure coincides with the connection r
asso-
ciated with the Riemannian structure which solves the relation
Thus
(7.21).
asso-
the components of the symplectic 2-form w and the metric tensor g are proportional to each other in local complex isothermal coordinates for g. The
proportionality
factor must be
a
constant
so
existence of local Darboux coordinates for
that this be consistent with the
[157].
In other
words, w and g together locally phase space M. Conversely, suppose that r is the Levi-Civita connection associated to some generic, globally-defined metric tensor g on M, and consider the rank define
(1,1)
a
Khhler structure
w
on
the
tensor field
J1,
dettwgg"Aw' et w
:::
A
(7.27)
(7.27) defines a linear isomorphism J general, if such a linear transformation J exists then it is called an almost complex structure of the manifold M [41, 61]. This means that there is a local basis of tangent vectors in which the only non-vanishing components of J are given by (5.9), so that there is "almost" a separation of the tangent bundle into holomorphic and anti-holomorphic components. However, an almost complex -structure does not necessarily lead to a complex structure there are certain sufficiency requirements to be met before J can be used to define local complex coordinates in which the overlap transition functions can be taken to be holomorphic [22]. One such case is when J is covariantly constant, VJ 0 actually this condition only ensures that a sub-collection of subsets of the differentiable structure determine a local complex structure (but recall that any Riemann surface is a complex manifold). Again in 2-dimensions this means that then VW 0 In
2-dimensions, it is easily --+ TM satisfying j2
TM
seen
=
that
-1. In
-
=
-
=
and the pair (g, w) define a Kdhler structure on M (again note that any 2dimensional symplectic manifold is automatically a Khhler manifold for some metric defined
Given these
by w). facts,
suppose
now
that g and
w
the 2n-dimensional manifold M with respect to
define an
a
Kdhler structure
almost
complex
on
structure
242
J,
Beyond
7.
i.e. det w
det g, g is Hermitian with respect to
=
9tiv and
w
means
Approximation
the Semi-Classical
=
JA 9AP JP A
(7.28)
1
V
by (7.27).
is determined from 9
J,
In the local coordinates
(5.9),
and w,,F, 0, gir, -ig,,p. g*,, gpr, F/ action of the Hamiltonian vector field =
=
(,Cvg),,,,
=
=
this
we encountered before, i.e. g,', In this case, the flows of g under the
the usual Kdhler conditions that
=
V,
gj,,xV,,w-XPo9pH + g,,.xVtwAPapH + wAP(gj,.XV,,VpH + g,,.XVt'VpH) (7.29)
can
be written
using the almost complex
,Cvg Thus if V is
a
global Killing
=
structure
as
the anti-commutator
[VVH, J]+
vector of
a
(7.30)
Kiffiler metric
covariant Hessian of H is also Hermitian with respect to
(2,0)
metric
by
on as
M, then the
in
(7.28).
Since
essentially
the unique Hermitian tensor, it follows that the covariant Hessian is related to the Kiffiler a transformation of the form
the Kiffiler metric of rank
K6hler manifold is
J,
a
VI,V,H
=
KP K\g,\ P it
(7.31)
V
non-singular (1,I) tensor which commutes with J. In 2Hermiticity conditions imply that both the Hessian and g dimensions, have only 1 degree of freedom and (7.31) gets replaced by the much simpler condition (7.21). From the fundamental equivariant localization principle we know that this implies the vanishing of the 2-loop correction term, i.e. the Duistermaat-Heckman theorem. Indeed, from the analysis of the last 2 Chapters we have seen that most of the localizable examples fall into these Kiffiler-type, scenarios. Notice that the covariant Hessian determined from the Hamiltonian equations is
where K is
some
the
V,hV,,H so
=
VAVAwv,\
that in the Kdhler case, when Vw
=
+
0, the proportionality function 9 is
determined in terms of the Riemann moment map AV
9(x) On
a
=
(7.32)
wv,\Vl-IV,
Vd_et;1_tv(x)
=
VV
as
(7.33)
homogeneous Kdhler manifold, when g is integrated to be (7.25), this generalizes that observed for the height function of the sphere in
relation
Section 5.5.
particularly interesting here is that conversely the localization partition function determines this sort of Kdhler structure on M. The Lie derivative (7.29) of the metric (7.21) is easily seen to be zero in a neighbourhood of the critical point. Conversely, if the Lie derivative of the What is
of the
7.1 Geometrical Characterizations of the
Loop Expansion
243
metric in (7-21) vanishes on M, then it induces a Khhler structure (i.e. VW 0). Recalling from Section 3.3 the proof of the Duistermaat-Heckman theorem =
using solely symplectic geometry arguments, we see that the main feature of possibility of simultaneously choosing harmonic and Darboux coordinates. This same feature occurs similarly above, when we map onto local Darboux coordinates [157]. The new insight gained here is the geometric manner in which this occurs the vanishing of the loop expansion beyond leading order gives the dynamical system a local Kdhler structure (see (7.19)). Whether or not this extends to a global Kdhler geometry depends on many things. If the topology of M allows this to be globally extended away from MV, then the Riemannian geometry so introduced induces a global Kiffiler structure with respect to the canonical symplectic structure of M 3. Furthermore, the coefficient function 9(x) in (7.20) must be so that the the localization is the
-
metric defined
by that equation
has
signature on the whole of M. impossible to choose the function g in (7.20) such that, say, g has a uniform Euclidean signature on the whole of M. But if the correction terms above vanish, then Kirwan's theorem implies that H has only even Morse indices and it may be possible to extend this geometry globally. In this way, the examination of the vanishing of the loop expansion beyond leading order gives insights into some novel geometrical structures on the phase space representing symmetries of the localization. Moreover, if such a metric is globally-defined on M, then Lvg 0, and If H has odd Morse indices
A(p),
a
constant
then it is
=
these classes of localizable systems fall into the studied before.
same
framework
as
those
we
The second general geometric symmetry of the loop expansion that we wish to point out here is based on the observation that the symplectic connection (7.8) is reminescent of the connection that appears when one constructs the L
Fubini-Study
-*
M
bundle of this
[41].
over
If
metric
we
using the geometry of
choose such
a
line bundle
M and view the Liouville
bundle,
then from it
one can
as
a
density (7.9)
construct
a
where
we
-i(o9 + 5)-yg
have restricted to the
nents of the connection
is the
=
Fubini-Study
The existence of
(7.8).
=
as a
metric in the fibers
Kdhler structure
the curvature 2-forms of the associated connections Q
holomorphic line bundle symplectic line
the standard
(7.8),
on
-05 log hL
(7.34)
holomorphic and anti-holomorphic compoinstance, the symplectic structure (5.208)
For
metric associated with the natural line bundle L
an
almost
M from
i.e.
complex
structure J for which the
-->
CpN
symplectic 2-form
is Hermitian and for which the associated Khhler metric J w is positiveg definite is not really an issue for a symplectic: manifold [171]. Such a J always w
exists
=
(and
is
unique
up to
homotopy)
-
because the Siegal upper-half plane is
contractible. Thus the existence of a Khhler structure for which A is not
a
problem. However, for the Killing equation for
g
=
J
-
W
must be invariant under the flows of the Hamiltonian vector field
=
to
V,
V log hL 0 hold, J itself =
i.e.
Cv J
=
0.
244
which is a
Beyond the Semi-Classical Approximation
7.
a
point p
sub-bundle of the trivial bundle (CpN
E
CpN
is
just the
set of
(zl,
points
.
.
.
(CN+l, i.e. the fiber of L over ZN+1) ECN+1 which belong
X ,
to the line p. In that case, the natural fiber metric induced
ical
com p lex
Euclidean metric of
(CN+l
is
h(p; zl,..., ZN+I)
by the =
canon-
EN+1 'U=1 JZttJ2
symplectic reduction of h as in (7.34). This holomorphic line bundle once a fiber metric has been chosen. In the general case, if M itself is already a Kdhler manifold, then whether or not (7.34) agrees with the original Kiffiler 2-form will depend on where these 2-forms sit in the DeRham cohomology of M. If we further adjust the Christoffel connection.P so that it is related to the Liouville connection by the boundary condition FA i.e. A 0, then the correction term (7.14) will involve only derivatives of the Hamiltonian, but now the vanishing of the correction term can be related to the geometry of a and
(5.208)
is determined
generalizes
construction
the
as
to any
=
line bundle L
M.
--+
a general geometric interpretation of loop expansion. The above discussion shows what deep structures could be uncovered from further development of such analyses. It is in this way that the localization of the partition function probes the geometry and topology of the phase space M and thus can lead to interesting topological quantum field theories.
This is
far
as
as we
shall go with
localization from the covariant
7.2 Conformal and Geodetic Localization
Symmetries
geometric symmetries which a dynamical system and a which extend the fundamental symmetry requirement of the localization theIn this Section
orems we
tensor
we
shall examine
some
alternative
localization of the partition function of
lead to
on
on [84, 134]. Let g be a globally-defined metric and consider its flows under the Hamiltonian vector field V.
encountered earlier
M,
Instead of the usual assumption that V be an infinitesimal isometry generator for g, we weaken this requirement and assume that instead V is globally an
infinitesimal generator of
conformal transformations with respect LVg
where
Y(x)
is
some
Cl-function
=
on
to g, i.e.
(7.35)
Yg
M.
Intuitively,
this
means
that the
diffbomorphisms generated by V preserve angles in the space, but not lengths. The function Y can be explicitly determined by contracting both sides of
(7.35)
with
g-'
to
get T
which This
we
note vanishes
implies,
which
case
in
V is
=
on
particular, an
V,,Vl'/n
=
V,,w1",9,H1n
(7.36)
the critical point set MV of the Hamiltonian H. that either Y =- 0 almost everywhere on M (in
isometry of g)
or
Y(x)
is
a
non-constant function
on
M
7.2 Conformal and Geodetic Localization
Symmetries
245
corresponding to non-homothetic or 'special conformal' transformations (i.e. constant rescalings, or dilatations, of g are not possible under the flow of a Hamiltonian vector field). Ordinary Killing vector fields in this context arise as those which are covariantly divergence-free, tr /,tv VtVA 0. We shall show that this conformal symmetry requirement on the dynamical system also leads to the Duistermaat-Heckman integration formula. First, we establish this at the level of the perturbative loop expansion of the last Section by showing that the (infinitesimal) corrections to the I-loop term in the stationary phase series vanish. For simplicity we restrict our attention here to the case of n I degree of freedom. The extension to arbitrary freedom of is degrees immediate, and indeed we shall shortly see how the condition (7.35) explicitly implies the vanishing of the correction terms to the Duistermaat-Heckman formula in arbitrary dimensions and in a much more general framework. We insert everywhere into the covariant connectionindependent expression (7.14) the covariant derivative V associated with g in (7.35). Using the covariant Hessian (7.32) and the conformal Killing equation =
=
=
(7.35)
we can
field. Notice
solve for the covariant derivatives of the Hamiltonian vector
that,
in contrast to the
ordinary Killing equations,
one
of the 3
components of the conformal Killing equation (7.35) will be an identity since one of the Killing equations tells us that V is covariantly divergence-free with
respect
to g
(symmetric)
(see (5.51)). In 2-dimensions, after Hessian (7.32) can be written as
VI,V,H where
we
=
-S? Z gA, +
+
algebra we
V,H V/, log
find that the
0) /2
(7.37)
have introduced the Cl-functions
Z(X)
det
VV(x)
-
(VxVA (x)/2)2
The covariant derivatives
(7.37)
(Vj_tH V, log S?
some
appearing
0
in
(7.14)
Ldet W(X! rEet
are now
et 6 e
g (x)
(7.38)
easily found
from
to be
VpV/,V,H
J?Z +
Vp(f2Z)V,,V,H
+
1V,, log 12VpV,H
2
1V, log f2VpV,,H +.F(VH) 2
(7.39)
246
Beyond
7.
the Semi-Classical
Approximation
V,xV,,,V,,V,H I
f2Z
(V,\V,,(S?Z)V,,V,H
1V,\(ZV,S?)VpV,,H
-
+
2
2
V,\(ZVAS?)VpV,H
IV,\VpHV/-, log S?V, log Q
(7.40)
2
1V,\V,H (21V1, log f?VP log S2 + vpvt, log Q
+
2
IV,\V/,H (2IV, log QVp log S? + VpV, log S?
+
2
F(VH)
where
I +
+.F(VH)
involving single derivatives
denotes terms
H, and which contracting (7.23) with H(x)AA leads to a relationship between the symplectic structure and the flows generated by the Hamiltonian vector field,
therefore vanish
on
I(H). Substituting (7.39)
V,, log Q
+
2V,, log Z
into
(7.23)
of
and then
F(VH)
=
(7.41)
now use a covariant derivative of (7-23) to rewrite the curvature term (7.14) in terms of covariant derivatives of the Hamiltonian, substitute (7.39) and (7.40) into (7.14), and use everywhere the identity (7.41). Since , Aj, V,, log Q for the metric connection.P, it follows after some algebra that the generally-covariant expression (7.14) for the first-order correction to the Duistermaat-Heckman formula is F(VH). This establishes the claim above.
We in
=
We shall
now
symmetry is in a
establish quite generally that this conformal localization the most general one that can be construed for
some sense
classical partition function [134]. For this, we return to the general localprinciple of Section 2.5 and re-evaluate the function Z(S) in (2.128)
ization
generic (not necessarily equivariant) metric-dual 1-form,3 ivg of the (i.e. with no assumptions for now about the symmetries of g) and the equivariantly-closed form a e iT(H+w)1(iT)n with 0, but LV,8 =h 0 in general. We also assume, for full respect to V, i.e. Dva generality, that the manifold M can have a boundary 9M. The derivative for
a
==
Hamiltonian vector field V
=
=
(2.129) using
need not vanish
now
and the second
Stokes' theorem and the Cartan-Weil
equality there can be evaluated identity. Then using the identity
00
Z(O)
=
lim S
00
Z(S)
-
j
ds
d ds
Z(S)
(7.42)
0
it follows that the
partition function Z(T)
=
fm
a
can
be determined in
general by 00
Z (T)
=
lim 5
00
f M
a
e- sDv,3
+
f 0
ds
tMi
a,3 e-,Dv,3
_
s
f aO(Lvo) e-sDv)3 M
(7.43)
7.2 Conformal and Geodetic Localization
The
large-s
limit
integral
(7.43)
in
did in Section 2.6 to arrive at the
S?V
the 2-form
=
d,3 there
be worked out in the
=
2g
-
same
way
247
as we
expression (2.142), except that
same
given quite arbitarily
is
Qv so
can
Symmetries
pv
-
now
as
(7.44)
Lvg
that
lim S
00
1
a
e- DvO
=
a
(-2,,)n/2 x
Pfaff
(dV(p)
(P)
det dV (p)
PEMv
M
(0)
-
(7.45)
g(p)-1Lvg(p)12)
Using (3.59) to rewrite derivatives of V in (7.45) in the usual way, substituting in the definitions in Section 2.5 for the geometric quantities in (7.43), and then expanding exponentials in the 2-forms w and S?V to the highest degrees that the manifold integrations pick up, we arrive at an expression for the partition function in terms of geometrical characteristics of the phase space
Z(T)
[134]
(T )n 21ri
=
(P) pEMv
6det (p)eiTH(p) jet
(p)
x,\/-det (1 -H-1wg-1LVg12) (p) 00
+
J TZT__)n I ds
0
0
(n
-
g(V,
A
(iTw
-g(V, 1)!
A
(Lvg)(V, -)
1)!
-
s0v)
n-1
am
00
(iT)n J
eiTH-sKV
ds
s
I
eiTH-sKV
(n
-
A
(iTw
-
sQV)
1
M
(7.46) arbitrary Riemannian metric g on M. From the expression (7.46) and the fact that g(V,.)A2 =- 0, we can see explicitly now how the conformal Lie derivative condition (7.35) collapses this expression down to the Duistermaat-Heckman formula (3.63) when aM 0, as we saw by more explicit means above (This conformal localization property has also been explained within a more general BRST approach recently in [83]). Note that we cannot naively carry out the s-integrations quite yet, because the function KV g(V, V) has zeroes on M. The expression (7.46), although quite complicated, shows explicitly how the Lie derivative conditions make the semi-classical approximation to the partition function exact. This is in contrast to the loop-expansion we studied earlier, where the corrections to which holds for
an
=
=
the Duistermaat-Heckman formula
derivatives.
(7.46)
were
therefore represents
expansion that explicitly takes into
a
not
just
some
combinations of Lie
sort of resummation of the
account the
loopgeometric symmetries that
248
7.
Beyond the Semi-Classical Approximation
1-loop approximation exact. We shall see soon that it is quite conpredicted from the loop-expansion, and moreover that it gives many new insights. There are several points to make at this stage. First of all we note the appearence of the boundary contribution in (7.43). If we assume that LV,8 0, that the group action represented by the flows of the Hamiltonian vector field V preserves the boundary of M (Le. g aM W) and that the action is free on o9M, then the s-integral in the boundary term in (7.43) can be carried out explicitly and we find the extra contribution to the DuistermaatHeckman formula for manifolds with boundary, make the
sistent with the results
=
=
-
3 A
Zam(T)
a
(7.47)
DV,8 am
In this context 3 action
on
the
=
ivg is the
boundary aM,
ment map for this action. This
Jeffrey-Kirwan-Kalkman
connection 1-form for the induced group as we have seen earlier dp is the mo-
because
boundary
residue that
term
was
can
be determined
using the 3.8, i.e. the
introduced in Section
coefficient of in the quantity (,0 A a)IDV,3, where 0 is the element of the symmetric algebra S(g*) representing the given circle action [82]. Secondly, notice that the conformal localization symmetry gives an explicit realization in (7.37) of the Hessian-metric ansatz which was discussed in the last Section. In particular, (7.46) establishes that with this Hessian-metric substitution and the appropriate extension away from the critical points of H (the terms proportional to VH in (7-37)) the corrections to the DuistermaatHeckman integration formula vanish to all orders of the loop-expansion (and not just to 2-loop order as was established in the last Section). Notice, how-
(7.36) any conformal Killing vector on a Kdhler manifold automatically an isometry. In fact, the generic case of a non-vanishing scaling function Y(x) in (7.35) is similar to the isometry case from the point of view of the equivariant localization priniciple. Note that away from the critical points of H we can rescale 0 =- g(V, -) lg(V, V). With this choice for the localization 1-form 0 it is easy to show from (7.35) that away from the critical point set of the Hamiltonian it satisfies CV,6 0, i.e. '3 is then an equivariant differential form on M MV. Thus away from the subset MV C M the conformal Killing condition can be cast into the same equivariant cohomological framework as the isometry condition by a rescaling of ever, that because of
is
--+
=
-
the metric tensor in
(7.35),
The rescaled metric
G,,,(x)
gttv is
--->
Gl,,,
=
only defined
g1,,1g(V, V), on
M
-
for which
MV, but
we
LvG
=
0.
recall from
Section 2.5 that all that is needed to establish the localization of the
zeroes
of the vector field V
(i.e.
the equivariant localization
(3.52) onto principle) is
(or equivalently an equivariant differential form everywhere on M except possibly in an arbitrarily small neighbourhood of MV. The fact that the weaker conformal symmetry condition is equivalent to the isometry condition in this respect is essentially a an
,3)
invariant metric tensor
which is defined
7.2 Conformal and Geodetic Localization
Symmetries
249
consequence of the fact that the differential form 3 ivg above is still a connection 1-form that specifies a splitting of the tangent bundle into a com=
MV (represented by the discrete sum over MV in (7.46)) and a to MV (represented by the Lie derivative integral in (7.46)). This is in fact implicit in the proof by Atiyah and Bott in [9] using
ponent
over
component orthogonal the Weil
algebra.
