Equivariant Cohomology and Localization of Path Integrals (Lecture Notes in Physics)

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Lecture Notes in

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Monographs Editorial Board R. Beig, Wien, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zfirich, Switzerland

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Richard J. Szabo

Equivariant Cohomology

and Localization of Path Inte

4

-

-

Springer

ra s

Author Richard J. Szabo The Niels Bohr Institute

Blegdamsvej 17 2100 Copenhagen 0, Denmark and

Department of Physics University of Oxford i

-

Theoretical

Physics

Keble Road

Oxford OX1 3NP, United

Kingdom

Library of Congress Cataloging-in-Publication Data. Die Deutsche Bibliothek

-

CIP-Einheitsaufnahme

Szabo, Richard J.:

Equivariant cohomology and localization of path integrals / Richard J. Szabo. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan ; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in physics : N.s. M, Monographs ; 63)

-

ISBN 3-540-67126-9 ISSN 0940-7677 (Lecture Notes in Physics. Monographs) ISBN 3-540-67126-9 Springer-Verlag Berlin Heidelberg New York

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5 43

2 10

Preface

This book reviews equivariant localization techniques for the evaluation of Feynman path integrals. It develops systematic geometric methods for study-

ing the semi-classical properties

of

phase

space

path integrals for dynamical

systems, emphasizing the relations with integrable and topological quantum field theories.

Beginning with a detailed review of the relevant mathematbackground equivariant cohomology and the Duistermaat-Heckman theorem, it demonstrates how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. The book emphasizes the common symmetries that such localizable models always possess and uses these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemarm surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented and used to discuss the general relationship between topological field theories and more conventional physical models. ical

-

Copenhagen, September

Acknowledgements:

1999

Richard J. Szabo

I would like to thank I.

Semenoff for advice and encouragement book and for

helpful discussions.

I

during

Kogan,

G.

Landi,

F. Lizzi and G.

various stages of the

writing

the various calculational and

grateful conceptual aspects of Section 7,

portant historical remarks

the manuscript, and to 0. Tirkkonen for

on

am

to L. Paniak for his

of this

participation

in

to E. Gozzi for im-

illuminating

VIII

Preface

discussions. I would also like to thank D. Austin and R.

suggestions supported Canada.

on some

in part

of the

more

Douglas for

comments and

mathematical aspects of this book. This work

was

by the Natural Sciences and Engineering Research Council of

Contents

1.

Introduction

2.

..............................................

1.2 1.3

Outline

............

1

..........................

5

...............................................

Equivariant Cohomology Principle 2.1 Example: The Height Function of the Sphere 2.2 A Brief Review of DeRham Cohomology 2.3 The Cartan Model of Equivariant Cohomology and the Localization

2.4

Fiber Bundles

2.5

The

2.6

The

and

3.

7

............................

11

...............

11

..................

13

.............

19

....................

24

....................

33

.............................

38

Equivariant Characteristic Classes Equivariant Localization Principle Berline-Vergne Theorem

Fin ite-Dimensional Localization for 3.1 3.2 3.3

Theory Dynamical Systems Symplectic Geometry Equivariant Cohomology on Symplectic Manifolds Stationary-Phase Approximation

...................................

43

...................................

44

..........

47

and the Duistermaat-Heckman Theorem 3.4 3.5

3.7

...................

51

.....................

56

...................................

58

and Kirwan's Theorem

Theory Examples: The Height Function Morse

of 3.6

a

Riemann Surface

Equivariant Localization Degenerate Version

and Classical

of the Duistermaat-Heckman Theorem

4.

I

Path

Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory Equivariant Localization Theory

1.1

....................

67

70

............................

74

The Witten Localization Formula

3.9

The Wu Localization Formula

Quantum 4.1

62

.........................

3.8

Localization

Theory Space Path Integrals Phase Space Path Integrals

for Phase

Integrability

.........

...........................

77

..............................

78

X

Contents

4.2

4.3 4.4

Example: Path Integral Derivation of the Atiyah-Singer Index Theorem Loop Space Symplectic Geometry and Equivariant Cohomology Hidden Supersymmetry and the Loop Space Localization Principle

......................

84

............................

95

.................

99

..........................

105

4.5

The WKB Localization Formula

4.6

Degenerate Path Integrals

4.7

Connections with the Duistermaat-Heckman

and the Niemi-Tirkkonen Localization Formula

4.8 4.9 4.10 5.

.............

Integration Formula Equivariant Localization and Quantum Integrability Localization for Functionals of Isometry Generators Topological Quantum Field Theories

107

....................................

112

........

114

.........

117

......................

121

Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems 127 5.1 Coadjoint Orbit Quantization .........................................

and Character Formulas 5.2

5.3

5.4 5.5

5.6 5.7

.................................

Isometry Groups of Simply Connected Riemannian Spaces Euclidean Phase Spaces and Holomorphic Quantization Coherent States on Homogeneous Kiffiler Manifolds and Holomorphic Localization Formulas Spherical Phase Spaces and Quantization of Spin Systems Hyperbolic Phase Spaces Localization of Generalized Spin Models

..................

137

...........................

146

...................

153

........................................

158

................................

169

and Hamiltonian Reduction 5.8 5.9 6.

Quantization Quantization

of on

130

..............................

Isospin Systems Non-Homogeneous Phase Spaces

..........................

...........

Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations 6.1 Isometry Groups of Multiply Connected Spaces 6.2 Equivariant Hamiltonian Systems in Genus One 6.3 Homology Representations and Topological Quantum Field Theory 6.4 Integrability Properties and Localization Formulas 6.5 Holomorphic Quantization and Non-Symmetric Coadjoint Orbits 6.6 Generalization to Hyperbolic Riemann Surfaces

171 180 191

...............

203

............

205

............

207

...................

210

..........

213

.....................

217

............

226

Contents

7.

Beyond 7.1

the Semi-Classical

................

233

..................................

234

Geometrical Characterizations of the

Loop Expansion

7.2

Conformal and Geodetic Localization

7.3

Corrections to the Duistermaat-Heckman Formula:

Symmetries

..........

244

..................................

253

..............................................

259

A Geometric

Approach

7.4

Examples

7.5

Heuristic Generalizations to Path

Supersymmetry Breaking 8.

Approximation

Integrals:

................................

Equivariant Localization in Cohomological. Field Theory Two-Dimensional Yang-Mills Theory: Equivalences 8.1 Between Physical and Topological Gauge Theories 8.2 Symplectic Geometry of Poincar6 Supersymmetric Quantum Field Theories 8.3

8.4

269

..........

270

.........................................

276

Supergeometry and the Batalin-Radkin-Vilkovisky Formalism Equivariant Euler Numbers, Thom Classes and the Mathai-Quillen Formalism The Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles

A: BRST

.............

Appendix

10.

Appendix B: Other Models of Equivariant Cohomology 10.1 The Topological Definition 10.2 The Weil Model

.........................

295

...............................

299

...............................

299

......................................

300

304

..................................

306

....................................................

309

Loop Space

References

287 291

........................................

10.3 The BRST Model

281

...................

Quantization

9.

10.4

266

...........................

........................

8.5

XI

Extensions

1. Introduction

In this book

shall review the systematic situations where

approaches and applications of a Feynman path integrals of physical systems can be evaluated exactly leading to a complete understanding of the quantum physics. These mathematical formalisms are in large part motivated by the symmetries present in integrable systems and topological quantum field theories which make these latter examples exactly solvable problems. Besides providing conceptual understandings of the solvability features of these special classes of problems, this framework yields geometric approaches to evaluating the quantum spectrum of generic quantum mechanical and quantum field theoretical partition functions. The techniques that we shall present here in fact motivate an approach to studying generic physical problems by relating their properties to those of integrable and topological field theories. In doing so, we shall therefore also review some of the more we

theory which investigates

modern quantum field theoretical and mathematical ideas which have been at the forefront of theoretical physics over the past two decades.

1.1 Path

Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory The idea of as a

novel

integral

path integration was introduced by Feynman [46] in the 1940's approach to quantum theory. Symbolically, the fundamental path

formula is IC (q', q;

eiTL[Cqqll

T)

(1. 1)

Cqqt where the 'sum' is

over all paths Cqq, between the points q and q, on the configuration space of a physical system, and L[Cqqll is the length of the path. The quantity on the left-hand side of (1.1) represents the probability amplitude for the system to evolve from a state with configuration q to one with configuration q' in a time span T. One of the great advantages of the path integral formulation is that it gives a global (integral) solution of the quantum problem in question, in contrast to the standard approach to quantum mechanics based on the Schr8dinger equation which gives a local (differential) formulation of the problem. Of utmost significance at the time was Feynman's

R. J. Szabo: LNPm 63, pp. 1 - 9, 2000 © Springer-Verlag Berlin Heidelberg 2000

1. Introduction

generalization of the path integral

to quantum electrodynamics from which systematic derivation of the famous Feynman rules, and hence the basis of most perturbative calculations in quantum field theory, can be carried out a

[75]. The problem of quantum integrability, i.e. the possibility of solving analytically for the spectrum of a quantum Hamiltonian and the corresponding energy eigenfunctions, is a non-trivial problem. This is even apparent from the point of view of the path integral, which describes the time evolution of wavefunctions. Relatively few quantum systems have been solved exactly and even fewer have had an exactly solvable path integral. At the time that the functional integration (1.1) was introduced, the only known examples where it could be evaluated exactly were the harmonic oscillator and the free particle. The path integrals for these 2 examples can be evaluated using the formal functional analog of the classical Gaussian integration formula [176] n

fl dx -00

k

(2-7r e' .7r/2)

el1E.

i

e

I

Ei,jAi(M-YjAj

v1det M

k=1

(1.2) where M

=

[Mij]

is

a

non-singular symmetric

matrix. In this way, be evaluated formally for any field theory n x n

Feynman propagator (1.1) can quadratic in the field variables. If this is not the case, then one can expand the argument of the exponential in (1.1), approximate it by a quadratic form as in (1.2), and then ,take the formula (1.2) as an approximation for the integral. For a finite-dimensional integral this is the well-known stationary phase (otherwise known as the saddle-point or steepest-descent) approximation [64]. In the framework of path integration, it is usually referred to as the Wentzel-Kramers-Brillouin (or WKB for short) approximation [101, 147]. Since the result (1.2) is determined by substituting into the exponential integrand the global minimum (Le. classical value) of the quadratic form and multiplying it by a term involving the second variation of that form (i.e. the fluctuation determinant), this approach to functional integration is also called the semi-classical approximation. In this sense, (1.1) interprets quantum mechanics as a sum over paths fluctuating about the classical trajectories (those with minimal length L[Cqqll) of a dynamical system. When the semi-classical approximation is exact, one can think of the Gaussian integration formula (1.2) as a 'localization' of the complicated looking integral on the left-hand side of (1.2) onto the global minimum of the quadratic form the

which is at most

there. For

long time, these

the

only examples of exactly solvable path a path integral describing the [146] precession of a spin vector was given exactly by its WKB approximation. This was subsequently generalized by Dowker [38] who proved the exactness of the semi-classical approximation for the path integral describing free geodesic a

integrals.

motion

were

In 1968 Schulman

on

compact

found that

group manifolds. It

was

not until the late 1970's that

1.1 Path

Integrals

3

general methods, beyond the restrictive range of the standard WKB method, were developed. In these methods, the Feynman path integral is calculated rigorously in discretized form (i.e. over piecewise-linear paths) by a careful regularization prescription [93], and then exploiting information provided by functional analysis, the theory of special functions, and the theory of differential equations (see [31] and references therein). With these tricks the list of exactly solvable path integrals has significantly increased over the last 15 years, so that today one is able to essentially evaluate analytically the path integral for any quantum mechanical problem for which the Schr6dinger equation can be solved exactly. We refer to [91] and [62] for an overview of these methods and a complete classification of the known examples of exactly solved quantum mechanical path integrals to present date. The situation is somewhat better in quantum field theory, which represents the real functional integrals of interest from a physical standpoint. There are many non-trivial examples of classically integrable models (i.e. ones whose classical equations of motion are 'exactly solvable'), for example the sine-Gordon model, where the semi-classical approximation describes the exact spectrum of the quantum field theory [176]. Indeed, for any classically integrable dynamical system one can canonically transform the phase space variables so that, using Hamilton-Jacobi theory [55], the path integral can be formally manipulated to yield a result which if taken naively would imply the exactness of the WKB approximation for any classically integrable system [147]. This is not really the case, because the canonical transformations used in the phase space path integral do not respect the ordering prescription used for the properly discretized path integral and consequently the integration measure is not invariant under these transformations [30]. However, as these problems stem mainly from ordering ambiguities in the discretization of the path integral, in quantum field theory these ordering ambiguities could be removed by a suitable renormalization, for instance by an operator commutator ordering prescription. This has led to the conjecture that properly interpreted results of semi-classical approximations in integrable field theories reproduce features of the exact quantum spectrum [176]. One of the present motivations for us is to therefore develop a systematic way to implement more

realizations of this conjecture. Another class of field theories where the

path integral is exactly solvable in most cases is supersymmetric topological quantum field theories for concise a review). Topological field theories have lately been of (see [22] much interest in both the mathematics and physics literature. A field theory is topological if it has only global degrees of freedom. This means, for example, that its classical equations of motion eliminate all propagating degrees of freedom from the problem (so that the effective quantum action vanishes). In particular, the theory cannot depend on any metric of the space on which theories and

the fields

are

fore describe

defined. The observables of these quantum field theories theregeometrical and topological invariants of the spacetime which

1. Introduction

computable by the conventional techniques of quantum field theory and of prime interest in mathematics. Physically, topological quantum field theories bear resemblances to many systems of longstanding physical interest and it is hoped that this special class of field theories might serve to provide insight into the structure of more complicated physical systems and a testing ground for new approaches to quantum field theory. There is also a conjecture that topological quantum field theories represent different (topological) phases of their more conventional counterparts (e.g. 4-dimensional Yang-Mills theory). Furthermore, from a mathematical point of view, these field theories provide novel representations of some global invariants whose properties are frequently transparent in the path integral approach. Topological field theory essentially traces back to the work of Schwarz [148] in 1978 who showed that a particular topological invariant, the RaySinger analytic torsion, could be represented as the partition function of a certain quantum field theory. The most important historical work for us, however, is the observation made by Witten [166] in 1982 that the supersymmetry algebra of supersymmetric quantum mechanics describes exactly the DeRham complex of a manifold, where the supersymmetry charge is the exterior derivative. This gives a framework for understanding Morse theory in terms of supersymmetric quantum mechanics in which the quantum partition function computes exactly the Euler characteristic of the configuration manifold, i.e. the index of the DeRharn complex. Witten's partition function computed the so-called Witten index [167], are

are

the difference between the number of bosonic and fermionic states. In order for the

of

supersymmetry

to be broken in the

zero

energy

ground

state

supersymmetric model, the Witten index must vanish. As supersymmetry, i.e. a boson-fermion symmetry, is not observed in nature, it is necessary to have some criterion for dynamical supersymmetry breaking if supersymmetric theories are to have any physical meaning. Witten's construction was subsequently generalized by Alvarez-Gaum6 [5], and Friedan and Windey [48], to give supersymmetric field theory proofs of the Atiyah-Singer index theorem [41]. In this way, the partition function is reduced to an integral over the configuration manifold M. This occurs because the supersymmetry of the action causes only zero modes of the fields, i.e. points on M, to contribute to the path integral, and the integrals over the remaining fluctuation a

modes

are

Gaussian. The

resulting integral encodes topological information a huge reduction of the original infinite-

about the manifold M and represents dimensional path integration. This field

began to draw more attention around 1988 when Witten introtopological field theories in a more general setting [169] (see also [170]). A particular supersymmetric non-abelian gauge theory was shown by Witten to describe a theory with only global degrees of freedom whose observables duced

are

the

the Donaldson invariants, certain differential invariants which are used for study of differentiable structures on 4-manifolds. Subsequent work then

1.2

put these ideas into

a

Equivariant Localization Theory

general framework

oretic structures of Witten's actions

are

so

that

today

the formal field the-

well-understood

[22]. Furthermore,

because of their topological nature, these field theories have become the focal point for the description of topological effects in quantum systems using

quantum field theory, for instance for the description of holonomy effects in physical systems arising from the adiabatic transports of particles [152] and extended objects such as strings [181 (i.e. Aharonov-Bohm type effects). In this way the functional integral has in recent years become a very popular tool lying on the interface between string theory, conformal field theory and topological quantum field theory in physics, and between topology and algebraic geometry in mathematics. Because of the consistent reliability of results that path integrals of these theories can produce when handled with care, functional integration has even acquired a certain degree of respectability among mathematicians.

1.2 The

Equivariant Localization Theory feature of

topological field theories is that their path integrals exactly by the semi-classical approximation. It would be nice to put semi-classically exact features of functional integrals, as well as the features which reduce them to integrals over finite-dimensional manifolds as described above, into some sort of general framework. More generally, we would are

common

described

like to have certain criteria available for when

we

expect partition functions

of quantum theories to reduce to such

simple expressions, or 'localize'. This motivates an approach to quantum integrability in which one can systematically study the properties of integrable field theories and their conjectured semi-classical "exactness" that we mentioned before. In this approach we focus on the general features and properties that path integrals appearing in this context have in

large

number of

Foremost among these is the existence of a (super-) symmetries in the underlying dynamical theory, so

that these functional

common.

integrals reduce

sent finite-dimensional

integrals'.

to Gaussian ones and essentially repreThe transition between the functional and

finite-dimensional

integrals can then be regarded as a rather drastic localoriginal infinite-dimensional integral, thereby putting it into a form that is useable for extracting physical and mathematical information. The mathematical framework for describing these symmetries, which turn out to be of a topological nature, is equivariant cohomology and the approach discussed in this paragraph is usually called 'equivariant localization ization of the

The exact solvability features of path integrals in this context is similar to the solvability features of the Schr6dinger equation in quantum mechanics when there is a large symmetry group of the problem. For instance, the 0(4) symmetry of the 3-dimensional Coulomb problem is what makes the hydrogen atom an exactly

solvable quantum system

[1011.

1. Introduction

theory'. This approach introduces an equivariant cohomological framework as a tool for developing geometric techniques for manipulating path integrals and examining their localization properties. Historically, this subject originated in the mathematics literature in 1982 with the Duistermaat-Heckman theorem

[39], which established the exactness

of the semi-classical

approximation for finite-dimensional oscillatory integrals (i.e. finite-dimensional versions of (1.1)) over compact symplectic manifolds

in certain instances. The Duistermaat-Heckman theorem

applies

to classical

systems whose trajectories all have a common period, so that the symmetry responsible for the localization here is the existence of a global Hamiltonian torus action

on

the manifold.

Duistermaat-Heckman localization ization

property of

action in the

case

Atiyah

and Bott

[9]

showed that the

special case of a more general localequivariant cohomology (with respect to the torus group was a

of the Duistermaat-Heckman

theorem).

This fact

was

used

by Berline and Vergne [19, 20] at around the same time to derive a quite general integration formula valid for Killing vectors on general compact Riemannian manifolds.

The first infinite-dimensional

generalization of the Duistermaat-Heckman Atiyah and Witten [8], in the setting of a supersymmetric path integral for the index (i.e. the dimension of the space of zero modes) of a Dirac operator. They showed that a formal application of the Duistermaat-Heckman theorem on the loop space LM of a manifold M to

theorem is due to

the partition function of N

.1

supersymmetric quantum mechanics (i.e. a 2 supersymmetric spinning particle in a gravitational background) reproduced the well-known

=

Atiyah-Singer

index theorem

correctly.

The crucial idea

was

the interpretation of the fermion bilinear in the supersymmetric action as a loop space symplectic 2-form. This approach was then generalized by Bismut

[23, 24], within a mathematically rigorous framework, to twisted Dirac oper(i.e. the path integral for spinning particles in gauge field backgrounds), and to the computation of the Lefschetz number of a Killing vector field V (a measure of the number of zeroes of V) acting on the manifold. Another nice ators

infinite-dimensional generalization of the Duistermaat-Heckman theorem was suggested by Picken [139] who formally applied the theorem to the space of loops over a group manifold to localize the path integral for geodesic motion on the group, thus establishing the well-known semi-classical properties of these systems. It was the beautiful paper by Stone [156] in 1989 that brought the Duistermaat-Heckman theorem to the attention of a wider physics audience. Stone for

presented a supersymmetric derivation of the Weyl character formula SU(2) using the path integral for spin and interpreted the result as a

Duistermaat-Heckman localization. This supersymmetric derivation was exby Alvarez, Singer and Windey [4] to more general Lie groups using

tended

fiber bundle very

closely

theory,

and the

supersymmetries in both of these approaches are cohomology. At around this time, other im-

related to equivariant

1.3 Outline

7

portant papers concerning the quantization of spin appeared. Most notably,

[1171 (see also [50]) viewed the path integral for spin geometrical point of view, using as action functional the solid angle swept out by the closed orbit of the spin. This approach was related more closely to geometric quantization and group representation theory by Alekseev, Faddeev and Shatashvili [2, 3], who calculated the coadjoint orbit path integral for unitary and orthogonal groups, and also for cotangent bundles of compact groups., Kae-Moody groups and the Virasoro group. The common feature is always that the path integrals are given exactly by a semi-classical Nielsen and Rohrlich from

a more

localization formula that resembles the Duistermaat-Heckman formula. The connections between supersymmetry and equivariant

cohomology

in

the quantum mechanics of spin were clarified by Blau in [26], who related the Weinstein action invariant [165] to Chern-Simons gauge theory using the

Duistermaat-Heckman

integration formula. Based

on

this interpretation, and

[57]-[59] of the hidden supersymmetry underclassical dynamical system, Blau, Keski-Vakkuri and Niemi [30] lying any introduced a general supersymmetric (or equivariant cohomological) frame-

the observation of Gozzi et al.

investigating Duistermaat-Heckman (or WKB) localization formulas generic (non-supersymmetric) phase space path integrals, leading to the fair amount of activity in this field which is today the foundation of equivariant localization theory. They showed formally that the partition function for the quantum mechanics of circle actions of isometries on symplectic manifolds localizes. Their method of proof involves formal techniques of Becchi-RouetStora-Tyupin (or BRST for short) quantization of constrained systems [16]. BRST-cohomology is the fundamental structure in topological field theories, and such BRST supersymmetries are always the symmetries that are responwork for for

sible for localization in these models.

1.3 Outline

primarily explore the geometric features of the localiza, phase space path integrals. In particular, we shall focus on how these models can be used to extract information about integrable and topological quantum field theories. In this sense, the path integrals we study for the most part can be thought of as "toy models" serving as a testing ground for ideas in some more sophisticated field theories. The main idea behind this reasoning is that the localization in topological field theories is determined by their kinematical properties. The path integrals we shall focus on allow us to study their kinematical (i.e. geometrical and topological) aspects in isolation from their dynamical properties. These models are typically dynamically linear (i.e. free field theories) in some sense and the entire non-triviality of their path integrals lie in thelarge kinematical non-linearity that these theories possess. The appropriate relationship between topological field theories and more conventional, physical interacting quantum field In this Book

we

shall

tion formalism for

1. Introduction

(which are kinematically linear but dynamically highly non-linear) then, in principle at least, allow one to incorporate the approaches and techniques which follow to generic physical models. Indeed, one of the central themes in what follows will be the interplay between physical, integrable and topological field theories, and we shall see that the equivariant localization formalism implies connections between these 3 classes of models and thus a sort of unified description of functional integration which provides alternative approximation techniques to the usual perturbative expansion in quantum field theory. We shall therefore approach the localization formalism for path integrals in the following manner. Focusing on the idea of localizing a quantum paxtition function by reducing it using the large symmetry of the dynamical system to a sum or finite-dimensional integral in analogy with the classical Gaussian integration formula (1.2), we shall first analyse the symmetries responsible for the localization of finite-dimensional integrals (where the symmetry of the dynamics is represented by an equivariant cohomology). The main focus of this Book will then be the formal generalizations of these ideas to phase space path integrals, where the symmetry becomes a "hidden" supersymmetry of the dynamics representing the infinite-dimensional analog of equivariant cohomology. The subsequent generalization will be then the extension of these theories

would

notions to both Poincar6

supersymmetric quantum field theories (where the symmetry is represented by the supersymmetry of that model) and topological quantum field theories (where the symmetry is represented by a gauge

symmetry).

The hope is that these serve as testing grounds for the more sophisticated quantum field theories of real physical interest, such as quantum chromodynamics (QCD). This gives a geometric framework for studying quantum integrability, as well as insights into the structure of topological and supersymmetric field theories, and integrable models. In particular, from this analysis we can hope to uncover systematically the reasons why some quantum problems are exactly solvable, and the reasons why others aren't. Briefly, the structure of this Book is as follows. In Chapter 2 we go through the main mathematical background for localization theory, i.e. equivariant cohomology, with reviews of some other mathematical ideas that will be important for later Chapters as well. In Chapter 3 we present the DuistermaatHeckman theorem and its generalizations and discuss the connections they imply between equivariant cohomology and the notion of classical integrability of a dynamical system. Chapter 4 then goes through the formal supersymmetry and loop space equivariant cohomology arguments establishing the localization of phase space path integrals when there is a Riemannian structure on the phase space which is invariant under the classical dynamics of the system. Depending on the choice of localization scheme, different sets of phase space trajectories are lifted to a preferred status in the integral. Then all contributions to the functional integral come from these preferred paths along with a term taking into account the quantum fluctuations about these

1.3 Outline

9

selected

loops. Chapters 5 and 6 contain the main physical and mathematical applications of equivariant localization. There we use the fundamental isometry condition to construct numerous examples of localizable path integrals. In each case we evaluate and discuss the localization formulas from both physical and mathematical standpoints. Here we shall encounter numerous examples and gain much insight into the range of applicability of the localization formalism in general. We will also see here many interesting features of the localizable partition functions when interpreted as topological field theories, and

discuss in detail various other issues

(e.g. coherent state quantization coadjoint orbit character formulas) which are common to all the localizable examples that we find within this setting and which have been of interest in the more modern approaches to the quantization of dynamical systems. Chapter 7 then takes a slightly different approach to analysing localizable systems, this time by some geometric constructs of the full loop-expansion on the phase space. Here we shall discuss how the standard localization symmetries could be extended to more general ones, and we shall also show how the localization ideas could be applied to the formulation of a geometric approach to obtaining corrections to the standard WKB approximation for non-localizable partition functions. The analysis of this Chapter is a first step towards a systematic, geometric understanding of the reasons why the localization formulas may not apply to a given dynamical system. In Chapter 8 we turn our attention to field theoretical applications of equivariant localization and discuss the relationships that are implied between topological field theories, physical quantum field theories, and the localization formalism for dynamical systems. For completeness, 2 Appendices at the end of the book are devoted to an overview of some ideas in the BRST quantization formalism and some more mathematical ideas of equivariant cohomology, all of which play an important role in the development of the ideas in the main body of we

and

this Book. We close this

introductory Chapter with some comments about the style Although we have attempted to keep things self-contained and at places where topics aren't developed in full detail we have included ample references for further reading, we do assume that the reader has a relatively solid background in many of the mathematical techniques of modern theoretical physics such as topology, differential geometry and group theory. All of the group theory that is used extensively in this Book can be found in [162] (or see [53] for a more elementary introduction), while most of the material discussing differential geometry, homology and cohomology, and index theoof this Book.

rems can

For

be found in the books

a more

[61, 32, 111]

detailed introduction to

and the review articles

algebraic topology,

see

[98].

[22, 411.

The basic

reference for quantum field theory is the classic text [75). Finally, for a discussion of the issues in supersymmetry theory and BRST quantization, see

[16, 22, 69, 118, 155]

and references therein.

Equivariant Cohomology and the Localization Principle 2.

2.1

The

Example:

Function of the

Height

Sphere

some of the abstract and technical formalism which follows, by considering the evaluation of a rather simple integral. Consider 3 the 2-sphere S2 of unit radius viewed in Euclidean 3-space R as a sphere and centered at z a symmetrically about standing on end on the xy-plane the z-axis. We introduce the usual spherical polar coordinates x sin 0 cos 0, for 0 the of the 0 sin and sin 0 z cos a sphere in 3-space embedding y

To

help

we

start

motivate

=

=

=

as

=

S2

=:

f (x, y, z)

0 height function 0
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


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


pairing between Hq (M; R) and Hq (M; R) and isomorphism. In particular, (2-34) generalizes to theorem,

I

dw

w

m

which relates the

(n

closed

-

w

c-

19

is the basis of the DeRham the

global

version of Stokes'

A'-'M

(2.35)

am

integral

of

an

exact form

over

M to

an

integral

over

the

I)-dimensional boundary

ifold is defined

by

aM of M. Here integration over a manpartitioning the manifold up into open sets homeomorphic

n

integrating a top form (i.e. a differential form of highest degree n on M) locally over Rn as usual', and then summing up all of these contributions. In this way, we see how the DeRham cohomology of a manifold measures its topological (or global) features in an analytic way suited for the differential calculus of C' manifolds. We refer to [98] and [32] for a more complete and leisurely introduction to this subject. to R

,

2.3 The Cartan Model of We shall

now

where there is whose group

the constructions of the last Section to the

generalize a


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.



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)



21

Wewant to extend this notion of equivariance to differential forms. Consider the symmetric polynomial functions from the Lie algebra g of G =- exp(g) into the exterior algebra AM of the manifold M. These maps form the algebra

S(g*) (&AM,

where

S(g*)

is called the

symmetric algebra over the dual vector

space g* of g and it corresponds to the algebra of polynomial functions g. The action of g E G on an element a E S(g*) 0 AM is given by

(g a) (X) -

where X

Here

g

=

(a (g-'Xg))

-

have used the natural

on

(2.39)

coadjoint

action of G

on g* by the tensor transformation law (2.12) with x'(x) g x. From this it follows immediately that the equivariance condition (2.38) is satisfied for the polynomial maps a: g --> AM in the G-invariant subalgebra

E g.

we

and the induced G-action

AM from that

on

=

A,M where the

(2.40)

are

Elements of G

are

g are

as

dictated

0

AM)'

(infinitesimal)

(2.40) G-invariant part. The

called equivariant differential forms

represented through the exponential map,

where &

M

-

(S(g*)

G denotes the

superscript

elements of

=

on

constants and Xa

are

[19, 21].

in terms of elements of the Lie

=

ec

aXa

algebra

g

(2.41)

the generators of g

obeying the

Lie bracket

algebra

[Xa, Xb] with

ing

fabc

we

the

shall

=

fabcXc

(2.42)

antisymmetric structure constants of g. Here and in the followimplicit sum over the Lie algebraic indices a, b, c,.

assume an

...

The space where the ca,s in (2.41) lie defines the group manifold of G. The generators Xa can be written as Xa gl,=o and so the Lie algebra g =

be

a

ac,

the tangent space to the identity on the group manifold regarded of the Lie group G. The strucutre constants in (2.42) define a natural representation of G of dimension dimG, called the adjoint representation, whose can

as

(Hermitian) generators

have matrix elements

The smooth G-action

on

M

can

be

(ad

Xa )be

=

represented locally

if abc. as

the continuous

flow 9t

-

X

=

X(t)

t E

R+

(2.43)

where gt is a path in G starting at the identity gt=o. The induced action differential forms is defined by pullback, i.e. as

(gt a)(x) -

For

==

a(x(t))

on

(2.44)

example, we can represent the group action on C' functions by diffeomorphisms on M which are connected to the identity, i.e.

2.

22

Equivaxiant Cohomology and the Localization Principle

(gt f ) (x)

=

-

The action where

V(x)

element. It

f (x (t))

etv ('(t)) f (x)

=

f

,

AO M

E

(2.45)

(2.45) represents the flow of the group on C' functions on M, VA(x) ax" a is a vector field on M representing a Lie algebra is related to the flows (2.43) on the manifold by =

- 'W which defines

a

set of

=

VIWO)

in M which

curves

we

(2.46)

will refer to

as

the

integral

curves

of the group action. If VI is the vector field representing the generator X1 of g, then the Lie algebra (2.42) is represented on C' functions by

[Va' Vb] (h)

fabcVc(h)

=

Vh

E

AOM

(2.47)

represented by the ordinary commutator bracket. This derepresentation of G by vector fields in the tangent bundle TM. In this setting, the group G is represented as a subgroup of the (infinite-dimensional) connected diffeomorphism group of M whose Lie algebra is generated by all

with Lie bracket fines

a

vector fields of M with the commutator bracket.

The infinitesimal

(t

--*

0)

action of the group

on

AOM

can

be

expressed

as

V(f)

(2.48)

ivdf

=

where

AkM

---+

operator,

or

iv is the

nilpotent

contraction

to V and it is defined

iva

locally I

=

(k

-

1)!

:

on

V111

interior

k-forms

Wcellvl

Ak-IM

(2.49) multiplication, with respect

(2-15) by

...

M,

(X)dXIA2

A

...

A dx Ilk

(2.50)

The operator iv is a graded derivation (c.f. (2.22)) and the quantity ivT represents the component of a tensor T along the vector field V. The infinitesimal G-action the Lie derivative

on

along

the

higher-degree differential forms

is

generated by

V

Lv: A kM

--+

AkM

(2.51)

where

Lv

=

generates the induced action of G

Lva(x(O)) This

can

be verifed

using (2.12)

and

by

direct

(2.46),

and

(2.52)

div + ivd

on

AM, d

=

i.e.

wt- a(x(t))

lt=O

computation from expanding (2.44) about by noting that

(2.53) t

=

0


[,CV. "CVb ] (a) Thus the Lie derivative in

=


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.


derivations D 9 thereon defines the

G-equivariant complex A*(M) G

of the

manifold M.

Thus, just

as

in the last

Section,

we can

proceed

to define the

of the operator Dg. The space of equivariantly closed modulo the space of equivariantly exact forms, i.e. a

G-equivariant cohomology TTk "G(M)

With this

definition,

free G-action space

MIG,

known

(2.60)

as

=

in

(2.37).

Dg,8,

Dga

=

0,

is called the

DgIA M/im DglAk-im

cohomology

k a

(2.62)

a

of the operator

(2.62) [21, 33, 99]. Note

The definition

the Cartan model

resembles

ker

=

cohomology

i.e.

M,

M reduces to the DeRham

on

as

the

group of

forms,

Dg

for

a

fixed-point

cohomology

of the quotient of equivariant cohomology is that the definition of

Dg

in

gauge-covariant derivative. We close this Section with a few remarks concerning the above construction. First of all, it follows from these definitions that HGk (M) coincides with the ordinary DeRham cohomology of M if G is the trivial group consisting of only the identity element (i.e. V =- 0 in the above), and that the Gequivariant cohomology of a point is the algebra of G-invariant polynomials on g, HG(pt) S(g*)G, of the given degree. Secondly, if a form a c AGM is equivariantly exact, a Dg,8, then its top-form component a(') E AIM a

=

=

ordinary DeRham sense. This follows because the iv part of Dg lowers the form-degree by I so there is no way to produce a toP-form by acting with iv. Finally, in what follows we shall have occasion to also consider of AGM to include arbitrary G-invariant smooth the C' extension AIM G functions from g to AM. In this extension we lose the Z-grading described above, but we are left with a Z2-grading corresponding to the differential form being of even or odd degree [21] (Z2 Z/2Z is the cyclic group of order is exact in the

=

2).

2.4 Fiber Bundles

and

Equivariant Characteristic Classes

The 'bundles'

examples

we

introduced in Section 2.2

(tangent, cotangent, etc.)

are

all

general geometric entity known as a fiber bundle. The geometry and topology of fiber bundles will play an important role in the development of equivariant localization theory, and in this Section we briefly review the essential features that we shall need (see [22, 41, 341 for more detailed discussions). A fiber bundle consists of a quadruple (E, M, F,7), where E is a topological space called the total space of the fiber bundle, M is a topological space called the base space of the fiber bundle (usually we take M to be a manifold), F is a topological space called the fiber, and 1 7r : E --> M is a suriective continuous map with 7r- (x) F, Vx E M, which is called the projection of the fiber bundle. A fiber bundle is also defined so of

a more

=

2.4 Fiber Bundles and

that

locally it is trivial, neighbourhood U C M


25

locally the bundle is a product U x F of an open fibers, and 7r : U x F -+ U is the 3. the In first coordinate onto the case of the tangent bundle, for projection the fibers T R' F and the projection map is defined -_ are .,R' instance, on TM --+ M by 7r: T ,M --+ x. In fact, in this case the fibration spaces are vector spaces, so that the tangent bundle is an example of a vector bundle. If the fiber of a bundle is a Lie group G, then the fiber bundle is called a principal fiber bundle with structure group G. It has a right, smooth and i.e.

of the base and the

=

free action of G

the total space E and the base M which gives

on

representation of the group in the fibers. This action also embeds the group'G inside E. The vector and tensor fields introduced in Section 2.2 should be defined

cisely s

:

M

-+

Although

as

'sections' of the associated

E which take

shall be

a

point

bit abusive in

a

local

copy of

more

pre-

i.e. smooth maps

M into the fiber

E

X

bundles,

a

7r-'(x)

over

discussion

by considering M, for simplicity and ease of notation, it should be "kept in mind that it is only locally where these objects admit such a functional interpretation. Thus, for instance, the tangent bundle is TM J(x, V) : x E M, V E TxMJ and locally a section of TM can be x.

these

as

we

a

genuine functions

our

on

=

(x, V4 (x) .0. ). (i.e. bases) on the tangent bundle TM form a principal GL(n,R)-bundle over M, called the frame bundle, whose points are (x; (ei,..., en)) where x E M and (ei,..., en) is a linear basis for TxM. If M has a.metric (i.e. a globally defined inner product on each tangent space TxM), then we can restrict the basis to an orthonormal basis and obtain a principal 0 (n N, N)-bundle, where (n N, N) is the signature of the metric. written

as

X

The set of frames

-

-

If M is furthermore

orientable, then we can further restrict to an oriented ordefined basis, by the equivalence classes with respect to the equivthonoi'tmal alence relation e =- M f where det M > 0, and get a principal SO (n N, N)bundle. When M is a space-time manifold, the Lie group SO (n N, N) is -

-

-

as the local Lorentz group of M (or of the tangent bundle The associated spin group spin(n N, N) is defined as a double cover of the local Lorentz group, i.e. SO(n N, N) -- spin(n N, N)/Z2 (for

then referred to

TM).

-

-

instance, spin(2) bundle

over

=

U(1)

and

spin(3)

M whose fibers form

orthqnormal

frame bundle is called

structure

the manifold M.

on

Conversely,

to any

principal'

=

double

a

a

-

SU(2)).

principal spin(n

A

cover

spin bundle and

G-bundle P

--+

-

N, N)-

of those of the oriented is said to define

M there is

an

a

spin

associated

vector bundle. Let W be the

G. Since G acts

representation space for a representation p of smoothly and freely on the right of P, locally on P x W

(p, v) bundle (P

there is the G-action

associated vector 3

The

topology

taken

as

on

(p g 1, p(g) v) where g E G. This defines the x W)IG PIG for the representation p, which

--+

-

-

the total spaces E

the induced

topology

-

-4

=

ILEM 7r-'(x)

from the erection of

of fiber bundles is

points from M.

usually

26

2.

Equivaxiant Cohomology and the Localization Principle

has fiber the vector space W. For instance, for the trivial representation p, (P x W)IG PIG x W. In this way, we can naturally identify sections (e.g. =

(P x W)IG with equivariant functions f : P W, i.e. p(g-1) f (p). Notice that for a vector bundle E M, the bundle

differential

f (p g) -

=

forms)

on

--+

-+

-

of differential forms

on

M with values in E is defined

Ak (M, E)

=-

as

A kM 0 E

(2.63)

products and direct (Whitney) sums of bundles are defined locally by corresponding algebraic combination of their fiber spaces. Intuitively, a fiber bundle 'pins' some geometrical or topological object where tensor

the

over

each point of

bundle).

a

manifold M

vector space W

bundle is

=

(e.g.

a

vector space in the

case

of

a

vector

R', then the tangent bundle TR' associates the R' to each point of R'. In fact in this case, the tangent

For instance, if M

=

globally given by TR'

R'

=

x

W, the product

of its base and

fibers. We then say that the bundle is trivial, in that the erecting of points into vector spaces is done without any 'twistings' of the fibers. However, a

general

vector bundle is

only locally

trivial and

globally

the fibers

can

twist

very complicated fashion. One way to characterize the non-triviality of fiber bundles is through special cohomology classes of the base manifold M

in

a

[104].

called characteristic classes

signifies

sense

As

the

non-triviality

A non-trivial characteristic class in this

of the vector bundle.

shall see, all of these notions can be generalized to the case of the equivariant cohomology of a manifold which signifies the non-triviality of an we

equivariant bundle. First,

we

define what

we mean

by

an

equivariant bundle

[19, 20]. We say that a fiber bundle E ") M is a G-equivariant bundle if there are G-actions on both E and M which are compatible with each other in the

sense

that g

This

means

.7r(x)

=

that the bundle

ir(g x)

VxEE

-

projection

7r

is

a

,

(2.64)

VgEG

G-equivariant

map. The action

of the group G on differential forms with values in the bundle is by the Lie derivatives LV..

generated

ordinary DeRham case, when there are 'twists' in the given bundle specify how to 'connect' different fibers. This is done using a connection.V which is a geometrical object (such as a 1-form) defined over M with values in E whose action on sections of the bundle specifies their parallel transport along fibers, as required. The parallel transport is generated by the covariant derivative associated with F, In the

one

needs to

V

:--

d +.V

(2.65)

The derivative operator (2.65) is a linear derivation which associates to each section of the given vector bundle a 1-form in Al (M, E). If V is a tangent vector on M, its action on a section s is defined in local coordinates by


(VS)"M where

(.P,,)Pdx"

the

path

is

1-form

a

E).

space indices in

on

If x(t) is

a

=


V109AS"

+

(FOPSO)

M with values in

path

(2.66)

End(E) (a,,3

M, then: (t)

in

27

is

a

are

tangent

the vector

vector

along

and the equation

Ts)(: M) determines

parallel transport along

the

=

(2.67)

0

path allowing

us

to connect different

fibers of the bundle. The first order differential equation (2.67) can admit topologically non-trivial solutions if either the space M is multiply-connected

(Hi (M; Z) 54- 0), or if the connection F has non-trivial curvature F 54 0 (see below). The latter condition characterizes the non-triviality of the bundle, so that F 0 on a trivial bundle and the solutions to (2.67) are straight lines. At each point p of the total space E, there is a natural vertical tangent space Vp in the tangent space TpE along the fiber of E. A choice of connection above is just equivalent to a choice of horizontal component Hp in the tangent space so that TpE Vp Hp. =

=

When the bundle P

that this

(i.e. Hp

splitting

Hp.,

--*

connection.P is r E

A' (P,

g).

Horizontality

The horizontal

and

ker subspace can then be taken to be Hp F that satisfies G-equivariance mean, respectively, =

=

Cvl'

X on

P

adjoint finite, total adjoint

r,.g

on

briefly

G and G

look at

then ---

pt

=

a

=_

E g

-[x, ri and

action of the group

Ad(g-')r

=

(2.68)

ad(X)

on

can

r

denotes the

be

exponenti-

as

g-irg

(2.69)

examples. If G is a matrix group (e.g. SU(N), regard matrix elements of g E G as functions principal G-bundle. Then the unique solution to

some

as

a

(2.68)

for

a

Lie group is called the Cartan-Maurer

1-form F

For

ad(x)r

F.

can

we

the connection conditions

(matrix)

-

representing X

ated to give the

us

=

action of X. This infinitesimal action

SO(N), etc.),

require further G-equivariant

we

-

where V is the vector field

Let

principal G-bundle,

a

under the action of g E G). In this case, Vp g so that the 1-form with values the Lie in a globally-defined algebra g, i.e.

iVI'

infinitesimal

M is

-->

into horizontal and vertical components be

general G-bundle

P

=

") M, in

g-ldg a

(2.70)

local trivialization U

x

G

-*

ir-'(U),

the connection must look like

r(.,,g) where A which is

=

A,-,dxA

usually

=

A'Xa

g-ldg + g-'A,,gdx4 dxA is

a

Lie

refered to in this context

as

algebra a

(2.71) valued 1-form

gauge connection

on

or

M,

gauge

28


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

=


AG(U, E)


(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.


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.


DgP(Fg) and


rP(VgFg)

-

=

31

(2.92)

0

the

resulting equivariant characteristic classes P(Fg) of the given G-equivariant bundle are elements of the algebra AGM. These are denoted by Pg(F), or when evaluated on an element X E g with associated vector field V E TU (9 W, we write now

Pg(F)(X)

P(Fv)

=:

=_

PV(F)

(2.93)

The equivariant cohomology class of Pg(F) is independent of the chosen connection on the bundle. Consequently, on a trivial vector bundle M x W choose

we can

flat connection 5, F

a

Pmxw(Fg)(X)

=

0, and then

=

PMxW(I-tv)

=

P(p(X))

(2-94)

where p is the representation of G defined by the G-action on the fibers W. There are 4 equivariant characteristic classes that commonly appear in the localization formalism for be understood and

as

extensively discussed polynomial tr ec, and is used one

topological field theories, all of which are to completion A M. These can all be found

elements of the

are

in which the fibers

are

in

for

[211.

The first

one

is related to the invariant

G-equivariant complex vector bundles (i.e. over the complex numbers C). It is

vector spaces

called the G-equivariant Chern character

chg(F) The other 3

axe

equivariant real

tr

=

eFg

(2.95)

given by determinants of specific polynomials. On we define the equivariant Dirac A-genus

a

G-

vector bundle

Ag(F)

det sinh(!Fg)] 2

=

F

9

(2.96)

2

where the inverse of

always

an

inhomogeneous polynomial

of differential forms is

to be understood in terms of the power series

(1

+

X)-1

=

1:(_I)kXk

(2-97)

k=O

On

a

is the

complex fiber bundle, the complex equivariant Todd class

tdg(F) 5

A n6n-trivial vector bundle with

a

can

non-trivial curvature F

applications

to

topological

:34

=

det

always

[

version of the

eFg

-

11

be considered

0. This

point of

gauge theories.

equivariant

A-genus

(2.98) as a

view is

trivial

one

quite useful

endowed

in certain

32

2.



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


discuss

a

very

interesting property of equivariant cohomology which

is the fundamental feature of all localization theorems. It also introduces

the fundamental

constraint that will be

'geometric

in what follows. In most of

following

situation. Let M be

ary and let V be

circle group G

0

E

S(u(l)*), U(1),

bra of

a

=

a

vector field

U(1)

-

which is

will not be

a

applications

our

S'

we

one

of the issues of focus

will be concerned with the

compact orientable manifold without boundM corresponding to some action of the

over

M. In this

on

linear functional

case on

the role of the

multiplier algefollows. Indeed,

the 1-dimensional Lie

important for the discussion that

regard 0 as just some external parameter in'this case and 'localize' -1. As shown in [9] (see also Appendix B), algebraically by setting 0 the operations of evaluating 0 on Lie algebra elements and the formation of equivariant cohomology commute for abelian group actions, so that all results below will coincide independently of the interpretation of 0. In particular, for we can

=

a

free

U(1)-action

on

(S(u(l)*) so

that in this

case

M

0

we

have

AM)U(1)

the

=

S(u(l)*)

(9

A(MIU(1))

(2.106)

multipliers 0 play no cohomological role and the restricts to the cohomology of the quotient space

equivariant cohomology just

34

Equivariant Cohomology and

2.

MIU(1).

The

the Localization

corresponding equivariant

Principle

exterior derivative is

now

denoted

as

D,,,(:L) and it is

now

considered

as an

AvM It

was

Atiyah

Dv

=

operator

fa

=

[9]

and Bott

-=

d + iv

on

the

algebra

AM:,Cva

E

(2.107)

01

=

and Berline and

(2.108)

Vergne [19, 20] who first point locus

noticed that equivariant cohomology is determined by the fixed of the G-action. In our simplified case here, this is the set

MV

fX

=

E

M:

V(X)

=

01

(2.109)

This fact is at the very heart of the localization theorems in both the finite dimensional case and in topological field theory, and it is known as the equivariant localization

principle.

analytic ways. For a the Weil algebra and the in 2

see

In this Section

shall establish this property description of this principle using we

algebraic topological definition

more

of

equivariant cohomology,

[9]. Our first argument for localization involves

an explicit proof at the level of integral fM a over M of an equivariantly closed differential form a E AVM, Dva 0, we wish to show that this integral depends only on the fixed-point set (2.109) of the U(I)-action on M. To show this, we shall explicitly construct a differential form A on M MV a. This is just the equivariant version of the Poincar6 satisfying DVA lemma. Thus the form a is equivariantly exact away from the zero locus MV, and we recall that this implies that the top-form component of a is exact. Since integration over M picks up the top-form component of any differential 0 by hypothesis here, it follows from Stokes' theorem form, and since o9M (2.35) that the integral fm a only receives contributions from an arbitrarily small neighbourhood of Mv in M, i.e. the integral 'localizes' onto the smaller subspace MV of M. To construct A, we need to impose the following geometric restriction on the manifold M. We assume that M has a globally-defined U(1)-invariant Riemannian structure on it, which means that it admits a globally-defined

differential forms. Given

an

=

-

=

=

metric tensor I 9

which is invariant under the

=

2

g,,, (x) dx" 0 dx'

U(1)-action generated by V, LVg

or

in local coordinates

on

(2.110)

=

i.e. for which

(2.111)

0

M,

gM,X-q,V-1 + g ,,Xa,,VA + Oaxgl"

=

0

(2.112)

2.5 The

Alternatively,


this Lie derivative constraint

g"XV1.V1'

gA,\VVV,1

+

(2.65)

where V is the covariant derivative

be written

can

35

as

(2.113)

0

=

constructed from the Levi-Civita-

Christoffel connection

_VA

/w

associated with g

on

2

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

(2.114)

'9PgA')

-

the tangent bundle TM. Here gAV is the matrix inverse on the vector field V in the usual

of gA, and the covariant derivative acts way

as

V A VV

19AVV

=

VM VA

+ -PA

(2.115)

I

plus sign for (0, k)-tensors and a minus sign for (k, 0)-tensors in front in (2.115). Notice that by construction the Levi- Civita-Christoffel r,\ and it is compatible with the the metric connection is torsion-free, rA V117 AV 0, which together mean that V preserves the inner product in g, V,\gAv

with

a

of F,

as

=

=

the fibers of the tangent bundle. The equivalent equations (2.111)-(2.113) and in this

case we

say that V is

a

Killing

are

called the

Killing equations

vector field of the metric g. Since

the map V --+ LV is linear, the space of Killing vectors of a Riemannian manifold (M, g) generate the Lie algebra of a Lie group acting on M by

diffeomorphisms

isometry group of (M, g). We shall deChapters 5 and 6. The Killing equations here

which is called the

scribe this group in detail in are assumed to hold globally

over

the entire manifold M. If both M and G

are compact, then such a metric can always be obtained from an arbitrary Riemannian metric h on M by averaging h over the group manifold of G in its (G-invariant) Haar measure, i.e. g = D (j h). However, we shall have

fG

occasion to also consider

more

general

-

vector field flows which aren't

neces-

or when the manifold M isn't compact, as are the cases in many applications. In such cases the Lie derivative constraint (2.111) is a

closed

sarily physical very

stringent

one on

the manifold. This feature of the localization

that the manifold admit

globally

a

formalism,

defined metric with the property (2.111) are globally-defined C' functions on M,

whose components gAv (x) g,, (x) is the crux of all finite- and infinite-dimensional localization formulas and will =

be

analysed

in detail later

on

in this Book. For now,

we

content ourselves with

assuming that such a metric tensor has been constructed. Any metric tensor defines a duality between vector fields and differential 1-forms, i.e. we can consider the metric tensor (2.110) as a map g: TM

which takes

a

--+

T*M

(2.116)

vector field V into its metric dual 1-form

#

=-

g (V,

-)

=

gA, (x) V' (x) dxA

(2.117)

36

2.


