Equivariant cohomology and localization of path integrals

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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

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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)

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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 A Brief Review of DeRham Cohomology 2.2 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'

at

infinity

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

spaces of the

tangent

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 space

(5 T 9

dx t'

The

Tx*M

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

exact form

an

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

(2.46)

will refer to

we

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

in the last

as

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).

manifold M

a

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

the fibers

globally

twist

can

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

r

denotes the

be

can

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

[211.

in

for

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

=

F9

(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

=

operator

fa

=

[9]

and Bott

Dv

-=

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

9" (.9Ag" + '9vg"

2

(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

a

particular representative of the cohomolTaking the equivariant differential integral

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

emphasize the

we

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

canonical momentum and

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

correction terms in

(3.53)

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

construction,

its

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

Riemann Surface

a

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

expected

as

be written

can

Z_ro (T)

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"'

=

(3.87)

0

and from the Hamilton equations of motion

i,, (t) 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

VA

I

1

0

(3.116)

"FS)

G'V)1(Xo)(S?V)P'(Xo)X"X' I I +0(11 'Fs) 1A

-

2

(Qv)O,(xo)R' P(xo)xl'x',q' no

+

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.

large equivariant cohomological symmetry

Notice also the role that the

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

s

oo)

--+

is

given by

a

[171]

formula

(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)

into its

freely

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

(H a)2

iT

e

n!

M dim

1

vol(G)

=

f

W

-

vol(G)

Gd Oa

gj.

II

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)