One may ask as to the possibilities of using other localization forms to carry out the localization onto MV, but it is readily seen that, up to components orthogonal to V, 3 ivg is the most general localization form up to multiplication by some strictly positive function. This follows from the fact =
that in order to obtain finite results in the limit
s
oo
-*
(7.45)
in
we
need to
that the form
Dv,3 has a 0-form component to produce an exponential damping factor, since higher-degree forms will contribute only polynomially in the Taylor expansion of the exponential. This is guaranteed only if '3 has a 1-form component. Thus it is only the 1-form part of 3 that is relevant to the localization formula, and so without loss of generality the most general localization principle follows from choosing,3 to be a 1-form. Furthermore, the 0-form part Vi',3,, of DV,3 must attain its global minimum at zero so that the large-s limit in (7.45) yields a non-zero result. This boundedness requirement is equivalent to the condition that the component of 3 along V has the same orientation as V, i.e. that,3 be proportional to the metric dual 1-form of V with respect to a globally-defined Euclidean signature Riemannian structure on the phase space M. In addition, for a compact phase space M the conformal group is in general non-compact, so that conformal Killing vectors need not automatically generate circle actions in these cases as opposed to isometry generators where this would be immediate. To explore whether this larger conformal group symmetry of the Duistermaat-Heckman localization leads to globally different sorts of dynamical systems, one would like to construct examples of systems with non-trivial (i.e. T =h 0) conformal symmetry. For this, one has ensure
to look at spaces which have
nian vector field V is
a
a
Riemannian metric 9 for which the Hamilto-
generator of both the conformal group Conf (M, g)
and the the
symplectomorphism group Sp(M, w) of canonical transformations on phase space. From the analysis thus far we have a relatively good idea
of what the latter group looks like. The conformal group for certain Riemannian manifolds is also well-understood [54]. For instance, the conformal group of flat Euclidean space of dimension d > 3 is
SO(d + 1, 1). was
already
The
locally isomorphic
to
(global)
conformal group of the Riemann sphere C U tool encountered in Chapter 6 above (in a different context), namely
the group SL(2, (C)/Z2 f:-- SO(3, 1) of projective conformal transformations. In these cases, the conformal group consists of the usual isometries of the space,
in
z
=
along r
e'0)
with dilatations
or
scale transformations
and the d-dimensional
transformations.
subgroup
(e.g.
of so-called
translations of
special
r
conformal
250
7.
Beyond the Semi-Classical Approximation
An interesting
example is provided by the flat complex plane C. Here algebra is infinite-dimensional and its Lie algebra is just the classical Virasoro algebra [54]. Indeed, the conformal Killing equations in this case are just the first set of Cauchy-Riemann equations in (5.51) (the other one represents the divergence-free condition tr Av 0). This means that the conformal Killing vectors in this case are the holomorphic functions V' The Hamiltonian flows of these vector fields are f (z), V2 therefore the arbitrary analytic coordinate transformations the conformal
=
=
=
40 As a
an
=
f WO)
ZW
I
=
A2(0)
(7.48)
explicit example, consider the conformal Killing vector which describes n + I distinct stationary points,
Hamiltonian system with
V' at
z
=
(7.35)
0 and
z
=
=
i,3z(l
I /ai, where
-
a1z)
3, ai
E C
-
z)
an
(7.49)
The associated
-
scaling
function in
is then
T(Z'; ) The
(1
...
a'Vz
symplecticity condition LVw
a2VO
+
(7.50)
0 leads to the first-order linear
partial
differential equation
VZ09ZWZ2
+
V29 WZ2
-T(Z" )WZ2
=
(7.51)
easily solved by separation of the variables z,; . The solution symplectic 2-form with arbitrary separation parameter A E R is
which is
=
Z2
W'\(Z)iv-'\(' )/Vzv
for the
(7.52)
where
w), (z)
iA
=
e'
f dz/V'
.
((1
-
z
a,Z)AI
a,,, Z)
...
A,
(7.53)
and the constants
Ai(ai,...' an)
==
(ai )n-1 i0i
are
the coefficients of the
partial
(VZ )_ 1=
1
!)_3
aj
(7.54)
decomposition n
1+ E
_z
single-valued phase, so that Ai (a,,
-
Ai 1
-
aiz
(7.55)
C, we restrict the ak'S same On) E R, and the parameter 3 be real-valued. The Hamiltonian equations (5.68) can now be integrated To
ensure
that
to all have the to
(7.52)
fraction
ai
is
a
function
.
-
-
,
on
7.2 Conformal and Geodetic Localization
up with the vector field we
find the
family
(7.49)
and the
sYmplectic 2-form, (7.52),
( (I
1
-#'ai (Z"
from which
/P
Z -
a,Z)AI
-
anZ)An
...
(7.56)
An)
X
that this Hamiltonian has
ensure
251
of Hamiltonians
W(IN)
To
Symmetries
only non-degenerate
critical points
we
3. This also guarantees that the level (constant energy) curves of this Hamiltonian coincide with the curves which are the solutions of the equations set A
=
of motion
(7.48) [134].
Since the Hamiltonian
(7.56)
either vanishes
or
is infinite
on
its critical
it is easy to show that the partition function (3.52) is independent of ak and coincides with the anticipated result from the Duistermaat-Heckman
point set,
integration formula, namely Z(T) 2-7ri,3/T. This partition function coincides with that of the simple harmonic oscillator z. 1,3, as expected since for =
H,,( ,)
the harmonic oscillator 0, A) becomes the Darboux 2-form and '0 Z Hamiltonian. In fact, we can integrate up the flow equation (7.48) in the ai
=
general case and by the equation
e'0('0)
we
=
find that the classical
w,3(Z(0)
=
(1
-
trajectories z(t)
Z(t) a,Z(t))Al (1 -
-
-
-
are
determined
(7.57)
anz(t))An
change z --+ wo(z) is just the finite conformal transforma, generated by the vector field (7.49) and it maps the dynamical system
The coordinate tion
(wlqT, Ho( ),) 'C'
onto the harmonic oscillator H
circular classical trajectories
w(t)
=
e'O('-'O)
oc
w Fv,
W OC
WD with the usual
associated with
a
U(1)
gener-
isometry. This transformation is in general multicorresponding valued and has singularities at the critical points z 1/ai of the Hamiltonian ator
to
an
=
27(o)
ho,a..
It is therefore not
a
diffeomorphism,
(R W;(02), H(3) ) 2
Hamiltonian system , oscillator. For
of the
plane
for ai
7-
0 and the
globally isomorphic to the simglobal differences between these systems and those associated with isometry generators, see [134]. The conformal group structures on phase spaces like S2 yield novel generalizations of the localizable systems which are associated with coadjoint orbits of isometry groups as we discussed in Chapter 5. The appearence of the larger (non-compact) conformal group may lead to interesting new structures in other instances which usually employ the full isometry group of the phase ple harmonic
space, such
as
more
is not
details about the
the Witten localization formalism of Section 3.8.
Other geometric alternatives to the Lie derivative condition LVg 0 have also been discussed by Mirki and Niemi in [84]. For instance, consider the =
alternative condition
252
Beyond the Semi-Classical Approximation
7.
V"V.Xv/A
=
(7.58)
0
to the
VA are Killing equation, which means that the Hamiltonian flows _V, geodetic to g (see Section 2.4). From (7.58) it follows that the Lie derivative =
of the localization
LV,3
=
1-form,3
VP(gpXVIV\
+
=
ivg
can
be written
g1,,\VpV\)dx1'
=
as
VPgp,\V,,V-Xdxl'
div Comparing this with the Cartan-Weil identity LV derivative acting on differential forms we find the relation =
ivd,3
=
-
(7.60)
[841 (7.61)
0
=
dynamical systems (I2 KV, OV)
Hamiltonian structure. Moreover, in this
(7.59)
iVd for the Lie
+
1div,3
Dv (Kv /2 + f2v) that the
divo
2
which leads to the equivariance condition
so
2
and
(H, w)
it is also
case
determine
possible
to
a
bi-
explicitly
solve the equivariant Poincar6 lemma [84], just as we did in Section 3.6. Thus + OV is an equivariantly-closed differential form, the condigiven that -!KV 2 tion (7.58) has the potential of leading to possibly new localization formulas. However, there are 2 things to note about the geometric condition (7.58). The first is its connection with a non-trivial conformal Killing equation LVg Tg, which follows from the identity
VV 9 aA(Lvg)A,,
Contracting both sides
of
(7.62) YKv
(7.35)
=
V\V,\V'
with
==
+
g,pVP
(7.62)
g'AVIKvl2
leads to
2g,,,,V'V,\VXVA (7.58)
(7.63)
satisfied, then T -= 0 away everywhere on M and so the be condition can compatible with the Killing equation, only geometric (7.58) and not the inhomogeneous conformal Killing equation. Secondly, the exact 2-form S2V =- divg is degenerate on M, because an application of the Leibniz rule and Stokes' theorem gives
when
from the
n!
holds. This
zeroes
d 2nX
implies that
of V. Thus T
-
V/det Qv (x)
=
if
-=
0 almost
f
on V
0. Thus det S-2v (x)
=
0
Hamiltonian system determined by in Section 3.6, this isn't so crucial a
f
d
(ivg A
V
=
0
(7.64)
M
M
when aM
=
is
on some 1
submanifold of M of dimension at least 2
M, and so the degenerate. As mentioned
submanifold of
KV, f2v) 2 so long as
is
the support of det
(so
S?v(x)
is
that there exists at least
1 degree of freedom from the classical equations of motion). It would be interesting to investigate these geometric structures in more detail and see what localization schemes they lead to.
7.3 Corrections to the Duistermaat-Heckman Formula
253
7.3 Corrections to the Duistermaat-Heckman Formula: A Geometric Approach The integration formula
(7.46) suggests
a
geometric approach
tion of corrections to the Duistermaat-Heckman formula in the
to the evaluacases
where it
is known to fail. Recall that there is
for which V is
3.6). there
a
Killing
vector
always locally a metric tensor on M M V (see the discussion at the beginning of Section -
For the systems where the semi-classical approximation is not exact, are global obstructions to extending these locally invariant metric ten-
to globally-defined geometries on the phase space which are invariant under the full group action generated by the Hamiltonian vector field V on M, i.e. there are no globally defined single-valued Riemannian geometries on sors
M for which V is
globally a Killing vector. This means that although the Killing equation LVg 0 can be solved for g locally on patches covering the manifold, there is no way to glue the patches together to give a single-valued ==
invariant geometry on the whole of M (c.f. Section 5.9). In this Section we shall describe how the expression (7.46) could be used in this sense to evaluate the corrections to the
sum over
critical points there
that not
only does this method encompass much more than the term-by-term analysis of Section 7.1 above,
[157], of the
and
we
shall
see
loop-expansion
but it also character-
izes the non-exactness of the Duistermaat-Heckman formula in
a
much
more
transparent and geometric way than Kirwan's theorem. In this way we can obtain an explicit geometric picture of the failure of the Duistermaat-Heckman theorem and in addition a systematic, geometric method for approximating the integral (3.52). Furthermore, this analysis will show explicitly the reasons why for certain dynamical systems there are no globally defined Riemannian metrics on the given symplectic manifold for which any given vector field with isolated zeroes is a Killing vector, and as well this will give another geometric description of the integrability properties of the given dynamical system. The analysis presented here is by no means complete and deserves a more careful, detailed investigation. The idea is to define solve the
a
set of
patches covering M
in each of which
we can
Killing equations for g, but for which the gluing of these patches to give a globally defined metric tensor is highly singular. The non-
together triviality that
occurs
when these subsets
are
patched back together will then
represent the corrections to the Duistermaat-Heckman formula, and from our earlier arguments we know that this will be connected with the integrability of the Hamiltonian system. We introduce a set of preferred coordinates x" for the vector field V following Section 5.2. In general, this diffeomorphism
only be defined locally on patches over M and the failure of this coorproducing globally-defined C'-coordinates on M gives an analytic picture of why the Hamiltonian vector field fails to generate global isometries. Notice in particular that these coordinates are only defined on M MV. In this way we shall see geometrically how Kirwan's theorem restricts dynamical systems whose phase spaces have non-trivial odd-degree can
dinate transformation in
-
254
Beyond the Semi-Classical Approximation
7.
homology
explicitly
and
what type of flow the Hamiltonian vector field gen-
erates.
Recall that the coordinate functions x" map the constant coordinate lines 2n-1 onto the integral curves of the isometry defined by the E R X2n) 0
(X2'. 0
classical Hamilton equations of motion :P(t) VA(x(t)), i.e. in the coordinates x"(x), the flows generated by the Hamiltonian vector field look like =
X111(t)
X,0
X""(t)
+ t
sociated with the fact that there is
no
formation functions
were
M to be any of x"I thereby
on
independent
(7.46)
from
X" 0
,
singularities
Otherwise, MV, then
0 holds.
=
on
M
-
on
as-
M for
if these transwe
could take
whose components in the x"-coordinates are solving the Killing equations directly, and hence
one
approximation would be
the WKB
must therefore be
system, there
(7.65)
p > 1
Riemannian metric tensor
Cvg globally defined
which the Lie derivative condition the metric
=
this coordinate transformation function will have
general,
In
=
some
exact. For
a
non-integrable defining the
sort of obstructions to
globally over M. In light of the above comments, these partition the manifold up into patches P, each of which is
x"-coordinate system
singularities a
will
2n-dimensional contractable, submanifold of M with boundaries 09P which
are some
other
(2n
-
l)-dimensional
by the conBy dropping some that these patches
submanifolds of M induced
stant coordinate line transformation from R
2n- 1
above.
of these coordinate surfaces if necessary, we can assume induced from the singularities of the above coordinate transformation form
disjoint
cover
function
of the
manifold, M
=
4. Then
JJp p
we can
write the
a
partition
as
(7.66)
alp
Z(T) p
p
equivariantly-closed differential form (3.57). patches P, in their interior there is a well-defined (bounded) translation action generated by V"". Since the patches P are 2n we can place a Euclidean metric on diffeomorphic to hypercubes in R
where
as
By
usual
a
is the
the choice of the
,
them, gp
where the conformal factor function
x"',
on
P. If
we
then the metric
patch P, by
e" ("') dx" (D dx" ' A
=
op(x")
choose it
(7.67)
so
a
globally-defined real-valued C'independent of the coordinate
that it is
Killing equation on P. Thus on each coordinates, we can solve the Lie deriva-
satisfies the
choice of
the
is
(7.67)
given though this cannot be extended to the whole of M. Then each integral over P in (7.66) can be written using the formula (7.46), restricted to the patch P, to get
tive
constraint,
Here
even
we assume
that M is compact, but
also be extended to the
phase
space R
2n
we
shall
see
that this formalism
can
7.3 Corrections to the Duistermaat-Heckman Formula
f
(P)
T
f
here,
e'iTH-sKv
ds
(n
1)!
-
g(VI.)
SOV)n-1
(jTW
A
lap
ap
0
The first term
iTH(p)
(7.68)
00
(iT)n
e
(D) P)
et t
pEMvnP
p
+
a_dTet(p)
n
27ri
alp
255
when
(7.68)
is substituted back into
(7-66), represents
the lowest-order term in the semi-classical expansion of the partition function over M, i.e. the Duistermaat-Heckman term Zo(T) in (3.63), while the
boundary terms give the general corrections to this formula and represent non-triviality that occurs rendering inexact the stationary-phase approx-
the
imation. The result is
Z(T)
Zo(T)
=
+
JZ(T)
(7.69)
where 00
(iT)n I
JZ(T)
ds
p
0
(n
patch
geometric approach
conserved
charges
we can
nian, which
take
we
(7.70)
JZ(T),
-8QV)n-1
lap
(7.70)
therefore represent
an
al-
we
(eqs.
recall from Section 5.2
X,4(x)
for tz
2,...,2n
=
are
local
of the Hamiltonian system, i.e.
one
of
V'o9,X4
say X 2, to be
them,
X112(X)
choose to be
=
=
X2(X)
irrelevant constant to H
we
(compact)
Then, using the
we
-) A (iTw
loop-expansion of Section 7.1 above.
that the coordinate functions
IX", Hj,, Thus
g (V,
terms in
to the
To evaluate the correction term
(5.39),(5.43))
1)!
-
ap
The contributions from the ternative
e7,TH-sKv
Ei
manifold M.
find that the metric
may
(7.67)
assume
=
=
a
(7.71)
0
functional of the Hamilto-
VF(-x),
that it is
a
where
by adding
positive function
on
an
the
law, original (unprimed)
metric tensor transformation
when written back into the
coordinates has the form
gp
=
eWP (x)
1
(V,\ ax X 1)2
09'4X1aI'X1
1 +
4H
o9jHa,,H
(7.72) +
E,9jx',9,,x'
dxl' (g dxv
a>2 so
that the
metric-dependent quantities appearing
in
(7.70)
can
be written
as
91:' V.
ewP(x)_ aA Xl(x) dX
V'(X)aAX1(X)
Kv(x)lp
=
gp(V, V)
=
ewP(
(7.73)
7.
256
Beyond the Semi-Classical Approximation
QVIP
ewP(II)
=
al\Xl (OAV' 2(V,\aAT_2 f
a X1
-
a V
+VAa,xxl (91, ppaxl -,O, op,01,xl) I dxl'
A
A0140 dx'
(7.74)
When these expressions are substituted back into the correction term (7.70), find that the integrands of JZ(T) depend only on the coordinate function
we
X1 (x). This is not surprising, since the only effect of the other coordinate functions, which define local action variables of the dynamical system, is to make the effect of the partitioning of M into patches above non-trivial, reflecting the fact that the system is locally integrable, but not globally (otherwise, the partition function localizes). In general, the correction term (7.70) is extremely complicated, but we recall that there is quite some freedom left in the choice of X1. All that is required of this function is that it have no critical points in the given neighbourhood. We can therefore choose it appropriately so as simplify the correction JZ(T) somewhat. Given this choice, in general singularities will appear from the fact that it cannot be defined globally on M, and we can use these identifications to identify the specific regions P above. The form of the function X1 is at the very heart of this approach to evaluating corrections to the Duistermaat-Heckman formula. We shall see how this works in some explicit examples in the next Section. Notice that a the similar phenomenon to what occured in Section 5.9 has happened here coordinate
to
-
function Kv in
(7.73)
is non-zero,
as
the
zeroes
of the vector field V have been
gp(V, -) thereby making it singular. We can therefore now carry out the explicit s-integral in (7.70), as the singularities on absorbed into the metric term
already present in the integrand there. Although this may seem to everything hopelessly singular, we shall see that they can be regulated with special choices of the function X1 thereby giving workable forms. We shall see in fact that when such divergences do occur, they are related to those predicted by Kirwan's theorem which we recall dictates also when the full stationary-phase series diverges for a given function H. There does not seem to be any immediate way of simplifying the patch corrections 6Z(T) above due to the complicated nature of the integrand forms. However, as usual in 2-dimensions things can be simplified rather nicely and the analysis reveals some very interesting properties of this formalism which could be generalized to higher-dimensional symplectic manifolds. To start, we notice that in 2-dimensions, if M is a compact manifold, then the union above over all of the patch boundaries 9P C M will in general form a sum over 1-cycles ae E Hi (M; Z). Next, we substitute (7.73) and (7.74) into (7.70) with n 1, and after working out the easy s-integration we find that the MV
are
make
=
2-dimensional correction terms
JZ(T)
=
-1
E iT e
can
1
be written e
as
iTH(x)
V1(49,\X1(X)
49jX'(x)dxt'
(7.75)
7.3 Corrections to the Duistermaat-Heckman Formula
XI,
As for the function
need to choose
we
one
which is
independent
257
of the
other coordinate transformation function X 2 to ensure that these 2 functions truly do define a (local) diffeomorphism of M. The simplest choice, as far as the evaluation of
(7.75)
is to choose
concerned,
is
x1
as
the solution of the
first-order linear partial differential equation
Vl(X)a,XI(X) X1, the ajX'(x) 7 0,
With this choice of
V2(X),g2X1 (X)
=
X1
functions
X2
and
(7.76)
are
independent
of each
1, 2, which follows from working out the Jacobian for the coordinate transformation defined by X/I and using their defining partial differential equations above. With this and the Hamiltonian equations dH -ivw, the correction other wherever
terms
(7-75)
p,
v
=
become
6Z(T) where
we
Fl,,,
2iT
(7.77)
have introduced the 1-form F
=
W12(x)
(7.77)
e
(,02H(x)
iTH(x)
dxl
-
c9jH(x)
dX2
)
(7.78)
geometric interpretation of the correchomology H, (M; Z), there corresponds a cohomology class 77j E H1 (M; R),
The expression
leads to
nice
a
tions above to the Duistermaat-Heckman formula. To each of the
cycles
at E
called their Poincar6 dual
grals
of 1-forms
a
E
A'M
[32],
which has the property that it localizes inte-
to at, i.e.
i
a
I a,
(7.79)
A 77t
a
M
at
Defining E
we see
that the correction term
(7.77)
6Z(T)
H(M; R) can
1 =
-
2iT
be written
1
(7.80) as
(7.81)
F Aq
M
Noting also
that the
original partition
Z(T)
I =
2
function itself
j
FAdH
M
it then follows from
Z(T)
=
Zo(T)
+
6Z(T)
that
can
be written
as
(7.82)
7.