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


on

map

=

Hn (.A4;

R)

for

-+

HO (pt; R)

=

R. The

same

is true for

equivariant

integration. Since, general M, integration of a differential form picks up the top-form component which for an equivariantly exact form is exact, for any equivariantly-closed differential form a we can again invoke G-action

a

on

Stokes' theorem to deduce

f (a

DgA)

+

(2.126)

a

M

M

that the

integral of an equivariantly closed form depends only on the equivcohomology class defined by it, and not on the particular representative. Note, however, that equivariant integration for general Lie groups G takes a far richer form. In analogy with the DeRham case above, the integration of equivariant differential forms defines a map on HG(M) HG(pt) S(g*)G. This we define by so

ariant

-->

I

(X)

a

I a(X)

=

(2.127)

X E g

,

M

with

the AM part of

ordinary DeRham. sense. algebra elements 01 in a more 'dynamical' situation where they are a more integral part of the cohomological description above. We shall see then how this definition of integration should be accordingly modified. In any case, the arguments below which lead to the equivariant localization principle generalize immediately to the non-abelian integration

Later on,

case as

we

over

in the

a

shall also consider the dual Lie

well.

Given that the class defined

by

a,

fm a depends only on the equivariant cohomology

integral we can

choose

particular representative of the cohomolTaking the equivariant differential integral

a

ogy class making the localization manifest. form,3 defined in (2.117), we consider the

Z(S)

f

=

a

e- Dv,6

(2.128)

M

viewed of

S E

as a

function of

R+ and that

its

s

E

s

--+

R+.

We

0 and

s

-*

oo

that

(2.128)

is

a

regular

limits exist. Its

s

--

assume

function

0 limit is the

interest, fM a, while from the identities (2.119) and (2.120) we see integrand of (2.128) is an increasingly sharply Gaussian peaked form around Mv C M as s --+ oo. The crucial point here is that the equivariant differential form which is the integrand of (2.128) is equivariantly cohomologous to a for all s E R+. This can be seen by applying Stokes' theorem to

integral

that the

get

of

38


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.



dnq el'7"M111"7' d?7' dq'-1

_-

Principle

Pfaff M

(2.135)

d771. Note that under a local change of basis qA algebra the antisymmetry property (2.131) and the Berezin rules (2.134) imply that f dni7 det A f dnq. It follows from this that the Berezin integral in (2.135) is invariant under similarity transformations. (2.135) is the fermionic analog of the Gaussian integration formula (1.2). The differentiation of Grassmann variables, for which the integration in (2.134) is the antiderivative thereof, is defined by the anticommutator where

dn,

=

...

A Vlq' of the Grassmann

--+

19

aqA With these

,77V

I

=

R

(2.136)

A

+

definitions, the integration by parts formula

f dq"

d

d

d7711

f(ntt)

=

0

f dqA f(,qli) &71A gqji). definitions, we can now alternatively write the integral of any differential form over M as an integral over the cotangent bundle M OA1M. Thus given the localization formula (2.130) with the I-formfi in (2.117) and always holds,

since

=

Given these

the identities

j

a

=

(2.119)-(2.122),

M

d'x

have

d',q a(x, 77)

MOAIM x

where the the

j

lim 5--+oo

we

exp,

measure

measures

d'x

(2.137) S

S9tIV (X) V, (X) V, (X) d'x =

dlq

dxl

other. To evaluate the

on

A

...

large-s

(&A' M

M

-

2

Wn"n')

is

coordinate-independent because dnq transform inversely to each

A dXn and

limit of

(QV) w,

(2.137),

we

use

the delta-function

representations 5 (V)

lim.

=

s

Jim 5-00

as can on

be

seen

directly

00

S Ir

)

n/2

1

( -S)-n/2 Pfaff

is

2

e

from the respective

integrations in local coordinates Killing equations (2.113), the matrix

given by

(S'2v),,, Thus

S?v V

(2.138)

(2.139)

-

-

M and A' M. Notice that 'from the

(Qv),,,

Vld-etg e-'9- V" V'

using (2.138) and (2.139)

=

2g7,,\V,,VA

we can

write

(2.140) (2.137)

n

f j1d#4 d?714 jA=1

e-fI"M-`7'

=

detM

as

2.6 The

f

a

f

(-,r)n/2

=

M

dnx

dnq a(x, 71)

Pfaff f2v (

M(DA1M

where

VFd-etg(x)

A'M

Theorem

6(V(x))6(,q)

41

(2.141)

Sn/2 between (2.138) and

note the cancellation of the factors of

we

(2.139).

Berline-Vergne

(2-141)

kills off all k-form components of the form a except its Cl-function part a(') (x) =- a(x, 0), while the integration over M localizes it onto a sum over the points in MV. This yields The

integration

J

over

in

(0)(p) Pfaff S?v(p) (-Ir)n/2 E IdetdV(p)l VFd-etg(p) pEMv C,

a

=

M

(2.142)

I det dV(p) I comes from the Jacobian of the coordinate transV(x) used to transform J(V(x)) to a sum of delta-functions EPEM, J(x -p) localizing onto the zero locus MV. Substituting in the idendV(p), the tity (2.140) and noting that at a point p E Mv we have VV(p) expression (2.142) reduces to

where the factor formation

x

--

=

1

a=

PEMv

M

where

we

emphasize the

a

(-2-7r)n/2

manner

(0)

Pfaff

in which the

(p) W(p)

(2.143)

dependence of orientation

in the

Pfaffian has been transfered from the numerator to the denominator in going from (2.142) to (2.143). This is the (non-degenerate form of the) Berline-

Vergne integration formula, a

and it is

localization formula. It reduces the

space M to

example of what we shall call original integral over the n-dimensional our

first

discrete set of points in M and it is valid for any differential form a on a manifold with a globally-defined

a sum over a

equivariantly-closed

diffeomorphism. Killing vector). In general, the localization formulas we shall encounter will always at least reduce the dimensionality of the integration of interest. This will be particularly important for path integrals, where we shall see that localization theory can be used to reduce complicated infinitedimensional integrals to finite sums or finite-dimensional integrals. We close this Chapter by noting the appearence of the operator in the denominator of the expression (2.143). For each p E MV, it is readily seen circle action

(and

generator is

a

Riemannian metric for which the associated

that the operator dV(p) appearing in the argument of the Pfaffian in (2.143) is just the in vertible linear transformation Lv(p) induced by the Lie derivative the tangent spaces TpM, i.e. by the induced infinitesimal group the tangent bundle (see (2.55)). Explicitly, this operator is defined vector fields W W"(x) .9x" Jx=p E TpM by

acting

on

action

on

on

2.

=

Lv(p)W Note however that

right

on

dV(p)

=

aV"(p)W'(p)

is not covariant in

the tangent space

TpM

a ax"

(2.144) X=P

general and so this is only true general on the entire tangent

and not in

42

2.


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.


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)


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


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)


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


If, 0", it

obeys

Theory

for

Dynamical Systems

1 g' f 1"'

(3.14)

the Leibniz property

I f, ghl,, and it satisfies the Jacobi

gf f, hj,,

=

+

hf f, gl,,

+

I h, I f, gj,, j,,,

(3.15)

identity

I f, Ig, hl,, J,,

+

I g, I h, f j,, 1,,

=

(3.16)

0

This latter property follows from the fact (3-3) that w is closed. These 3 properties of the Poisson bracket mean that it defines a Lie bracket. Thus the Poisson bracket makes the space of C'-functions we call the Poisson algebra of (M, w)

which

M into

on

a

Lie

algebra

-

The connection with the

elementary formulation of classical mechangiven by a result known as Darboux's theorem [65], which states that this connection is always possible locally. More precisely, Darboux's theorem states that locally there exists a system of coordinates (pj, qll)'=, on M in which the symplectic 2-form looks like ics discussed above is

w

so

that

they have

dpl,

=

A

Poisson brackets

fpj,,p,j,,,=fqA,q"j,=0 These coordinates

(3.18)

from

we see

are

called canonical

that

they

fp/,, q'l,,,

,

0

(3.8)

0

where

(P,,, Ql')'=, A

=

are

pjdqA

J"

(3.18)

A

Darboux coordinates

or

position variables on the phase the symplectic potential is

and the transformation

=

on

M and

be identified with the usual canonical

can

mentum and nates

(3.17)

dql'

space M

[551.

pjdqA

=

mo-

In these coordi-

(3.19)

becomes F )

0 + dF

=

OF

=

also canonical coordinates

(3.20)

PldQ" according

to

(3.10).

It fol-

lows that

pjzdqA

-

PjdQ1'

=

(3.21)

dF

position vari(p,,, qA) (Pl,,-Qll) are the usual canonical form is of transformation determined on a (3.21) the function F by generating [55]. Smooth real-valued functions H on M (i.e. elements of AOM) will

where both ables

and

canonical momentum and

M.

be called classical observables. Exterior

products

mine non-trivial closed 2k-forms

(i.e.

[Wk]

E

on

M

H2k (M; R)). In particular, the 2n-form

of

non-zero

W

with itself deter-

cohomology

classes

3.2


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)


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


,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


(3.28)

to the definition


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


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


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


53

was essentially born in 1982 general class of Hamiltonian the stationary-phase approximation

The field of equivariant localization theory [39] found a

when Duistermaat and Heckman

systems for which the leading-order of gives the exact result for the partition function

the

O(I/Tn+l)

the

(3.53)

correction terms in

Duistermaat-Heckman theorem goes

all

as

(3.52) (i.e. for which vanish). Roughly speaking,

follows. Let M be

a

compact sym-

plectic manifold. Suppose that the vector field V defined by (3.29) generates the global Hamiltonian action of a torus group T (S')' on M (where we 1 for simplicity) Since the critical point set of shall usually assume that m the Hamiltonian H coincides with the fixed-point set MV of the T-action on M we can apply the equivariant Darboux theorem to the Hamiltonian sys=

=

[65].

tem at hand

This

-

of Darboux's theorem tells

generalization

us

that not

local canonical system of coordinates in a neighbourhood of each critical point in which the symplectic 2-form looks like (3.17), but

only

find

can we

a

these coordinates coordinate

can

0 of the so that the origin p,, q1' the fixed point p of the given compact that in these canonical coordinates the torus

further be chosen

=

=

neighbourhood represents

group action

M. This

on

means

(locally) linear and has the form (rotations in each (p,,, ql') plane) [39]

action is

V

(P

=

a -

/t

i qA

-

ql-t

of

n

canonical rotation generators

PEMV

apl,

(3.54)

A,(p) are weights that will be specified shortly. From the Hamilton equations (3.29) it follows that the Hamiltonian near each critical point p

where

can

be written in the

quadratic

form n

H(x)

=

H(p)

+

2

(P2

+

q 2)

(3.55)

by the Hamilton equations of mot e"\, about the critical points, which tion (3.34) are the circles p,, (t), qt(t) gives an explicit representation of the Hamiltonian T-action locally on M and the group action preserves the Darboux coordinate neighbourhood. Thus each neighbourhood integration above is purely Gaussian and so all higher-order terms in the stationary-phase evaluation of (3.52) vanish and the partition function is given exactly by the leading term in (3.53) of its stationary-phase series2. This theorem therefore has the potential of supplying a large class of dynamical systems whose partition function (and hence all thermodynamic and statistical observables) can be evaluated exactly. Atiyah and Bott [9] pointed out that the basic principle underlying the Duistermaat-Heckman theorem is not that of stationary-phase, but rather of In these coordinates the fiows determined -

2

Of T

course

the

proof

contributing

is

in this

completed by showing case

to

(3.53)

-

that there is

for details

we

no

refer to

regular

[39].

function of

54


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)


77('H(p)) Using

the

identity

1

eif(11-1)

=

2n

=

55

(3.61)

2A(p)

-

it follows that

n

sgn Pfaff

14 sgn iA,, (p)

dV(p)

e'i -1'q (71 (p)) 2

=

-

2

n)

-i)'\(P)

=e-'12F\(P)

JL=1

(3.62) substituting (3.59) and (3.62) into (3.58) Duistermaat-Heckman integration formula and

so

27ri

Z(T)

T

)n

(-i)\(P)

e

we

iTH(p)

pEMv

Recall from Section 2.6 that

dV(p)

finally

arrive

at the

d-et(p) aet

(3.63)

(P)

is associated with the

anti-self-adjoint

linear operator LV (p) which generates the infinitesimal circle (or torus) action on the tangent space TpM. From the above it then follows that the complex numbers

4(p)

the Cartan

generators)

are just the weights (i.e. eigenvalues of complex linear representation of the circle (or

(3.54)

introduced in of the

action in the tangent space at p and the determinant factors from appear in terms of them as the products

torus) (3.58)

n

e(p)

as

=

(_,),\(p)/2

if each unstable mode contributes

a

(3.64)

At, (p)

factor of i to the

for

integral

Z(T)

fact, dV(p) which appears in (3.58) is none other Pfaff dV(p) (see than the equivariant Euler characteristic class Ev(Yp) critical each of in M normal bundle the of point p E MV. The Arp (2.100)) the Pfaffian Pfaff

above. In

=

the bundle of points normal to the directions Mv, so that in a neighbourhood near MV we can

normal bundle is defined of the critical

point

set

as

write the local coordinates

By

its

construction,

the terms in

(3-58)

Pfaff define

x

as

dV(p)

=

p + pj_ with p E

is taken

over

MV and

A(p (see

Section

equivariant cohomology class

an

A(p

in

p

i

2.6).

(-=

Arp.

Thus

T-T2n,) U( (.A4).

..'

through nontrivial irreducible representations and we can therefore decompose the normal bundle at p E MV into a direct (Whitney) sum of 2-plane bundles with respect to this group action, From

(3.54)

it follows that the induced circle action

on

is

n

Ar,P

NO')

=

P

(3.65)

A=J

is simply Ev (Np(t,)) (3.54) then implies that the equivariant Euler class of N(t') P of the orientation into account JVp induced by the proper iA1, (p) /2. Taking

Hamiltonian vector field

near x

=

p and the Liouville measure, and

using the

56


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


Theory

for

Dynamical Systems

which localize

topological integrals such as (3.71) onto the zero locus of a Mathai-Quillen formalism and its application to the construction of topological field theories [27, 29, 34, 81, 99, 121]. We shall discuss some of these ideas in Chapter 8. In regards to the Duistermaat-Heckman theorem, there is a very interesting Morse theoretical result due to Kirwan [881. Kirwan showed that the only vector field is the basis of the

Morse functions for which the stationary phase approximation can be exact those which have only even Morse indices A(p). This theorem includes the cases where the Duistermaat-Heckman integration formula is exact, and are

under the assumptions of the Duistermaat-Heckman theorem it is a consequence of the circle action (see the previous Section). However, this result is

stronger

even

phase uniformly

series in

as

-

it

means

that when

one

constructs the full

described in the last Section

11T

to the exact

stationary-

[134], if that series partition function Z(T), then the

converges

Morse in-

dex of every critical point of H must be even. From the Morse inequalities mentioned above this furthermore gives a relation between equivariant localization and the

topology of the phase space of interest if the manifold M has cohomology groups of odd dimension, then the stationary phase series diverges for any Morse function defined on M and in particular the Duistermaat-Heckman localization formula for such phase spaces can never give the exact result for Z(T). In this way, Kirwan's theorem rules out a large number of dynamical systems for which the stationary phase approximation could be exact in terms of the topology of the underlying phase space where the dynamical system lives. Moreover, an application of the Morse lacunary principle [111] shows that, when the stationary-phase approximation is exact so that H has only even Morse indices, H is in fact a perfect Morse function and its Morse inequalities become equalities. We shall not go into the rather straightforward proof of Kirwan's theorem here, but refer to [88] for the details. In the following we can therefore use Kirwan's theorem as an initial test using the topology of the phase space to determine which dynamical -

non-trivial

systems will localize in the

sense

of the Duistermaat-Heckman theorem. In

Chapter 7 we shall see the direct connection between the higher order terms in the saddle-point series for the partition function and Kirwan's theorem, and more generally the geometry and topology of the manifold M.

Examples: The Height Rinction

3.5

of We

a

Riemann Surface

some concrete examples of the equivariant localization presented above. One of the most common examples in both Morse theory and localization theory is the dynamical system whose phase

present

now

formalism space is

with g

a

compact Riemann surface Z9 of genus g

'handles')

and whose Hamiltonian hZg is the

(i.e. height

a

closed surface

function

on

Z9

3.5

Examples: The Height

Function of

a

Riemann Surface

59

[29, 85, 111, 153, 157].

For instance, we have already encountered the case S2 in Section 2.1 with the height function h_ro sphere ZO given by (2.1). The symplectic 2-form. is the usual volume form of the Riemann

=

wzo

dcos 0 A

=

(3.72)

do

metric g,,, = J,,, of R in 3-dimensional space. The partition function

induced

by the Euclidean

Z_,o (T)

f

=

e

wzo

3

from the

of S2

embedding

iThzo

(3.73)

ro

is

given by the expression (2.3) which

is

precisely the

value

anticipated

from

the Duistermaat-Heckman theorem. The relative minus sign in the last line -7r of (2.3) comes from the fact that the Morse index of the maximum 0 =

is 2 while that of 0 action

on

=

S2 associated with rigid rotations

corresponding

generating the compact group of the sphere is V a,-, and the ao

0 is 0. The vector field

moment map is

The Poincar6 series of the

just hzo. 3 2-sphere is

2

PS2 (t; 3F)

1: dimiF H (S2; IF)tk k

=

=

I +

t2

(3.74)

k=O

which coincides with the Morse series consistent with Kirwan's

with

even

theorem,

(3.67)

we see

for the

that

hzo

height function h-Vo. Thus, a perfect Morse function

is

Morse indices. Notice that the Hamiltonian vector field V

-

-2ao

here generates an isometry of the standard round metric dO(gdO+sin 20 doodo induced by the flat Euclidean metric of R 3. The differential form (2.123) with this metric is

=

do, which

Now the partition function

as

can

Z_ro (T)

expected be written I

=

-

T

f

d

is ill-defined at the 2

poles

of

S2

as

(

e

iTh zrO

do)

(3.75)

zo

0, 7r, the endreceiving contributions from only the critical points 0 the with for in the of cos explicit eval0, agreement points integration range uation in Section 2.1. The partition function (2.3) represents the equivariant 4 cohornology classes in

thus

3

In

=

general,

if M is

path-connected,

Z and if M is closed, then 4

as we

H2,,(M; Z)

=

always

assume

here, then Ho (M; Z)

Z. The intermediate homology groups

depend on whether or not M has 'holes' in it or not. Equivariant cohomology groups are usually computed using so-called classifying bundles of Lie groups (the topological definition of equivariant cohomology) see [111], for example.

-

60

3. Finite-Dimensional Localization TT2

11u(j) (S

Intuitively, (3.76)

2

)

are

=

for

Dynamical Systems

Z (D Z

(3.76)

follows from the fact that the

single

Lie

algebra generator linearly independent, for any 2 functions fl, f2 E S(u(l)*), the equivariant cohomology classes spanned by the linearly independent generators f, (fl (wzo + Ohzo) and

P E R and the invariant volume form i.e.

Theory

f2 (!P)

of S2

(3.72)

are

-

As

later on, the above example for the Riemann sphere is essentially only Hamiltonian system to which the geometric equivariant localization constraints apply on a simply-connected phase space (i.e. we

shall

see

the

H,(M;Z)

=

0).

The situation is much different

on

a

multiply-connected

phase space, which as we shall see is due to the fact that the non-trivial first homology group of the phase space severely restricts the allowed U(1) group actions on it and hence the Morse functions thereon. For example, consider the case of a genus I Riemann surface [85, 153, 157], i.e. Z' is the 2-torus T 2= S1 x S'. The torus can be viewed as a in the

parallelogram

complex edge the line segment from 0 to 1 along the real axis and the other slanted edge the line segment from 0 to some complex number -r in the complex plane. The number plane

-r

with its

opposite edges identified. We take

as

is called the modular parameter of the torus and

the upper

horizontal

we can

take it to lie in

complex half-plane C+

Geometrically,

-r

=

fz

C: Im

E

>

z

01

(3.77)

determines the inner and outer radii of the 2 circles of the

torus, and it labels the inequivalent complex structures of Z' '. We view the torus embedded in

3-space as a doughnut standing on end on xy-plane and centered symmetrically about the z-axis. If (01) 02) are the angle coordinates on S' x S', then the height function on Z' can be written

the

as

hEi (01, 02) where ri IRe -rl + Im radii of the torus. The =

:.-::

that there is

one

=

do,

a

A

d02

(3.79)

would therefore say that

C+

is the Teichmiiller space

a

in genus 0. This is

Riemann uniformization theorem. We refer to introduction to Teichmiiller spaces in can

(3.78)

simply-connected Riemann surface is a unique complex structure (i.e. a unique way of defin-

ing complex coordinates)

treatment

01) COS 02

2-form

of the torus. The TeichmUller space of so

TCOS

andr2 IRe -rl + 2 Im -r label the inner and outer symplectic volume form on T 2 is just that induced Z' as a parallelogram in the plane with its opposite

WD

point,

+ IM

-r

by the identification of edges identified, i.e. the Darboux

5In algebraic geometry

(r,

r2

be found in

[74].

a

consequence of the celebrated

[111]

and

[145]

for

algebraic geometry, while

an

elementary

a more

extensive

3.5

Examples:

The

Height Function

The associated Hamiltonian vector field for this

of

a

Riemann Surface

61

dynamical system has

com-

ponents T,r 1

,,_r,=-(rj+Im-rcos0j)sin02

Tr2l =Im-rsin0jC0S02 ,z

i

(3.80)

The Hamiltonian

(3.78) has 4 isolated non-degenerate critical points on (011 02) (0, 7r) (top of the outer circle), a minimum at (0, 0) (bottom of the outer circle), and 2 saddle points at (ir, 0) and (ir, ir) (corresponding to the bottom and top of the inner circle, respectively). The S1

x

S1

-

a

maximum at

=

Morse index of the maximum is 2, that of the minimum is 0, and those of the 2 saddle points are both 1. According to Kirwan's theorem, the appearence of odd Morse indices, or equivalently the fact that

H,(Zl; Z)

=

Z E) Z

(3.81)

with each Z

labelling the windings around the 2 independent non-contractable loops associated with each Sl-factor, implies that the Duistermaat-Heckman integration formula should fail in this case. Indeed, evaluating the right-hand side of the Duistermaat-Heckman formula (3.63) gives eiTh_,l (p)

27ri

e(p)

T

PEMv_,, 27ri =

T vII-m

-r

[r

1/2

(1+ e 2iTr2)

+

which for the parameter values iT value 21r

On the other

hand,

e3

an

( v'32

IRe rl-112

e

2iTImr

e

2iTIRe-rl

)1 (3.82)

=

1 and

I + i

-r

sinh 3 + 2 cosh 1

explicit evaluation

gives the numerical

(3.83)

1849.33

of the

partition function gives

21r 21r

ZZ, (T)

=

I I dol d02 0

e

iTh_,l (01,02)

0

(3.84)

27r =

27r

e

r2j do, Jo (iT(r,

+ Im

-r cos

01))

0

with J0 the

regular Bessel function of order 0 [60]. For the parameter values 2117.13 6 contradicting integration in (3.84) gives Z_ri the result (3.83). Thus even though in this case the Hamiltonian h_rl is a above,

a

numerical

All numerical

software

-

,

integrations

package

in this Book

MATHEMATICA.

were

performed using the mathematical


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


generalized,

coordinate

Liouville theorem that dictates when

63

independent version of the classical

given Hamiltonian system will have whose solutions can be explicitly found by integrating a

equations of motion by quadratures [35, 55]. The Liouville-Arnold theorem version of Darboux's theorem and it states that

if

one can

find

defined almost

everywhere

H(1)

functional of

=

is

themselves

essentially a global integrable

a

are

supposed

the

on

phase

JV

=

(3.86)

A

M,

space

such that the Hamiltonian

only [6]. The action variables functionally-independent and in involution, the action variables

to be

f 1,,, 1, J"'

=

i,, (t)

=

(3.87)

0

and from the Hamilton equations of motion

so

a

canonically conjugated action-angle variables (I,,,

f 1'4' OV 10 H

is

Hamiltonian is

(3.32)

f I,,, H (1) J,,

=

it follows that

(3.88)

0

that the time-evolution of the action variables is constant.

Consequently,

that the action variables generate a Cartan subalgebra (SI)l of the Poisson algebra of the phase space, and the 1,, therefore label a set of

(3.87) implies

canonically therefore

on the phase space which are called Liouville tori. is constrained to the Liouville tori, and the system is

invariant tori

The motion of

H(I)

integrable

in the

that

have found

n independent degrees of problems are conserved simple 1,,,'s quantities such as the total energy or angular momentum which generate a particular symmetry of the dynamics, such as time-independence or radial symmetry. The symplectic 2-form in the action-angle variables is

sense

we

freedom for the classical motion. The

w

and the as

=

dI1,

in

do"

A

(3.89)

corresponding symplectic potential which generates the Hamiltonian a global U(1) group action on M as in (3.47) is

the moment map of

OF The connection between

=-

0 + dF

=

integrability

11,do" and

(3.90)

equivariant localization

now

becomes rather transparent. The above integrability requirement that H be a functional of some torus action generators is precisely the requirement of the Duistermaat-Heckman theorem. The

equations of

motion in this

I,, W where

wl'(I)

tem therefore

=

=

1,, (0)

49I,,H(1).

move

along

global

solutions to the Hamilton

case are

1

00 (t)

The classical

=

00 (0)

+ W/' (I)t

trajectories

of the

the Liouville tori with constant

(3.91)

dynamical Sysangular velocity

64


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


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


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


Theory

for

Dynamical Systems

Under the usual assumptions used in deriving the equivariant localization principle, it follows that the tangent bundle, equipped with a Levi-Civitaa U(1)-invariant metric tensor g as vector bundle. Recall that the Lie derivative Lv

Christoffel connection r associated with in

(2.114),

induces

is

an

equivariant

non-trivial action of the group on the fibers of the tangent bundle which is mediated by the matrix dV. More precisely, this action is given by a

,Cv and

so

W'al,

dV"iv,.

+

the moment map associated with this

moment map

mann

=

-

(3.108)

dV

equivariant bundle

is the Rie-

[21] /.IV

=

(3.109)

VV

which

as always is regarded as a matrix acting on the fiber spaces. Given the Killing equations for V, this moment map is related to the 2-form S?V by

(S?V)i,,

=

2g,,,\(AV),A

(3.110)

and the equivariant curvature of the bundle is

Rv

=

(3.111)

R + /-tv

where the Riemann curvature 2-form of the tangent bundle is 1

R',',

2

R'

(3.112)

and

RPI,,,

=

0'

are

01,FP,

-

al_ P,

+ FP F\ 14A vo-

-

V'\ FP VA

(3.113)

tta

-

the components of the associated Riemann curvature tensor R dr+rA the normal bundle from the inherits a decomposition that, (3.106), =

.P. Note

U(I)-invariant

TM, and the curvature and moment map on corresponding objects defined on TM. Given these features of the 2-form S?V, it follows that the generators 7710' AMV satisfy

TjVV of

are

connection from

just the

restrictions of the

(f2V)I,v(xo)?7v0

=

2(gI,.\o9vV\)(xo)77v0

=

(3.114)

0

Mv. For large s E R+ in a neighbourhood of Mv, exponentially and so, in the linearization (3.105) of the coordinates perpendicular to MV wherein we approximate this neighbourhood with a neighbourhood of the normal bundle JVV, we can extend the integration over all values of x 1 there. We now introduce the scaled change of integration variables dxI lie ql0 0 the integral (2.137)

since

xg

-

=

in

a

direction cotangent to

will localize

xA xA1 --4x' + xI 0 0 + 1 /-vFs

to

,

q"

=

q' 0 +,ql 1

-->

ql0

+

77'1 1- Fs

(3.115)

expand the argument of the large-s exponential in (2.137) using the decompositions (3.115). The Jacobian determinants from the anticommuting and

Degenerate Version of the Duistermaat-Heckman Theorem

3.7

69

variables and the commuting x1f variables cancel each other, and so the integral (2.137) remains unchanged under this coordinate rescaling. A tedious but straightforward calculation using observations such as (3.114) shows that the large-s expansion of the argument of the exponential in (2.137) is given by [121]

nl-z

S

S_+00

-(S?V)A,(xo)nl L n _L 2

V

2

+ 0 (1 /

I

S-00

SnV

-

2

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

+

VA

I

1

0

(3.116)

"FS)

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

expanded the C'-functions in (3.116) in their respective Taypoints. Thus with the coordinate change (3.115), the integration over the normal part of the full integration domain

where

have

we

lor series about the critical

A'M

M 0 i.e.

over

(x 1711 )

The result is

an

in

=

(Mv (& A'Mv)

(2.137),

integral

over

(Xv (& A'Arv)

is Gaussian and

(3.117)

be carried out

can

explicitly.

the critical submanifold

27ri

Z(T)

II

d'xo d'770 eiT(H(xo)+w(xo,?7o))

T MvOA'Mv X

Pfaff S?v (xo)

.\/de-t QV (xo) (MV (xo)

+

(3.118)

R(xo, no))

chv(iTw)jm, Ev (R) JArv

27ri

T MV

where

we

have identified the

equivariant Chern and Euler characters (2.95)

and (2.100) of the respective fiber bundles. In (3.118) the equivariant Chern and Euler characters are restricted to the critical submanifold MV, and the determinant and Pfaffian there

taken

are

over

the normal bundle

jvv. Note Mv is

that the above derivation has assumed that the critical submanifold

connected. If MV consists of several connected components, then the formula (3.118) means a sum over the contributions from each of these components. Notice also the role that the

large equivariant cohomological symmetry

of the

dynamical system has played here it renders the Jacobian for the rescaling transformation (3.115) trivial and reduces the required integrations to Gaus-

ones. This symmetry appears as a sort of supersymmetry here symmetry between the scalar x1' and Grassmann 779 coordinates).

sian

There

are

several comments in order here. First of

of discrete isolated

points,

so

that

r

=

2n, then,

all, if MV

(i.e.

a

consists

since the curvature of the

70


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


Theory

for

Dynamical Systems

The right-hand side of (3.121) localizes onto the extrema of the square of the moment map E.(H')'. The absolute minima of this function are the solutions to H 0. The contribution of the absolute minimum to 0' 0 H' =

=

ZG (the dominant contribution for formula

s

oo)

--+

is

given by

a

[171] (2-7r) di G12 vol(G) 'm

ZNin

lim

=

f

-

s-oo

IMO

e' +

Mo

where Mo

=

H-1(0)IG

(or symplectic quotient)

projection we

that takes

is the Marsden-Weinstein reduced

phase space [97] global minima onto A40 is integration in (3.120) (as one can

X

E H-

have assumed that G acts

the over

1

(0)

freely

into its on

-4

H-1(0)

it is the

H- 1 (0) IG, with bundle

equivalence class [x]

H- 1 (0), and (9 is

cohomology group H 4(Mo ; R) that is defined the O's in (3.120) we note that E.(O,)2 /2

restricted to

(3.122)

and the localization of the

a consequence of the G-equivariance of the then integrate over the fibers of the bundle H- 1 (0) -7r

simple cohomological

pullback

of

Mo).

E

Here

certain element of

follows. In

as

E

a

HG4(M),

(9 E H4 (MO;

integrating

so

that when

R).

Therefore the equivariant cohomology class of Ea (0a)2 /2 E HG4(M) is determined by this form e which then serves as a characteristic class of the principal Gbundle H-1(0) ---> Mo. The Witten localization formula can in this way be used to describe the

cohomology

given symplectic G-action

on

some

of the reduced

M. We refer to

[171]

phase

space

Mo of the

for further details of this

construction.

However, the contributions from the other local extrema of Ea(Ha)2, which correspond to the critical points of H as in the Duistermaat-Heckman integration formula, are in general very complicated functions of the limiting parameter s E R+. For instance, in the simple abelian example of Sections 2.1 and 3.5 above where G S2 and H U(1), M hzo is the height function (2.1) of the sphere, the Witten localization formula (3.121) above =

=

=

becomes +1

ZZO

lim

=

s-oo

1/2

_0 f (2-7r

e-s(a-cos 0)2 /2

dcosO

=

liM 8__ 00

(1

_

I+ (S)

_

I- (s))

1

where

(3.123) we

functions

have assumed that

Jal

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


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)

:


for Phase


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


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


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


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)


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


4.

for Phase


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)


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


4.

for Phase


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


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.


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.


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.


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


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


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


4.

decomposition (4.124) is essentially some complete sets of states JX1J'(t)JkEZ k The

(t)

k

k


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


(4.129)

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


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



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.


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


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)

=


=

=

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.


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.


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


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 =


128

5.


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


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


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)


W

4AZ" O In

general,

is called

a

a

complex manifold

=

192 F(')(z,.

)

(5.14)

-

azt'09' Fl with

a

symplectic

Kdhler manifold. The closed 2-forms

structure such

(5.10)

are

as

(5. 10)

then refered to

as Kiffiler classes or Khhler 2-forms, the associated metrics (5.11) are called Khhler metrics, and the locally-defined functions F(') in (5.14) are called Kdhler potentials. For an elementary, comprehensive introduction to com-

plex manifolds and Kiffiler structures,

we

refer to

[41]

and

[61].

In the

case

5.1 at hand

here, the above

which act

on

133

yields a G-action on MG by symplecholomorphic functions f (z) on MG the Khhler potentials by

(canonical)

tic

Coadjoint Orbit Quantization and Character Formulas construction

transformations

[68],

i.e.

F(')(z,, )

(5.15)

This follows from the fact that the cotangent bundle of G is

T*G

=

G

(5.16)

g*

x

on T*G is g A) (g j, A). w() define G-invariant integral symplectic 2 structures on MG- Since H (G) 0, the 2-cocycles in (3.39) vanish and this G-action determines group isomorphisms into the Poisson algebras of M. This also follows directly from the property (5.5) of the Kirillov-Kostant 2-form above. Notice that the only non-vanishing components (up to permutation of indices and complex conjugation) of the Riemannian connection

so

that the natural

Consequently,

symplectic

action of G

=

the closed 2-forms

=

(5.11)

and curvature associated with the Kdhler metric

r"'A

=

RA

gAP'9Vg'X'0

g

=

-49-IA P

-

-

h(

by

(5.17)

JZV

AVP

The Cartan basis of g is defined

are

the root space

decomposition

(ED g,,,)

(5.18)

ci

of g, where tan

a

=

generators

subspaces of

(a,,..., a,)

are the roots of g (i.e. the eigenvalues of the Caradjoint representation of G) and g,, are one-dimensional [53, 162]. In this basis, the generators have the non-vanishing

in the

g

Lie brackets

N,,,,,aE,,+,3

[Hi, E,,]

where

a,,3

=

are

aiE,,

,

[E, E,3]

the roots of g,

Hi

=

a

+,3 54

0

(5.19)

r

3

aiHi

Hil,

i

_-

1,

.

.

.

,

r,

are

=

-a

the generators of the

El are the step operators of subalgebra h 0 C of g (9 C, and E,, which, for each a, span g,, in (5.18) and which act as raising operators by a > 0 (relative to some Weyl chamber hyperplane in root space) on the representation states I A) which diagonalize the Cartan generators (the weight states), i.e. HiJA) oc JA) and E,,,IA) oc IA+a) for a > 0. The unitary irreducible r. For 1, representations of G are characterized by highest weights Ai, i each i, Ai is an eigenvalue of Hi whose eigenvector is annihilated by all the E, for a > 0. Corresponding to each highest weight vector A (A,.... Ar) we introduce the G-invariant symplectic 2-form Cartan

=

,

g0C

=

=

.

.

.

,

I

134

5.


on

Simply Connected

Phase

Spaces

r

w(A) The

symplectic potentials associated

O(A) To construct

=

E ( \i

i9F(') azil

Aiw(') with

dz"

(5.20)

(5.20)

are

i9F(') -

a

A

+ dF

(5.21)

topological path integral from this symplectic structure, we a Hamiltonian satisfying (3.27), i.e. a Hamiltonian which is given by generators of the subalgebra of g 0 C which leave the symplectic potential (5.21) invariant. These are the canonical choices that give welldefined functions on the coadjoint orbit (5.1). As remarked at the end of Section 3.2, there usually exists a choice of function F(z,, ) in (5.21) for which this subalgebra, contains the Cartan subalgebra h 0 C of g (9 C. Let a

need to construct

H(A)

be the generators of h 0 C in the representation with

highest weight

vector A. Then the Hamiltonian

H(A)

hiH A)

(5.22)

satisfies the

required conditions and the corresponding path integral will adtopological form (4.157). Note that this is also consistent with the integrability arguments of the previous Chapters, which showed that the localizable Hamiltonians were those given by the Cartan generators of an isometry group G. Thus the path integral for the above dynamical system determines a cohomological. topological quantum field theory which depends only on the second cohomology class of the symplectic 2-form (5.20), i.e. on the representation with highest weight vector A (A,, A,). To apply the equivariant localization formalism to these dynamical systems, we note that since the Kiihler metrics g(i) above are G-invariant, the mit the

=

.

.

.

,

metric

9('\) obeys

Aig(')

the usual localization criteria. We shall

soon see

(5.23) that these group theo-

implied by the localization constraints, in that they are the only equivariant Hamiltonian systems associated with homogeneous symplectic manifolds as above. Through numerous examples of such systems and others we shall verify the localization formulas of the last Chapter and retic structures

discuss the now,

common

however,

resent for the

space

in fact

are

we

features that these quantum theories all represent. For just explore what the localization formulas will rep-

will

propagators

path integral

tr

eiTH( ') assuming

that they admit the phase Chapter with the above symplectic strucimportant group theoretic notions, and later

form of the last

ture. This will introduce

some

Coadjoint Orbit Quantization

5.1

we

shall show

precise1v

more

tions and discuss

and Character Formulas

path integral representaevaluating the localization

how to arrive at these

of the intricacies involved in

some

135

formulas.

apply the Niemi-Tirkkonen localization formula (4.130) to the dynamabove, we first observe that the tangent and normal bundles of related by [21] in are 0A1 g* To

ical system

TA, g*

=

Rom the construction of the

MON

TON (D MON

coadjoint

=

ON

(5.24)

9*

X

orbit it follows that the normal bundle

on the fibers, and the with the coadjoint action of G g* g* in the fibers. Then using (2.94) and the multiplicativity property (2.99), we can write the G-equivariant A-genus of the orbit ON as

g* product ON in

is

a

trivial bundle with trivial G-action is

x

a

Av

trivial bundle

over

rdet[ sinh(ad Y) I ad X

=

-1

-

=

-

where ad X is the Cartan element X E h in the g. We

now

vector

A,

choose the radius of the orbit to be the

i.e. A'

=

Weyl regarded as or

adjoint representation of Weyl shift of the weight

A + p where

P

is the

(5.25)

V/37(-adX)

2

(5.26)

a

"'>0

half-sum of positive roots of G), where A and a are linear functions on g by returning the total value of the weight

vector

(the

root associated to X E h. Then the localization formula

other than the celebrated Kirillov character formula

tr,\

eiTX

n!

(4.130)

is

none

[87, 1401 e

iTH(A)

(5.27)

0X+P

where tr,\ denotes the trace in the representation with highest weight vector A and H01) is the Hamiltonian (5.22) associated with the Cartan element X E h. If

we

further

apply

the finite-dimensional Duistermaat-Heckman theorem

to the Fourier transform of the orbit


(4.135))

we

on

the

right-hand

side of

(5.27) (i.e.

the

arrive at the famous Harish-Chandra formula

[21, 67, 140]. The

resulting

character formula associated with the Harish-Chandra for-

mula for the Fourier transform of the orbit is the classical

Weyl

character

N(Hc) lHc W(Hc) [2]-[4], [51, 87, 117, 143, 156, 1621. Weyl group of Hc, where N(Hc) is the normalizer subgroup of Hc, i.e. the subgroup of g E G with hgHC gHC, Vh E HC, so that N(HC) is the left of HC on the orbit MG action of the fixed of GlHc. points subgroup Let

formula of G

=

be the

=

=

136


5.

Given

w

=

nHC

W(Hc),

E

with

Simply Connected Phase Spaces

on

n

=

the respective adjoint representation formula can then be written as

tr.x

iN

E

N(Hc),

e x(_)

WEW(Hc)

a>O

let

X(')

=

n-'Nn be

n-1 e'xn. The Weyl character

eiT(A+p)(X('))

eiTX

a(X(w))

e

2i sin

2:a(X(-)) 2

(5.28)

XW. We shall how these character formulas arise from the equivariant localization formulas of the last Chapter, but for now we simply note here the deep group theoretical significance that the localization formulas will where see

are

explicitly later

the roots associated to the Cartan elements

on

represent for the path integral representations of the characters tr,\ e iTX in that the equivariant localization formalism reproduces some classical results of group theory. Note that the Weyl character formula writes the character a Cartan group element as a sum of terms, one for each element of the

of

Weyl

group, the group of

symmetries of the roots of the Lie algebra g. In the Chapter 4, the Weyl character formula will follow

context of the formalism of

from the

coadjoint orbit path integral over LO,\+p. It was Stone [156] who first Weyl character formula to the index of a Dirac operator from a supersymmetric path integral and hence to the semi-classical WKB evaluation of the spin partition function, as we did quite generally in Section 4.2 above. The path integral quantization of the coadjoint orbits of semi-simple Lie groups is essential to the quantization of spin systems. One important feature of the above topological field theories is that there is a one-to-one correspondence between the points on the orbits GIHC and the related this derivation of the

so-called coherent states associated with the Lie group G in the representation highest weight vector A [137]. The above character formulas can therefore be represented in complex polarizations using coherent state path integrals.

with

We shall discuss these and other aspects of the path integral representations as we go along in this Chapter.

of character formulas

Rom the

point of

performing Weyl Weyl shift problem a

view of

shift A

---

path integral quantization, the necessity of unsatisfactory. This

A + p in the above is rather

has been a point of some controversy in the literature shall [49]. see, the Weyl character formula follows directly from the WKB formula for the spin partition function [156], and a proper discretization

As

we

(5.27) really does give the path integral over the orbit 0,\ [3, 117]. The Weyl shift is in fact an artifact of the regularization procedure [3, 102, 117, 143, 160] discussed in Section 4.2 in evaluating the fluctuation

of the trace in

determinant there which led to the Niemi-Tirkkonen localization formula (4.130) and which leads directly to the Kirillov character formula (5.27). As

A-genus is inherently related to tangent bundles of real manifolds, the problem here essentially is that the regularization discussed in Section 4.2 does not respect the complex structure defined on the orbit. We shall see later on how a coherent state formulation avoids this problem and leads to a

the

5.2

Isometry Groups of Simply Connected Riemannian Spaces

correct localization formula without the need to introduce

an

137

explicit Weyl

shift.

Isometry Groups Simply Connected

5.2

of

Riemannian

Spaces

large class of localizable dynamical systems of the last Section topological and group theoretical properties, we now turn to an opposite point of view and begin examining what Hamiltonian systems in general fit within the framework of equivariant localization. For this we shall analyse the fundamental isometry condition on the physical theory in a quite general setting, and show that the localizable systems "essentially" all fall into the general framework of the coadjoint orbit quantization of the last Section. Indeed, this will be consistent with the integrability features implied by the equivariant localization criteria. We consider a simply-connected, connected and orientable Riemannian manifold (M, g) of dimension d (not necessarily symplectic for now) and with Given the

and their novel

metric g of Euclidean signature, for definiteness. The isometry group _T(M, g) is the diffeomorphism subgroup of C' coordinate transformations x ---> x'(x)

which preserve the metric distance on M, i.e. for which The generators VI of the connected component of I(M, field Lie

g)

=

g,,,(x').

form the vector

algebra

IC(M, g) and

g,,(x')

obey

=

IV

E

the commutation relations

TM: Lvg

(2.47).

For

=

a

(5.29)

0}

generic simply-connected

space, the Lie group T(M, g) is locally compact in the compact-open topology induced by M [681. In particular, if M is compact then so is _T(M, g). When

dim IC(M,

g) 0 0,

symmetric

space.

we

shall say that the Riemannian manifold

(M, g)

is

a

quickly run through some of the-basic facts concerning simply-connected Riemannian manifolds, all of whose proofs can be found in [42, 43, 68, 159, 164]. First of all, by analysing the possible solutions of the first order linear partial differential equations Lvg 0, it is possible to show that the number of linearly independent Killing vectors (i.e. We shall

now

isometries of

=

generators of

(5.29))

is bounded

as

dim IC(M,

g)

:5

d(d + 1)/2

(5.30)

d, so that the infinitesimal isometries of (M, g) are therefore characterized by finitely-many linearly independent Killing vectors in IC(M, g). There are 2 important classes of metric spaces (M, g) characterized by their possible isometries. We say that a metric space (M, g) is homogeneous if there exists infinitesimal isometries V that carry any given point x E M to any other point in its immediate neighbourhood. (M,g) is

when M has dimension

138


5.

on


said to be

isotropic about a point X E M if there exists infinitesimal isomepoint x fixed, and, in particular, if (M, g) is isotropic about all of its points then we say that it is isotropic. The homogeneity con-

tries V that leave the

dition

that the metric g must admit Killing vectors that at any given on all possible values (i.e. any point on M is geometrically like any other point). The isotropy condition means that an isotropic point x0 of M is always a fixed point of an -T(M, g)-action on M, V(xo) 0 for means

point of M take

=

V E

some

IC(M, g),

subject only

but whose first derivatives take

to the

on

all

possible values,

0. Killing equation Lvg dim M homogeneous metric space always admits d linearly independent Killing vectors (intuitively generating translations in the d directions), and a space that is isotropic about some point admits d(d 1) /2 Killing vector fields (intuitively generating rigid rotations about that point). The connection between isotropy and homogeneity of a metric space lies in the fact that any metric space which is isotropic is also homogeneous. The spaces which have the maximal number d(d + 1)/2 of linearly independent Killing vectors enjoy some very special properties, as we shall soon see. We shall refer to such spaces as maximally symmetric spaces. The above dimension counting shows that a homogeneous metric space which is isotropic about some point is maximally symmetric, and, in particular, any isotropic space is maximally symmetric. The converse is also true, i.e. a maximally symmetric space is homogeneous and isotropic. In these cases, there is only one orbit under the -T(M, g)-action on M, i.e. M can be represented as the orbit M I(M, g) -x of any element x E M, and the space of orbits M/T(M, g) consists of only a single point. In this case we say that the group I(M, g) acts transitively on

It follows that

=

a

=

-

=

M.