258
1
Beyond
FA
Approximation
the Semi-Classical
(iTdH +,q)
=
-47r
1: PEMv
M
(_,),\(P)Fdeet (p) a-t T_ et
( (p)
e
iTH(p)
(7.83)
Thus in this sense, the partition function represents intersection numbers of M associated to the homology cycles at. This last
equation
is
particularly interesting.
It shows that the
correc-
tions to the Duistermaat-Heckman formula generate the Poincar6 duals to
homology cycles which signify that the Hamiltonian vector field does not generate a globally well-defined group action on M. When the correction the
1-form
qliT
is added to the 1-form dH
of the Hamiltonian vector field
on
M,
=
the
-w(V, -)
resulting
means
that
although
enough
to
ren-
"effective" partition the initial Hamiltonian flow dH doesn't
der the Duistermaat-Heckman formula exact for the function. This
which defines the flow
1-form is new
required for the Duistermaat-Heckman satisfy enough' theorem, adding the cohomological Poincar6 dual to the singular homology cycles of the flow is enough to close the flows so that the partition function is now given exactly by the lowest-order term Zo(T) of its semi-classical expansion. One now can solve for the vector field W satisfying the "renormalized" Hamiltonian equations 'close
the conditions
to
dH We
consider W
can
which renders the
as a
+,qliT
=
-w(W,
(7.84)
"renormalization" of the Hamiltonian vector field V
stationary-phase
series
convergent and the Duistermaat-
symplectic form W defines a cohojust corresponds to choosing a different, possi-
Heckman formula exact. Note that since the
H'(M; R), mology bly non-trivial representative in H1 (M; R) for w (V, -) (recallq C- H1 (M; R)). Thus in our approach here, the corrections to the Duistermaat-Heckman formula compute (possibly) non-trivial cohomology classes of the manifold M and express geometrically what symmetry is missing from the original dynamical system that prevents its saddle-point approximation from being exact. The explicit constructions of the Poincar6 duals above are well-known [32] M of S1 in M which corresponds to the one takes the embedding at : S1 current which is the Dirac delta-function DeRham. its and constructs loop at, 1-form P 1) (x, at (y)) E A' M (x) 0 A' M (y) with the property (7.79) [18]. this
class in
-
--+
1
one crucial point that needs to be addressed before we turn explicit examples. In general we shall see that there are essentially 2 types of homology cycles that appear in the above when examining the singularities of the diffeomorphisms X/1 that prevent them from being global coordinate transformations of M. The first type we shall call 'pure singular cycles'. These arise solely as a manifestation of the choice of equation satisfied by X1. The second type shall be refered to as 'critical cycles'. These are the cycles on which at least one of the components of the Hamiltonian vector field 1 or 2. On these latter cycles the above integrals in 0 for M vanish, VA(x) 6Z(T) become highly singular and require regularization. Notice in particular
There is
to
some
=
=
7.4
that
if,
(7.71)
say,
Vl(x) (7.76)
and
=
0 but
V2(X) :
0
Examples
259
cycle at, then the equations X/1 imply that a2XA(x)
on some
which determine the functions
=
leaving the derivatives 491X/I(x) undetermined. Recall that it was precisely at these points where the Jacobian of the coordinate transformation defined by X/' vanished. In this case one must regulate the 1-form F defined above by letting a1X1 and o92X' both approach zero on this cycle aj in a correlated manner so as to cancel the resulting divergence in the integrand of (7.75). Note that this regularization procedure now requires that x1 and x 2 transform identically, particularly under rescalings, so that the tensorial properties of the differential form F are unaffected by this definition. In this case, the Morm F which appears above gets replaced by the 1-form 0 while
Flat
1 =
-
V2(X)
(dxl
ing
can
thought higher-loop
be
tion for the
e
iTH(x)
=
O,H(x)
(dxl
general expression (7.75).
which follows from the F
2)
+ dx
of
+
This
dX2 ) eiTH(x) (7.85)
procedure
for defin-
quantum field theoretic ultraviolet regularizacorrections to the partition function. In general, we as a
shall
always obtain such singularities corresponding to the critical points of because, as mentioned before, the diffeomorphism equations above become singular at the points where Vl'(x) 0. Note that (7,85) will also diverge when the cycle ae crosses a critical point, i.e. on aj n Mv. Such singularities, as we shall see, will be just a geometric manifestation of Kirwan's theorem and the fact that in general the stationary-phase expansion does not converge for the given Hamiltonian system. We shall also see that in general the pure singular cycles do not contribute to the corrections, as anticipated, as they are only a manifestation of the particular coordinate system used, of which the covariant corrections should be independent. It is only the critical cycles that contribute to the corrections and mimick in some the Hamiltonian
=
sense
7.4
the
sum over
critical
points
series for the
partition function.
Examples
In this Section
we illustrate some of the formalism of this Chapter with 2 explicit examples. The first class we shall consider is the height function of a Riemann surface, a set of examples with which we have become well-acquainted. In the case of the Riemann sphere we have little to add at this point since the height function (2.1) localizes. The only point we wish
classes of
to make here is that the covariant Hessian in this
standard Kdhler geometry of S2 metric gS2 by
VVhzo
=
2
1
(I
(see
Z,
+
Z )3
Section
dz o d,
=
5.5)
2(1
case
with respect to the
is related to the Khhler
-
hzo)gS2
(7.86)
260
Beyond the Semi-Classical Approximation
7.
which is in agreement with the analysis of Section 7.1 above. This shows the precise mechanism (i.e. the Hessian of hzo generates covariantly the Khhler
S2)
structure of
that makes the
loop
corrections vanish.
An interesting check of the above formalisms is provided by a modified version of the height function h_ro which is the quadratic functional (2)
hZo
2
hzo
=
-
hZ
which has the
for H
COS
2z.
0)2
I + Zf
+ Zf
hEo.
as
2zf
Now
gz (z,, )
,
Z
=
)
2
(7.87)
find that the metric
we
the isothermal solution
(2) (2) a2 h ZO /o9zh Zo
9Z9Z2
[157]
H'(zf)
(7.88)
(2)
2zf
(2) VVh Zo
(7.89)
As
_
by taking
solved
are
ZZ
(I
_
hZo ,which follows from (7.24) written in local isothermal coordinates implicitly defined metric. Thus the solution to (7.21) is
=
for the
9
0)
critical behaviour
same
equations (7.21)
F;ZZ
COS
0
(z,
-
+
Z )3
-
1) dz (9 df
=
gz2dz
does not coincide with the standard Khhler
(7.89)
0 d,
geometry of
S2,
the
partition function in this case is not exact, as I-loop approximation the expected. However, partition function still localizes, in the sense that it to the
can
be
computed
via the Gaussian
integral transform
00
f dILL
Z(T)
iT(H-H 2) e
do e-i02 J -,/-27r-i f dl-IL /2
=
M
e
i(T-2iv TO)H
M
-00
of the usual equivariant characteristic classes. Thus since
tional of as
an
isometry generator (i.e.
anticipated
a
charge),
conserved
(7.87)
it is still
is
func-
a
localizable,
from the discussions in Sections 4.8 and 4.9. This is also
sistent with the formalism of the
previous Section. In this
coordinates for the Hamiltonian vector field
Although
(7.90)
these coordinates
are
singular
0 and
are
at the
poles
of
case, the x
=
0/(I
con-
preferred -
COS
0).
S2 (i.e. the critical
points of (7.87)), the correction terms JZ(T) do not localize onto any cycles and just represent the terms in the characteristic class expansion for Z(T) here. This just reflects the fact that S2 is simply connected, and also that the geometric terms JZ(T) detect the integrability features of a dynamical system
(as (7.87)
Next,
we
is
an
integrable Hamiltonian). height function on the torus, with the Khhler ge0 in (6.9) and v I in (6.39). adjusted so that o
consider the
ometry in Section 6.2
=
The covariant Hessian of the Hamiltonian
'H(01 02) 1
=
IM
7- COS
01
+ (rl + IM
COS
02do,
T COS
(&
do,
=
(3-78) -
01) COS 02dO2
in this
2 Im (9
-r
case
is
sin 01 sin 02do, (9
d02
d02
(7.91)
7.4
In the
complex coordinatization used
Examples
261
to define the Khhler structure this Hes-
analysis used to show stationary phase approximation in the case of the height S' using the loop-expansion will not work here. Indeed, we do not
sian is not of the standard Hermitian form and the
the exactness of the function
on
expect that
any metric
on
T 2 will be defined from the covariant Hessian here
already know that the Duistermaat-Heckman example. This is because of the saddle-points at (017 02) (0, ir) and (ir, 7r). The Hessian at these points will always determine an indefinite metric which is not admissible as a globally-defined geometry
as we
did in Section 7. 1, and
we
formula is not exact for this =
on
the torus.
This is also apparent from examination of the connection (7.8) and its Fubini-Study geometry defined by (7.34). In this case ^/ =- 0 and
associated
(7.34)
the curvature
is trivial. The 2-form Q does not determine the
same
cohomology class as the Kiihler 2-form of Z' does, so that there is not enough "mixing" of the Hessian and Liouville terms in the loop expansion to cancel out higher-order corrections. For the sphere, the Fubini-Study metric coincides with the standard Kdhler metric and thus the
dynamics integrable (recall
there to make the of formation of
a
appropriate mixing
that CP1
non-trivial Kiffiler structure
=
S2).
is
It is the lack
the torus here that makes
on
dynamical systems on it non-integrable. Although the failure of the Duistermaat-Heckman theorem in this case 2 via Kircan be understood in terms of the non-trivial first homology of T in the obstructions examine wan's theorem, we can extending the analytically Khhler standard the of Hamiltonian vector field (3.80) to a global isometry 2 for Riemannian the metric (6.9) of T which defines equivgeometry unique almost all
ariant localization
defined
by the
on
the torus. We shall find that the local translation action
vector field
(3.80)
cannot be extended
way to the whole 4 T 2. The set of coordinates the components of the Hamiltonian vector field
(x, y) are
globally on
V'
in
a
smooth
the torus in which
=
1 and VY
=
0
as
by taking X 2(01, 02) prescribed of the height function (3.78) and X1 (011 02) to be the Cl-function with nonvanishing first order derivatives which is the solution of the partial differential equation (7.76). In the case at hand (7.76) can be written as before
-
are
(r,
to be the square root
first defined
+ Im
-r COS
01)
ax
1 =
Im
-r
sin
01
Cot
02
axi
(7.92)
which is solved by
X1 (011 02) integrating (7.93) This gives and
as
=
in
log(r,
+ IM
(5.43) yields
T COS
01)
-
109(COS 02)
the desired set of coordinates
(7.93) (x, y).
262
7.
Beyond
the Semi-Classical
Approximation
2 Imr
X(01 02)
V r2 lRe
M'r
(t
log
an
-rJ sin 02
arctan
FRe -7
tan
r2
01 2
02
(7.94)
2
Cos
Vr
Y(01) 02)
(rl
IM
T Cos
01) COS 02
which hold
provided that Re -r: k 0. by the diffeomorphism. (7.94) the Hamiltonian vector field generates the local action of the group R1 of translations in x. However, this diffeomorphism cannot be extended globally to the whole of T2 because it has singularities along the coordinate circles In the coordinates defined
a,
=
bi This
means
geometry
that
on
T 2.
J(-7r/2, 0)
E
{(O,O)
E
cannot
globally generate
=
Vz1
Although
T21
a2
=
f (3-7r/2, 0)
21
b2
=
f(O,ir)
T
E
E T
T21
21
(7.95)
(7.96)
isometries of any Riemannian
translations in the coordinate
x
represent
some
unusual local symmetry of the torus, it shows that the existence of nontrivial homology cycles on T 2lead to singularities in the circle action of the on T2. These singularities do not appear on the sphere because any closed loop on S2 is contractable, so that the singular circles above collapse to points which can be identified with the critical points of the Hamiltonian function. In fact, as we saw in Chapter 6, the only equivariant Hamiltonians on the torus are precisely those which generate translations along the homology cycles of T2, and so we see that the Hamiltonian (3.78) generates a circle action that is singular along those cycles which are exactly the ones required for a globally equivariantly-localizable system on the torus. This is equivalent to the fact that the flow generated by V_ri bifurcates at the saddle points of h_vl (like the equations of motion for a pendulum), and the above shows analytically why there is no singlevalued, globally-defined Riemannian geometry on the torus for which the height function h_ri generates isometries. The local circle action defined by the diffeomorphism (7.94) however partitions the torus into 4 open sets Pi which are the disjoint sets that remain when one removes the 2 canonical homology cycles discussed above. Each of these sets Pi is diffeomorphic to an open rectangle in R2on which the Hamiltonian vector field Vzi generates a global Rl-action. Thus the above formalism implies that the corrections to the Duistermaat-Heckman formula for the partition function in this case is given by (7.75) evaluated on the pure singular cycles a, and a2 above, and on the critical cycles bi and b2 (see the previous Section). Summing the 2 contributions from the 1-form F in (7.78) along the pure homology cycles shows immediately that Fl,,, + 07 Fla2
Hamiltonian vector field Riemann
=
7.4
as
anticipated. As
Examples
263
integrals along the critical cycles, taking proper by the contractable patches, we find that the
for the
care
of orientations induced
contributions from b, and b2 written
the
are
and that the corrections
same
can
be
as
6ZT2(T)
eiT(r2-rl)
iT Im
-r
f
e-iT
do
Im
rcoso
sin
0 7r
f do
eiT(r2+rl)
-
e
(7.97)
iT Im
r cos
0
sin
0
After
a
change of variables we find that the integrals in (7.97) exponential integral function [60]
can
expressed
be
in terms of the
00
Ei(x) which
diverges
for
integration. After
JZT2(T)
< 0.
x
[
1 =
-
iT Imr -
Here the
algebra
some
2iT Im
e-iT
Im
e
-r cos
7-
Ei
2
-
integral
e
Im
r
Ei 2
a
Cauchy principal value
iT Im
-r
(Ei(-2iT
Im
r)
2))
Y
Im
-r
(-2iT
sin2
6)
Ei(2iT
-
2
Im
-r)) I
-r
Ei(2iT
2
eiT
denotes
2
(2iT Im
e-iT
iT(r2+rl)
2
(7.98)
find
we
eiT(r2-ri)
Ei
et
dt
Im
7- sin
2
Im
2)
7-)
-
-
Ei
2iT Im
Ei(-2iT
Im
-r)
r cos
2
2))
Y
) 11 (7.99)
where Y 7r in (7.97) at 0 =
-
r:
=
and
c
0 and
The correction term is
a sum
0 is used to =
the
divergence
of the
integrals
7r.
(7.99)
of 4 terms which
regulate
can
tells
us
quite
a
bit. First of
all,
note that it
be identified with the contributions from the
critical points of the Hamiltonian h_rl However, these terms are resummed, since the above correction terms take into account the full loop corrections to -
Next, the terms involving 6 are divergent, divergence Of JZT2(T) is anticipated from Kirwan's theorem, which says that the full saddle-point series for this Hamiltonian diverges. The exponential integral function can be expanded as the series [60] the Duistermaat-Heckman formula.
and the overall
264
7.
Beyond
the Semi-Classical
Approximation 00
Ei(x)
7 +
=
log x
+
n
E nn!
(7.100)
n=1
for
x
-y is the Euler-Mascheroni constant. Thus the
small, where
divergent
1, giving a much T simpler way to read off the coefficients of the loop-expansion (note the enormous complexity of the series coefficients in (7.3) for this Hamiltonian a direct signal of the messiness of its stationary-phase series). Finally, the finite terms (those independent of the regulator e), can be evaluated for pieces
in
(7.99)
be
can
explicitly expanded
in powers of
-
find JZT2 123.086. In Section 3.5 we partition function for this dynamical system 1849.327. was 2117.12, while the Duistermaat-Heckman formula gave Zo Thus Zo + 6ZT2 1972.41, which is a better approximation to the partition function than the Duistermaat-Heckman formula. Of course, given the large divergence of the stationary phase series, we do not expect that the finite contributions in (7.99) will give the exact result for the partition function, but we certainly do get much closer. As the function X1 which generates the set of preferred coordinates is by no means unique, perhaps a refined definition of it could lead to a better approximation Zo + 6Z. Then, however, we lose a lot of the geometrical interpretation of the corrections that we gave in T
=
saw
-i and
-r
I +
=
i, and
we
=
that the exact value of the
=
=
the last Section.
examples we consider here 2 plane R ,where U(q) is
The second set of
(5.273)
defined
on
the
function. In this
non-degenerate
case
-
i9q
proceeding
as
(7-101)
=
P2 /2
(7.102)
Thus here there
)(qU'(q)
-
are
P
-
U(q)
above the local coordinates
I =
only
=
2)
-
critical
f (O,q)
E R
( t, 9)
are
,
Y(q, p)
=
'cycles' given by
21
,
in which the Hamilto-
VT2 _T
+
U(q)
(7.103)
the infinite lines
Uj=f(p,qj)ER 21
(7.104)
where qj are the extrema of the potential U(q). Since for the Darboux. Hamiltonian (5.273), VP and Vq vanish the renormalized version of
use Uj respectively, we 'cycles' namely (7.85). Combining (7.85) with (7.77) function, that the corrections are
P and
a
-U'(q)-- P-
nian vector field generates translations
(q, p)
potential problems
by
X1 (q, p) Then
the
a C' potential which is the equation (7.76) becomes
09XI
,9 X1
p-
which is solved
are
must
we
find, for
an even
on
the
(7.78),
potential
7.4 00
6ZR2 (T)
eiTU(q)
dq
-;-T
-
Uf (q)
00
(E
f
eiTU(qi)
qj
0
Examples
dp
eiTp
2
265
/2
P
0
(7.105) and
we
note the
manner
divergences
in which the
are
cancelled here. From
aq2, potential U(q) integration measures in (7.105) contain implicit factors Of W12 that maintain covariance). Similarly, it is easily verified, by a simple change of variables, that for a potential of the 2 form U(q) aq + bq these correction terms vanish, again as expected. Fi-
this
we
immediately see that
the corrections
(7.105)
for the harmonic oscillator
vanish
(note
=
that the
=
nally,
for
quartic potential U(q)
a
q2
=
2
q4
+
a numerical integration of (7.105) for T Duistermaat-Heckman formula yields Zo
4
(the
anharmonic
oscillator),
-0.538 and the gives 6ZR2 2,7r. A numerical integration of
i
=
4.851, which differs from the value original partition function gives Z do not The 5.745. corrections Zo + 6ZR2 give the exact value here, but again at least they are a better approximation than the Duistermaat-Heckman formula. Again, a refinement of the preferred coordinates could lead to a better approximation. The method of the last Section has therefore "stripped" off any potentially divergent contributions to the loop-expansion and at the same time approximated the partition function in a much better way. These last few examples illustrate the applicability and the complete consistency of the geometric approach of the last Section to the saddle-point expansion. Indeed, we see that it reproduces the precise analytic features of the loop-expansion but avoids many of the cumbersome calculations in evaluating (7.3). It would be interesting to develop some of these ideas further. We would next like to check, following the analysis of Section 5.9, if there are any conformally-invariant geometries for this dynamical system when the potential U(q) > 0 is bounded from below. In the harmonic-polar coordinates (5.274), the conformal Killing equations (7.35) can be determined by setting the right-hand sides of the Killing equations (5.276) equal to instead (aoVO + FOOOVO)gm,. After some algebra, we find that they (V0V0)gj,, generate the 2 equations the
=
=
=
00 log
(
,00 log
(7.107)
can
be
)
grr
(
formally solved
2
groOr log Vo
(7.106)
grr
V0 900
goo
gro
gro
ar log V0
(7.107)
as
0
gro
=
-Vogoo
f
dO'
o9,Vo'
+
(7.108)
f (r)
00
from which
we see
only when (5.279)
that again
go (r, 0) holds g,0 (r, 0 + 27r) is the harmonic oscillator potential
single-valuedness
is true, i.e. when
U(q)
=
266
with
Beyond
7.