Conversely, if a Lie group G acts transitively on a C'-manifold M, then a homogeneous space and the stabalizer Gx Ig E G g x x} of h x defines any point x E M is a closed subgroup of G. The map hG,, a homeomorphism GlGx -- M with the quotient topology on GlGx induced by the natural (continuous and surjective) projection map ir : G GlGx. On the other hand, if G is locally compact and H is any closed subgroup of G, then there is a natural action of G on GIH defined by g -7r(h) =,7r(gh), M is

=

=

-

-

--+

g, h E

G, which is transitive and for which H is the stabalizer of the point words, homogeneous spaces are essentially coadjoint orbits of Lie groups [68], with H GA the stabalizer group of a point A E g* under the coadjoint action OA Ad*(G)A C g* of G on g*. A sufficient

7r(l). GIH

In other

condition for the coset space GIH fgH : g E GI to be a symmetric space a reductive decomposition, i.e. an orthogonal decomposition =

is that g admit g

=

h (D

h-L such that [h

introduce Khhler structures

1 ,

h-L]

(the

C

h

[68]. Furthermore,

it is

possible

to

Kirillov-Kostant 2-form introduced in the previous Section) on the group orbits for which G is the associated isometry group. These spaces therefore generalize the maximal coadjoint orbit models of the last Section where H was taken as the Cartan subgroup HC and which


5.2

maximally symmetric [68]. We shall

in fact be shown to be

can

examples

later

We shall

see

139

explicit

on.

now

manifold if and

maximally symmetric spaces. uniquely characterized by a special curva(M, g) is a maximally symmetric Riemannian

describe the rich features of

It turns out that these spaces ture constant K. Specifically,

only

curvature tensor of g

are

if there exists can

R,\p,,

a

be written

=

9AMRp1,,,

constant K E R such that the Riemann

locally

=

In dimension d > 3, Schur's lemma

everywhere

almost

K(g,pg,\,

-

as

(5.31)

gpg,\,)

states that the existence of such

[68]

a

the constancy of K. For d 2, however, this is not the case, and indeed dimension counting shows that the curvature of a Riemann surface always takes the form (5.31). In form for the curvature tensor

automatically implies

-

=

this

(M, g)

K is called the Gaussian curvature of

case

implies

not constant. The above result

and it is in

general

that the Gaussian curvature K of

maximally symmetric simply connected

a

Riemann surface is constant.

amazing result here is the isometric correspondence between maximally symmetric spaces. Any 2 maximally symmetric spaces (M 1, gi) and The

(M2) 92)

of the

same

dimension and with the

same

curvature constant K

are

M2 between the 2 isometric, i.e. there exists a diffeomorphism f : MI Thus manifolds relating their metrics by gi (x) given any maxi92 (f (x)). onto it isometrically any other one with mally symmetric space we can map --+

=

the

same

curvature tensor

(5.31).

We

can

therefore model

maximally

sym-

metric spaces by some "standard" spaces, which we now proceed to describe. Consider a flat (d + l)-dimensional space with coordinates W', z) and metric I

?ld+ 1

dx,,

=

JKJ

(9 dxP +

1

K

(5.32)

dz (9 dz

real-valued constant. A d-dimensional space can be embedded into this larger space by restricting the variables x" and z to the surface of

where K is

a

a

(pseudo-)sphere,

Using (5.33) induced

on

(dx,-, JKJ (dx,, 1

1

9K

=

(5.33)

1

z(x) and substituting by this embedding is then

0 dx"

for

XAX, &A (& dxv X2

1

-

for

(5.32),

the metric

K > 0

-

xxv2 dxl' (& dx'

I

-

for

(5.34)

K < 0

X

K =0

represent, respectively, the standard metrics

of radius K- 1/2, the curvature

Z2

this into

(& dx" +

jdx,,0dxA cases

+

to solve for

the surface

k

These 3

sgn(K)x2

hyperbolic Lobaschevsky

K, and Euclidean d-space

R

d

space

on

the

d-sphere Sd

'Hd.,of constant negative

with its usual flat metric 77Ed

-

140

5.


on

Simply Connected

Phase

Spaces

(5.33) and the manifest invariances of the it is straightforward to show that the above (5.32) space all admit a + d(d 1)/2-parameter group of isometries. These consist spaces about the of d(d rotations and d (quasi-)translations. The rigid origin 1)/2 Rom the

embedding

embedding

condition

geometry

-

first set of isometries:

always leave

points

some

on

the manifold

fixed, while

the second set translate any point on M to any other point in its vicinity. The 3 spaces above are therefore the 3 unique (up to isometric equivalence) maxi-

mally symmetric

spaces in

d-dimensions,

and any other

maximally symmetric

0, space will be isometric to one of these spaces, depending on whether K K > 0 or K < 0. It is this feature of maximally symmetric spaces that allows =

complete

the rather

vector fields that

VK

isometric

correspondence which will follow. The Killing are, respectively,

generate the above stated isometries

(Q,Axv

+ a '

sgn(K)X2]1/2 )

[I

09 aXA

for

K

:A

0

(5.35)

=

(s2AXV V

where S?A V

=

-S?,v,

+

a") -5 X/"

and a"

are

for

K

=

0

real-valued parameters. These

Killing

vectors

generate the respective isometry groups

_T(Sd)

=

SO (d +

1)

,

_T(lid)

=

SO (d,

1)

,

T(R d)

=

E

d

(5.36)

where E d denotes the Euclidean group in d-dimensions, i.e. the semi-direct product of the rotation and translation groups in R d' SO(d + 1) is the rotad+1 and SO (d, 1) is the Lorentz group in (d + l)-dimensional tion group of R ,

Minkowski space. Rom this we see therefore what sort of group actions should be considered within the localization framework for maximally symmetric spaces. Note that the maximal

symmetry of the spaces Sd and 'Hd are acSd can be regarded as the one-point

that of R d, because

tually implied by d R U fool (also known as stereographic compactification of Rd, i.e. Sd projection), and Hd can be obtained from Sd by Wick rotating one of its coordinates to purely imaginary values. ==

The next situation of interest is the

case

where

(M, g)

is not itself maxi-

mally symmetric, but contains a smaller, dH-dimensional maximally symmetric subspace Mo (e.g. a homogeneous but non-isotropic space). The general theorem that governs the structure of such spaces is as follows. We can distinguish Mo from M Mo by d dH coordinates V", and locate points within -

-

subspace Mo with dH coordinates u . It can then be shown [68, 1641 that is possible to choose the local u-coordinates so that the metric of the entire

the it

space M has the form

g

=

Ig,,,, (x) dx" 0 dx'

2

2

g,,O(v)dv'

0

dv'9

+

If(v)jij(u)du 0dui

2

(5.37)

where g,,o (v) and f (v) are functions of the v-coordinates alone, and ij (u) is a function of only the u-coordinates that is itself the metric of Mo. As (MO, j)


5.2

141

a dH-dimensional maximally symmetric space, it is isometric to one of the 3 standard spaces in dH-dimensions above and can be represented in one of the forms given in (5.34) depending on the curvature of the maximally

is

symmetric subspace Mo.

general result concerning Killing vectors on generic d-dimensional

Our final

connected manifolds is for the

simply

where the isometry group of

cases

(M, g) has the opposite feature of maximal symmetry, i.e. when _T(M, g) is 1-dimensional. Consider a 1-parameter group of isometries acting on the met-

8

V" (x) ric space (M, g). Let V E TM be a generator of I(M, g), and ax" let X/-'(x) be differentiable functions on M such that the change of variables =

x"j,

X/-(x)

=

has non-trivial Jacobian

det

For /.t

=

2,..., d

-

d-1 .

diffeornorphisms XA(x)

=

VV49,X/'

constant coordinate lines

given by the from R

LVXI'

=

The functions

(5.38)

0

to in addition be

solutions of the first order linear

1

V(X1)

=A

aXV

choose the

we can

linearly independent partial differential equation

the d

I']

X/(x)

0

=

IL

X/(x)

for M

=

(5.39)

2,..., d

constant embedded into M

2,...,d also have

=

homogeneous

an

invertible

Jacobian matrix since then

I I "X/'

rank2 2. It follows, under the simple-connectivity 0 or K > 0 the phase space assumption of this Chapter, that when K M can be either compact or non-compact, but when K < 0 it is necessarily Thus

a

=

non-compact. The other extremal

case

is where

(M, g)

general

of isometries. FYom the above

X1

there exist 2 differentiable functions

x1

on

09

09X/1

X2 (X1, X2)

and in these coordinates the 0.

ator

1-parameter

group case

and X 2on M and local coordinates

Moreover,

V,

on a

912("rf) f

Xf2

0

=

Killing

the characteristic

the initial data surfaces of the

be chosen to be

can

lie

=

by

fined

a

M such that VIZ

1, V/2

only

admits

discussion it follows that in this

i.e.

we can

orthogonal

to the

0. Thus in this g

and from

(5.45)

2(X1,X2)

vector field has curves

(5.55)

components V`

of the coordinate X'2

case

partial paths defined by the isometry means

the metric

glldx" 1

=

0

dx"

==

differential equation in

choose the initial conditions for the solutions of

non-characteristic surface. This =

X

=

+

can

that in these

be written

9212dx

/2

0 dx

new

locally

X2, de(5.55), gener-

(5.55)

to

coordinates

as

/2

(5.56)

it follows that gil and 92'2 are functions only of X'2 The phase a surface of revolution, for example a cylinder or .

space therefore describes

the

'cigar-shaped' geometries

that

are

described in

typical black hole theories

[164]. The

only other

case

left to consider here is when

dimensional isometry group. In this

case we

have 2

(M, g)

independent

has

a

2-

vector fields


5.2

145

obey the'Lie algebra (2.47) with possibilities 1, algebra either the isomea, b, it is or is fabc 0, abelian, non-abelian, fabc 4 0 for some a, b, c. try group Since V, and V2 cannot have the same path in M, we can choose paths for V,

=

c

8

Vi" axi,

and V2

2. There

=

V21 8XA

=

which

for this Lie

2

are

-

=

the constant coordinate lines

V12

that

so

commutativity

[V1 V21 implies that T

V2 2

rl

l

ric

0. In the abelian case, the

(5.57)

V22

is a function only Of X2 everywhere on M in which these coordinates, the Killing equations imply that the met-

V11

is

1. In

a

x1 alone and

function of

.

choose local coordinates almost

we can --

=

0

=

7

As above,

V21

=

of V, and V2

components g,,, (x)

are

all constant. Thus in this

case

(M, g)

is isometric

to flat Euclidean space, which contradicts the standard maximal

symmetry

arguments above. In the non-abelian case, we can choose linear combinations of the so that their Lie algebra is

isometry

generators V, and V2

[V1 V21 i

which

implies

2

V22

1 and

V11

a291,tv

=

a2 109 V11

0

1/

choose local coordinates almost

so we can

=

(5.58)

V1

that

'91V2 and

7--

=

e-X2

191911

=

Killing equations

The

.

=

0

191912

7

=

911

V22

(5.59)

everywhere

on

M in which

then become

7

191922

=

2g12

(5.60)

which have solutions gil

=

a

,

912

=

aX

1

+0

922

1

=

a(xl)2

+

20xl

where a, 6 and -y are real-valued constants. It is then compute the Gaussian curvature of g from the identity

K(x)

=

-R1212(x)/det;g(x)

which gives K(x) as the constant K a/()32 maximal symmetry theorems quoted above. =

Thus

a

2-dimensional

phase

-

space is either

+ -y

(5.61)

straightforward

to

(5.62)

a-y), again contradicting

the

maximally symmetric with

a

3-dimensional isometry group, or it admits a I-parameter group of isometries (or, equivalently, has a single 1-dimensional maximally symmetric subspace), because the above arguments show that it clearly cannot have a 2-dimensional isometry group. The fact that there are only 2 distinct classes of isometries in 2 dimensions is another very appealing feature of these cases for the analysis which follows. For the remainder of this Chapter we shall analyse the

146


5.

Simply Connected

on

Phase

Spaces

equivariant Hamiltonian systems which can be studied on the various isometric types of spaces discussed in this Section and discuss the features of the integrable quantum models that arise from the localization formalism. We shall primarily develop these systems in 2 dimensions, and present higherdimensional examples in Sections 5.7 and 5.8. This will provide a large set of explicit examples of the formalism developed thus far and at the same time

clarify

some

other issues that arise within the formalism of

path integral

quantization.

5.3 Euclidean Phase

and

Spaces Holornorphic Quantization

begin

We

study of general localizable Hamiltonian systems with the case 0. The conformal factor phase space M is locally flat, i.e. K (5.49) and (5.52) then satisfies the 2-dimensional Laplace equation our

where the in

=

v2 whose

general solutions

aaw(z" )

(M, g)

(5.63)

0

are

O(Z' o ) where

=

=

fW

+

A0

(5.64)

f (z)

is any holomorphic function on M. The Riemannian manifold is isometric to the flat Euclidean space (R 'nE2) and from the metric

2,

tensor transformation law it follows that this coordinate

the metric

aw a'CV az

(5.49)

09' aw +

_FZ

=

-5-z- wz=

It follows from

change

z --+ w

taking

to dw 0 dfv- satisfies

(5.65)

ew(z,2)

=

ef (z) el(2)

49W &cv

aw afv=

,

-

19Z az

a' ;

_FZ

=

0

(5.65)

that this isometric transformation is the 2-dimensional

conformal transformation

z -4 wf (z) (i.e. an analytic dard flat Euclidean metric of the plane) where

Wf W

=

f d6

ef (

rescaling

of the stan-

)

(5.66)

C.

and Cz C M is

a

simple

curve

(eq. (5.35)) we complex coordinates (w, fv-)

from

fixed

basepoint in M to z. Rom 2 Killing vectors of (R ,77E2) in the general form

some

the last Section

know that the

the

take

VRW2 where S?

E

R and

a

=

E

-iS?W + C

are

C1

on

VJ:2

constants. The

=

(5.67)

iJ?fV + 5V

Killing

vectors

(5.67)

follow

0 there, and they generate the groups of 2directly (5.51) with W dimensional rotations w --- eis?w and translations w ---> w + a whose semidirect product forms the Euclidean group E 2 of the plane.

from

=

5.3 Euclidean Phase

In these local

Spaces and Holomorphic Quantization

147

2 complex coordinates on R the Hamiltonian equations dH

-ivw take the form aH

Z

=

-w(w, iv-)Vw

o9H

,

2

=

-

-w(w, fv- )VW

(5.68)

2

where w

The

symplectic:

2-form

(5.69)

=

-w(w, Cv)dw 2

can

be

A

(5.69)

dCv

explicitly

by recalling symplectic so that

determined here

that the Hamiltonian group action on the phase space is 0. In local coordinates this means that LVw =

'9j'(V1'W"\) -,9,(V'\W,,,\)

=

(5.70)

0

for each p and v. Requiring this symplecticity condition for the full isometry 2 2 group action of E on R , we substitute into (5.70) each of the 3 linearly

0, represented by (5.67) (corresponding to Q 0 there). The differential equations (5.70) for the function 2 easily imply that it is constant on R with these substitutions.

independent Killing a'

=

0 and a2

(w, fv-) now Thus w(w, Cv)

w

vectors

=

=

is the Riemannian volume

R 2.

(and

in this

case

the

Darboux)

2-

w(w, -) globally on Substituting Killing vectors (5.67) into the Hamiltonian equations above and integrating them up to get H(w, fv-), we see that the most general equivariant Hamiltonian on a planar phase space M is form

Ho(z,, ) where CO E R is formation

(5.66)

the Darboux value

=

S?wf (z),Cvf (2)

+

dwf (z)

+

afv-f

iv

+

Co

=

1 and the

(5.71)

constant of integration and wf (z) is the conformal transfrom the flat Euclidean space back onto the original phase

a

space.

uniquely determined to be the phase space geometry is a general feature of any homogeneous symplectic manifold. Indeed, when a Lie group G acts transitively on a symplectic manifold there is a unique G-invariant measure [68], i.e. a unique solution for the d(d 1)/2 functions w,', from the d(d 1) d(d + 1)/4 differential equations (5.70). Thus Wn/n! is necessarily the maximally symmetric volume form of (M, g) and the phase space is naturally a Kdhler manifold, as in Section 5. 1. We shall soon see the precise connection between maximally symmetric phase spaces and the coadjoint orbit models of Section 5. L In the present context, this is one of the underlying distinguishing features between the maximally symmetric and inhomogeneous cases. In the latter case w is not uniquely determined from the requirement of symplecticity of the isometry group action on M, leading to numerous possibilities for the equivariant Hamiltonian systems. In the case at hand here, the Darboux 2form on R2 is the unique 2-form which is invariant under the full Euclidean The fact that the

symplectic

2-form here is

volume form associated with the

-

-

-

148

5.


on

Simply Connected

Phase

Spaces

group, i.e. invariant under rotations and translations in the plane, and it is the Kdhler form associated with the Kdhler metric (5.49) and

The form

on

M

(5.64).

(5.71)

planar equivariant Hamiltonian systems illustrates how the integrable dynamical systems which obey the localization criteria depend on the phase space geometry which needs to be introduced in this formalism. These systems are all, however, holomorphic: copies of the same initial dynamical system on R 2 defined by the Darboux Hamiltonian for the

HOD( z,. )

=

Qz.

+ dz + a. +

CO

;

(5.72)

C

E

z

or identifying z,. iq with (p, q) canonical momentum p variables, these dynamical Hamiltonians are of the form =

D

H0

Thus the in the

(p, q)

dependence

on

=

p(P

2

+ q

phase

the

2)

+ CelP + C92q +

and position

(5.73)

CO

space Riemannian

geometry is trivial

that these systems all lift to families of

holomorphic copies of the planar dynamical systems (5.72). This sort of trivial dependence is to be expected since the (classical or quantum) dynamical problem is initially independent of any Riemannian geometry of the phase space. It is also anticipated from the general topological field theory arguments that we presented earlier. Nonetheless, the general functions Ho(z,, ) in (5.71) illustrate how the geometry required for equivariant localization is determined by the different dynamical systems, and vice versa, i.e. the geometries that make these dynamical systems integrable. This probes into what one may consider to be the geometry of the classical or quantum dynamical system, and it illustrates the strong interplay between the Hamiltonian and Riemannian symmetries that are responsible for localization. Thus essentially the only equivariant Hamiltonian system on a planar symplectic manifold is the displaced harmonic oscillator Hamiltonian sense

HOD and in this

=

Q(z

case

action with the

+

a)(.

+

a)

=

Q

J(p + a, )2 + (q + a2 )21

+

(5.74)

CO

we can replace the requirement that H generate a circle requirement that it generate a semi-bounded group action.

To compare the localization formulas with

ementary quantum mechanics,

we

some

well-known results from el-

note that the Hamiltonian

(5.73)

can

only

describe 2 distinct 1-dimensional quantum mechanical models. These are the 1 harmonic oscillator Iz. (P2+ q2) wherein we take Q .12 and a 0 in 2 2 (5.72) and apply either the WKB or the Niemi-Tirkkonen localization formu=

las of the last a

=

Chapter,

112V2- in (5.72)

=

and the free

and

where particle jp2 2

we

=

take S?

=

0 and

apply quadratic localization formula (4.150) (or In fact, these are the original classic examples, which the

equivalently (4.142)). were for a long time the only known examples, where the Feynman path integral can be evaluated exactly because then their functional (and classical statistical mechanical) integrals are Gaussian. For the same reasons, these

Spaces and Holomorphic Quantization

5.3 Euclidean Phase

are

also the basic

exact

examples

where the WKB

149

is known to be

approximation

[147]. straightforward

It is

(4.130)

to

verify the Niemi-Tirkkonen localization formula r eiO with r C R+ polar coordinates z

for the harmonic oscillator. In

and 0 E

[0, 2-r],

that the

integral

we

in

have w,O

=

=

(S?V)O,

r,

=

-2r and R

0

=

on

flat R 2,

so

(4.130) gives 00

Zharm(T)

T

dr

can

be

1,

2

k E

2i sin

seen

by the Schr6dinger equation Z+ [1011, so that

determined k +

1

-iTr2/2

2

0

That this is the correct result

e

2 sin 1:

(5.75)

T 2

by noting that the energy spectrum for the harmonic oscillator is Ek 00

trJJ

e-

iT(02+42)/211

E

e-

iTEk

=

E

2 e- iT(k+.!)

2i sin

k=O

k

(4.115)

This result also follows from the WKB formula

regularized jectories determined by the flows

fluctuation determinant

orbits on as

z

(t)

=

z

(0)

it/2 e

as

of the vector field V'

only

loop space LC is to regard z(t) independent complex variables. This the

=

should be evaluated in

a

working

after

out the

described there. Here the classical tra-

Note that the

.

(5.76)

T 2

=

iz/2

way these orbits it/2 and e

z(O)

(t)

means

are

the circular

can

=

be defined i(T-t)/2 e

2(T)

that the functional

holomorphic polarization.

integral

We shall return to this

point shortly.

Alternatively, we note that for T: 27rn the only T-periodic critical trajecdynamical system are the critical points z,2 0 of the harmonic oscillator Hamiltonian z2 and (5.76) also follows from (4.135) which gives

tories of this

=

Pfaff For the discretized values T to

1

1

Zharm(T)

=

(

1

2i

0 -1

1)

sin

2i sin

2

(5.77)

T 2

0

2-7rn any initial condition z(O) E C leads space of critical trajectories is non-

T-periodic orbits, and the moduli

2

R In that case phase space M the degenerate path integral formula (4.122) yields the correct result. These results therefore all agree with the general assertions made at the beginning of Section 4.6 concerning the structure of the moduli space of T-periodic classical trajectories for a Hamiltonian circle action on the phase space. For the free particle partition function, we have R OV 0, and so The 001. the localization formula contributes in term theA-genus (4.150) and find Gaussian in trivial thus is one we a integral (4.150)

isolated and coincides with the entire

=

=

00

Zftee(T) 00

J -00

=

00

00

dp dq

.

doo e' T02-iTOop/2v 2_ 0

,

f dp dq -00

e-'TP2/2 (5.78)

150

5.


Simply Connected

on

Phase

Spaces

which also coincides with the exact propagator trII e-iTp2 /2 11 in the phase 2 2 space representation. In this case the Hamiltonian 1p is degenerate on R 2

that the WKB localization formula is unsuitable for this

dynamical system (4.122) by noting 2 that LMs R in this case. Notice also that (5.78) coincides exactly with the classical partition function of this dynamical system as there are no quantum so

and the result

(5.78)

follows from the

degenerate

formula

=

fluctuations. There is another way to look at the path integral quantization of the Darboux Hamiltonian system (5.72) which ties in with some of the general ideas of Section 5.1 above. The Heisenberg-Weyl algebra 9HW [101] is the

algebra generated by

the usual harmonic oscillator creation and annihilation

operators

a, ait in the canonical

(4.1).

ata

6, and

=

The Lie

I(p2

2

+

42

(infinite-dimensional)

space R

2

and the

algebra generated by the operators with the commutation relations 1) 9HW is

_

[at,a] The

(5.79)

quantum theory associated with the phase

operator algebra

at,

id)

,,F2

=

(5.80)

1

Hilbert space which defines

a

representation of

these operators is spanned by the bosonic number basis In), n E Z+, which form the complete orthonormal system of eigenstates of the number operator with eigenvalue n,

NIn) and

which tit and 6, act

on

at In) We

now

=

as

=

ataln)

aln)

define the canonical coherent states

Iz)

=_

Heisenberg-Weyl Zn

e'at 10) n=0

These states

are

normalized

-=

Iz)t,

and

(5-81)

V/n_In

-

[45, 101, 137]

group

In)

=

;

GHW

z

E

1)

(5.82)

associated with

as

C

v n.

(5.83)

as

(z I z)

(zl

n1n)

raising and lowering operators, respectively,

v/n-_+1 In+ 1)

this representation of the

with

=

=

e'2

they obey the completeness relation

(5.84)

5.3 Euclidean Phase

d2Z

d2 Z

IZ)XZ1

21r

21r

Spaces and Holomorphic Quantization Zn,

e-'2

m

Vn--!m!

In) (ml 21r

00

f

dr

r

e-r

2

n+m

E Vn--!m! f dO ei(n-m)0In)(mI

n, m

0

0

00

f

151

dr

r

e-

r2

E

E In)(nI

-21rJnmIn)(mI

_n _ ! M!

n,m

0

00

rn+m

=

1

n=0

(5.85) where

we

have

as

usual written

=

r

e'0 and

Iz)IV(zlz)

Iz)) are

z

e-

=

z2/21Z)

(5.86)

the normalized coherent states. The normalized matrix elements of the

algebra generators

in these states

are

((ZIataIZ)) Thus the 3

=

independent

Z

((ZI&'tIZ))

((ZIetIZ))

,

terms in the Darboux Hamiltonian

=

(5.87)

Z

(5.72)

are none

other than the normalized canonical coherent state matrix elements of the group generators. These 3 observables represent the PoisLie group action of the Euclidean group E 2 on the coadjoint orbit GHWIHc = GHWIU(I) C' with the Darboux Poisson bracket

Heisenberg-Weyl son

=

IZI *"D

=

(5.88)

1

algebra representation of the Heisenberg-Weyl algebra correspondence with the coset space GHWIHC and the general framework of Section 5.1 is not entirely surprising, since homogeneous symplectic manifolds are in general essentially coadjoint orbits of Lie groups [681, i.e. they can be represented as the quotient of their isometry groups by a maximal torus according to the general discussion of the last Section. The integrable Hamiltonian systems in this case are functionals of Cartan elements Of 9HW (e.g. the harmonic oscillator at & or the free particle (a + at)2) which is the Poisson

(5.80).

This

.

The canonical coherent states

(5.83)

are

those quantum states which min' Aq -Ap ! [101], because they

Heisenberg uncertainty principle diagonalize the annihilation operator 6,, 6.1z) imize the

eralized to

arbitrary Lie

groups

[137],

as

we

2

=

z1z),

shall

and

soon

can be genThe Darboux

they

see.

2-form Z

WD

=

2

globally on C and, since R 2is contractable, and hence be be generated globally by the symplectic potential

is defined

0, it

can

OD

(5.89)

-dz A d,

2

(2dz

-

zd2)

H 2 (R

2;Z) (5.90)

152

Equivariant

5.

The canonical 1-form on

R

2

can

Localization

(5.90)

on

and the

=

dz (9 d,

11 d1z)) 11

=

globally-defined

2

tr

e-M

f

=

Spaces

11 d1z)) 11

(9

Kdhler

(5.83)

(5.89)

as

((zjdjz)) -

(5.91)

((zjdjz))

(9

((zjdjz))*

potential associated with (5.90)

FR2 (Z, ) The path integral path integral

Phase

and the flat Kdhler metric associated with

be written in terms of the coherent states

OD 9D

Simply Connected

=

(5.92) is

(5.93)

Z

here then coincides with. the standard coherent state

d2Z

((zl e`? I Z))

2-7r

T

dz (t) d, LR2 tE[O,T]

(t)

exp

27r

i

f

dt

(z'!-Z

-

i

-

H (z,

2))

0

(5.94) where

H(z,. )

=

((zJ7 jz))

is the coherent state matrix element of

some

(5.95)

operator 7

=

7 (6, at)

on

the

underlying representation space of the Heisenberg-Weyl algebra. The derivation of (5.94) is identical to that in Section 4.1 except that now we use the completeness relation (5.85) for the coherent state representation. of describing the quantum dynamics goes under the names of holomorphic, coherent state or Kdhler polarization. One of its nice features in general is that it provides a natural identification of the path integral and loop space Liouville measures. We recall from (4.21) that in the former measure there is one unpaired momentum in general and, besides the periodic boundary conditions, there is a formal analog between the measures in (4.21) and (4.25) only if the initial configuration of the propagator depends on the position variables q and the final configuration on the momentum variables p, or vice versa. In the holomorphic polarization above, however, the initial configuration depends on the z variables, the final one on the ' variables, and the dzi d, j/27r. Since path integral measure is the formal N -* OC) limit of IIN j= 1 modified This

manner

the number of

z

and,

integrations

are

the same,

we

obtain the desired formal

analog between the path integral localization formulas and the Duistermaat-Heckman theorem and its generalizations, this enables one to also ensure that the loop space supersymmetry encountered in Section 4.4, which is intimately connected with the definition of the path integral measure (as are the boundary conditions for the propagator), is consistent with the imposed boundary conditions. identifications. Besides

providing

one

with

a

formal

5.4

Holomorphic Localization Formulas

153

Thus on a planar phase space essentially the only equivariant Hamiltonian systems are harmonic oscillators, generalized as in (5.71) to the inclusion of a generic flat geometry so that the remaining Hamiltonian systems are merely holomorphic copies of these displaced oscillators defined by the analytic coordinate transformation (5.66). These systems generate a topological quantum theory of the sort discussed in Sections 4.10 and 5.1, with the Darboux Hamiltonian

topological

symplectic potential (5.90) by the usual HOD 'VR2 OD reflecting the fact that (5.90) is invariant of the rotation group of the plane. It is not, however, invari-

(5.72)

related to the

condition

under the action

:_

ant under the translation group not determine

a

action,

so

Witten-type topological

lator Hamiltonians do. This

means

that the translation generators do theory like the harmonic oscil-

field

that there

the

i.e. it is

E 2-invariant

are no

to find

impossible potentials plane, gives an invariant potential simultaneously for all on

a

function F in

3 of the

symplectic

(5.21)

independent

that

gener-

(5.72).

The harmonic oscillator nature of these systems is consistent with their global integrability properties. The holomorphic polarization of ators in

theory associates the canonical quantum theory above with the topological coadjoint orbit quantum theory of Section 5.1 and the coherent state path integral (5.94) yields character formulas for the isometry group of the phase space. This will be the general characteristic feature of all localizable systems we shall find. In the case at hand, the character formulas are associated with the Cartan elements of the Heisenberg-Weyl group. the quantum

5.4 Coherent States

and

Holomorphic

Before

carrying

on

with

on

Homogeneous

Kdhler Manifolds

Localization Formulas our

geometric determination of the localizable dy-

namical systems and their path integral representations, we pause to briefly discuss how the holomorphic quantization introduced above on the coadjoint orbit R2

be

can

generalized

to the action of

an

arbitrary semi-simple

Lie group

[17, 90, 1371. This representation of the quantum dynamics proves to be the most fruitful on homogeneous spaces GIHC, and later on we shall generG

alize this construction to apply to non-homogeneous phase spaces and even non-symmetric multiply connected phase spaces. As the coherent states are those which are closest to "classical" states, in that they are the most tightly peaked ones about their locations, they are the best quantum states in which to study the semi-classical localizations for quantum systems. We shall also see that they are related to the geometric quantization of dynamical systems

[1721. Given any irreducible unitary representation D(G) of the group G and normalized state 10) in the representation space, we define the (normal-

some

ized)

state

1g) by 1g)

=

D(g)10)

(5.96)

154

5.


on


Dg denotes Haar measure of G, then Schur's lemma [162] and the pleteness of the representation D(G) implies the completeness relation If

dimD(G) vol(G)

f Dg Ig)(gl

=

com-

(5.97)

1

G

Following

the derivation of Section 4.1, it follows that the

associated with can

be

operator 7

an

acting

on

partition function

the representation space of

D(G)

represented by the path integral

f Dg (gl e-M Ijg)

trD(G) e-iT) i

G

(5.98)

T

LG

dimD(G) Dg(t) Vo l(G)

tE[O,T]

i

exp,

j (gldlg) if -

dt

(gJ7 jg)

0

-t(g)

10) to be a simultaneous eigenstate of the generators weight state), then the 'coherent' states 1g) associated with any one coset of GIRC are all phase multiples of one another. Thus the set of coherent states form a principal Hc-bundle L -- MG over GIHC and the coherent state path integral (5.98) is in fact taken over paths in the homogeneous space GIHC. This geometrical method for constructing irreducible representations of semi-simple Lie groups as sections of a line bundle L --4MG associated to the principal fiber bundle G -+ GIHC is known as the Borel-Weil-Bott method [142]. The holomorphic sections of this complex line bundle (the coherent states) form a basis for the irreducible representation. What is most interesting about the character representation (5.98) is that it is closely related to the Kiffiler geometry of the homogeneous space GIHC. To see this, we first define the Borel subgroups B of G which are the exponentiations of the subalgebras B spanned by Hi E h 0 C and E,, for a > 0 and a < 0, respectively (see (5.18)). The complexification of the coadjoint orbit MG is then provided by the isomorphism GIHC GIIB, where G' is the complexification of G [162]. Almost any g E G can be factored as a Gauss decomposition (+h(_ g (5.99) However, of HC c

if

we

G

take

(i.e.

a

-

=

where h E

Hb

and

(+ Here z' E C, weight state, states

are

in

=

and if we then a

(_

eF-,->o z'E,,,

eE

" (w(z,. ), Cv(z,, )) which accomplishes this isometric correspondence. First of all, we rewrite the spherical metric in (5.34) in complex coordi1 nates w, fvx iX2, with x1' the spherical coordinates defined in (5.34), to =

get iV-

gS2

_K

1

+2

-

2+

2

wCv

)

W Cv

I

-

W

dw 0 dw +

WiD

1

W Cv

-

dw 0 AD

2

dCv & div-

(5.118)

I

3

sphere as centered in the x'y'-plane in R and symmetrically about the z'-axis, then we can map S2 onto the complex -1, plane via the standard stereographic projection from the south pole z'

where wCv < 1. If

we

view the unit

=

2w/ 1 + W/V

This

gives

a

i

-

1/1

/ =

I

diffeomorphism of S2

-

with the

WfV

-

W11-V1

compactified plane CU fool.

(5.117)

(5.118),

the metric tensor transformation law and

algebra

that the coordinate transformation above must

( aw,

1

(1

+

WliV-1)2

aw, afv-,

afv-, 92

+

'92

K

'9Z

(5.119)

1 + W'fV_1

e '(',2)

find after

we

From some

satisfy

af (Z)51(2)

K

(K4

+

f (Z) f 0

))

2

(5.120) From

(5.120)

(5.119) it (M' g) to S2

and

formation from

then follows that the desired coordinate trans-

(5.118)

with the standard round metric

by

W(Z" )

4K1 +

1/2f(Z)

is

given

(5.121)

4K-1f (z)l(. )

The mapping (5.121) is just a generalized stereographic projection from the south pole of S' where f (z) maps (M, g) onto the entire complex plane with the usual Kiffiler geometry of S2 defined

4AFS2 (Z,

gS2

WS2

205FS2 (Z,

where the associated Kiffiler

(w,,Fv)

E

)dz

17;7Z ) 2 2i

A J

space

dz (& d2

(5.122)

-dz A d,

J + Zj)2

10g(l

+

(5.123)

Z, )

diffeomorphism (5.121) obeys

,

(5.119),

is

S2 and that the Khhler metric

original phase

the coordinates in 4

0 d5

potential

FS2 Notice that the

)dz

by

wiv-

gS2 in

geometry (5.49) when f (z)

=

SU(2)1U(1) bundle. the basic For C of U(1), monopole representation space W the associated vector bundle (SU(2) x C)/U(l) is the usual symplectic line bundle over S2 The Borel-Weil-Bott wavefunctions in the presence of a magThe

=

netic

=

.

monopole take values in this bundle. Notice that, comparing (5.131) with the stereographic

netic

coordinates

(5.119),

that these observables just describe the Larmor precession of a classical spin vector of unit length J 1. The coadjoint orbit path integral

we see

=

associated with the observables

dynamics

(5.131)

will therefore describe the quantum

classical spin system, e.g. the system with Hamiltonian H J3 describes the Pauli magnetic moment interaction between a spin J and a uniof

a

form magnetic field directed along the z-axis. Thus in this

=

case

S2 is actually

162


5.


on

naturally the configuration space for a spin system [156], which has on it a sYmplectic structure and so the corresponding path integral can be regarded as one for the Lagrangian formulation of the theory, rather than the Hamiltonian one [139, 156]. This is also immediate from noting that the stereographic complex coordinates above can be written as natural

z

in terms of the usual

J3

in

(5.131)

to

an

additive

=

e- O tan(0/2)

spherical polar coordinates (0, 0), height function (2.1)

so

that the observable

of S2 with

coincides with the

a

=

I

(up

constant),

the Khhler geometry above becomes the standard and the kinetic term in the action is

round geometry of S2 ,

01, (X).t" after

(5.135)

=

Cos

O

(5.136)

integration by parts over t. The classical partition function, evaluated 2.1, yields the usual Langevin formula for the classical statistical mechanics of a spin system. Alternatively, the motion of the precessing spin can be reduced to that of a charged particle around a monopole which is isomorphic to the problem for the motion of an excited diatomic molecule where the electrons are in a state with angular momentum j about the axis joining the nuclei (i.e. a rigid rotator with fixed angular momentum j about an

in Section

axis) [156].

on the particle, dO of the monopole located at the magnetic field w center of the sphere, and the potential force on the moment, due to the real magnetic field, that leads to the characteristic Larmor precession about the

its

It is the balance between the the Lorentz force

due to the fictitious

=

direction of the field. To construct

Section

j

=

where .

.

we

topological

.

,

Hamiltonian

along the lines

of the

theory of

an irreducible spin-j representation of SU(2), where The state space for this representation with heighest

.1, 1, 3,2.... [162]. 2 2

weight 1,

5.1,

a

consider

spanned by the complete set of orthonormal basis states li,m), the magnetic quantum numbers with the range m -j, -j + 1, j. The SU (2) generators act on these states as

3 is

m are

j

-

J3 Ji7 M)

-=

=

M

1i) M)

J Jj7 m)

I

=

V(j

::F

m) (j

m

+

1) 1j, m 1) (5.137)

Following the last Section, we define the SU(2) coherent states by successive applications of the raising operator j+ to the lowest weight (vacuum) state

1j, -j) [45, 137], 3

Jz)

=

e-3P

e'J+ 1j, -j)

=

e-'jP

E M=-j

1/2Z

2j i

+

M

j+M

lj,m)

;

zEC

(5.138) where for n,

m

E

Z+ with

n

>

m

(n) M

the binomial coefficient is defined n! =

m! (n

-

m)!

by

(5.139)

5.5

Spherical Phase Spaces and Quantization

of

Spin Systems

p(z,, ) is an arbitrary phase which as we F(z, ) in (5.21). It is easily verified that

and where the function

is related to the function

SU( )

generators

J3, J

in the coherent states

(5.131)

are

(Z2 I ZI)

see

then the

the normalized matrix elements of the operators

=

(I

2)2j e j[P(Z2,22)-P(Z1,21)1

+ Zl'

(5.140)

have used the binomial theorem

we

n

(X

Y),

+

(n) Xkyn-k

F-

=

They obey

the

completeness relation

f dl_i(j)(z,. ) Iz))((zl 10)

(5.141)

k

k=O

where

shall

(5.138), respectively 3 (5.138) are normalized as

The coherent states

where

163

is the

identity operator

the coherent state

measure

in the

=

1(j)

(5.142)

spin-j representation of SU(2) and

is

2j+1

djL(j) (z,. ) -i

27r

(1

+

zf) 2

(5.143)

dz A J

symplectic 2-form of the spin system above. The identity (5.142) follows from a calculation analogous to that in (5.85). Note that, as explained in the last Section, the Kdhler structure is generated e2jFS2 (Z,2). through the identity (zlz) which coincides with the

=

We want to evaluate the propagator K (Z2 , zi; for

some

T)

SU(2) operator ? given

=

((Z21

e

-M

I Z1))

the one-to-one

(5.144)

correspondence between

S2 -- C U foo} and the the points on the coadjoint orbit SU(2)IU(l) SU(2) coherent states (5.138). Dividing the time interval in (5.144) up into =

N segments and letting N ---> oo, following the analogous steps as in Section 4.1 using the completeness relation (5.142) we arrive at the coherent state

path integral IC (Z2 zi;

T)

=

JV

Vdet 1100)

dz(t) d. (t)

LR2 tE 0,T]

x

exp

j 109(l

+

Z2f2)

+

i 109(l

+

Zlfl)

T

+if [ dt

ij I + Z'

(fi

-

Zzi-)

-

ii

(azLei

+

LP 02

Z) -

-

H(z, f) j

0

(5.145) 3The dimension dimRj

=

2j +

1 of the

spin-j representation of SU(2)

be derived from the index theorems of the last Section

(see (5.108)).

can

also

164


5.

on


where N-1

11

M= lim IV--+OO

is

a

normalization constant and

(5.138).

H(z,. )

Here

(5.146)

2jir

k=1

the coherent states

+

denotes the matrix elements

(5.95)

in

again the formal equivalence of defined by the Kdhler polarization

we see once

the

path integral and Liouville measures above. In particular, the local symplectic potential generating the Kiihler

structures

(5.132)

are

0(j)

(. Wz

1 + Z'

-

zd. )

(5.147)

ijdp

-

they coincide with the standard coherent state canonical 1-forms (5.91). Similarly, the Khhler structure (5.122) can be represented in the standard and

coherent state form

(5.92).

The Wess-Zumino-Witten quantization condition (4.161) applied to WW implies that j must be a half-integer, since 4,7r, corresponding to the WS2

fS2

=

unitary irreducible representations of G SU(2) [162]. This is the topological of The spin. (or Dirac) quantization quantization of the magnetic quantum =

numbers

m

above then follows from

an

application

of the semi-classical Bohr-

a quantization [101] topological quantum theory (or equivalently an integrable quantum system) as described in Sections 4.10 and 5.1, we need to choose the phase function p(z,. ) in the above so that ivOW H. This problem was analysed in detail by Niemi and Pasanen [123] who showed that it is impossible to satisfy this integrability requirement simultaneously for all 3 of the generators in (5.131). Again, this means that there are no SU(2)-invariant symplectic potentials on the sphere S2 However, such 1-forms do exist on the cylindrical representation of SU(2) [123], i.e. the complex plane with the origin removed, which is conformally equivalent to the Kdhler representation of S2 above under the

Sommerfeld

condition

for the spin system. To construct

=

.

transformation In this latter

z

=

e'l +iS2

which maps

(s 1,82)

representation, the Hamiltonian

in

E R x

(5.145)

S'

to

can

z

E

C

-

101.

be taken to be

arbitrary linear combination of the SU(2) generators, and the coherent path integral (5.145) determines a topological quantum field theory with p 0 in (5.147). This is not true, however, in the Khhler representation above, but we do find, for example, that the symplectic invariance condition can be fulfilled by choosing the basis H(z,. J3(j) (z,. ) of the Cartan sub) .1 and The algebra u(1) p(z,, ) ensuing topological path integral 2 log(z/. ). (5.145) then describes the quantization of spin. To evaluate this spin partition function, we set p 0 above. Although the ensuing quantum theory now does not have the topological form in terms of a BRST-exact action, it still maintains the Schwarz-type topological form described in Section 4.10, since the Hamiltonian then satisfies (3.45) with C j and the function K in (3.46) is an

state

=

=

=

=

=

5.5

Spherical

Spaces and Quantization

Phase

i

K(z,. )

log

=

2

of

Spin Systems

(Z7)

165

(5.148)

-

(5.145) is a topological path integral of the form (4.165), i.e. the quantheory determines a Schwarz-type topological field theory, as opposed to a Witten-type one as above. We first analyse the WKB localization formula (4.115) for the coadjoint orbit path integral (5.145). We note first of all that the boundary conditions in (5.145) are z(O) 2. In particu(T) zi and lar, the final value z (T) and the initial value (O) are not specified, and the boundary terms in (5.145) ensure that with these boundary conditions there is no boundary contribution to the pertinent classical equations of motion so

that

tum

=

+ iz

In

general,

are no

sphere

if

z(t)

#)

and

are

.L.

0

=

Z

7

-

=

i'

=

(5.149)

0

complex conjugates of each other,

classical trajectories that connect z(O) zi with S2 But if we view the path integral (5.145) instead =

.

(T)

then there

=

as a

2

on

the

matrix ele-

configurations in different polarizations, then there is always following solution to the equations of motion (5.149) with the required boundary conditions for arbitrary z, and, 2, ment between 2

the

z(t) (5.150)

=

zi

e-'t

#)

7

e-i(T-t)

2

=

complex, and hence z(t) and

#)

(5.150)

regarded one independent the holomorphic quantization formalism that makes it suitable to describe topological field theories. The trajectories (5.150) are therefore regarded as describing a complex saddle-point of the path integral [44, 85, 143]. We shall The solution

is

as

see

other forms of this feature later

Substituting the

on.

(5.150)

the solutions

must be

of the characteristic features behind

variables. This is

into the WKB formula

(4.115)

we

find

propagator4 IC (Z2 , zi;

The exact propagator from

((Z21 e-iTJ3 I ZO)

a

+

Z04)j (1

+

+

(5.151)

Z2, 2)j

direct calculation is

1

(I

e-ijT

+ ZO 2 e- iT)2j

T)

Z121)j(l

+

Z222)j

M=

"j (i

2j +

M

) (zl, 2

e-

iT

)j+' e ijT (5.152)

which coincides with

(5.141). ing

In

particular, setting

the coherent state

In this

(5.151)

case

measure

upon zi

=

Z2

(5.143),

=

z

we

the fluctuation determinant in

non-periodic boundary conditions

of the binomial theorem

application

and integrating

over z

E

C

us-

find the partition function

(4.115)

is

discussed above.

regulated using the generic

166

5.


f dy(i)(z,,

ZSU(2) (T)

on

Simply Connected

Phase

Spaces

) ((zl e-iTj3lZ))

00

dr

(2j

r

1)(1

+

+

r

2e-iT)2j

e

-ijT

sin -

(1

0

+

r2)2j

sin

i

e-iTm M=-j

e

1

-

2

(5.153)

which also coincides with the exact result

trj e-iTJ3

21 (2i + 1))

iTj

e-'Ti +

e-iTi

eiTi

(5.154)

The

right-hand side of (5.154) is precisely what one anticipates from the Weyl character formula (5.28). The roots of SU(2) are a 1 [162], and the Cartan subalgebra is u(1) consisting of the single element j3. The Weyl group is W Z2 and it has 2 elements, the identity map and the reflection map 4 -+ -j3. Thus the formula (5.154) is simply the Weyl character formula (5.28) for the spin-j representation of SU(2). Within the framework of the Duistermaat-Heckman theorem, the terms summed in (5.154) are each associated with one of the poles of the sphere S2, i.e. with the critical points of the height function on S2. Indeed, since this Hamiltonian is a perfect Morse function with even Morse indices, we expect that the Weyl character formula above coincides with the pertinent stronger =

=

(4.135) of the localization formulas. Because of the Khhler structure (5.122) on S' (see (5.53)), the Riemann moment map has the non-vanishing

version

components

G'Vw)'Z and

consequently

the Dirac

=

iJO) (Z" O/i 3

-(1LV(i))2

A-genus

A(TS?V(j))

(5.155)

is

JU) 3

T

2i

sin

(5.156)

(2j Jw) T

3

Substituting these into the localization formula (4.135) yields precisely the Weyl character formula (5.154). This localization onto the critical points of the Hamiltonian, as for the harmonic oscillator example of Section 5.3, agrees with the general arguments at the beginning of Section 4.6. Substituting the stereographic projection map (5.135) into the classical equations of motion (5.149) gives

sinO For T

=

0

+ 1

=

(5.157)

0

:A 2-7rn,

n E Z, the only T-periodic critical trajectories coincide with cos 0), i.e. 0 points of the Hamiltonian j (I 0, 7r, and in this case the critical point set of the action is isolated and non-degenerate. How27rn, n E Z, we find T-periodic classical solutions for any initial ever, for T

the critical

-

=

=

value of 0 and

0

in

(5.157)

and the critical point set of the classical action

5.5

Spherical

Phase

Spaces and Quantization of Spin Systems

167

original phase space S'. Thus the moduli space of classical S2 and the localization onto this moduli is LMS space is now easily verified from (4.122) to give the correct anticipated result above. From the discussion of Section 4.10, it also follows that the sum of the terms in (5.154) describes exactly the properly normalized period group of the symplectic 2-form w(j) on the sphere [85], i.e. the integer-valued surface integrals of w(j) as in (4.161). We shall see in the next Chapter that quantizations of the propagation time T as above lead to interesting quantum coincides with the solutions in this

case

==

,

theories in certain other instances of the localization framework. It is

an

instructive exercise to work out the Niemi-Tirkkonen localization

(4.130)

formula

dynamical system.

for the above

S2,

because of the Kiihler geometry of

For this

note

we

that, again

the Riemann curvature 2-form has the

non-vanishing components R'z and

so

combined with

(5.155)

AV(j) (TR)

The equivariant extension of

J(j) 3 where

-

we

W(j)

j

1

-

-iW(j)

-RzZ

=

we see

that the

J(j) 3

T =

(5.158)

equivariant A-genus here

-

W(j)

w(j)

sin

(-L2j (J(j) 3

-

)

2i

Z-)2

1 + Z'

=j

(I

have redefined the Grassmann variables

ZSU(2) (T)

(4.130)

f

i -

7rT

,W))

is

Z'

Tirkkonen localization formula

(5.159)

-

2j

is

can

dz d.

Z`

+ Z

ql"

-+

(5.160)

77

+

Vi7- n/.

then be written

The Niemi-

as

d77 d L(z, +,q )

(5.161)

R20AIR2

where T 2

L(y) sin

Using

1-Y I+y

exp

(T (1-y)) f

1+y

[-ijT (I Y)l

the Parisi-Sourlas integration formula

(5.162)

1 + Y

[136]

00

1 7r

1

d 2x

d77 d

L(X2

+

7A

R20A-IR2 we

obtain from

du

=

du

L(oo)

-

L(O)

(5.163)

0

(5.161)

the partition function

ZSU(2) (T)

-

sin(Ti) / sin(T/2)

(5.164)

168

5.


Introducing

the

Weyl shift j

Phase

Spaces

in + .1 2

(5.164) then yields the correct Weyl (5.153) for SU(2) 5 Note that (5.163) shows explicitly localization in (5.161) comes directly from the extrema of the height j

---

character formula how the

function at

As

Simply Connected

on

z

=

oo

.

and

z

0.