V
=
=
Approximation
oscillator, the equations (7.106) and radially-symmetric solutions g,,,, g,,(r) so global isometry of g. Thus, even though we lose the third
1. Even for the harmonic
(7.107) only that VO
the Semi-Classical
to admit
seem
1 is
a
=
equation in (5.276) which established the results of Section 5.9 using the Killing equations, we still arrive at the conclusion that there are no singlevalued metric tensors obeying the conformal Lie derivative requirement for
essentially all potentials which are bounded from below (and the harmonic oscillator only seems to generate isometries). Thus the conformal symmetry requirement in the case at hand does not lead to any new localizable systems. Finally, we examine what can be learned in these cases from the vanishing of the
2-loop,
(7.19) in harmonic coordinates. (7.8) has components
correction
the connection 1-form
^YP and the condition
(7.19)
=
0
'YY
In these
coordinates,
q
(7.109)
dy
reads d
71yy 'U Y
=
_'Y
2
(7.110)
0, in which case U(q) is the (7-110). Either -y. potential, or -(., (y + a)-', where a is an integration constant. This latter solution, however, yields q(y) Cly 2+ ay + C0, which gives a potential U(q) which is not globally defined as a COO-function on R 2. Thus the only potential which is bounded from below that leads to a localizable partition function is that of the simple-harmonic oscillator. This example illustrates how the deep geometric analyses of this Chapter serve of use in examining the localizability properties of dynamical systems. As for these potential problems, it could prove of use in examining the localization features of other more complicated integrable systems [56]. There
are
2 solutions to
harmonic oscillator
=
=
7.5 Heuristic Generalizations to Path
Integrals:
Supersymmetry Breaking generalization of the loop expansion to functional integrals is known, although some formal suggestive techniques for carrying out
The
not
yet
the full
semi-classical expansion can be found in [931 and [147]. It would be of utmost interest to carry out an analysis along the lines of this Chapter for
path integrals for several reasons. There the appropriate loop space expansion should again be covariantized, but this time the functional result need not be fully independent of the loop space coordinates. This is because the quantum corrections could cause anomalies for many of the symmetries of the classical theory (i.e. of the classical partition function). In particular, the larger conformal dynamical structures discussed in Section 7.2 above could
7.5 Heuristic Generalizations to Path
Integrals: Supersymmetry Breaking
267
important role in path integral localizations which are expressed in trajectories on the phase space [134]. It would be very interesting to see if these general conformal symmetries of the classical theory remain unbroken by quantum corrections in a path integral generalization. The absence of such a conformal anomaly could then lead to a generalization of the above extended localizations to path integral localization formulas. As this symmetry in the finite-dimensional case is not represented by a nilpotent operator, such as an exterior derivative, one would need some sort of generalized supersymmetry arguments to establish the localization with these sorts of symmetries. When these supersymmetries are globally present, the vanishing of higher-loop terms in the path integral loop expansion is a result of the usual non-renormalizations of I-loop quantities in supersymmetric quantum
play
an
terms of
field theories that arise from the mutual cancellations between bosonic and
loops in perturbation theory (where the fermionic loops have an sign compared to the bosonic ones). Quite generally though, one also has to keep in mind that the loop space
fermionic
extra minus
localization formulas
are
rather formal. We have overlooked several formal
functional aspects, such as difficulties associated with the definition of the path integral measure. There may be anomalies associated with the argument
path integral is independent of the limiting parameter R, for instance the supersymmetry may be broken in the quantum theory A g). (e.g. by a scale anomaly in the rescaling of the phase space metric g The same sort of anomalies could also break the larger conformal symmetry we have found for the classical theory above. However, even if the localization formulas are not correct as they stand, it would then be interesting to uncover the reasons for that. This could then provide one with a systematic geometric method for analysing corrections to the WKB approximation. The ideas presented in this Chapter are a small step forward in this direction. In particular, it would be interesting to generalize the construction of Section 7.3, as this is the one that is intimately connected to the integrability features of the dynamical system. The Poincar6 duality interpretation there is one possible way that the construction could generalize to path integrals. For path integrals, we would expect the feature of an invariant metric tensor that cannot be extended globally to manifest itself as a local (i.e. classical) supersymmetry of the theory which is dynamically broken globally on the loop space. This has been discussed by Niemi and Palo [122] in the context of the supersymmetric non-linear sigma-model (see Chapter 8). Another place where the metric could enter into a breakdown of the localization formulas is when the localization 1-form V) iwg does not lead to a homotopically-trivial element under the (infinitesimal) supersymmetry transformation described by QS. Then additional input into the localization formalism should be required on a topologically non-trivial phase space to ensure that QSO indeed does reside in the trivial homotopy class. These inputs could follow from an appropriate loop space extension of the correction in Section 4.4 that the
,\ E
-->
-
-
268
terms
7.
Beyond the Semi-Classical Approximation
JZ(T)
discussed
above,
which will then
always reflect global properties
of the quantum theory. Other directions could also entail examining the connections between equivariant localization and other ideas we have discussed in this Book. One is the Parisi-Sourlas supersymmetry that
we
encountered
height sphere (Section 5.5), although this feature seems to be more intimately connected to the Kdhler geometry of S', as we showed above. The Kiffiler symmetries we found in Section 7.1 would be a good probe of the path integral correction formulas, and it would interesting to see if they could also be generalized to some sort of supersymmetric structure. in the evaluation of the Niemi-Tirkkonen localization formula for the
function
on
the
8.
in
Equivariant Localization Cohomological Field Theory
seen that the equivariant localization formalism is an excellent, conceptual geometric arena for studies of supersymmetric and topological field theories, and more generally of (quantum) integrability. Given that the Hamiltonians in an integrable hierarchy are functionals of action variables alone [106], the equivariant localization formalism might yield a geometric characterization of quantum integrability, and perhaps some deeper connection between quantum-integrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory. In this final Chapter we shall discuss some of the true field theoretical models to which the equivariant localization formalism can be applied. We
We have
that the quantum field theories which fall into this framework al-
shall
see
ways
have,
gauge
as
anticipated,
symmetry
cohomological
or a
some
large symmetry
supersymmetry)
structure
on
that
group
serves
to
the space of fields that
(such
as a
provide
can
an
topological equivariant
be understood
as a
"hidden" supersymmetry of the theory. Furthermore, the configuration spaces of these models must always admit some sort of (pre-)symplectic structure
properties of phase space path integrals can applied. Because of space considerations we have not attempted to give a detailed presentation here and simply present an overview of the various constructions and applications, mainly just presenting results that have been
in order that the localization
be
obtained. The interested reader is refered to the extensive list of references
Chapter for m ore details. We shall emphasize throughout this Book between the localizaformalism for dynamical systems and genuine topological quantum field tion theories. Exploring the connections between the topological field theories and more conventional physical quantum field theories will then demonstrate how the equivariant localization formalism for phase space path integrals serves as the correct arena for studying the (path integral) quantization of real physical that
are
cited
throughout
this
here the connections eluded to
systems.
R. J. Szabo: LNPm 63, pp. 269 - 294, 2000 © Springer-Verlag Berlin Heidelberg 2000
270
8.
Equivaxiant Localization
8.1 Two-Dimensional
Between
Physical
In Section 3.8
we
first
and
Cohomological
in
Field
Theory
Yang-Mills Theory: Equivalences Topological Gauge Theories
pointed
that, instead
out
consider the Poisson action of
of circular
actions,
one can
non-abelian Lie group acting on the phase space. Then the non-abelian generalizations of the equivariant localization formulas, discussed in Sections 3.8, 4.9 and 5.8, lead to richer structures some
in the quantum
representations discussed earlier and one obtains intriguing path integral representations of the groups involved. In this Section we shall demonstrate how a formal application of the Witten localization formalism can be used to study a cohomological. formulation of 2-dimensional QCD (equivalently the weak-coupling limit of 2-dimensional pure Yang-Mills theory). This leads to interesting physical and mathematical insights into the structures of these theories. We shall also discuss how these results
can
be
generalized to topological field theory limits of other models. First, we briefly review some of the standard lore of 2-dimensional QCD. The action for pure Yang-Mills theory on a 2-dimensional surface Zh of genus h is
Sym[A]
=
e2
-
-
f
tr
FA
(8.1)
FA
*
Eh a gauge connection of a trivial prinicipal G-bundle over Zh, FA is its curvature 2-form (c.f. Section 2.4), and e2 is the coupling constant
where A is
theory. Since 0 =- *FA is a scalar field in 2-dimensions, (8.1) depends on the metric of Zh only through its dependence
of the gauge field
the action on
the
area
A(Zh)
*1 of the surface. A deformation of the metric
can
constant
e2. The action
is invariant under the gauge transformations (2.72). The quantum field theory is described by the path integral
corresponding
therefore be
compensated by
a
change
in the
coupling
(8.1)
ZZh(e 2)
f [dA]
=
e sym [A]
(8.2)
A
where A is the space of gauge connections We can write the partition function in
by treating the g-valued multiplier to write Z-Vh (e 2)
scalar field
[dA] A
In the weak
logical
one
and which
coupling
a
limit e2
fh tr iOFA (i.e.
-+
one
0, the
simpler (first order) form g) as a Lagrange
Coo (_ph,
e_'f-htr( OFA+22 0*0) f
(8.3)
action in
(8.4)
reduces to the topo-
independent of the metric Of Zh cohomological. 4uantum field theory)'.
that is
topological field theory
Schwarz-type topological
E
(Eh,g)
consequently determines
This sort of of
C-
a
much
*FA
[do]
_Th.
over
a
is called
gauge
a
'BF
theory [22].
theory'
and it is the prototype
8.1 Two-Dimensional
Yang-Mills Theory
271
The gauge invariance of the action S[o, A] appearing in (8.3) is expressed as S[g-log, Ag] S[o, A] where g E g and 0 transforms under the adjoint =
gauge group. Because of this gauge invariance, it is necgauge and restrict the integration in (8.3) to the equivalence of gauge connections modulo gauge transformations. This can
representation of the essary to fix
AIG
classes
a
done
by
brief
account).
an
the standard BRST gauge
fixing procedure (see Appendix
A for
a
For this, we introduce an auxilliary, g-valued fermion field 0", which is anti-commuting 1-form in the adjoint representation of g, and write (8.3)
as
I
1
(e2)
ZZ ph
Vol Coo (Zh'
g)
I
[dA] [dO]
AOA'A
C-
[do]
(Zh,g) N
f
xexp -ftr
OFA-
V)
A
0
ie2
(8.4)
f 0*0-if *IQ, Tl} I tr
Zh
Zh
Zh
(8.4)
Faddeev-Popov gauge fixing by the graded BRST commutator of a gauge fermion TV ,0A17,,(x) with the usual BRST charge Q. The square of Q is Q2 _ijo where J,p is the generator of a gauge transformation with infinitesimal parameter 0. Thus Q is nilpotent on the space of physical (i.e. gauge-invariant) states of the quantum field theory. The system of fields (A,,O, 0) is the basic multiplet of cohomological Yang-Mills theory. The (infinitesimal) gauge invariance of (8.4) In
we
have introduced the usual BRST and
terms defined
=
=
is manifested in its invariance under the infinitesimal BRST
supersymmetry
transformations
JAI-t
=
icolt
1
601z
=
-E(VA)AO
=
+
[A,,, 0])
,
JO
=
0
(8.5)
anticommuting parameter. The supersymmetry transformations (8.5) are generated by the graded BRST commutator &P -ijQ, Pj for each field 4i in the multiplet (A,,O, 0). The ghost quantum numbers (Z-gradings) of the fields (A, 0, 0) are (0, 1, 2). We shall not enter here into a discussion of the physical characteristics of 2-dimensional Yang-Mills theory. It is a super-renormalizable quantum field theory which is exactly solvable and whose simplicity therefore allows one to explore the possible structures of more complicated non-abelian gauge theories such as higher-dimensional cohomological field theories and other physical models such as 4-dimensional QCD. It can be solved using group character expansion methods [34] or by diagonalization of the functional integration in (8.3) onto the Cartan subalgebra using the elegant Weyl integral formula [29]. Here we wish to point out the observation of Witten [171] that the BRST gauge-fixed path integral (8.4) is an infinite-dimensional version of the partition function in the last line of (3.125) used for non-abelian localization. Indeed, the integration over the auxilliary fermion fields 0 acts to
where
e
is
an
=
272
8.
produce
Equivariant Localization
in
Cohomological
Field
Theory
field theoretical
analog of the super-I oop, space Liouville measure Chapter 4. The "Hamiltonian" here is the field strength tensor FA while the Lagrange multiplier fields 0 serve as the dynamical generators of the symmetric algebra S(g*) used to generate the G-equivariant cohomology. The "phase space" M is now the space A of gauge connections, and the Cartan equivariant exterior derivative a
introduced in
D
I (01'_6 6AI-i *
=
-
io'V"'
6
60,4
_V h
in this
case
coincides with the action of the BRST
)
(8.6)
charge Q,
i.e. DO
=
f Q, 01. The gauge fermion TI thus acts as the localization 1-form A and, by the equivariant localization principle, the integration will localize onto the -
field
configurations
vector fields
where
generating
A(V')
=
VI,4ff,,
=
0 where V'
=
V',/'
'9 ax"
are
the
g.
equivalence between the first and last lines of (3.125) is the basis of the mapping between "physical" Yang-Mills theory with action (8.1) and the cohomological Yang-Mills theory with actionf5-htr iOFA which is defined essentially by the steps which lead to the non-abelian localization principle, but now in reverse. The extrema of the action (8.1) are the classical YangMills field equations FA 0. Thus the localization of the partition function will be onto the symplectic quotient MO which here is the moduli space The
=
of flat gauge connections modulo gauge transformations associated with the gauge group G. This mapping between the physical gauge theory and the
cohomological quantum field theory is the basis for the localization of the 2-dimensional Yang-Mills partition function. Thus the large equivariant cohomological symmetry of this theory explains its strong solvability properties that have been known for quite sometime now. More generally, as mentioned at the end of Section 5.8, the equivariant localization here also applies to the basic integrable models which are related to free field theory reductions of 2-dimensional Yang-Mills theory, such as Calegoro-Moser integrable models
[56]. To carry out the localization onto MO explicitly, we choose a G-invariant metric g on Zh and take the localization 1-form in (3.125) to be
I\
=
I
tr
0
A
*Df
Eh
where
f
=
*FA. The localization
=
f dvol(g(x))
tr
O"DIJ
(8.7)
_r h
onto
A(V)
ization onto the solutions of the classical
=
0 is then identical to local-
Yang-Mills equations. We shall not enter into the cumbersome details of the evaluation of the partition function (8.4) at weak coupling e 2 --+ 0 (the localization limit) using the Witten nonabelian localization formalism. For details, the reader is refered to [171]. As in Section 5.8, the final integration formula can be written as a sum over the unitary irreducible representations of G,
8.1 Two-Dimensional
ZZh (e 2)
2 -
=
P?
-
e
11 a(P)2-2h a>O
(dim
Yang-Mills Theory
RX)2-2h
e2 e
_-T
273
(8.8)
AEZ'
expanding the various (G-invariant) physical quanin the localization formula in characters of the group G (c.f.
This result follows from tities
appearing
Section 5.8). From a physical standpoint the localization formula (8.8) is interesting because although it expresses the exactness of a loop approximation to the partition function, it is a non-polynomial function of the coupling constant e2. This non-polynomial dependence arises from the contributions of the unstable classical solutions to the functional integral as described in Chapter 3. Such behaviours are not readily determined using the conventional perturbative techniques of quantum field theory. Thus the mapping provided above between the physical and topological gauge theories (equivalently the generalization of the Duistermaat-Heckman integration formula to problems with non-abelian symmetries) provides an unexpected and new insight into the structure of the partition function of 2-dimensional Yang-Mills theory. This simple mapp ing provides a clearer picture of this quantum field theoretical equivalence which is analogous to the more mysterious equivalence of topological and physical gravity in 2 dimensions. Rom a mathematical perspective, the quantity (8.8) is the correct one to use for determining the intersection numbers of the moduli space of flat G-connections on Zh [78, 82, 171]. This approach to 2-dimensional Yang-Mills theory has also been studied for genus h
=
0 in
[105].
The
intriguing mapping between a physical gauge theory, with propagating particle-like local degrees of freedom, and a topological field theory with only global degrees of freedom has also been applied to more complicated models. In [28], similar considerations were applied to the non-linear cousin of 2-dimensional topological Yang-Mills theory, the gauged GIG WessZumino-Witten model. This model at level k E Z is defined by the action
SGIG [g, A]
k
f
T7r
tr
9_1VA9 A *9_1VA9 +
k 127r
Eh
k -
47r
f tr(g-ldg)A3 M
I
tr
(A A dgg-1
+ A A
(8.9)
Ag)
Zh
where g (=- C, (Zh' g) is a smooth group-valued field, M is a 3-manifold with boundary the surface Eh , and A is a gauge field for the diagonal G subgroup
GR symmetry group of the ordinary (ungauged) Wess-Zuminoby the action SG [g] SGIG [g, A 0] [54]. Since the is * when acting on 1-forms, invariant Hodge duality operator conformally the action (8.9) depends only on the chosen complex structure of Z h. As for the Yang-Mills theory above, the geometric interpretation of the theory comes from adding to the bosonic action (8.9) the term Q(O) =-L 042 21r fEh quadratic in Grassmann-odd variables 0 which represents the symplectic form of the
GL
x
Witten model defined
=
=
274
Equivariant Localization
8.
in
Cohomological
Field
Theory
JAJA on the space A of gauge fields on _Vh. Again the resulting theory supersymmetric and the infinitesimal supersymmetry transformations are
f,, is
JA,
=
0,
with the
6V) ,
,
=
Ag
-
Z
A,
;
supplemental condition Jg
JA2 =
02
=
J,02
,
0. Unlike its
=
A
-
(Aq)
-'
Z
(8. 10)
Yang-Mills theory counter-
5' does not generate infinitesimal part, the square of this supersymmetry,6 transformations but rather 'global' gauge transformations (generated gauge the of the elements cohomological by gauge group which are not connected =
to the
identity).