=

final application for the above dynamical system, we examine the quadratic localization formula (4.150). Now the (degenerate) Hamiltonian is a

.F(j(j)) 3

Following written

the

same

steps

as

-

(j(j))2 3

above,

(1+ Zf) 1

j2

=

-

Z'

(5.165)

the localization formula

(4.150)

can

be

as 00

W

ZS U (2) (T I (J3' ) )

f doo f 00

i

2

-

-v

47riT

dz d2

dq d L(Oo, z2 +,q )

R20A1R2

-00

(5.166)

where T-00 1-Y 2 1+y

L(00, y) sin

and

we

formula

010 (1-0) 2

1+y

have redefined

n4

(5.163) again

and

ZSU(2) Prp(j))2) 3

exp

14002 0

-

ijToo

(I Y)] 1 + Y

(5.167)

V/i7/_0o -7f. Using the Parisi-Sourlas integration introducing the Weyl shift j we find j + .1, 2

--+

--->

T4T7ri J

doo e'T 020/4

sin[(j + 25')Too] sin(Too/2)

--

__

-00

3

E M=-i

FT_

V 41ri

-

00

f doo

e- iTmoo

e

iTO2/4 0

L

e-

iTM2

M=-j

-00

(5.168)

which is again the correct character trj e- iTj32 Thus on a spherical phase space geometry the equivariant Hamiltonian systems provide a rich example of the topological quantum field theories discussed in Section 4.10, and they are the natural framework for the study of the quantum properties of classical spin systems. The character formula path integrals above describe the quantization of the harmonic oscillator on the

sphere, and therefore the only integrable quantum system, up to holomorphic equivalence (i.e. modification by the general geometry of the phase space), that exists within the equivariant localization framework on a general spherical geometry is the harmonic oscillator defined on the reduced compact phase space D 2. 5

Of course,

we

could

alternatively obtain the Weyl character formula using instead

the G-index localization formula

shift

[102].

(5.114)

without

having

to

perform this Weyl

Hyperbolic Phase Spaces

5.6

5.6

Hyperbolic

The situation for the

Phase case

169

Spaces

where the

space is endowed with

phase

a

Rieman-

parallels that negative differthe essential of the last Section, and we only therefore briefly discuss ences [1591. The phase space M is now necessarily a non-compact manifold, and we can map it onto the maximally symmetric space V, the Lobaschevsky plane (or pseudo-sphere) of constant negative curvature, with its standard curved hyperbolic metric g'H2 [42, 43, 68]. The Killing vectors of this metric have the general form nian

V;

geometry of

constant

2

a(l

==

-iQw +

+

Gaussian curvature K < 0

WiV-)112

V4

2

=

iQiV +

d(l

+

WfV-)112 (5.169)

they generate the isometry group SO(2, 1). The rest of the analysis at the beginning of the last Section now carries through analogously to the case at hand here, where we replace the K factors everywhere by -JKJ and the K1/2 factors by JK11/2. In particular, with these changes the generalized stereographic coordinate transformation (5.121) is the same except that now the holomorphic function -1 1/2 f (z) there maps the phase space onto the Poincar6 disk of radius 2 JKJ and

i.e. the disk

D2 with the Poincar6 metric 4 gIH2 -

(5.170)

dz (& d;

Khhler geometry on the disk for which the associated sym2-form is the unique invariant volume form under the transitive

which defines

plectic

Z )2

a

stereographic projection image for the Lobaschevsky plane when we regard through its embedding in R 3as the it by pseudo-spherical coordinates pseudo-sphere, so that we can represent cosh -r. The x2 sinh -r sin 0 and z xi sinh -r cos 0, (-r, 0) E R x [0, 27r] as z' the center from taken -1, projection stereographic projection is again at to disc Poincar6 the and the boundary of infinity of the corresponds points hyperboloid H2. The pseudo-sphere itself is represented by the interior of the disk. The explicit transformation in terms of pseudo-spherical coordinates is

SO(2, l)-action.

The Poincar6 disk is the

it

=

=

=

=

W1 Z

T7Z

=

e-'O tanh(-r/2)

(5.171)

conformally equivalent to the i)/( + i) which ( plane C+ via the Cayley transform --+ z C+ onto the Poincar6 disk, and the Poincar6 metric (5.170) on the

We also note here that the Poincar6 disc is upper half

takes

E

=

(Poincar6) upper-half plane

is

g, 12

The

path integral

over

such

studies of quantum chaos

-

=

IM(6)-2d6 (9 d

hyperbolic geometries

[311.

(5.172) arises in

string theory and

170

5.


The most

general

on

Simply Connected

localizable Hamiltonian in

Phase

Spaces

hyperbolic phase

a

space

geometry is therefore

(

S? H_ (z,

2)

=

IKI 4

f

+

(Z)1(2))

al(2)

+

IK-I- AZ)A O 4

]KI

_

4

The transformation to Darboux coordinates

df(Z)

+

(5.173)

f(Z)I(' ).+ CO

on

M is

now

accomplished by

diffeornorphism

the

f(Z)

V(Z,,9) IKI -

4

(5.174)

1/2

f(z)!(

which maps M onto the complement of the unit disk C general Darboux Hamiltonians are therefore

B

(z,. )

=

Qz2 +

(az + a. ) (1

We note that here there

+

Z )1/2

C

E

z

;

-

-

int(D 2)

in

R2

int(D 2)

The

.

(5.175)

inequivalent Hamiltonians, corresponding to "spacelike" Killing vectors, but the generic hyall Hamiltonians are holomorphic copies of one another, again perbolic again harmonic oscillator. However, given that the to a quasi-displaced reducing Darboux phase space is now non-compact, we can again weaken the requirement of a global circle action on the phase space to a semi-bounded group a

2

are

and "timelike"

choice of

action.

Considering therefore the quantum problem defined on the Poincar6 disk radius, we write the 3 independent observables in (5.173) as

of unit

S(") (z, ) 3

=

kI + Z2 1

-

,

Z2

S(k) + (Z" )

=

Z

2k

I

-

Z2

S(k) (Z" )

7

=2k

z

1

-

Z'

(5.176) kW'H2,we see that the associated Defining the Khhler 2-form w(k) algebra of these observables is just the SU(I, 1) Lie algebra =

f S(k) 3

1

S(k)

I

IS(k)

-

7

L" (k)

The Hamiltonians in

(5.173)

are

f S(k), S(k) IW(k) +

therefore functions

SU(I' I)IU(I) of the non-compact Lie group

-

-

-

,

on

the

2S(k) 3

Poisson

(5.177)

coadjoint orbit

H2

(5.178)

,

SU(1, 1), and the generators (5.176) are the SU(I, 1) generators in the SU(I, 1) coher-

normalized matrix elements of the ent states c'O

Jz)

=

e' S+

Ik,O)

=

E n=O

(2k+n+ 1)1/2 n

z' I k,

n)

;

z

c

int (D

2) (5.179)

5.7 Localization of Generalized

171

Spin Models and Hamiltonian Reduction

representation of SU(1, 1) characterized by k representation spaces are now infinite-dimensional

for the discrete irreducible

1,

[137]. -32 , 2, 55,... 2

The

=

because of the non-compactness of the group manifold of SU(1, 1), and the representation states I k, n) defined here are the eigenstates of the generator

S3 with eigenvalues

S31k, n) The coherent states

(5.179)

have the normalization

(Z21ZO where

we

(5.180)

(k + n)lk, n)

=

=

(1

_

Zl' 2)-2k

expansion

have used the binomial series 00 =

X)n

+

m

E

(5.181)

n

-

1

(5.182)

XM

M

Tn=0

n E Z+ and JxJ < 1. integrable Hamiltonian systems are obtained by taking H S(k) which is the height function on H2 and the corresponding coherent state 3 path integral describes the quantization of the harmonic oscillator on the these are the only open infinite space R' (and up to holomorphic equivalence is straightforward It a on space). phase hyperbolic general integrable systems to analyse the localization formulas for the coherent state path integral just

which is valid for

Again,

the

,

,

as

in the last Section. For

coadjoint

instance, the WKB localization formula for the

path integral

orbit

T

[d cosh -r] [do]

Zsu(,,,)(T)

exp

i

dt

-

k(l

+ cosh

-r))

0

LIV

can

I (k coshr

(5.183)

be shown to coincide with the exact

Weyl character formula for SU(1, 1)

[51, 143] 00

Zsu(,,,)(T)

=

trk

e

e-iT(k+n)

-iTS3 n=O


=

2i

e

-iT(ksin

T

2

(5.184)

2

Spin Models

and Hamiltonian Reduction The

explicit examples

have given thus far of the localization formalism quantum cases have, for simplicity, focused on dy-

we

in both the classical and

namical systems with 2-dimensional phase spaces. Our main examples have been the harmonic oscillator, where the localization is trivial because the Hamiltonian is a quadratic function, and the spin partition function, where

172


5.


on

the exactness of the

stationary-phase approximation is a consequence of the conspiracy between the phase space volume and energy which makes this dynamical system resemble a harmonic oscillator. In Chapter 8 we shall present some true field theoretical applications of equivariant localization, but in this Section and the next

we

wish to overview the results which

actness of the localization formulas for

concern

the

ex-

higher-dimensional coadjoint orbit models which can be considered as generalizations of the spin models of this Chapter (and the previous ones) to larger Lie groups. We have already established quite generally that these are always examples of localizable dynamical systems, and here we shall explicitly examine their features in some special instances. The generalization of the classical partition function for SU(2) is what is commonly refered to as the Itzykson-Zuber integral [76]

I[X, Y; TJ

f

=

some

DU

e

iT

tr(UXUtY)

(5.185)

U(N)

where IV

DU

N

11

=

dUij

J

i,j=l

is Haar

(or N

on

x

SU(N)

N matrices

(5.186)

JiJ

-

U-1 U(N) of N x N unitary matrices Ut Here Hermitian are (Xt, Yt) (X, Y) U(N)I(U(l) Z2)). (i.e. elements of the U(N) Lie algebra) and which can be the group

measure on =

E UikU*k k=1

=

X

=

eigenvalues xi, yi E R by unitary transformations (VtXV, WtYW). By the invariance of the Haar measure in (5.185) (X, Y) WUVt of U(N), we can thus assume without under the left-right action U loss of generality that the matrices X and Y in (5.185) are diagonal so that therefore

with

diagonalized

---*

--*

N

I[X, Y; TI

DU exp

iT

U(N)

The

E

X, Yj

1Uj12 i

(5.187)

i,j=l

Itzykson-Zuber integral is a fundamental object that appears in mastring theories, low dimensional quantum gravity and higher-

trix models of

dimensional lattice gauge theories [36, 92, 106]. The integration over unitary matrices in (5.185) the Duistermaat-Heckman theorem via the

following

can

be carried out

using

observation. If we define

the Hermitian matrix

A then

explicitly

we can

variables U

--

A

in

=_

UYUt

(5.188)

compute the Jacobian for the

(5.185)

to

get

change

of

integration

[36, 106]

N

DA

=-

11 dAii i=1

fl I<j

now

into

angles.

SU(2)IU(l)

S2 which effectively

stituting these identifications and

the usual Euler

SU(2)

(5.199)

2

The

projection Hopf

is then the

map

map

0 in (5.199). Sub0 the Itzykson-Zuber integral (5.185), we

follows from

sets

(5.196) (for

=

a

=

0 in Section

2.1).


There

which a

are

unitary


various extensions of the above

are

relevant in matrix model theories matrix

generalized

[92].

classical

First of

all,

175

spin model

we can

consider

integral of the form

I[A; T]

=

N

f

DU exp

iT

1:

Aij I Ujj 12

(5.200)

i,j=l

U(N)

The stationary conditions for the Hamiltonian in

(5.200)

are

N

EUjj(Ajj-Ajk)Utjk =0

i, k

(5.201)

N

=

j=1

whose solutions

expansion of

again the permutation matrices (5.200) thus yields (c.f. eq. (3.53)) are

U

=

P. The

saddle-point

(I

0(11T A))

eiT E. Ak,P(k)

I[A; T] PESN

r1j<j (Ai,p(j)

+

Aj,p(j)

-

Aj,p(j)

-

Aj,p(j))

+

-

(5.202) In

general, the higher-order corrections in (5.202) to the lowest-order stationary phase approximation do not vanish because the Hamiltonian in (5.200) is not defined on any coadjoint orbit in general but on the entire group manifold which is not even a symplectic space. The corrections do, however, vanish in some interesting exceptions where (5.200) is a coadjoint orbit integral. One case is that discussed above, namely when Aij is of rank 1, so that Aij xjyj. Another interesting case is when N 2, so that (5.200) is slight modification of the spin partition function. For the case of SU(2) one can check explicitly that the leading order term in (5.201) is the exact result for the integral so =

=

that

ISU(2) [A; T] The unitary matrix

eiT(Ajj+A22) All

+

A22

-

-

e

iT(A12+A21)

A12

-

(5.203)

A21

integrals of the form (5.200) are generating functions Itzykson-Zuber model (5.185),

for the correlation functions in the I [X,

Y; T]

k-j1j***kIII

=

f

DU

Uj1j1

...

U(N) x

which

U iPjP

(Ut)ki I,

...

(Ut)kplp (5.204)

eiT tr(UXUtY)

only non-vanishing correlators because of the U(1) phase ine"9Uij of the integral (5.185). The evaluation of the unitary matrix integrals (5.204) is very important for matrix models of induced gauge theories and string theories [92] and has been a difficult problem that has are

variance

the

Ujj

--+

received much attention in the last few years. For instance, using GelfandTseytlin coordinates on the group manifold, Shatashvili [1541 has shown that

176


5.

some

(5.204)

of the correlators

mulas, such


on

explicitly given by

are

complicated for-

very

as

I[X,Y;T]'IJ1JJ2, "" k2,1...k,,l IJP *

Jil ki Ji2 k2

*

*

00

f! riN-1 R=1

Jjp kp

(iT)(N-2)(N-1)

6 [X1,1A (Y2 i

...

7

f

YN,

_cO

(k=1E N

X

iTy,

exp

N-I

Xk

E

-

k=1

Ak)

N-1

11

n

dAk

k=1

11 1=1

XjI)

-

-

XjI)

eM\iyj

det 1 I the spaces CP N are not homeomorphic: to any of the maximally symmetric spaces discussed in Section 5.2 above and this symplectic manifold leads to our first example of localization on a homogeneous space which is not maximally symmetric. This space for N > 1 has N independent isometries which are the rotations in each of the N 2-planes (see below). In the next Section we shall see how to explicitly relate the space CpN to a (non-maximal) coadjoint orbit of SU(N + 1) so that, physically, this example describes the d'ynamics of a particle with internal SU(N + 1) isospin degrees of freedom in an external magnetic field. There we shall also encounter other examples of this situation, i.e. where the phase space is not maximally symmetric but has a maximally symmetric subspace. For now, we shall concentrate

on

the properties of the SU(N + an integrable model.

1)

classical spin partition

function associated with

(Sl)N+l on CN+1 is given by z1' generated by the Darboux. Hamiltonian

An action of the torus TN+1 P C-

[0, 2-7r),

which is

e O"' ZA'

-+

=

N+1

-(N+l) BD (z

0)

=

2J

1: 01'ZA'

'a

(5.210)

that describes the

dynamics of N + 1 independent simple harmonic oscillators correspond to a conserved charge of this integrable system. The classical partition function is of course given by a trivial Gaussian integration

which each

N+1

ri

(T)

D

R2N+2

A=1

2J dzl' d, A e

iTH

(N+I)(Z,2;0)

D

(-iT)N+l

7r

01, rjN+1 t1=1

(5.211) which coincides with the Duistermaat-Heckman formula

the

only fixed point

interesting though

generated by (5.210)

of the torus action

is the Hamiltonian T N -action

on

always

as

is at z/'

CpN' "

--+

because

=

each tt, the Hamiltonian function which generates the circle action A-plane in (CpN is the conserved charge

2JO It4 1

and

an

integrable

functions

(5.212),

Hamiltonian

can

on

For

the

(5.212)

+EN V=1

be constructed

0. More

e'6" A.

as

the

sum

of the N

height-

178

5.


on

Simply Connected

Phase

Spaces

N

(5.213)

HN

The conserved charges (5.212) are the action variables of the above integrable model and the associated angle variables OA c- [0, 27r] are the usual polar angles of the A-planes (as for the (CN+l example above). The LiouvilleArnold integrability of this dynamical system can be made more explicit by considering the generalized stereographic projection onto CpN in terms of the spherical coordinates (OA, 0") E S2N+1 (where 0 < 0/1 < ir),

tan(O'/2) COS(02 /2) e-'O'

2

,

N-l

=

tan(Ol/2) sin(02 /2)

...

N

=

tan(Ol/2) sin(02 /2)

...

=

tan(Ol/2) sin(02 /2) cos(O'/2)

2

e

sin(ON- 1/2) COS(ON /2) e-ioN-'

sin(ON- 1/2) sin(ON /2) e-ioN (5.214)

The action variables

It,

=

(5.212)

2J A sin 2(01/2) IN

...

in these

sin

are

2(0/'/2) COS2 (0"+1/2)

V N sin2(01/2)

=

spherical coordinates

sin

...

then

p < N

2(ON- 1/2) sin (ON /2) 2

(5.215)

straightforward to see that the symplectic 2-form (5.208) takes the z11,01 as usual). integrable form dI,, A dol' (for the oscillator above 1,, explicit torus action on (CpN generated by the Hamiltonian vector field

and it is usual The

-

associated with

(5.213)

(N

=

is thus

E i /-' 14=1

Vt

'9

a -

5p-A

E /I=

A

aol-t

(5.216)

integrable system on the phase space (CpN is therefore once again isomorphic to a linear combination of independent harmonic oscillators as above, generalizing the dynamics of our standard spin example. The relation between such generalized spin systems and systems of harmonic oscillators, and hence the exactness of the localization formulas, can be understood from a slightly different perspective than the usual localization symmetries provided by the underlying equivariant cohomological or supersymmetric structures of the dynamical system. To see this, we evaluate the classical partition function explicitly for the above dynamical system by extending the integration over (CpN to the whole of (CN+l via a Lagrange multiplier which enforces the constraint (5.206). This leads to This


ZN (T)


2J dtf' d ' IIN p=j N(j + EN qrl)N+l

eiTHN( , ;O)

-

Ir

S2N+IISI

,,=,

00

f

dA

N+1

j

27r

d2F1 &A

ri

R2N+2

-00

(5.217)

7r

tj=1

1)

N+1 x

where the Gaussian

Tl.(N+ T "D

exp

179

1)

(Z

1

7

0)

E

+ j'X

angular parameters in (5.217) are related by OA integral over CN+1 in (5.217) can be performed 00

ZN (T)

f

=

01'

=

to

_

ON+1 The .

yield

N+1

dA

%+

e-'A

TOA

(5.218)

A

00

remaining integration in (5.218) can be carried out by continuing the integration over a large contour in the complex plane and using the residue theorem to pick up the N + 1 simple poles of the integrand at A -TOA, each of which have residue 1. Thus the classical partition function is The

=

N+1

ZN (T)

=

E

e

iT01'

j1=1

It is

readily verified

that

(5.219)

11 V014

i

T(Ov

-

(5.219)

P)

coincides the Duistermaat-Heckman integra(5.213) which has precisely N + 1 stationary

tion formula for the Hamiltonian

points [52]. (5.219) therefore represents the TN -equivariant cohomology of the manifold CpN. The

explicit evaluation above illustrates

an

interesting feature of the lo-

calization in these cases, as was first pointed out in [44, 52]. This is seen in (5.217) the dynamical system describing the SU(N + 1) isospin is the -

Hamiltonian reduction of

quadratic

Hamiltonian

on

a

larger dynamical system which is described by a 2N+2 and for symplectic manifold R

the Darboux

which the localization is therefore trivial. The reduction constraint function

(5.206)

commutes with the Darboux oscillator Hamiltonian

P(z,, ), HD( +')I N

so

that the constraint function

P(z,, )

=

(,+ )

0

(5.210),

i.e.

(5.220)

WD

determines

a

first class constraint

on

dynamical system and is therefore a symmetry (or conserved of the classical dynamics. This commutativity property is in fact the

the Darboux

charge)

correspondence between the integrations in (5.217), CN+1 is restricted to the symplectic subspace dedynamics termined by the constant values of the conserved charges P(z,, ) (according crucial mechanism for the as

then the

on

180

5.


[7]

to the reduction theorem

-

on

see


(4.144)).

reduction mechanism here is the action of the usual matrix-vector

representation of

SU(N

1).

the Darboux system above

on

and hence restricts the

U(1)

defining (vector) symplectic Hamiltonian action leaves invariant the sphere S2N+1

multiplication by +

matrices in the

This defines

[441

dynamics

which

Another important feature of the SU(N + 1) on CN+1 defined by a

to this submanifold of R

invariance of the harmonic oscillator Hamiltonian

restricts further to the coset space

charges under (CN+l become bundle of the

U(1)

(CpN

=

on

S2N+11U(j)

2N+2

From the

.

(C N+1

above, this

and the conserved

the mapping by the symplectic constraint function P(z,. the components of the projection map of the generalized

S2N+1

_4

_-,

(CpN [44]. It is easily

quadratic (5.217) exactly with the poles in (5.218) Hamiltonian in

coincide

seen

in the space of the A and so

that

)

on

Hopf that the critical points

(5.219)

z

variables

coincides

precisely

with the Duistermaat-Heckman formula for this Darboux system. The localization of the isospin partition function is thus the reduction of that from R 2N+2

We recall that such

a

reduction method in equivariant localization is also

that which is used to derive the Callias-Bott index theorems from dimensional Section

4.2).

Atiyah-Singer

index theorems

[70] (see

higher-

comments at the end of

The above reduction

procedure is a standard method of inteby grating dynamical systems associating to a given Hamiltonian system a related one on a lower dimensional symplectic manifold [1]. In this scheme one

reduces the rank of the differential equations of motion

or

the number of

degrees of freedom using invariant relations and constants of the motion onto a coadjoint orbit as above. Furthermore, in the context of generic integrable models, there is the conjecture that every integrable system is the Hamiltonian reduction of a larger linear dynamical system by first class constraints [176]. If this conjecture were true, then the localization of generic integrable systems could be cast in another formalism giving an even stronger connection between the Duistermaat-Heckman theorem and the integrability properties of a dynamical system. Conversely, as the integration formulas we have encountered also always correspond to a sort of Hamiltonian reduction of the original dynamical model, they yield realizations of this reduction conjecture

as

-well

as

the quantum mechanical

conjecture mentioned earlier about description of the

the exactness of the semi-classical approximation for the quantum dynamics of generic integrable systems.

5.8 Let

Quantization of Isospin Systems us

now

quickly describe the quantum generalizations of the results of First, we note that, for a general group G, a point particle

the last Section. in

its

irreducible representation R of the internal symmetry group G has time-dependent isospin vectors R(t) living in a fixed orbit of G in the

an

adjoint representation [121. As described earlier, this fixed coadjoint orbit

Quantizat on

5.8

determines

a

of

Isospin Systems

181

unitary irreducible representation of G and if we let A' denote a fixed fiducial point on this orbit, then the internal isospin

(time-independent) vectors

be written

can

as

R(t) with

g(t)

Ad*(A')g(t)

=

=

g(t)A'g(t)-'

(5.221)

time-dependent group elements. In the absence of any "exter0. charged particle, the isospin vector is conserved, 7 (t) Thus the point particle action describing its motion in a generalized external magnetic field B B'R(X') is [12] E G

nal" motion of the

=

=

T

S[g, A]

=

I

dt

( tr(A'g-' )

+

tr(IZ B))

(5.222)

-

0

When A' is taken to be

Cartan

element, this dynamics of

generalizes the spin particle moving in both monopole and external fields (Hamiltonianly reduced from a free particle action on R 4) [12, 44]. Thus again we see that the kinetic term in (5.222) is of first order in time derivatives and consequently the action (5.222) is already cast in phase space. The phase space variables are the isospin vectors R(XI) and the Poisson algebra of them is just the Lie algebra of the internal symmetry group G [12]. Thus these generalized dynamical systems all fall into the class of those systems whose configuration and phase spaces coincide so that their path integral interpretations on the respective spaces are the a

action before which described the

action a

same.

To describe the quantum dynamics of these models, we start with the quantization of the classical SU(N + 1) isospin model of the last Section over the phase space (CpN. Since [143]

SU(N

+

1)

S2N+1

2,

x

it follows that this

phase

1)1(SU(N)

of

will be

a

x

U(1))

SU(N)

_

space is the

SU(N + 1) [68],

space is not

a

flag

_

(Sl)N

manifold

as

C

Aj(f)

and the

S2N-1

is

sum

10)

canonically

a

highest weight

over

is

vector

=

x

U(1) (so

associated with and

so

OJN,1J

that

10)

is

S3

X

...

(5.223)

coadjoint orbit OJN,1J SU(N an integrable Hamiltonian on =

SU(N

1)

+

+

it

which leave

Although the orbit possible to generalize the Section 5.4. They are constructed by invariant.

it is still

Ee neAe(f)

where nj is

vector of the fundamental

t is taken in such

SU(N)

I)ISU(N + 1)AI)

10)

X

and

OfN,1J

before,

construction of the coherent states in

taking the highest weight

group of

X

combination of the N Cartan elements of

the maximal torus HC

integer,

S2N+1

a

non-negative

a

representation,

way that the maximum stabalizer

that the

loop space path integral will be Ad* (SU(N + 1))A'-- SU(N + OA1 appropriate to the geometry of the given

=

-

coadjoint orbit (e.g. the normalization of the coherent Kdhler potentials of the orbits in the manner described

states

generates the

in Section

5.4).

The

182

5.


existence of such

weight

a

state is in

Simply Connected

Phase

Spaces

general always guaranteed by the Borel-

[68, 142].

Weil-Bott theorem We

on

define the coherent states associated with the irreducible [2J] representation of the SU(N + 1) group, where 2J E Z+, 2J < N + 1 [53, 871

(i.e.

now

the representation with fundamental

highest weight J corresponding to Young tableau representation by a single column of 2J boxes). In this representation, the highest weight vector is denoted as I J; N + 1), there are N (i.e. roots which cannot be decomprecisely N simple roots a 1, 2, a

=

E,(,

N +1) of 2 other positive roots) with 1 J; N + 1) 0, and the Cartan generators in the Gell-Mann basis have the (diagonal) matrix elements

posed

[H,,,]ij

into the

sum

1 =

V2-m(m+ 1) -

(km=l1:

6ikJjk

-

=

MJi,m+lJj,m+l

M

N

=

!jab The 2

(5.224) where

have normalized the generators dimension of this representation is we

dim

The

1)

[2J]

=

( N+1)

so

that

tr(XaXb)

(N + 1)! (2J)!(N TI.- 2J)!

2J

.

(5.225)

generalized coherent states associated with this representation of SU(N+

are

1)

then

e

[137]

EN

-1

-E

(N+1)

IJ;N+l) 2J)! nN+11

nl+...+nN+1=2J

N)

=

(61)nl

...

C

CpN

(6N)nN JJnk 1N+1 k=1 ;N+1) (5.226)

These coherent states have the normalizations

(1 and

they obey

the

IV

+

2J

E V 2'5

(5.227)

a=1

completeness relation

dy(J)

1(j)

(5.228)

where

dl,P)

rIN I &P (2J + N)! (2J)! 7rN (I + EN

is the associated coherent state

C'=

measure.

Consider the quantum propagator

A

d6c'

(5.229)

Quantization

5.8

IC(J) ( 2, 61; T)

(( 21 e-M 161))

=

SU(N + 1) operator (5.224)

defined in terms of the invariant nation of the Cartan

generators

Isospin Systems

of

183

(5.230) that is

a

linear combi-

N

7

=

mHm

2J

(5.231)

2JCO

+

M=1

which leads

usual to

as

IC (J) (62 7

topological and integrable quantum theory. Then yields the coherent state path integral

a

the standard calculation

61; T) N

[dl-P) (6,

2J log

exp

1 +

L(S2N+IISI) T

N

+2J log

E V2 2ei a=1

1 +

E 61c'M

+ i

a=1

I

N

2U(4 6 11 1 + El'

1:

dt

-

CX=1

0

1"-) 61 6a -

qA

H(6, e)

(5.232) where the

boundary

path integral

conditions in the

(O)

are

and

(T)

and

2JCo

+ 2J

E

(5.233)

generalized height function (5.213) on (CpN. The WKB approximation path integral has been discussed extensively in [52, 131]. In fact, all of the localization formulas of Chapter 4 can be explicitly verified as the above dynamical system is just a multi-dimensional generalization of the Kdhler polarization for the spin propagator in Section 5.5 above. The classical equations of motion are is the

for the coherent state

+

2U ' '

whose solutions in the

periodic

=

2UP '

0

stereographic coordinates (5.214)

=

are

0

the

(5.234) conditionally

motions

0'(t)

=

0'(0)

0'(t)

generalizing the results of Section In any case,

1CM

we

=

0'(0)

+

2Jdt

(5.235)

5.5.

arrive at the propagator

+

T)

=

EN

1 1 '=

'Q

1

-2iJi"T

)2J

-

+

rk E,, 11 14) (1 + E,, 2 t2 ) J J

e

2iJCoT

(5.236)

184


5.

on

Simply Connected

Phase

Spaces

and the associated quantum partition function

f dy(j)(C,t) (( j e_i? TI6))

ZSU(N+1)(T)

N+1

E

e-2iJO'T

a=1

11 '301a

(5.237)

1

I

-

e-2iJ(d#-d1)T

N+l =- Co. These generalize the previous results for SU(2) and, in particular, (5.237) coincides with the anticipated Weyl character formula for the [2J] representation of G SU(N + 1). What is also interesting here is where

=

that the Hamiltonian reduction mechanism of the last Section has

counterpart which

a

quantum

be used to interpret the quantum localization of this dynamical system. To see this, we recall from the last Section that the Hamiltonian (5.233) is the reduction of an (N + I)-dimensional simple harmonic oscillator

can

Hamiltonian,

which in the quantum

case

is the Hermitian operator

N+1

, j (D)

=

2J

E O'at"a"

(5.238)

a=1

Heisenberg-Weyl algebra E)N+1 gHW, where mutually commuting copies of the raising and lowering operators (5.79). Relating the coordinates on CN+1 and (CpN as usual via the constraint function (5.206), it is straightforward to show that the projection acting etta, bic,

on a

multi-dimensional

N + 1

are

operator in

(5.228)

is

[52]

27r

d'

pj

In,.... nN+1)(nj,...,nN+1I

27r nl+...+nN+1=2J

0

(5.239) where

Ini.... nN+l) ,

_(at)ni.... (,itN +J)nN+110) 1 -.nN+I! Vn-,!

-

-

are

the states of the orthonormal number basis for

the

weight

states of the

In,,..., nN+l) of

(5.240)

-

This identifies N+1 a=1 9HW.

SU(N+ 1) representation above as If nk jN+1; N+1) k=1 (5.240). This method

in terms of the bosonic Fock space states

constructing coherent

states is known

as

the

Schwinger

boson formalism

[49]-[52]. The trace formula

(5.237)

can now

be

represented

in terms of the canonical

coherent states N+'1

I z)) as

-=

e

=1

0) /

e'! 2

Z

=

(Zl'... zN+l) ,

E

(CN+l (5.241)

Quantization

5.8

ZSU(N+1)(T) Then

N+1

fl,,,=,

f

=

Isospin Systems

185

-

d.V dz"

-, j (D) T e-' I Z))

-

((zlPj

7rN+l

(5-242)

using the important property

I)q (D), f J] it is

of

straightforward

to show that

(5-243)

=0

(5.242)

becomes

[52]

ZSU(N+l) (T) 21r

dA

f Tir

e

d,01 (t) dzll (t)

-2iJ,

T x

rN+l

LR2N+2 tE[O,T]

0

exp

-

f

N+1

dt

we

states

(5.241).

e-

2iJTj'+i,\

(2aja

Z Za-La)

-

Z

a

a=1

0

where

(5.244)

have also used the resolution of The Gaussian functional

to carry out and

we

unity for the canonical coherent integration in (5.244) is now trivial

find 21r

ZSU(N+I)(T)

N+1

I

dA 2-7r

I

J1

e-2iJA

-

-2iJ61T+iA e

0

J

d aw 21r

W2J+N

we

integral

have transformed the

up N + I

simple poles each

w

-

e-2iJj11T

angular integration

S1. Carrying

the unit circle

over

J1 a=1

S1

where

(5.245)

N+1

in

(5-245)

out the contour

of residue I and leads to

into

a

contour

integration picks

Weyl character formula

(5.237). Thus the classical Hamiltonian reduction mechanism discussed in the last Section also

summed

implies localization

over

plectic reduction of CN+1 straint

at the

quantum level, and again the terms

in the WKB formula represent the

algebra (5.243)

coincides with the

to

CpN.

is determined

identity operator

singular points

In the quantum

case

of the sym-

the first class

by the projection operator

on

161

con-

which

the group representation space when as above. This Hamiltonian reduc-

restricted to the SU (N + 1) coherent states

always just a manifestation of the fact that these localizable dynamical are always in some way just a set of harmonic oscillators, because of the large "hidden" supersymmetry in these problems. The above analysis can also be straightforwardly generalized to the non-compact hyperbolic NJ space D (the complex open (N + l)-dimensional Poincar6 ball [68]) with the associated SU(N, 1) coherent states, generalizing the results of Section tion is

systems

186

5.

5.6 for

Weyl


SU(1, 1) [52].

In

on

Simply Connected

Spaces

particular, the localization formulas all lead

to the

SU(N, 1)

character formula for

N

ZSU(N,I)(T)

=

1

]1

e-iCoKT

I

Oe=1

The

Phase

CpN model above

(5.246)

e-ij"T

-

special case of the more general Kdhler space Geometrically, this is defined as the k (N k) complex-dimensional space Gr(N, k) of k-planes through the origin of (CN (note that Gr(N, 1) CpN-1). Algebraically, it is the space of N x N Hermitian matrices obeying a quadratic constraint, called

a

is

a

Grassmann manifold.

-

-

=

Gr(N, k) from which it

lp : pt

=

be shown to be

can

=

pj p2

p,

=

isomorphic

tr P

to the

=

kj

(5.247)

U(N) coadjoint

orbit

[68, 143] Gr(N, k)

-

U(N)I(U(k)

x

U(N

that the

eigenvalues

k))

UPUt

U(N) (coadjoint) action P quadratic (5.247) implies

with the transitive

-

--->

(5.248) on

Gr(N, k).

Note

that the operators P have therefore be interpreted as fermionic occupation

constraint in

0

or

1.

They

can

number operators and the trace condition in (5.247) can be interpreted as the total number of fermions in a given state. The Grassmann manifolds are

intimately related to an underlying free fermion theory [143]. The symplectic structure on (5.247) is the associated Kirillov-Kostant 2-form

therefore

W

(N,k)

=

i

tr(PdP A dP)

(5.249)

explicit form can be written using the local coordinatization provided diffeomorphism (5.223) and U(N) -- SU(N) x S' [49]. Similarly, the non-compact hyperbolic analog of the Grassmann manifold is known as the Siegel disc D N,k [68]. It is defined algebraically as whose

by

the

D N,k

=

IfD:

j5t

=

16) -Pt?7(N,k)P

=

77(N,k)}

(5.250)

where ??(Nk) is the flat Minkowski metric with diagonal elements consisting k entries of -1 and k entries of +1. It is isomorphic to the coadjoint of N -

orbit D N,k

--

of the non-compact Lie group

.,(N,k) ':Z

=

The Grassmann and

tr

U (N,

k) / (U (N)

U(N, k),

x

U (k))

(5.251)

and its Kdhler structure is defined

(-Pn(N,k)df3 ?7(Nk)dA7(N,k)) A

by

(5.252)

Siegel spaces above are the representative spaces homogeneous manifolds [68]. The coherent state quantization and semiclassical exactness of these dynamical systems has been discussed extensively for

Quantization of Isospin Systems

5.8

in

[49, 143].

constant N

The x

integrable Hamiltonians

N Hermitian matrix X and

Hx(P) so

that the partition function algebra is

=

(5.247)

on are

parametrized by

are

defined

187

a

by

(5.253)

tr(XP)

generalizes the Itzykson-Zuber integral (5.185).

Their Poisson

f Hx, Hy J,,,(N, k) so

(5.254)

H[x,y]

=

that the associated partition functions define

topological

theories. The

Duistermaat-Heckman formula Gr(N,k)

Z.'I

2,7ri

(T)

k(N-k),

e

iT

Elk..

k

T

1

hil

(hj

-

hi,)

(5.255) eigenvalues of the Hermitian matrix X in (5.253) parametrizing the Hamiltonian, has been verified for this dynamical system and shown

with

hi

E

R the

to be associated with

a

Hamiltonian reduction mechanism

as

described above

of (CpN. The construction of coherent states via the

algebraic Schwinger boson formalisms parallels that above for the (CpN coherent states, although in the present case it is somewhat more involved. We shall not go into the technical details here, but refer to [49] where it was also shown for the

case

or

that the WKB localization formula e

Gr(N k)

Z (N)' (T)

-ikT

Ek

nk1=111joijl-

1 Xni+l Xni+2 Xn2 > The complexification of any element g E SU(N) N column vectors Zi E (CN, i N, with X1

=

(5.221))

eq.

Ad*(X)g

=

=

.

.

.

EN j=1 xi

=

0 and

Xnt-1 Xnt, be parametrized

=

can

(5.258)

=

by

=

Zil Zj The canonical I-form

=

on

det[Zl,..., ZNI

6ij

=

(5.259)

1

the orbit is then N-I

0Inij where Ji

=

x, +...

constraint in

tr

(Xg-ldg)

=

JiZtdZ.. i

i

(5.260)

+XN-1 ! 0 and we have used the determinant The associated Kdhler 2-form is

+2xi

(5.259).

N-1

.1nij

and the

=

dO {nil

isospin element (5.258)

can

N-1

Q

i

=

forward to

=

ji -

N

tr(QX')

A

dZj

(5.261)

as

1)

(5.262)

with the normalization of

above for the (CpN case, it is straightthat these functions generate the SU(N) Lie algebra in the

generators defined see

expressed

(Jizizil

Defining the isospin generators Q`

SU(N)

be

i

E i=1

the

JjdZj

-i

as

Poisson bracket generated by the set of orthonormal constraints in lations. These functions

symplectic

(5.259)

are

structure

(5.261)

once

the first

substituted into the above

re-

generalize the angular momentum generators in the SU(2) case (where the only coadjoint orbit is the maximal one). The constraint functions in (5.259) are easily seen to be first class constraint functions [49, 130] so that the coadjoint orbit is determined again as the symplectic

Quantization of Isospin Systems

5.8

reduction of

a

larger

SU(ni)

x

SU(ne) x U(1) -'

x

...

space. These constraint functions

of

189

generate the subgroup

SU(N).

The localization properties of the the orbit functions (5-262) can there-

topological Hamiltonians generated by fore be interpreted again in terms of generalized dynamical systems, the characteristic feature of

sorts of harmonic oscillator

the

equivariant localization

mechanism. The coherent states in these generic cases are defined analogously to those over CpN above, with the restrictions discussed at the beginning of this Section. The structure of the

integrable models defined

on

these coset spaces has

[66, 86]. The Poisson-Lie relations of the SU(N) symmetry of each orbit (5.257) lead to a maximal number N 2 N of mutually commuting functions, which is equal to half the (real) dimension of the maximal been discussed in

-

coadjoint orbit.

particular, the dynamical properties of the flag manifold theory of non-commutative integrability. The coherent state quantization of this flag manifold in the case In

SU(N) IU(I)N- 1

of

SU(3)

have been related to ideas in the

has also been worked out in detail in

that case, the Kdhler

FSU(3) (Zi 2)

:-*z

potential

109 K1

is

+ Zl, l +

[66, 86] (see

[79, 100]).

also

In

[139]

Z2, 2) P(I

+ Z3, 3 +

(Z2

-

Z1

Z3) (' 2

-

'l

3))'71 (5.263) Q3 and

where p and q are integers. Using the usual integrable Hamiltonians Q8 defined as above, it is straightforward to construct a coherent state

path integral representation for the quantum dynamics of this localizable system and verify the localization formulas of Chapter 4 above. For instance, the semi-classical approximation has been verified for the dynamical system with topological Hamiltonian function H Ei=:,: jQj, where Q Q3 , 'F3Q8 the circle actions symplectic generate (Z1 Z2) Z3) -+ ( e'01 zi, e'01 Z2 Z3) and ei02 e-i02 -(Zl) Z2) Z3) ( Z3), respectively [66, 86] (so that H generates Z1, Z2, the symplectic action of the 2-torus group T 2 S' x S' on SU(3) /U(1)2). =

=

i

7

=

The WKB localization formula for the quantum propagator in the coherent state representation for this integrable spin model can then be worked out to

be

ICSU(3) (. I, z; T)

=

(I +. jzj ei(i++j_)T X

(1

+

+

o 21Z2

eij+T)p

213 Z3 e-id- T+ (212 - 1, 31)(Z2

-

ZlZ3)

e'j+T)q (5.264)

which

can

be shown to coincide with the exact result from

Finally,

we

-point

out that

one can

also

formalisms of Sections 3.8 and 4.9 to these

a

direct calculation.

apply the nonabelian localization generalized spin models regarding

the associated partition functions as defined on the coordinate manifolds. As such, they can be applied to problems such as geodesic motion on group

manifolds, and

in

particular

the Hamiltonian framework

we can

[161].

reproduce the results of Picken [139] in precisely, we note that in these cases

More

there is the natural G-invariant metric

190

5.

Equivariant Localization g


on

tr(F

=

Ff)

0

(5.265)

h-ldh is the adjoint repredefined on the group manifold of G, where sentation of the Cartan-Maurer 1-form which takes values in the Lie algebra

geodesic motion on G, the Hamiltonian operator Schr6dinger polarization is given by the Laplace-Beltrami operator

g of G. For free

1

fi(O)

=

-

2

V2

(5.266)

9

with respect to the metric (5.265). The invariant be the Riemannian volume form associated with

trajectories

to show that the space of

manifold

G

over

in the

measure on

(5.265),

G is taken to

and it is

be made into

can

by absorbing the Riemannian volume form

into

an

possible a

Kdhler

effective action

[1391.

in the usual way

The generic classical trajectories generated by the Hamiltonian (5.266) are straight lines in the Weyl alcove A HcIW(Hc) of the Lie group G =

because of the

decomposition [162] 9192

1 =

vhv-

1

V91) 92

E

(5.267)

G

HC and the elements v can be parametrized by the coset space GlHc. The geodesic distance between g, and 92 defined by (5.265) is independent of v. Then the sum over extrema, in the semi-classical approximation is given by a sum over the lattice Z' in r-dimensional Euclidean root space generated by the simple roots of g. The time-evolution kernel is therefore a

where h E

function of the r-dimensional vector

formulas for it

IC (a;

T)

can

(

be shown to

2-7riT

analog

a(a

+

of the

tion formula

which is

a

21rk)

=

[96]

.

.

,

ar)

anticipated

E

A and the localization

result

the

[38, 139, 146, 161]

iE' (aj+21rkj)2/2T

(5.268)

+

Ej=, aj(aj

leads to over

.

2,7rk) '2 a (a + 21rk)

Weyl character

series

(a,,

k=(k-l,...,kr)EZ'

2 sin

ce>O

yield

=

the

G12

a(a

X

where

)

dim

a

+

27rkj). (5.268) is the configuration space (5.28). Applying the Poisson resumma-

formula

a spectral expansion of the quantum propagator unitary irreducible representations of G given by

IC(a; T)

E

dim R,\

(tr,\a) e-ic(,\)T

(5.269)

p?)

(5.270)

AEZ'

where

C(A)

+

p,)2

_

Quantization

5.9

on

Non-Homogeneous

Phase

Spaces

191

eigenvalues of the quadratic Casimir operator E,,(Xa)2 in the reprehighest weight vector A. Similar considerations also apply to n-spheres Sn SO(n + 1)/SO(n) [96, 100] and their hyperbolic counterparts Rn -_ SO(n 1, 1)/SO(n) obtained by the usual analytical continuation of Sn [31]. The case of S' we saw was associated with the Dirac monopole, that of S3 -- SU(2) describes the emergence of spin, and S4 corresponds to the BPST instanton-antiinstanton pair with 2 chiral spins [100]. Notice that these localizations also apply to the basic integrable models which are well-known to be equivalent to the group geodesic motion problems above, such as 2-dimensional Yang-Mills theory, supersymmetric quantum mechanics and Calegoro-Moser type theories. These describe the quantum mechanics of integrable models related to Hamiltonian reduction of free field theories [44, 56] and will be discussed again in Chapter 8. Note that these free theory reductions again illustrate the isomorphism the

are

sentation with

-

-

between the localizable models and

5.9

Quantization

on

(trivial)

harmonic oscillator type theories.

Non-Homogeneous Phase Spaces

Thus far in this

Chapter we have examined the localizable dynamical systems phase spaces which are maximally symmetric and those with multi-dimensional maximally-symmetric subspaces. This exhausts all spaces both those

on

with constant curvature and which

are

symmetric, and

we

outlined both the

physical

and group theoretical features of these dynamical systems. In this final Section of this Chapter we consider the final remaining possible class of Riemannian

geometries

K(x) /C(M,g)

which is

on

the

phase

space

M,

i.e. those with

a

Gaussian

non-constant function of the coordinates

on M, again to 2dimensional phase spaces. The geometries which admit only a single Killing vector are far more numerous than the maximally symmetric or homogeneous ones and it is here that one could hope to obtain more non-trivial applica-

curvature so

that dim

=

a

1. For

simplicity

we

restrict attention

tions of the localization formulas. Another nice feature of these spaces is that

the

corresponding

Hamiltonian Poisson

algebra will be abelian, so that the automatically be Cartan elements, in contrast to the previous cases where the Lie algebra IC(M, g) was non-abelian. Thus the abelian localization formulas of the last Chapter can be applied straightforwardly, and the resulting propagators will yield character formulas for the isometry group elements defined in terms of a topological field theory type path integral describing the properties of integrable quantum systems corresponding to Cartan element Hamiltonians. Given a I-parameter isometry group GM acting on (M, g), we begin by introducing a set of prefered coordinates (X11' X12 ) defined in terms of 2 Hamiltonians

so

obtained will

differentiable functions

x1

coordinates the

vector V has

the function

X1

Killing

and

X2

as

described in Section 5.2,

components V"

is any non-constant function

on

M,

=

so

1, V/2

but

we

that in these

=

0. For now,

shall

soon see

192

5.


Simply Connected

on

Phase

Spaces

how,

once a given isometry of the dynamical system is identified, it can be fixed to suit the given problem. For a Hamiltonian system (M, W, H) which generates the flows of the given isometry in the usual way via Hamilton's

equations,

the

defining

(5.55)

condition

for the coordinate function X 2

now

reads

JH, X21,0

=

LVX2

=

(5.271)

0

which is assumed to hold away from the critical point set of H (i.e. the zeroes of V) almost everywhere on M. This means that X2 is a conserved charge of

the given dynamical system, i.e. a GM-invariant function of action variables. In higher dimensions there would be many such possibilities for the conserved

charges depending

the integrability properties of the system. However, requirement fixes the action variable to be simply a functional of the Hamiltonian H, on

in 2-dimensions this

X

and

so even

in the

2

(5.272)

non-homogeneous

cases we see

the intimate connection

here between the equivariant localization formalism and the

(classical

integrability

of

quantum) dynamical system. We note that this only fixes the requirement (5.271) that the coordinate transformation function be constant a

along

the

or

integral

Killing vector field V. The isometry consymplectic 2-form now only implies that, in the new xl-coordinates, w,,,(x') is independent of x" (just as for the metric). The Hamiltonian equations with V" 0 must be solved consistently 1, V/2 and associated now using (5.272) an symplectic structure. Notice that this construction is explicitly independent of the other coordinate transformation function x1 used in the construction of the prefered coordinates for V (c.f. dition

(5.70)

on

curves

of the

the

=

=

Section 5 .2).

Thus for

a

general

metric

(5.49)

that admits

a

sole isometry, the

general

"admissible" Hamiltonians within the framework of equivariant localization are given by the functionals in (5.272) determined by the transformation x

-+

x' to coordinates in which the

(circle

or

translation)

action of the

cor-

now arises because responding Killing explicit. the integrability condition LVw 0 for the Hamiltonian equations does not uniquely determine the symplectic 2-form w, as it did in the case of a homoge-

The rich structure

vector is

=

neous

symmetric geometry. The above

construction could therefore be started

with any given symplectic 2-form obeying this requirement, with the hope of being able to analyse quite general classes of Hamiltonian systems. This has

the

possibility

of

largely expanding the

known

examples

of quantum systems

where the Feynman path integral could be evaluated exactly, in contrast to the homogeneous cases where we saw that there was only a small number of

few-parameter Hamiltonians which fit the localization framework. However, it argued that the set of Hamiltonian systems in general for which the localization criteria apply is still rather small [40, 1591. For instance, we could 2 from the onset take w to be the Darboux 2-form on M R and hope to has been

=

Quantization

5.9

Non-Homogeneous

on

Phase

Spaces

193

obtain localizable

examples of 1-dimensional quantum mechanical problems potentials. These are defined by the Darboux Hamiltonians

with static

HQM (p, q) where

1P2 + U(q)

=

(5.273)

2

U(q)

is some potential which is a C'-function of the position q E Dykstra, Lykken and Raiten [40] who first pointed out that the formalism in Chapter 4 above, which naively seems like it would imply the exact solvability of any phase space path integral, does not work for arbitrary potentials U(q). To see this, we consider a generic potential U(q) which is bounded from below. By adding an irrelevant constant to the Hamiltonian (5.273) if necessary, we can assume that U(q) > 0 without loss of generality. We introduce a "harmonic" coordinate y E R and polar coordinates (r, 0) E R+ x S, by

R1.

It

was

p

where

we

further

=

r

in

=

Ir 2, 2

(5.274)

these

polar

so

and

that

assume

(5.273)

nates the Hamiltonian

form H

U(q)

rsinO

U(q)

is

IY2 2

1 =

2

r2 Cos 2o

(5.274)

monotone function. In these coordi-

takes the usual

integrable harmonic oscillator

X2 above defines the radial coordinate (5-272). The Hamiltonian vector field in

that the function

F(H)

vr2_H_

=

coordinates has the

in

single non-vanishing component

Vo The metric tensor

a

=

(5.49)

will have in

=

_dy

general have

gor under the coordinate transformation (2.112) become

Voo9ogoo + 2gooo9oVo

=

0

V00909rr

7

+

(5.275)

dq

(5.274),

190(groVO)

2goo9rVo

=

3 components grr, goo and

and the

+

Killing equations

900-9rVo

=

0

(5.276)

0

The 3 equations in (5.276) can be solved in succession and the general solution has the form

by integrating

them

0

goo

f (r)

gro

f (r) VO

I 00

dO' ar

( Vo,I-)+

h

Vo

grr

f (r)

+k(r)

(5.277) f (r), h(r) and k(r) are arbitrary C'-functions that are independent of the angular coordinate 0. Note that, as expected, there is no unique solution for the conformal factor W in (5.49), only the requirement that it be radially symmetric (i.e. independent of 0). However, the equations (5.277) impose a much stronger

where

194

5.