Thus the action
less,
(8.9)
here admits
supersymmetry which does not mani-
a
the local gauge symmetry of the quantum field theory. Nonethethis implies a supersymmetric structure for equivariant cohomology
fest itself
which
as
can
be used to obtain
localization of the
a
corresponding path integral
in the usual way. The localization formula of [23] for equivariant Kdhler geometry has a field theoretic realization in this model [28] and the fixed-point
localization formula is the
algebraic
Verlinde formula for the dimension of
the space of conformal blocks of the ordinary Wess-Zumino-Witten model. For example, in the case G SU(2) the localization formula gives =
Z_r h(SU(2), k)
f
=
[dg]
)
2
k + 2 Thus the some
2-2h
k+1
1: e=1
(sin k+2) 7r
equivariant localization formalism
of the
more
[138]
e SsU(2)ISU(2)[g,A]
A
(Zh,su(2))
C-
f [dA]
can
2-2h
also be used to shed
(8.11)
light
on
formal structures of 2-dimensional conformal field theories.
path integral for a version of the ordinary Wessgive a field theoretical generalization of Stone's derivation for the Weyl-Kac character formula for Kac-Moody algebras (i.e. loop groups). This is done by exploiting the GL x GR Kac-Moody symmetry associated with the quantum field theory with action SG[g] as a supersymmetry of the model along the same lines as in Chapter 5 before. Let us now briefly describe Perret's derivation. The Kac-Moody group d is a central extension of the loop group S' --+ d -4 LG of a compact semi-simple Lie group G, and it looks locally like the direct product LG (& S', i.e. an element j E G looks locally like j (g(x), c) where g : S1 ---* G and C E S1. The coherent state path integral for the character is Perret
has used the
Zumino-Witten model to
=
tr,\
e
iH
q
Lo
f [dj]
=
e
f
jd+i(H+-rLo)dtjj)
(8.12)
L6
where H of the
=
loops.
Ej hjHj
E
HC,
The action in
q
=
eir and LO is the generator of rotations
(8.12) depends
on
2
coordinates, the coordinate
8.1 Two-Dimensional
x
along
central
the
loop
SI-part
in
Lg and the
Yang-Mills Theory
275
time coordinate t of the
of the coherent states in the
path integral. The path integral drops out due to
invariance, and thus the character representation (8.12) will define a theory on the torus T 2 (i.e. the quotient of loops
gauge
2-dimensional quantum field in the loop group LG). It
can
be shown
[138]
that the coherent state
path integral (8.12)
is the
quantum field theory with action
SKM
-1
:-`
21r
f
dx dt tr
(Ag- (6 1
+
H)g
k +
g-1 ax (a +
T2
(8.13)
k +
2H)g)
tr(g- ldg)A3
121r M
Here k E Z is the
weight
of
given central extension of the loop group, A is a dominant at -ro9., so that the Cartan angle -r becomes the G, and 5 =
-
modular parameter of the torus. When A H 0 the action (8.13) becomes the chiral Wess-Zumino-Witten model with the single Kac-Moody symmetry g --+ k(z)g. This symmetry is still present for generic A = - 0, and we can gauge =
the action with
a
SKM with respect
vector field
(see (8.9)).
to
an
One
=
arbitrary subgroup
can now
of G
by replacing
H
evaluate the infinite-dimensional
Duistermaat-Heckman integration formula for this path integral. The critical points of the action (8.13) are in one-to-one correspondence with the affine Weyl group Waff (Hc) W(Hc) z) kc, where Ac is the set of co-roots. Let =
r(w)
denote the rank of an element
w
W(Hc),
E
and let h be the dual Coxeter
number of the Lie group G [162]. Using modular i nvariance of the character to fix the conventional zero-point energy (associated with the usual SL(2, R)invariance of the conformal field theory vacuum [54]), it can be shown that the WKB localization formula for the coherent state
path integral (8.12)
with the
notation
Weyl-Kac
Lo tr,\ eW q
character formula
1:
[138] (for
see
coincides
Section
5.1)
(- 1)'(') q E,((A+p)i+(k+h)77i2-pi2)/2(k+h)
wEW(Hc),??Ef1c x
e
i(A+p+(k+h)77)(H(') )
e-ip(H(-))
qn)-r n>O
X
e
C >o
ia(H(-))
qn)
e-ia(H(')) qn-1 (8.14)
which arises from the expansions of various quantities defined on the torus in terms of Jacobi theta-functions. Thus infinite dimensional analogs of the
localization formulas in quantum field theory can also lead to interesting generalizations of the character formulas that the topological field theories of earlier Chapters represented.
276
8.
Equivariant Localization
Finally, localization
it is as
possible
well. For
to
Cohomological Field Theory
use some
instance,
Scsp [A]
in
k =
87r
in
of these ideas in the context of abelian
[1261
the abelian gauge
j AAdA--m j 2
M
defined
on
a
3-manifold M
framework. The first term in
topological
field
theory 2,
theory with
(8.15)
AA*A
M
studied within the equivariant; localization (8.15) is the Chern-Simons action which defines a was
while the second term is the Proca,
mass
term for the
gauge field which gives a propagating degree of freedom with mass breaks the topological invariance of the quantum field theory. In
formalism where M
=
Zh
x
action
R1,
one can
naturally write
m a
the model
and thus
canonical
(8.15)
as a
h
quantum mechanics problem on the phase space _T and apply the standard abelian equivariant localization techniques to evaluate the path integral from the ensuing supersymmetry generated by the gauge invariance of (8.15) (in the Lorentz gauge c9AA,_, 0) [26]. The path integral localization formula coincides with that of a simple harmonic oscill ator of frequency wh 8-7rm/k, =
=
i.e. Z
field
=
1/2 sin(T 4,7rm/k), indicating -
theory using
a
mapping
once
again
to
a
topological
the equivariant localization framework. The infrared limit
0 of the model leads to the usual
topological quantum mechanical theory [261 and the supersymmetry, which is determined by a loop space equivariant cohomology, emerges from the symplectic structure of the theory on _T h and could yield interesting results in the full 3-dimensional quantum field theory defined by (8.15). m
-4
models associated with Chern-Simons
Symplectic Geometry Quantum Field Theories
8.2
In the last Section
showed how
of Poincare
loop
Supersymmetric
equivariant localization progauge theories and topological ones which makes manifest the localization properties of the physical models. In these cases the original theory contains a large gauge symmetry which leads to a localization supersymmetry and a limit where the model becomes topological that gives the usual localization limit. It is now natural to ask what happens when the original quantum field theory is explicitly defined with a supersymmetry (i.e. one that is not "hidden"). In Section 4.2 -1 mechanics admits a loop space we saw that N 2 supersymmetric quantum equivariant cohomological structure as a result of the supersymmetry which provides an alternative explanation for the well-known localization properties of this topological field theory (where the topological nature now arises because the bosonic and fermionic degrees of freedom mutually cancel each vides
a
we
correspondence between
certain
space
physical
=
The Cherns-Simons action is in fact another prototype of
logical
field
theory [221.
a
Schwarz-type topo-
8.2 Poincar6
other
Supersymmetric Quantum
Field Theories
277
out).
From the point of view of path integration, this approach in fact nice, geometric interpretation on the functional loop space of the features of this theory. It has been argued [107, 108, 133] that this interpretation can be applied to generic quantum field theories with Poincar6
led to
a
supersymmetry. In this Section
we shall briefly discuss how this works and techniques of equivariant localization could lead to new geometrical interpretations of such models. To start, let us quickly review some of the standard ideas in Poincar6
how the
supersymmetric quantum field theories. The idea of supersymmetry was first used to relate particles of different spins to each other (e.g. the elementary representation theory of SU(3) which groups particles of the same spin into multiplets) by joining the internal (isospin) symmetries and space-time (Poincar6) symmetries into one large symmetry group (see [155] for a com-
prehensive introduction). This is not possible using bosonic commutation relations because then charges of internal symmetries have to commute with space-time transformations so that dynamical breaking of these symmetries (required since in nature such groupings of particles are not observed) is not possible. However, it is possible to consider anti-commutation relations, i.e. supersymmetries, and then the imposition of the Jacobi identity for the symmetry
group leads to
shall be concerned
a
very restricted set of commutation relations. Here
only
with tho
se
anti-commutation relations satisfied
we
by
the infinitesimal supersymmetry generators
f Q', Qi4 I a
where Z"
=
20 Z'
"4PA + ZV. a,3
.
;
N
.
%13
=
(8.16)
-1"C with -yA the Dirac matrices and C the charge conjugation matrix, P,, -0,, is the generator of space-time translations, and Pi is an antisymmetric matrix of operators proportional to the generators of the internal symmetry group. For the rest of the relations of the super-Poincar6 group, see [155]. We assume here that the spacetime has Minkowski signature. We shall be interested in using the relations of the super-Poincar6 algebra to obtain a symplectic structure on the space of fields. For this, it turns out that only the i i terms in (8.16) are relevant. It therefore suffices to consider an N 1 suPersymmetry with no internal Z'i symmetry group terms. The most expedient way to construct supersymmetric field theories (i.e. those with actions invariant under the full super-Poincar6 group) is to use a superspace formulation. We introduce 2 Weyl spinors 0' and #& which =
=
=
==
parametrize the infinitesimal supersymmetry transformations. Then the N I supersymmetry generators in 4 dimensions can be written as a
a
-
Q&
.
=
Kinetic terms in the supersymmetric action ant
_
aO& are
+
'A -
0111 a
(8.17)
constructed from the covari-
superderivatives a a0a
+
iZa6,PaA
,
D&
aO&
iZ.1'6,0"a4
(8.18)
278
8.
Equivariant Localization
in
Cohomological Field Theory
readily verified that with the representation (8.17) the relations of super-Poincar6 algebra are satisfied. In this superspace notation, eO"Q-+&6,Q6 and, a general group element of the supersymmetry algebra is using the supersymmetry algebra along with the Baker-Campbell-Hausdorff eo"Q- e66'06A(y) formula, its action on a field A(x) is e0'Q-+6 6Q6A(x) where y" x4 + WZ,",&P are coordinates in superspace. Thus the superIt
be
can
the N
=
I
=
=
symmetry transformation parameters live in superspace and the fields of the
supersymmetric field theory are defined on superspace. To incorporate the supersymmetry algebra as the symmetry algebra of a physical system, we need some representation of it in terms of fields defined over the space-time. The lengthy algorithm to construct supermultiplets associated with a given irreducible or reducible spin representation of the super-Poincar6 algebra can be found in [155]. For instance, in 4 dimensions chiral superfields (satisfying D&!P 0) are given by =
V(x, 0, 9) (0,,0)
where
are
spin (0,
=
0"(y)
fields, .1) 2
+
0'0 '(y)
+
a
and F
are
0'0,,F4(y)
(8.19)
auxilliary fields
use
to close
the supersymmetry representation defined by the chiral superfields. In the following we shall consider multiplets of highest spin 1. Other multiplets be obtained
by imposing some additional constraints. The most general supermultiplet in 4 space-time dimensions consists of a scalar field 3 M, pseudoscalar fields C, N and D, a vector field A,-, and 2 Dirac spinor fields X and A. The supersymmetry charges Q and Q' respectively raise and lower the (spin) helicity components of the mulitplets by 1. The super2 Poincar6 algebra can be represented in the Majorana representation where 0 2 iorl (g 1and y3 -io.3 (& 0.3, with a' -ia 30 a 1, 72 -9 o 1, 71 ly can
N
1
=
=
=
=
the usual Pauli spin matrices. Then the 4 side of (6fNsusyalg) are
-TO Z2 The
X
=
(
( 0
i
-
-i 1
) 1)
0
1
*
0
=
x
4 E-matrices
T,
T3
=
on
the
right-hand
(0 1) 1
0
0
-1
(8.20)
Majorana representation selects the preferred light-cone coordinates X2
XO for the translation generators P,, in (8.16) above. Different representations of the Dirac gamma-matrices would then define different pre=
light-cone directions. I supergeneral supersymmetry transformations of the complex N the and of the action M, A; multiplet V N, At,; (C; X; D) supersymmetry on V can be found in [133]. There it was shown that the incharges Q, finitesimal supersymmetry transformations can be written in a much simpler form using the auxilliary fields
ferred
The
=
Supersymmetric Quantum
8.2 Poincax6
M'
=
M +
Al
A3 +491C =
Al
N'
,
a3X1
-
A12
i
A,
N +
==
A2
=
-
-
Field Theories
D'
a3C A'3
101XI
D +
=
2A3
=
-
aIA3
-
279
o93AI
49-Xl
(8.21) precisely the (non-standard) auxilliary fields introduced in [107, 108] which, as we discussed in Section 4.9, form the basis for equivariant localization in supersymmetric quantum field theories and phase space path integrals whose Hamiltonians are functionals of isometry generators. Using these auxilliary fields we now define 2 functional derivative operators on the space of fields, These
are
T
d3X
D
i
X2
dt
6
9C
iA+
+
6
JXJ
iM'
+
6 +
iN'
3
5
JX4
0
-A/3
A/2
3-A-
A'
A_
iD'
Aq3
(8.22)
4
A
T
j
d3X
-TV+
dt
O+C
10+X3
+
6X2
6
j
6M,
+
19+X4 N' +,9+xi 6A+
0
-a+A3
a+Aj
JA'
5 -
JA'
a+A-
6
49+A4
6A'
6 JD1
(8.23) Here
we
imposed the boundary conditions on the fields P(x, t) that they xO, spatial infinity and that they be periodic in time t
have
vanish at
=
lim 0 (X, IXI-00 so
t)
=
P(x, t
0
+
T)
P(x, t)
=
(8.24)
that the space of fields can be thought of as a loop space. The operators (8.22) and (8.23) are nilpotent. If we now define
Q+ then it
can
=
D
(8.25)
+IV+
be checked that T
Q2+
DIV+
+ IV+ E)
=
LV+
d 3X
i
dt
ia+
(8.26)
0
I supersymmetry transThe operator (8.25) generates the appropriate N formations on the field multiplet and the supersymmetry algebra (8.26) co=
supersymmetry algebra (8.16). The above geometric representation of the superalgebra (8.16) on the space of fields of the supersymetric field theory. A different representation of Z" leads to a different choice of preferred Q, in (8.26) and a different decomposition of the fields into loop space coordinates and 1-forms
incides with the
pertinent N provides
construction therefore
=
I
a
280
in
Equivaxiant Localization
8.
(8.22)
and
(8.23).
It is
in
Cohomological
possible decompose
now
ric quantum field theories
Field
Theory
to examine how various
supersymmet-
with respect to the above equivariant cohomological structure on the space of fields. The canonical choice is the Wess-Zumino model which is defined by the sigma-model action T
SWZ
d3X
1 f dt
dO d
D P + W[
2
P]
(8.27)
0
where
W[4i]
is some super-potential. With respect to the above decomposipossible to show that the supersymmetric action decomposes into a sum of a loop space scalar H and a loop R + Q. space 2-form 0, S Because of the boundary conditions on the fields of the theory, the supersymmetry charges, which geometrically generate translations in the chosen light-cone direction, are nilpotent on the spaces of fields. By separating the loop space forms of different degrees, we find that the supersymmetry of the 0 and DH action, Q+S 0, implies separately that DS-2 -.TV+ Q. Thus the supersymmetric model admits a loop space symplectic structure and the corresponding path integral can be written as a super-loop space (i.e. phase 0 on the space of space) functional integration. Furthermore, because Q2 Q obeys lv+?9 fields, the symplectic potential 0 with DO H, so that the action is always locally a supersymmetry variation, S Q+,O (D + lv+),O. Thus, any generic quantum field theory with Poincar6 supersymmetry group admits a loop space symplectic structure and a corresponding U(1) equivariant cohomology responsible for localization of the supersymmetric path integral. The key feature is an appropriate auxilliary field formalism
tions it is
=
=
=
=
=
=
=
=
which defines
a
splitting
=
of the fields into
loop space "coordinates" and their general although the fields of the evenly into loop space coordinates
associated "differentials" 3. Notice that in
supersymmetric theory always split and
1-forms, the coordinates and fields. It is only in the simplest
mechanics)
up
1-forms involve both bosonic and fermionic cases
(e.g.
N
=
-1 2
supersymmetric
quan-
that the pure bosonic fields are identified as coordinates and the pure fermionic ones as 1-forms. In the auxilliary field formalism outlined above, the supersymmetry of the model is encoded within the model inde-
turn
pendent loop space equivariant cohomology defined by Q+. In this way, one a geometric interpretation of general Poincare supersymmetric quantum field theories and an explicit localization of the supersymmetric path integral onto the constant modes (zeroes of 0+). We shall not present any explicit examples of the above general constructions here. They have been verified in a number of cases. To check this formalism in special instances one needs to impose certain additional constraints on the multiplets [133] (e.g. the chirality condition mentioned above). The obtains
3Note
that the
problem of choosing an appropriate set of auxilliary fields is the analog of finding a preferred set of coordinates for an isom-
infinite-dimensional
etry generator
(c.f.
Section
5.2).
8.3
Supergeometry
and the
Batalin-aadkin-Vilkovisky
Formalism
281
above constructions have been in this way explictly carried out in [107, 108] 1 supersymmetric quantum mechanics (i.e. the (0 + l)-dimensional Wess-Zumino model (8.27)), the Wess-Zumino model (8.27) in both 2 and 4 dimensions, and 4-dimensional N 1 supersymmetric SU(N) Yang-Mills for N
=
=
theory
defined
by
the action
d4X
sym where
0
(_4 1
Fa 011F a,ttv +
Majorana fermion fields
are
gauge group. The model
(8.28)
in the
2
VA'O)
adjoint representation
be reduced to
(8.28) of the
Wess-Zumino model
by eliminating the unphysical degrees of freedom and representing the theory directly in terms of transverse (physical) degrees of freedom. The equivariant can
a
localization framework has also been applied to the related supersymmetries quantization [136] in [108]. Palo [133] applied these constructions to the 2-dimensional supersymmetric non-linear sigmaof Parisi-Sourlas stochastic
(i.e. the Wess-Zumino Section).
model
model
(8-27)
with
a
curved target space
-
see
the next
8.3
Supergeometry Batalin-Fradkin-Vilkovisky
and the
Formalism
The role that the Cartan exterior derivative of equivariant cohomology plays in localization resembles a Lagrangian BRST quantization in terms of the
gauge-fixing In
of
a
Lagrangian
[112, 113] equivariant
field
theory
localization
was
over
a gauge field 0,' [22]. in terms of the Batalin-
M with
interpreted
Vilkovisky Langrangian anti-field formalism. In this formalism, a theory with first-stage reducible constraints or with open gauge algebras is quantized by introducing an antibracket [13]-[15] which naturally introduces a new supersymmetry element into the BRST quantization scheme. This formulation is especially important for the construction of the complete quantum actions for
gauge theories (especially those of Schwarz-type) [22]. In this shall sketch some of the basic conceptual and computational ideas
topological
Section of this
we
formalism, which
lead to
more
in the context of localization for
dynamical systems topological field
direct connections with super symmetric and
theories. We return to the
(M, w, H). a
We have
formulation of
and Grassmann
simpler
situation of
a
generic Hamiltonian system
that the localization prescription naturally requires defined over a super-manifold (i.e. one with bosonic
seen
objects
coordinates), namely the cotangent
bundle MS =- MOT*M. supersymmetric quantum field theories that we considered in the previous Section, our fields where defined on a superspace and the path integral localizations were carried out over a superloop space. We would now like to try to exploit the mathematical characteristics of a super-manifold In the
case
of the
282
8.
Equivariant Localization
in
Cohomological
Theory
Field
and reformulate the localization concepts in the more rigorous framework of supergeometry. This is particularly important for some of the other localization features of
field theories that
topological
shall discuss in the next 2
we
Sections.
First, we shall incorporate the natural geometrical objects of the BatalinVilkovisky formalism into the equivariant localization framework. The local coordinates on the super-manifold MS are denoted as ZA (xA"qA). We define a Grassmann-odd degree symplectic structure on Ms by the nondegenerate odd symplectic 2-form =
S21
=
The 2-form
B d ZA A C)i 'ABdz
(8.29)
wlzvdx"
=
determines
bracket. It is defined
aA
dn'
4a
aA
qZB
'9xA
+
A(x,,q)
and
77\dn"
ax,\
odd Poisson bracket
awt"'
where
19W/1-
+
on
B(x,,q)
tisymmetry properties bracket,
gA, BC11
are
are
gA,BJ1 =
),
opposite
a?7v
aA
super-functions
09XII
B
anti-
'0771,
on
(8.30)
anv Ms. The grading and
to those of the
ordinary graded
an-
Poisson
_(_1)(p(A)+1)(p(13)+1)J13, All
=
gA, L311C
+
(_ 1)p(13) (p(A)+I) BgA, CR g gA, BI 1, C1 (8.31) super-function A(x, 77) with the
=
-
the
super-coordinate
are
9XA, X, I i
=
9x"' 77, 11
0
1,,', 71, We define
a
mapping
!2W,11(x)?7Aq"
=
on
-V7v,','h
COO(M)
-+
-
Qf (Z)
=
=
-
W, x" I I
=
W/4v
(8.32)
49WAV =
5- 77
C'(Ms) using
If, W11
the
super-function
f A, f },,.
In
=
-2-f-77 ,gxA
A
(8.33)
original Poisson bracket of the phase particular, the dynamical systems (M, w, H)
Then the antibracket coincides with the
I A, f I,
=
by fW
space,
A
+
p(A) is the Grassmann degree of the p(A) + p(B) + I In particular, property p(JA, BI)
=
(8.29)
MS called the
M
B
71 ji-A x anA
where
W(Z)
dqv
1) (P(4)+') (P(C)+') 9B, 9A, C1 1 1
gA, gB, C) I 1
antibrackets
A
by
f2ABL3
5Z-A
an
A
Supergeometry
8.3
and
(MS,fll, QH)
tions of motion
:VZ
=
determine
a
bi-Hamiltonian pair. The
Formalism
283
corresponding
equa-
are
JXO) QHJ1
Generally,
Batalin-Fradkin-Vilkovisky
and the
=
JxO, HJ,,
V1'
=
W'
,
=
1?7t7 QHJ1
=
IOX177V (8.34)
readily seen that the operation Jw, .11 acts as exterior difd, JH, .11 acts like interior multiplication iv with respect to the Hamiltonian vector field V, and I QH 1 1 acts as the Lie derivative Lv along V. The antibracket provides an equivalent supersymmetric generalization of the ordinary Hamiltonian dynamics. The key feature is that the supersymmetry of the odd Hamiltonian system (MS 011 QH) is equivalent to the equivariant cohomology determined by the equivariant exterior derviative DV d + iv. If M admits an invariant Riemannian metric tensor g (the equivariant localization constraints), then the super-function it is
ferentiation
-
7
1
=
TH is
an
integral
JQHJH11 structure
=
g"VVIJI 77V
(8-35)
of motion for the Hamiltonian system (MS, S?1 7 QH) , i.e. 0. Furthermore, TH determines the usual bi-Hamiltonian
M because
on
2
With these observations
JwJH11
=
!(Qv)4vqAqv 2
easily
one can
now
ariant localization written
as an
principle. The usual localization integral over the super-manifold MS,
Z(s)
f
1 =
(iT)n
d
4n
z
KV. JHJH11 equiv(classical) integral (2.128) can be and
=
establish the
eiT(H+w)-sJH-w,1HR1
(8.36)
MS
where
as
always
form d4nZ
=
the classical partition function is Z(T) Z(O). The volume is invariant under the equivariant transformations of =
d2nx d 2n,,
DV and LV determined by the anti-brackets. Furthermore,
JH
-
w,
e
iT(H+w)-sJH-w,.THJ1
9QH,IH
e
11
=
JQH,
e
we
have
iT(H+w)-sJH-w,TffJ1
iT(H+w)-sJH-w,_THR11
I,
=
0
0
(8-37) The first 2
vanishing
conditions
just represent the invariance of the integrand in (8.36) under the actions of the operators DV and CV, so that the usual equivariant cohomological structure for localization in (8.36) is manifested in the supersymmetry of the antibracket formalism. With the identities (8.37), it is straightforward to establish that jd Z(s) 0, and hence the localization ds =
principle (i.e. the Duistermaat-Heckman theorem). Thus the lifting of the original Hamiltonian system to the odd one defined over a supermanifold has provided another supersymmetric way to interpret the localization, this time in terms of the presence of supersymmetric biHamiltonian dynamics with even and odd symplectic structures which is the usual Batalin-Vilkovisky procedure for the evaluation of BRST gauge-fixed
284
8.
Equivariant Localization in Cohomological Field Theory
path integrals. The representation (8.36) of the canonical localization integral formally coincides with the representation of differential forms in the case where the original space M is a supermanifold. In [150], Schwarz and Zaboronsky derived some general localization formulas for integrals over a finite-dimensional supermanifold M where the integrand is invariant under the action of an odd vector field W. Using the supergeometry of M, they formulated sufficient conditions which generalize those above under which the integral localizes onto the zero locus of the c-number part W(,q 0) of W. Their theo rems quite naturally generalize the usual equivariant localization principles and could apply to physical models such as those where the Batalin-Vilkovisky formalism is applicable [149] or in the dimensional reduction mechanism of the Parisi-Sourlas model [136]. A generalization to more general (non-linear) supermanifolds can be found in [175]. Nersessian has also demonstrated how to incorporate the anti-bracket structure into the other models of equivariant cohomology (other than the Cartan model see Appendix B) in [115] and how the usual equivariant =
-
characteristic class representations of the localization formulas appear in the Batalin-Vilkovisky formalism in [113]. The superspace structure of cohomo-
logical
field theories in this context has been studied
by
Niemi and Tirkkonen
[129]. They discussed the role of the BRST model of equivariant cohomology in non-abelian localization (see Appendix B) and topological field theories in
and showed how these
an
appropriate superfield formulation
equivariant cohomological
structures to the
can
be used to relate
Batalin-Radkin-Vilkovisky
Hamiltonian quantization of constrained systems with first stage reducible constraints. This suggests a geometric (superspace) picture of the localization
properties of some topological quantum field theories such as 4-dimensional topolog ical Yang-Mills theory (defined by the action f tr FA A FA). This picture is similar to those of the Poincar6 supersymmetric theories described in the previous Section in that the BRST charge of the cohomological field theory can be taken to generate translations in the q-direction in superspace and then the connection between equivariant cohomology and BatalinFradkin-Vilkovisky quantization of 4-dimensional topological Yang-Mills theory becomes transparent. These superfield formalisms therefore describe both equivariant cohomology in the symplectic setting relevant for localization and the BRST structure of cohomological field theories. This seems to imply the existence of a unified description of localization in the symplectic loop space, the supersymmetric loop space of the last Section, and in cohomological. field theory. Indeed, it is conjectured that all lower dimensional integrable models are obtainable as dimensional reductions of 4-dimensional self-dual YangMills theory (i.e. FA *FA) which is intimately connected to topological Yang-Mills theory. Finally, the incorporation of the Batalin-Vilkovisky formalism into loop space localization has been discussed recently by Miettinen in [103]. For path integral quantization, one needs an even Hamiltonian and symplectic struc=
Supergeometry
8.3
ture. This
An
can
be done
symplectic
even
9%
1 =
2
1
Batalin-Radkin-Vilkovisky
and the
Formalism
285
provided that M has on it a Riemannian structure. on MS is given by the super-symplectic 2-form
structure
(w"V + Rjv,\p?7AqP) dx" A dxv + 2gjvD9?7mA D977
V
(8.38)
where
D9771' is the covariant derivative
=
dqm
+
rv",\?7vdx'\
(8.39)
M, and the subscript i 0, 2 labels the Hamilw' and Ho S?v, H2 H) w, (M, Kv) The corresponding symplectic I-forms, with S?' &9 , are then tonian
systems
(M, wo
on
=
=
=
=
=
-
=
eo
0A dx,4
=
+
gI'ZVq1iDg,v
The 2-forms 0' determine the
A,Bjj
=
192
,
=
g,,,Vvdxm
+
Poisson brackets
even
(VjA)[w'tv
gj,,,qmDgij'
on
+gm'
(8.40)
MS
aA
a'q1-I
B
15Oqv
(8.41)
where
v,
=
a,
r-N jj'V?7V-
-
Then the equations of motion for the odd and
coincide, jz A, jj, ,
=
(8.42)
0977A
IZA, QHJ1,
even
Poisson brackets
on
Ms
where Hi Hi + f2v. Thus the odd and bi-Hamiltonian structures also on the super=
Poisson brackets provide symplectic manifold MS and therefore the Hamiltonian system is also integrable on Ms. Given this integrability feature on MS, we can now examine the corresponding localizations [150]. We write the partition functions Zj(T) strIl e -Mii 11 as path integrals over a super-loop space corresponding to LMS by absorbing the Liouville measure factors associated with 0' into the argument of the action in the usual way (c.f. Chapter 4). Then the Hamiltonian even
=
systems
(Ms, S-2',Hi)
have the quantum actions
[103]
T
So
dt
(01,--bm
-
H +
-177"g,,, 2
1
dt
+
2
(.f2V)1,vqI'?7'
+
Iw,,,,AI'Av
2
0
+
1Rmv,\pA"Av?7A 77
P
2
+
1g,,vFMF'
2
T
S2
1
dt
g,,,Vv&"
-
KV +
1 2
77A9pV
dt
+
2
(Qv)mvqlqv
0
+
I 1(S?v)tvA"Av + R,,,,\pAI'Avq'\?7 2 2
P
1 +
2
gtzvFmF' (8.43)
286
Equivariant Localization
8.
Cohomological
in
where the quantum partition functions
Zi(T)
I
=
Field
Theory
are
2n 2n FA [d 2nX1 [d 2n,7] [d F] [d A] e'si [ x,,q; 1
(8.44)
LMsOLA'Ms
and
we
have introduced
bosonic variables F11 the usual way. If H
LTdt 2'g Av
-
=
dx" and auxilliary anticommuting variables A" dqll to exponentiate the determinant factors in 0 (the topological limit), then the action So + -
(0
is that of the
+
I)-dimensional
N
=
supersymmetric
I
1 DeRham supersymmetric sigma-model, i.e. the action of N 4 quantum mechanics in background gravitational and gauge fields One can now develop the standard machinery to evaluate these path integrals using super-loop space equivariant cohomology. This has been done explicitly in [122]. The super-loop space equivariant exterior derivative is
non-linear
=
T
I (
Q
dt
AA
J
TX_A
J
+ F"
+
Jql,
(: A
-
VA)
J +
JAA
0/4
+
alj'Vv?7,)
JFA
0
(8.45) Q2
LV where S is the classical action associated with the Hamiltonian original system (M, w, H). Notice that a canonical conjugation dl does --+ not alter the cohomology groups of the derivative =_ C Q e-Q the loop space functional operator Q. Choosing with
=
Ls
=
Lb
-
T
dt
0
R\
AV
7 7A A
J
V
(8.46)
T_ FTA
0
can be explicitly worked out (see (topologically equivalent) operator [122]). With regards to this supersymmetry charge, the pertinent action So in (8.43) can be obtained from the 2-dimensional N 1 supersymmetric sigma-
the
=
by partial localization of it to a I-dimensional model. This is done by breaking its (left-right) (1, 1) supersymmetry explicitly by the Hamiltonian flow. In this procedure the usual boson kinetic term (in light-cone coordinates) g,,,,a+0A,9_0v drops out. The path integral Zo(T) can be evaluated by adding an explicit gauge-fixing term ,o to the action for an appropriate
model
gauge 4
fermion,0. Taking 0
The action for N
=
1
T
=
L
dt
(g1,vFAq1
+
1 2
resulting
V/)A)
-
supersymmetric quantum mechanics
can
localizes
be obtained
auxilliary (8.27) by integrating superfields (8.19) and integrating over the 0 .1 The action for N supersymmetric quantum 2 out the
from the Wess-Zumino model action field F"
gAv(V'
from the chiral
coordinates of the superspace.
mechanics discussed in Section 4.2
setting AA =,q" above.
=
can
be obtained from the N
=
1 model
by
drops
ut
Notice that the Riemann curvature term then
because of its symmetry
properties.
o
8.4 The
the
path integral in the limit original action S,
s
--+
oo
Mathai-Quillen Formalism
onto the
287
T-periodic classical trajectories
of the
Zo (T)
=
E
sgn[det p2S(X(t)) 11] e'sl'(01
(8.47)
X(t)ELMs T
On the other
dt (g1,,FA?71' + !g,,,.t/1A1') we find that hand, selecting 2 the path integral localizes onto an ordinary integral over equivariant characteristic classes of the phase space M,
Zo (T)
=
f
chv (-iTw)
A
Ev (R)
(8.48)
M
The
equality of these 2 expressions for the quantum partition function be thought of as an equivariant, loop space generalization of the relation (3.71) in Morse theory between the Gauss-Bonnet-Chern and Poincar6Hopf theorems for the representation of the Euler characteristic X(M) of the manifold M. Indeed, in the limit H, 0 --- 0 the quantities (8.47) and A (8.48) reproduce exactly the relation (3.71). These relations have been used to study the set of Hamiltonian systems which satisfy the Arnold conjecture on the space of T-periodic classical trajectories for (time-dependent) classical Hamiltonians [119, 122]. Thus, the path integrals associated with the supermanifolds defined by the Batalin-Vilkovisky formalism for dynamical systems lead to loop space and equivariant generalizations of other familiar topological invariants. Furthermore, these models are closely related to supersymmetric non-linear sigma-mode Is which ties together the "hidden" supersymmetry of the given Hamiltonian system with the Poincare supersymmetry of the localizable quantum field theories. These ideas lead us naturally into the final topic of this Book which emphasizes these sorts of relations between equivariant cohomology and topological quantum field theories. The discussion of this Section then shows how this next topic is related to the equivariant localization formalism for dynamical systems. Zo (T)
8.4
can
Equivariant
and the We have
Numbers, Thom Classes Mathai-Quillen Formalism Euler
almost completed the connections between the equivariant loformalism, cohomological field theories and their relations to physical systems, thus uniting most of the ideas presented in this Book. The last Section showed how the localization formalism connects phase space path integrals of dynamical systems to some basic topological field theory models (namely supersymmetric sigma-models). Conversely, in Section 8.2 we demonstrated that arbitrary field theoretical models of these types could be placed into the loop space equivariant localization framework so that there now
calization
288
is
Equivariant Localization
8.
in
Cohomological
Field
Theory
equivalence between dynamical systems and field theory modgeometric context. The discussion of Section 8.1 then illustrated genuine, geometrical equivalences between physical and topological
sort of
a
els in this certain
gauge theories which demonstrates the power of the formalisms of both topological field theory and equivariant localization of path integrals in describing
the quantum characteristics of physical systems There is one final step for this connection which is the .
Atiyah-Jeffrey
geometric interpretation of generic cohomological field theories [10] which is based on the Mathai-Quillen construction of Gaussian-shaped Thom forms
[99].
This construction is the natural
arena
properties of topological field theories, and it
can
based
be used to build up
study
of the localization
gauge models. This approach, although is rather different in spirit than the equiv-
topological
equivariant cohomology,
on
for the
in its infinite dimensional versions
ariant localization formalisms
we
have discussed thus far and
we
therefore
briefly highlight the details for the sake of completeness. At the end of this Chapter, we will discuss a bit the connections with the other ideas of this Book. More detailed reviews of the Mathai-Quillen formalism in topological field theory can be found in [27, 29, 34]. The basic idea behind the Mathai-Quillen formalism is the relation (3.71) between the Poincar6-Hopf and Gauss-Bonnet-Chern representations of the Euler characteristic. It represents the localization of an explicit differential form representative of the Euler class of a vector bundle onto the zero locus of some section of that bundle. The original idea for the application and generalization of this relation to cohomological field theories traces back to Witten's connection between supersymmetric quantum mechanics and Morse theory [166]. Let us start with a simple example in this context. Given the local coordinates (x,,q) on the cotangent bundle M (9 T*M of some phase '9 -2, In the spirit of the previous and ,, space M, we denote p,, 8XII a77P Section, we then interpret (x, q) as local coordinates on a supermanifold S*M and qA dx",p,, ft, as the local basis for the cotangent bundle of S*M. The nilpotent exterior derivative operator on S*M can be written as
only
very
=
=
-
-
-
d
77/-t
-5aiT X
a + P/,
(8.49)
= 710
e-'Pd e'P produces another linear derivation conjugation d same cohomology as d. If M has metric g, then we can (as in (8.46)) so that (8.49) conjugates to
The invertible
which generates the select P
d
=
779
49XA
+
The action of
(PM
r;",?7Vt\)
+
(8.50)
supermanifold S*M
on
a
a ,,
+
( rX A,P,\77-
1
-
A MP,77 2'
V
77 77,\ ) P-
a
5PI,
__
(8.50)
the local coordinates of the cotangent bundle of the
8.4 The
dx '
=
d?71'
=
77 '
dpl,
=
0
d4t,
=
Mathai-Quillen Formalism
r,\ /J,VP,\77V P, +
1
IJA
-
2
tZPA
V
P
289
-
77 77A
(8.51)
r,\ ?7vq,\ /'tV
coincides with the standard infinitesimal transformation laws of N
=
1 DeR-
ham supersymmetric quantum mechanics. We now consider the following integral Z*
f
=
d
2n
d2n p d2n 77
x
d2nq
e
dO[x,p;?j,fj]
(8.52)
S*MOT*(S*M)
Since d 2= 0 the
integral (8.52) is formally independent of the function '0 on 0. We can therefore evaluate super-manifold S*M 0 T* (S*M), i.e. (8.52) in 2 equivalent ways. First, we introduce a Hamiltonian vector field V on M and take 0 so that Ov -!V"q,, 2 the
=
=
dov
pjV"
=
+
(8.53)
77"V,,V'q,
integration in (8.52) can then be carried out explicitly. The integration p,, produces a delta-function J(V) localizing the integral onto the zero locus MV of the vector field V. The integration over the Grassmann coordinates in (8.53) yields a determinant of VV. Computing the relevant Jacobian for the transformation x -- V(x) (c.f. Section 2.6), we arrive finally at The
over
Z*
E
=
sgn det R (p)
(8.54)
pEMv
Next,
we
take
0
=
Og
=
dog Evaluating
gl"p,, ,
=
the Gaussian
so
1
gl"plp,
integrals
Z*
d
2n
that
-
2 over
x
RA"'n p,, and
d2nq
Pfaff
Me)T*M
which
we
equality
recognize
of
(8.54)
as
and
P-
n nAg
q,,
in
(8.55)
1"
n'
(8.52)
leads to
(1RI,',,P?7)'?7P)
(8.56)
2
the Euler class of the tangent bundle TM. Thus the leads immediately to the relation (3.71).
(8.56)
The exterior derivative operator
(8.50) produces
the
Mathai-Quillen
rep-
resentative of the Euler class of the tion is
a
special
case
of
a more
tangent bundle of M. The above derivageneral construction of explicit differential
form representatives for the Euler numbers of vector bundles E
representatives tions
are
are
so-called
Gaussian-shaped
--+
M. These
Thom forms whose construc-
best understood within the framework of equivariant cohomology. one realizes the vector bundle E -* M as its associated prin-
The idea is that
cipal G-bundle
P
x
W
(with
W the standard fiber space of
E)
and constructs
290
Equivariant Localization in Cohomological Field Theory
8.
particular representative (the Thom class) of the G-equivariant cohomology x W. Given a section V : M E, the regularized Euler class EV (FA) A of the bundle) is then the pullback of the connection curvature a (with FA of the Thom class to M under this section. It can be expressed as a
of P
--+
J d,q
Ev (FA) where
m
respect
=
to
dim M and
are
fixed fiber metric
a
IIVI12 +21
e
FAP '?7,
,
+ i V A V " 77 1,
Grassmann variables. The E and VA is
on
a
(8.57) V 11 2 is with
norm
compatible
connection. We
shall not go into the details of the construction of Thom classes using equivariant cohomology, but refer the reader to [27, 34, 81] for lucid accounts of
Mathai-Quillen representative general V, integrating out the Grassmarm variables 2m-form, and the fact that it is closed follows from
this formalism. The important features of the
(8.57)
are as
follows. For
EV(FA)
shows that
is
a
the invariance of the exponent in mations
JVA
::=
(8.57)
under the supersymmetry transfor-
VAV11
with the additional condition 6xl'
=
Jq" 71"
when
=
(8.57)
(8.58)
ivi, is
integrated
over X
E
M.