Simply Connected

on

Phase

Spaces

requirement, this time on the actual coordinate transformation (5.274). If impose the required single-valuedness property on the metric components above, then the requirement that g,O (r, 0) g,O (r, 0 + 27r) is equivalent to

we

=

the condition 27r

a

dO

f V-0

09r

0

(5.278)

constant

(5.279)

=

0 or

equivalently

that 27r

f

dO

_q

=

dy

0

However,

only

the

solution to

(5.279)

is when the function

dy

is

independent

of the radial coordinate r, which from (5.274) is possible only when y so that U(q) -!q 2 and HQm is the harmonic oscillator Hamiltonian. =

=

-q,

Thus,

with the exception of the harmonic oscillator, equivariant localization fails for all 1-dimensional quantum mechanical Hamiltonians with static potentials which

are

bounded

satisfying the

below, due

to the non-existence of

Lie derivative constraint in this

a

single-valued

metric

case.

It is instructive to examine the localization formulas for the harmonic

oscillator, theory, to

which is considered trivial from the point of view of localization what role is played by the degree of freedom remaining in the

see

metric tensor which is not determined

generates

a

global SI-action on M

R

=

by

the

(5.275)

straints. The Hamiltonian vector field

2

equivariant localization con1 which case is Vo

in this

=

given by translations of the angle

co-

ordinate 0. Thus the localization formulas should be exact for the harmonic

radially symmetric geometry (5.49) to make manifest principle. This is certainly true of the WKB formula (4.115) which does not involve the metric tensor at all, but the more general localization formulas, such as the Niemi-Tirkkonen formula (4.130), are explicitly metric dependent through, e.g. theA-genus terms, although not manifestly so. Explicitly, the non-vanishing components of the metric tensor (5.49) under oscillator using any

the localization

the coordinate transformation grr

and it is

ew(-)

in the

goo

I

case

=

r

2

at hand

are

ew(r)

straightforward to work out the Riemann moment map and VO I lead to the non-vanishing components

tensor which with

where

=

(5.274)

we

(5.280) curvature

=

(00or

=

Roror

=

r

-(f2V)ro 1 -

2

S?V

=

2

d Or

dr

have introduced the function

ew(r) log

A (r)

2 +

r

dr

(5.281)

5.9

Quantization

A(r)

=

Non-Homogeneous

on

Phase

Spaces

e- '(')(Qv)o,/2r

195

(5.282)

Substituting the above quantities into the Niemi-Tirkkonen formula (4.130) r and working out the Grassmann and 0 integrals there, after some algebra we find the following expression for the harmonic oscillator partition function,

with w,O

=

00

Zharm(T)

-

-1 t

I

dr

d

sinTA(r)

dr

e-iTr 2/2

1 i

lim r,o

A(r) sin TA (r)

(5.283)

0

Comparing with (5.75),

we see

for the harmonic oscillator

behaves at the origin

r

=

0

that this result coincides with the exact result

partition function only if the function (5.282) as

lim r-O

which

isfy,

using (5.281) and (5.282)

in addition to the radial

(5.284)

2

that the

phase space metric must satthe additional constraint constraint, symmetry means

lim

d r

dr

r-O

The

A(r)

W(T)

=

(5.285)

0

requirement (5.285) means that the conformal factor W(r) of the Riegeometry must be an analytic function of r about r 0, and this

mannian

restriction

=

on

the

general

form of the metric

(5.49) (i.e.

on

the functional

properties of the conformal factor W) ensures that the partition function is independent of this phase space metric, as it should be. This analyticity requirement, however, simply means that the metric should be chosen so as to eliminate the singularity at the origin of the coordinate transformation to polar coordinates (r, 0) on the plane. That this transformation is singular at p 0 is easily seen by computing the Jacoq bian for the change of variables (5.274) with the harmonic oscillator potential (or by noting that w and g are degenerate at r 0 in these coordinates). Since the equivariant Atiyah-Singer index which appears as the Niemi-Tirkkonen formula for the quantum mechanical path integral is an integral over charac=

=

=

teristic

classes,

it is

manifestly invariant

under CI deformations of the metric

M. The transformation to polar coordinates is a diffeomorphism only on 2 the punctured plane R 101, which destroys the manifest topological invarion

_

of the partition function (at r 0 anyway). For A(O) 4 0, the metric a conical geometry [102] for which the parameter A(O) represents the tip angle of the cone. This example shows that for the localization ance

=

tensor describes

to work the choice of metric tensor is not

completely arbitrary,

since it has to

respect the topology of the phase space on which the problem is defined. As discussed in [40] and [159], this appears to be a general feature of the generalized localization

formalisms,

in that

they detect explicitly the topology

of the

196


5.

on


phase

space and this can be used to eliminate some of the arbitrariness of the metric (5.49). Indeed, in the set of prefered coordinates for V it has no zeroes

and

so the critical points are "absorbed" into the symplectic 2-form W and general also the metric g. Thus the prefered coordinate transformation for V is a diffeomorphism only on M Mv in general. Nonetheless, this simple example illustrates that quite general, non-homogeneous geometries can still be used to carry out the equivariant localization framework for path integrals and describe the equivariant Hamiltonian systems which lead to topological quantum theories in terms of the generic phase space geometry. Although the above arguments appear to have eliminated a large number of interesting physical problems, owing to the fact that their Hamiltonian vector fields do not generate well-defined orbits on the O-circle, it is still possible that quantum mechanical Hamiltonians with unbounded static potentials could fit the localization framework. Such dynamical systems indeed do represent a rather large class of physically interesting quantum systems. The first such attempt was carried out by Dykstra, Lykken and Raiten [40]

in

-

who showed that the Niemi-Tirkkonen localization formula for such models be reduced to

can

a

relatively simple

contour

the equivariant localization formalism atom Hamiltonian [95]

Hh(p, q) The

=

applied

_1P2 2

integral.

For

example, consider hydrogen

to the 1-dimensional

1

(5.286)

_

JqJ

eigenvalues of the associated quantum Hamiltonian energies

form

a

discrete spec-

trum with

E,,

=

-1/2n

2

n

=

1,2....

(5.287)

which resembles the bound state spectrum of the more familiar 3-dimensional hydrogen atom [1011. What is even more interesting about this dynamical

system is that the classical bound state orbits all coalesce at the phase space 2 oo on R so that a localization onto classical trajectories points q 0, p =

(like

=

the WKB

,

formula)

highly unsuitable for this quantum mechanical problem could therefore provide an example wherein although the standard WKB approximation cannot be employed, the more general localization formulas, like the Niemi-Tirkkonen formula, which seem to have no constraints on them other than the usual isometry restrictions on the phase space M, could prove of use in describing the exact quantum theory of the dynamical system. The key to evaluating the localization formulas for the Darboux Hamiltonian (5.286) is the transformation to the hyperbolic coordinates (r, -r) with problem.

-oo

< r,

This

-r

< oo,

p so

is

=

Irl sinh T

that the Hamiltonian is again Hh

field has the

q =

=

2/rJrJ cosh'T

_1r2 and the Hamiltonian 2

single non-vanishing component

(5.288) vector

5.9

Quantization

on

Non-Homogeneous

1r3 cosh3

V

Phase

Spaces

(5.289)

-r

4

197

Killing equations have precisely the same form as in (5.276), with (r, 0) replaced by (r,,r) there, and thus the general solutions for the metric tensor have Precisely the same form as in (5.277). However, because of the non-compact range of the hyperbolic coordinate r in the case at hand, we do not encounter a single-valuedness problem in defining the components grr 2 and from (5.277) and (5.289) we find that it is given as C' functions on R the explicitly by perfectly well-defined function Now the

gr'r

_

12f (r)

=

sinh -r

V'r

In the context of

cosh

our

2

coordinates

are

X1

1arctan(sinh-r)

=

X"'

2

so

that.F(H)

is determined

the x'-coordinates in

(5.43)

=

we

h(r)

+

2

isometry analysis above,

nate transformation function X

coordinate function

+ _r

(5.290)

Vr

again choose the coordi-

,/----2H in (5.272).

by noting

from which

The other

that the above we

(r, -r)

wish to define the

prefered set of x"-coordinates for the Hamiltonian vector field V. There we identify (X11,X12) (-r, r) according to that prescription. Carrying out the x" r using (5.289), and then substituting in the over explicit integration transformation (5.288) back to the original Darboux coordinates, after some algebra we find =

=

X'(p,q)

2 _

jqj

P

-3/2

2

plql

1/2

2 _

jqj

P

2

P

+ 2 arctan 2

-

Jqj

P

2

11/2 (5.291)

Thus the Hamiltonian sor

(5.49)

(5.286)

is associated with the

phase

which is invariant under the translations

(5.291).

above shows

X1

space metric ten-

--+

x1

+ ao of the

explicitly that the phase

analysis space globally well-defined metric which is translation invariant in the variable (5.291). It is also possible to evaluate the Niemi-Tirkkonen localization formula for this quantum problem in a similar fashion as the harmonic oscillator example above. We shall not go into this computation here, but refer to [40] for the technical details. The only other point we wish to make here is that one needs to impose again certain regularity requirements coordinate

indeed does admit

on

The

a

(5.49).

the conformal factor of the metric

These conditions

are

far

more

complicated than above because of the more complicated form of the translation function (5.291), but they are again associated with the cancelling of the coordinate singularities in (5.288) which make the equivariant Atiyah-Singer index in (4.130) an explicitly metric dependent quantity. With these appropriate geometric restrictions it is enough to argue that the quantum partition function for the Darboux Hamiltonian

(5.286)

has the form

[40]

00

Zh (T)

-

E n=1

e

iT/2n2

(5.292)

198

5.


which from

(5.287)

we see

1-dimensional


on

is indeed the exact

spectral propagator for the

hydrogen [95]. example shows that more complicated quantum systems can be studied within the equivariant localization framework on a simply connected phase space, but only for those phase spaces which admit Riemannian geometries which have complicated and unusual symmetries, such as translations in the coordinate (5.291) above. Thus besides having to find a metric tensor appropriate to the geometry and topology of a phase space, there is the further general problem as to whether or not a geometry can in fact possess the required symmetry (e.g. for Hamiltonians associated with bounded potentials, there is no such geometry). It is not expected, of course, that any Hamiltonian will have an exactly solvable path integral, and from the point of view of this Chapter the cases where the Feynman path integral fails to be effectively computable within the framework of equivariant localization will be those cases where a required symmetry of the phase space geometry does not lead to a globally well-defined metric tensor appropriate to the given topology. Nonetheless, the analysis in [40] for the 1-dimensional hydrogen atom is a highly non-trivial success of the equivariant localization formulas for path integrals which goes beyond the range of the standard WKB method. We conclude this Chapter by showing that it is possible to relate the path integrals for generic dynamical systems on non-homogenous phase spaces which fall into the framework of loop space equivariant localization to character formulas for the associated I-parameter isometry groups GM [159]. For this, we need to introduce a formalism for constructing coherent states assoatom

This

ciated with non-transitive group actions on manifolds [89, 159]. We consider (5.49) in the prefered x-coordinates for a Hamilto-

the isothermal metric nian vector field V

Using these coordinates, we define the complex analogy with the case where V defines a rotationally symmetric geometry (as for the harmonic oscillator) Let f (z, ) be a G(l)-invariant analytic solution of the ordinary differential equation coordinates

z

=

x"

on

M.

e x'l,

in

-

d

d

T(TZ) For the

ZZ_

d(z. )

log f (Z )

symplectic 2-form of the phase

volume form associated with

WM

0'(')

2

space,

we

take the

d

d

-

dz A

Z

whose associated

symplectic potential .

Om This definition turns the fold with Kiihler

=

(5.293) GM-invariant

(M, g),

log f (z. ) % -( zi) Z (Zi)) .

=

1

-1 2

phase potential

d

_F)

(5.294)

is

log f (z. ) (. dz

space into

a

-

zd. )

non-homogeneous

(5.295) Kiffiler mani-

5.9

Quantization

F( ') (z, 2) w(w) determines

such that

bundle L(w) Let

Np,

Taylor

the

Non-Homogeneous Phase Spaces

on

log f (z2)

=

199

(5.296) symplectic

the first Chern class of the usual

line

M.

--+

Nw

0
I is non-trivial and the hyperbolic Riemann surface is non-abelian. Its first

6.6 Generalization to

homology

group is

Hyperbolic

given by (3.85), and, using 1,

denote its generators as well by ai, bi, i basis of homology cycles for Zh.

an

Riemann Surfaces

227

abusive notation, we shall a canonical

h and call them

=

According to the Riernann uniformization theorem [111], there are only 3 (compact or non-compact) simply-connected Riemann surfaces the 2-sphere S2, the plane C and the Poincare upper half-plane H2, each equipped with -

their standard metrics

discussed in the last

Chapter. The sphere is its having a unique complex structure), while C is the universal cover of the torus. The hyperbolic plane 7j2 is always the universal cover of a Riemann surface of genus h > 2, which is represented as Zh =,H2/Fh (6.81) own

universal

as

(being simply-connected

cover

and

where Fh 7r, (_rh) is in this context refered to as a discrete FVchsian group. The quotient in (6.81) is by the fixed-point free bi-holomorphic action of =

Fh jj2

on

?j2 The

is

PSL(2,R)

group of

-

transformations

=

as

analytic automorphisms

of the upper

half-plane

SL(2,R)/Z2, the group of projective linear fractional in (6.73) except that now the coefficients a,,3,-Y,J are

7r,(Zh) is taken as a discrete subgroup of this -H2 and the different isomorphism. classes of complex analytic structures of _rh are essentially the different possible classes of discrete subgroups. Note that this generalizes the genus 1 situation above, where the automorphism group of C was the group PSL(2, C) of global conformal taken to be real-valued. Then

PSL(2,R)-action

on

transformations in 2-dimensions and 7rl (Zl)

was

taken to be the lattice sub-

Indeed, it is possible to regard Zh as a 4h-gon in the plane with edges identified appropriately to generate the h 'holes' in Zh. It is difficult to generalize the explicit constructions of the last few Sections because of the complicated, abstract fashion in (6.81) that the complex coordinatization of Zh occurs. For the various ways of describing the Teichmfiller space and Fuchsian groups of hyperbolic Riemann surfaces without the explicit introduction of local coordinates, see [74]. The Teichmiiller space of Zh can be naturally given the geometric structure of a non-compact complex manifold which is homeomorphic: to (CM-3, so that the coordinatization of _Vh is far more intricate for h > 2 because it now involves 3h 3 complex before. I to still it is as as parameters, opposed just Nonetheless, possible to deduce the unique localizable Hamiltonian system on a hyperbolic Riemann surface and deduce some general features of the ensuing topological quantum field theory just as we did above. We choose a complex structure on Zh for which the universal bundle projection in (6.81) is holomorphic (as for the torus), and then the metric gZh induced on Zh by this projection involves a globally-defined conformal group.

-

factor

p

as

in

(6.9)

and

a

constant

negative

curvature Khhler metric

(the

see Section 5.6). The condition now that the hyperbolic Poincar6 metric Killing vectors of this metric be globally-defined on _rh means that they must be single-valued under windings around the canonical homology cycles -

228


6.

ae,

bt I h

E

1

Hi (_Th; Z),

Multiply Connected

on

equivalently

or

dV1'

dV"

Spaces

that

(6.82)

h

0

=

Phase

bt

at

Using this single-valued

condition and the g_Vh (dV,

-)

Killing equations

-iVdg_rh

=

(6.83)

deduce the general form of the Killing Hodge decomposition theorem [22, 32]

Zh. For this,

we can now

vectors of

apply

to the metric-dual 1-form

g(V, -)

the

we

Al _Th'

E

9(V7

dX

+

*d

+

(6.84)

4

where X and are Cl-functions on Zh and Ah is a harmonic solution of the zero-mode Laplace equation for 1-forms,

A'AlAh In the

above,

*

_=

(*d* d + d* d*)Ah

denotes the

i.e.

a

(6.85)

0

Hodge duality operator and,

on

a

general

(M, g),

d-dimensional Riemannian manifold

it encodes the Riemannian geinto the DeRham cohomology. It is defined as the map

ometry directly

*:

which is given

=

1-form,

AkM

Ad-kM

---*

(6.86)

locally by I

*ce

=

(d Xf

-

A1

k)! ...

Vdetg(X)

Ad-kil

...

ik

a l

gj,,X,

...

ik

W

-

(x) dxjl

9jd-kAd-k A

...

A

W

(6.87)

dXjd- k

and satisfies

(-,)(d-I)k on

AkM.

Using

product fm a A *,8 on each vector space AkM. product possible to show that a differential form Ah as harmonic if and only if It defines

an

inner

this inner

above is

(6.88)

it is

dAh

=

d

*

Ah

=

(6.89)

0

and the Hodge decomposition theorem (6.84) (which can be generalized to arbitrary degree differential forms in the general case) implies that the DeRham cohomology groups of M are spanned by a basis of harmonic forms. The Hodge decomposition (6.84) is unique and the components involved there are explicitly given by I X

=

V2Eh

*

d*

g(V,

V2Eh

dg (V,

(6.90)


Laplacians V2

where the scalar

*d

h

Hyperbolic

(6.90)

d in

*

Riemann Surfaces

229

assumed to have

are

modes removed. The 1-form Ah in (6.84) can be written as a linear combination of basis elements of the DeRham cohomology group H'(Zh; R)

their

zero

-

According choose which

an

are

to the

Poinear6-Hodge duality theorem [32],

orthonormal basis of harmonic 1-forms

f at

f at, &1h

homology

Poincar6-dual to the chosen canonical

H, (M; Z) above,

we can 1

basis

in

particular

H'(Z h; R) f at, bilht= 1 E

E

i.e.

at,

=

J 0j,

=

6et,

at,

=

bt

bt

f3t,

=

(6.91)

0

at

We remark here that the local parts of the decomposition (6.84) simply form the decomposition of the vector gZh (V7 -) into its curl-free, longitudinal and

divergence-free,

transverse

pieces

as

VZhX

+

VZh

We

can now

write the

general

. The harmonic part

x

Ah accounts for the fact that this 1-form may sit in cohomology class of H1 (Zh; R).

a

non-trivial DeRham

form of the isometries of Zh

_r hinherits 3 local isometries via the bundle

in

As before, (6.81) from the .

projection maximally symmetric Poincar6 upper half-plane. However, only the 2 quasitranslations on H2 become global isometries of Zh and they can be expressed in terms of the canonical homology basis using the above relations. This global isometry condition along with (6.82) and (6.83) imply that Hodge decomposition (6.84) of the metric-dual 1-form to the Hamiltonian vector field VZh is simply given by its harmonic part which can be written as ,

h

E (Vl'ae + 4,8f)

gZh (VZh,

(6.92)

t=1

decomposition (6.92) is the generalization of (6.18). Indeed, on the torus we can identify the canonical harmonic forms above as a do, /27r and 0 d02/2-7r. The Killing vectors dual to (6.92) generate translations along the homology cycles of _Th and the isometry group of Zh is 11 2hJ U(1). 0 on the symplectic 2-form of _Th The usual equivariance condition 'C Vv CO The harmonic

=

=

,

=

h

now

becomes h

div_,h where

O(x)

d

is the C'-function

(6.93) implies

that it is constant

Vl at

(D on

on

h

+

V2,Ot

defined

Zh7 just

as

by w,,,(x) in

(6.93)

0

=

CD(x),E1,

and

(6.19).

Integrating up the Hamiltonian equations we see therefore that the unique equivariant (Darboux) Hamiltonians have the form

H_r h (X)

=

E I (Mae + W2,3e) 1

t=1C.

(6.94)

230

6.


hle

on

Multiply Connected

Phase

Spaces

real-valued constants and Q, C Zh is a simple curve from basepoint to x. The Hamiltonian (6.94) is multi-valued because it depends explicitly on the particular representatives at,0i of the DeRham 2h R As before, single-valuedness of the cohomology classes in H'(Zh; R) time-evolution operator requires that V W h for some W E Z and h E R, A A A and the propagation times are again the discrete intervals T NTh-'. Thus the Hamiltonian (6.94) represents the windings around the non-trivial homology cycles of'Eh and the partition function defines a topological quantum field theory which again represents the homology classes of Z h through a family of homomorphisms from 2h Z into U(j)(92h Again, the partition function path integral should be properly defined in the homologically-invariant form (6.27) to make the usual quantities appearing in the associated action S welldefined by restricting the functional integrations to homotopically equivalent loops. We note that again the general conformal factor involved in the metric gZh obeys Riemannian restrictions from the Gauss-Bonnet-Chern theorem where

.

are

fixed

some

=

.

=

7

=

.

and

a

ume

of

volume constraint similar to those in Section 6.2 above. When the vol-

parameter is quantized

physical

states will be

as

(6.55),

in

(VIV2)3h-3

we

expect that the Hilbert space

(one copy of the genus I Hilbert spaces for each of the 3h 3 modular degrees of freedom in this case) and the coherent state wavefunctions, which can be expressed in terms of D 3 dimensional Jacobi theta functions (6.59), will in addition 3h carry dimensional

-

=

-

(V2)3h-3

explicit

dimensional projective representation of the discrete first homol_ph (i.e. of the equivariant localization constraint algebra). The proofs of all of the above facts appear to be difficult, because of the

lack of

complex coordinatization

a

ogy group of

for these manifolds which is

the definition of coherent states associated with the

n2h i=1 U(1)

=

S' 112h i=1

on

the non-symmetric space

isometry Zh (S1 X =

required for group action

Sl)#h.

Thus the general feature of abelian equivariant localization of path integrals on multiply connected compact Riemann surfaces is that it leads to a

topological quantum theory whose

associated topologically invariant partition function represents the non-trivial homology classes of the phase space. The coherent states in the finite-dimensional Hilbert space also carry a multidimensional representation of the discrete first homology group, and the lo-

calizable Hamiltonians

these

phases spaces are rather unusual and even simply-connected cases. The invariant symplectic 2-forms in these cases are non-trivial elements of H 2(Zh; Z) Z, as in the maximally-symmetric cases, and it is essentially the global topological features of these multiply-connected phase spaces which leads to these rather more

on

restricted than in the

=

severe

restrictions. The coherent state

quantization of these systems shows

that the path

integral describes the coadjoint orbit quantization of an unusual spin system described by the Riemann surface. These spin systems are exactly solvable both from the point of view of path integral quantization on the loop space and of canonical holomorphic quantization in the Schr6dinger polarization. The localizable systems that

one

obtains in these

cases are

rather


Hyperbolic

Riemann Surfaces

231

trivial in appearence and are associated with abelian isometry groups acting on the phase spaces. However, these quantum theories probe deep geometric and

topological

features of the

phase

spaces, such

as

their

complex algebraic

geometry and their homology. This is in contrast with the topological quanwe found in the simply-connected cases, where at best topological path integral could only represent the possible non-trivial co2 homology classes in H (M; Z). It is not completely clear though how these path integral representations correspond to analogs of the standard character formulas on homogeneous symplectic manifolds which are associated with semi-simple Lie groups, since, for instance, the usual Khhler structure between the Riemannian and symplectic geometries is absent in these non-

tum field theories that

the

symmetric

cases.

Beyond

7.

In this

Chapter

ization

[1571.

the Semi-Classical

we

shall examine

We return to the

3 and consider

a

Approximation

a different approach to the problem of localgeneral finite-dimensional analysis of Chapter

Hamiltonian system whose Hamiltonian function is a Morse we will construct the full -Lexpansion for the Classical

function'. From this

T

partition function, as we described briefly in Section 3.3. A proper covariantization of this expansion will then allow us to determine somewhat general geometrical characteristics of dynamical systems whose partition functions localize, which in this context will be the vanishing of all terms in the perturbative loop expansion beyond I-loop order. The possible advantages of this analysis are numerous. For instance, we can analyse the fundamental isometry condition required for equivariant localization and see more precisely what mechanism or symmetry makes the higher-order terms disappear. This could then expand the set of localizable systems beyond the ones we have

already

predicted from localization theory, and at the same geometrical structures of the phase space probe deeper the whole thus or providing richer examples of topological dynamical system field theories. Indeed we shall find some noteworthy geometrical significances of when a partition function is given exactly by its semi-classical approximation as well as new geometric criteria for localization which expand the encountered that

are

into the

time

previous isometry conditions.

approach to the Duistermaat-Heckman integration formula using the perturbative loop-expansion has been discussed in a different context recently in [174] where the classical partition function was evaluated for the dynamical system describing the kinematics of thin vortex tubes in a 3-dimensional fluid This

whose Hamiltonian is similar to that considered at the end of Section 5.8 for

geodesic motion on group manifolds. For such fluid mechanics problems, the phase space M is neither finite-dimensional nor compact and the Hamiltonian flows need not be periodic, but the dynamical system admits an infinite sequence of constants of motion which

localizable and

one

[1741.

are

in involution

The standard localization

needs to resort to

The extension to

an

analysis

so

that it should be

analyses therefore do

not

apply

of the sort which will follow here. We

degenerate Hamiltonians

is

fairly straightforward.

In what fol-

lows all statements made concerning the structure of the discrete critical Point set

Mv of H will then apply

to the full critical submanifold.


234

Beyond the Semi-Classical Approximation

7.

shall indeed find extensions of the localization formalisms which such

cover

certain

cases.

Recalling that the isometry condition can always be satisfied at least locally on M, we then present some ideas towards developing a novel geometric method for systematically constructing corrections to the DuistermaatHeckman formula. Given that a particular system does not localize, the idea is that we can "localize" in local neighbourhoods on M where the Killing equation can be satisfied. The correction terms are then picked up when these open sets are patched back together on the manifold, as then there are non-trivial singular contributions to the usual I-loop term owing to the fact that the Lie derived metric tensor cannot be defined globally in a smooth way over the entire manifold M. Recalling from Section 3.6 that the properties of such a metric tensor are intimately related to the integrability properties of the dynamical system, we can explore the integrability problem again in a (different) geometric setting now by closely examining these correction terms. This will provide a highly non-trivial geometric classification of the localizability of a dynamical system which is related to the homology of M, the integrability of the dynamical system, and is moreover completely consistent witl Kirwan's theorem. Although these ideas are not yet fully developed, they do provide a first step to a full analysis of corrections to path integral localization formulas

(e.g.

corrections to the WKB

approximation),

and to

uncovering systematically the reasons why these approximations aren't exact for certain dynamical systems [157]. The generalizations of these ideas to path integrals are not yet known, but we discuss the situation somewhat

heuristically

Chapter.

at the end of this

7.1 Geometrical Characterizations

of the

Loop Expansion

Throughout

this

Chapter

the Hamiltonian H is manifold M. For manifolds with

a

return to the situation of Section 3.3 where

we

Morse function

now we assume

boundary.

We

that 49M

now

phase series whose construction we first expand the C'-function H in point p E MV in a Taylor series

H(x)

=

H(p)

+

on a =

0,

(usually compact) symplectic but later

we

shall also consider

explicitly work out the full stationary briefly outlined in Section 3.3 [72]. We a neighbourhood U. of a given critical

+ g(x; p) 7i(p),,,x1'x'/2 P P

x

E

U,,

(7.1)

where xp x p E Up are the fluctuation modes about the extrema of H and g(x;p) is the Gaussian deviation of H(x) in the neighbourhood Up (i.e. =

-

all terms in the the

symplectic

Taylor

series

beyond quadratic order). The determinant (3.52) is similarly expanded in Up

2-form which appears in

of as

7.1 Geometrical Characterizations of the 00

VI'det (x) w

V-det (p)

-

=

+

w

1

!

where

x

/-t 1

ii XP

k=1

...

XAk o91,1

...

P

Loop Expansion

c9/,, V

_det (x) w

IX=P

235

(7.2)

U,.

E

We substitute

(7.1)

and

(7.2)

into

(3.52), expand the exponential function

there in powers of the Gaussian deviation function, and then integrate over each of the neighbourhoods Up R 2n. In this way we arrive at a series -

expansion of (3.52) for large-T in terms of Gaussian moment integrals over the P P fluctuations x., with Gaussian weight e iTW(V),,x1'x'12 associated with each , X11k) open neighbourhood Up for p E MV. The Gaussian moments (xAl ...

be found from the Gaussian integration formula (1.2) in the usual way '9 '9 to both sides of (1 -2) and then setting by applying the operator ax" a'\Ik all the A's equal to 0. The odd-order moments vanish, since these integrands can

-

-

-

odd functions, and the 2k-th order moment contributes

are

0(11Tn+k) Rearranging

terms

.

carefully, taking

of the Hessian at each critical point, and noting that for will localize around each of the disjoint neighbourhoods

standard

stationary-phase expansion 21ri

Z(T)

T

)n

a

term of order

into account the

signature

large-T the integral U, we arrive at the

2

w

i)

A (P)

eiTH(p)

pEMv

At(p)

E (-2T)t

(7.3)

i=O

where

A (p)

1 =

,,/det'H (p) x

and

7i(x)11v

jE

=0

(-1)3 2ij! (f + j)!

(g(x;p)jV1detw(x))

is the matrix inverse of

'H(x)Av

N(P) AV19AC9J,)t+3*

(7.4)

Jx=P -

stationary-phase series diverges (e.g. applying Kirwan's theorem in appropriate instances), then (7.3) is to be understood formally as an asymptotic expansion order by order in 1. Borrowing terminology from quantum T field theory, we shall refer to the series (7.3) as the loop-expansion of this zero-dimensional quantum field theory, because each of the 2f + I terms in (i.e. a loop) as(7.4) can be understood from pairing fluctuation modes x"xv p P sociated with each derivative operator there. Indeed, the expansion (7.3),(7.4) is just the finite-dimensional version of the perturbation expansion (for largeT) in quantum field theory. We shall call the O(11T1+i) contribution to the If the

.

series

(7.3)

the

(t + 1)-loop

dynamical 2

See

(72]

for the

function.

term.

shall be interested in extracting information about the system under consideration from the loop-expansion with the hope

In this Section

we

generalization

of this formula to the

case

where H is

a

degenerate

236

of


7.

vanishing or non-vanishing of the k-loop congeometrical and topological features of the experience now with the Duistermaat-Heckman theo-

to understand the

being able

tributions for k > I in terms of

phase

space. Given

will

our

requirements on the flows of the Hamiltonian vector quite arbitrary for now. When these orbits describe tori, we already have a thorough understanding of the localization in terms of equivariant cohomology, and we shall therefore look at dynamical systems which do not necessarily obey this requirement. Thus any classification that we obtain below that is described solely by the vanishing of higher-loop contributions will for the most part be of a different geometrical nature than the situation that prevails in Duistermaat-Heckman localization. This then has the possibility of expanding the cohomological symmetries usually resposible rem,

we

any

remove

field and leave these

as

for localization.

The perturbative series before

we can

put

it to

(7.3), however, The

use.

must be

appropriately modified

(3.52) is invariant under (i.e. Z(T) is manifestly a be reflected order by order in the 1IT-

partition

function

arbitrary changes topological invariant), and this should expansion (7.3). This is explicitly observed in the lowest-order DuistermaatHeckman term Ao(p) above, but the higher-order terms (7.4) in the loop expansion are not manifestly scalar quantities under local diffeomorphisms of the coordinates. This is a result of having to pick local coordinates on M 2n At each order of the to carry out explicitly the Gaussian integrations in R coordinate should have a we independent manifestly quantity, '-expansion smooth

of local coordinates

M

on

.

i.e.

scalar. To write the contributions

a

manifestly a

invariant under local

Christoffel connection

r;, ,

(7-4)

in such

diffeomorphisms

of

a

M,

fashion

we

so

as

to be

have to introduce

of the tangent bundle of M which makes the

derivative operators appearing in (7.4) manifestly covariant objects, i.e. d + r. Because dH(p) write them in terms of covariant derivatives V =

we =

0

point p E Mv, the Hessian evaluated at a critical point is automatically covariant, Le. VVH(p) H(p). This process, which we shall call 'covariantization', will then ensure that each term (7.4) is manifestly a scalar. We note that the Morse index of any critical point is a topological at

a

critical

=

invariant in this

First,

we

sense.

cycle

out the

symplectic factors

V

Ae (p)

=

Ao (p)

1.y

1: 2ij!

+

i=O

j)!

in

(7.4)

to

get

(H(p)"'D,,EI,)'+i g(x; p)j

(7.5) X=P

where

Ao(p) is the Duistermaat-Heckman

=

(1-loop) D

=

(T __det(p) jet

contribution to

d+-y

(7.6)

(p)

(7.3),

and

(7.7)

7.1 Geometrical Characterizations of the

where

we

Loop Expansion

237

have introduced the one-component connection

hL'dhL

-y

(7.8)

and

-,Idetw(x)

hL(-'r) is the Liouville volume

(7.9)

density. The derivative operator D transforms like diffeomorphisms x --+ x'(x) of M,

an

abelian gauge connection under local A

Dy W

)

I DI,(x)

=

A'(x) [E),(x') A

+ tr

A-l(x')o9' A(x')]

(7.10)

where

A-1(x)

=

I

ax"

I

GL(2n, R)

E

(7.11)

is the induced

change of basis transformation on the tangent bundle. 1. We expand out the example, consider the expression (7.5) for t 3 terms there in higher-order derivatives of H and the connection -/, noting that only third- and higher-order derivatives of g(x; p) when evaluated at x p are non-vanishing. After some algebra, we arrive at For

=

=

A 1 (P)

Ao(p) 2

li(p) 'v

D,(x)7,(x)

+4-y,,(x)a,\a,apH(x)

+

-

12

4

o9,, a,\ o9, o9p H (x)

[3a,,a,\,9,,H(x)o9pao9,oH(x)

(7.12)

+2a,,i9,\49,,H(x),O,,api9,oH(x)]) I Jx=P readily checked, after some algebra, that this expression is indeed indiffeomorphisms of M. To manifestly covariantize it, we introduce an arbitrary torsion-free connection r ',\, of the tangent bundle TM.

It is

variant under local

For now, we need not assume that r is the Levi-Civita connection associated a Riemannian metric on M. Indeed, as the original dynamical problem

with

is defined

only

symplectic geometry, not a Riemannian geommanifestly covariant in its own right without reference to any geometry that is external to the problem. All that is required is some connection that specifies parallel transport along the fibers in terms of

etry, the expression

(7.12)

a

should be

of the tangent bundle and allows

us

to extend derivatives of

quantities

to

an

neighbourhood, rather than just at a point, in a covariant; way. The Hessian of H can be written in terms of this connection and the

entire

associated covariant derivative

7i(x),,, and, using d r, appearing in (7.12) in

=

as

V,,V,,H(x)

we can

+

rA (x)49,\H(x) Mv

(7.13)

write the third and fourth order derivatives

terms of V and r

by taking derivatives

of

(7.13).

238

7.


Substituting

these

complicated derivative expressions into (7.12) and using 0 on MV, after a long and quite tedious calculation we arrive at a manifestly covariant and coordinate-independent expression for the 2-loop correction, the fact that dH

Aj(p)

=

H(P)"PH(PY113

7AW,

8

3

[3V,,V,V,\H(x)V,,VoVpH(x)

+2V,,V,\V,,H(x)V,VpV,3H(x)] +4

(VI, + A,,

-

-

H(p)APVpV,,V,V,\H(x)

X=P

H(p),\PVpV,,V,\H(x)) A, (x) + R,(r) I

(7-14) where

R,,, (r)

= -

R\gAv

-

aV r,\, /4"

-

a,\r,\ ttv

+

r, r,',A rc,A rc,, ,lov

(symmetric) Ricci curvature tensor of F and =A, (x)dxA with the local components

is the

we

(7-15)

have introduced the

1-form A

,6,, (x) It is

intriguing that the

=

-

r,, ,\ (x)

=

V,, log hL (X)

covariantization of the

replacing ordinary

volves

y,, (x)

(7.16)

2-loop expression simply inones V, non-covariant

derivatives d with covariant

connection terms -y with the 1-form A, and then the remainder terms from this process are simply determined by the curvature of the Christoffel connection 1' which realizes the covariantization. Note that if r is in addition chosen

as

the Levi-Civita connection compatible with a metric g, i.e. 0,, log vqe-tg and the 1-form components (7.16) become

Vg

=

0, then P\ JLA

A ,

=

a,, log Vdet(g-1 w).

=

-

The covariantization of the

higher-loop terms is hopelessly complicated. if, however, the 2-loop, correction Al (x) vanishes in an entire neighbourhood Up C M of each critical point p, then this is enough to imply the vanishing of the corrections to the Duistermaat-Heckman formula to all orders in the loop-expansion. To see this, we exploit the topological invariance of (7.14) and apply the Morse lemma [111] to the correction terms (7.5). This says that there exists a sufficiently small neighbourhood Up about each critical point p in which the Morse function H looks like a "harmonic oscillator", We note that

H(x) so

=

H(p) _(XI)2 _(X2)2_..._(X,\(p))2+(X,X(p)+1)2+...+(X2n)2

,

Up (7.17)

x

E

that the critical point p is at x 0 in this open set in M We shall call these coordinates', and this result simply means that the symmetric ma=

.

'harmonic trix

H(x)

critical nates

can be diagonalized constantly in point. Given that the quantity (7.5)

(although

not

manifestly),

we can

an

neighbourhood of the independent of coordi-

entire

must be

evaluate it in

a

harmonic coordinate


system. Then the Gaussian deviation function the the

neighbourhood Up and only series (7.3) is simply 2-7ri

Z(T)

T

)n

E (-i))(P)

the j

e

=

iTH (P)

g (x;

p)

Loop Expansion vanishes

identically

Ao (p)

(

in

(7.5).

Thus

neighbourhood

of the

0 term contributes to

PEMv

239

e-

(7.18) It follows that if the

2-loop term vanishes in the (and not just at x p), i.e.

critical point p

=

'H(p)P'D,,(x)D,(x) then,

as

all

entire

-=

0

for

x

E

Up

(7.19)

higher-loop terms in these coordinates can be written as derivative on the 2-loop contribution A, (x) as prescribed by (7.18), all

operators acting

corrections to the semi-classical

approximation vanish. In the

case

of the

Duistermaat-Heckman theorem, it is for this reason that the Lie derivative condition Cvg 0 is generally required to hold globally on M. In general, =

not the vanishing of A, (p) implies the vanishing of all loop clear, because there is a large ambiguity in the structure of a function in a neighbourhood of MV which vanishes at each critical point p (any functional of VH will do). The above vanishing property in an entire neighbourhood therefore need not be true. It is hard to imagine though that the vanishing of the 2-loop correction term would not imply the vanishing of all higher-orders, because then there would be an infinite set of conditions that a dynamical system would have to obey in order for its partition function to be WKB-exact. This would then greatly limit the possibilities for localization. In any case, from the point of view of localization, we can consider the vanishing of the 2-loop contribution in (7.14) as an infinitesimal Duistermaat-

though,

whether

or

orders is not that

Heckman localization of the partition function. The expression (7.14) in genextremely complicated. However, besides being manifestly indepen-

eral is

dent of the choice of

coordinates, (7.14) is independent of the chosen connecV, because by construction it simply reduces to the original connectionindependent term (7.12). We can exploit this degree of freedom by choosing a connection that simplifies the correction (7.14) to a form that is amenable to explicit analysis. We shall now describe 2 geometrical characteristics of the loop-expansion above which can be used to classify the localizable or non-localizable dynamical systems 1157]. The first such general geometric localization symmetry is a symmetry implied by (7.14) between Hessian and metric tensors, i.e. that the Hessian essentially defines a metric on M. This is evident in the correction term tion

(7.14), form

where the inverse Hessians contract with the other tensorial terms to

scalars,

i.e. the Hessians in that expression act just like metrics. This suggests that the non-degenerate Hessian of H could be used to define a

metric which is

compatible with the

connection V used in

(7.14).

This in

7.

240

Beyond

the Semi-Classical

Approximation

globally on the manifold M, because the signature of 'H(x) varies over M in general, but for a C' Hamiltonian H it can at least be done locally in a sufficiently small neighbourhood surrounding each critical point. For now, we concentrate on the case of a 2-dimensional phase space. We define a Riemannian metric tensor g that is proportional to the covariant Hessian in the neighbourhood Up of each critical point p, general

cannot be done

where

!9(x)

is

some

globally-defined

(7.20)

=!Pg

VVH

C'-function

on

M, and for which the

connection F used in the covariant derivatives V is the Levi-Civita connection

for g. This

means

coupled

the

that, given

!g(x)g,,,(x) rA

MV

consistently

a

Hamiltonian H

system of non-linear

for g

2

=

partial

0,,OH(x)

gAP Oitgpv

+

M,

on

we

try

to

locally

solve

differential equations -

r\ (x)a,\H(x) MV

CUP/I

-

(7.21)

19p9tiv)

(or F).

This sort of "metric ansatz" may seem somewhat peculiar, and indeed impossible to solve in the general case. The covariant constancy condition on g in

(7.21) implies

that

')/,!g

c

where R

identity

=

gII`R,_,,

=

R,\a,\H M

=

is the scalar curvature of g.

for the Riemann curvature tensor R

V,,V,,V,\H

=

(7.22)

Ro9lH

V,V/-,V,\H

(7.22) =

follows from the

dF + F A

+ RP

A/m/

defining

-P,

VP H

(7.23)

H, (7.22) determines g locally in terms of g. This means that the can be written as an equation for the associated connection coefficients F\

Given

above ansatz

/4V

V,\V,,V,H The existence of

a

local solution to

=

Rl,,V,\H

(7.24)

'almost' all of the time

(7.24) can now

set of differential

equations in local isothermal coordinates (5.49) for the connection.P above [157]. Notice, in particular, that if the metric (7-20) actually has a constant curvature R, then the equation (7.22) can be integrated in each neighbourhood U., to give be

argued by analysing this

!P(x)

=

Co

+ R

-

H(x)

(7.25)

examples and other evidences for this sort of geometric structure throughout this Chapter. The main advantage of using the inductively-defined metric in (7.21) is that it allows a relatively straightforward analysis of the 2-loop, correction We shall

see


Loop Expansion

terms to the Duistermaat-Heckman formula. With this

plies that and hence

definition, (7.22) impoint p E Mv,

the first order derivatives of 9 vanish at each critical so

do all third order covariant derivatives of H in

(7.14).

The fourth

order covariant derivatives contribute curvature terms

to

which

in

are

241

then cancelled

The final result is

by

according already present

the curvature tensor

(7.24), (7.14).

expression involving only the Liouville and Levi-Civita expressed in terms of the 1-form. A, which after some algebra we be written in the simple form an

connections

find

can

Aj(p) Requiring

9" IVA

=

29

+

AA W I A, W

this correction term to vanish in

an

entire

JX=P

(7.26)

neighbourhood

of each

critical point implies, from the definition (7.16), that the connection y ciated with the symplectic structure coincides with the connection r

asso-

ciated with the Riemannian structure which solves the relation

Thus

(7.21).

asso-

the components of the symplectic 2-form w and the metric tensor g are proportional to each other in local complex isothermal coordinates for g. The

proportionality

factor must be

a

constant

so

existence of local Darboux coordinates for

that this be consistent with the

[157].

In other

words, w and g together locally phase space M. Conversely, suppose that r is the Levi-Civita connection associated to some generic, globally-defined metric tensor g on M, and consider the rank define

(1,1)

a

Khhler structure

w

on

the

tensor field

J1,

dettwgg"Aw' et w

:::

A

(7.27)

(7.27) defines a linear isomorphism J general, if such a linear transformation J exists then it is called an almost complex structure of the manifold M [41, 61]. This means that there is a local basis of tangent vectors in which the only non-vanishing components of J are given by (5.9), so that there is "almost" a separation of the tangent bundle into holomorphic and anti-holomorphic components. However, an almost complex -structure does not necessarily lead to a complex structure there are certain sufficiency requirements to be met before J can be used to define local complex coordinates in which the overlap transition functions can be taken to be holomorphic [22]. One such case is when J is covariantly constant, VJ 0 actually this condition only ensures that a sub-collection of subsets of the differentiable structure determine a local complex structure (but recall that any Riemann surface is a complex manifold). Again in 2-dimensions this means that then VW 0 In

2-dimensions, it is easily --+ TM satisfying j2

TM

seen

=

that

-1. In

-

=

-

=

and the pair (g, w) define a Kdhler structure on M (again note that any 2dimensional symplectic manifold is automatically a Khhler manifold for some metric defined

Given these

by w). facts,

suppose

now

that g and

w

the 2n-dimensional manifold M with respect to

define an

a

Kdhler structure

almost

complex

on

structure

242

J,

Beyond

7.

i.e. det w

det g, g is Hermitian with respect to

=

9tiv and

w

means

Approximation

the Semi-Classical

=

JA 9AP JP A

(7.28)

1

V

by (7.27).

is determined from 9

J,

In the local coordinates

(5.9),

and w,,F, 0, gir, -ig,,p. g*,, gpr, F/ action of the Hamiltonian vector field =

=

(,Cvg),,,,

=

=

this

we encountered before, i.e. g,', In this case, the flows of g under the

the usual Kdhler conditions that

=

V,

gj,,xV,,w-XPo9pH + g,,.xVtwAPapH + wAP(gj,.XV,,VpH + g,,.XVt'VpH) (7.29)

can

be written

using the almost complex

,Cvg Thus if V is

a

global Killing

=

structure

as

the anti-commutator

[VVH, J]+

vector of

a

(7.30)

Kiffiler metric

covariant Hessian of H is also Hermitian with respect to

(2,0)

metric

by

on as

M, then the

in

(7.28).

Since

essentially

the unique Hermitian tensor, it follows that the covariant Hessian is related to the Kiffiler a transformation of the form

the Kiffiler metric of rank

K6hler manifold is

J,

a

VI,V,H

=

KP K\g,\ P it

(7.31)

V

non-singular (1,I) tensor which commutes with J. In 2Hermiticity conditions imply that both the Hessian and g dimensions, have only 1 degree of freedom and (7.31) gets replaced by the much simpler condition (7.21). From the fundamental equivariant localization principle we know that this implies the vanishing of the 2-loop correction term, i.e. the Duistermaat-Heckman theorem. Indeed, from the analysis of the last 2 Chapters we have seen that most of the localizable examples fall into these Kiffiler-type, scenarios. Notice that the covariant Hessian determined from the Hamiltonian equations is

where K is

some

the

V,hV,,H so

=

VAVAwv,\

that in the Kdhler case, when Vw

=

+

0, the proportionality function 9 is

determined in terms of the Riemann moment map AV

9(x) On

a

=

(7.32)

wv,\Vl-IV,

Vd_et;1_tv(x)

=

VV

as

(7.33)

homogeneous Kdhler manifold, when g is integrated to be (7.25), this generalizes that observed for the height function of the sphere in

relation

Section 5.5.

particularly interesting here is that conversely the localization partition function determines this sort of Kdhler structure on M. The Lie derivative (7.29) of the metric (7.21) is easily seen to be zero in a neighbourhood of the critical point. Conversely, if the Lie derivative of the What is

of the


Loop Expansion

243

metric in (7-21) vanishes on M, then it induces a Khhler structure (i.e. VW 0). Recalling from Section 3.3 the proof of the Duistermaat-Heckman theorem =

using solely symplectic geometry arguments, we see that the main feature of possibility of simultaneously choosing harmonic and Darboux coordinates. This same feature occurs similarly above, when we map onto local Darboux coordinates [157]. The new insight gained here is the geometric manner in which this occurs the vanishing of the loop expansion beyond leading order gives the dynamical system a local Kdhler structure (see (7.19)). Whether or not this extends to a global Kdhler geometry depends on many things. If the topology of M allows this to be globally extended away from MV, then the Riemannian geometry so introduced induces a global Kiffiler structure with respect to the canonical symplectic structure of M 3. Furthermore, the coefficient function 9(x) in (7.20) must be so that the the localization is the

-

metric defined

by that equation

has

signature on the whole of M. impossible to choose the function g in (7.20) such that, say, g has a uniform Euclidean signature on the whole of M. But if the correction terms above vanish, then Kirwan's theorem implies that H has only even Morse indices and it may be possible to extend this geometry globally. In this way, the examination of the vanishing of the loop expansion beyond leading order gives insights into some novel geometrical structures on the phase space representing symmetries of the localization. Moreover, if such a metric is globally-defined on M, then Lvg 0, and If H has odd Morse indices

A(p),

a

constant

then it is

=

these classes of localizable systems fall into the studied before.

same

framework

as

those

we

The second general geometric symmetry of the loop expansion that we wish to point out here is based on the observation that the symplectic connection (7.8) is reminescent of the connection that appears when one constructs the L

Fubini-Study

-*

M

bundle of this

[41].

over

If

metric

we

using the geometry of

choose such

a

line bundle

M and view the Liouville

bundle,

then from it

one can

as

a

density (7.9)

construct

a

where

we

-i(o9 + 5)-yg

have restricted to the

nents of the connection

is the

=

Fubini-Study

The existence of

(7.8).

=

as a

metric in the fibers

Kdhler structure

the curvature 2-forms of the associated connections Q

holomorphic line bundle symplectic line

the standard

(7.8),

on

-05 log hL

(7.34)

holomorphic and anti-holomorphic compoinstance, the symplectic structure (5.208)

For

metric associated with the natural line bundle L

an

almost

M from

i.e.

complex

structure J for which the

-->

CpN

symplectic 2-form

is Hermitian and for which the associated Khhler metric J w is positiveg definite is not really an issue for a symplectic: manifold [171]. Such a J always w

exists

=

(and

is

unique

up to

homotopy)

-

because the Siegal upper-half plane is

contractible. Thus the existence of a Khhler structure for which A is not

a

problem. However, for the Killing equation for

g

=

J

-

W

must be invariant under the flows of the Hamiltonian vector field

=

to

V,

V log hL 0 hold, J itself =

i.e.

Cv J

=

0.

244

which is a


7.

a

point p

sub-bundle of the trivial bundle (CpN

E

CpN

is

just the

set of

(zl,

points

.

.

.

(CN+l, i.e. the fiber of L over ZN+1) ECN+1 which belong

X ,

to the line p. In that case, the natural fiber metric induced

ical

com p lex

Euclidean metric of

(CN+l

is

h(p; zl,..., ZN+I)

by the =

canon-

EN+1 'U=1 JZttJ2

symplectic reduction of h as in (7.34). This holomorphic line bundle once a fiber metric has been chosen. In the general case, if M itself is already a Kdhler manifold, then whether or not (7.34) agrees with the original Kiffiler 2-form will depend on where these 2-forms sit in the DeRham cohomology of M. If we further adjust the Christoffel connection.P so that it is related to the Liouville connection by the boundary condition FA i.e. A 0, then the correction term (7.14) will involve only derivatives of the Hamiltonian, but now the vanishing of the correction term can be related to the geometry of a and

(5.208)

is determined

generalizes

construction

the

as

to any

=

line bundle L

M.

--+

a general geometric interpretation of loop expansion. The above discussion shows what deep structures could be uncovered from further development of such analyses. It is in this way that the localization of the partition function probes the geometry and topology of the phase space M and thus can lead to interesting topological quantum field theories.

This is

far

as

as we

shall go with

localization from the covariant

7.2 Conformal and Geodetic Localization

Symmetries

geometric symmetries which a dynamical system and a which extend the fundamental symmetry requirement of the localization theIn this Section

orems we

tensor

we

shall examine

some

alternative

localization of the partition function of

lead to

on

on [84, 134]. Let g be a globally-defined metric and consider its flows under the Hamiltonian vector field V.

encountered earlier

M,

Instead of the usual assumption that V be an infinitesimal isometry generator for g, we weaken this requirement and assume that instead V is globally an

infinitesimal generator of

conformal transformations with respect LVg

where

Y(x)

is

some

Cl-function

=

on

to g, i.e.

(7.35)

Yg

M.

Intuitively,

this

means

that the

diffbomorphisms generated by V preserve angles in the space, but not lengths. The function Y can be explicitly determined by contracting both sides of

(7.35)

with

g-'

to

get T

which This

we

note vanishes

implies,

which

case

in

V is

=

on

particular, an

V,,Vl'/n

=

V,,w1",9,H1n

(7.36)

the critical point set MV of the Hamiltonian H. that either Y =- 0 almost everywhere on M (in

isometry of g)

or

Y(x)

is

a

non-constant function

on

M


Symmetries

245

corresponding to non-homothetic or 'special conformal' transformations (i.e. constant rescalings, or dilatations, of g are not possible under the flow of a Hamiltonian vector field). Ordinary Killing vector fields in this context arise as those which are covariantly divergence-free, tr /,tv VtVA 0. We shall show that this conformal symmetry requirement on the dynamical system also leads to the Duistermaat-Heckman integration formula. First, we establish this at the level of the perturbative loop expansion of the last Section by showing that the (infinitesimal) corrections to the I-loop term in the stationary phase series vanish. For simplicity we restrict our attention here to the case of n I degree of freedom. The extension to arbitrary freedom of is degrees immediate, and indeed we shall shortly see how the condition (7.35) explicitly implies the vanishing of the correction terms to the Duistermaat-Heckman formula in arbitrary dimensions and in a much more general framework. We insert everywhere into the covariant connectionindependent expression (7.14) the covariant derivative V associated with g in (7.35). Using the covariant Hessian (7.32) and the conformal Killing equation =

=

=

(7.35)

we can

field. Notice

solve for the covariant derivatives of the Hamiltonian vector

that,

in contrast to the

ordinary Killing equations,

one

of the 3

components of the conformal Killing equation (7.35) will be an identity since one of the Killing equations tells us that V is covariantly divergence-free with

respect

to g

(symmetric)

(see (5.51)). In 2-dimensions, after Hessian (7.32) can be written as

VI,V,H where

we

=

-S? Z gA, +

+

algebra we

V,H V/, log

find that the

0) /2

(7.37)

have introduced the Cl-functions

Z(X)

det

VV(x)

-

(VxVA (x)/2)2

The covariant derivatives

(7.37)

(Vj_tH V, log S?

some

appearing

0

in

(7.14)

Ldet W(X! rEet

are now

et 6 e

g (x)

(7.38)

easily found

from

to be

VpV/,V,H

J?Z +

Vp(f2Z)V,,V,H

+

1V,, log 12VpV,H

2

1V, log f2VpV,,H +.F(VH) 2

(7.39)

246

Beyond

7.

the Semi-Classical

Approximation

V,xV,,,V,,V,H I

f2Z

(V,\V,,(S?Z)V,,V,H

1V,\(ZV,S?)VpV,,H

-

+

2

2

V,\(ZVAS?)VpV,H

IV,\VpHV/-, log S?V, log Q

(7.40)

2

1V,\V,H (21V1, log f?VP log S2 + vpvt, log Q

+

2

IV,\V/,H (2IV, log QVp log S? + VpV, log S?

+

2

F(VH)

where

I +

+.F(VH)

involving single derivatives

denotes terms

H, and which contracting (7.23) with H(x)AA leads to a relationship between the symplectic structure and the flows generated by the Hamiltonian vector field,

therefore vanish

on

I(H). Substituting (7.39)

V,, log Q

+

2V,, log Z

into

(7.23)

of

and then

F(VH)

=

(7.41)

now use a covariant derivative of (7-23) to rewrite the curvature term (7.14) in terms of covariant derivatives of the Hamiltonian, substitute (7.39) and (7.40) into (7.14), and use everywhere the identity (7.41). Since , Aj, V,, log Q for the metric connection.P, it follows after some algebra that the generally-covariant expression (7.14) for the first-order correction to the Duistermaat-Heckman formula is F(VH). This establishes the claim above.

We in

=

We shall

now

symmetry is in a

establish quite generally that this conformal localization the most general one that can be construed for

some sense

classical partition function [134]. For this, we return to the general localprinciple of Section 2.5 and re-evaluate the function Z(S) in (2.128)

ization

generic (not necessarily equivariant) metric-dual 1-form,3 ivg of the (i.e. with no assumptions for now about the symmetries of g) and the equivariantly-closed form a e iT(H+w)1(iT)n with 0, but LV,8 =h 0 in general. We also assume, for full respect to V, i.e. Dva generality, that the manifold M can have a boundary 9M. The derivative for

a

==

Hamiltonian vector field V

=

=

(2.129) using

need not vanish

now

and the second

Stokes' theorem and the Cartan-Weil

equality there can be evaluated identity. Then using the identity

00

Z(O)

=

lim S

00

Z(S)

-

j

ds

d ds

Z(S)

(7.42)

0

it follows that the

partition function Z(T)

=

fm

a

can

be determined in

general by 00

Z (T)

=

lim 5

00

f M

a

e- sDv,3

+

f 0

ds

tMi

a,3 e-,Dv,3

_

s

f aO(Lvo) e-sDv)3 M

(7.43)


The

large-s

limit

integral

(7.43)

in

did in Section 2.6 to arrive at the

S?V

the 2-form

=

d,3 there

be worked out in the

=

2g

-

same

way

247

as we

expression (2.142), except that

same

given quite arbitarily

is

Qv so

can

Symmetries

pv

-

now

as

(7.44)

Lvg

that

lim S

00

1

a

e- DvO

=

a

(-2,,)n/2 x

Pfaff

(dV(p)

(P)

det dV (p)

PEMv

M

(0)

-

(7.45)

g(p)-1Lvg(p)12)

Using (3.59) to rewrite derivatives of V in (7.45) in the usual way, substituting in the definitions in Section 2.5 for the geometric quantities in (7.43), and then expanding exponentials in the 2-forms w and S?V to the highest degrees that the manifold integrations pick up, we arrive at an expression for the partition function in terms of geometrical characteristics of the phase space

Z(T)

[134]

(T )n 21ri

=

(P) pEMv

6det (p)eiTH(p) jet

(p)

x,\/-det (1 -H-1wg-1LVg12) (p) 00

+

J TZT__)n I ds

0

0

(n

-

g(V,

A

(iTw

-g(V, 1)!

A

(Lvg)(V, -)

1)!

-

s0v)

n-1

am

00

(iT)n J

eiTH-sKV

ds

s

I

eiTH-sKV

(n

-

A

(iTw

-

sQV)

1

M

(7.46) arbitrary Riemannian metric g on M. From the expression (7.46) and the fact that g(V,.)A2 =- 0, we can see explicitly now how the conformal Lie derivative condition (7.35) collapses this expression down to the Duistermaat-Heckman formula (3.63) when aM 0, as we saw by more explicit means above (This conformal localization property has also been explained within a more general BRST approach recently in [83]). Note that we cannot naively carry out the s-integrations quite yet, because the function KV g(V, V) has zeroes on M. The expression (7.46), although quite complicated, shows explicitly how the Lie derivative conditions make the semi-classical approximation to the partition function exact. This is in contrast to the loop-expansion we studied earlier, where the corrections to which holds for

an

=

=

the Duistermaat-Heckman formula

derivatives.

(7.46)

were

therefore represents

expansion that explicitly takes into

a

not

just

some

combinations of Lie

sort of resummation of the

account the

loopgeometric symmetries that

248

7.


1-loop approximation exact. We shall see soon that it is quite conpredicted from the loop-expansion, and moreover that it gives many new insights. There are several points to make at this stage. First of all we note the appearence of the boundary contribution in (7.43). If we assume that LV,8 0, that the group action represented by the flows of the Hamiltonian vector field V preserves the boundary of M (Le. g aM W) and that the action is free on o9M, then the s-integral in the boundary term in (7.43) can be carried out explicitly and we find the extra contribution to the DuistermaatHeckman formula for manifolds with boundary, make the

sistent with the results

=

=

-

3 A

Zam(T)

a

(7.47)

DV,8 am

In this context 3 action

on

the

=

ivg is the

boundary aM,

ment map for this action. This

Jeffrey-Kirwan-Kalkman

connection 1-form for the induced group as we have seen earlier dp is the mo-

because

boundary

residue that

term

was

can

be determined

using the 3.8, i.e. the

introduced in Section

coefficient of in the quantity (,0 A a)IDV,3, where 0 is the element of the symmetric algebra S(g*) representing the given circle action [82]. Secondly, notice that the conformal localization symmetry gives an explicit realization in (7.37) of the Hessian-metric ansatz which was discussed in the last Section. In particular, (7.46) establishes that with this Hessian-metric substitution and the appropriate extension away from the critical points of H (the terms proportional to VH in (7-37)) the corrections to the DuistermaatHeckman integration formula vanish to all orders of the loop-expansion (and not just to 2-loop order as was established in the last Section). Notice, how-

(7.36) any conformal Killing vector on a Kdhler manifold automatically an isometry. In fact, the generic case of a non-vanishing scaling function Y(x) in (7.35) is similar to the isometry case from the point of view of the equivariant localization priniciple. Note that away from the critical points of H we can rescale 0 =- g(V, -) lg(V, V). With this choice for the localization 1-form 0 it is easy to show from (7.35) that away from the critical point set of the Hamiltonian it satisfies CV,6 0, i.e. '3 is then an equivariant differential form on M MV. Thus away from the subset MV C M the conformal Killing condition can be cast into the same equivariant cohomological framework as the isometry condition by a rescaling of ever, that because of

is

--+

=

-

the metric tensor in

(7.35),

The rescaled metric

G,,,(x)

gttv is

--->

Gl,,,

=

only defined

g1,,1g(V, V), on

M

-

for which

MV, but

we

LvG

=

0.

recall from

Section 2.5 that all that is needed to establish the localization of the

zeroes

of the vector field V

(i.e.

the equivariant localization

(3.52) onto principle) is

(or equivalently an equivariant differential form everywhere on M except possibly in an arbitrarily small neighbourhood of MV. The fact that the weaker conformal symmetry condition is equivalent to the isometry condition in this respect is essentially a an

,3)

invariant metric tensor

which is defined


Symmetries

249

consequence of the fact that the differential form 3 ivg above is still a connection 1-form that specifies a splitting of the tangent bundle into a com=

MV (represented by the discrete sum over MV in (7.46)) and a to MV (represented by the Lie derivative integral in (7.46)). This is in fact implicit in the proof by Atiyah and Bott in [9] using

ponent

over

component orthogonal the Weil

algebra.

One may ask as to the possibilities of using other localization forms to carry out the localization onto MV, but it is readily seen that, up to components orthogonal to V, 3 ivg is the most general localization form up to multiplication by some strictly positive function. This follows from the fact =

that in order to obtain finite results in the limit

s

oo

-*

(7.45)

in

we

need to

that the form

Dv,3 has a 0-form component to produce an exponential damping factor, since higher-degree forms will contribute only polynomially in the Taylor expansion of the exponential. This is guaranteed only if '3 has a 1-form component. Thus it is only the 1-form part of 3 that is relevant to the localization formula, and so without loss of generality the most general localization principle follows from choosing,3 to be a 1-form. Furthermore, the 0-form part Vi',3,, of DV,3 must attain its global minimum at zero so that the large-s limit in (7.45) yields a non-zero result. This boundedness requirement is equivalent to the condition that the component of 3 along V has the same orientation as V, i.e. that,3 be proportional to the metric dual 1-form of V with respect to a globally-defined Euclidean signature Riemannian structure on the phase space M. In addition, for a compact phase space M the conformal group is in general non-compact, so that conformal Killing vectors need not automatically generate circle actions in these cases as opposed to isometry generators where this would be immediate. To explore whether this larger conformal group symmetry of the Duistermaat-Heckman localization leads to globally different sorts of dynamical systems, one would like to construct examples of systems with non-trivial (i.e. T =h 0) conformal symmetry. For this, one has ensure

to look at spaces which have

nian vector field V is

a

a

Riemannian metric 9 for which the Hamilto-

generator of both the conformal group Conf (M, g)

and the the

symplectomorphism group Sp(M, w) of canonical transformations on phase space. From the analysis thus far we have a relatively good idea

of what the latter group looks like. The conformal group for certain Riemannian manifolds is also well-understood [54]. For instance, the conformal group of flat Euclidean space of dimension d > 3 is

SO(d + 1, 1). was

already

The

locally isomorphic

to

(global)

conformal group of the Riemann sphere C U tool encountered in Chapter 6 above (in a different context), namely

the group SL(2, (C)/Z2 f:-- SO(3, 1) of projective conformal transformations. In these cases, the conformal group consists of the usual isometries of the space,

in

z

=

along r

e'0)

with dilatations

or

scale transformations

and the d-dimensional

transformations.

subgroup

(e.g.

of so-called

translations of

special

r

conformal

250

7.


An interesting

example is provided by the flat complex plane C. Here algebra is infinite-dimensional and its Lie algebra is just the classical Virasoro algebra [54]. Indeed, the conformal Killing equations in this case are just the first set of Cauchy-Riemann equations in (5.51) (the other one represents the divergence-free condition tr Av 0). This means that the conformal Killing vectors in this case are the holomorphic functions V' The Hamiltonian flows of these vector fields are f (z), V2 therefore the arbitrary analytic coordinate transformations the conformal

=

=

=

40 As a

an

=

f WO)

ZW

I

=

A2(0)

(7.48)

explicit example, consider the conformal Killing vector which describes n + I distinct stationary points,

Hamiltonian system with

V' at

z

=

(7.35)

0 and

z

=

=

i,3z(l

I /ai, where

-

a1z)

3, ai

E C

-

z)

an

(7.49)

The associated

-

scaling

function in

is then

T(Z'; ) The

(1

...

a'Vz

symplecticity condition LVw

a2VO

+

(7.50)

0 leads to the first-order linear

partial

differential equation

VZ09ZWZ2

+

V29 WZ2

-T(Z" )WZ2

=

(7.51)

easily solved by separation of the variables z,; . The solution symplectic 2-form with arbitrary separation parameter A E R is

which is

=

Z2

W'\(Z)iv-'\(' )/Vzv

for the

(7.52)

where

w), (z)

iA

=

e'

f dz/V'

.

((1

-

z

a,Z)AI

a,,, Z)

...

A,

(7.53)

and the constants

Ai(ai,...' an)

==

(ai )n-1 i0i

are

the coefficients of the

partial

(VZ )_ 1=

1

!)_3

aj

(7.54)

decomposition n

1+ E

_z

single-valued phase, so that Ai (a,,

-

Ai 1

-

aiz

(7.55)

C, we restrict the ak'S same On) E R, and the parameter 3 be real-valued. The Hamiltonian equations (5.68) can now be integrated To

ensure

that

to all have the to

(7.52)

fraction

ai

is

a

function

.

-

-

,

on


up with the vector field we

find the

family

(7.49)

and the

sYmplectic 2-form, (7.52),

( (I

1

-#'ai (Z"

from which

/P

Z -

a,Z)AI

-

anZ)An

...

(7.56)

An)

X

that this Hamiltonian has

ensure

251

of Hamiltonians

W(IN)

To

Symmetries

only non-degenerate

critical points

we

3. This also guarantees that the level (constant energy) curves of this Hamiltonian coincide with the curves which are the solutions of the equations set A

=

of motion

(7.48) [134].

Since the Hamiltonian

(7.56)

either vanishes

or

is infinite

on

its critical

it is easy to show that the partition function (3.52) is independent of ak and coincides with the anticipated result from the Duistermaat-Heckman

point set,

integration formula, namely Z(T) 2-7ri,3/T. This partition function coincides with that of the simple harmonic oscillator z. 1,3, as expected since for =

H,,( ,)

the harmonic oscillator 0, A) becomes the Darboux 2-form and '0 Z Hamiltonian. In fact, we can integrate up the flow equation (7.48) in the ai

=

general case and by the equation

e'0('0)

we

=

find that the classical

w,3(Z(0)

=

(1

-

trajectories z(t)

Z(t) a,Z(t))Al (1 -

-

-

-

are

determined

(7.57)

anz(t))An

change z --+ wo(z) is just the finite conformal transforma, generated by the vector field (7.49) and it maps the dynamical system

The coordinate tion

(wlqT, Ho( ),) 'C'

onto the harmonic oscillator H

circular classical trajectories

w(t)

=

e'O('-'O)

oc

w Fv,

W OC

WD with the usual

associated with

a

U(1)

gener-

isometry. This transformation is in general multicorresponding valued and has singularities at the critical points z 1/ai of the Hamiltonian ator

to

an

=

27(o)

ho,a..

It is therefore not

a

diffeomorphism,

(R W;(02), H(3) ) 2

Hamiltonian system , oscillator. For

of the

plane

for ai

7-

0 and the

globally isomorphic to the simglobal differences between these systems and those associated with isometry generators, see [134]. The conformal group structures on phase spaces like S2 yield novel generalizations of the localizable systems which are associated with coadjoint orbits of isometry groups as we discussed in Chapter 5. The appearence of the larger (non-compact) conformal group may lead to interesting new structures in other instances which usually employ the full isometry group of the phase ple harmonic

space, such

as

more

is not

details about the

the Witten localization formalism of Section 3.8.

Other geometric alternatives to the Lie derivative condition LVg 0 have also been discussed by Mirki and Niemi in [84]. For instance, consider the =

alternative condition

252


7.

V"V.Xv/A

=

(7.58)

0

to the

VA are Killing equation, which means that the Hamiltonian flows _V, geodetic to g (see Section 2.4). From (7.58) it follows that the Lie derivative =

of the localization

LV,3

=

1-form,3

VP(gpXVIV\

+

=

ivg

can

be written

g1,,\VpV\)dx1'

=

as

VPgp,\V,,V-Xdxl'

div Comparing this with the Cartan-Weil identity LV derivative acting on differential forms we find the relation =

ivd,3

=

-

(7.60)

[841 (7.61)

0

=

dynamical systems (I2 KV, OV)

Hamiltonian structure. Moreover, in this

(7.59)

iVd for the Lie

+

1div,3

Dv (Kv /2 + f2v) that the

divo

2

which leads to the equivariance condition

so

2

and

(H, w)

it is also

case

determine

possible

to

a

bi-

explicitly

solve the equivariant Poincar6 lemma [84], just as we did in Section 3.6. Thus + OV is an equivariantly-closed differential form, the condigiven that -!KV 2 tion (7.58) has the potential of leading to possibly new localization formulas. However, there are 2 things to note about the geometric condition (7.58). The first is its connection with a non-trivial conformal Killing equation LVg Tg, which follows from the identity

VV 9 aA(Lvg)A,,

Contracting both sides

of

(7.62) YKv

(7.35)

=

V\V,\V'

with

==

+

g,pVP

(7.62)

g'AVIKvl2

leads to

2g,,,,V'V,\VXVA (7.58)

(7.63)

satisfied, then T -= 0 away everywhere on M and so the be condition can compatible with the Killing equation, only geometric (7.58) and not the inhomogeneous conformal Killing equation. Secondly, the exact 2-form S2V =- divg is degenerate on M, because an application of the Leibniz rule and Stokes' theorem gives

when

from the

n!

holds. This

zeroes

d 2nX

implies that

of V. Thus T

-

V/det Qv (x)

=

if

-=

0 almost

f

on V

0. Thus det S-2v (x)

=

0

Hamiltonian system determined by in Section 3.6, this isn't so crucial a

f

d

(ivg A

V

=

0

(7.64)

M

M

when aM

=

is

on some 1

submanifold of M of dimension at least 2

M, and so the degenerate. As mentioned

submanifold of

KV, f2v) 2 so long as

is

the support of det

(so

S?v(x)

is

that there exists at least

1 degree of freedom from the classical equations of motion). It would be interesting to investigate these geometric structures in more detail and see what localization schemes they lead to.

7.3 Corrections to the Duistermaat-Heckman Formula

253

7.3 Corrections to the Duistermaat-Heckman Formula: A Geometric Approach The integration formula

(7.46) suggests

a

geometric approach

tion of corrections to the Duistermaat-Heckman formula in the

to the evaluacases

where it

is known to fail. Recall that there is

for which V is

3.6). there

a

Killing

vector

always locally a metric tensor on M M V (see the discussion at the beginning of Section -

For the systems where the semi-classical approximation is not exact, are global obstructions to extending these locally invariant metric ten-

to globally-defined geometries on the phase space which are invariant under the full group action generated by the Hamiltonian vector field V on M, i.e. there are no globally defined single-valued Riemannian geometries on sors

M for which V is

globally a Killing vector. This means that although the Killing equation LVg 0 can be solved for g locally on patches covering the manifold, there is no way to glue the patches together to give a single-valued ==

invariant geometry on the whole of M (c.f. Section 5.9). In this Section we shall describe how the expression (7.46) could be used in this sense to evaluate the corrections to the

sum over

critical points there

that not

only does this method encompass much more than the term-by-term analysis of Section 7.1 above,

[157], of the

and

we

shall

see

loop-expansion

but it also character-

izes the non-exactness of the Duistermaat-Heckman formula in

a

much

more

transparent and geometric way than Kirwan's theorem. In this way we can obtain an explicit geometric picture of the failure of the Duistermaat-Heckman theorem and in addition a systematic, geometric method for approximating the integral (3.52). Furthermore, this analysis will show explicitly the reasons why for certain dynamical systems there are no globally defined Riemannian metrics on the given symplectic manifold for which any given vector field with isolated zeroes is a Killing vector, and as well this will give another geometric description of the integrability properties of the given dynamical system. The analysis presented here is by no means complete and deserves a more careful, detailed investigation. The idea is to define solve the

a

set of

patches covering M

in each of which

we can

Killing equations for g, but for which the gluing of these patches to give a globally defined metric tensor is highly singular. The non-

together triviality that

occurs

when these subsets

are

patched back together will then

represent the corrections to the Duistermaat-Heckman formula, and from our earlier arguments we know that this will be connected with the integrability of the Hamiltonian system. We introduce a set of preferred coordinates x" for the vector field V following Section 5.2. In general, this diffeomorphism

only be defined locally on patches over M and the failure of this coorproducing globally-defined C'-coordinates on M gives an analytic picture of why the Hamiltonian vector field fails to generate global isometries. Notice in particular that these coordinates are only defined on M MV. In this way we shall see geometrically how Kirwan's theorem restricts dynamical systems whose phase spaces have non-trivial odd-degree can

dinate transformation in

-

254


7.

homology

explicitly

and

what type of flow the Hamiltonian vector field gen-

erates.

Recall that the coordinate functions x" map the constant coordinate lines 2n-1 onto the integral curves of the isometry defined by the E R X2n) 0

(X2'. 0

classical Hamilton equations of motion :P(t) VA(x(t)), i.e. in the coordinates x"(x), the flows generated by the Hamiltonian vector field look like =

X111(t)

X,0

X""(t)

+ t

sociated with the fact that there is

no

formation functions

were

M to be any of x"I thereby

on

independent

(7.46)

from

X" 0

,

singularities

Otherwise, MV, then

0 holds.

=

on

M

-

on

as-

M for

if these transwe

could take

whose components in the x"-coordinates are solving the Killing equations directly, and hence

one

approximation would be

the WKB

must therefore be

system, there

(7.65)

p > 1

Riemannian metric tensor

Cvg globally defined

which the Lie derivative condition the metric

=

this coordinate transformation function will have

general,

In

=

some

exact. For

a

non-integrable defining the

sort of obstructions to

globally over M. In light of the above comments, these partition the manifold up into patches P, each of which is

x"-coordinate system

singularities a

will

2n-dimensional contractable, submanifold of M with boundaries 09P which

are some

other

(2n

-

l)-dimensional

by the conBy dropping some that these patches

submanifolds of M induced

stant coordinate line transformation from R

2n- 1

above.

of these coordinate surfaces if necessary, we can assume induced from the singularities of the above coordinate transformation form

disjoint

cover

function

of the

manifold, M

=

4. Then

JJp p

we can

write the

a

partition

as

(7.66)

alp

Z(T) p

p

equivariantly-closed differential form (3.57). patches P, in their interior there is a well-defined (bounded) translation action generated by V"". Since the patches P are 2n we can place a Euclidean metric on diffeomorphic to hypercubes in R

where

as

By

usual

a

is the

the choice of the

,

them, gp

where the conformal factor function

x"',

on

P. If

we

then the metric

patch P, by

e" ("') dx" (D dx" ' A

=

op(x")

choose it

(7.67)

so

a

globally-defined real-valued C'independent of the coordinate

that it is

Killing equation on P. Thus on each coordinates, we can solve the Lie deriva-

satisfies the

choice of

the

is

(7.67)

given though this cannot be extended to the whole of M. Then each integral over P in (7.66) can be written using the formula (7.46), restricted to the patch P, to get

tive

constraint,

Here

even

we assume

that M is compact, but

also be extended to the

phase

space R

2n

we

shall

see

that this formalism

can


f

(P)

T

f

here,

e'iTH-sKv

ds

(n

1)!

-

g(VI.)

SOV)n-1

(jTW

A

lap

ap

0

The first term

iTH(p)

(7.68)

00

(iT)n

e

(D) P)

et t

pEMvnP

p

+

a_dTet(p)

n

27ri

alp

255

when

(7.68)

is substituted back into

(7-66), represents

the lowest-order term in the semi-classical expansion of the partition function over M, i.e. the Duistermaat-Heckman term Zo(T) in (3.63), while the

boundary terms give the general corrections to this formula and represent non-triviality that occurs rendering inexact the stationary-phase approx-

the

imation. The result is

Z(T)

Zo(T)

=

+

JZ(T)

(7.69)

where 00

(iT)n I

JZ(T)

ds

p

0

(n

patch

geometric approach

conserved

charges

we can

nian, which

take

we

(7.70)

JZ(T),

-8QV)n-1

lap

(7.70)

therefore represent

an

al-

we

(eqs.

recall from Section 5.2

X,4(x)

for tz

2,...,2n

=

are

local

of the Hamiltonian system, i.e.

one

of

V'o9,X4

say X 2, to be

them,

X112(X)

choose to be

=

=

X2(X)

irrelevant constant to H

we

(compact)

Then, using the

we

-) A (iTw

loop-expansion of Section 7.1 above.

that the coordinate functions

IX", Hj,, Thus

g (V,

terms in

to the

To evaluate the correction term

(5.39),(5.43))

1)!

-

ap

The contributions from the ternative

e7,TH-sKv

Ei

manifold M.

find that the metric

may

(7.67)

assume

=

=

a

(7.71)

0

functional of the Hamilto-

VF(-x),

that it is

a

where

by adding

positive function

on

an

the

law, original (unprimed)

metric tensor transformation

when written back into the

coordinates has the form

gp

=

eWP (x)

1

(V,\ ax X 1)2

09'4X1aI'X1

1 +

4H

o9jHa,,H

(7.72) +

E,9jx',9,,x'

dxl' (g dxv

a>2 so

that the

metric-dependent quantities appearing

in

(7.70)

can

be written

as

91:' V.

ewP(x)_ aA Xl(x) dX

V'(X)aAX1(X)

Kv(x)lp

=

gp(V, V)

=

ewP(

(7.73)

7.

256


QVIP

ewP(II)

=

al\Xl (OAV' 2(V,\aAT_2 f

a X1

-

a V

+VAa,xxl (91, ppaxl -,O, op,01,xl) I dxl'

A

A0140 dx'

(7.74)

When these expressions are substituted back into the correction term (7.70), find that the integrands of JZ(T) depend only on the coordinate function

we

X1 (x). This is not surprising, since the only effect of the other coordinate functions, which define local action variables of the dynamical system, is to make the effect of the partitioning of M into patches above non-trivial, reflecting the fact that the system is locally integrable, but not globally (otherwise, the partition function localizes). In general, the correction term (7.70) is extremely complicated, but we recall that there is quite some freedom left in the choice of X1. All that is required of this function is that it have no critical points in the given neighbourhood. We can therefore choose it appropriately so as simplify the correction JZ(T) somewhat. Given this choice, in general singularities will appear from the fact that it cannot be defined globally on M, and we can use these identifications to identify the specific regions P above. The form of the function X1 is at the very heart of this approach to evaluating corrections to the Duistermaat-Heckman formula. We shall see how this works in some explicit examples in the next Section. Notice that a the similar phenomenon to what occured in Section 5.9 has happened here coordinate

to

-

function Kv in

(7.73)

is non-zero,

as

the

zeroes

of the vector field V have been

gp(V, -) thereby making it singular. We can therefore now carry out the explicit s-integral in (7.70), as the singularities on absorbed into the metric term

already present in the integrand there. Although this may seem to everything hopelessly singular, we shall see that they can be regulated with special choices of the function X1 thereby giving workable forms. We shall see in fact that when such divergences do occur, they are related to those predicted by Kirwan's theorem which we recall dictates also when the full stationary-phase series diverges for a given function H. There does not seem to be any immediate way of simplifying the patch corrections 6Z(T) above due to the complicated nature of the integrand forms. However, as usual in 2-dimensions things can be simplified rather nicely and the analysis reveals some very interesting properties of this formalism which could be generalized to higher-dimensional symplectic manifolds. To start, we notice that in 2-dimensions, if M is a compact manifold, then the union above over all of the patch boundaries 9P C M will in general form a sum over 1-cycles ae E Hi (M; Z). Next, we substitute (7.73) and (7.74) into (7.70) with n 1, and after working out the easy s-integration we find that the MV

are

make

=

2-dimensional correction terms

JZ(T)

=

-1

E iT e

can

1

be written e

as

iTH(x)

V1(49,\X1(X)

49jX'(x)dxt'

(7.75)


XI,

As for the function

need to choose

we

one

which is

independent

257

of the

other coordinate transformation function X 2 to ensure that these 2 functions truly do define a (local) diffeomorphism of M. The simplest choice, as far as the evaluation of

(7.75)

is to choose

concerned,

is

x1

as

the solution of the

first-order linear partial differential equation

Vl(X)a,XI(X) X1, the ajX'(x) 7 0,

With this choice of

V2(X),g2X1 (X)

=

X1

functions

X2

and

(7.76)

are

independent

of each

1, 2, which follows from working out the Jacobian for the coordinate transformation defined by X/I and using their defining partial differential equations above. With this and the Hamiltonian equations dH -ivw, the correction other wherever

terms

(7-75)

p,

v

=

become

6Z(T) where

we

Fl,,,

2iT

(7.77)

have introduced the 1-form F

=

W12(x)

(7.77)

e

(,02H(x)

iTH(x)

dxl

-

c9jH(x)

dX2

)

(7.78)

geometric interpretation of the correchomology H, (M; Z), there corresponds a cohomology class 77j E H1 (M; R),

The expression

leads to

nice

a

tions above to the Duistermaat-Heckman formula. To each of the

cycles

at E

called their Poincar6 dual

grals

of 1-forms

a

E

A'M

[32],

which has the property that it localizes inte-

to at, i.e.

i

a

I a,

(7.79)

A 77t

a

M

at

Defining E

we see

that the correction term

(7.77)

6Z(T)

H(M; R) can

1 =

-

2iT

be written

1

(7.80) as

(7.81)

F Aq

M

Noting also

that the

original partition

Z(T)

I =

2

function itself

j

FAdH

M

it then follows from

Z(T)

=

Zo(T)

+

6Z(T)

that

can

be written

as

(7.82)

7.

258

1

Beyond

FA

Approximation

the Semi-Classical

(iTdH +,q)

=

-47r

1: PEMv

M

(_,),\(P)Fdeet (p) a-t T_ et

( (p)

e

iTH(p)

(7.83)

Thus in this sense, the partition function represents intersection numbers of M associated to the homology cycles at. This last

equation

is

particularly interesting.

It shows that the

correc-

tions to the Duistermaat-Heckman formula generate the Poincar6 duals to

homology cycles which signify that the Hamiltonian vector field does not generate a globally well-defined group action on M. When the correction the

1-form

qliT

is added to the 1-form dH

of the Hamiltonian vector field

on

M,

=

the

-w(V, -)

resulting

means

that

although

enough

to

ren-

"effective" partition the initial Hamiltonian flow dH doesn't

der the Duistermaat-Heckman formula exact for the function. This

which defines the flow

1-form is new

required for the Duistermaat-Heckman satisfy enough' theorem, adding the cohomological Poincar6 dual to the singular homology cycles of the flow is enough to close the flows so that the partition function is now given exactly by the lowest-order term Zo(T) of its semi-classical expansion. One now can solve for the vector field W satisfying the "renormalized" Hamiltonian equations 'close

the conditions

to

dH We

consider W

can

which renders the

as a

+,qliT

=

-w(W,

(7.84)

"renormalization" of the Hamiltonian vector field V

stationary-phase

series

convergent and the Duistermaat-

symplectic form W defines a cohojust corresponds to choosing a different, possi-

Heckman formula exact. Note that since the

H'(M; R), mology bly non-trivial representative in H1 (M; R) for w (V, -) (recallq C- H1 (M; R)). Thus in our approach here, the corrections to the Duistermaat-Heckman formula compute (possibly) non-trivial cohomology classes of the manifold M and express geometrically what symmetry is missing from the original dynamical system that prevents its saddle-point approximation from being exact. The explicit constructions of the Poincar6 duals above are well-known [32] M of S1 in M which corresponds to the one takes the embedding at : S1 current which is the Dirac delta-function DeRham. its and constructs loop at, 1-form P 1) (x, at (y)) E A' M (x) 0 A' M (y) with the property (7.79) [18]. this

class in

-

--+

1

one crucial point that needs to be addressed before we turn explicit examples. In general we shall see that there are essentially 2 types of homology cycles that appear in the above when examining the singularities of the diffeomorphisms X/1 that prevent them from being global coordinate transformations of M. The first type we shall call 'pure singular cycles'. These arise solely as a manifestation of the choice of equation satisfied by X1. The second type shall be refered to as 'critical cycles'. These are the cycles on which at least one of the components of the Hamiltonian vector field 1 or 2. On these latter cycles the above integrals in 0 for M vanish, VA(x) 6Z(T) become highly singular and require regularization. Notice in particular

There is

to

some

=

=

7.4

that

if,

(7.71)

say,

Vl(x) (7.76)

and

=

0 but

V2(X) :

0

Examples

259

cycle at, then the equations X/1 imply that a2XA(x)

on some

which determine the functions

=

leaving the derivatives 491X/I(x) undetermined. Recall that it was precisely at these points where the Jacobian of the coordinate transformation defined by X/' vanished. In this case one must regulate the 1-form F defined above by letting a1X1 and o92X' both approach zero on this cycle aj in a correlated manner so as to cancel the resulting divergence in the integrand of (7.75). Note that this regularization procedure now requires that x1 and x 2 transform identically, particularly under rescalings, so that the tensorial properties of the differential form F are unaffected by this definition. In this case, the Morm F which appears above gets replaced by the 1-form 0 while

Flat

1 =

-

V2(X)

(dxl

ing

can

thought higher-loop

be

tion for the

e

iTH(x)

=

O,H(x)

(dxl

general expression (7.75).

which follows from the F

2)

+ dx

of

+

This

dX2 ) eiTH(x) (7.85)

procedure

for defin-

quantum field theoretic ultraviolet regularizacorrections to the partition function. In general, we as a

shall

always obtain such singularities corresponding to the critical points of because, as mentioned before, the diffeomorphism equations above become singular at the points where Vl'(x) 0. Note that (7,85) will also diverge when the cycle ae crosses a critical point, i.e. on aj n Mv. Such singularities, as we shall see, will be just a geometric manifestation of Kirwan's theorem and the fact that in general the stationary-phase expansion does not converge for the given Hamiltonian system. We shall also see that in general the pure singular cycles do not contribute to the corrections, as anticipated, as they are only a manifestation of the particular coordinate system used, of which the covariant corrections should be independent. It is only the critical cycles that contribute to the corrections and mimick in some the Hamiltonian

=

sense

7.4

the

sum over

critical

points

series for the

partition function.

Examples

In this Section

we illustrate some of the formalism of this Chapter with 2 explicit examples. The first class we shall consider is the height function of a Riemann surface, a set of examples with which we have become well-acquainted. In the case of the Riemann sphere we have little to add at this point since the height function (2.1) localizes. The only point we wish

classes of

to make here is that the covariant Hessian in this

standard Kdhler geometry of S2 metric gS2 by

VVhzo

=

2

1

(I

(see

Z,

+

Z )3

Section

dz o d,

=

5.5)

2(1

case

with respect to the

is related to the Khhler

-

hzo)gS2

(7.86)

260


7.

which is in agreement with the analysis of Section 7.1 above. This shows the precise mechanism (i.e. the Hessian of hzo generates covariantly the Khhler

S2)

structure of

that makes the

loop

corrections vanish.

An interesting check of the above formalisms is provided by a modified version of the height function h_ro which is the quadratic functional (2)

hZo

2

hzo

=

-

hZ

which has the

for H

COS

2z.

0)2

I + Zf

+ Zf

hEo.

as

2zf

Now

gz (z,, )

,

Z

=

)

2

(7.87)

find that the metric

we

the isothermal solution

(2) (2) a2 h ZO /o9zh Zo

9Z9Z2

[157]

H'(zf)

(7.88)

(2)

2zf

(2) VVh Zo

(7.89)

As

_

by taking

solved

are

ZZ

(I

_

hZo ,which follows from (7.24) written in local isothermal coordinates implicitly defined metric. Thus the solution to (7.21) is

=

for the

9

0)

critical behaviour

same

equations (7.21)

F;ZZ

COS

0

(z,

-

+

Z )3

-

1) dz (9 df

=

gz2dz

does not coincide with the standard Khhler

(7.89)

0 d,

geometry of

S2,

the

partition function in this case is not exact, as I-loop approximation the expected. However, partition function still localizes, in the sense that it to the

can

be

computed

via the Gaussian

integral transform

00

f dILL

Z(T)

iT(H-H 2) e

do e-i02 J -,/-27r-i f dl-IL /2

=

M

e

i(T-2iv TO)H

M

-00

of the usual equivariant characteristic classes. Thus since

tional of as

an

isometry generator (i.e.

anticipated

a

charge),

conserved

(7.87)

it is still

is

func-

a

localizable,

from the discussions in Sections 4.8 and 4.9. This is also

sistent with the formalism of the

previous Section. In this

coordinates for the Hamiltonian vector field

Although

(7.90)

these coordinates

are

singular

0 and

are

at the

poles

of

case, the x

=

0/(I

con-

preferred -

COS

0).

S2 (i.e. the critical

points of (7.87)), the correction terms JZ(T) do not localize onto any cycles and just represent the terms in the characteristic class expansion for Z(T) here. This just reflects the fact that S2 is simply connected, and also that the geometric terms JZ(T) detect the integrability features of a dynamical system

(as (7.87)

Next,

we

is

an

integrable Hamiltonian). height function on the torus, with the Khhler ge0 in (6.9) and v I in (6.39). adjusted so that o

consider the

ometry in Section 6.2

=

The covariant Hessian of the Hamiltonian

'H(01 02) 1

=

IM

7- COS

01

+ (rl + IM

COS

02do,

T COS

(&

do,

=

(3-78) -

01) COS 02dO2

in this

2 Im (9

-r

case

is

sin 01 sin 02do, (9

d02

d02

(7.91)

7.4

In the

complex coordinatization used

Examples

261

to define the Khhler structure this Hes-

analysis used to show stationary phase approximation in the case of the height S' using the loop-expansion will not work here. Indeed, we do not

sian is not of the standard Hermitian form and the

the exactness of the function

on

expect that

any metric

on

T 2 will be defined from the covariant Hessian here

already know that the Duistermaat-Heckman example. This is because of the saddle-points at (017 02) (0, ir) and (ir, 7r). The Hessian at these points will always determine an indefinite metric which is not admissible as a globally-defined geometry

as we

did in Section 7. 1, and

we

formula is not exact for this =

on

the torus.

This is also apparent from examination of the connection (7.8) and its Fubini-Study geometry defined by (7.34). In this case ^/ =- 0 and

associated

(7.34)

the curvature

is trivial. The 2-form Q does not determine the

same

cohomology class as the Kiihler 2-form of Z' does, so that there is not enough "mixing" of the Hessian and Liouville terms in the loop expansion to cancel out higher-order corrections. For the sphere, the Fubini-Study metric coincides with the standard Kdhler metric and thus the

dynamics integrable (recall

there to make the of formation of

a

appropriate mixing

that CP1

non-trivial Kiffiler structure

=

S2).

is

It is the lack

the torus here that makes

on

dynamical systems on it non-integrable. Although the failure of the Duistermaat-Heckman theorem in this case 2 via Kircan be understood in terms of the non-trivial first homology of T in the obstructions examine wan's theorem, we can extending the analytically Khhler standard the of Hamiltonian vector field (3.80) to a global isometry 2 for Riemannian the metric (6.9) of T which defines equivgeometry unique almost all

ariant localization

defined

by the

on

the torus. We shall find that the local translation action

vector field

(3.80)

cannot be extended

way to the whole 4 T 2. The set of coordinates the components of the Hamiltonian vector field

(x, y) are

globally on

V'

in

a

smooth

the torus in which

=

1 and VY

=

0

as

by taking X 2(01, 02) prescribed of the height function (3.78) and X1 (011 02) to be the Cl-function with nonvanishing first order derivatives which is the solution of the partial differential equation (7.76). In the case at hand (7.76) can be written as before

-

are

(r,

to be the square root

first defined

+ Im

-r COS

01)

ax

1 =

Im

-r

sin

01

Cot

02

axi

(7.92)

which is solved by

X1 (011 02) integrating (7.93) This gives and

as

=

in

log(r,

+ IM

(5.43) yields

T COS

01)

-

109(COS 02)

the desired set of coordinates

(7.93) (x, y).

262

7.

Beyond

the Semi-Classical

Approximation

2 Imr

X(01 02)

V r2 lRe

M'r

(t

log

an

-rJ sin 02

arctan

FRe -7

tan

r2

01 2

02

(7.94)

2

Cos

Vr

Y(01) 02)

(rl

IM

T Cos

01) COS 02

which hold

provided that Re -r: k 0. by the diffeomorphism. (7.94) the Hamiltonian vector field generates the local action of the group R1 of translations in x. However, this diffeomorphism cannot be extended globally to the whole of T2 because it has singularities along the coordinate circles In the coordinates defined

a,

=

bi This

means

geometry

that

on

T 2.

J(-7r/2, 0)

E

{(O,O)

E

cannot

globally generate

=

Vz1

Although

T21

a2

=

f (3-7r/2, 0)

21

b2

=

f(O,ir)

T

E

E T

T21

21

(7.95)

(7.96)

isometries of any Riemannian

translations in the coordinate

x

represent

some

unusual local symmetry of the torus, it shows that the existence of nontrivial homology cycles on T 2lead to singularities in the circle action of the on T2. These singularities do not appear on the sphere because any closed loop on S2 is contractable, so that the singular circles above collapse to points which can be identified with the critical points of the Hamiltonian function. In fact, as we saw in Chapter 6, the only equivariant Hamiltonians on the torus are precisely those which generate translations along the homology cycles of T2, and so we see that the Hamiltonian (3.78) generates a circle action that is singular along those cycles which are exactly the ones required for a globally equivariantly-localizable system on the torus. This is equivalent to the fact that the flow generated by V_ri bifurcates at the saddle points of h_vl (like the equations of motion for a pendulum), and the above shows analytically why there is no singlevalued, globally-defined Riemannian geometry on the torus for which the height function h_ri generates isometries. The local circle action defined by the diffeomorphism (7.94) however partitions the torus into 4 open sets Pi which are the disjoint sets that remain when one removes the 2 canonical homology cycles discussed above. Each of these sets Pi is diffeomorphic to an open rectangle in R2on which the Hamiltonian vector field Vzi generates a global Rl-action. Thus the above formalism implies that the corrections to the Duistermaat-Heckman formula for the partition function in this case is given by (7.75) evaluated on the pure singular cycles a, and a2 above, and on the critical cycles bi and b2 (see the previous Section). Summing the 2 contributions from the 1-form F in (7.78) along the pure homology cycles shows immediately that Fl,,, + 07 Fla2

Hamiltonian vector field Riemann

=

7.4

as

anticipated. As

Examples

263

integrals along the critical cycles, taking proper by the contractable patches, we find that the

for the

care

of orientations induced

contributions from b, and b2 written

the

are

and that the corrections

same

can

be

as

6ZT2(T)

eiT(r2-rl)

iT Im

-r

f

e-iT

do

Im

rcoso

sin

0 7r

f do

eiT(r2+rl)

-

e

(7.97)

iT Im

r cos

0

sin

0

After

a

change of variables we find that the integrals in (7.97) exponential integral function [60]

can

expressed

be

in terms of the

00

Ei(x) which

diverges

for

integration. After

JZT2(T)

< 0.

x

[

1 =

-

iT Imr -

Here the

algebra

some

2iT Im

e-iT

Im

e

-r cos

7-

Ei

2

-

integral

e

Im

r

Ei 2

a

Cauchy principal value

iT Im

-r

(Ei(-2iT

Im

r)

2))

Y

Im

-r

(-2iT

sin2

6)

Ei(2iT

-

2

Im

-r)) I

-r

Ei(2iT

2

eiT

denotes

2

(2iT Im

e-iT

iT(r2+rl)

2

(7.98)

find

we

eiT(r2-ri)

Ei

et

dt

Im

7- sin

2

Im

2)

7-)

-

-

Ei

2iT Im

Ei(-2iT

Im

-r)

r cos

2

2))

Y

) 11 (7.99)

where Y 7r in (7.97) at 0 =

-

r:

=

and

c

0 and

The correction term is

a sum

0 is used to =

the

divergence

of the

integrals

7r.

(7.99)

of 4 terms which

regulate

can

tells

us

quite

a

bit. First of

all,

note that it

be identified with the contributions from the

critical points of the Hamiltonian h_rl However, these terms are resummed, since the above correction terms take into account the full loop corrections to -

Next, the terms involving 6 are divergent, divergence Of JZT2(T) is anticipated from Kirwan's theorem, which says that the full saddle-point series for this Hamiltonian diverges. The exponential integral function can be expanded as the series [60] the Duistermaat-Heckman formula.

and the overall

264

7.

Beyond

the Semi-Classical

Approximation 00

Ei(x)

7 +

=

log x

+

n

E nn!