Note that setting V 0 in (8.57) and integrating out the Grassmann coordinates we see that it coincides with the usual Euler characteristic class =
E(FA)
=
pendent
any V. This can
means
be evaluated
limit
s
oo
--+
V,
EV(FA)
i.e.
is
that the Euler characteristic
by rescaling
onto the
this limit does not
zeroes
by
V
s
thus
Poincar6-Hopf
vector bundles
over
E R
and
of the section V
contribute),
relation between the
generic
closed, (8.57) is indecohomologous to E(FA) for
Pfaff (FA). Since the associated Thom class is
of the chosen section
X(E
--+
localizing
(note
reproducing
M) the
=
fm Ev(FA)
integral
in the
the curvature term in
in this way the standard
and Gauss-Bonnet-Chern theorems for
M. Thus the Thom class not
only yields a repproduces the
but it also
bundle, given section of the bundle. The use of equivariant cohomology and localization techniques therefore also reproduce some classical results from geometry and topology. Niemi and Palo [1211 have shown how to construct equivariant generalizations of the Mathai-Quillen formalism. For this one considers the usual Cartan equivariant exterior derivative DV and the associated Lie derivative D'V on the super-manifold S*M. If the Christoffel connection satisfies ,CV 0 then the conjugation by !P introduced above of these operators Lvl' produces an action on the local coordinates of the cotangent bundle of S*M for LV which generates the usual covariant coordinate transformation laws with respect to coordinate change defined by the Hamiltonian vector field V. Thus the integral resentative of the Euler class of
Poincar6-dual form of t he
zero
a
vector
locus of
a
=
=
Z
,
=
J S*MOT*(S*M)
d2nx d2np
d2n?7 d2n
e'00(H+w)+DvO
(8.59)
8.5 The
is
Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles
formally independent of any generally covariant
function
0
on
291
the cotangent
bundle of S*M. This is again just the equivariant localization principle. The measure in (8.59) is the invariant Liouville measure on the extended phase
Evaluating (8.59) using
space.
the 2 choices
for'O mentioned above,
we
arrive
at the relation
E
eiOOH(p)
sgn det R (p)
PEMv
f
(8.60)
1
d2n x d2n 77 e'O(&(H+w) pfaff V,V/,'
RlAP77A77P 2
+
MOT*M can be recognized as an equivariant generalization of (3.71). Thus an appropriate equivariantization leads to a Mathai-Quillen representative for the equivariant Euler number of an equivariant vector bundle. In the limit V, 0 -- 0, (8.60) reduces to the usual relation. The non-degenerate cases are also possible to treat in this way [121].
which
Mathai-Quillen Formalism
8.5 The
for Infinite-Dimensional Vector Bundles In this final Section of this tion to
cohomological
Book,
we
shall discuss briefly the
field theories. This will make
explicit
explicit
connec-
the relations of
localization quite generically to topological field theory that we have mentioned through out this Book. As originally pointed out by Atiyah and Jeffrey
[10], although
the Euler number itself does not make
mensional vector define
bundle,
the
Mathai-Quillen
regularized Euler numbers
dles for those choices of V whose
xv (E zero
-+
form
M)
=
sense
for
EV(FA)
an
can
fm Ev (FA)
infinite di-
be used to
of such bun-
locus is finite-dimensional
so
that the
localization makes these quantities well-defined. Although these numbers are not independent of V as in the finite-dimensional cases, they are naturally associated with M for certain choices. The functional
integrals
which arise in
this way are equivalent to ordinary finite-dimensional integrals and represent the fundamental property of topological field theories, i.e. that t heir path
integrals represent characteristic classes. This has been noted throughout this book as our central theme, and indeed most topological field theories can be obtained or interpreted in terms of the infinite-dimensional Mathai-Quillen formalism The
LM
--+
[34].
simplest example M
over a
this bundle is
The
regularized Euler
number of the
loop
space
V-'(t).
loop space version of the integral (8.52) is a superloop space, and we replace the extederivative d there by the equivariant, loop space,one Q& on L(S*M). Lie derivative L& Q? as before is the generator of time translations,
path integral rior
is the
manifold M. The canonical vector field associated with
over
Now the
an
extended
=
X
292
8.
Equivariant Localization
in
Cohomological
Field
Theory
and
employing the standard conjugation above the path integral can be localized using any single-valued functional io on L(S*M). This follows from the equivariant localization principle for the model independent S'-action.
Choosing
loop
the natural
space extensions of the functionals
finite-dimensional calculations
(8.54) and (8.56) for (8.57 ) with V
of
=
above,
we
the Euler characteristic of b
yields the
action of N
quantum mechanics. In Section 4.2
'0 used in the precisely the same results M. The path integral analog
arrive at
we saw
1 DeRham
=
that the
supersymmetric 1 path integral for N 2 =
Dirac supersymmetric quantum mechanics localized onto constant modes and yielded the index of the twisted spin complex of M. In the present case the
localization of this Witten index onto constant DeRham,
complex
of M
(i.e.
mental observation of Witten
the Euler
[166]
loops yields
characteristic).
and
was one
the index of the
This
was
of the main
the funda-
ingredients
in
the birth of topological field
theory. If the target space manifold has a Kdhler structure then the sigma-model actually has 2 independent (holomorphic and anti-holomorphic) supersymmetries. Restricting the computation of the supersymmetric quantum mechanics partition function to the anti-holomorphic sector of the Hilbert space
described earlier leads to the representation of complex in terms of the Todd class. Equivariant generalizations of this simple example are likewise possible. In (8.59) the path integration now involves the action S rather than the Hamiltonian, and Dv gets replaced by QS in the usual routine of Chapter 4. Now we conjugate the relevant operators and find that the localization priniciple requires the localization functionals 0 to be generally covariant and singlevalued. The resulting path integrations yield precisely the computation at the end of Section 8.3 above. For Hamiltonians which generate circle actions, the right-hand side of (8.47) coincides with the left-hand side of (8.60) because of the structure of the set LMS discussed at the beginning of Section 4.6. Thus in this case we again obtain the ordinary finite-dimensional relation (8.60). These relations play a deeper role when the Hamiltonian depends explicitly on time t. Then the right-hand side of (8.47) represents a regularized measure of the number of T- periodic classical trajectories of the given dynamical system [119, 122]. Thus the classical dynamics of a physical system can be characterized in this way via the localization properties of supersymmetric non-linear sigma-models. In [124], these relations were related to a functional Euler character in the quantum cohomology defined by the topological nonlinear sigma-model and also to a loop space generalization of the Lefschetz fixed point theorem. Besides supersymmetric quantum mechanics the localization features of more complicated topological gauge theories can be studied by the computing as
the index of the Dolbeault
the Euler numbers of vector bundles
over
the infinite-dimensional space
AIG
of gauge connections modulo gauge transformations of a principal G-bundle. One can either start with a given topological field theory and analyse its lo-
calization characteristics using the
techniques
of this
Chapter,
or
conversely
8.5 The
Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles
293
by applying the Mathai-Quillen formalism to some vector bundle over AIG and reconstructing the action of the corresponding topological gauge theory from there. The resulting path integrals always compute sorts of intersection numbers on moduli space. A discussion of these models is beyond the scope of this book and
we
refer to
[34]
for
an
extensive discussion of the theories
can be viewed in this way. The basic example is Donaldson theory [22] which is the prime example of a cohomological field theory and is used to
which
calculate intersection numbers of moduli spaces of instantons for the study of 4-manifolds. Topological Yang-Mills theory in 4-dimensions is another in-
teresting application of this formalism. The field theoretic generalization of supersymmetric quantum mechanics, i.e. the topological sigma-model [22], is the appropriate setting for studying the quantum symmetries of string theory and more generally super-conformal field theories. The Mathai-Quillen formalism applied to 2-dimensional topological gravity could presumably shed light on its equivalence with physical gravity in 2 dimensions. The coupling of the topological sigma-model to topological gravity can be interpreted as topological string theory and studied using these methods. Finally, viewing 2-dimensional Yang-Mills theory as a topological field theory (see Section 8.1
above) leads in this way to a localization onto the rather complicated Hurwitz space of branched covers of the Riemann surface. This construction has been exploited recently as a candidate for a string theoretical realization of Yang-Mills theory [34]. Thus, the Mathai-Quillen formalism serves as the natural arena for the localization properties of cohomological field theories. However, the connection between the localization formalisms of the earlier Chapters of this Book (i.e. the stationary-phase formula) and the constructive Mathai-Quillen formalism above has yet to be completely clarified, as the latter relies on quite different cohomological symmetries than the ordinary BRST supersymmetries responsible for equivariant localization [1211. Recall these models all possess 2-dimensional
a
Grassmann-odd symmetry J that defines a supersymmetry transformation which resembles the usual BRST supersymmetries of equivariant local-
(8.58)
possible to argue [29, 34] that the 5-action is not free and that the path integral receives contributions from some arbitrarily small J-invariant tubular neighbourhood of the fixed point set of J. The integration over the ization. It is
directions normal to this fixed point set
can
be calculated in
a
stationary-
phase approximation. One readily sees from (8.58) that the fixed point set of J is the precisely the moduli space MV described by the zero locus of V 0. In this and its tangents 0 satisfying the linearized equation VAV(O) reduces field to an integration of theory path integral way the topological still obtain It remains to differential forms over Mv. a more precise though connection between these BRST fixed points, localization, and the interpretation of the geometrical and topological features of path integrals in terms of the Mathai-Quillen formalism which shows how such infinite-dimensional integrations are a pTioTi designed to represent finite-dimensional integrals. The =
294
8.
Equivariant Localization
in
Cohomological
Field
Theory
antibracket formalism
developed in Section 8.3 above is a key stepping stone Mathai-Quillen localization features of topological field theory path integrals, and the path integral localizations of generic Hamiltonian systems. The supersymmetric formulation of equivariant cohomology developed in [129], and its connections with 4-dimensional topological Yang-Mill s theory, could serve as another approach to this connection. This might give a between the
more
direct connection between localization and
some
of the
more
modern
theories of quantum integrability [35], such as R-matrix formulations and the Yang-Baxter equation. This has been discussed somewhat in [56]. These are
derstandings theories, and space
all
important and should be found in order to have full untopological and integrable quantum field hence generic physical models, from the point of view of loop equivariant localization.
connections
of the structures of
9.
Appendix
Quantization
A: BRST
BRST quantization was first introduced in the quantization of Yang-Mills theory as a useful device for proving the renormalizability of non-abelian gauge theories in 4 dimensions. It
was
shown that
present after Yang-Mills gauge fixing invariance of the model and
global fermionic symmetry was incorporated the original gauge to straightforward derivations of
a
which
ultimately led
the Ward identities associated with the gauge symmetry in both quantum electrodynamics and quantum chromodynamics. New impetus came when
quantization of Hamiltonian systems completeness, in this Appendix we shall [69]. outline the essential features of the BRST quantization scheme of which the loop space localization principle can be thought of as a special instance. Consider any physical system with symmetry operators K' that (possibly locally) generate a closed Lie algebra g, the BRST
theory
was
applied
to the
For
with first class constraints
[Ka, K b]
fabcK'
=
(9.1)
'9 which are Faddeev-Popov ghost and anti-ghost fields 0a, #a a0a anticommuting Grassmann variables that transform in the adjoint representation of g. They have the canonical anticommutator
Introduce
-
[0a, Ob 1+ We define the
ghost number operator U
whose We
eigenvalues now
are
=
=
jab
(9.2)
as
Oap
integers running from
(9.3) 0 to dim g.
introduce the operator
Q
=
OaKa
1 -
2
f abcoaob c
(9.4)
as the BRST charge, while algebra coboundary operator that computes the cohomology of the Lie algebra g with values in the representation defined by the operators Ka. The crucial property of Q is that it is 0, which can be seen from (9.1) and the identity nilpotent, Q2
In the
physics literature the operator Q
is known
in the mathematics literature it is the Lie
=
R. J. Szabo: LNPm 63, pp. 295 - 298, 2000 © Springer-Verlag Berlin Heidelberg 2000
296
9.
Appendix A:
BRST
f abef
Quantization
cde
f bdcf
+
cae
+
f dacf
cbe
(9.5)
0
=
which follows from
(9.1) via the Jacobi identity for the Lie bracket. Let Rk be the Hilbert space of states of ghost number k, i.e. UTI k T1 for T1 E I& We say that a state T1 E -Hk is BRST invariant if it is annihilated by Q, =
QT1
=
0, where
number
by
general
in
Any
1.
the action of
other state TV
T1 +
=
Q QX
-
any state raises the ghost ghost number k is regarded X E -Hk-1. The space of Q-
on
of
as equivalent to T1 E Hk for any other state equivalence classes of ghost number k is called the BRST-cohomology in the physics literature. Mathematically, it forms the k-th cohomology group Hk (g; R) of the Lie algebra g with values in the representation R carried by
the symmetry operators K'. Of particular interest from states of
ghost
0 must be annihilated
Q
on
such
a
a
number 0. From
by
all of the
=
cannot be annihilated =
0 is
a
the BRST-invariant
are
state TV of
anti-ghost fields P,
OaK a Tf
The anticommutation relations
QT1
it follows that
so
ghost number
that the action of
state is
QT,
P
physical standpoint
(9.3)
equivalent Ka,p=o
by
T E
(9.2) imply
any of the,
that
Ho a
(9.6)
state annihilated
ghost fields 0a,
and
so
by
all
the condition
to
,
a=1,...'dimg
T1 E
HO(g; R)
(9.7)
Therefore
a state T1 of ghost number 0 is BRST-invariant if and only if it g-invariant, and thus the cohomology group HI(g; R) coincides with the space of g-invariant states that do not contain any ghosts, i.e. the physical
is
states.
In
theory with gauge group G, the partition function must be dim G, which always with gauge-fixing functions ga, a 1, specify representatives of the gauge equivalence classes of the theory and restrict the functional integration to a subsPace U0 of the original configuration space of the field theory defined by the zeroes of the functions ga. Then the path integral can be written symbolically as a
gauge
evaluated
as
=
.
eis
=
vol(G)
U0
where VI
are
as
, U0
.
.
,
dimG
11
j(ga) det IlVb(gc)ll e's
(9.8)
a=1
usual vector fields associated with
an
orthonormal basis
of g (i.e. tr(XaXb) jab). Here S is the classical G-invariant gauge field action and the volume factor vol(G) is infinite for a local gauge field
fXal
=
theory. Modulo this infinite factor, the right-hand side of (9.8) is what is taken as the definition of the quantum gauge theory partition function. Introducing Faddeev-Popov ghost fields and additional auxilliary fields 0a, we can absorb
Appendix
9.
the additional factors
on
the
right-hand
A: BRST
side of
(9.8)
Quantization
into the
297
exponential
to
write
f e's
=
vol(G)
UO
f e's,
(9.9)
UO
where
Sq is the
=
S +
Oaga
+
gauge-fixed, quantum action. BRST-symmetry of this model is
The
s(4j)=Va( p)0a
S(Oa)=O
,
64Va(gb)Ob defined
(9.10)
by the following differential,
S(Oa)=_1fabcObOc 2
;
,
S(P)=_Oa (9.11)
where 0 is any scalar-valued functional of the gauge fields of the theory. With 0 and the quantum action (9.10) can this definition we have s 2 0, s(S) =
=
be written
as
Sq= S + Thus
s
is
a
S(_gaja)
BRST operator that determines
an
(9.12) N
=
dimG supersymmetry
gauge-fixed field theory, and the statement that the partition function (9.8) is independent of the choice of gauge-fixing functions ga is equivalent to the fact that the path integral depends only on the BRST-cohomology class of the action S, not on its particular representative. Thus the BRSTsupersymmetry here represents the local gauge symmetry of the theory. The gauge variation of any functional 0 of the fields of the theory is then represented as a graded commutator of the
is, 0} charge
with the fermionic
(i.e. gauge-invariant)
=
SO
-
(-1)POS
(9.13)
ghost-degree of 0. The physical gauge theory is the space of BRST-
s, where p is the
Hilbert space of the 0.
cohomology classes of ghost number
explicit relationship between equivariant localization of path integrals quantization (see the localization the next Appendix we shall make this connection principle in Section 4.4). In As have we a bit more explicit. mentioned, BRST-cohomology is the fundamental structure in topological field theories [22]. By definition, a topological action is a Witten-type action if the the classical action S is BRST-exact, while it is a Schwarz-type action if the gauge-fixed, quantum action S. is
(9.8)
and
(9.12)
demonstrate the
and BRST
BRST-exact
(but
not the classical
one).
In the
case
of the localization for-
BRST-operator is identified with phase space path integrals, of the underlying equivariant derivative exterior loop space equivariant of physical states consists "Hilbert and the space" cohomological. structure, of loop space functionals which are invariant under the flows of the loop space Hamiltonian vector field. This BRST-supersymmetry is always the symmetry that is responsible for localization in these models. The BRST formalism can
malism for the
the
Appendix
298
9.
also be
applied
A: BRST Quantization
systems with first-class constraints, i.e. those generate a Poisson subalgebra representation of a Lie algebra (9.1). The supersymmetric states then represent the observables which respect the constraints of the dynamical system (as in a gauge to Hamiltonian
whose constraint functions KI
theory).
Appendix B: Other Models Equivariant Cohomology
10.
of
In this
els for
Appendix we shall briefly outline the G-equivariant cohomology of
some a
of the other standard mod-
differentiable manifold M and
used extensively throughcompare them with the Cartan model which was out this Book. We shall also discuss how these other models apply to the derivation of
some
of the
more
general localization formulas which
briefly sketched in Section 4.9, as topological quantum field theory.
10.1 The
well
as
were
just
their importance to other ideas in
Topological Definition
ordinary cohomology, equivariant cohomology has a somewhat direct interpretation in terms of topological characteristics of the manifold M (and in this case also the Lie group G) [9]. This can be used to develop which in the usual way an axiomatic formulation of equivariant cohomology the characterize cohomology groups [341. provides properties that uniquely This topological definition resides heavily in the topology of the Lie group G through the notion of a classifying space [73]. A classical theorem of topology tells us that to G we can associate a very special space EG which is characAs with
terized
by
the fact that it is contractible and that G acts
it without fixed
on
is called the universal G-bundle. The
classifying
space points. The space EG BG for G-bundles is then defined as the base space of a universal bundle whose total space is EG. The space BG is unique up to homotopy and EG is unique up to equivariant homotopy (i.e. smooth continuous equivariant BG has 2 remarkable univerdeformations of the space). The bundle EG sal properties. The first one is that any given principal G-bundle E --+ M --+ BG. The over a manifold M has an isomorphic copy sitting inside EG -->
isomorphism classes of principal G-bundles are therefore in one-to-one corBG. The second respondence with the homotopy classes of maps f : M M can of E the of natural all that topology is measuring ways property --
be obtained from For
while for G and
Bu(j)
gauge
H*(BG)Z,
example, when G =
=
U(1) CP 00
we
get
U-
transformations,
EU(1)
we
S(H)
have Ez. =
100 Un=0
R' and BZ.
=
q2n+l
-
(the
Hilbert
(Sl)n, sphere)
.