(7.100)

n=1

for

x

-y is the Euler-Mascheroni constant. Thus the

small, where

divergent

1, giving a much T simpler way to read off the coefficients of the loop-expansion (note the enormous complexity of the series coefficients in (7.3) for this Hamiltonian a direct signal of the messiness of its stationary-phase series). Finally, the finite terms (those independent of the regulator e), can be evaluated for pieces

in

(7.99)

be

can

explicitly expanded

in powers of

-

find JZT2 123.086. In Section 3.5 we partition function for this dynamical system 1849.327. was 2117.12, while the Duistermaat-Heckman formula gave Zo Thus Zo + 6ZT2 1972.41, which is a better approximation to the partition function than the Duistermaat-Heckman formula. Of course, given the large divergence of the stationary phase series, we do not expect that the finite contributions in (7.99) will give the exact result for the partition function, but we certainly do get much closer. As the function X1 which generates the set of preferred coordinates is by no means unique, perhaps a refined definition of it could lead to a better approximation Zo + 6Z. Then, however, we lose a lot of the geometrical interpretation of the corrections that we gave in T

=

saw

-i and

-r

I +

=

i, and

we

=

that the exact value of the

=

=

the last Section.

examples we consider here 2 plane R ,where U(q) is

The second set of

(5.273)

defined

on

the

function. In this

non-degenerate

case

-

i9q

proceeding

as

(7-101)

=

P2 /2

(7.102)

Thus here there

)(qU'(q)

-

are

P

-

U(q)

above the local coordinates

I =

only

=

2)

-

critical

f (O,q)

E R

( t, 9)

are

,

Y(q, p)

=

'cycles' given by

21

,

in which the Hamilto-

VT2 _T

+

U(q)

(7.103)

the infinite lines

Uj=f(p,qj)ER 21

(7.104)

where qj are the extrema of the potential U(q). Since for the Darboux. Hamiltonian (5.273), VP and Vq vanish the renormalized version of

use Uj respectively, we 'cycles' namely (7.85). Combining (7.85) with (7.77) function, that the corrections are

P and

a

-U'(q)-- P-

nian vector field generates translations

(q, p)

potential problems

by

X1 (q, p) Then

the

a C' potential which is the equation (7.76) becomes

09XI

,9 X1

p-

which is solved

are

must

we

find, for

an even

on

the

(7.78),

potential

7.4 00

6ZR2 (T)

eiTU(q)

dq

-;-T

-

Uf (q)

00

(E

f

eiTU(qi)

qj

0

Examples

dp

eiTp

2

265

/2

P

0

(7.105) and

we

note the

manner

divergences

in which the

are

cancelled here. From

aq2, potential U(q) integration measures in (7.105) contain implicit factors Of W12 that maintain covariance). Similarly, it is easily verified, by a simple change of variables, that for a potential of the 2 form U(q) aq + bq these correction terms vanish, again as expected. Fi-

this

we

immediately see that

the corrections

(7.105)

for the harmonic oscillator

vanish

(note

=

that the

=

nally,

for

quartic potential U(q)

a

q2

=

2

q4

+

a numerical integration of (7.105) for T Duistermaat-Heckman formula yields Zo

4

(the

anharmonic

oscillator),

-0.538 and the gives 6ZR2 2,7r. A numerical integration of

i

=

4.851, which differs from the value original partition function gives Z do not The 5.745. corrections Zo + 6ZR2 give the exact value here, but again at least they are a better approximation than the Duistermaat-Heckman formula. Again, a refinement of the preferred coordinates could lead to a better approximation. The method of the last Section has therefore "stripped" off any potentially divergent contributions to the loop-expansion and at the same time approximated the partition function in a much better way. These last few examples illustrate the applicability and the complete consistency of the geometric approach of the last Section to the saddle-point expansion. Indeed, we see that it reproduces the precise analytic features of the loop-expansion but avoids many of the cumbersome calculations in evaluating (7.3). It would be interesting to develop some of these ideas further. We would next like to check, following the analysis of Section 5.9, if there are any conformally-invariant geometries for this dynamical system when the potential U(q) > 0 is bounded from below. In the harmonic-polar coordinates (5.274), the conformal Killing equations (7.35) can be determined by setting the right-hand sides of the Killing equations (5.276) equal to instead (aoVO + FOOOVO)gm,. After some algebra, we find that they (V0V0)gj,, generate the 2 equations the

=

=

=

00 log

(

,00 log

(7.107)

can

be

)

grr

(

formally solved

2

groOr log Vo

(7.106)

grr

V0 900

goo

gro

gro

ar log V0

(7.107)

as

0

gro

=

-Vogoo

f

dO'

o9,Vo'

+

(7.108)

f (r)

00

from which

we see

only when (5.279)

that again

go (r, 0) holds g,0 (r, 0 + 27r) is the harmonic oscillator potential

single-valuedness

is true, i.e. when

U(q)

=

266

with

Beyond

7.

V

=

=

Approximation

oscillator, the equations (7.106) and radially-symmetric solutions g,,,, g,,(r) so global isometry of g. Thus, even though we lose the third

1. Even for the harmonic

(7.107) only that VO

the Semi-Classical

to admit

seem

1 is

a

=

equation in (5.276) which established the results of Section 5.9 using the Killing equations, we still arrive at the conclusion that there are no singlevalued metric tensors obeying the conformal Lie derivative requirement for

essentially all potentials which are bounded from below (and the harmonic oscillator only seems to generate isometries). Thus the conformal symmetry requirement in the case at hand does not lead to any new localizable systems. Finally, we examine what can be learned in these cases from the vanishing of the

2-loop,

(7.19) in harmonic coordinates. (7.8) has components

correction

the connection 1-form

^YP and the condition

(7.19)

=

0

'YY

In these

coordinates,

q

(7.109)

dy

reads d

71yy 'U Y

=

_'Y

2

(7.110)

0, in which case U(q) is the (7-110). Either -y. potential, or -(., (y + a)-', where a is an integration constant. This latter solution, however, yields q(y) Cly 2+ ay + C0, which gives a potential U(q) which is not globally defined as a COO-function on R 2. Thus the only potential which is bounded from below that leads to a localizable partition function is that of the simple-harmonic oscillator. This example illustrates how the deep geometric analyses of this Chapter serve of use in examining the localizability properties of dynamical systems. As for these potential problems, it could prove of use in examining the localization features of other more complicated integrable systems [56]. There

are

2 solutions to

harmonic oscillator

=

=

7.5 Heuristic Generalizations to Path

Integrals:

Supersymmetry Breaking generalization of the loop expansion to functional integrals is known, although some formal suggestive techniques for carrying out

The

not

yet

the full

semi-classical expansion can be found in [931 and [147]. It would be of utmost interest to carry out an analysis along the lines of this Chapter for

path integrals for several reasons. There the appropriate loop space expansion should again be covariantized, but this time the functional result need not be fully independent of the loop space coordinates. This is because the quantum corrections could cause anomalies for many of the symmetries of the classical theory (i.e. of the classical partition function). In particular, the larger conformal dynamical structures discussed in Section 7.2 above could

7.5 Heuristic Generalizations to Path

Integrals: Supersymmetry Breaking

267

important role in path integral localizations which are expressed in trajectories on the phase space [134]. It would be very interesting to see if these general conformal symmetries of the classical theory remain unbroken by quantum corrections in a path integral generalization. The absence of such a conformal anomaly could then lead to a generalization of the above extended localizations to path integral localization formulas. As this symmetry in the finite-dimensional case is not represented by a nilpotent operator, such as an exterior derivative, one would need some sort of generalized supersymmetry arguments to establish the localization with these sorts of symmetries. When these supersymmetries are globally present, the vanishing of higher-loop terms in the path integral loop expansion is a result of the usual non-renormalizations of I-loop quantities in supersymmetric quantum

play

an

terms of

field theories that arise from the mutual cancellations between bosonic and

loops in perturbation theory (where the fermionic loops have an sign compared to the bosonic ones). Quite generally though, one also has to keep in mind that the loop space

fermionic

extra minus

localization formulas

are

rather formal. We have overlooked several formal

functional aspects, such as difficulties associated with the definition of the path integral measure. There may be anomalies associated with the argument

path integral is independent of the limiting parameter R, for instance the supersymmetry may be broken in the quantum theory A g). (e.g. by a scale anomaly in the rescaling of the phase space metric g The same sort of anomalies could also break the larger conformal symmetry we have found for the classical theory above. However, even if the localization formulas are not correct as they stand, it would then be interesting to uncover the reasons for that. This could then provide one with a systematic geometric method for analysing corrections to the WKB approximation. The ideas presented in this Chapter are a small step forward in this direction. In particular, it would be interesting to generalize the construction of Section 7.3, as this is the one that is intimately connected to the integrability features of the dynamical system. The Poincar6 duality interpretation there is one possible way that the construction could generalize to path integrals. For path integrals, we would expect the feature of an invariant metric tensor that cannot be extended globally to manifest itself as a local (i.e. classical) supersymmetry of the theory which is dynamically broken globally on the loop space. This has been discussed by Niemi and Palo [122] in the context of the supersymmetric non-linear sigma-model (see Chapter 8). Another place where the metric could enter into a breakdown of the localization formulas is when the localization 1-form V) iwg does not lead to a homotopically-trivial element under the (infinitesimal) supersymmetry transformation described by QS. Then additional input into the localization formalism should be required on a topologically non-trivial phase space to ensure that QSO indeed does reside in the trivial homotopy class. These inputs could follow from an appropriate loop space extension of the correction in Section 4.4 that the

,\ E

-->

-

-

268

terms

7.


JZ(T)

discussed

above,

which will then

always reflect global properties

of the quantum theory. Other directions could also entail examining the connections between equivariant localization and other ideas we have discussed in this Book. One is the Parisi-Sourlas supersymmetry that

we

encountered

height sphere (Section 5.5), although this feature seems to be more intimately connected to the Kdhler geometry of S', as we showed above. The Kiffiler symmetries we found in Section 7.1 would be a good probe of the path integral correction formulas, and it would interesting to see if they could also be generalized to some sort of supersymmetric structure. in the evaluation of the Niemi-Tirkkonen localization formula for the

function

on

the

8.

in

Equivariant Localization Cohomological Field Theory

seen that the equivariant localization formalism is an excellent, conceptual geometric arena for studies of supersymmetric and topological field theories, and more generally of (quantum) integrability. Given that the Hamiltonians in an integrable hierarchy are functionals of action variables alone [106], the equivariant localization formalism might yield a geometric characterization of quantum integrability, and perhaps some deeper connection between quantum-integrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory. In this final Chapter we shall discuss some of the true field theoretical models to which the equivariant localization formalism can be applied. We

We have

that the quantum field theories which fall into this framework al-

shall

see

ways

have,

gauge

as

anticipated,

symmetry

cohomological

or a

some

large symmetry

supersymmetry)

structure

on

that

group

serves

to

the space of fields that

(such

as a

provide

can

an

topological equivariant

be understood

as a

"hidden" supersymmetry of the theory. Furthermore, the configuration spaces of these models must always admit some sort of (pre-)symplectic structure

properties of phase space path integrals can applied. Because of space considerations we have not attempted to give a detailed presentation here and simply present an overview of the various constructions and applications, mainly just presenting results that have been

in order that the localization

be

obtained. The interested reader is refered to the extensive list of references

Chapter for m ore details. We shall emphasize throughout this Book between the localizaformalism for dynamical systems and genuine topological quantum field tion theories. Exploring the connections between the topological field theories and more conventional physical quantum field theories will then demonstrate how the equivariant localization formalism for phase space path integrals serves as the correct arena for studying the (path integral) quantization of real physical that

are

cited

throughout

this

here the connections eluded to

systems.


270

8.


8.1 Two-Dimensional

Between

Physical

In Section 3.8

we

first

and

Cohomological

in

Field

Theory

Yang-Mills Theory: Equivalences Topological Gauge Theories

pointed

that, instead

out

consider the Poisson action of

of circular

actions,

one can

non-abelian Lie group acting on the phase space. Then the non-abelian generalizations of the equivariant localization formulas, discussed in Sections 3.8, 4.9 and 5.8, lead to richer structures some

in the quantum

representations discussed earlier and one obtains intriguing path integral representations of the groups involved. In this Section we shall demonstrate how a formal application of the Witten localization formalism can be used to study a cohomological. formulation of 2-dimensional QCD (equivalently the weak-coupling limit of 2-dimensional pure Yang-Mills theory). This leads to interesting physical and mathematical insights into the structures of these theories. We shall also discuss how these results

can

be

generalized to topological field theory limits of other models. First, we briefly review some of the standard lore of 2-dimensional QCD. The action for pure Yang-Mills theory on a 2-dimensional surface Zh of genus h is

Sym[A]

=

e2

-

-

f

tr

FA

(8.1)

FA

*

Eh a gauge connection of a trivial prinicipal G-bundle over Zh, FA is its curvature 2-form (c.f. Section 2.4), and e2 is the coupling constant

where A is

theory. Since 0 =- *FA is a scalar field in 2-dimensions, (8.1) depends on the metric of Zh only through its dependence

of the gauge field

the action on

the

area

A(Zh)

*1 of the surface. A deformation of the metric

can

constant

e2. The action

is invariant under the gauge transformations (2.72). The quantum field theory is described by the path integral

corresponding

therefore be

compensated by

a

change

in the

coupling

(8.1)

ZZh(e 2)

f [dA]

=

e sym [A]

(8.2)

A

where A is the space of gauge connections We can write the partition function in

by treating the g-valued multiplier to write Z-Vh (e 2)

scalar field

[dA] A

In the weak

logical

one

and which

coupling

a

limit e2

fh tr iOFA (i.e.

-+

one

0, the

simpler (first order) form g) as a Lagrange

Coo (_ph,

e_'f-htr( OFA+22 0*0) f

(8.3)

action in

(8.4)

reduces to the topo-

independent of the metric Of Zh cohomological. 4uantum field theory)'.

that is

topological field theory

Schwarz-type topological

E

(Eh,g)

consequently determines

This sort of of

C-

a

much

*FA

[do]

_Th.

over

a

is called

gauge

a

'BF

theory [22].

theory'

and it is the prototype

8.1 Two-Dimensional

Yang-Mills Theory

271

The gauge invariance of the action S[o, A] appearing in (8.3) is expressed as S[g-log, Ag] S[o, A] where g E g and 0 transforms under the adjoint =

gauge group. Because of this gauge invariance, it is necgauge and restrict the integration in (8.3) to the equivalence of gauge connections modulo gauge transformations. This can

representation of the essary to fix

AIG

classes

a

done

by

brief

account).

an

the standard BRST gauge

fixing procedure (see Appendix

A for

a

For this, we introduce an auxilliary, g-valued fermion field 0", which is anti-commuting 1-form in the adjoint representation of g, and write (8.3)

as

I

1

(e2)

ZZ ph

Vol Coo (Zh'

g)

I

[dA] [dO]

AOA'A

C-

[do]

(Zh,g) N

f

xexp -ftr

OFA-

V)

A

0

ie2

(8.4)

f 0*0-if *IQ, Tl} I tr

Zh

Zh

Zh

(8.4)

Faddeev-Popov gauge fixing by the graded BRST commutator of a gauge fermion TV ,0A17,,(x) with the usual BRST charge Q. The square of Q is Q2 _ijo where J,p is the generator of a gauge transformation with infinitesimal parameter 0. Thus Q is nilpotent on the space of physical (i.e. gauge-invariant) states of the quantum field theory. The system of fields (A,,O, 0) is the basic multiplet of cohomological Yang-Mills theory. The (infinitesimal) gauge invariance of (8.4) In

we

have introduced the usual BRST and

terms defined

=

=

is manifested in its invariance under the infinitesimal BRST

supersymmetry

transformations

JAI-t

=

icolt

1

601z

=

-E(VA)AO

=

+

[A,,, 0])

,

JO

=

0

(8.5)

anticommuting parameter. The supersymmetry transformations (8.5) are generated by the graded BRST commutator &P -ijQ, Pj for each field 4i in the multiplet (A,,O, 0). The ghost quantum numbers (Z-gradings) of the fields (A, 0, 0) are (0, 1, 2). We shall not enter here into a discussion of the physical characteristics of 2-dimensional Yang-Mills theory. It is a super-renormalizable quantum field theory which is exactly solvable and whose simplicity therefore allows one to explore the possible structures of more complicated non-abelian gauge theories such as higher-dimensional cohomological field theories and other physical models such as 4-dimensional QCD. It can be solved using group character expansion methods [34] or by diagonalization of the functional integration in (8.3) onto the Cartan subalgebra using the elegant Weyl integral formula [29]. Here we wish to point out the observation of Witten [171] that the BRST gauge-fixed path integral (8.4) is an infinite-dimensional version of the partition function in the last line of (3.125) used for non-abelian localization. Indeed, the integration over the auxilliary fermion fields 0 acts to

where

e

is

an

=

272

8.

produce


in

Cohomological

Field

Theory

field theoretical

analog of the super-I oop, space Liouville measure Chapter 4. The "Hamiltonian" here is the field strength tensor FA while the Lagrange multiplier fields 0 serve as the dynamical generators of the symmetric algebra S(g*) used to generate the G-equivariant cohomology. The "phase space" M is now the space A of gauge connections, and the Cartan equivariant exterior derivative a

introduced in

D

I (01'_6 6AI-i *

=

-

io'V"'

6

60,4

_V h

in this

case

coincides with the action of the BRST

)

(8.6)

charge Q,

i.e. DO

=

f Q, 01. The gauge fermion TI thus acts as the localization 1-form A and, by the equivariant localization principle, the integration will localize onto the -

field

configurations

vector fields

where

generating

A(V')

=

VI,4ff,,

=

0 where V'

=

V',/'

'9 ax"

are

the

g.

equivalence between the first and last lines of (3.125) is the basis of the mapping between "physical" Yang-Mills theory with action (8.1) and the cohomological Yang-Mills theory with actionf5-htr iOFA which is defined essentially by the steps which lead to the non-abelian localization principle, but now in reverse. The extrema of the action (8.1) are the classical YangMills field equations FA 0. Thus the localization of the partition function will be onto the symplectic quotient MO which here is the moduli space The

=

of flat gauge connections modulo gauge transformations associated with the gauge group G. This mapping between the physical gauge theory and the

cohomological quantum field theory is the basis for the localization of the 2-dimensional Yang-Mills partition function. Thus the large equivariant cohomological symmetry of this theory explains its strong solvability properties that have been known for quite sometime now. More generally, as mentioned at the end of Section 5.8, the equivariant localization here also applies to the basic integrable models which are related to free field theory reductions of 2-dimensional Yang-Mills theory, such as Calegoro-Moser integrable models

[56]. To carry out the localization onto MO explicitly, we choose a G-invariant metric g on Zh and take the localization 1-form in (3.125) to be

I\

=

I

tr

0

A

*Df

Eh

where

f

=

*FA. The localization

=

f dvol(g(x))

tr

O"DIJ

(8.7)

_r h

onto

A(V)

ization onto the solutions of the classical

=

0 is then identical to local-

Yang-Mills equations. We shall not enter into the cumbersome details of the evaluation of the partition function (8.4) at weak coupling e 2 --+ 0 (the localization limit) using the Witten nonabelian localization formalism. For details, the reader is refered to [171]. As in Section 5.8, the final integration formula can be written as a sum over the unitary irreducible representations of G,

8.1 Two-Dimensional

ZZh (e 2)

2 -

=

P?

-

e

11 a(P)2-2h a>O

(dim

Yang-Mills Theory

RX)2-2h

e2 e

_-T

273

(8.8)

AEZ'

expanding the various (G-invariant) physical quanin the localization formula in characters of the group G (c.f.

This result follows from tities

appearing

Section 5.8). From a physical standpoint the localization formula (8.8) is interesting because although it expresses the exactness of a loop approximation to the partition function, it is a non-polynomial function of the coupling constant e2. This non-polynomial dependence arises from the contributions of the unstable classical solutions to the functional integral as described in Chapter 3. Such behaviours are not readily determined using the conventional perturbative techniques of quantum field theory. Thus the mapping provided above between the physical and topological gauge theories (equivalently the generalization of the Duistermaat-Heckman integration formula to problems with non-abelian symmetries) provides an unexpected and new insight into the structure of the partition function of 2-dimensional Yang-Mills theory. This simple mapp ing provides a clearer picture of this quantum field theoretical equivalence which is analogous to the more mysterious equivalence of topological and physical gravity in 2 dimensions. Rom a mathematical perspective, the quantity (8.8) is the correct one to use for determining the intersection numbers of the moduli space of flat G-connections on Zh [78, 82, 171]. This approach to 2-dimensional Yang-Mills theory has also been studied for genus h

=

0 in

[105].

The

intriguing mapping between a physical gauge theory, with propagating particle-like local degrees of freedom, and a topological field theory with only global degrees of freedom has also been applied to more complicated models. In [28], similar considerations were applied to the non-linear cousin of 2-dimensional topological Yang-Mills theory, the gauged GIG WessZumino-Witten model. This model at level k E Z is defined by the action

SGIG [g, A]

k

f

T7r

tr

9_1VA9 A *9_1VA9 +

k 127r

Eh

k -

47r

f tr(g-ldg)A3 M

I

tr

(A A dgg-1

+ A A

(8.9)

Ag)

Zh

where g (=- C, (Zh' g) is a smooth group-valued field, M is a 3-manifold with boundary the surface Eh , and A is a gauge field for the diagonal G subgroup

GR symmetry group of the ordinary (ungauged) Wess-Zuminoby the action SG [g] SGIG [g, A 0] [54]. Since the is * when acting on 1-forms, invariant Hodge duality operator conformally the action (8.9) depends only on the chosen complex structure of Z h. As for the Yang-Mills theory above, the geometric interpretation of the theory comes from adding to the bosonic action (8.9) the term Q(O) =-L 042 21r fEh quadratic in Grassmann-odd variables 0 which represents the symplectic form of the

GL

x

Witten model defined

=

=

274


8.

in

Cohomological

Field

Theory

JAJA on the space A of gauge fields on _Vh. Again the resulting theory supersymmetric and the infinitesimal supersymmetry transformations are

f,, is

JA,

=

0,

with the

6V) ,

,

=

Ag

-

Z

A,

;

supplemental condition Jg

JA2 =

02

=

J,02

,

0. Unlike its

=

A

-

(Aq)

-'

Z

(8. 10)

Yang-Mills theory counter-

5' does not generate infinitesimal part, the square of this supersymmetry,6 transformations but rather 'global' gauge transformations (generated gauge the of the elements cohomological by gauge group which are not connected =

to the

identity).

Thus the action

less,

(8.9)

here admits

supersymmetry which does not mani-

a

the local gauge symmetry of the quantum field theory. Nonethethis implies a supersymmetric structure for equivariant cohomology

fest itself

which

as

can

be used to obtain

localization of the

a

corresponding path integral

in the usual way. The localization formula of [23] for equivariant Kdhler geometry has a field theoretic realization in this model [28] and the fixed-point

localization formula is the

algebraic

Verlinde formula for the dimension of

the space of conformal blocks of the ordinary Wess-Zumino-Witten model. For example, in the case G SU(2) the localization formula gives =

Z_r h(SU(2), k)

f

=

[dg]

)

2

k + 2 Thus the some

2-2h

k+1

1: e=1

(sin k+2) 7r

equivariant localization formalism

of the

more

[138]

e SsU(2)ISU(2)[g,A]

A

(Zh,su(2))

C-

f [dA]

can

2-2h

also be used to shed

(8.11)

light

on

formal structures of 2-dimensional conformal field theories.

path integral for a version of the ordinary Wessgive a field theoretical generalization of Stone's derivation for the Weyl-Kac character formula for Kac-Moody algebras (i.e. loop groups). This is done by exploiting the GL x GR Kac-Moody symmetry associated with the quantum field theory with action SG[g] as a supersymmetry of the model along the same lines as in Chapter 5 before. Let us now briefly describe Perret's derivation. The Kac-Moody group d is a central extension of the loop group S' --+ d -4 LG of a compact semi-simple Lie group G, and it looks locally like the direct product LG (& S', i.e. an element j E G looks locally like j (g(x), c) where g : S1 ---* G and C E S1. The coherent state path integral for the character is Perret

has used the

Zumino-Witten model to

=

tr,\

e

iH

q

Lo

f [dj]

=

e

f

jd+i(H+-rLo)dtjj)

(8.12)

L6

where H of the

=

loops.

Ej hjHj

E

HC,

The action in

q

=

eir and LO is the generator of rotations

(8.12) depends

on

2

coordinates, the coordinate

8.1 Two-Dimensional

x

along

central

the

loop

SI-part

in

Lg and the

Yang-Mills Theory

275

time coordinate t of the

of the coherent states in the

path integral. The path integral drops out due to

invariance, and thus the character representation (8.12) will define a theory on the torus T 2 (i.e. the quotient of loops

gauge

2-dimensional quantum field in the loop group LG). It

can

be shown

[138]

that the coherent state

path integral (8.12)

is the

quantum field theory with action

SKM

-1

:-`

21r

f

dx dt tr

(Ag- (6 1

+

H)g

k +

g-1 ax (a +

T2

(8.13)

k +

2H)g)

tr(g- ldg)A3

121r M

Here k E Z is the

weight

of

given central extension of the loop group, A is a dominant at -ro9., so that the Cartan angle -r becomes the G, and 5 =

-

modular parameter of the torus. When A H 0 the action (8.13) becomes the chiral Wess-Zumino-Witten model with the single Kac-Moody symmetry g --+ k(z)g. This symmetry is still present for generic A = - 0, and we can gauge =

the action with

a

SKM with respect

vector field

(see (8.9)).

to

an

One

=

arbitrary subgroup

can now

of G

by replacing

H

evaluate the infinite-dimensional

Duistermaat-Heckman integration formula for this path integral. The critical points of the action (8.13) are in one-to-one correspondence with the affine Weyl group Waff (Hc) W(Hc) z) kc, where Ac is the set of co-roots. Let =

r(w)

denote the rank of an element

w

W(Hc),

E

and let h be the dual Coxeter

number of the Lie group G [162]. Using modular i nvariance of the character to fix the conventional zero-point energy (associated with the usual SL(2, R)invariance of the conformal field theory vacuum [54]), it can be shown that the WKB localization formula for the coherent state

path integral (8.12)

with the

notation

Weyl-Kac

Lo tr,\ eW q

character formula

1:

[138] (for

see

coincides

Section

5.1)

(- 1)'(') q E,((A+p)i+(k+h)77i2-pi2)/2(k+h)

wEW(Hc),??Ef1c x

e

i(A+p+(k+h)77)(H(') )

e-ip(H(-))

qn)-r n>O

X

e

C >o

ia(H(-))

qn)

e-ia(H(')) qn-1 (8.14)

which arises from the expansions of various quantities defined on the torus in terms of Jacobi theta-functions. Thus infinite dimensional analogs of the

localization formulas in quantum field theory can also lead to interesting generalizations of the character formulas that the topological field theories of earlier Chapters represented.

276

8.


Finally, localization

it is as

possible

well. For

to

Cohomological Field Theory

use some

instance,

Scsp [A]

in

k =

87r

in

of these ideas in the context of abelian

[1261

the abelian gauge

j AAdA--m j 2

M

defined

on

a

3-manifold M

framework. The first term in

topological

field

theory 2,

theory with

(8.15)

AA*A

M

studied within the equivariant; localization (8.15) is the Chern-Simons action which defines a was

while the second term is the Proca,

mass

term for the

gauge field which gives a propagating degree of freedom with mass breaks the topological invariance of the quantum field theory. In

formalism where M

=

Zh

x

action

R1,

one can

naturally write

m a

the model

and thus

canonical

(8.15)

as a

h

quantum mechanics problem on the phase space _T and apply the standard abelian equivariant localization techniques to evaluate the path integral from the ensuing supersymmetry generated by the gauge invariance of (8.15) (in the Lorentz gauge c9AA,_, 0) [26]. The path integral localization formula coincides with that of a simple harmonic oscill ator of frequency wh 8-7rm/k, =

=

i.e. Z

field

=

1/2 sin(T 4,7rm/k), indicating -

theory using

a

mapping

once

again

to

a

topological

the equivariant localization framework. The infrared limit

0 of the model leads to the usual

topological quantum mechanical theory [261 and the supersymmetry, which is determined by a loop space equivariant cohomology, emerges from the symplectic structure of the theory on _T h and could yield interesting results in the full 3-dimensional quantum field theory defined by (8.15). m

-4

models associated with Chern-Simons

Symplectic Geometry Quantum Field Theories

8.2

In the last Section

showed how

of Poincare

loop

Supersymmetric

equivariant localization progauge theories and topological ones which makes manifest the localization properties of the physical models. In these cases the original theory contains a large gauge symmetry which leads to a localization supersymmetry and a limit where the model becomes topological that gives the usual localization limit. It is now natural to ask what happens when the original quantum field theory is explicitly defined with a supersymmetry (i.e. one that is not "hidden"). In Section 4.2 -1 mechanics admits a loop space we saw that N 2 supersymmetric quantum equivariant cohomological structure as a result of the supersymmetry which provides an alternative explanation for the well-known localization properties of this topological field theory (where the topological nature now arises because the bosonic and fermionic degrees of freedom mutually cancel each vides

a

we

correspondence between

certain

space

physical

=

The Cherns-Simons action is in fact another prototype of

logical

field

theory [221.

a

Schwarz-type topo-

8.2 Poincar6

other

Supersymmetric Quantum

Field Theories

277

out).

From the point of view of path integration, this approach in fact nice, geometric interpretation on the functional loop space of the features of this theory. It has been argued [107, 108, 133] that this interpretation can be applied to generic quantum field theories with Poincar6

led to

a

supersymmetry. In this Section

we shall briefly discuss how this works and techniques of equivariant localization could lead to new geometrical interpretations of such models. To start, let us quickly review some of the standard ideas in Poincar6

how the

supersymmetric quantum field theories. The idea of supersymmetry was first used to relate particles of different spins to each other (e.g. the elementary representation theory of SU(3) which groups particles of the same spin into multiplets) by joining the internal (isospin) symmetries and space-time (Poincar6) symmetries into one large symmetry group (see [155] for a com-

prehensive introduction). This is not possible using bosonic commutation relations because then charges of internal symmetries have to commute with space-time transformations so that dynamical breaking of these symmetries (required since in nature such groupings of particles are not observed) is not possible. However, it is possible to consider anti-commutation relations, i.e. supersymmetries, and then the imposition of the Jacobi identity for the symmetry

group leads to

shall be concerned

a

very restricted set of commutation relations. Here

only

with tho

se

anti-commutation relations satisfied

we

by

the infinitesimal supersymmetry generators

f Q', Qi4 I a

where Z"

=

20 Z'

"4PA + ZV. a,3

.

;

N

.

%13

=

(8.16)

-1"C with -yA the Dirac matrices and C the charge conjugation matrix, P,, -0,, is the generator of space-time translations, and Pi is an antisymmetric matrix of operators proportional to the generators of the internal symmetry group. For the rest of the relations of the super-Poincar6 group, see [155]. We assume here that the spacetime has Minkowski signature. We shall be interested in using the relations of the super-Poincar6 algebra to obtain a symplectic structure on the space of fields. For this, it turns out that only the i i terms in (8.16) are relevant. It therefore suffices to consider an N 1 suPersymmetry with no internal Z'i symmetry group terms. The most expedient way to construct supersymmetric field theories (i.e. those with actions invariant under the full super-Poincar6 group) is to use a superspace formulation. We introduce 2 Weyl spinors 0' and #& which =

=

=

==

parametrize the infinitesimal supersymmetry transformations. Then the N I supersymmetry generators in 4 dimensions can be written as a

a

-

Q&

.

=

Kinetic terms in the supersymmetric action ant

_

aO& are

+

'A -

0111 a

(8.17)

constructed from the covari-

superderivatives a a0a

+

iZa6,PaA

,

D&

aO&

iZ.1'6,0"a4

(8.18)

278

8.


in

Cohomological Field Theory

readily verified that with the representation (8.17) the relations of super-Poincar6 algebra are satisfied. In this superspace notation, eO"Q-+&6,Q6 and, a general group element of the supersymmetry algebra is using the supersymmetry algebra along with the Baker-Campbell-Hausdorff eo"Q- e66'06A(y) formula, its action on a field A(x) is e0'Q-+6 6Q6A(x) where y" x4 + WZ,",&P are coordinates in superspace. Thus the superIt

be

can

the N

=

I

=

=

symmetry transformation parameters live in superspace and the fields of the

supersymmetric field theory are defined on superspace. To incorporate the supersymmetry algebra as the symmetry algebra of a physical system, we need some representation of it in terms of fields defined over the space-time. The lengthy algorithm to construct supermultiplets associated with a given irreducible or reducible spin representation of the super-Poincar6 algebra can be found in [155]. For instance, in 4 dimensions chiral superfields (satisfying D&!P 0) are given by =

V(x, 0, 9) (0,,0)

where

are

spin (0,

=

0"(y)

fields, .1) 2

+

0'0 '(y)

+

a

and F

are

0'0,,F4(y)

(8.19)

auxilliary fields

use

to close

the supersymmetry representation defined by the chiral superfields. In the following we shall consider multiplets of highest spin 1. Other multiplets be obtained

by imposing some additional constraints. The most general supermultiplet in 4 space-time dimensions consists of a scalar field 3 M, pseudoscalar fields C, N and D, a vector field A,-, and 2 Dirac spinor fields X and A. The supersymmetry charges Q and Q' respectively raise and lower the (spin) helicity components of the mulitplets by 1. The super2 Poincar6 algebra can be represented in the Majorana representation where 0 2 iorl (g 1and y3 -io.3 (& 0.3, with a' -ia 30 a 1, 72 -9 o 1, 71 ly can

N

1

=

=

=

=

the usual Pauli spin matrices. Then the 4 side of (6fNsusyalg) are

-TO Z2 The

X

=

(

( 0

i

-

-i 1

) 1)

0

1

*

0

=

x

4 E-matrices

T,

T3

=

on

the

right-hand

(0 1) 1

0

0

-1

(8.20)

Majorana representation selects the preferred light-cone coordinates X2

XO for the translation generators P,, in (8.16) above. Different representations of the Dirac gamma-matrices would then define different pre=

light-cone directions. I supergeneral supersymmetry transformations of the complex N the and of the action M, A; multiplet V N, At,; (C; X; D) supersymmetry on V can be found in [133]. There it was shown that the incharges Q, finitesimal supersymmetry transformations can be written in a much simpler form using the auxilliary fields

ferred

The

=

Supersymmetric Quantum

8.2 Poincax6

M'

=

M +

Al

A3 +491C =

Al

N'

,

a3X1

-

A12

i

A,

N +

==

A2

=

-

-

Field Theories

D'

a3C A'3

101XI

D +

=

2A3

=

-

aIA3

-

279

o93AI

49-Xl

(8.21) precisely the (non-standard) auxilliary fields introduced in [107, 108] which, as we discussed in Section 4.9, form the basis for equivariant localization in supersymmetric quantum field theories and phase space path integrals whose Hamiltonians are functionals of isometry generators. Using these auxilliary fields we now define 2 functional derivative operators on the space of fields, These

are

T

d3X

D

i

X2

dt

6

9C

iA+

+

6

JXJ

iM'

+

6 +

iN'

3

5

JX4

0

-A/3

A/2

3-A-

A'

A_

iD'

Aq3

(8.22)

4

A

T

j

d3X

-TV+

dt

O+C

10+X3

+

6X2

6

j

6M,

+

19+X4 N' +,9+xi 6A+

0

-a+A3

a+Aj

JA'

5 -

JA'

a+A-

6

49+A4

6A'

6 JD1

(8.23) Here

we

imposed the boundary conditions on the fields P(x, t) that they xO, spatial infinity and that they be periodic in time t

have

vanish at

=

lim 0 (X, IXI-00 so

t)

=

P(x, t

0

+

T)

P(x, t)

=

(8.24)

that the space of fields can be thought of as a loop space. The operators (8.22) and (8.23) are nilpotent. If we now define

Q+ then it

can

=

D

(8.25)

+IV+

be checked that T

Q2+

DIV+

+ IV+ E)

=

LV+

d 3X

i

dt

ia+

(8.26)

0

I supersymmetry transThe operator (8.25) generates the appropriate N formations on the field multiplet and the supersymmetry algebra (8.26) co=

supersymmetry algebra (8.16). The above geometric representation of the superalgebra (8.16) on the space of fields of the supersymetric field theory. A different representation of Z" leads to a different choice of preferred Q, in (8.26) and a different decomposition of the fields into loop space coordinates and 1-forms

incides with the

pertinent N provides

construction therefore

=

I

a

280

in


8.

(8.22)

and

(8.23).

It is

in

Cohomological

possible decompose

now

ric quantum field theories

Field

Theory

to examine how various

supersymmet-

with respect to the above equivariant cohomological structure on the space of fields. The canonical choice is the Wess-Zumino model which is defined by the sigma-model action T

SWZ

d3X

1 f dt

dO d

D P + W[

2

P]

(8.27)

0

where

W[4i]

is some super-potential. With respect to the above decomposipossible to show that the supersymmetric action decomposes into a sum of a loop space scalar H and a loop R + Q. space 2-form 0, S Because of the boundary conditions on the fields of the theory, the supersymmetry charges, which geometrically generate translations in the chosen light-cone direction, are nilpotent on the spaces of fields. By separating the loop space forms of different degrees, we find that the supersymmetry of the 0 and DH action, Q+S 0, implies separately that DS-2 -.TV+ Q. Thus the supersymmetric model admits a loop space symplectic structure and the corresponding path integral can be written as a super-loop space (i.e. phase 0 on the space of space) functional integration. Furthermore, because Q2 Q obeys lv+?9 fields, the symplectic potential 0 with DO H, so that the action is always locally a supersymmetry variation, S Q+,O (D + lv+),O. Thus, any generic quantum field theory with Poincar6 supersymmetry group admits a loop space symplectic structure and a corresponding U(1) equivariant cohomology responsible for localization of the supersymmetric path integral. The key feature is an appropriate auxilliary field formalism

tions it is

=

=

=

=

=

=

=

=

which defines

a

splitting

=

of the fields into

loop space "coordinates" and their general although the fields of the evenly into loop space coordinates

associated "differentials" 3. Notice that in

supersymmetric theory always split and

1-forms, the coordinates and fields. It is only in the simplest

mechanics)

up

1-forms involve both bosonic and fermionic cases

(e.g.

N

=

-1 2

supersymmetric

quan-

that the pure bosonic fields are identified as coordinates and the pure fermionic ones as 1-forms. In the auxilliary field formalism outlined above, the supersymmetry of the model is encoded within the model inde-

turn

pendent loop space equivariant cohomology defined by Q+. In this way, one a geometric interpretation of general Poincare supersymmetric quantum field theories and an explicit localization of the supersymmetric path integral onto the constant modes (zeroes of 0+). We shall not present any explicit examples of the above general constructions here. They have been verified in a number of cases. To check this formalism in special instances one needs to impose certain additional constraints on the multiplets [133] (e.g. the chirality condition mentioned above). The obtains

3Note

that the

problem of choosing an appropriate set of auxilliary fields is the analog of finding a preferred set of coordinates for an isom-

infinite-dimensional

etry generator

(c.f.

Section

5.2).

8.3

Supergeometry

and the

Batalin-aadkin-Vilkovisky

Formalism

281

above constructions have been in this way explictly carried out in [107, 108] 1 supersymmetric quantum mechanics (i.e. the (0 + l)-dimensional Wess-Zumino model (8.27)), the Wess-Zumino model (8.27) in both 2 and 4 dimensions, and 4-dimensional N 1 supersymmetric SU(N) Yang-Mills for N

=

=

theory

defined

by

the action

d4X

sym where

0

(_4 1

Fa 011F a,ttv +

Majorana fermion fields

are

gauge group. The model

(8.28)

in the

2

VA'O)

adjoint representation

be reduced to

(8.28) of the

Wess-Zumino model

by eliminating the unphysical degrees of freedom and representing the theory directly in terms of transverse (physical) degrees of freedom. The equivariant can

a

localization framework has also been applied to the related supersymmetries quantization [136] in [108]. Palo [133] applied these constructions to the 2-dimensional supersymmetric non-linear sigmaof Parisi-Sourlas stochastic

(i.e. the Wess-Zumino Section).

model

model

(8-27)

with

a

curved target space

-

see

the next

8.3

Supergeometry Batalin-Fradkin-Vilkovisky

and the

Formalism

The role that the Cartan exterior derivative of equivariant cohomology plays in localization resembles a Lagrangian BRST quantization in terms of the

gauge-fixing In

of

a

Lagrangian

[112, 113] equivariant

field

theory

localization

was

over

a gauge field 0,' [22]. in terms of the Batalin-

M with

interpreted

Vilkovisky Langrangian anti-field formalism. In this formalism, a theory with first-stage reducible constraints or with open gauge algebras is quantized by introducing an antibracket [13]-[15] which naturally introduces a new supersymmetry element into the BRST quantization scheme. This formulation is especially important for the construction of the complete quantum actions for

gauge theories (especially those of Schwarz-type) [22]. In this shall sketch some of the basic conceptual and computational ideas

topological

Section of this

we

formalism, which

lead to

more

in the context of localization for

dynamical systems topological field

direct connections with super symmetric and

theories. We return to the

(M, w, H). a

We have

formulation of

and Grassmann

simpler

situation of

a

generic Hamiltonian system

that the localization prescription naturally requires defined over a super-manifold (i.e. one with bosonic

seen

objects

coordinates), namely the cotangent

bundle MS =- MOT*M. supersymmetric quantum field theories that we considered in the previous Section, our fields where defined on a superspace and the path integral localizations were carried out over a superloop space. We would now like to try to exploit the mathematical characteristics of a super-manifold In the

case

of the

282

8.


in

Cohomological

Theory

Field

and reformulate the localization concepts in the more rigorous framework of supergeometry. This is particularly important for some of the other localization features of

field theories that

topological

shall discuss in the next 2

we

Sections.

First, we shall incorporate the natural geometrical objects of the BatalinVilkovisky formalism into the equivariant localization framework. The local coordinates on the super-manifold MS are denoted as ZA (xA"qA). We define a Grassmann-odd degree symplectic structure on Ms by the nondegenerate odd symplectic 2-form =

S21

=

The 2-form

B d ZA A C)i 'ABdz

(8.29)

wlzvdx"

=

determines

bracket. It is defined

aA

dn'

4a

aA

qZB

'9xA

+

A(x,,q)

and

77\dn"

ax,\

odd Poisson bracket

awt"'

where

19W/1-

+

on

B(x,,q)

tisymmetry properties bracket,

gA, BC11

are

are

gA,BJ1 =

),

opposite

a?7v

aA

super-functions

09XII

B

anti-

'0771,

on

(8.30)

anv Ms. The grading and

to those of the

ordinary graded

an-

Poisson

_(_1)(p(A)+1)(p(13)+1)J13, All

=

gA, L311C

+

(_ 1)p(13) (p(A)+I) BgA, CR g gA, BI 1, C1 (8.31) super-function A(x, 77) with the

=

-

the

super-coordinate

are

9XA, X, I i

=

9x"' 77, 11

0

1,,', 71, We define

a

mapping

!2W,11(x)?7Aq"

=

on

-V7v,','h

COO(M)

-+

-

Qf (Z)

=

=

-

W, x" I I

=

W/4v

(8.32)

49WAV =

5- 77

C'(Ms) using

If, W11

the

super-function

f A, f },,.

In

=

-2-f-77 ,gxA

A

(8.33)

original Poisson bracket of the phase particular, the dynamical systems (M, w, H)

Then the antibracket coincides with the

I A, f I,

=

by fW

space,

A

+

p(A) is the Grassmann degree of the p(A) + p(B) + I In particular, property p(JA, BI)

=

(8.29)

MS called the

M

B

71 ji-A x anA

where

W(Z)

dqv

1) (P(4)+') (P(C)+') 9B, 9A, C1 1 1

gA, gB, C) I 1

antibrackets

A

by

f2ABL3

5Z-A

an

A

Supergeometry

8.3

and

(MS,fll, QH)

tions of motion

:VZ

=

determine

a

bi-Hamiltonian pair. The

Formalism

283

corresponding

equa-

are

JXO) QHJ1

Generally,

Batalin-Fradkin-Vilkovisky

and the

=

JxO, HJ,,

V1'

=

W'

,

=

1?7t7 QHJ1

=

IOX177V (8.34)

readily seen that the operation Jw, .11 acts as exterior difd, JH, .11 acts like interior multiplication iv with respect to the Hamiltonian vector field V, and I QH 1 1 acts as the Lie derivative Lv along V. The antibracket provides an equivalent supersymmetric generalization of the ordinary Hamiltonian dynamics. The key feature is that the supersymmetry of the odd Hamiltonian system (MS 011 QH) is equivalent to the equivariant cohomology determined by the equivariant exterior derviative DV d + iv. If M admits an invariant Riemannian metric tensor g (the equivariant localization constraints), then the super-function it is

ferentiation

-

7

1

=

TH is

an

integral

JQHJH11 structure

=

g"VVIJI 77V

(8-35)

of motion for the Hamiltonian system (MS, S?1 7 QH) , i.e. 0. Furthermore, TH determines the usual bi-Hamiltonian

M because

on

2

With these observations

JwJH11

=

!(Qv)4vqAqv 2

easily

one can

now

ariant localization written

as an

principle. The usual localization integral over the super-manifold MS,

Z(s)

f

1 =

(iT)n

d

4n

z

KV. JHJH11 equiv(classical) integral (2.128) can be and

=

establish the

eiT(H+w)-sJH-w,1HR1

(8.36)

MS

where

as

always

form d4nZ

=

the classical partition function is Z(T) Z(O). The volume is invariant under the equivariant transformations of =

d2nx d 2n,,

DV and LV determined by the anti-brackets. Furthermore,

JH

-

w,

e

iT(H+w)-sJH-w,.THJ1

9QH,IH

e

11

=

JQH,

e

we

have

iT(H+w)-sJH-w,TffJ1

iT(H+w)-sJH-w,_THR11

I,

=

0

0

(8-37) The first 2

vanishing

conditions

just represent the invariance of the integrand in (8.36) under the actions of the operators DV and CV, so that the usual equivariant cohomological structure for localization in (8.36) is manifested in the supersymmetry of the antibracket formalism. With the identities (8.37), it is straightforward to establish that jd Z(s) 0, and hence the localization ds =

principle (i.e. the Duistermaat-Heckman theorem). Thus the lifting of the original Hamiltonian system to the odd one defined over a supermanifold has provided another supersymmetric way to interpret the localization, this time in terms of the presence of supersymmetric biHamiltonian dynamics with even and odd symplectic structures which is the usual Batalin-Vilkovisky procedure for the evaluation of BRST gauge-fixed

284

8.

Equivariant Localization in Cohomological Field Theory

path integrals. The representation (8.36) of the canonical localization integral formally coincides with the representation of differential forms in the case where the original space M is a supermanifold. In [150], Schwarz and Zaboronsky derived some general localization formulas for integrals over a finite-dimensional supermanifold M where the integrand is invariant under the action of an odd vector field W. Using the supergeometry of M, they formulated sufficient conditions which generalize those above under which the integral localizes onto the zero locus of the c-number part W(,q 0) of W. Their theo rems quite naturally generalize the usual equivariant localization principles and could apply to physical models such as those where the Batalin-Vilkovisky formalism is applicable [149] or in the dimensional reduction mechanism of the Parisi-Sourlas model [136]. A generalization to more general (non-linear) supermanifolds can be found in [175]. Nersessian has also demonstrated how to incorporate the anti-bracket structure into the other models of equivariant cohomology (other than the Cartan model see Appendix B) in [115] and how the usual equivariant =

-

characteristic class representations of the localization formulas appear in the Batalin-Vilkovisky formalism in [113]. The superspace structure of cohomo-

logical

field theories in this context has been studied

by

Niemi and Tirkkonen

[129]. They discussed the role of the BRST model of equivariant cohomology in non-abelian localization (see Appendix B) and topological field theories in

and showed how these

an

appropriate superfield formulation

equivariant cohomological

structures to the

can

be used to relate

Batalin-Radkin-Vilkovisky

Hamiltonian quantization of constrained systems with first stage reducible constraints. This suggests a geometric (superspace) picture of the localization

properties of some topological quantum field theories such as 4-dimensional topolog ical Yang-Mills theory (defined by the action f tr FA A FA). This picture is similar to those of the Poincar6 supersymmetric theories described in the previous Section in that the BRST charge of the cohomological field theory can be taken to generate translations in the q-direction in superspace and then the connection between equivariant cohomology and BatalinFradkin-Vilkovisky quantization of 4-dimensional topological Yang-Mills theory becomes transparent. These superfield formalisms therefore describe both equivariant cohomology in the symplectic setting relevant for localization and the BRST structure of cohomological field theories. This seems to imply the existence of a unified description of localization in the symplectic loop space, the supersymmetric loop space of the last Section, and in cohomological. field theory. Indeed, it is conjectured that all lower dimensional integrable models are obtainable as dimensional reductions of 4-dimensional self-dual YangMills theory (i.e. FA *FA) which is intimately connected to topological Yang-Mills theory. Finally, the incorporation of the Batalin-Vilkovisky formalism into loop space localization has been discussed recently by Miettinen in [103]. For path integral quantization, one needs an even Hamiltonian and symplectic struc=

Supergeometry

8.3

ture. This

An

can

be done

symplectic

even

9%

1 =

2

1

Batalin-Radkin-Vilkovisky

and the

Formalism

285

provided that M has on it a Riemannian structure. on MS is given by the super-symplectic 2-form

structure

(w"V + Rjv,\p?7AqP) dx" A dxv + 2gjvD9?7mA D977

V

(8.38)

where

D9771' is the covariant derivative

=

dqm

+

rv",\?7vdx'\

(8.39)

M, and the subscript i 0, 2 labels the Hamilw' and Ho S?v, H2 H) w, (M, Kv) The corresponding symplectic I-forms, with S?' &9 , are then tonian

systems

(M, wo

on

=

=

=

=

=

-

=

eo

0A dx,4

=

+

gI'ZVq1iDg,v

The 2-forms 0' determine the

A,Bjj

=

192

,

=

g,,,Vvdxm

+

Poisson brackets

even

(VjA)[w'tv

gj,,,qmDgij'

on

+gm'

(8.40)

MS

aA

a'q1-I

B

15Oqv

(8.41)

where

v,

=

a,

r-N jj'V?7V-

-

Then the equations of motion for the odd and

coincide, jz A, jj, ,

=

(8.42)

0977A

IZA, QHJ1,

even

Poisson brackets

on

Ms

where Hi Hi + f2v. Thus the odd and bi-Hamiltonian structures also on the super=

Poisson brackets provide symplectic manifold MS and therefore the Hamiltonian system is also integrable on Ms. Given this integrability feature on MS, we can now examine the corresponding localizations [150]. We write the partition functions Zj(T) strIl e -Mii 11 as path integrals over a super-loop space corresponding to LMS by absorbing the Liouville measure factors associated with 0' into the argument of the action in the usual way (c.f. Chapter 4). Then the Hamiltonian even

=

systems

(Ms, S-2',Hi)

have the quantum actions

[103]

T

So

dt

(01,--bm

-

H +

-177"g,,, 2

1

dt

+

2

(.f2V)1,vqI'?7'

+

Iw,,,,AI'Av

2

0

+

1Rmv,\pA"Av?7A 77

P

2

+

1g,,vFMF'

2

T

S2

1

dt

g,,,Vv&"

-

KV +

1 2

77A9pV

dt

+

2

(Qv)mvqlqv

0

+

I 1(S?v)tvA"Av + R,,,,\pAI'Avq'\?7 2 2

P

1 +

2

gtzvFmF' (8.43)

286


8.

Cohomological

in

where the quantum partition functions

Zi(T)

I

=

Field

Theory

are

2n 2n FA [d 2nX1 [d 2n,7] [d F] [d A] e'si [ x,,q; 1

(8.44)

LMsOLA'Ms

and

we

have introduced

bosonic variables F11 the usual way. If H

LTdt 2'g Av

-

=

dx" and auxilliary anticommuting variables A" dqll to exponentiate the determinant factors in 0 (the topological limit), then the action So + -

(0

is that of the

+

I)-dimensional

N

=

supersymmetric

I

1 DeRham supersymmetric sigma-model, i.e. the action of N 4 quantum mechanics in background gravitational and gauge fields One can now develop the standard machinery to evaluate these path integrals using super-loop space equivariant cohomology. This has been done explicitly in [122]. The super-loop space equivariant exterior derivative is

non-linear

=

T

I (

Q

dt

AA

J

TX_A

J

+ F"

+

Jql,

(: A

-

VA)

J +

JAA

0/4

+

alj'Vv?7,)

JFA

0

(8.45) Q2

LV where S is the classical action associated with the Hamiltonian original system (M, w, H). Notice that a canonical conjugation dl does --+ not alter the cohomology groups of the derivative =_ C Q e-Q the loop space functional operator Q. Choosing with

=

Ls

=

Lb

-

T

dt

0

R\

AV

7 7A A

J

V

(8.46)

T_ FTA

0

can be explicitly worked out (see (topologically equivalent) operator [122]). With regards to this supersymmetry charge, the pertinent action So in (8.43) can be obtained from the 2-dimensional N 1 supersymmetric sigma-

the

=

by partial localization of it to a I-dimensional model. This is done by breaking its (left-right) (1, 1) supersymmetry explicitly by the Hamiltonian flow. In this procedure the usual boson kinetic term (in light-cone coordinates) g,,,,a+0A,9_0v drops out. The path integral Zo(T) can be evaluated by adding an explicit gauge-fixing term ,o to the action for an appropriate

model

gauge 4

fermion,0. Taking 0

The action for N

=

1

T

=

L

dt

(g1,vFAq1

+

1 2

resulting

V/)A)

-

supersymmetric quantum mechanics

can

localizes

be obtained

auxilliary (8.27) by integrating superfields (8.19) and integrating over the 0 .1 The action for N supersymmetric quantum 2 out the

from the Wess-Zumino model action field F"

gAv(V'

from the chiral

coordinates of the superspace.

mechanics discussed in Section 4.2

setting AA =,q" above.

=

can

be obtained from the N

=

1 model

by

drops

ut

Notice that the Riemann curvature term then

because of its symmetry

properties.

o

8.4 The

the

path integral in the limit original action S,

s

--+

oo

Mathai-Quillen Formalism

onto the

287

T-periodic classical trajectories

of the

Zo (T)

=

E

sgn[det p2S(X(t)) 11] e'sl'(01

(8.47)

X(t)ELMs T

On the other

dt (g1,,FA?71' + !g,,,.t/1A1') we find that hand, selecting 2 the path integral localizes onto an ordinary integral over equivariant characteristic classes of the phase space M,

Zo (T)

=

f

chv (-iTw)

A

Ev (R)

(8.48)

M

The

equality of these 2 expressions for the quantum partition function be thought of as an equivariant, loop space generalization of the relation (3.71) in Morse theory between the Gauss-Bonnet-Chern and Poincar6Hopf theorems for the representation of the Euler characteristic X(M) of the manifold M. Indeed, in the limit H, 0 --- 0 the quantities (8.47) and A (8.48) reproduce exactly the relation (3.71). These relations have been used to study the set of Hamiltonian systems which satisfy the Arnold conjecture on the space of T-periodic classical trajectories for (time-dependent) classical Hamiltonians [119, 122]. Thus, the path integrals associated with the supermanifolds defined by the Batalin-Vilkovisky formalism for dynamical systems lead to loop space and equivariant generalizations of other familiar topological invariants. Furthermore, these models are closely related to supersymmetric non-linear sigma-mode Is which ties together the "hidden" supersymmetry of the given Hamiltonian system with the Poincare supersymmetry of the localizable quantum field theories. These ideas lead us naturally into the final topic of this Book which emphasizes these sorts of relations between equivariant cohomology and topological quantum field theories. The discussion of this Section then shows how this next topic is related to the equivariant localization formalism for dynamical systems. Zo (T)

8.4

can

Equivariant

and the We have

Numbers, Thom Classes Mathai-Quillen Formalism Euler

almost completed the connections between the equivariant loformalism, cohomological field theories and their relations to physical systems, thus uniting most of the ideas presented in this Book. The last Section showed how the localization formalism connects phase space path integrals of dynamical systems to some basic topological field theory models (namely supersymmetric sigma-models). Conversely, in Section 8.2 we demonstrated that arbitrary field theoretical models of these types could be placed into the loop space equivariant localization framework so that there now

calization

288

is


8.

in

Cohomological

Field

Theory

equivalence between dynamical systems and field theory modgeometric context. The discussion of Section 8.1 then illustrated genuine, geometrical equivalences between physical and topological

sort of

a

els in this certain

gauge theories which demonstrates the power of the formalisms of both topological field theory and equivariant localization of path integrals in describing

the quantum characteristics of physical systems There is one final step for this connection which is the .

Atiyah-Jeffrey

geometric interpretation of generic cohomological field theories [10] which is based on the Mathai-Quillen construction of Gaussian-shaped Thom forms

[99].

This construction is the natural

arena

properties of topological field theories, and it

can

based

be used to build up

study

of the localization

gauge models. This approach, although is rather different in spirit than the equiv-

topological

equivariant cohomology,

on

for the

in its infinite dimensional versions

ariant localization formalisms

we

have discussed thus far and

we

therefore

briefly highlight the details for the sake of completeness. At the end of this Chapter, we will discuss a bit the connections with the other ideas of this Book. More detailed reviews of the Mathai-Quillen formalism in topological field theory can be found in [27, 29, 34]. The basic idea behind the Mathai-Quillen formalism is the relation (3.71) between the Poincar6-Hopf and Gauss-Bonnet-Chern representations of the Euler characteristic. It represents the localization of an explicit differential form representative of the Euler class of a vector bundle onto the zero locus of some section of that bundle. The original idea for the application and generalization of this relation to cohomological field theories traces back to Witten's connection between supersymmetric quantum mechanics and Morse theory [166]. Let us start with a simple example in this context. Given the local coordinates (x,,q) on the cotangent bundle M (9 T*M of some phase '9 -2, In the spirit of the previous and ,, space M, we denote p,, 8XII a77P Section, we then interpret (x, q) as local coordinates on a supermanifold S*M and qA dx",p,, ft, as the local basis for the cotangent bundle of S*M. The nilpotent exterior derivative operator on S*M can be written as

only

very

=

=

-

-

-

d

77/-t

-5aiT X

a + P/,

(8.49)

= 710

e-'Pd e'P produces another linear derivation conjugation d same cohomology as d. If M has metric g, then we can (as in (8.46)) so that (8.49) conjugates to

The invertible

which generates the select P

d

=

779

49XA

+

The action of

(PM

r;",?7Vt\)

+

(8.50)

supermanifold S*M

on

a

a ,,

+

( rX A,P,\77-

1

-

A MP,77 2'

V

77 77,\ ) P-

a

5PI,

__

(8.50)

the local coordinates of the cotangent bundle of the

8.4 The

dx '

=

d?71'

=

77 '

dpl,

=

0

d4t,

=


r,\ /J,VP,\77V P, +

1

IJA

-

2

tZPA

V

P

289

-

77 77A

(8.51)

r,\ ?7vq,\ /'tV

coincides with the standard infinitesimal transformation laws of N

=

1 DeR-

ham supersymmetric quantum mechanics. We now consider the following integral Z*

f

=

d

2n

d2n p d2n 77

x

d2nq

e

dO[x,p;?j,fj]

(8.52)

S*MOT*(S*M)

Since d 2= 0 the

integral (8.52) is formally independent of the function '0 on 0. We can therefore evaluate super-manifold S*M 0 T* (S*M), i.e. (8.52) in 2 equivalent ways. First, we introduce a Hamiltonian vector field V on M and take 0 so that Ov -!V"q,, 2 the

=

=

dov

pjV"

=

+

(8.53)

77"V,,V'q,

integration in (8.52) can then be carried out explicitly. The integration p,, produces a delta-function J(V) localizing the integral onto the zero locus MV of the vector field V. The integration over the Grassmann coordinates in (8.53) yields a determinant of VV. Computing the relevant Jacobian for the transformation x -- V(x) (c.f. Section 2.6), we arrive finally at The

over

Z*

E

=

sgn det R (p)

(8.54)

pEMv

Next,

we

take

0

=

Og

=

dog Evaluating

gl"p,, ,

=

the Gaussian

so

1

gl"plp,

integrals

Z*

d

2n

that

-

2 over

x

RA"'n p,, and

d2nq

Pfaff

Me)T*M

which

we

equality

recognize

of

(8.54)

as

and

P-

n nAg

q,,

in

(8.55)

1"

n'

(8.52)

leads to

(1RI,',,P?7)'?7P)

(8.56)

2

the Euler class of the tangent bundle TM. Thus the leads immediately to the relation (3.71).

(8.56)

The exterior derivative operator

(8.50) produces

the

Mathai-Quillen

rep-

resentative of the Euler class of the tion is

a

special

case

of

a more

tangent bundle of M. The above derivageneral construction of explicit differential

form representatives for the Euler numbers of vector bundles E

representatives tions

are

are

so-called

Gaussian-shaped

--+

M. These

Thom forms whose construc-

best understood within the framework of equivariant cohomology. one realizes the vector bundle E -* M as its associated prin-

The idea is that

cipal G-bundle

P

x

W

(with

W the standard fiber space of

E)

and constructs

290

Equivariant Localization in Cohomological Field Theory

8.

particular representative (the Thom class) of the G-equivariant cohomology x W. Given a section V : M E, the regularized Euler class EV (FA) A of the bundle) is then the pullback of the connection curvature a (with FA of the Thom class to M under this section. It can be expressed as a

of P

--+

J d,q

Ev (FA) where

m

respect

=

to

dim M and

are

fixed fiber metric

a

IIVI12 +21

e

FAP '?7,

,

+ i V A V " 77 1,

Grassmann variables. The E and VA is

on

a

(8.57) V 11 2 is with

norm

compatible

connection. We

shall not go into the details of the construction of Thom classes using equivariant cohomology, but refer the reader to [27, 34, 81] for lucid accounts of

Mathai-Quillen representative general V, integrating out the Grassmarm variables 2m-form, and the fact that it is closed follows from

this formalism. The important features of the

(8.57)

are as

follows. For

EV(FA)

shows that

is

a

the invariance of the exponent in mations

JVA

::=

(8.57)

under the supersymmetry transfor-

VAV11

with the additional condition 6xl'

=

Jq" 71"

when

=

(8.57)

(8.58)

ivi, is

integrated

over X

E

M.

Note that setting V 0 in (8.57) and integrating out the Grassmann coordinates we see that it coincides with the usual Euler characteristic class =

E(FA)

=

pendent

any V. This can

means

be evaluated

limit

s

oo

--+

V,

EV(FA)

i.e.

is

that the Euler characteristic

by rescaling

onto the

this limit does not

zeroes

by

V

s

thus

Poincar6-Hopf

vector bundles

over

E R

and

of the section V

contribute),

relation between the

generic

closed, (8.57) is indecohomologous to E(FA) for

Pfaff (FA). Since the associated Thom class is

of the chosen section

X(E

--+

localizing

(note

reproducing

M) the

=

fm Ev(FA)

integral

in the

the curvature term in

in this way the standard

and Gauss-Bonnet-Chern theorems for

M. Thus the Thom class not

only yields a repproduces the

but it also

bundle, given section of the bundle. The use of equivariant cohomology and localization techniques therefore also reproduce some classical results from geometry and topology. Niemi and Palo [1211 have shown how to construct equivariant generalizations of the Mathai-Quillen formalism. For this one considers the usual Cartan equivariant exterior derivative DV and the associated Lie derivative D'V on the super-manifold S*M. If the Christoffel connection satisfies ,CV 0 then the conjugation by !P introduced above of these operators Lvl' produces an action on the local coordinates of the cotangent bundle of S*M for LV which generates the usual covariant coordinate transformation laws with respect to coordinate change defined by the Hamiltonian vector field V. Thus the integral resentative of the Euler class of

Poincar6-dual form of t he

zero

a

vector

locus of

a

=

=

Z

,

=

J S*MOT*(S*M)

d2nx d2np

d2n?7 d2n

e'00(H+w)+DvO

(8.59)

8.5 The

is

Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles

formally independent of any generally covariant

function

0

on

291

the cotangent

bundle of S*M. This is again just the equivariant localization principle. The measure in (8.59) is the invariant Liouville measure on the extended phase

Evaluating (8.59) using

space.

the 2 choices

for'O mentioned above,

we

arrive

at the relation

E

eiOOH(p)

sgn det R (p)

PEMv

f

(8.60)

1

d2n x d2n 77 e'O(&(H+w) pfaff V,V/,'

RlAP77A77P 2

+

MOT*M can be recognized as an equivariant generalization of (3.71). Thus an appropriate equivariantization leads to a Mathai-Quillen representative for the equivariant Euler number of an equivariant vector bundle. In the limit V, 0 -- 0, (8.60) reduces to the usual relation. The non-degenerate cases are also possible to treat in this way [121].

which


8.5 The

for Infinite-Dimensional Vector Bundles In this final Section of this tion to

cohomological

Book,

we

shall discuss briefly the

field theories. This will make

explicit

explicit

connec-

the relations of

localization quite generically to topological field theory that we have mentioned through out this Book. As originally pointed out by Atiyah and Jeffrey

[10], although

the Euler number itself does not make

mensional vector define

bundle,

the

Mathai-Quillen

regularized Euler numbers

dles for those choices of V whose

xv (E zero

-+

form

M)

=

sense

for

EV(FA)

an

can

fm Ev (FA)

infinite di-

be used to

of such bun-

locus is finite-dimensional

so

that the

localization makes these quantities well-defined. Although these numbers are not independent of V as in the finite-dimensional cases, they are naturally associated with M for certain choices. The functional

integrals

which arise in

this way are equivalent to ordinary finite-dimensional integrals and represent the fundamental property of topological field theories, i.e. that t heir path

integrals represent characteristic classes. This has been noted throughout this book as our central theme, and indeed most topological field theories can be obtained or interpreted in terms of the infinite-dimensional Mathai-Quillen formalism The

LM

--+

[34].

simplest example M

over a

this bundle is

The

regularized Euler

number of the

loop

space

V-'(t).

loop space version of the integral (8.52) is a superloop space, and we replace the extederivative d there by the equivariant, loop space,one Q& on L(S*M). Lie derivative L& Q? as before is the generator of time translations,

path integral rior

is the

manifold M. The canonical vector field associated with

over

Now the

an

extended

=

X

292

8.


in

Cohomological

Field

Theory

and

employing the standard conjugation above the path integral can be localized using any single-valued functional io on L(S*M). This follows from the equivariant localization principle for the model independent S'-action.

Choosing

loop

the natural

space extensions of the functionals

finite-dimensional calculations

(8.54) and (8.56) for (8.57 ) with V

of

=

above,

we

the Euler characteristic of b

yields the

action of N

quantum mechanics. In Section 4.2

'0 used in the precisely the same results M. The path integral analog

arrive at

we saw

1 DeRham

=

that the

supersymmetric 1 path integral for N 2 =

Dirac supersymmetric quantum mechanics localized onto constant modes and yielded the index of the twisted spin complex of M. In the present case the

localization of this Witten index onto constant DeRham,

complex

of M

(i.e.

mental observation of Witten

the Euler

[166]

loops yields

characteristic).

and

was one

the index of the

This

was

of the main

the funda-

ingredients

in

the birth of topological field

theory. If the target space manifold has a Kdhler structure then the sigma-model actually has 2 independent (holomorphic and anti-holomorphic) supersymmetries. Restricting the computation of the supersymmetric quantum mechanics partition function to the anti-holomorphic sector of the Hilbert space

described earlier leads to the representation of complex in terms of the Todd class. Equivariant generalizations of this simple example are likewise possible. In (8.59) the path integration now involves the action S rather than the Hamiltonian, and Dv gets replaced by QS in the usual routine of Chapter 4. Now we conjugate the relevant operators and find that the localization priniciple requires the localization functionals 0 to be generally covariant and singlevalued. The resulting path integrations yield precisely the computation at the end of Section 8.3 above. For Hamiltonians which generate circle actions, the right-hand side of (8.47) coincides with the left-hand side of (8.60) because of the structure of the set LMS discussed at the beginning of Section 4.6. Thus in this case we again obtain the ordinary finite-dimensional relation (8.60). These relations play a deeper role when the Hamiltonian depends explicitly on time t. Then the right-hand side of (8.47) represents a regularized measure of the number of T- periodic classical trajectories of the given dynamical system [119, 122]. Thus the classical dynamics of a physical system can be characterized in this way via the localization properties of supersymmetric non-linear sigma-models. In [124], these relations were related to a functional Euler character in the quantum cohomology defined by the topological nonlinear sigma-model and also to a loop space generalization of the Lefschetz fixed point theorem. Besides supersymmetric quantum mechanics the localization features of more complicated topological gauge theories can be studied by the computing as

the index of the Dolbeault

the Euler numbers of vector bundles

over

the infinite-dimensional space

AIG

of gauge connections modulo gauge transformations of a principal G-bundle. One can either start with a given topological field theory and analyse its lo-

calization characteristics using the

techniques

of this

Chapter,

or

conversely

8.5 The

Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles

293

by applying the Mathai-Quillen formalism to some vector bundle over AIG and reconstructing the action of the corresponding topological gauge theory from there. The resulting path integrals always compute sorts of intersection numbers on moduli space. A discussion of these models is beyond the scope of this book and

we

refer to

[34]

for

an

extensive discussion of the theories

can be viewed in this way. The basic example is Donaldson theory [22] which is the prime example of a cohomological field theory and is used to

which

calculate intersection numbers of moduli spaces of instantons for the study of 4-manifolds. Topological Yang-Mills theory in 4-dimensions is another in-

teresting application of this formalism. The field theoretic generalization of supersymmetric quantum mechanics, i.e. the topological sigma-model [22], is the appropriate setting for studying the quantum symmetries of string theory and more generally super-conformal field theories. The Mathai-Quillen formalism applied to 2-dimensional topological gravity could presumably shed light on its equivalence with physical gravity in 2 dimensions. The coupling of the topological sigma-model to topological gravity can be interpreted as topological string theory and studied using these methods. Finally, viewing 2-dimensional Yang-Mills theory as a topological field theory (see Section 8.1

above) leads in this way to a localization onto the rather complicated Hurwitz space of branched covers of the Riemann surface. This construction has been exploited recently as a candidate for a string theoretical realization of Yang-Mills theory [34]. Thus, the Mathai-Quillen formalism serves as the natural arena for the localization properties of cohomological field theories. However, the connection between the localization formalisms of the earlier Chapters of this Book (i.e. the stationary-phase formula) and the constructive Mathai-Quillen formalism above has yet to be completely clarified, as the latter relies on quite different cohomological symmetries than the ordinary BRST supersymmetries responsible for equivariant localization [1211. Recall these models all possess 2-dimensional

a

Grassmann-odd symmetry J that defines a supersymmetry transformation which resembles the usual BRST supersymmetries of equivariant local-

(8.58)

possible to argue [29, 34] that the 5-action is not free and that the path integral receives contributions from some arbitrarily small J-invariant tubular neighbourhood of the fixed point set of J. The integration over the ization. It is

directions normal to this fixed point set

can

be calculated in

a

stationary-

phase approximation. One readily sees from (8.58) that the fixed point set of J is the precisely the moduli space MV described by the zero locus of V 0. In this and its tangents 0 satisfying the linearized equation VAV(O) reduces field to an integration of theory path integral way the topological still obtain It remains to differential forms over Mv. a more precise though connection between these BRST fixed points, localization, and the interpretation of the geometrical and topological features of path integrals in terms of the Mathai-Quillen formalism which shows how such infinite-dimensional integrations are a pTioTi designed to represent finite-dimensional integrals. The =

294

8.


in

Cohomological

Field

Theory

antibracket formalism

developed in Section 8.3 above is a key stepping stone Mathai-Quillen localization features of topological field theory path integrals, and the path integral localizations of generic Hamiltonian systems. The supersymmetric formulation of equivariant cohomology developed in [129], and its connections with 4-dimensional topological Yang-Mill s theory, could serve as another approach to this connection. This might give a between the

more

direct connection between localization and

some

of the

more

modern

theories of quantum integrability [35], such as R-matrix formulations and the Yang-Baxter equation. This has been discussed somewhat in [56]. These are

derstandings theories, and space

all

important and should be found in order to have full untopological and integrable quantum field hence generic physical models, from the point of view of loop equivariant localization.

connections

of the structures of

9.

Appendix

Quantization

A: BRST

BRST quantization was first introduced in the quantization of Yang-Mills theory as a useful device for proving the renormalizability of non-abelian gauge theories in 4 dimensions. It

was

shown that

present after Yang-Mills gauge fixing invariance of the model and

global fermionic symmetry was incorporated the original gauge to straightforward derivations of

a

which

ultimately led

the Ward identities associated with the gauge symmetry in both quantum electrodynamics and quantum chromodynamics. New impetus came when

quantization of Hamiltonian systems completeness, in this Appendix we shall [69]. outline the essential features of the BRST quantization scheme of which the loop space localization principle can be thought of as a special instance. Consider any physical system with symmetry operators K' that (possibly locally) generate a closed Lie algebra g, the BRST

theory

was

applied

to the

For

with first class constraints

[Ka, K b]

fabcK'

=

(9.1)

'9 which are Faddeev-Popov ghost and anti-ghost fields 0a, #a a0a anticommuting Grassmann variables that transform in the adjoint representation of g. They have the canonical anticommutator

Introduce

-

[0a, Ob 1+ We define the

ghost number operator U

whose We

eigenvalues now

are

=

=

jab

(9.2)

as

Oap

integers running from

(9.3) 0 to dim g.

introduce the operator

Q

=

OaKa

1 -

2

f abcoaob c

(9.4)

as the BRST charge, while algebra coboundary operator that computes the cohomology of the Lie algebra g with values in the representation defined by the operators Ka. The crucial property of Q is that it is 0, which can be seen from (9.1) and the identity nilpotent, Q2

In the

physics literature the operator Q

is known

in the mathematics literature it is the Lie

=


296

9.

Appendix A:

BRST

f abef

Quantization

cde

f bdcf

+

cae

+

f dacf

cbe

(9.5)

0

=

which follows from

(9.1) via the Jacobi identity for the Lie bracket. Let Rk be the Hilbert space of states of ghost number k, i.e. UTI k T1 for T1 E I& We say that a state T1 E -Hk is BRST invariant if it is annihilated by Q, =

QT1

=

0, where

number

by

general

in

Any

1.

the action of

other state TV

T1 +

=

Q QX

-

any state raises the ghost ghost number k is regarded X E -Hk-1. The space of Q-

on

of

as equivalent to T1 E Hk for any other state equivalence classes of ghost number k is called the BRST-cohomology in the physics literature. Mathematically, it forms the k-th cohomology group Hk (g; R) of the Lie algebra g with values in the representation R carried by

the symmetry operators K'. Of particular interest from states of

ghost

0 must be annihilated

Q

on

such

a

a

number 0. From

by

all of the

=

cannot be annihilated =

0 is

a

the BRST-invariant

are

state TV of

anti-ghost fields P,

OaK a Tf

The anticommutation relations

QT1

it follows that

so

ghost number

that the action of

state is

QT,

P

physical standpoint

(9.3)

equivalent Ka,p=o

by

T E

(9.2) imply

any of the,

that

Ho a

(9.6)

state annihilated

ghost fields 0a,

and

so

by

all

the condition

to

,

a=1,...'dimg

T1 E

HO(g; R)

(9.7)

Therefore

a state T1 of ghost number 0 is BRST-invariant if and only if it g-invariant, and thus the cohomology group HI(g; R) coincides with the space of g-invariant states that do not contain any ghosts, i.e. the physical

is

states.

In

theory with gauge group G, the partition function must be dim G, which always with gauge-fixing functions ga, a 1, specify representatives of the gauge equivalence classes of the theory and restrict the functional integration to a subsPace U0 of the original configuration space of the field theory defined by the zeroes of the functions ga. Then the path integral can be written symbolically as a

gauge

evaluated

as

=

.

eis

=

vol(G)

U0

where VI

are

as

, U0

.

.

,

dimG

11

j(ga) det IlVb(gc)ll e's

(9.8)

a=1

usual vector fields associated with

an

orthonormal basis

of g (i.e. tr(XaXb) jab). Here S is the classical G-invariant gauge field action and the volume factor vol(G) is infinite for a local gauge field

fXal

=

theory. Modulo this infinite factor, the right-hand side of (9.8) is what is taken as the definition of the quantum gauge theory partition function. Introducing Faddeev-Popov ghost fields and additional auxilliary fields 0a, we can absorb

Appendix

9.

the additional factors

on

the

right-hand

A: BRST

side of

(9.8)

Quantization

into the

297

exponential

to

write

f e's

=

vol(G)

UO

f e's,

(9.9)

UO

where

Sq is the

=

S +

Oaga

+

gauge-fixed, quantum action. BRST-symmetry of this model is

The

s(4j)=Va( p)0a

S(Oa)=O

,

64Va(gb)Ob defined

(9.10)

by the following differential,

S(Oa)=_1fabcObOc 2

;

,

S(P)=_Oa (9.11)

where 0 is any scalar-valued functional of the gauge fields of the theory. With 0 and the quantum action (9.10) can this definition we have s 2 0, s(S) =

=

be written

as

Sq= S + Thus

s

is

a

S(_gaja)

BRST operator that determines

an

(9.12) N

=

dimG supersymmetry

gauge-fixed field theory, and the statement that the partition function (9.8) is independent of the choice of gauge-fixing functions ga is equivalent to the fact that the path integral depends only on the BRST-cohomology class of the action S, not on its particular representative. Thus the BRSTsupersymmetry here represents the local gauge symmetry of the theory. The gauge variation of any functional 0 of the fields of the theory is then represented as a graded commutator of the

is, 0} charge

with the fermionic

(i.e. gauge-invariant)

=

SO

-

(-1)POS

(9.13)

ghost-degree of 0. The physical gauge theory is the space of BRST-

s, where p is the

Hilbert space of the 0.

cohomology classes of ghost number

explicit relationship between equivariant localization of path integrals quantization (see the localization the next Appendix we shall make this connection principle in Section 4.4). In As have we a bit more explicit. mentioned, BRST-cohomology is the fundamental structure in topological field theories [22]. By definition, a topological action is a Witten-type action if the the classical action S is BRST-exact, while it is a Schwarz-type action if the gauge-fixed, quantum action S. is

(9.8)

and

(9.12)

demonstrate the

and BRST

BRST-exact

(but

not the classical

one).

In the

case

of the localization for-

BRST-operator is identified with phase space path integrals, of the underlying equivariant derivative exterior loop space equivariant of physical states consists "Hilbert and the space" cohomological. structure, of loop space functionals which are invariant under the flows of the loop space Hamiltonian vector field. This BRST-supersymmetry is always the symmetry that is responsible for localization in these models. The BRST formalism can

malism for the

the

Appendix

298

9.

also be

applied

A: BRST Quantization

systems with first-class constraints, i.e. those generate a Poisson subalgebra representation of a Lie algebra (9.1). The supersymmetric states then represent the observables which respect the constraints of the dynamical system (as in a gauge to Hamiltonian

whose constraint functions KI

theory).

Appendix B: Other Models Equivariant Cohomology

10.

of

In this

els for

Appendix we shall briefly outline the G-equivariant cohomology of

some a

of the other standard mod-

differentiable manifold M and

used extensively throughcompare them with the Cartan model which was out this Book. We shall also discuss how these other models apply to the derivation of

some

of the

more

general localization formulas which

briefly sketched in Section 4.9, as topological quantum field theory.

10.1 The

well

as

were

just

their importance to other ideas in

Topological Definition

ordinary cohomology, equivariant cohomology has a somewhat direct interpretation in terms of topological characteristics of the manifold M (and in this case also the Lie group G) [9]. This can be used to develop which in the usual way an axiomatic formulation of equivariant cohomology the characterize cohomology groups [341. provides properties that uniquely This topological definition resides heavily in the topology of the Lie group G through the notion of a classifying space [73]. A classical theorem of topology tells us that to G we can associate a very special space EG which is characAs with

terized

by

the fact that it is contractible and that G acts

it without fixed

on

is called the universal G-bundle. The

classifying

space points. The space EG BG for G-bundles is then defined as the base space of a universal bundle whose total space is EG. The space BG is unique up to homotopy and EG is unique up to equivariant homotopy (i.e. smooth continuous equivariant BG has 2 remarkable univerdeformations of the space). The bundle EG sal properties. The first one is that any given principal G-bundle E --+ M --+ BG. The over a manifold M has an isomorphic copy sitting inside EG -->

isomorphism classes of principal G-bundles are therefore in one-to-one corBG. The second respondence with the homotopy classes of maps f : M M can of E the of natural all that topology is measuring ways property --

be obtained from For

while for G and

Bu(j)

gauge

H*(BG)Z,

example, when G =

=

U(1) CP 00

we

get

U-

transformations,

EU(1)

we

S(H)

have Ez. =

100 Un=0

R' and BZ.

=

q2n+l

-

(the

Hilbert

(Sl)n, sphere)

.

(Cpn. In gauge theories G is the group of local that EG is the space A of Yang-Mills potentials

n= 0

so

=


300

Appendix

10.

B: Other Models of


while BG AIG is the space of gauge orbits. In string theory G is the semidirect product of the difleomorphism and Weyl groups of a Riemann surface _Th of genus h, EG is the space of metrics on _T h, and BG is the moduli =

space MEh of _Th From this point of view, one can define topological field theory and topological string theory as the study of H*(BG) and related cohomologies using the language of local quantum field theory. .

Given and M M

x

a

on

smooth G-action

M, on

we

thus have 2 spaces EG

the Cartesian

product

space

Eg via the diagonal action G

x

(M

Like the G-action

on

EG, this

Eg)

x

(g'X' e)

-

--+

(g

M

X'g

-

Eg

x

-

(10.1)

e)

action is also free and thus the

MG is

manifold

on a

which G acts. Thus G also acts

:--

(M

x

quotient

EG)IG

space

(10.2)

smooth manifold called the

homotopy quotient of M by G. Since Eg is EG is homotopic to M. Furthermore, if the G-action on M is free then MG is homotopic to MIG, so that both spaces have the same (ordinary) cohomology groups. In the general case we can regard MG as a bundle over BG with fiber M. These observations motivate the topological definition of equivariant cohomology as a

contractible, M

x

HGk,t.P(M)

=

Hk(MG)

(10.3)

Notice that if M is the space consisting of a single point, then MG BG. Thus H k't.p (pt)

and

the

=

H

k

(BG)

fl--

EGIG

(10.4)

G-equivariant cohomology point ordinary cohomology classifying space BG. This latter cohomology can be quite complicated [9], and the topological definition therefore shows that the equivariant cohomology measures much more than simply the cohomology of a manifold so

of

a

is the

of the

modulo

a

group action

on

calization formalisms of

it. It is this feature that makes the non-abelian lo-

topological field theories

and

integrable models

very

powerful techniques.

10.2 The Weil Model The topological definition of the G-equivariant cohomology above can be reformulated in terms of nilpotent differential operators [9, 81]. In this formulation, the equivariant cohomology is obtained in a more algebraic

by exploiting differential properties

of the Lie

algebra

way g of the group G.

10.2 The Weil Model

calculus of the

precisely, to describe the exterior differential algebra S(g*), we introduce the Weil algebra

More

W(g)

on

01,

a

usual

S(g*)

(10.5) algebraically

As in Section 2.6,

0

symmetric

(10.5)

Ag*

describes the exterior bundle of

consists of multilinear

algebra Ag* generated by an anti-commuting

where the exterior g and it is

=

301

antisymmetric

S(g*) forms

basis of Grassmann numbers

d0a). The Weil algebra has the (i.e. the 1-forms Oa the i.e. generators Oa of S(g*) have degree 2 Z-grading (ghost number), dim G

=

-

while the generators Oa of Ag* have degree 1. Both of these sets of generators of g. are dual to the same fixed basis jXajdimG a=1 There are 2 differential operators of interest acting on W(g). The first is the "abelian" exterior derivative

do where Ia

identifies

is the interior

a0a

Oa

as

we

Oa

=

multiplication

=

Oa

do

La which generate the

coadjoint La Ob

yield

a

=

-fabc

(01

a =

action of G

we

representation

define

on

of G

our

on

=

(10.8)

W(g) explicitly by =

fabcoc

=

Oa La +

1

2

(10.9)

W(g),

f abcLc

(10.10)

second differential operator, the

erator

dg

0

OIIC

La Ob

[L, Lb] Using (10.8),

+

(10.7)

-Ia

degree

0

fabcoc

=

degree -1 on W(g). (10.6) non-vanishing actions are

of

,90a

introduce the linear derivations of

and which

(10.6)

Ja

g

the superpartner of Oa and its

do0a Next,

Ag*

on

coboundary

f abcoaobic

op-

(10.11)

which computes the W(g)-valued Lie algebra cohomology of g, i.e. it is the BRST operator associated with the constraint operators La acting on W(g)

(see (9.4)). The

sum

of

(10.6)

and

(10.11) dw

whose action

on

the generators of

is known

do

+

W(g)

is

=

as

dg

the Weil

differential,

(10.12)

302

10.

Appendix

B: Other Models of

1

dw oa=oa_

fabcoboc

2

These 3 differential operators

and

they

act

d-pvoa

7

all

are

d2W


exterior derivatives

_fabcoboc

nilpotent derivations

d20

=

=

=

d

2 =

9

of

(10.13) degree 1,

(10.14)

0

W(g). (10.12)

makes the Weil

algebra cohomology of the Weil differential dw on W(g) is trivial. This can be seen by redefining the basis of W(g) by the shift 0a _-, 0a In the new basis, 0a Ifabcoboc. dWO' are 2 exact and the cohomology of d,,v coincides with that of do on W(g) so that into

as

exterior differential

an

on

algebra. However,

the

-

H The

=

k(W(g), dw)

=

R

(10.15)

cohomology

can be made non-trivial using the 2 derivations la and introduced above. We notice first of all the analogy between W(g) (10.13) and the algebra of connections and curvatures. The first relation in

L,,

on

(10.13)

is the definition of the curvature of

principal G-bundle

E

the curvature 2-form F

dA

=

M, while 0a, i.e.

-->

we

one

F

1 -

-

[A

A

A]

dF

g),

which

is the Bianchi

=-[A A, F]

recall that the characteristic classes of E

A E Al (E,

field

connection 1-form A

-

0a

on a

identity

for

-

2

Here

a

the second

can

be

regarded

F E A2 (E,

g),

which

as a

--+

map A:

M

g*

are

--+

(10.16)

constructed from

A1E,

and from the

regarded as a map F : g* --+ A2E. These maps generate a differential algebra homomorphism (W(g), dw) --+ (AE, d) which is called the Chern-Weil homomorphism. This homomorphism is unique and it maps the algebraic connection and curvature (0a, Oa) to the geometric ones (A, F) [99]. In this setting, a connection on a principal Gbundle E --+ M is just the same thing as a homomorphism W(g) --> AE. Thus the Weil algebra is an algebraic analog of the universal G-bundle EG. Like EG, it possesses universal properties, and therefore it provides a universal model of connections on G-bundles. In particular, the contractibility of EG is the analog of the triviality (10.15) of the cohomology of the Weil algebra. Pursuing this analogy between EG and W(g), we can find non-trivial and universal cohomology classes by considering the so-called "basic forms" [9]. First, we note that the operators I,, and La above are the algebraic analogues of the interior multiplication and Lie derivative of differential forms with respect to the infinitesimal generators 0a of the G-action on W(g). Indeed, the operator (10.8) can be expressed in terms of the Weil differential as strength

La and furthermore

we

have

=

Iadw

can

+

be

dwI,,

=-

[dg, la]+

(10.17)

10.2 The Weil Model

[1a, Ibl

[L,,, Ib]

0

==

=

fabc IC

Thus the derivation La has the natural structure of that commutes with all the derivatives

[dw, Lal

=

a

303

(10.18)

Lie derivative

on

W(g)

above,

[dg, Lal

=

[do, Lal

=

(10.19)

0

particular, the (anti-) commutation relations above among dw, La and Ia all independent of the choice of basis of g. As we saw in Section 2.3, these relations also reflect the differential geometric situation on a manifold M with a G-action on it, and the Chern-Weil homomorphism above maps (dw, La, Ia) --+ (d, CVa, jVa) between the differential algebras W(g) --+ AM. We can finally define the Weil model for equivariant cohomology. For this we consider the tensor product W(g) 0 AM of the Weil algebra with the exterior algebra over the manifold M. The replacement AM --+ W(g) 0 AM for the description of the equivariant cohomology is the algebraic equivalent of the replacement M EG x M in Section B.1 above. The basic subalgebra In

are

--*

W(g) (9 AM consists of those forms which have no vertical component (i.e. the horizontal forms) and which are G-invariant (i.e. have no vertical variation). These 2 conditions mean, respectively, that the

(W(g)

(9

AM)basic

basic forms

are

of

by all the operators 1,,,

those annihilated

LaOl+JOLVa (recall that iva is the component of vector field V E TM), so that

0 1 + 10

along

a

the

iv. and

(vertical)

dim G

(W(g)

0

(D (dimnker

AM)basic

ker

(1a

0 1 + 10

iv.)

a=1

(10.20)

G

n

(La

(9 1 + 10

LV

b=1

This

subalgebra

is stable under the action of the extended DeRham exterior

derivative

dT

=

dw

cohomology of (10.21) on (10.20) G-equivariant cohomology of M, and the

k

HG,alg(M)

=

H

k

(10.21)

0 1 + 10 d

((W(g)

algebraic

definition of the

AM)basic, dT)

(10.22)

is the

0

an

AE with E homomorphism W(g) if G is compact cohomology groups isomorphism

we

have

The Chern-Weil

--+

of

k

HG,alg(M) so

H k'tOP (M)

EG then reduces

and connected

[91

to

and

(10.23)

algebraic and topological definitions of equivariant cohomology equivalent.

that the

are

-

=

304

Appendix

10.

B: Other Models of

We close this Section with of

W(g) (VI that the

remark

0 in

Since

vanishes

=

translates into invariance under the

Hk (Bg d-pv) I

and

concerning the basic subcomplex Bg

on this basic subcomplex, we dw (10.20)). in this case means oa-independence, d-,V) Bg. Horizontality basic forms in W(g) lie in only S(g*). G-invariance in this case =

have Hk (Bg, so

a


coadjoint =

Bg

=

action of G

S(g*)

on

g*. Thus

G

(10.24)

the basic

subalgebra of the Weil algebra coincides with the algebra polynomial functions on the Lie algebra g, i.e. Bg is the algebra of corresponding Casimir invariants. It is known [9] that if G is a compact connected Lie group, then Hk(BG) Hk(Bg, dw), and so comparing (10.24) with (10.4) we find that the G-equivariant cohomology of a point is simply the algebra of G-invariant polynomials on g. This is in agreement with what so

of invariant

=

we

found in Section 2.3 from the Cartan model. In the next Section

indeed find that this

correspondence

is

no

we

shall

accident.

10.3 The BRST Model The final model for the

G-equivariant cohomology of a manifold M intermodels, and it therefore relates the Cartan model to the topological characteristics of this cohomology theory. It also ties in with the BRST quantization ideas that are directly related to the localization formalisms and it is the model of equivariant cohomology which arises naturally in the physical context of topological field theories. The (unrestricted) BRST algebra B of topological models on quotient spaces polates between

is the

same as

the Cartan and Weil

that for the Weil model

[81, 132]. Now, however, 5

which is

=

dw

the differential

vector space, B W(g) (9 AM it is the BRST operator

as a on

(9 1 + 1 o d + 0a

=

(gCV.

-

0a

(&

(10.25)

iV.

nilpotent graded derivation of degree 1 on B. The BRST operator is nilpotent extension of the Cartan equivariant exterior derivative defined in (2.59). The Weil differential dw in (10.25) takes care of the Dg non-nilpotency (2.61) of the Cartan model derivative. The Kalkman parametric model for the equivariant cohomology is defined by the Kalkman differential [81] a

the natural

Jt

=

e

t0a G)iVa

where t E

1

dT e-to' OiVa =dT+t0a(&'CV.-t0a (&iVa + t(l -t)f abcoaob(&iV, 2

[0, 1].

Notice that

Jo

=

dT and 51

=

J,

so

that

(10.26) (10.26) interpo-

lates between the differentials of the Weil and BRST models. it satisfies

[Jt, Ia

0 1 +

(I

-

t)l

0

iv.]

=

La

(9 1 + 1

,

(&,CV-

Furthermore,

(10.27)

10.3 The BRST Model

so

that

obtain in this way

we

a

family

of Lie

305

super-algebras acting on W(g) (9

AM. Notice also that

dT and thus the of d

on

-' =

I

e

cohomology

a fabcoaob _071

of 5

on

(d +do)

2fabcoaob 00 a

" e

(10.28)

B coincides with the DeRham

cohomology

AM.

Thus the BRST operator J does not capture the G-equivariant cohomology of M, because the W(g) part of (10.25) can be conjugated to the

cohomologically trivial operator (10.6) (equivalently

the

cohomology

of

dw

is trivial). We have to accompany 9 with a restriction of its domain in the same way that (2.59) computes the equivariant cohomology when restricted to the G-invariant subspace (2.40). The appropriate restriction is to the Oa-

independent and G-invariant subalgebra of B. This reduction maps B to AGM and 6 to Dg, so that the mapping Oa --+ 0, B --> S(g*) (9 AM induces the isomorphism of complexes

(Bbasic, d-,v)

_,

(10-29)

(AGM Dg) I

between the Weil and Cartan models. This restriction

can

be formulated

nilpotent operator W whose kernel is the desired G-invariant and Oa-independent subalgebra [129, 132]. For this, we introduce another copy W(g) of the Weil algebra. It is generated by Oa and Oa which are the g*-valued coefficients corresponding, respectively, to Oa-independence (generated by Ia) and G-invariance (generated by L, (& 1 + 1 (& LVa). A nilpotent operator with kernel AGM is

by introducing

VV where

dg

=

dg

another

& 10 1 +

is the Lie

6'

(9

(La

0 1+ 10

CVa )

-

a

(&Ia (& 1

(10.30)

algebra coboundary operator (10.11) on W(g) which (10.30) nilpotent. The action of J on W(g)

makes the overall combination in is taken

as

that of the abelian differential

(10.6)

affect its 5

=

so

that J commutes with

cohomology of do on W(g) is trivial, this alteration of 6 does cohomology. The equivariant BRST operator is therefore

W. As the

not

1(gd,(2)1+1(91(gd+do(gl(&1+1(goa (& LVa _100a Oiva (10.31)

and 6 and VV satisfy the

nilpotent algebra 62

=

[-yV, j]+

=

W2

=

0

(10.32)

G-equivariant cohomology of M is isomorphic to the cohomology of 6 subalgebra of VV(g) (9 W(g) 0 AM which is annihilated by W. This defines the BRST model of equivariant cohomology. We remark that it is also possible to formulate the restriction onto the basic subcomplex. in the Weil model using the nilpotent operator

The on

the

Ww

=

dg 0 10 1 +P 0 (La (9 1 + 1 OLVa) _ a(&(Ja(&1+1(&iV.) (10-33)

306

10.

and the

Appendix

B: Other Models of

corresponding


extension of the Weil differential is

dw

=

1 (&

dT

do

+

(& 1 o 1

(10.34)

The operators

(10.33) and (10.34) are similarity transforms of (10.30) and (10.31), respectively, just as in (10.28). They therefore obey the algebra (10.32) as well, as required for the appropriate restriction process above.

It is this model that is relevant for the construction of non-abelian generalizations of the Duistermaat-Heckman integration formula, such as the Witten localization formalism (see Section 3.8). Modelling the equivariant

cohomological

structure as above is the correct way to incorporate the idea equivariant integration that we discussed earlier. In these models the generators 01 of the symmetric algebra S(g*) generate dynamics of their own and only after they are fully incorporated as above can one define properly the required equivariant localization. All 4 equivariant derivations discussed in this Book the Cartan, Weil, Kalkman and BRST differentials, have been related by Nersessian [114, 115] to the geometric anti-bracket structure of the Batalin-Vilkovisky formalism (see Section 8.3). In these formalisms, one constructs antibrackets for W(g) in addition to the usual ones for the cotangent bundle M 0 A'M.

of

-

Loop Space Extensions

10.4

Loop

generalizations

of the constructions above have been

presented by dynamical localizations of Section 4.9. For this, we introduce superpartners 01(t) for the multipliers 0'(t) and make them dynamical by adding a kinetic term for the Grassmann coordinates 0a to the action, space

Tirkkonen in

[161]

in the context of the

T

S-+

ST

=

S+

f

dt

oa(t) a(t)

(10.35)

0

The circle action generator in

(4.151)

is also extended to include

0a,

i.e.

T

VS1

-->

VS1

+

f

dt

a (t)

0

Thus the

path integral

coordinates

is

(x, 0) (recall

now

from

formulated

(10.7)

over a

that 0a

(10.36)

joa(t) superloop

space with local

coordinates and

do0a Oa corresponding 1-forms in the Weil algebra). To exploit the loop space isometry generated by the semi-direct product LGS)S1, we need to construct the corresponding equivariant operators. Because the fields Oa and 0a are dynamical and are therefore an important part of the path integration, this are

are

the

has to be done in terms of the BRST model above.

=

Loop Space

10.4

307

Extensions

corresponding equivariant BRST operator, the part corresponding (4.151) generated by VLG on LM is just lifted from the corresponding equivariant BRST operator (10.31) for the G-action on M. The part associated with the circle action in (4.151) as generated by (10.36) dL i& on the superloop space is given by the Cartan model operator Q& In the

to the LG-action in

=

-

T

the operator Qj is nilpotent on the loop space LM. L dt A dt) yJ equivariant BRST operator for the semi-direct product LGZ)S then combines (10.31) lifted to the loop space with Q_;, Since

Q?

=

X

The total

T

QT

=

dL

(L)

+ dW +

I (O',Cv. dt

-

O'iv.

30)

+

J'qA

(10.37)

0

(L) where dW is the Weil differential

(10.12)

superloop

lifted to the

(10.30)

standard way. Furthermore, to the restriction operator corresponding to the Sl-action [161]

space in the

we

add

a

piece

T

)1VT

WL

=

+

1

dt 0

(10.38)

(t)

0

with WL the lifting of the restriction operator The algebra Of QT and WT is then

(10.30)

to the

superloop

space.

T

Q2T

d

dt

2

V4

dt

=

[QTi WTI+

=

0

(10.39)

0

The function

F(0)

discussed in Section 4.9 Weil space

=

!(Oa)2

effectively added to the action as interpreted as a symplectic 2-form on the interpreted as super 1-forms). The superloop

2

can now

algebra W(g) (as 0a symplectic 2-form is

are

which is

be

then T

S?T

=

Q +

1

dt

1 2

(Oa(t))2

(10.40)

0

equivariance properties by the equations

and the rized

QT(ST The first

+

vanishing condition

of the

S?T) in

=

phase

WT(ST

(10.41)

space

+

path integral

S?T)

=

identifies the

0

superloop

are summa-

(10.41) space action

the moment map for the action of the semi-direct product LGS)S' on LM, while the second one states that this group action is symplectic. To carry out the Niemi-Tirkkonen localization procedure over the superloop

ST

as

308

10.

Appendix

B: Other Models of

space with the BRST

(4.123)

to the Weil

operator QT,


we

generalize the

gauge fermion field

algebra, T

OT

I

=

dt

(10.42)

+

0

where the

we

have left the

same

localization

mode part of 0'0' Out Of'OT- We can now carry out procedure which led to the Niemi-Tirkkonen formula,

zero

(4.153) [161].

and thus arrive at the localization formula

non-abelian generalization of the modelindependent auxilliary field formalism which we discussed in Section 4.9. In the abelian case where we can a pTiori fix any function F(0) for the loThe above construction is

calization,

(10.40),

a

modifying the loop space symplectic structure as in model-independent formalism appears as the functional

instead of

which in the

110t1j,

determinant det

we

shift the gauge fermion field

(10.42)

as

T

OT

--*

dt

OT

(0,,,77"

-

2

(10.43)

0

The standard Niemi-Tirkkonen localization discussed even

above,

procedure then leads

to the lo-

(4.149) [128].

For the generic, non-abelian group actions the situation for loop space localization is different, and

calization formula

the second Weil

algebra

copy

IFV(g)

is made

dynamical. For discussions cohomology and

of the relations between the BRST model of equivariant

Witten-type topological field theories,

see

[27, 34, 129].

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