(Cpn. In gauge theories G is the group of local that EG is the space A of Yang-Mills potentials
n= 0
so
=
R. J. Szabo: LNPm 63, pp. 299 - 308, 2000 © Springer-Verlag Berlin Heidelberg 2000
300
Appendix
10.
B: Other Models of
Equivariant Cohomology
while BG AIG is the space of gauge orbits. In string theory G is the semidirect product of the difleomorphism and Weyl groups of a Riemann surface _Th of genus h, EG is the space of metrics on _T h, and BG is the moduli =
space MEh of _Th From this point of view, one can define topological field theory and topological string theory as the study of H*(BG) and related cohomologies using the language of local quantum field theory. .
Given and M M
x
a
on
smooth G-action
M, on
we
thus have 2 spaces EG
the Cartesian
product
space
Eg via the diagonal action G
x
(M
Like the G-action
on
EG, this
Eg)
x
(g'X' e)
-
--+
(g
M
X'g
-
Eg
x
-
(10.1)
e)
action is also free and thus the
MG is
manifold
on a
which G acts. Thus G also acts
:--
(M
x
quotient
EG)IG
space
(10.2)
smooth manifold called the
homotopy quotient of M by G. Since Eg is EG is homotopic to M. Furthermore, if the G-action on M is free then MG is homotopic to MIG, so that both spaces have the same (ordinary) cohomology groups. In the general case we can regard MG as a bundle over BG with fiber M. These observations motivate the topological definition of equivariant cohomology as a
contractible, M
x
HGk,t.P(M)
=
Hk(MG)
(10.3)
Notice that if M is the space consisting of a single point, then MG BG. Thus H k't.p (pt)
and
the
=
H
k
(BG)
fl--
EGIG
(10.4)
G-equivariant cohomology point ordinary cohomology classifying space BG. This latter cohomology can be quite complicated [9], and the topological definition therefore shows that the equivariant cohomology measures much more than simply the cohomology of a manifold so
of
a
is the
of the
modulo
a
group action
on
calization formalisms of
it. It is this feature that makes the non-abelian lo-
topological field theories
and
integrable models
very
powerful techniques.
10.2 The Weil Model The topological definition of the G-equivariant cohomology above can be reformulated in terms of nilpotent differential operators [9, 81]. In this formulation, the equivariant cohomology is obtained in a more algebraic
by exploiting differential properties
of the Lie
algebra
way g of the group G.
10.2 The Weil Model
calculus of the
precisely, to describe the exterior differential algebra S(g*), we introduce the Weil algebra
More
W(g)
on
01,
a
usual
S(g*)
(10.5) algebraically
As in Section 2.6,
0
symmetric
(10.5)
Ag*
describes the exterior bundle of
consists of multilinear
algebra Ag* generated by an anti-commuting
where the exterior g and it is
=
301
antisymmetric
S(g*) forms
basis of Grassmann numbers
d0a). The Weil algebra has the (i.e. the 1-forms Oa the i.e. generators Oa of S(g*) have degree 2 Z-grading (ghost number), dim G
=
-
while the generators Oa of Ag* have degree 1. Both of these sets of generators of g. are dual to the same fixed basis jXajdimG a=1 There are 2 differential operators of interest acting on W(g). The first is the "abelian" exterior derivative
do where Ia
identifies
is the interior
a0a
Oa
as
we
Oa
=
multiplication
=
Oa
do
La which generate the
coadjoint La Ob
yield
a
=
-fabc
(01
a =
action of G
we
representation
define
on
of G
our
on
=
(10.8)
W(g) explicitly by =
fabcoc
=
Oa La +
1
2
(10.9)
W(g),
f abcLc
(10.10)
second differential operator, the
erator
dg
0
OIIC
La Ob
[L, Lb] Using (10.8),
+
(10.7)
-Ia
degree
0
fabcoc
=
degree -1 on W(g). (10.6) non-vanishing actions are
of
,90a
introduce the linear derivations of
and which
(10.6)
Ja
g
the superpartner of Oa and its
do0a Next,
Ag*
on
coboundary
f abcoaobic
op-
(10.11)
which computes the W(g)-valued Lie algebra cohomology of g, i.e. it is the BRST operator associated with the constraint operators La acting on W(g)
(see (9.4)). The
sum
of
(10.6)
and
(10.11) dw
whose action
on
the generators of
is known
do
+
W(g)
is
=
as
dg
the Weil
differential,
(10.12)
302
10.
Appendix
B: Other Models of
1
dw oa=oa_
fabcoboc
2
These 3 differential operators
and
they
act
d-pvoa
7
all
are
d2W
Equivariant Cohomology
exterior derivatives
_fabcoboc
nilpotent derivations
d20
=
=
=
d
2 =
9
of
(10.13) degree 1,
(10.14)
0
W(g). (10.12)
makes the Weil
algebra cohomology of the Weil differential dw on W(g) is trivial. This can be seen by redefining the basis of W(g) by the shift 0a _-, 0a In the new basis, 0a Ifabcoboc. dWO' are 2 exact and the cohomology of d,,v coincides with that of do on W(g) so that into
as
exterior differential
an
on
algebra. However,
the
-
H The
=
k(W(g), dw)
=
R
(10.15)
cohomology
can be made non-trivial using the 2 derivations la and introduced above. We notice first of all the analogy between W(g) (10.13) and the algebra of connections and curvatures. The first relation in
L,,
on
(10.13)
is the definition of the curvature of
principal G-bundle
E
the curvature 2-form F
dA
=
M, while 0a, i.e.
-->
we
one
F
1 -
-
[A
A
A]
dF
g),
which
is the Bianchi
=-[A A, F]
recall that the characteristic classes of E
A E Al (E,
field
connection 1-form A
-
0a
on a
identity
for
-
2
Here
a
the second
can
be
regarded
F E A2 (E,
g),
which
as a
--+
map A:
M
g*
are
--+
(10.16)
constructed from
A1E,
and from the
regarded as a map F : g* --+ A2E. These maps generate a differential algebra homomorphism (W(g), dw) --+ (AE, d) which is called the Chern-Weil homomorphism. This homomorphism is unique and it maps the algebraic connection and curvature (0a, Oa) to the geometric ones (A, F) [99]. In this setting, a connection on a principal Gbundle E --+ M is just the same thing as a homomorphism W(g) --> AE. Thus the Weil algebra is an algebraic analog of the universal G-bundle EG. Like EG, it possesses universal properties, and therefore it provides a universal model of connections on G-bundles. In particular, the contractibility of EG is the analog of the triviality (10.15) of the cohomology of the Weil algebra. Pursuing this analogy between EG and W(g), we can find non-trivial and universal cohomology classes by considering the so-called "basic forms" [9]. First, we note that the operators I,, and La above are the algebraic analogues of the interior multiplication and Lie derivative of differential forms with respect to the infinitesimal generators 0a of the G-action on W(g). Indeed, the operator (10.8) can be expressed in terms of the Weil differential as strength
La and furthermore
we
have
=
Iadw
can
+
be
dwI,,
=-
[dg, la]+
(10.17)
10.2 The Weil Model
[1a, Ibl
[L,,, Ib]
0
==
=
fabc IC
Thus the derivation La has the natural structure of that commutes with all the derivatives
[dw, Lal
=
a
303
(10.18)
Lie derivative
on
W(g)
above,
[dg, Lal
=
[do, Lal
=
(10.19)
0
particular, the (anti-) commutation relations above among dw, La and Ia all independent of the choice of basis of g. As we saw in Section 2.3, these relations also reflect the differential geometric situation on a manifold M with a G-action on it, and the Chern-Weil homomorphism above maps (dw, La, Ia) --+ (d, CVa, jVa) between the differential algebras W(g) --+ AM. We can finally define the Weil model for equivariant cohomology. For this we consider the tensor product W(g) 0 AM of the Weil algebra with the exterior algebra over the manifold M. The replacement AM --+ W(g) 0 AM for the description of the equivariant cohomology is the algebraic equivalent of the replacement M EG x M in Section B.1 above. The basic subalgebra In
are
--*
W(g) (9 AM consists of those forms which have no vertical component (i.e. the horizontal forms) and which are G-invariant (i.e. have no vertical variation). These 2 conditions mean, respectively, that the
(W(g)
(9
AM)basic
basic forms
are
of
by all the operators 1,,,
those annihilated
LaOl+JOLVa (recall that iva is the component of vector field V E TM), so that
0 1 + 10
along
a
the
iv. and
(vertical)
dim G
(W(g)
0
(D (dimnker
AM)basic
ker
(1a
0 1 + 10
iv.)
a=1
(10.20)
G
n
(La
(9 1 + 10
LV
b=1
This
subalgebra
is stable under the action of the extended DeRham exterior
derivative
dT
=
dw
cohomology of (10.21) on (10.20) G-equivariant cohomology of M, and the
k
HG,alg(M)
=
H
k
(10.21)
0 1 + 10 d
((W(g)
algebraic
definition of the
AM)basic, dT)
(10.22)
is the
0
an
AE with E homomorphism W(g) if G is compact cohomology groups isomorphism
we
have
The Chern-Weil
--+
of
k
HG,alg(M) so
H k'tOP (M)
EG then reduces
and connected
[91
to
and
(10.23)
algebraic and topological definitions of equivariant cohomology equivalent.
that the
are
-
=
304
Appendix
10.
B: Other Models of
We close this Section with of
W(g) (VI that the
remark
0 in
Since
vanishes
=
translates into invariance under the
Hk (Bg d-pv) I
and
concerning the basic subcomplex Bg
on this basic subcomplex, we dw (10.20)). in this case means oa-independence, d-,V) Bg. Horizontality basic forms in W(g) lie in only S(g*). G-invariance in this case =
have Hk (Bg, so
a
Equivariant Cohomology
coadjoint =
Bg
=
action of G
S(g*)
on
g*. Thus
G
(10.24)
the basic
subalgebra of the Weil algebra coincides with the algebra polynomial functions on the Lie algebra g, i.e. Bg is the algebra of corresponding Casimir invariants. It is known [9] that if G is a compact connected Lie group, then Hk(BG) Hk(Bg, dw), and so comparing (10.24) with (10.4) we find that the G-equivariant cohomology of a point is simply the algebra of G-invariant polynomials on g. This is in agreement with what so
of invariant
=
we
found in Section 2.3 from the Cartan model. In the next Section
indeed find that this
correspondence
is
no
we
shall
accident.
10.3 The BRST Model The final model for the
G-equivariant cohomology of a manifold M intermodels, and it therefore relates the Cartan model to the topological characteristics of this cohomology theory. It also ties in with the BRST quantization ideas that are directly related to the localization formalisms and it is the model of equivariant cohomology which arises naturally in the physical context of topological field theories. The (unrestricted) BRST algebra B of topological models on quotient spaces polates between
is the
same as
the Cartan and Weil
that for the Weil model
[81, 132]. Now, however, 5
which is
=
dw
the differential
vector space, B W(g) (9 AM it is the BRST operator
as a on
(9 1 + 1 o d + 0a
=
(gCV.
-
0a
(&
(10.25)
iV.
nilpotent graded derivation of degree 1 on B. The BRST operator is nilpotent extension of the Cartan equivariant exterior derivative defined in (2.59). The Weil differential dw in (10.25) takes care of the Dg non-nilpotency (2.61) of the Cartan model derivative. The Kalkman parametric model for the equivariant cohomology is defined by the Kalkman differential [81] a
the natural
Jt
=
e
t0a G)iVa
where t E
1
dT e-to' OiVa =dT+t0a(&'CV.-t0a (&iVa + t(l -t)f abcoaob(&iV, 2
[0, 1].
Notice that
Jo
=
dT and 51
=
J,
so
that
(10.26) (10.26) interpo-
lates between the differentials of the Weil and BRST models. it satisfies
[Jt, Ia
0 1 +
(I
-
t)l
0
iv.]
=
La
(9 1 + 1
,
(&,CV-
Furthermore,
(10.27)
10.3 The BRST Model
so
that
obtain in this way
we
a
family
of Lie
305
super-algebras acting on W(g) (9
AM. Notice also that
dT and thus the of d
on
-' =
I
e
cohomology
a fabcoaob _071
of 5
on
(d +do)
2fabcoaob 00 a
" e
(10.28)
B coincides with the DeRham
cohomology
AM.
Thus the BRST operator J does not capture the G-equivariant cohomology of M, because the W(g) part of (10.25) can be conjugated to the
cohomologically trivial operator (10.6) (equivalently
the
cohomology
of
dw
is trivial). We have to accompany 9 with a restriction of its domain in the same way that (2.59) computes the equivariant cohomology when restricted to the G-invariant subspace (2.40). The appropriate restriction is to the Oa-
independent and G-invariant subalgebra of B. This reduction maps B to AGM and 6 to Dg, so that the mapping Oa --+ 0, B --> S(g*) (9 AM induces the isomorphism of complexes
(Bbasic, d-,v)
_,
(10-29)
(AGM Dg) I
between the Weil and Cartan models. This restriction
can
be formulated
nilpotent operator W whose kernel is the desired G-invariant and Oa-independent subalgebra [129, 132]. For this, we introduce another copy W(g) of the Weil algebra. It is generated by Oa and Oa which are the g*-valued coefficients corresponding, respectively, to Oa-independence (generated by Ia) and G-invariance (generated by L, (& 1 + 1 (& LVa). A nilpotent operator with kernel AGM is
by introducing
VV where
dg
=
dg
another
& 10 1 +
is the Lie
6'
(9
(La
0 1+ 10
CVa )
-
a
(&Ia (& 1
(10.30)
algebra coboundary operator (10.11) on W(g) which (10.30) nilpotent. The action of J on W(g)
makes the overall combination in is taken
as
that of the abelian differential
(10.6)
affect its 5
=
so
that J commutes with
cohomology of do on W(g) is trivial, this alteration of 6 does cohomology. The equivariant BRST operator is therefore
W. As the
not
1(gd,(2)1+1(91(gd+do(gl(&1+1(goa (& LVa _100a Oiva (10.31)
and 6 and VV satisfy the
nilpotent algebra 62
=
[-yV, j]+
=
W2
=
0
(10.32)
G-equivariant cohomology of M is isomorphic to the cohomology of 6 subalgebra of VV(g) (9 W(g) 0 AM which is annihilated by W. This defines the BRST model of equivariant cohomology. We remark that it is also possible to formulate the restriction onto the basic subcomplex. in the Weil model using the nilpotent operator
The on
the
Ww
=
dg 0 10 1 +P 0 (La (9 1 + 1 OLVa) _ a(&(Ja(&1+1(&iV.) (10-33)
306
10.
and the
Appendix
B: Other Models of
corresponding
Equivariant Cohomology
extension of the Weil differential is
dw
=
1 (&
dT
do
+
(& 1 o 1
(10.34)
The operators
(10.33) and (10.34) are similarity transforms of (10.30) and (10.31), respectively, just as in (10.28). They therefore obey the algebra (10.32) as well, as required for the appropriate restriction process above.
It is this model that is relevant for the construction of non-abelian generalizations of the Duistermaat-Heckman integration formula, such as the Witten localization formalism (see Section 3.8). Modelling the equivariant
cohomological
structure as above is the correct way to incorporate the idea equivariant integration that we discussed earlier. In these models the generators 01 of the symmetric algebra S(g*) generate dynamics of their own and only after they are fully incorporated as above can one define properly the required equivariant localization. All 4 equivariant derivations discussed in this Book the Cartan, Weil, Kalkman and BRST differentials, have been related by Nersessian [114, 115] to the geometric anti-bracket structure of the Batalin-Vilkovisky formalism (see Section 8.3). In these formalisms, one constructs antibrackets for W(g) in addition to the usual ones for the cotangent bundle M 0 A'M.
of
-
Loop Space Extensions
10.4
Loop
generalizations
of the constructions above have been
presented by dynamical localizations of Section 4.9. For this, we introduce superpartners 01(t) for the multipliers 0'(t) and make them dynamical by adding a kinetic term for the Grassmann coordinates 0a to the action, space
Tirkkonen in
[161]
in the context of the
T
S-+
ST
=
S+
f
dt
oa(t) a(t)
(10.35)
0
The circle action generator in
(4.151)
is also extended to include
0a,
i.e.
T
VS1
-->
VS1
+
f
dt
a (t)
0
Thus the
path integral
coordinates
is
(x, 0) (recall
now
from
formulated
(10.7)
over a
that 0a
(10.36)
joa(t) superloop
space with local
coordinates and
do0a Oa corresponding 1-forms in the Weil algebra). To exploit the loop space isometry generated by the semi-direct product LGS)S1, we need to construct the corresponding equivariant operators. Because the fields Oa and 0a are dynamical and are therefore an important part of the path integration, this are
are
the
has to be done in terms of the BRST model above.
=
Loop Space
10.4
307
Extensions
corresponding equivariant BRST operator, the part corresponding (4.151) generated by VLG on LM is just lifted from the corresponding equivariant BRST operator (10.31) for the G-action on M. The part associated with the circle action in (4.151) as generated by (10.36) dL i& on the superloop space is given by the Cartan model operator Q& In the
to the LG-action in
=
-
T
the operator Qj is nilpotent on the loop space LM. L dt A dt) yJ equivariant BRST operator for the semi-direct product LGZ)S then combines (10.31) lifted to the loop space with Q_;, Since
Q?
=
X
The total
T
QT
=
dL
(L)
+ dW +
I (O',Cv. dt
-
O'iv.
30)
+
J'qA
(10.37)
0
(L) where dW is the Weil differential
(10.12)
superloop
lifted to the
(10.30)
standard way. Furthermore, to the restriction operator corresponding to the Sl-action [161]
space in the
we
add
a
piece
T
)1VT
WL
=
+
1
dt 0
(10.38)
(t)
0
with WL the lifting of the restriction operator The algebra Of QT and WT is then
(10.30)
to the
superloop
space.
T
Q2T
d
dt
2
V4
dt
=
[QTi WTI+
=
0
(10.39)
0
The function
F(0)
discussed in Section 4.9 Weil space
=
!(Oa)2
effectively added to the action as interpreted as a symplectic 2-form on the interpreted as super 1-forms). The superloop
2
can now
algebra W(g) (as 0a symplectic 2-form is
are
which is
be
then T
S?T
=
Q +
1
dt
1 2
(Oa(t))2
(10.40)
0
equivariance properties by the equations
and the rized
QT(ST The first
+
vanishing condition
of the
S?T) in
=
phase
WT(ST
(10.41)
space
+
path integral
S?T)
=
identifies the
0
superloop
are summa-
(10.41) space action
the moment map for the action of the semi-direct product LGS)S' on LM, while the second one states that this group action is symplectic. To carry out the Niemi-Tirkkonen localization procedure over the superloop
ST
as
308
10.
Appendix
B: Other Models of
space with the BRST
(4.123)
to the Weil
operator QT,
Equivariant Cohomology
we
generalize the
gauge fermion field
algebra, T
OT
I
=
dt
(10.42)
+
0
where the
we
have left the
same
localization
mode part of 0'0' Out Of'OT- We can now carry out procedure which led to the Niemi-Tirkkonen formula,
zero
(4.153) [161].
and thus arrive at the localization formula
non-abelian generalization of the modelindependent auxilliary field formalism which we discussed in Section 4.9. In the abelian case where we can a pTiori fix any function F(0) for the loThe above construction is
calization,
(10.40),
a
modifying the loop space symplectic structure as in model-independent formalism appears as the functional
instead of
which in the
110t1j,
determinant det
we
shift the gauge fermion field
(10.42)
as
T
OT
--*
dt
OT
(0,,,77"
-
2
(10.43)
0
The standard Niemi-Tirkkonen localization discussed even
above,
procedure then leads
to the lo-
(4.149) [128].
For the generic, non-abelian group actions the situation for loop space localization is different, and
calization formula
the second Weil
algebra
copy
IFV(g)
is made
dynamical. For discussions cohomology and
of the relations between the BRST model of equivariant
Witten-type topological field theories,
see
[27, 34, 129].
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